---
abstract: |
    A model with a singular forward scattering amplitude for particles with opposite spins in d spatial dimensions is proposed and solved by using the bosonization transformation. This interacting potential leads to the spin-charge separation. Thermal properties at low temperature for this Luttinger liquid are discussed. Also, the explicit form of the single-electron Green function is found; it has square-root branch cut. New fermion field operators are defined; they describe holons and spinons as the elementary excitations. Their single particle Green functions possess pseudoparticle properties. Using these operators the spin-charge separated Hamiltonian for an ideal gases of holons and spinons is derived and reflects an inverse (fermionization) transformation.\
    PACS Nos.71.10.+x, 71.27.+a
address: |
    (a) Institute of Theoretical Physics, Warsaw University, ul. Hoża 69, 00-681 Warszawa, Poland\
    (b) Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland 
author:
- 'Krzysztof Byczuk$^{a}$ and Jozef Spałek$^{b,a}$ [^1]'
title: 'Spin-Charge Separated Luttinger Liquid in Arbitrary Spatial Dimensions '
---

It was suggested [@and1] that the properties of normal state of high-temperature superconductors are properly described by Luttinger liquid, where the spin and charge degrees of freedom are separated. In one-dimensional systems this phenomenon is well understood [@rew]. However, in two and three dimensions the present understanding of the spin-charge separation is rather poor. In this letter we formulate and solve exactly a d-dimensional model exhibiting the spin-charge separation, as well as discuss its thermal and dynamic properties.

A natural approach to study spin-charge decoupling phenomena is the bosonization transformation, generalized recently to the multidimensional space situation [@hal]. Here we adopt the operator version of the bosonization developed in Ref.[@how]. The starting assumption in this method is the existence of the Fermi surface (FS) defined as a collection of points at which the momentum distribution function has singularities at zero temperature ($T=0$). These points are parameterized by vectors $\bf S$ and ${\bf T}$, which label a finite and a locally flat (rectangular in shape) mesh of grid points on FS with spacing $\Lambda \ll k_F$ between them [@how; @hal]. Introducing coarse-grained density fluctuation operators $J_{\sigma}({\bf S},{\bf q})$, defined in boxes centered at each FS point and having surface area $\Lambda^{d-1}$ and the thicknesses $\lambda/2$ both above and below it, one can transform the effective Hamiltonian for interacting fermions into an effective Hamiltonian for free bosons. Explicitly, it takes the general form $$H = \frac{1}{2} \sum_{{\bf S}, {\bf T}} \sum_{\bf q} \sum_{\sigma \sigma'}
\Gamma_{\sigma \sigma'} ({\bf S}, {\bf T}, {\bf q}) J_{\sigma}({\bf S}, {\bf
q})J_{\sigma'}({\bf T}, - {\bf q}),
\label{e7}$$ where $
\Gamma_{\sigma \sigma'} ({\bf S}, {\bf T}, {\bf q}) = v_F( {\bf S})
\frac{1}{\Omega} \delta_{\sigma, \sigma'}
\delta^{d-1}_{{\bf S}, {\bf T}} +
\frac{1}{L^d} V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q})
%\label{e8}
$ is the positive defined matrix element. The first term corresponds to the kinetic energy part of the original fermionic Hamiltonian with linearized dispersion relation close to FS, whereas the second term is the effective (low-energy) interaction between the particles with spins $\sigma$ and $\sigma'$. The geometrical factor $\Omega = \Lambda^{d-1} (\frac{L}{2 \pi})^d$ depends on the system dimension $d$.

The explicit expression for $V_{\sigma \sigma'}({\bf S},{\bf T}, {\bf q})$ is generally derived by transforming out the high energy modes in the fermionic Hamiltonian [@shankar]. Obviously, this procedure can also change the Fermi velocity $v_F({\bf S})$. Therefore, we take $v_F({\bf S})$ as an effective value obtained after removing the high-energy degrees of freedom. As shown below, we can characterize the universal properties of fermions knowing only the asymptotic behavior of the interaction potential $V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q})$ in the thermodynamic limit.

