title: Orthotics

text: "KAFO with locked knee joint -" The mechanical knee joint of a KAFO with locked knee joint is locked when standing and also locked when walking in both the stance phase and the swing phase in order to achieve the necessary stability when standing and walking. To sit, the user can unlock the knee joint. When walking with a KAFO with a locked knee joint, it is difficult for the user to swing the leg forward in swing phase. In order not to stumble, the leg must be swung forward laterally to the direction of movement in the form of a circular arc (circumduction) or the hip of the affected leg must be raised unnaturally in order to achieve swing of the stiff leg. Each of these incorrect gait patterns can lead to secondary diseases in the bone and muscle system. In any case, such compensatory movement patterns lead to increased energy consumption when walking. The film Forrest Gump impressively shows how the main character Forrest Gump is additionally hindered in his urge to move by such orthoses. For centuries, a KAFO was built with mechanical knee joints that stiffened the knee of the paralyzed leg. Even today, such orthotic fittings are still common. Typical designations for a KAFO with a locked knee joint include "KAFO with Swiss lock" or "KAFO with drop lock lock".


title: Orthotics

text: "KAFO with unlocked knee joint -" The mechanical knee joint of a KAFO with an unlocked knee joint can move freely when standing and can also move freely when walking, both in the stance phase and in the swing phase, in order to ensure that the leg swings through without stumbling, with the necessary knee flexion of approx 60° to allow. The user does not need to unlock the knee joint to sit. A KAFO with a non-locked knee joint can only compensate to a small extent for paralysis-related uncertainties when standing and walking. In order to increase the safety function, an orthotic knee joint with a rearward displacement of the pivot point can be installed in the KAFO. However, a KAFO with a non-locked knee joint, even with a backward displacement of the pivot point, should only be used in the case of minor paralysis of the knee and hip extensors. With more severe paralysis with low levels of strength in these muscle groups, there is a significant risk of falling. A typical designation for a KAFO with a non-locked knee joint is, among other things, "KAFO with knee joint for movement control".


title: Orthotics

text: "KAFO with locked and unlocked knee joint -" The mechanical knee joint of a KAFO with locked and unlocked knee joint is locked when walking in the stance phase. In the stance phase, the necessary stability and security is achieved for the user. The knee joint is automatically unlocked in the swing phase. By unlocking it in the swing phase, the leg can be swung through without stumbling. In order to be able to walk energy-efficiently and without stumbling without compensating mechanisms, the mechanical knee joint used should allow a knee flexion of approx. 60° in the swing phase.The first promising developments of automatic knee joints or stance phase locking knee joints emerged in the 1990s. In the beginning there were automatic mechanical constructions that took over the locking and unlocking. Automatic electromechanical and automatic electrohydraulic systems are now available that make standing and walking safer and more comfortable. Various terms are used for a KAFO with a locked and unlocked knee joint. Typical designations are "KAFO with automatic knee joint" or "KAFO with stance phase control knee joint". In scientific articles, the English term Stance Control Orthoses SCO is often used. This term differ from the ICS classification, which is why one of the first two terms is preferable.


title: Orthotics

text: Different functional elements to compensate for a weakness of the "dorsiflexors" or a weakness of the "plantar flexors" can be integrated into the ankle joint of the orthosis depending on the combination of the degree of paralysis of the two muscle groups. It is of great advantage if the resistances for these two functional elements can be set separately.


title: Orthotics

text: The functional elements to compensate for paralysis of the knee-securing muscle groups, the "knee extensors" and the "hip extensors", are integrated into the knee joint of the orthosis via knee-securing functional elements.


title: Orthotics

text: A KAFO can compensate for the limitations of the named muscle groups through the variety of combinations of different variants in the stiffness of the foot shell, the different variants of the functional elements of a dynamic ankle joint, the variants in the shape of the lower leg shell and the functional elements of a knee joint.


title: Orthotics

text: HKAFO is the abbreviation for ankle-foot orthoses; English name for an orthosis that spans the hip, the knee, the ankle and the foot. In the treatment of paralyzed patients, a HKAFO is used when there is a weakness of the pelvic stabilizing trunk muscles.


title: Orthotics

text: Relief Orthoses are used both in degeneration and after injuries to joints. These includes torn ligaments and wear and tear of joints. Relief orthoses are also used after operations such as a complete replacement of a joint or operations on the joint structure or other operations on bony, muscular structures or operations on the joint ligaments.


title: Orthotics

text: A custom-made ankle/foot orthosis for the treatment of patients having plantar ulcers is disclosed, which comprises a rigid L-shaped support member and a rigid anterior support shell articulated on a hinge to the L-shaped support member. The plantar portion of the L-shaped member further comprises at least one ulcer-protecting hollow spatially located for fitted placement in inferior adjacency to a user's plantar ulcer, thus allowing the user to transfer the user's weight away from the plantar ulcer and facilitating plantar ulcer treatment. The anterior support shell is designed for lateral hinged attachment to the L-shaped member to take advantage of medial tibial flare structure for enhancing the weight-bearing properties of the disclosed orthosis. A flexible, polyethylene hinge member attaches by hinge to the anterior support shell to the L-shaped member and securing straps securely attach the anterior support shell in fixed, weight-bearing relation about the proximal, anterior portion of the user's lower leg.


title: Orthotics

text: Foot orthoses (commonly called "orthotics") are devices inserted into shoes to provide support for the foot by redistributing ground reaction forces acting on the foot joints while standing, walking or running. They may be either pre-moulded (also called pre-fabricated) or custom made according to a cast or impression of the foot. A great body of information exists within the orthotic literature describing their medical use for people with foot problems as well as the impact orthotics can have on foot, knee, hip, and spine deformities. They are used by everyone from athletes to the elderly to accommodate biomechanics deformities and a variety of soft tissue conditions. Foot orthoses are effective at reducing pain for people with painful high-arched feet, and may be effective for people with rheumatoid arthritis, plantar fasciitis, First metatarsophalangeal (MTP) joint pain or hallux valgus (bunions). For children with juvenile idiopathic arthritis (JIA) custom-made and pre-fabricated foot orthoses may also reduce foot pain. Foot orthoses may also be used in conjunction with properly fitted orthopedic footwear in the prevention of foot ulcers in the at-risk diabetic foot. A real-time weight bearing orthotic can be created using a neutral position casting device and the Vertical Foot Alignment System VFAS.


title: Orthotics

text: An AFO is also used to immobilize the ankle and lower leg in the presence of arthritis or fracture.Ankle–foot orthoses are the most commonly used orthoses, making up about 26% of all orthoses provided in the United States. According to a review of Medicare payment data from 2001 to 2006, the base cost of an AFO was about $500 to $700. A recent study looked at the prescription practices amongst the clinical practitioners in the United Kingdom to explore the material used, the thickness of the material, positive cast rectification, AFO reinforcement, footplate design, padding, strapping system, and the height of an AFO. This study found that there is a consensus in practice.


title: Orthotics

text: A knee orthosis (KO) or knee brace is a brace that extends above and below the knee joint and is generally worn to support or align the knee. In the case of diseases causing neurological or muscular impairment of muscles surrounding the knee, a KO can prevent flexion or extension instability of the knee. In the case of conditions affecting the ligaments or cartilage of the knee, a KO can provide stabilization to the knee by replacing the function of these injured or damaged parts. For instance, knee braces can be used to relieve pressure from the part of the knee joint affected by diseases such as arthritis or osteoarthritis by realigning the knee joint into valgus or varus. In this way a KO may help reduce osteoarthritis pain, however, there is no clear evidence to advise people with osteoarthritis of the knee about the most effective orthosis to use or the best approach to rehabilitation. A knee brace is not meant to treat an injury or disease on its own, but is used as a component of treatment along with drugs, physical therapy and possibly surgery. When used properly, a knee brace may help an individual to stay active by enhancing the position and movement of the knee or reducing pain.


title: Orthotics

text: Prophylactic braces are used primarily by athletes participating in contact sports. Evidence about prophylactic knee braces, the ones football linemen wear that are often rigid with a knee hinge, indicates they are ineffective in reducing anterior cruciate ligament tears, but may be helpful in resisting medial and lateral collateral ligament tears.


title: Orthotics

text: Functional braces are designed for use by people who have already experienced a knee injury and need support to recover from it. They are also indicated to help people who have pain associated with arthritis. They are intended to reduce the rotation of the knee and support stability. They reduce the chance of hyperextension, and increase the agility and strength of the knee. The majority of these are made of elastic. They are the least expensive of all braces and are easily found in a variety of sizes.


title: Orthotics

text: Rehabilitation braces are used to limit the movement of the knee in both medial and lateral directions- these braces often have an adjustable range of motion stop potential for limiting flexion and extension following ACL reconstruction. They are primarily used after injury or surgery to immobilize the leg. They are larger in size than other braces, due to their function.


title: Orthotics

text: A soft brace is sometimes also called soft support or a bandage. Bandages, which belong to the field of orthoses, are so-called support bandages, which are supposed to protect the joints from excessive loads. Soft braces are also classified according to regions of the body. In sport, bandages are used to protect bones and joints. One of the purposes of these braces is injury protection. They mostly consist of textiles, some of which have supportive elements. The supporting functions are low compared to paralysis and relief orthoses. Bandages should also achieve proprioceptive. Equally, orthoses are sometimes used prophylactically or to optimize performance in sport. At present, the scientific literature does not provide sufficient high quality research to allow strong conclusions on their effectiveness and cost-effectiveness.


title: Orthotics

text: Upper-limb (or upper extremity) orthoses are mechanical or electromechanical devices applied externally to the arm or segments thereof in order to restore or improve function or structural characteristics of the arm segments encumbered by the device. In general, musculoskeletal problems that may be alleviated by the use of upper limb orthoses include those resulting from trauma or disease (arthritis for example). They may also be beneficial in aiding individuals who have a neurological impairment such as stroke, spinal cord injury, or peripheral neuropathy. Disease is the standard respiratory rate.


