Charge ordering of spinless fermions on the Bethe lattice at general coordination number: exact inhomogeneous solution, phase diagram, and thermodynamics

Spinless fermions (equivalently, a hard-core lattice gas) on the Bethe lattice with repulsive nearest-neighbor interaction V>0 were exactly solved in the fixed-density ensemble by Mancini, Mancini and Naddeo only for coordination z=3, where the self-consistency equations factorize into quadratics, the discriminant condition Q=b^2-4ac>0 delimits the charge-ordered region in the (n,T) plane, the zero-temperature critical densities are 1/3 and 2/3, and the transition to a two-sublattice charge-ordered state is second order. The open problem is the completion of this analysis at general coordination z: solving the degree-(z-1) fixed-point system, establishing the exact charge-ordering boundary T_c(n,z) and its zero-temperature endpoints, the chemical-potential plateau structure, the full thermodynamic suite (internal energy, free-energy difference, specific heat, compressibility, entropy) with its z-dependence, and the order of the transition for every z. The authors' own 2010 exact solution of the spinful atomic-limit extended Hubbard model on the same lattice settles part of the z-dependence of the ground-state windows (ordering requires n>1/z), but the general-z canonical characterization of the spinless model has not been published.

Background

A Bethe lattice is an infinite regular tree: every site has exactly $z$ neighbors, and the coordination number $z$ plays the role that dimension plays on ordinary lattices (the $z\to\infty$ limit with $zV$ held fixed underlies dynamical mean-field theory, the standard exact treatment of lattice fermions in infinite dimension). The model in question is the simplest charge-ordering lattice gas: spinless fermions localized on the sites, so each site is empty or singly occupied, with a repulsive nearest-neighbor interaction $V>0$ between occupied adjacent sites and chemical potential $\mu$. The occupation numbers $n(i)\in\{0,1\}$ make it a hard-core lattice gas, and a particle-hole transformation maps it exactly onto the spin-1/2 antiferromagnetic Ising model in a uniform field on the same lattice ($\mu=2(h-zJ)$, $V=-4J$ with $J<0$, in the notation of the source paper's Eq. (2.21)). Charge ordering (CO) means a two-sublattice state in which the two bipartition classes of the tree carry different densities $n_1\neq n_2$ with $n=(n_1+n_2)/2$ — the tree analogue of checkerboard order.

Mancini, Mancini and Naddeo solved this model exactly with the equations-of-motion / Green's-function method, in which the Heisenberg equations of motion of the fermionic field operators close on a finite set of eigenoperators of the Hamiltonian, so that all thermodynamics reduces to a finite self-consistency problem: Eq. (2.19), $(1-Y)(1+AX)^{z-1}=e^{-\beta\mu}Y$ and $(1-X)(1+AY)^{z-1}=e^{-\beta\mu}X$ with $A=e^{-\beta V}-1$ and $\beta=1/T$, for the two sublattice parameters $X$ and $Y$; Eq. (2.20) for the sublattice densities $n_1=(1+AX)^z/[e^{-\beta\mu}+(1+AX)^z]$ and $n_2=(1+AY)^z/[e^{-\beta\mu}+(1+AY)^z]$; and the canonical constraint (2.22), $n=(n_1+n_2)/2$, which fixes the chemical potential at prescribed density. Working at fixed $n$ rather than fixed $\mu$ is the natural fermionic ensemble (the system adjusts $\mu$ to the imposed density); on trees this ensemble distinction is not vacuous, because at coexistence and at the zero-temperature chemical-potential plateaus several Gibbs measures (equilibrium states) compete.

Executed for $z=3$ only ('for the sake of simplicity, on the particular case $z=3$'), the analysis factorizes Eq. (2.19) into a homogeneous branch and a quadratic branch $aX^2+bX+c=0$ whose discriminant $Q=b^2-4ac$ is positive exactly inside the ordered region; the numerical study of $Q>0$ yields the $(n,T)$ phase diagram — a CO dome below a critical line $T_c(n)$ maximal near $n=1/2$, with zero-temperature critical densities $n=1/3$ and $n=2/3$, chemical-potential plateaus $\mu=0$ for $n<1/2$, $\mu=3V/2$ at $n=1/2$, $\mu=3V$ for $n>1/2$, and a second-order transition signaled by specific-heat and compressibility jumps, with internal energy, free-energy difference, entropy, and compressibility given in closed form. The fixed-field (grand-canonical) physics of the isomorphic antiferromagnetic Ising model on general-$z$ Bethe lattices is classical — Katsura and Takizawa, Muller-Hartmann and Zittartz, Thompson, and for the hard-core gas Runnels — but those transfer-matrix and activity-based treatments do not deliver the canonical, sublattice-resolved characterization at imposed density. The companion exact solution of the homogeneous sector (Mancini and Naddeo 2006) is valid for any $z$ but covers only the non-ordered branch, its closed forms being case-by-case because the degree of the fixed-point polynomial grows with $z$. The authors' later exact solution of the spinful atomic-limit extended Hubbard model on the same lattice (Mancini and Mancini 2010) — the spin-1/2 variant with an onsite interaction $U$ added and with hopping set to zero, which is what 'atomic limit' means — is likewise formal for any $z$ and establishes several general-$z$ ground-state features of the charge-ordering sector — ordering requires $n>1/z$, and the zero-temperature complete-charge-order windows have edges scaling as $1/z$, $2/z$, $2(z-1)/z$, $(2z-1)/z$ — but its detailed thermodynamics and phase diagrams are computed at $z=3$, and the spinless model itself was never re-analyzed at general $z$.

Problem Statement

Extend the exact solution of the repulsive spinless-fermion model on the Bethe lattice from coordination $z=3$ to arbitrary coordination number $z$ (all $z\ge 3$; the chain limit $z=2$ is exactly solvable and serves as a consistency check). At fixed density $n$ and temperature $T$ (in units $V=1$), determine, as an explicit or exactly implicit function of $z$:

(i) the charge-ordered solution of the self-consistency system of the source paper — Eqs. (2.19), (2.20), (2.22) — for the sublattice parameters $X$, $Y$ and the chemical potential $\mu$; at $z=3$ the system reduces to the quadratic $aX^2+bX+c=0$, while for general $z$ it is a coupled polynomial system of degree $z-1$;

(ii) the exact condition generalizing the discriminant requirement $Q=b^2-4ac>0$ — the existence boundary of the charge-ordered branch, i.e. where the real inhomogeneous self-consistency solution ceases to exist — and, separately, the thermodynamic transition curve $T_c(n,z)$ defined as the equal-free-energy line $\Delta F=0$ between the homogeneous and charge-ordered phases, with its zero-temperature endpoints $n_c(z)$; both curves are required and are to be reported separately, since they coincide only where the transition is second order, so wherever item (iv) yields a first-order segment the existence boundary is a spinodal limit distinct from $T_c(n,z)$; together with the chemical-potential plateau structure $\mu(n,z)$;

(iii) the full thermodynamic characterization — internal energy, Helmholtz free-energy difference between the homogeneous and inhomogeneous phases, specific heat, compressibility, and entropy — with its dependence on $z$;

(iv) the order of the homogeneous-to-charge-ordered transition as a function of $z$ (second order at $z=3$): does this persist for every $z$, or do first-order segments or reentrant behavior appear at larger coordination.

These items constitute one program — the Sections 3-4 analysis of the source paper executed at general coordination number — and all four items (i)-(iv) are required for a complete answer; establishing only a subset (for instance (i)-(ii) alone) is a partial result, not a resolution. Any answer must reproduce the published $z=3$ results identically as the special case.

References

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