On the particle-hole symmetry of the fermionic spinless Hubbard model in $D=1$

We revisit the particle-hole symmetry of the one-dimensional ($D=1$) fermionic spinless Hubbard model, associating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 $XXZ$ Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temperature. Upon comparing two regimes of that chain model so that the number of particles in one regime equals the number of holes in the other, one finds that, in general, their thermodynamics is similar, but not identical: both models share the specific heat and entropy functions, but not the internal energy per site, the first-neighbor correlation functions, and the number of particles per site. Due to that symmetry, the difference between the first-neighbor correlation functions is proportional to the $z$-component of magnetization of the $XXZ$ Heisenberg model. The results presented in this paper are valid for any value of the interaction strength parameter $V$, which describes the attractive/null/repulsive interaction of neighboring fermions.

The one-band Hubbard model [1,2] partially describes quantum magnetic phenomena; the complexity of real materials, however, imposes severe limitations on the direct comparison of experimental and theoretical results. It is not always clear which missing terms should be included in the fermionic Hamiltonian to account for the diversity of phenomena in a strongly correlated electron system.
The development of optical lattices over the last two decades has made the experimental simulation of chain models possible. The three-dimensional Hubbard model at low temperatures has been simulated by a fermionic quantum gas trapped in an optical lattice [3,4]. A review of the simulation of the Fermi-Hubbard model with fermionic atoms in optical lattices can be found in [5]. The simulation of a onedimensional spin-1/2 Ising model by a degenerate Bose gas of rubidium atoms confined in an optical lattice can be found in [6]. The simplest one-dimensional fermionic model is the fermionic spinless Hubbard model, the generalizations of which have been applied to the description of Verwey metal-insulator transitions and charge-ordering phenomena of the Fe 3 O 4 , Ti 4 O 7 , LiV 2 O 4 and other d-metal compounds [7,8].
In this paper we revisit the consequences of the particle-hole symmetry on the thermodynamics of the one-dimensional fermionic spinless Hubbard model in the whole range of temperatures, by mapping it into the exactly solvable D = 1 spin-1/2 X X Z Heisenberg model. Appendix A shows the β-expansion of the Helmholtz free energy (HFE) of this model, up to the order β 6 [9].
The spinless fermionic Hubbard model in D = 1 is a very simple anti-commutative model the Hamiltonian of which is [10]: in which (c i , c † i ), with i ∈ {1, 2, . . . , N }, are fermionic annihilation and creation operators, respectively, and N is the number of sites in the chain. These operators satisfy anticommutation relations, {c i , c † j } = δ i j 1l i and {c i , c j } = 0, in which t is the hopping integral, V is the strength of the repulsion (V > 0) or attraction (V < 0) between first-neighbour fermions, µ is the chemical potential and n i = c † i c i is the operator number of fermions at the i th site of the chain.
Sznajd and Becker [10] have shown that the Hamiltonian (1) has a particle-hole symmetry; consequently, the HFE of this model, W (t ,V, µ; β), satisfies the relation in which β = 1 kT , k is the Boltzmann's constant and T is the absolute temperature in Kelvin. The relation (2) is valid for any values (positive, null or negative) of V and µ. Equation (2) provides the condition for having the same number of particles and holes at the same potential V but at distinct chemical potentials, in which 〈n i 〉 is the average number of fermionic particles at each site of the chain at temperature T .
Haldane [11] showed the equivalence of the model (1) and the spin-1/2 X X Z Heisenberg model in D = 1. More recently, Sznajd and Becker [10] also used the inverse Wigner-Jordan transformation to show that the Hamiltonian (1) is mapped onto the Hamiltonian of the one-dimensional spin-1/2 X X Z Heisenberg model with a longitudinal magnetic field, in which S l = σ l 2 , l ∈ {x, y, z}, and σ l are the Pauli matrices. The norm of the spin operator S is || S|| = 3 2 . The Hamiltonians (1) and (4) and 1l is the identity operator of the chain. This relation shows a constant shift between the energy spectrum of these two models.
Let W S=1/2 (J , ∆, h; β) be the HFE associated to the Hamiltonian (4) of the D = 1 spin-1/2 X X Z Heisenberg model. A direct consequence of (5) is that At finite temperature (T 0), the HFE of the one-dimensional S = 1/2 X X Z Heisenberg model is an even function of the longitudinal magnetic field h, Such invariance of W S=1/2 comes from the symmetry of the Hamiltonian (4) upon reversal of the external magnetic field, h → −h, and of the spin operators, Consider, for a given magnetic field h and a fixed value (positive, null or negative) of V , the chemical potential µ so that h = µ − V . For a reversed magnetic field, the corresponding chemical potential µ 2 for The identity (7) and the condition (8) recover the result (2) satisfied by the HFE of the spinless Hubbard model for any values of V and µ. Notice that in the half-filling condition (µ = V ), we have µ 2 = µ, and there is no visible consequence of the symmetry (7).
We point out that the quantity −µ+2V , which appears on the r.h.s. of (8), also appears as an argument of W (the HFE of D = 1 spinless fermionic Hubbard model) in the r.h.s. of (2), which in its turn comes from the particle-hole symmetry of the Hamiltonian (1). On the other hand, (7) comes from the fact that the HFE of the D = 1 spin-1/2 X X Z Heisenberg model is insensitive to a reversal of the longitudinal magnetic field.
Equation (3) can be interpreted as follows: the number of particles in the chain under a potential V and a chemical potential µ equals the number of holes in the chain under the same potential V and a chemical potential µ 2 given by (8). Those configurations correspond to distinct distributions of the fermionic particles in the chain, and certainly have some different thermodynamic functions at temperature T . In what follows, we explore the consequences of the equality (7) in the thermodynamic functions of those two configurations.
The specific heat C and the entropy S, both per site, are related to the HFE of the model (1) by C (t ,V, µ; β) = −β 2 ∂ 2 ∂β 2 β W (t ,V, µ; β) and S k = β 2 ∂ ∂β W (t ,V, µ; β), respectively. Due to equation (6)    The two-point correlation function G i,i+1 is related to the HFE by From relation (6), the symmetry condition (7) and the definition of the z-component of the magnetization of the D = 1 spin-1/2 X X Z Heisenberg model, in which i ∈ {1, 2, · · · , N } and 〈S z i 〉(J , ∆, h; T ) is the mean value of the z-component of the spin-1/2 operator at the i th site of the chain and at temperature T , we obtain thus, recovering equation (3).
In summary, we have verified that the particle-hole symmetry of the one-dimensional spinless fermionic Hubbard model (1)  In reference [9] we calculated the β-expansion of the HFE of the normalized one-dimensional spin-S of the X X Z Heisenberg model with single-ion anisotropy term in the presence of a longitudinal magnetic field up to the order β 6 , in terms of the rescaled spin operator s = S/ S(S + 1).
In the present work we have applied equation (6) to equation (B) of reference [9], with || S|| = 3 2 , to derive the β-expansion, up to the order β 6 , of the HFE of the one-dimensional fermionic spinless Hubbard model. We have obtained