If the system is invariant under the time reversal, the interaction part must be explicitly symmetric under this operation, which means that $
V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q}) = V_{\bar{\sigma}
\bar{\sigma}'}(-{\bf S}, -{\bf T}, - {\bf q})
$. Furthermore, if FS is also invariant under the reflections ${\bf S} \rightarrow - {\bf S}$ etc., the last condition becomes $
V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q}) = V_{\bar{\sigma}
\bar{\sigma}'}({\bf S}, {\bf T},  {\bf q})
$. In that case $V_{\sigma \sigma'}({\bf S}, {\bf T}, {\bf q})$ depends only on the relative orientation of the spins $\sigma$ and $\sigma'$; there are only two independent components: $V_{\sigma \sigma}$ for parallel spins and $V_{\sigma \bar{\sigma}}$ for antiparallel spins. It is convenient to introduce the symmetric and the antisymmetric combinations: $
V^{c,s}({\bf S},{\bf T},{\bf q}) \equiv \frac{1}{2} (V_{\sigma \sigma}({\bf S},
{\bf T}, {\bf q})
\pm V_{\sigma \bar{\sigma}}({\bf S}, {\bf T}, {\bf q}) ),
$ where $c$ and $s$ superscripts corresponds to “$\pm$” signs, respectively. Correspondingly, we define the currents $
J_{c,s}
({\bf S}, {\bf q}) \equiv \frac{1}{\sqrt{2}} ( J_{\uparrow}({\bf S}, {\bf q})
\pm J_{\downarrow}({\bf S}, {\bf q})),
%\label{e13}
$ which describe the charge and the spin density fluctuations, respectively. Then, the original Hamiltonian (\[e7\]) takes the form $$H = \sum_{\alpha = c,s}
\frac{1}{2} \sum_{{\bf S}} v_F({\bf S}) \frac{1}{\Omega} \sum_{{\bf q}}
J_{\alpha}({\bf S}, {\bf q}) J_{\alpha}({\bf S}, - {\bf q}) +
 \frac{1}{L^d} \sum_{{\bf S}, {\bf T}} \sum_{{\bf q}}
V^{\alpha}({\bf S}, {\bf T}, {\bf q}) J_{\alpha}({\bf S}, {\bf q}) J_{\alpha}(
{\bf T}, -{\bf q}).
\label{e15}$$ The $\alpha =c$ term describes the dynamics of the charge density fluctuations in the system, whereas the $\alpha = s$ term deals with the longitudinal spin density fluctuations. There are no terms which mix the degrees of freedom (i.e. $ \sim J_c \cdot J_s $) because the Hamiltonian is assumed to be invariant under the spin flip (i.e. $J_c \rightarrow J_c$, $J_s \rightarrow - J_s$). We can check out that for the noninteracting case, i.e. for $V^{\alpha}\equiv
0$, both the spin and the charge density fluctuations propagate with the same velocity $v_F({\bf S})$. The commutation relation for density fluctuation operators take the following form $
\left[
J_{\alpha}({\bf S}, {\bf q}) ,  J_{\beta}({\bf T}, {\bf p})
\right] = \delta_{\alpha \beta} \delta^{d-1}_{{\bf S}, {\bf T}} \delta^d_{ {\bf
p} + {\bf q},0}\:
\Omega \: {\bf q} \cdot \hat{n}_{\bf S},
%\label{e19}
$ where $\alpha , \beta = c,s$. Thus, the two branches of density fluctuations are independent of each other. The commutation relations become equivalent to those obeyed by the bosonic harmonic-oscillator creation and annihilation operators after rescaling them by the factor on the right hand side, i.e. by defining the creation ($a_{\alpha}^+$) and the annihilation ($a_{\alpha}$) operators according to $$J_{\alpha}({\bf S}, {\bf q}) = \theta( \hat{n}_{{\bf S}} \cdot {\bf q})
\sqrt{\Omega \hat{n}_{{\bf S}} \cdot {\bf q}}
\; a_{\alpha}({\bf S}, {\bf q}) +
\theta(- \hat{n} _{\bf S} \cdot {\bf q}) \sqrt{-\Omega \hat{n}_{{\bf S}} \cdot
{\bf q}}
\; a_{\alpha}^+({\bf S},- {\bf q}),$$ where $\theta(x) $ is the step function. Then $
\left[
a_{\alpha}({\bf S}, {\bf q}), a_{\beta}^+({\bf T}, {\bf p})
\right]
= \delta^{d-1}_{{\bf S}, {\bf T}} \delta_{{\bf p},{\bf q}}^d \delta_{\alpha
\beta}.