title: Orthotics

text: Scoliosis, a condition describing an abnormal curvature of the spine, may in certain cases be treated with spinal orthoses, such as the Milwaukee brace, the Boston brace, Charleston bending brace, and Providence brace. As this condition develops most commonly in adolescent females who are undergoing their pubertal growth spurt, compliance with wearing is these orthoses is hampered by the concern these individuals have about changes in appearance and restriction caused by wearing these orthoses.


title: Orthotics

text: A Jewett brace, for instance, may be used to facilitate healing of an anterior wedge fracture involving the T10 to L3 vertebrae. A body jacket may be used to stabilize more involved fractures of the spine. There are several types of orthoses for managing cervical spine pathology. The halo brace is a cervical thoracic orthosis used to immobilize the cervical spine, usually following fracture. The halo brace allows the least cervical motion of all cervical orthoses currently in use; it was first developed by Vernon L. Nickel at Rancho Los Amigos National Rehabilitation Center in 1955.


title: Orthotics

text: Orthotists are healthcare professionals who specialize in the provision of orthoses. In the United States, orthotists work by prescription from a licensed healthcare provider. Physical therapists are not legally authorized to prescribe orthoses in the U.S. In the U.K., orthotists will often accept open referrals for orthotic assessment without a specific prescription from doctors or other healthcare professionals.


title: Gray fox

text: The gray fox (Urocyon cinereoargenteus), or grey fox, is an omnivorous mammal of the family Canidae, widespread throughout North America and Central America. This species and its only congener, the diminutive island fox ("Urocyon littoralis") of the California Channel Islands, are the only living members of the genus "Urocyon", which is considered to be genetically basal to all other living canids. Its species name "cinereoargenteus" means "ashen silver".


title: Gray fox

text: It was once the most common fox in the eastern United States, and though still found there, human advancement and deforestation allowed the red fox to become the predominant fox-like canid. Despite this post-colonial competition, the gray fox has been able to thrive in urban and suburban environments, one of the best examples being southern Florida. The Pacific States and Great Lakes region still have the gray fox as their prevalent fox.


title: Gray fox

text: The genus "Urocyon" comes from the Latin 'uro' meaning tail, and 'cyon', meaning dog. The species epithet "cinereoargenteus" is a combination of 'cinereo' meaning ashen, and 'argenteus' (from argentum), meaning 'silver', referencing the color of the tail.


title: Gray fox

text: The gray fox is mainly distinguished from most other canids by its grizzled upper parts, black stripe down its tail and strong neck, ending in a black-tipped tail, while the skull can be easily distinguished from all other North American canids by its widely separated temporal ridges that form a ‘U’-shape. Like other canids, the fox's ears and muzzle are angular and pointed. Its claws tend to be lengthier and curved.


title: Gray fox

text: There is little sexual dimorphism, save for the females being slightly smaller than males. The gray fox ranges from in total length. The tail measures of that length and its hind feet measure . The gray fox typically weighs , though exceptionally can weigh as much as . The grey fox is readily distinguished from the red fox by its obvious lack of the "black stockings" that stand out on the red fox. The grey fox has a stripe of black hair that runs along the middle of its tail, and individual guard hairs that are banded with white, gray, and black. The gray fox displays white on the ears, throat, chest, belly, and hind legs. Gray foxes also have black around their eyes, on the lips, and on their noses.


title: Gray fox

text: In contrast to all "Vulpes" and related (Arctic and fennec) foxes, the gray fox has oval (instead of slit-like) pupils. The gray fox also has reddish coloration on parts of its body, including the legs, sides, feet, chest, and back and sides of the head and neck. The stripe on the fox's tail ends in a black tip as well. Their weight can be similar to that of a red fox, but gray foxes appear smaller because their fur isn't as long and they have shorter limbs.


title: Gray fox

text: The gray fox appeared in North America during the mid-Pliocene (Hemphillian land animal age) epoch ago (AEO) with the first fossil evidence found at the lower 111 Ranch site, Graham County, Arizona with contemporary mammals like the giant sloth, the elephant-like "Cuvieronius", the large-headed llama, and the early small horses of "Nannippus" and "Equus." Faunal remains at two northern California cave sites confirm the presence of the gray fox during the late Pleistocene. Genetic analysis has shown that the gray fox migrated into the northeastern United States post-Pleistocene in association with the Medieval Climate Anomaly warming trend.


title: Gray fox

text: Genetic analyses of the fox-like canids confirmed that the gray fox is a distinct genus from the red foxes ("Vulpes" spp.). The genus "Urocyon" is considered to be the most basal of the living canids. Genetically, the gray fox often clusters with two other ancient lineages: The east Asian raccoon dog ("Nyctereutes procyonoides") and the African bat-eared fox ("Otocyon megalotis").


title: Gray fox

text: The chromosome number is 66 (diploid) with a fundamental number of 70. The autosomes include 31 pairs of sub-graded subacrocentrics, but one only pair of metacentrics.


title: Gray fox

text: Recent mitochondrial genetic studies suggests divergence of North American eastern and western gray foxes in the Irvingtonian mid-Pleistocene into separate sister taxa.


title: Gray fox

text: The gray fox's dwarf relative, the island fox, is likely descended from mainland gray foxes. These foxes apparently were transported by humans to the islands and from island to island, and are descended from a minimum of 3–4 matrilineal founders.


title: Gray fox

text: The species occurs throughout most rocky, wooded, brushy regions of the southern half of North America from southern Canada (Manitoba through southeastern Quebec) to the northern part of South America (Venezuela and Colombia), excluding the mountains of northwestern United States. It is the only canid whose natural range spans both North and South America. In some areas, high population densities exist near brush-covered bluffs.


title: Gray fox

text: The gray fox is specifically adapted to climb trees. Its strong, hooked claws allow it to scramble up trees to escape many predators, such as the domestic dog or the coyote, or to reach tree-bound or arboreal food sources. It can climb branchless, vertical trunks to heights of 18 meters and jump from branch to branch. It descends primarily by jumping from branch to branch, or by descending slowly backwards like a domestic cat. The gray fox is primarily nocturnal or crepuscular and makes its den in hollow trees, stumps or appropriated burrows during the day. Such gray fox tree dens may be located 30 ft above the ground. For the most part, they rest on the ground rather than higher up in trees.


title: Gray fox

text: Prior to European colonization of North America, the red fox was found primarily in boreal forest and the gray fox in deciduous forest. With the increase in human populations in North America, their habitat selection has adapted: Gray foxes that live near human populations tend to choose areas near hardwood trees, locations used primarily by humans, or roads to utilize as their habitat.


title: Gray fox

text: The increase of coyote populations around North America has reduced certain fox populations, so gray foxes have to choose a habitat that will allow them to escape the coyote threat as much as possible, hence the choice of habitat nearer to areas where humans are active. The larger predators of the gray fox, like coyotes and bobcats, tend to avoid human-use areas and paved roads, making this habitat useful for the gray fox. They heavily utilize the edges of forests as a travel corridor, which is used for primary movement from place to place. Their choices do not change based on sex, the season, or the time of day. They also do the majority of their hunting in edges, and use them to escape from predators as well. Gray foxes are thus known as an “edge species”.


title: Gray fox

text: Gray foxes often hunt for the same prey as bobcats and coyotes who occupy the same region. To avoid interspecific competition, the gray fox has developed certain behaviors and habits to increase their survival chances. In regions where gray foxes and coyotes hunt for the same food, the gray fox has been observed to give space to the coyote, staying within its own established range for hunting. Gray foxes might also avoid their competitors by occupying different habitats than them. In California, gray foxes do this by living in chaparral where their competitors are fewer and the low shrubbery provides them a greater chance to escape from a dangerous encounter. It also has been suggested that gray foxes could be more active at night than during the day to avoid their larger, diurnal competitors.


title: Gray fox

text: Still, gray foxes frequently fall victim to bobcats and coyotes. When killed, the carcasses are often unconsumed, suggesting they are victims of intraguild predation. These gray foxes are often killed on or near the boundary of their established range, when they begin to interfere with their competitors. Gray foxes are known as mesopredators because they are mid-tier predators and their prey consists mostly of smaller mammals, while coyotes are known as de facto apex predators due to the removal of other apex predators like wolves in North America. This explains the gray fox's tendency to change behavior in response to the coyote threat, as they are essentially lower on the food chain.


title: Gray fox

text: The gray fox is assumed to be monogamous, like other foxes. The breeding season of the gray fox varies geographically; in Michigan, the gray fox mates in early March, in Alabama, breeding peaks occur in February. The gestation period lasts approximately 53 days. Litter size ranges from 1–7, with a mean of 3.8 young per female.


title: Gray fox

text: The sexual maturity of females is around 10 months of age. Kits begin to hunt with their parents at the age of 3 months. By the time that they are 4 months old, the kits will have developed their permanent dentition and can now easily forage on their own. The family group remains together until the autumn, when the young males reach sexual maturity, then they disperse. In a study of 9 juvenile gray foxes, only the males dispersed up to . The juvenile females stayed within proximity of the den within and always returned. On the other hand, adult gray foxes showed no signs of dispersion for either gender. The gray fox will typically live between six to ten years.


title: Gray fox

text: The annual reproductive cycle of males has been described through epididymal smears and become fertile earlier and remain fertile longer than the fertility of females.


title: Gray fox

text: Logs, trees, rocks, burrows, or abandoned dwellings serve as suitable den sites. Dens are used at any time during the year but mostly during whelping season. Dens are built in brushy or wooded regions and are better concealed than the dens of the red fox.


title: Gray fox

text: The gray fox is an omnivorous, solitary hunter. It frequently preys on the eastern cottontail ("Sylvilagus floridanus") in the eastern U.S., though it will readily catch voles, shrews, and birds. In California, the gray fox primarily eats rodents (such as deer mice, woodrats, and cotton rats), followed by lagomorphs, e.g. jackrabbit, brush rabbit, etc. When available, gray foxes may also feed on carrion. In some parts of the Western United States (such as in the Zion National Park in Utah), the gray fox is primarily insectivorous and herbivorous. Fruit is an important component of the diet of the gray fox and they seek whatever fruits are readily available, generally eating more vegetable matter than does the red fox ("Vulpes vulpes"). Generally, there is an increase in fruits and invertebrates (such as grasshoppers, beetles, butterflies, and moths) within the gray fox's diet in the transition from winter to spring. As nuts, grains, and fruits become more numerous, they are cached by foxes. Typically, they attempt to cover the area with their scent either through their scent glands or urine. This marking serves the dual purpose of allowing them to find the food again later and preventing other animals from taking it.


title: Gray fox

text: Since woodrats, cotton rats, and mice make up a large part of the gray fox's diet, they serve as important regulators of small rodent populations.


title: Gray fox

text: In addition to their beneficial predation on rodents, gray foxes are also less welcome hosts to some external and internal parasites, which include fleas, lice, nematodes, and tapeworms. In the United States, the most common parasite of the gray fox is a flea ("Pulex simulans"); however, several new parasitic arthropods were found in populations in central Mexico, and a warming climate may encourage them to migrate north.


title: Gray fox

text: Gray foxes are hunted in the U.S. The intensity of the hunting has correlated with the value of their pelts. Between the 1970–1971 and 1975–1976 hunting seasons, the price of gray fox pelts greatly increased and the number of individuals hunted jumped over six-fold from 26,109 to 163,458. It has been recently reported that over 500,000 gray foxes are killed every year for their fur.


title: Gray fox

text: Parasites of gray fox include trematode "Metorchis conjunctus". Other common parasites that were collected on gray foxes in Texas were a varity of tapeworms ("Mesocestoides litteratus", "Taenia pisiformis", "Taenia serialis") and roundworms ("Ancylostoma caninum", "Ancylostoma braziliense", "Haemonchus similis", "Spirocerca lupi", "Physaloptera rara", "Eucoleus aerophilus"). "T. pisiformis" was the most common parasite species and was associated with frequent impacts on health.


title: Supernatural (season 5)

text: The fifth season of "Supernatural", an American dark fantasy television series created by Eric Kripke, premiered September 10, 2009, and concluded on May 13, 2010, on The CW. Season five regular cast members include Jared Padalecki, Jensen Ackles and Misha Collins, who was promoted to series regular this season. After the conclusion of this season, series creator Eric Kripke stepped down as showrunner. The fifth season was released on DVD and Blu-ray in region 1 on September 7, 2010. The season follows Sam and Dean as they set out to take down Lucifer, whom Sam had inadvertently released from Hell at the end of the previous season.


title: Supernatural (season 5)

text: Mark Pellegrino plays the role of Lucifer. Paris Hilton guest stars in the episode "Fallen Idols", in which she plays Leshi, a god that takes the form of various well-known faces including herself. The season also features the return of Jo and her mother Ellen Harvelle, as well as hunter Rufus Turner, the Trickster and the prophet Chuck Shurley.


title: Supernatural (season 5)

text: In this table, the number in the first column refers to the episode's number within the entire series, whereas the number in the second column indicates the episode's number within that particular season. "U.S. viewers in millions" refers to how many Americans who watched the episode live or on the day of broadcast.


title: Supernatural (season 5)

text: The fifth season received critical acclaim, and is widely regarded as one of the show's best seasons. The review aggregator website Rotten Tomatoes reported a 90% approval rating for "Supernatural"'s fifth season, with an average rating of 7.83/10 based on 10 reviews. The critics consensus reads, ""Supernatural" continues to benefit from its charismatic leads and a solid script, creating a story that is almost supernaturally addictive."


title: Supernatural (season 5)

text: The episode "Hammer of the Gods" was controversial among Indian and Hindu audiences, as it depicted Kali and Ganesh in a negative light.


title: Paul Ronan

text: In 1995, Ronan's acting career started. Ronan has been a regular performer in TV and films since the mid-'90s, notably as Edso Dowling in the "Ballykissangel" TV series; other TV appearances include "Rebel Heart" and "The Tudors".


title: Fortitude (TV series)

text: Fortitude is a British horror psychological thriller television series created and written by Simon Donald. A 12-episode season was commissioned by Sky Atlantic in 2013, and started airing on 29 January 2015. The series is set in the fictional Arctic Norwegian settlement of Fortitude. On 9 April 2015, Sky Atlantic recommissioned the show for a second season consisting of 10 episodes, which premiered on 26 January 2017. The third and final season premiered on 6 December 2018 and concluded on 27 December, consisting of 4 episodes.


title: Fortitude (TV series)

text: On the Norwegian Arctic island of Fortitude, things appear calm on the surface, but a string of violent and increasingly strange deaths slowly exact a toll on the quiet international community where almost everyone appears to be keeping at least one secret.


title: Fortitude (TV series)

text: After talking to a parasitologist, writer Simon Donald based his parasite on ichneumon wasps, which lay their eggs in hosts.


title: Fortitude (TV series)

text: Fortitude is a fictional community located in Svalbard in Arctic Norway. It is described as an international community, with inhabitants from many parts of the world (population of 713 inhabitants and 4 police officers). The series was filmed in both the UK and in Reyðarfjörður, Iceland.


title: Fortitude (TV series)

text: American actor Stanley Tucci appears in his first British television role as Detective Chief Inspector Morton from London's Metropolitan Police, who has flown to the peaceful close-knit community of Fortitude, as the local sheriff, played by Richard Dormer, investigates a violent murder. Christopher Eccleston plays Charlie Stoddart, a British scientist who leads the arctic biology department at the Fortitude arctic research facility. Michael Gambon portrays Henry Tyson, a wildlife photographer who is dying of cancer. "The Killing" star Sofie Gråbøl plays Hildur Odegard, the governor of Fortitude, in her first UK television drama role.


title: Fortitude (TV series)

text: The first season received positive reviews from critics. On Review aggregator Rotten Tomatoes, the first season has an 86% rating with an average score rating of 7.4 out of 10 based on 28 critic reviews. The critical consensus reads ""Fortitude"s fine cast, deliberate pacing, and chilling setting provide the show with more than enough of the titular quality." On Metacritic, the first season has a score of 75 out of 100, based on 15 critic reviews, which indicates "generally favorable reviews".


title: Fortitude (TV series)

text: The second season received generally positive reviews. On Rotten Tomatoes, the season has an 88% approval rating with an average rating of 6.7 out of 10 based on 8 reviews.


title: Fortitude (TV series)

text: In the United States and Canada, the series premiered on 29 January 2015 (the same date as the UK) on Pivot and Super Channel, respectively. It was also dubbed in Persian language and aired by Manoto. The series debuted in Australia on 15 February 2015 on ABC. Ale Kino+ HD started showing series 3 on 10 February 2019 dubbed in Polish, with original English audio also available.


title: Fortitude (TV series)

text: When FX broadcast the series in Indonesia on 19 February 2017, the violent scenes led to the banning of the channel by Indonesian authorities from 10 to 14 April 2017.


title: Akshara Gowda

text: Akshara Gowda (born Harini Gowda in Bangalore, Karnataka) is an Indian actress who has appeared in Hindi, Tamil, Telugu and Kannada films.


title: Akshara Gowda

text: In April 2011, Akshara Gowda was linked to Indian cricketer Yuvraj Singh. However, Gowda immediately denied it and said it was mere media imagination.


title: Akshara Gowda

text: Akshara made her debut in the Tamil film "Uyarthiru 420" (2011) She played a role in the festival Hindi film "Chitkabrey - The Shades of Grey" the same year. She also made a special appearance in the Tamil film with Vijay in "Thuppakki" (2012) directed by AR Murugadoss. She made her Bollywood debut in Priyadarshan's "Rangrezz" in 2013. Gowda appeared in Ajith-Vishnuvardhan's Tamil film "Arrambam" (2013). She became popular by appearing in "Stylish Thamizhachi" song from the same film "Arrambam" which eventually became instant hit. She is also featured in films like "Irumbu Kuthirai" (2014), "Bogan" (2017), "Sangili Bungili Kadhava Thorae" (2017) and "Maayavan" (2017).


title: Braddock, Pennsylvania

text: Braddock is a borough located in the eastern suburbs of Pittsburgh in Allegheny County, Pennsylvania, upstream from the mouth of the Monongahela River. The population was 1,721 as of the 2020 Census.


title: Braddock, Pennsylvania

text: The borough is represented by the Pennsylvania State Senate's 45th district, the Pennsylvania House of Representatives' 34th district, and in the U.S. House of Representatives.


title: Braddock, Pennsylvania

text: The borough is named for General Edward Braddock (1695–1755), commander of American colonial forces at the start of the French and Indian War. The Braddock Expedition to capture Fort Duquesne (modern day Pittsburgh) from the French led to the British general's own fatal wounding and a sound defeat of his troops after crossing the Monongahela River on July 9, 1755. This battle, now called the Battle of the Monongahela, was a key event at the beginning of the French and Indian War.


title: Braddock, Pennsylvania

text: The area surrounding Braddock's Field was originally inhabited by the Lenape, ruled by Queen Alliquippa. In 1742, John Fraser and his family established the area at the mouth of Turtle Creek as the first permanent English settlement west of the Allegheny Mountains. George Washington visited the area in 1753-1754. It was the site of Braddock's Defeat on July 9, 1755.


title: Braddock, Pennsylvania

text: Braddock's first industrial facility, a barrel plant, opened in 1850. The borough was incorporated on June 8, 1867. The town's industrial economy began in 1873, when Andrew Carnegie built the Edgar Thomson Steel Works on the historic site of Braddock's Field in what is now North Braddock, Pennsylvania. This was one of the first American steel mills which used the Bessemer process. As of 2010, it continues operation as a part of the United States Steel Corporation. This era of the town's history is depicted in Thomas Bell's novel "Out of This Furnace".