$ In terms of the bosonic harmonic-oscillator creation and annihilation operators the Hamiltonian (\[e15\]) is $$H = \sum_{\alpha = c,s} \sum_{{\bf S}, {\bf T}} \frac{1}{2}
\sum_{{\bf q}} \left\{ \theta( \hat{n}_{{\bf S}} \cdot {\bf q})
\theta(\hat{n}_{{\bf T}}\cdot {\bf q})
\sqrt{ ({\bf v}_F({\bf S})  \cdot {\bf q}) ({\bf v}_F({\bf T}) \cdot {\bf q})}
\; \times
\right.
\label{e22}$$ $$\left( \delta_{{\bf S}, {\bf T}}^{d-1} +
 \frac{\Lambda^{d-1}}{(2 \pi)^d}
\frac{V^{\alpha}({\bf S},{\bf T},{\bf q})}{\sqrt{v_F({\bf S})v_F({\bf T})}}
\right)
a_{\alpha}^+({\bf S},{\bf q}) \: a_{\alpha}({\bf T}, {\bf q}) \; +$$ $$\theta(- \hat{n}_{{\bf S}} \cdot {\bf q}) \theta(-\hat{n}_{{\bf T}}\cdot {\bf
q})
\sqrt{ (-{\bf v}_F({\bf S})  \cdot {\bf q}) (-{\bf v}_F({\bf T}) \cdot {\bf
q})} \; \times$$ $$\left( \delta_{{\bf S}, {\bf T}}^{d-1}  +
 \frac{\Lambda^{d-1} }{(2 \pi)^d}
\frac{V^{\alpha}({\bf S},{\bf T},{\bf q})}{\sqrt{v_F({\bf S})v_F({\bf T})}}
\right)
a_{\alpha}^+({\bf S},-{\bf q})\: a_{\alpha}({\bf T},- {\bf q}) \; +$$ $$\theta( \hat{n}_{{\bf S}} \cdot {\bf q}) \theta(-\hat{n}_{{\bf T}}\cdot {\bf
q})
\sqrt{ ({\bf v}_F({\bf S})  \cdot {\bf q}) (-{\bf v}_F({\bf T}) \cdot {\bf q})}
\left(   \frac{\Lambda^{d-1}}{(2 \pi)^d}
\frac{V^{\alpha}({\bf S},{\bf T},{\bf q})}{\sqrt{v_F({\bf S})v_F({\bf T})}}
\right)
a_{\alpha}({\bf S},{\bf q})\: a_{\alpha}({\bf T}, -{\bf q}) \; +$$ $$\theta(- \hat{n}_{{\bf S}} \cdot {\bf q}) \theta(\hat{n}_{{\bf T}}\cdot {\bf
q})
\sqrt{ (-{\bf v}_F({\bf S})  \cdot {\bf q}) ({\bf v}_F({\bf T}) \cdot {\bf q})}
\left.