title: Braddock, Pennsylvania

text: Braddock is also the location of the first of Andrew Carnegie's 1,679 (some sources list 1,689) public libraries in the US, designed by William Halsey Wood of Newark, New Jersey, and dedicated on March 30, 1889. The Braddock Library included a tunnel entrance for Carnegie's millworkers to enter a bathhouse in the basement to clean up before entering the facilities (which originally included billiard tables). An addition in 1893, by Longfellow, Alden and Harlow (Boston & Pittsburgh, successors to Henry Hobson Richardson), added a swimming pool, indoor basketball court, and 964-seat music hall that included a Votey pipe organ. The building was rescued from demolition in 1978 by the Braddock's Field Historical Society, and is still in use as a public library. The bathhouse has recently been converted to a pottery studio; the music hall is currently under restoration.


title: Braddock, Pennsylvania

text: During the early 1900s many immigrants settled in Braddock, primarily from Croatia, Slovenia, and Hungary.


title: Braddock, Pennsylvania

text: Braddock lost its importance with the collapse of the steel industry in the United States in the 1970s and 1980s. This coincided with the crack cocaine epidemic of the early 1980s, and the combination of the two woes nearly destroyed the community. In 1988, Braddock was designated a financially distressed municipality. The entire water distribution system was rebuilt in 1990-1991 at a cost of $4.7 million, resulting in a fine system where only 5% of piped water is deemed "unaccounted-for.". From its peak in the 1920s, Braddock has since lost 90% of its population.


title: Braddock, Pennsylvania

text: John Fetterman, mayor of Braddock from 2006 until his 2019 inauguration as Lieutenant Governor of Pennsylvania, launched a campaign to attract new residents to the area from the artistic and creative communities. He also initiated various revitalization efforts, including the nonprofit organization Braddock Redux.


title: Braddock, Pennsylvania

text: Fetterman appeared in various media to discuss his vision of Braddock's needs, including PBS, "The Colbert Report" on Comedy Central, CNN, Fox News, CNBC, and "The New York Times". In the UK, "The Guardian" and the BBC have reported on him. He has also had his own episode on Hulu's original series "A Day in the Life".


title: Braddock, Pennsylvania

text: Since 1974, Braddock resident Tony Buba has made many films. One of his earlier films is "Justice League" centering on the borough and its industrial decline, including "Struggles in Steel." In September 2010, the IFC and Sundance television channels showed the film "Ready to Work: Portraits of Braddock", produced by the Levi Strauss corporation. This film interviews many of the local residents and shows their efforts to revitalize the town.


title: Braddock, Pennsylvania

text: According to the United States Census Bureau, the borough has a total area of , of which, is land and (13.85%) is water. Its average elevation is above sea level.


title: Braddock, Pennsylvania

text: Braddock has two land borders, with North Braddock from the north to the southeast, and Rankin to the northwest. Across the Monongahela River to the south, Braddock is adjacent to Whitaker and West Mifflin.


title: Braddock, Pennsylvania

text: "Note: the US Census treats Hispanic/Latino as an ethnic category. This table excludes Latinos from the racial categories and assigns them to a separate category. Hispanics/Latinos can be of any race."


title: Braddock, Pennsylvania

text: According to the American Community Survey, in 2020, Braddock has an employment rate of 34.2%, a median household income of $23,050, 3.7% of the population has no health care coverage, with 10.7% of the population possessing a Bachelor's degree or higher.


title: Braddock, Pennsylvania

text: As of the census of 2000, there were 2,912 people, 1,161 households, and 695 families residing in the borough. The population density was 5,159.9 people per square mile (2,007.7/km²). There were 1,624 housing units at an average density of 2,877.6 per square mile (1,119.7/km²). The racial makeup of the borough was 30.12% White, 66.52% African American, 0.14% Native American, 0.24% Asian, 0.69% from other races, and 2.30% from two or more races. Hispanic or Latino of any race were 1.51% of the population.


title: Braddock, Pennsylvania

text: There were 1,161 households, out of which 30.0% had children under the age of 18 living with them, 21.4% were married couples living together, 31.8% had a female householder with no husband present, and 40.1% were non-families. 37.0% of all households were made up of individuals, and 18.6% had someone living alone who was 65 years of age or older. The average household size was 2.47 and the average family size was 3.24.


title: Braddock, Pennsylvania

text: In the borough the population was spread out, with 31.5% under the age of 18, 6.3% from 18 to 24, 24.9% from 25 to 44, 19.1% from 45 to 64, and 18.3% who were 65 years of age or older. The median age was 36 years. For every 100 females, there were 84.3 males. For every 100 females age 18 and over, there were 78.1 males.


title: Braddock, Pennsylvania

text: The median income for a household in the borough was $18,473, and the median income for a family was $20,669. Males had a median income of $26,333 versus $19,867 for females. The per capita income for the borough was $13,135. About 34.4% of families and 35.0% of the population were below the poverty line, including 54.4% of those under age 18 and 14.5% of those age 65 or over.


title: Ising model

text: The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.


title: Ising model

text: The Ising model was invented by the physicist , who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by alone in his 1924 thesis; it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by . It is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory.


title: Ising model

text: In dimensions greater than four, the phase transition of the Ising model is described by mean-field theory.


title: Ising model

text: The Ising problem without an external field can be equivalently formulated as a graph maximum cut (Max-Cut) problem that can be solved via combinatorial optimization.


title: Ising model

text: Consider a set Λ of lattice sites, each with a set of adjacent sites (e.g. a graph) forming a "d"-dimensional lattice. For each lattice site "k" ∈ Λ there is a discrete variable σ such that σ  ∈ {+1, −1}, representing the site's spin. A "spin configuration", σ = (σ) is an assignment of spin value to each lattice site.


title: Ising model

text: For any two adjacent sites "i", "j" ∈ Λ there is an "interaction" "J". Also a site "j" ∈ Λ has an "external magnetic field" "h" interacting with it. The "energy" of a configuration σ is given by the Hamiltonian function


title: Ising model

text: where the first sum is over pairs of adjacent spins (every pair is counted once). The notation ⟨"ij"⟩ indicates that sites "i" and "j" are nearest neighbors. The magnetic moment is given by µ. Note that the sign in the second term of the Hamiltonian above should actually be positive because the electron's magnetic moment is antiparallel to its spin, but the negative term is used conventionally. The "configuration probability" is given by the Boltzmann distribution with inverse temperature β ≥ 0:


title: Ising model

text: The configuration probabilities "P"(σ) represent the probability that (in equilibrium) the system is in a state with configuration σ.


title: Ising model

text: The minus sign on each term of the Hamiltonian function "H"(σ) is conventional. Using this sign convention, Ising models can be classified according to the sign of the interaction: if, for a pair "i", "j"


title: Ising model

text: The system is called ferromagnetic or antiferromagnetic if all interactions are ferromagnetic or all are antiferromagnetic. The original Ising models were ferromagnetic, and it is still often assumed that "Ising model" means a ferromagnetic Ising model.


title: Ising model

text: In a ferromagnetic Ising model, spins desire to be aligned: the configurations in which adjacent spins are of the same sign have higher probability. In an antiferromagnetic model, adjacent spins tend to have opposite signs.


title: Ising model

text: The sign convention of "H"(σ) also explains how a spin site "j" interacts with the external field. Namely, the spin site wants to line up with the external field. If:


title: Ising model

text: Ising models are often examined without an external field interacting with the lattice, that is, "h" = 0 for all "j" in the lattice Λ. Using this simplification, the Hamiltonian becomes


title: Ising model

text: When the external field is zero everywhere, "h" = 0, the Ising model is symmetric under switching the value of the spin in all the lattice sites; a nonzero field breaks this symmetry.


title: Ising model

text: Another common simplification is to assume that all of the nearest neighbors ⟨"ij"⟩ have the same interaction strength. Then we can set "J" = "J" for all pairs "i", "j" in Λ. In this case the Hamiltonian is further simplified to


title: Ising model

text: A subset S of the vertex set V(G) of a weighted undirected graph G determines a cut of the graph G into S and its complementary subset G\S. The size of the cut is the sum of the weights of the edges between S and G\S. A maximum cut size is at least the size of any other cut, varying S.


title: Ising model

text: For the Ising model without an external field on a graph G, the Hamiltonian becomes the following sum over the graph edges E(G)


title: Ising model

text: Here each vertex i of the graph is a spin site that takes a spin value formula_14. A given spin configuration formula_15 partitions the set of vertices formula_16 into two formula_15-depended subsets, those with spin up formula_18 and those with spin down formula_19. We denote by formula_20 the formula_15-depended set of edges that connects the two complementary vertex subsets formula_18 and formula_19. The "size" formula_24 of the cut formula_20 to bipartite the weighted undirected graph G can be defined as


title: Ising model

text: where formula_27 denotes a weight of the edge formula_28 and the scaling 1/2 is introduced to compensate for double counting the same weights formula_29.


title: Ising model

text: where the total sum in the first term does not depend on formula_15, imply that minimizing formula_32 in formula_15 is equivalent to minimizing formula_34. Defining the edge weight formula_35 thus turns the Ising problem without an external field into a graph Max-Cut problem


title: Ising model

text: A significant number of statistical questions to ask about this model are in the limit of large numbers of spins:


title: Ising model

text: The most studied case of the Ising model is the translation-invariant ferromagnetic zero-field model on a "d"-dimensional lattice, namely, Λ = Z, "J" = 1, "h" = 0.


title: Ising model

text: In his 1924 PhD thesis, Ising solved the model for the "d" = 1 case, which can be thought of as a linear horizontal lattice where each site only interacts with its left and right neighbor. In one dimension, the solution admits no phase transition. Namely, for any positive β, the correlations ⟨σσ⟩ decay exponentially in |"i" − "j"|:


title: Ising model

text: and the system is disordered. On the basis of this result, he incorrectly concluded that this model does not exhibit phase behaviour in any dimension.


title: Ising model

text: The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. Namely, the system is disordered for small β, whereas for large β the system exhibits ferromagnetic order:


title: Ising model

text: The Ising model on a two-dimensional square lattice with no magnetic field was analytically solved by . Onsager showed that the correlation functions and free energy of the Ising model are determined by a noninteracting lattice fermion. Onsager announced the formula for the spontaneous magnetization for the 2-dimensional model in 1949 but did not give a derivation. gave the first published proof of this formula, using a limit formula for Fredholm determinants, proved in 1951 by Szegő in direct response to Onsager's work.


title: Ising model

text: A number of correlation inequalities has been derived rigorously for the Ising spin correlations (for general lattice structures), which enabled mathematicians to study the Ising model both on and off criticality.


title: Ising model

text: which means that spins are positively correlated on the Ising ferromagnet. An immediate application of this is that the magnetization of any set of spins formula_43 is increasing with respect to any set of coupling constants formula_44.


title: Ising model

text: The Simon-Lieb inequality states that for any set formula_45 disconnecting formula_46 from formula_47 (e.g. the boundary of a box with formula_46 being inside the box and formula_47 being outside),


title: Ising model

text: This inequality is proven first for a type of positively-correlated percolation model, of which includes a representation of the Ising model. It is used to determine the critical temperatures of planar Potts model using percolation arguments (which includes the Ising model as a special case).


title: Ising model

text: One of Democritus' arguments in support of atomism was that atoms naturally explain the sharp phase boundaries observed in materials, as when ice melts to water or water turns to steam. His idea was that small changes in atomic-scale properties would lead to big changes in the aggregate behavior. Others believed that matter is inherently continuous, not atomic, and that the large-scale properties of matter are not reducible to basic atomic properties.


title: Ising model

text: While the laws of chemical bonding made it clear to nineteenth century chemists that atoms were real, among physicists the debate continued well into the early twentieth century. Atomists, notably James Clerk Maxwell and Ludwig Boltzmann, applied Hamilton's formulation of Newton's laws to large systems, and found that the statistical behavior of the atoms correctly describes room temperature gases. But classical statistical mechanics did not account for all of the properties of liquids and solids, nor of gases at low temperature.


title: Ising model

text: Once modern quantum mechanics was formulated, atomism was no longer in conflict with experiment, but this did not lead to a universal acceptance of statistical mechanics, which went beyond atomism. Josiah Willard Gibbs had given a complete formalism to reproduce the laws of thermodynamics from the laws of mechanics. But many faulty arguments survived from the 19th century, when statistical mechanics was considered dubious. The lapses in intuition mostly stemmed from the fact that the limit of an infinite statistical system has many zero-one laws which are absent in finite systems: an infinitesimal change in a parameter can lead to big differences in the overall, aggregate behavior, as Democritus expected.


title: Ising model

text: In the early part of the twentieth century, some believed that the partition function could never describe a phase transition, based on the following argument:


title: Ising model

text: This argument works for a finite sum of exponentials, and correctly establishes that there are no singularities in the free energy of a system of a finite size. For systems which are in the thermodynamic limit (that is, for infinite systems) the infinite sum can lead to singularities. The convergence to the thermodynamic limit is fast, so that the phase behavior is apparent already on a relatively small lattice, even though the singularities are smoothed out by the system's finite size.


title: Ising model

text: Shortly after Lenz and Ising constructed the Ising model, Peierls was able to explicitly show that a phase transition occurs in two dimensions.


title: Ising model

text: To do this, he compared the high-temperature and low-temperature limits. At infinite temperature (β = 0) all configurations have equal probability. Each spin is completely independent of any other, and if typical configurations at infinite temperature are plotted so that plus/minus are represented by black and white, they look like television snow. For high, but not infinite temperature, there are small correlations between neighboring positions, the snow tends to clump a little bit, but the screen stays randomly looking, and there is no net excess of black or white.


title: Ising model

text: A bogus argument analogous to the argument in the last section now establishes that the magnetization in the Ising model is always zero.


title: Ising model

text: As before, this only proves that the average magnetization is zero at any finite volume. For an infinite system, fluctuations might not be able to push the system from a mostly plus state to a mostly minus with a nonzero probability.


title: Ising model

text: For very high temperatures, the magnetization is zero, as it is at infinite temperature. To see this, note that if spin A has only a small correlation ε with spin B, and B is only weakly correlated with C, but C is otherwise independent of A, the amount of correlation of A and C goes like ε. For two spins separated by distance "L", the amount of correlation goes as ε, but if there is more than one path by which the correlations can travel, this amount is enhanced by the number of paths.


title: Ising model

text: A bound on the total correlation is given by the contribution to the correlation by summing over all paths linking two points, which is bounded above by the sum over all paths of length "L" divided by


title: Ising model

text: At low temperatures (β ≫ 1) the configurations are near the lowest-energy configuration, the one where all the spins are plus or all the spins are minus. Peierls asked whether it is statistically possible at low temperature, starting with all the spins minus, to fluctuate to a state where most of the spins are plus. For this to happen, droplets of plus spin must be able to congeal to make the plus state.


title: Ising model

text: The energy of a droplet of plus spins in a minus background is proportional to the perimeter of the droplet L, where plus spins and minus spins neighbor each other. For a droplet with perimeter "L", the area is somewhere between ("L" − 2)/2 (the straight line) and ("L"/4) (the square box). The probability cost for introducing a droplet has the factor "e", but this contributes to the partition function multiplied by the total number of droplets with perimeter "L", which is less than the total number of paths of length "L":


title: Ising model

text: So that the total spin contribution from droplets, even overcounting by allowing each site to have a separate droplet, is bounded above by


title: Ising model

text: which goes to zero at large β. For β sufficiently large, this exponentially suppresses long loops, so that they cannot occur, and the magnetization never fluctuates too far from −1.


title: Ising model

text: So Peierls established that the magnetization in the Ising model eventually defines superselection sectors, separated domains not linked by finite fluctuations.


title: Ising model

text: Kramers and Wannier were able to show that the high-temperature expansion and the low-temperature expansion of the model are equal up to an overall rescaling of the free energy. This allowed the phase-transition point in the two-dimensional model to be determined exactly (under the assumption that there is a unique critical point).


title: Ising model

text: After Onsager's solution, Yang and Lee investigated the way in which the partition function becomes singular as the temperature approaches the critical temperature.


title: Ising model

text: The Ising model can often be difficult to evaluate numerically if there are many states in the system. Consider an Ising model with


title: Ising model

text: Since every spin site has ±1 spin, there are "2" different states that are possible. This motivates the reason for the Ising model to be simulated using Monte Carlo methods.


title: Ising model

text: The Hamiltonian that is commonly used to represent the energy of the model when using Monte Carlo methods is


title: Ising model

text: Furthermore, the Hamiltonian is further simplified by assuming zero external field "h", since many questions that are posed to be solved using the model can be answered in absence of an external field. This leads us to the following energy equation for state σ:


title: Ising model

text: Given this Hamiltonian, quantities of interest such as the specific heat or the magnetization of the magnet at a given temperature can be calculated.


title: Ising model

text: The Metropolis–Hastings algorithm is the most commonly used Monte Carlo algorithm to calculate Ising model estimations. The algorithm first chooses "selection probabilities" "g"(μ, ν), which represent the probability that state ν is selected by the algorithm out of all states, given that one is in state μ. It then uses acceptance probabilities "A"(μ, ν) so that detailed balance is satisfied. If the new state ν is accepted, then we move to that state and repeat with selecting a new state and deciding to accept it. If ν is not accepted then we stay in μ. This process is repeated until some stopping criterion is met, which for the Ising model is often when the lattice becomes ferromagnetic, meaning all of the sites point in the same direction.


title: Ising model

text: When implementing the algorithm, one must ensure that "g"(μ, ν) is selected such that ergodicity is met. In thermal equilibrium a system's energy only fluctuates within a small range. This is the motivation behind the concept of single-spin-flip dynamics, which states that in each transition, we will only change one of the spin sites on the lattice. Furthermore, by using single- spin-flip dynamics, one can get from any state to any other state by flipping each site that differs between the two states one at a time.


title: Ising model

text: The maximum amount of change between the energy of the present state, "H" and any possible new state's energy "H" (using single-spin-flip dynamics) is 2"J" between the spin we choose to "flip" to move to the new state and that spin's neighbor. Thus, in a 1D Ising model, where each site has two neighbors (left and right), the maximum difference in energy would be 4"J".


title: Ising model

text: Let "c" represent the lattice coordination number; the number of nearest neighbors that any lattice site has. We assume that all sites have the same number of neighbors due to periodic boundary conditions. It is important to note that the Metropolis–Hastings algorithm does not perform well around the critical point due to critical slowing down. Other techniques such as multigrid methods, Niedermayer's algorithm, Swendsen–Wang algorithm, or the Wolff algorithm are required in order to resolve the model near the critical point; a requirement for determining the critical exponents of the system.


title: Ising model

text: Specifically for the Ising model and using single-spin-flip dynamics, one can establish the following.


title: Ising model

text: Since there are "L" total sites on the lattice, using single-spin-flip as the only way we transition to another state, we can see that there are a total of "L" new states ν from our present state μ. The algorithm assumes that the selection probabilities are equal to the "L" states: "g"(μ, ν) = 1/"L". Detailed balance tells us that the following equation must hold:


title: Ising model

text: If "H" > "H", then "A"(ν, μ) > "A"(μ, ν). Metropolis sets the larger of "A"(μ, ν) or "A"(ν, μ) to be 1. By this reasoning the acceptance algorithm is:


title: Ising model

text: The change in energy "H" − "H" only depends on the value of the spin and its nearest graph neighbors. So if the graph is not too connected, the algorithm is fast. This process will eventually produce a pick from the distribution.