\left(  \frac{ \Lambda^{d-1}}{(2 \pi)^d}
\frac{V^{\alpha}({\bf S},{\bf T},{\bf q})}{\sqrt{v_F({\bf S})v_F({\bf T})}}
\right)
a_{\alpha}^+({\bf S},-{\bf q})\: a_{\alpha}^+({\bf T}, {\bf q})
\right\}.$$

We have shown previously [@my; @phd] that a universal behavior of the system takes place and depends on how the interaction part of the effective Hamiltonian (\[e22\]) behaves in the scaling (thermodynamic) limit $\Lambda \rightarrow 0$. If the interaction in this limit has a singular power-law behavior, i.e. $V_{\sigma,\bar{\sigma}}({\bf S},{\bf T},{\bf q})$ scales for ${\bf S} \rightarrow {\bf T}$ as $\Lambda^{d-1} / \Lambda^{\eta}$, then: (i) for $\eta < d-1$ the Landau Fermi liquid (FL) fixed point is stable; (ii) for $\eta > d-1$ the statistical spin liquid (SSL) is the stable fixed point; (iii) the case $\eta = d-1$ leads to the Luttinger liquid (LL) type of behavior. Properties of FL in $d=3$ and $2$ are well known [@lfl]. SSL was discussed in Refs.[@ssl; @my; @phd]. Here we concentrate on the properties of the spin-charge separated LL in an arbitrary spatial dimension.

To examine the basic properties of Luttinger liquid we assume the following form of the effective interaction in (\[e7\]): $$V_{\uparrow \uparrow} ({\bf S}, {\bf T}, {\bf q}) = f_{\uparrow \uparrow}({\bf
S}, {\bf T}, {\bf q}),
%\label{e33}$$ and $$V_{\uparrow \downarrow} = \left\{
\begin{array}{ccc}
\frac{1}{\Lambda^{d-1}} \tilde{g}({\bf S}, {\bf q}) & for & {\bf S} = {\bf T}
\\
f_{\uparrow \downarrow} ({\bf S}, {\bf T}, {\bf q}) & for & {\bf S} \neq {\bf
T} ,
\end{array}
\right.
\label{e34}$$ where $f_{\sigma \sigma'} ({\bf S}, {\bf T}, {\bf q})$ and $\tilde{g}({\bf S}, {\bf q}) $ are nonsingular functions. In this manner, we model the situation with a divergent forward scattering amplitude (taking place for ${\bf S} = {\bf T}$ and ${\bf q}=0$), which leads to the LL fixed point. Also, since we examine only the low-energy limit, we can expand the nonsingular part according to: $
\tilde{g}({\bf S}, {\bf q}) = g_0({\bf S}) + g({\bf S}) \hat{n}_{{\bf S}} \cdot
{\bf q} + ...
$, and omit the higher-order terms. Substituting this expansion into (\[e34\]) we obtain in the thermodynamic limit ($\Lambda \rightarrow 0$) the Hamiltonian with two branches of free bosons excitations, each with a different form of the kinetic energy. Namely, defining $
v_F^{c,s} ({\bf S}) \equiv v_F({\bf S}) \pm g({\bf S}),
$ the Hamiltonian (\[e22\]) simplifies to $$H = \sum_{\alpha = c , s}
\sum_{{\bf S}, {\bf q}> 0} ({\bf v}_F^{\alpha}({\bf S}) \cdot {\bf q} ) \;
a_{\alpha}^+ ({\bf S}, {\bf q})
\; a_{\alpha}({\bf S}, {\bf q}).
\label{e40}$$ The charge and the spin fluctuations propagate in the system with different velocities and express the separation of the corresponding degrees of freedom. One of the velocities diminishes and the other increases. In the extreme case one of them vanishes transforming into a soft mode, signalling a phase transition in the corresponding channel, $c$ or $s$. The character of this transition will be determined by the regular part. The spin-charge decoupling takes always place in the one-dimensional systems of interacting fermions in the low energy limit [@rew]. In a system of higher dimension the potential must be singular. Obviously, this is a simplified model. More appealing form would be $
V_{\uparrow \downarrow}({\bf S}, {\bf T}, {\bf q}) \sim \frac{1}{
|{\bf S} - {\bf T}|^{\eta} + |{\bf q}|^{\eta} },
$ which clearly diverges in the forward direction (for ${\bf q} \rightarrow 0$). Such a singular effective potential in $d=2$ was discussed in Refs.[@and2; @fqhe; @stamp] in the context of the high-temperature superconductors and the fractional quantum Hall effect. Our choice (\[e34\]) is modeled by the most divergent term of this potential, and physically means that the fermions with antiparalel spins interact through the forward scattering processes along the radial FS direction only.