title: Ising model

text: It is possible to view the Ising model as a Markov chain, as the immediate probability "P"(ν) of transitioning to a future state ν only depends on the present state μ. The Metropolis algorithm is actually a version of a Markov chain Monte Carlo simulation, and since we use single-spin-flip dynamics in the Metropolis algorithm, every state can be viewed as having links to exactly "L" other states, where each transition corresponds to flipping a single spin site to the opposite value. Furthermore, since the energy equation "H" change only depends on the nearest-neighbor interaction strength "J", the Ising model and its variants such the Sznajd model can be seen as a form of a voter model for opinion dynamics.


title: Ising model

text: In the nearest neighbor case (with periodic or free boundary conditions) an exact solution is available. The Hamiltonian of the one-dimensional Ising model on a lattice of "L" sites with periodic boundary conditions is


title: Ising model

text: where "J" and "h" can be any number, since in this simplified case "J" is a constant representing the interaction strength between the nearest neighbors and "h" is the constant external magnetic field applied to lattice sites. Then the


title: Ising model

text: where "C"(β) and "c"(β) are positive functions for "T" > 0. For "T" → 0, though, the inverse correlation length "c"(β) vanishes.


title: Ising model

text: If "h" = 0, it is very easy to obtain the free energy in the case of free boundary condition, i.e. when


title: Ising model

text: hence it decays exponentially as soon as "T" ≠ 0; but for "T" = 0, i.e. in the limit β → ∞ there is no decay.


title: Ising model

text: If "h" ≠ 0 we need the transfer matrix method. For the periodic boundary conditions case is the following. The partition function is


title: Ising model

text: The coefficients formula_75 can be seen as the entries of a matrix. There are different possible choices: a convenient one (because the matrix is symmetric) is


title: Ising model

text: The energy of the lowest state is −"JL", when all the spins are the same. For any other configuration, the extra energy is equal to 2"J" times the number of sign changes that are encountered when scanning the configuration from left to right.


title: Ising model

text: If we designate the number of sign changes in a configuration as "k", the difference in energy from the lowest energy state is 2"k". Since the energy is additive in the number of flips, the probability "p" of having a spin-flip at each position is independent. The ratio of the probability of finding a flip to the probability of not finding one is the Boltzmann factor:


title: Ising model

text: The problem is reduced to independent biased coin tosses. This essentially completes the mathematical description.


title: Ising model

text: From the description in terms of independent tosses, the statistics of the model for long lines can be understood. The line splits into domains. Each domain is of average length exp(2β). The length of a domain is distributed exponentially, since there is a constant probability at any step of encountering a flip. The domains never become infinite, so a long system is never magnetized. Each step reduces the correlation between a spin and its neighbor by an amount proportional to "p", so the correlations fall off exponentially.


title: Ising model

text: The partition function is the volume of configurations, each configuration weighted by its Boltzmann weight. Since each configuration is described by the sign-changes, the partition function factorizes:


title: Ising model

text: which is analytic away from β = ∞. A sign of a phase transition is a non-analytic free energy, so the one-dimensional model does not have a phase transition.


title: Ising model

text: To express the Ising Hamiltonian using a quantum mechanical description of spins, we replace the spin variables with their respective Pauli matrices. However, depending on the direction of the magnetic field, we can create a transverse-field or longitudinal-field Hamiltonian. The transverse-field Hamiltonian is given by


title: Ising model

text: The transverse-field model experiences a phase transition between an ordered and disordered regime at "J" ~ "h". This can be shown by a mapping of Pauli matrices


title: Ising model

text: obtained the following analytical expression for the free energy of the Ising model on the anisotropic square lattice when the magnetic field formula_88 in the thermodynamic limit as a function of temperature and the horizontal and vertical interaction energies formula_89 and formula_90, respectively


title: Ising model

text: From this expression for the free energy, all thermodynamic functions of the model can be calculated by using an appropriate derivative. The 2D Ising model was the first model to exhibit a continuous phase transition at a positive temperature. It occurs at the temperature formula_92 which solves the equation


title: Ising model

text: In the isotropic case when the horizontal and vertical interaction energies are equal formula_94, the critical temperature formula_92 occurs at the following point


title: Ising model

text: When the interaction energies formula_89, formula_90 are both negative, the Ising model becomes an antiferromagnet. Since the square lattice is bi-partite, it is invariant under this change when the magnetic field formula_88, so the free energy and critical temperature are the same for the antiferromagnetic case. For the triangular lattice, which is not bi-partite, the ferromagnetic and antiferromagnetic Ising model behave notably differently.


title: Ising model

text: Start with an analogy with quantum mechanics. The Ising model on a long periodic lattice has a partition function


title: Ising model

text: Think of the "i" direction as "space", and the "j" direction as "time". This is an independent sum over all the values that the spins can take at each time slice. This is a type of path integral, it is the sum over all spin histories.


title: Ising model

text: A path integral can be rewritten as a Hamiltonian evolution. The Hamiltonian steps through time by performing a unitary rotation between time "t" and time "t" + Δ"t":


title: Ising model

text: The product of the U matrices, one after the other, is the total time evolution operator, which is the path integral we started with.


title: Ising model

text: where "N" is the number of time slices. The sum over all paths is given by a product of matrices, each matrix element is the transition probability from one slice to the next.


title: Ising model

text: Similarly, one can divide the sum over all partition function configurations into slices, where each slice is the one-dimensional configuration at time 1. This defines the transfer matrix:


title: Ising model

text: The configuration in each slice is a one-dimensional collection of spins. At each time slice, "T" has matrix elements between two configurations of spins, one in the immediate future and one in the immediate past. These two configurations are "C" and "C", and they are all one-dimensional spin configurations. We can think of the vector space that "T" acts on as all complex linear combinations of these. Using quantum mechanical notation:


title: Ising model

text: Like the Hamiltonian, the transfer matrix acts on all linear combinations of states. The partition function is a matrix function of T, which is defined by the sum over all histories which come back to the original configuration after "N" steps:


title: Ising model

text: Since this is a matrix equation, it can be evaluated in any basis. So if we can diagonalize the matrix "T", we can find "Z".


title: Ising model

text: The contribution to the partition function for each past/future pair of configurations on a slice is the sum of two terms. There is the number of spin flips in the past slice and there is the number of spin flips between the past and future slice. Define an operator on configurations which flips the spin at site i:


title: Ising model

text: In the usual Ising basis, acting on any linear combination of past configurations, it produces the same linear combination but with the spin at position i of each basis vector flipped.


title: Ising model

text: Define a second operator which multiplies the basis vector by +1 and −1 according to the spin at position "i":


title: Ising model

text: where "A" and "B" are constants which are to be determined so as to reproduce the partition function. The interpretation is that the statistical configuration at this slice contributes according to both the number of spin flips in the slice, and whether or not the spin at position "i" has flipped.


title: Ising model

text: Just as in the one-dimensional case, we will shift attention from the spins to the spin-flips. The σ term in "T" counts the number of spin flips, which we can write in terms of spin-flip creation and annihilation operators:


title: Ising model

text: Ignore the constant coefficients, and focus attention on the form. They are all quadratic. Since the coefficients are constant, this means that the "T" matrix can be diagonalized by Fourier transforms.


title: Ising model

text: Onsager famously announced the following expression for the spontaneous magnetization "M" of a two-dimensional Ising ferromagnet on the square lattice at two different conferences in 1948, though without proof


title: Ising model

text: A complete derivation was only given in 1951 by using a limiting process of transfer matrix eigenvalues. The proof was subsequently greatly simplified in 1963 by Montroll, Potts, and Ward using Szegő's limit formula for Toeplitz determinants by treating the magnetization as the limit of correlation functions.


title: Ising model

text: At the critical point, the two-dimensional Ising model is a two-dimensional conformal field theory. The spin and energy correlation functions are described by a minimal model, which has been exactly solved.


title: Ising model

text: In three as in two dimensions, the most studied case of the Ising model is the translation-invariant model on a cubic lattice with nearest-neighbor coupling in the zero magnetic field. Many theoreticians searched for an analytical three-dimensional solution for many decades, which would be analogous to Onsager's solution in the two-dimensional case. Such a solution has not been found until now, although there is no proof that it may not exist.


title: Ising model

text: In three dimensions, the Ising model was shown to have a representation in terms of non-interacting fermionic strings by Alexander Polyakov and Vladimir Dotsenko. This construction has been carried on the lattice, and the continuum limit, conjecturally describing the critical point, is unknown.


title: Ising model

text: In three as in two dimensions, Peierl's argument shows that there is a phase transition. This phase transition is rigorously known to be continuous (in the sense that correlation length diverges and the magnetization goes to zero), and is called the critical point. It is believed that the critical point can be described by a renormalization group fixed point of the Wilson-Kadanoff renormalization group transformation. It is also believed that the phase transition can be described by a three-dimensional unitary conformal field theory, as evidenced by Monte Carlo simulations, exact diagonalization results in quantum models, and quantum field theoretical arguments. Although it is an open problem to establish rigorously the renormalization group picture or the conformal field theory picture, theoretical physicists have used these two methods to compute the critical exponents of the phase transition, which agree with the experiments and with the Monte Carlo simulations.


title: Ising model

text: This conformal field theory describing the three-dimensional Ising critical point is under active investigation using the method of the conformal bootstrap. This method currently yields the most precise information about the structure of the critical theory (see Ising critical exponents).


title: Ising model

text: In 2000, Sorin Istrail of Sandia National Laboratories proved that the spin glass Ising model on a nonplanar lattice is NP-complete. That is, assuming P ≠ NP, the general spin glass Ising model is exactly solvable only in planar cases, so solutions for dimensions higher that two are also intractable. Istrail's result only concerns the spin glass model with spatially varying couplings, and tells nothing about Ising's original ferromagnetic model with equal couplings.


title: Ising model

text: In any dimension, the Ising model can be productively described by a locally varying mean field. The field is defined as the average spin value over a large region, but not so large so as to include the entire system. The field still has slow variations from point to point, as the averaging volume moves. These fluctuations in the field are described by a continuum field theory in the infinite system limit.


title: Ising model

text: The field "H" is defined as the long wavelength Fourier components of the spin variable, in the limit that the wavelengths are long. There are many ways to take the long wavelength average, depending on the details of how high wavelengths are cut off. The details are not too important, since the goal is to find the statistics of "H" and not the spins. Once the correlations in "H" are known, the long-distance correlations between the spins will be proportional to the long-distance correlations in "H".


title: Ising model

text: For any value of the slowly varying field "H", the free energy (log-probability) is a local analytic function of "H" and its gradients. The free energy "F"("H") is defined to be the sum over all Ising configurations which are consistent with the long wavelength field. Since "H" is a coarse description, there are many Ising configurations consistent with each value of "H", so long as not too much exactness is required for the match.


title: Ising model

text: Since the allowed range of values of the spin in any region only depends on the values of "H" within one averaging volume from that region, the free energy contribution from each region only depends on the value of "H" there and in the neighboring regions. So "F" is a sum over all regions of a local contribution, which only depends on "H" and its derivatives.


title: Ising model

text: By symmetry in "H", only even powers contribute. By reflection symmetry on a square lattice, only even powers of gradients contribute. Writing out the first few terms in the free energy:


title: Ising model

text: On a square lattice, symmetries guarantee that the coefficients "Z" of the derivative terms are all equal. But even for an anisotropic Ising model, where the "Z"'s in different directions are different, the fluctuations in "H" are isotropic in a coordinate system where the different directions of space are rescaled.


title: Ising model

text: is a positive definite quadratic form, and can be used to "define" the metric for space. So any translationally invariant Ising model is rotationally invariant at long distances, in coordinates that make "Z" = δ. Rotational symmetry emerges spontaneously at large distances just because there aren't very many low order terms. At higher order multicritical points, this accidental symmetry is lost.


title: Ising model

text: Since β"F" is a function of a slowly spatially varying field, the probability of any field configuration is:


title: Ising model

text: The denominator in this expression is called the "partition function", and the integral over all possible values of "H" is a statistical path integral. It integrates exp(β"F") over all values of "H", over all the long wavelength fourier components of the spins. "F" is a Euclidean Lagrangian for the field "H", the only difference between this and the quantum field theory of a scalar field being that all the derivative terms enter with a positive sign, and there is no overall factor of "i".


title: Ising model

text: The form of "F" can be used to predict which terms are most important by dimensional analysis. Dimensional analysis is not completely straightforward, because the scaling of "H" needs to be determined.


title: Ising model

text: In the generic case, choosing the scaling law for "H" is easy, since the only term that contributes is the first one,


title: Ising model

text: This term is the most significant, but it gives trivial behavior. This form of the free energy is ultralocal, meaning that it is a sum of an independent contribution from each point. This is like the spin-flips in the one-dimensional Ising model. Every value of "H" at any point fluctuates completely independently of the value at any other point.


title: Ising model

text: The scale of the field can be redefined to absorb the coefficient "A", and then it is clear that "A" only determines the overall scale of fluctuations. The ultralocal model describes the long wavelength high temperature behavior of the Ising model, since in this limit the fluctuation averages are independent from point to point.


title: Ising model

text: To find the critical point, lower the temperature. As the temperature goes down, the fluctuations in "H" go up because the fluctuations are more correlated. This means that the average of a large number of spins does not become small as quickly as if they were uncorrelated, because they tend to be the same. This corresponds to decreasing "A" in the system of units where "H" does not absorb "A". The phase transition can only happen when the subleading terms in "F" can contribute, but since the first term dominates at long distances, the coefficient "A" must be tuned to zero. This is the location of the critical point:


title: Ising model

text: Since "t" is vanishing, fixing the scale of the field using this term makes the other terms blow up. Once "t" is small, the scale of the field can either be set to fix the coefficient of the "H" term or the (∇"H") term to 1.


title: Ising model

text: To find the magnetization, fix the scaling of "H" so that λ is one. Now the field "H" has dimension −"d"/4, so that "H""dx" is dimensionless, and "Z" has dimension 2 − "d"/2. In this scaling, the gradient term is only important at long distances for "d" ≤ 4. Above four dimensions, at long wavelengths, the overall magnetization is only affected by the ultralocal terms.


title: Ising model

text: There is one subtle point. The field "H" is fluctuating statistically, and the fluctuations can shift the zero point of "t". To see how, consider "H" split in the following way:


title: Ising model

text: The first term is a constant contribution to the free energy, and can be ignored. The second term is a finite shift in "t". The third term is a quantity that scales to zero at long distances. This means that when analyzing the scaling of "t" by dimensional analysis, it is the shifted "t" that is important. This was historically very confusing, because the shift in "t" at any finite "λ" is finite, but near the transition "t" is very small. The fractional change in "t" is very large, and in units where "t" is fixed the shift looks infinite.


title: Ising model

text: The magnetization is at the minimum of the free energy, and this is an analytic equation. In terms of the shifted "t",


title: Ising model

text: For "t" < 0, the minima are at "H" proportional to the square root of "t". So Landau's catastrophe argument is correct in dimensions larger than 5. The magnetization exponent in dimensions higher than 5 is equal to the mean-field value.


title: Ising model

text: When "t" is negative, the fluctuations about the new minimum are described by a new positive quadratic coefficient. Since this term always dominates, at temperatures below the transition the fluctuations again become ultralocal at long distances.


title: Ising model

text: To find the behavior of fluctuations, rescale the field to fix the gradient term. Then the length scaling dimension of the field is 1 − "d"/2. Now the field has constant quadratic spatial fluctuations at all temperatures. The scale dimension of the "H" term is 2, while the scale dimension of the "H" term is 4 − "d". For "d" < 4, the "H" term has positive scale dimension. In dimensions higher than 4 it has negative scale dimensions.


title: Ising model

text: This is an essential difference. In dimensions higher than 4, fixing the scale of the gradient term means that the coefficient of the "H" term is less and less important at longer and longer wavelengths. The dimension at which nonquadratic contributions begin to contribute is known as the critical dimension. In the Ising model, the critical dimension is 4.


title: Ising model

text: In dimensions above 4, the critical fluctuations are described by a purely quadratic free energy at long wavelengths. This means that the correlation functions are all computable from as Gaussian averages:


title: Ising model

text: valid when "x" − "y" is large. The function "G"("x" − "y") is the analytic continuation to imaginary time of the Feynman propagator, since the free energy is the analytic continuation of the quantum field action for a free scalar field. For dimensions 5 and higher, all the other correlation functions at long distances are then determined by Wick's theorem. All the odd moments are zero, by ± symmetry. The even moments are the sum over all partition into pairs of the product of "G"("x" − "y") for each pair.


title: Ising model

text: where "C" is the proportionality constant. So knowing "G" is enough. It determines all the multipoint correlations of the field.


title: Ising model

text: To determine the form of "G", consider that the fields in a path integral obey the classical equations of motion derived by varying the free energy:


title: Ising model

text: This is valid at noncoincident points only, since the correlations of "H" are singular when points collide. "H" obeys classical equations of motion for the same reason that quantum mechanical operators obey them—its fluctuations are defined by a path integral.


title: Ising model

text: At the critical point "t" = 0, this is Laplace's equation, which can be solved by Gauss's method from electrostatics. Define an electric field analog by


title: Ising model

text: since "G" is spherically symmetric in "d" dimensions, and "E" is the radial gradient of "G". Integrating over a large "d" − 1 dimensional sphere,


title: Ising model

text: When "t" does not equal zero, so that "H" is fluctuating at a temperature slightly away from critical, the two point function decays at long distances. The equation it obeys is altered:


title: Ising model

text: For "r" small compared with formula_133, the solution diverges exactly the same way as in the critical case, but the long distance behavior is modified.


title: Ising model

text: To see how, it is convenient to represent the two point function as an integral, introduced by Schwinger in the quantum field theory context:


title: Ising model

text: This is "G", since the Fourier transform of this integral is easy. Each fixed τ contribution is a Gaussian in "x", whose Fourier transform is another Gaussian of reciprocal width in "k".


title: Ising model

text: This is the inverse of the operator ∇ − "t" in "k"-space, acting on the unit function in "k"-space, which is the Fourier transform of a delta function source localized at the origin. So it satisfies the same equation as "G" with the same boundary conditions that determine the strength of the divergence at 0.


title: Ising model

text: The interpretation of the integral representation over the "proper time" τ is that the two point function is the sum over all random walk paths that link position 0 to position "x" over time τ. The density of these paths at time τ at position "x" is Gaussian, but the random walkers disappear at a steady rate proportional to "t" so that the Gaussian at time τ is diminished in height by a factor that decreases steadily exponentially. In the quantum field theory context, these are the paths of relativistically localized quanta in a formalism that follows the paths of individual particles. In the pure statistical context, these paths still appear by the mathematical correspondence with quantum fields, but their interpretation is less directly physical.


title: Ising model

text: The integral representation immediately shows that "G"("r") is positive, since it is represented as a weighted sum of positive Gaussians. It also gives the rate of decay at large r, since the proper time for a random walk to reach position τ is r and in this time, the Gaussian height has decayed by formula_136. The decay factor appropriate for position "r" is therefore formula_137.


title: Ising model

text: This is not an exact form, except in three dimensions, where interactions between paths become important. The exact forms in high dimensions are variants of Bessel functions.


title: Ising model

text: The interpretation of the correlations as fixed size quanta travelling along random walks gives a way of understanding why the critical dimension of the "H" interaction is 4. The term "H" can be thought of as the square of the density of the random walkers at any point. In order for such a term to alter the finite order correlation functions, which only introduce a few new random walks into the fluctuating environment, the new paths must intersect. Otherwise, the square of the density is just proportional to the density and only shifts the "H" coefficient by a constant. But the intersection probability of random walks depends on the dimension, and random walks in dimension higher than 4 do not intersect.