Since the Hamiltonian (\[e40\]) is diagonal we can calculate the internal energy of the system and then the specific heat. The energy is $$U = E_0 + \sum_{\alpha=c,s}
\sum_{{\bf S}} \sum_{{\bf q}} \frac{ {\bf v}_F^{\alpha}({\bf S}) \cdot {\bf q}}
{e^{\beta {\bf v}_F^{\alpha}({\bf S}) \cdot {\bf q}} - 1},
\label{e42}$$ where we utilize the fact that now we are dealing with bosons. $E_0$ is the ground state energy of the initial noninteracting system. This term must be incorporated because in the bosonization procedure the energy is measured with respect to the Fermi energy. Also, the chemical potential $\mu$ does not appear in (\[e42\]) because the number of bosons is not conserved. In other words, those fields describe the system particle-hole excitations. This is also the reason why those bosons cannot condense. For an isotropic system the velocities do not depend on the index labeling the point of FS, i.e. $v_F^{\alpha}({\bf S}) = v_F^{\alpha}$, and we find that $
U = \frac{\pi^2}{6} (k_B T)^2 \left( \frac{L}{2 \pi} \right) ^d
\frac{d \pi^{d/2}}{\Gamma(d/2 + 1)} k_F^{d-1} \sum_{\alpha}
\frac{1}{v_F^{\alpha}}
+ E_0.
$ Hence, the specific heat is $
C_V = \frac{1}{L^d} \frac{\partial U}{\partial T} =
\frac{\pi^2}{3} (k_B )^2 T \left( \frac{1}{2 \pi} \right) ^d
\frac{d \pi^{d/2}}{\Gamma(d/2 + 1)} k_F^{d-1} \sum_{\alpha}
\frac{1}{v_F^{\alpha}}.
%\label{e44}
$ Introducing the density of states for the charge and the spin excitations on FS: $
\rho_{\alpha}(\epsilon_F) = \left( \frac{1}{2 \pi} \right) ^d
\frac{d \pi^{d/2}}{\Gamma(d/2 + 1)} k_F^{d-1}  \frac{1}{v_F^{\alpha}},
$ we have that $
C_V = \frac{\pi^2}{3} (k_B )^2 T \sum_{\alpha} \rho_{\alpha} (\epsilon_F)
\equiv
\gamma_{LL} T.
$ The spin-charge separated liquid has a linear specific heat, with $
\gamma_{LL} \sim \frac{1}{v_F^c} + \frac{1}{v_F^s}.
$ The linear specific heat is thus a general characteristic following from the existence of the FS, independently of the statistical properties of the particles. In the limiting case $v_F^c = v_F^s = v_F$ we recover the FL result. Similarly, we find the free energy in the form $
F= E_0 - \frac{\pi^2}{6} (k_BT)^2 \sum_{\alpha} \rho_{\alpha}(\epsilon_F),
$ and the low-$T$ entropy $
S = \frac{U-F}{T} = \frac{\pi^2}{3} (k_B)^2 T \sum_{\alpha}
\rho_{\alpha}(\epsilon_F),
$ which coincides with the specific heat. Thus the low-T thermal properties of the present Luttinger spin-charge separated liquid are very similar to those of free fermions. The only difference is in the form of the density of states at FS. However, the dynamic properties in the LL case are quite unique, as we discuss next.

We define the fermion correlation function as $$G^>_{\sigma} ({\bf S}, {\bf x}, t > 0) = < \psi_{\sigma}({\bf S}, {\bf x}, t)
\psi_{\sigma}^+ ({\bf S}, 0, 0)>.
\label{e50}$$ To derive an explicit form of this function we substitute the bosonized Fermi field operators [@how] $
\psi_{\sigma} ({\bf S}, {\bf x}) = \sqrt{\frac{\Omega}{a}} e^{i {\bf k}_{{\bf
S}} \cdot {\bf x}}
e^{i \frac{\sqrt{4 \pi}}{\Omega} \phi_{\sigma} ({\bf S},{\bf x})} \hat{O}({\bf
S}),
$ and utilize the identity $
e^A e^B = :e^{A+B}: e^{<AB - \frac{1}{2}(A^2 + B^2)>},
$ where $::$ means that $:e^{A+B}:$ is a normal ordered product of operators. Then, we have that $
G^>_{\sigma} ({\bf S}, {\bf x}, t) = \frac{\Omega}{a} exp \left(
{\frac{4 \pi}{\Omega ^2} \frac{1}{2}
G_B^{\sigma}({\bf S}, {\bf x}, t)} \right),
$ where $G_B^{\sigma}$ is expressed via the Bose fields, namely $
G_B^{\sigma}({\bf S}, {\bf x}, t) = < (\phi_c({\bf S}, {\bf x}, t) + \sigma
\phi_s({\bf S}, {\bf x}, t))
(\phi_c({\bf S}, 0, 0) + \sigma \phi_s({\bf S}, 0, 0))> -
<(\phi_c({\bf S}, 0, 0) + \sigma \phi_s({\bf S}, 0, 0))^2 >.
$ Next, using the Heisenberg representation for the boson field operators $\phi_{\alpha} ({\bf S},{\bf x},t)$, and subsequently decomposing them into the Fourier components, we find the explicit form of the boson correlation function: $
G_B^{\sigma}({\bf S}, {\bf x}, t) = - \frac{\Omega^2}{4 \pi}
\ln \left( \frac{ (\hat{n}_{{\bf S}}\cdot {\bf x} - v_F^c t+ ia) (
\hat{n}_{{\bf S}} \cdot {\bf x} -
v_F^s t + ia)}{ (ia)^2} \right) .
$ Hence the fermion correlation function is $$G_{\sigma}^> ({\bf S}, {\bf x}, t) = i \Omega \frac{ e^{i {\bf k}_{{\bf S}}
\cdot {\bf x}}
}{ \sqrt{\hat{n}_{{\bf S}}\cdot {\bf x} - v_F^c t + ia }
\sqrt{ \hat{n}_{{\bf S}} \cdot {\bf x} -
v_F^s t + ia }
}.
\label{e55}$$ It is independent of the spin index $\sigma$. We see that instead of a quasiparticle pole, taking place in the FL case, we have now a branch cut ranging from $v_F^s$ to $v_F^c$. This branch cut survives when we transform $G^>_{\sigma}({\bf S}, {\bf x}, t)$ into $G^>_{\sigma}({\bf S}, {\bf k}, \omega)$ as can be easily seen by decomposing the argument into normal and transverse parts: ${\bf k} \cdot {\bf x} =
k ( \hat{n}_{{\bf S}} \cdot {\bf x} + {\bf t} \cdot {\bf x})$, where ${\bf t} k$ is the transverse part of ${\bf k}$, and noting that the part $\hat{n}_{{\bf S}} \cdot {\bf x}$ can be transformed in the same manner, as in the $d=1$ case [@and3]. The ${\bf t} \cdot {\bf x}$ part has a trivial form. This means that the analytic character of the LL Green function is the same in both $d=1$ and $d>1$ cases. This universal character follows from the relation $
G^>({\bf S}, {\bf k}, \omega) \equiv G^> ( {\bf S}, \hat{n}_{{\bf S}}\cdot {\bf
k}, \omega)
\delta_{
{\bf t} \cdot {\bf k}, 0}$. This very significant result tells us also that there are no quasiparticle excitations having a direct relation to the noninteracting particles in this charge-spin separated system. In other words, when we put a single electron forming a wave packet on FS, it dissociates into many wave packets propagating with velocities ranging form $v_F^s$ to $v_F^c$. This nonperturbative result means that there is no one-to-one correspondence between the dynamics of LL liquid system and the system of non-interacting fermions. Note that the distribution function $\bar{n}_{\bf k}$ is in our model situation at $T=0$ a step function with a jump at $k_F$, as in the FL case. The inclusion the nonsingular part of the interaction will change this step distribution for a finite-volume system.

The fundamental question arises if we can still define a proper fermionic pseudoparticles in this spin-charge separated liquid. To construct such a state we consider the field operators $\psi_c$ and $\psi_s$, defined through the following fermionization transformation $$\psi_{c,s} ({\bf S}, {\bf x}) = \sqrt{\frac{\Omega}{a}} e^{i {\bf k}_{{\bf S}}
\cdot {\bf x}}
e^{i \frac{\sqrt{4 \pi}}{\Omega} \phi_{c,s} ({\bf S},{\bf x})} \hat{O}({\bf
S}).
\label{e58}$$ The operators $\psi_{\alpha}({\bf S},{\bf x})$ obey proper anticommutation relations. By the procedure similar to that employed in deriving (\[e55\]), we now have the correlation function in the new fermionic variables in the form $$G^>_{\alpha} ( {\bf S}, {\bf x}, t>0)
\equiv <\psi_{\alpha}({\bf S}, {\bf x}, t) \psi_{\alpha}^+({\bf S},0,0)>
= i \Omega \frac{ e^{ i {\bf k}_{{\bf S}} \cdot {\bf x} }
}{ \hat{n}_{{\bf S}} \cdot {\bf x} - v_F^{\alpha} t + ia
},$$ which has simple poles. Also, one can show that $[ \psi^+_{\alpha} ({\bf S},{\bf x}), \int d {\bf y} J_{\alpha}({\bf S}, {\bf
y}) ]
=  \psi^+_{\alpha}({\bf S}, {\bf x})$, which means that $\psi^+_{\alpha}( {\bf S},{\bf x})$ changes the total number of either charge $(\alpha=c)$ or spin $(\alpha=s)$ of the system at the FS point ${\bf S}$ by one unit. Hence, the operators (\[e58\]) represent new fermionic pseudoparticles for the interacting non-Fermi liquid. Those single-particle excitations are called the [**holons**]{} for $\alpha = c$ and the [**spinons**]{} for $\alpha =s$. They are the only single-particle excitations across FS in our model system. Since they obey the fermion statistics, the contribution to the specific heat is of the fermionic type. In other words, the spin-charge separated liquid is composed of ideal gases of spinons and holons. They represent the exact eigenstates of the system. Effectively, we have the following Hamiltonian for noninteracting fermion pseudoparticles $$H = \sum_{\alpha = c,s} \sum_{{\bf S}} v_F^{\alpha}
({\bf S}) \int d {\bf x} \; \psi^+_{\alpha}({\bf S}, {\bf x}) \left(
\frac{\hat{n}_{{\bf S}} \cdot \nabla}{i} \right) \psi_{\alpha}({\bf S}, {\bf
x}) .
\label{nonham}$$ Since we have treated the interaction between fermions exactly in the thermodynamic limit ($\Lambda \rightarrow 0$), the spectrum of our new noninteracting Hamiltonian (\[nonham\]) is exactly the same as that of the interacting Hamiltonian (\[e7\]). To conclude, the two branches of elementary excitations in this LL can be represented either as boson (collective) or as fermion (single-particle) excitations.

Having determined the excitation spectrum of the singular part of interaction, which resulted in the spin-charge separated liquid, we can now include in (\[nonham\]) the nonsingular parts of the interaction, transformed to the fermionic representation (\[e58\]). Since for the former part we determined an exact state of the system, we can now treat the residual interaction among them as a perturbation, i.e. regard the system of interacting holons and spinons as being in one-to-one correspondence to the system of noninteracting holons and spinons. This statement can be proved by referring to the Gell-Mann and Low theorem which means that the evolution operator is well defined in all orders, since the interactions among the holons and the spinons are nonsingular functions. In other words, the adiabatic “switching on” procedure is justified. Here we discuss only a semiclassical approach to the interacting holons and spinons in the bosonic language. The equations of motion for $J_{\alpha}({\bf S},{\bf q})$ operators are $$i \frac{\partial}{\partial t} J_{\alpha}({\bf S}, {\bf q},t) =
{\bf v}_F^{\alpha}({\bf S}) \cdot {\bf q} \; J_{\alpha}({\bf S},{\bf q},t) +
{\bf q} \cdot \hat{n}_{{\bf S}} \; \Lambda^{d-1} \left( \frac{1}{2\pi}
\right)^d \sum_{{\bf T}} f_{\alpha}({\bf S} - {\bf T}) J_{\alpha}({\bf S}, {\bf
q},t),$$ where we supposed that the interaction $f_{\alpha}$ does not depend on ${\bf
q}$. In the semiclassical approach we take the expectation value of $J_{\alpha}$, i.e. define $
u_{\alpha}({\bf S},{\bf q},t) \equiv < J_{\alpha}({\bf S},{\bf q},t)>$. This quantity measures the shape deformation of FS with respect to the ground state form. Furthermore, focusing our attention on the single Fourier mode $u_{\alpha}(t)  = e^{-i \omega t}u_{\alpha}$, we find the integral equation for the collective excitation spectrum amplitude $u_{\alpha} \equiv u_{\alpha} (\tilde{\Omega},{\bf q})$, in the form $$(v_F^{\alpha} q \cos \theta - \omega ) u_{\alpha} (\tilde{\Omega}, q) =
q v_F^{\alpha} \cos \theta \int \frac{d \tilde{\Omega}'}{S_d}
F_{\alpha}(\tilde{\Omega}, \tilde{\Omega}')
u_{\alpha}(\tilde{\Omega}', q),
\label{e63}$$ where $S_d = \int d \tilde{\Omega}$, $F_{\alpha}(\tilde{\Omega},
\tilde{ \Omega}') = \rho(\epsilon_F)
f_{\alpha}(\tilde{\Omega},
\tilde{\Omega}')$, and $\tilde{\Omega} $ is the solid angle. This is an equation of motion for either the holon ($\alpha =c$) or the spinon ($\alpha=s$) sound-wave amplitudes. The introduced bosons represent the sound waves propagating around FS, here characterized by $\omega$ and ${\bf q}$. Eq.(\[e63\]) gives both the stable solution for collective modes and the solution with the imaginary frequency. Solution of Eq.(\[e63\]) is analogical to that considered in the FL theory [@lfl]; it will not be discussed in detail here.

In summary, we presented a model with the spin-charge separation in the space of arbitrary dimensions, as well as have discussed some of its basic properties. The next step would require a careful analysis of the nonsingular part of the interaction for finite-volume systems. In particular, the most important question is to construct a theory of interacting holons and spinons at low energies (in the spirit of the Landau FL), including also the effects coming from the presence of an applied magnetic field. One should also examine the stability of this liquid against charge or spin-density wave formation.

The paper was supported by the Committee of Scientific Research (KBN) of Poland. The work was performed in part at Purdue University (U.S.A.), where it was supported by the MISCON Grant No. DE-FG 02-90 ER 45427, and by the NSF Grant No. INT. 93-08323.

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[^1]: E-mail: byczuk@fuw.edu.pl, ufspalek@if.uj.edu.pl