title: Ising model

text: The fractal dimension of an ordinary random walk is 2. The number of balls of size ε required to cover the path increase as ε. Two objects of fractal dimension 2 will intersect with reasonable probability only in a space of dimension 4 or less, the same condition as for a generic pair of planes. Kurt Symanzik argued that this implies that the critical Ising fluctuations in dimensions higher than 4 should be described by a free field. This argument eventually became a mathematical proof.


title: Ising model

text: The Ising model in four dimensions is described by a fluctuating field, but now the fluctuations are interacting. In the polymer representation, intersections of random walks are marginally possible. In the quantum field continuation, the quanta interact.


title: Ising model

text: The numerical factors are there to simplify the equations of motion. The goal is to understand the statistical fluctuations. Like any other non-quadratic path integral, the correlation functions have a Feynman expansion as particles travelling along random walks, splitting and rejoining at vertices. The interaction strength is parametrized by the classically dimensionless quantity λ.


title: Ising model

text: Although dimensional analysis shows that both λ and "Z" are dimensionless, this is misleading. The long wavelength statistical fluctuations are not exactly scale invariant, and only become scale invariant when the interaction strength vanishes.


title: Ising model

text: The reason is that there is a cutoff used to define "H", and the cutoff defines the shortest wavelength. Fluctuations of "H" at wavelengths near the cutoff can affect the longer-wavelength fluctuations. If the system is scaled along with the cutoff, the parameters will scale by dimensional analysis, but then comparing parameters doesn't compare behavior because the rescaled system has more modes. If the system is rescaled in such a way that the short wavelength cutoff remains fixed, the long-wavelength fluctuations are modified.


title: Ising model

text: A quick heuristic way of studying the scaling is to cut off the "H" wavenumbers at a point λ. Fourier modes of "H" with wavenumbers larger than λ are not allowed to fluctuate. A rescaling of length that make the whole system smaller increases all wavenumbers, and moves some fluctuations above the cutoff.


title: Ising model

text: To restore the old cutoff, perform a partial integration over all the wavenumbers which used to be forbidden, but are now fluctuating. In Feynman diagrams, integrating over a fluctuating mode at wavenumber "k" links up lines carrying momentum "k" in a correlation function in pairs, with a factor of the inverse propagator.


title: Ising model

text: Under rescaling, when the system is shrunk by a factor of (1+"b"), the "t" coefficient scales up by a factor (1+"b") by dimensional analysis. The change in "t" for infinitesimal "b" is 2"bt". The other two coefficients are dimensionless and do not change at all.


title: Ising model

text: This equation is an identity inside any correlation function away from other insertions. After integrating out the modes with Λ < "k" < (1+"b")Λ, it will be a slightly different identity.


title: Ising model

text: Since the form of the equation will be preserved, to find the change in coefficients it is sufficient to analyze the change in the "H" term. In a Feynman diagram expansion, the "H" term in a correlation function inside a correlation has three dangling lines. Joining two of them at large wavenumber "k" gives a change "H" with one dangling line, so proportional to "H":


title: Ising model

text: The first part is not proportional to "t", and in the equation of motion it can be absorbed by a constant shift in "t". It is caused by the fact that the "H" term has a linear part. Only the second term, which varies from "t" to "t", contributes to the critical scaling.


title: Ising model

text: This new linear term adds to the first term on the left hand side, changing "t" by an amount proportional to "t". The total change in "t" is the sum of the term from dimensional analysis and this second term from operator products:


title: Ising model

text: So "t" is rescaled, but its dimension is anomalous, it is changed by an amount proportional to the value of λ.


title: Ising model

text: But λ also changes. The change in λ requires considering the lines splitting and then quickly rejoining. The lowest order process is one where one of the three lines from "H" splits into three, which quickly joins with one of the other lines from the same vertex. The correction to the vertex is


title: Ising model

text: The numerical factor is three times bigger because there is an extra factor of three in choosing which of the three new lines to contract. So


title: Ising model

text: and is proportional to the area of a three-dimensional sphere of radius λ, times the width of the integration region "b"Λ divided by Λ:


title: Ising model

text: In other dimensions, the constant "B" changes, but the same constant appears both in the "t" flow and in the coupling flow. The reason is that the derivative with respect to "t" of the closed loop with a single vertex is a closed loop with two vertices. This means that the only difference between the scaling of the coupling and the "t" is the combinatorial factors from joining and splitting.


title: Ising model

text: To investigate three dimensions starting from the four-dimensional theory should be possible, because the intersection probabilities of random walks depend continuously on the dimensionality of the space. In the language of Feynman graphs, the coupling does not change very much when the dimension is changed.


title: Ising model

text: The process of continuing away from dimension 4 is not completely well defined without a prescription for how to do it. The prescription is only well defined on diagrams. It replaces the Schwinger representation in dimension 4 with the Schwinger representation in dimension 4 − ε defined by:


title: Ising model

text: The solutions to this equation are the possible consistent mean fields. For β"J" < 1 there is only the one solution at "H" = 0. For bigger values of β there are three solutions, and the solution at "H" = 0 is unstable.


title: Ising model

text: The instability means that increasing the mean field above zero a little bit produces a statistical fraction of spins which are + which is bigger than the value of the mean field. So a mean field which fluctuates above zero will produce an even greater mean field, and will eventually settle at the stable solution. This means that for temperatures below the critical value β"J" = 1 the mean-field Ising model undergoes a phase transition in the limit of large "N".


title: Ising model

text: Above the critical temperature, fluctuations in "H" are damped because the mean field restores the fluctuation to zero field. Below the critical temperature, the mean field is driven to a new equilibrium value, which is either the positive "H" or negative "H" solution to the equation.


title: Ising model

text: For β"J" = 1 + ε, just below the critical temperature, the value of "H" can be calculated from the Taylor expansion of the hyperbolic tangent:


title: Ising model

text: The spontaneous magnetization "H" grows near the critical point as the square root of the change in temperature. This is true whenever "H" can be calculated from the solution of an analytic equation which is symmetric between positive and negative values, which led Landau to suspect that all Ising type phase transitions in all dimensions should follow this law.


title: Ising model

text: The mean-field exponent is universal because changes in the character of solutions of analytic equations are always described by catastrophes in the Taylor series, which is a polynomial equation. By symmetry, the equation for "H" must only have odd powers of "H" on the right hand side. Changing β should only smoothly change the coefficients. The transition happens when the coefficient of "H" on the right hand side is 1. Near the transition:


title: Ising model

text: Whatever "A" and "B" are, so long as neither of them is tuned to zero, the spontaneous magnetization will grow as the square root of ε. This argument can only fail if the free energy β"F" is either non-analytic or non-generic at the exact β where the transition occurs.


title: Ising model

text: But the spontaneous magnetization in magnetic systems and the density in gasses near the critical point are measured very accurately. The density and the magnetization in three dimensions have the same power-law dependence on the temperature near the critical point, but the behavior from experiments is:


title: Ising model

text: The exponent is also universal, since it is the same in the Ising model as in the experimental magnet and gas, but it is not equal to the mean-field value. This was a great surprise.


title: Ising model

text: In three dimensions, the perturbative series from the field theory is an expansion in a coupling constant λ which is not particularly small. The effective size of the coupling at the fixed point is one over the branching factor of the particle paths, so the expansion parameter is about 1/3. In two dimensions, the perturbative expansion parameter is 2/3.


title: Ising model

text: But renormalization can also be productively applied to the spins directly, without passing to an average field. Historically, this approach is due to Leo Kadanoff and predated the perturbative ε expansion.


title: Ising model

text: The idea is to integrate out lattice spins iteratively, generating a flow in couplings. But now the couplings are lattice energy coefficients. The fact that a continuum description exists guarantees that this iteration will converge to a fixed point when the temperature is tuned to criticality.


title: Ising model

text: Write the two-dimensional Ising model with an infinite number of possible higher order interactions. To keep spin reflection symmetry, only even powers contribute:


title: Ising model

text: By translation invariance, "J" is only a function of i-j. By the accidental rotational symmetry, at large i and j its size only depends on the magnitude of the two-dimensional vector "i" − "j". The higher order coefficients are also similarly restricted.


title: Ising model

text: The renormalization iteration divides the lattice into two parts – even spins and odd spins. The odd spins live on the odd-checkerboard lattice positions, and the even ones on the even-checkerboard. When the spins are indexed by the position ("i","j"), the odd sites are those with "i" + "j" odd and the even sites those with "i" + "j" even, and even sites are only connected to odd sites.


title: Ising model

text: The two possible values of the odd spins will be integrated out, by summing over both possible values. This will produce a new free energy function for the remaining even spins, with new adjusted couplings. The even spins are again in a lattice, with axes tilted at 45 degrees to the old ones. Unrotating the system restores the old configuration, but with new parameters. These parameters describe the interaction between spins at distances formula_156 larger.


title: Ising model

text: Starting from the Ising model and repeating this iteration eventually changes all the couplings. When the temperature is higher than the critical temperature, the couplings will converge to zero, since the spins at large distances are uncorrelated. But when the temperature is critical, there will be nonzero coefficients linking spins at all orders. The flow can be approximated by only considering the first few terms. This truncated flow will produce better and better approximations to the critical exponents when more terms are included.


title: Ising model

text: The simplest approximation is to keep only the usual "J" term, and discard everything else. This will generate a flow in "J", analogous to the flow in "t" at the fixed point of λ in the ε expansion.


title: Ising model

text: To find the change in "J", consider the four neighbors of an odd site. These are the only spins which interact with it. The multiplicative contribution to the partition function from the sum over the two values of the spin at the odd site is:


title: Ising model

text: where "N" is the number of neighbors which are ±. Ignoring the factor of 2, the free energy contribution from this odd site is: