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Journal of Physics A Mathematical and Theoretical

Mutual exclusion statistics and the infinite-range Hubbard model: critical behavior in the quantum regime

Gabriel F. O. Ramos, Weverton Lucas da Silva Rosendo, Ernesto P. Raposo, Mauricio D. Coutinho-Filho

Published Jan 10, 2025• 0 citations
Publisher

Abstract

Abstract We investigate the critical behavior in the quantum regime of the Hubbard model in d-dimensional hypercubic lattices with infinite-range Coulomb interaction. The Haldane–Wu mutual exclusion statistics approach, which comprises three species of exclusions, allows us to derive the singular part of the free energy, entropy, and specific heat in the quantum regime g > T, where g is the scaling interaction parameter and T is the temperature. We present results from the analytic series expansions of observables near the quantum critical point: T = g = μ = 0 , where µ is the chemical potential. The signature of the mutual exclusion statistics manifests through the appearance of polylogarithms related to Lerch functions in the coefficients of the free energy series in powers of T / g . The calculated critical exponents from the analytic series are consistent with our scaling analysis in the quantum regime. In particular, the specific heat is shown to display a variety of critical behaviors, C / T ∼ g − α g , g → 0 + , depending on the lattice dimension: from a singular behavior in d = 1 ( α g = 1 / 2 ) to a constant value in d = 2 ( α g = 0 ), and regular behavior for d > 2 ( α g = 1 − d / 2 < 0 ).

DOI
10.1088/1751-8121/ada8e9

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Abstract We investigate the critical behavior in the quantum regime of the Hubbard model in d-dimensional hypercubic lattices with infinite-range Coulomb interaction. The Haldane–Wu mutual exclusion statistics approach, which comprises three species of exclusions, allows us to derive the singular part of the free energy, entropy, and specific heat in the quantum regime g > T, where g is the scaling interaction parameter and T is the temperature. We present results from the analytic series expansions of observables near the quantum critical point: T = g = μ = 0 , where µ is the chemical potential. The signature of the mutual exclusion statistics manifests through the appearance of polylogarithms related to Lerch functions in the coefficients of the free energy series in powers of T / g . The calculated critical exponents from the analytic series are consistent with our scaling analysis in the quantum regime. In particular, the specific heat is shown to display a variety of critical behaviors, C / T ∼ g − α g , g → 0 + , depending on the lattice dimension: from a singular behavior in d = 1 ( α g = 1 / 2 ) to a constant value in d = 2 ( α g = 0 ), and regular behavior for d > 2 ( α g = 1 − d / 2 < 0 ).
Abstract The infinity-range Hubbard model on a d-dimensional hypercubic lattice is mapped onto an ideal gas with three species obeying the Haldane-Wu fractional exclusion statistics (FESs). Using this map, we study this system in the spin-incoherent regime and strong coupling limit. We have derived the polynomial series of the grand-canonical free, whose coefficients are Lerch functions, a signature of our FES description, as demonstrated for the entropy and specific heat in any dimension. For even dimensions the series are finite, and a table of coefficients versus dimensionality can be written, while for odd dimensions, the series diverges asymptotically, and can thus be circumvented by Borel summation.
The ordering of charges on half-filled hypercubic lattices is investigated numerically, where electroneutrality is ensured by background charges. This system is equivalent to the s=1/2s=1/2s=1/2 Ising lattice model with antiferromagnetic 1/r1/r1/r interaction. The temperature dependences of specific heat, mean staggered occupation, and of a generalized susceptibility indicate continuous order-disorder phase transitions at finite temperatures in two- and three-dimensional systems. In contrast, the susceptibility of the one-dimensional system exhibits singular behavior at vanishing temperature. For the two- and three-dimensional cases, the critical exponents are obtained by means of a finite-size scaling analysis. Their values are consistent with those of the Ising model with short-range interaction, and they imply that the studied model cannot belong to any other known universality class. Samples of up to 1400, 1122{112}^{2}1122, and 223{22}^{3}223 sites are considered for dimensions 1 to 3, respectively.
The ground state and low-temperature properties of the Hubbard model with infinite-range attractive interaction in the presence of a magnetic field are considered. This model is exactly solvable in any dimension and is mapped onto an ideal gas of three species of exclusons. The system is found to behave as a noninteracting Fermi gas whenever the correlation function $C(\mathbf{k})=⟨{n}_{\mathbf{k}\ensuremath{\uparrow}}{n}_{\mathbf{k}\ensuremath{\downarrow}}⟩\ensuremath{-}⟨{n}_{\mathbf{k}\ensuremath{\uparrow}}⟩⟨{n}_{\mathbf{k}\ensuremath{\downarrow}}⟩$ is negligible, otherwise it displays nontrivial correlated behavior which manifests as fractional statistical effects. The corresponding effective Hamiltonians and free energies governing the distinct regimes are presented. In one dimension, our results are analyzed in light of those derived using the on-site attractive Hubbard chain.
Even though the Hubbard model is one of the most fundamental models of highly correlated electrons, analytical and numerical data describing its thermodynamics at nonzero magnetization are relatively scarce. We present a detailed investigation of the thermodynamic properties for the one-dimensional repulsive Hubbard model in the presence of an arbitrary magnetic field for all values of the filling fraction and temperatures as low as $T\ensuremath{\sim}0.005t.$ Our analysis is based on the system of integral equations derived in the quantum transfer matrix framework. We determine the critical exponents of the quantum phase transitions and also provide analytical derivations for some of the universal functions characterizing the thermodynamics in the vicinities of the quantum critical points. Extensive numerical data for the specific heat, susceptibility, compressibility, and entropy are reported. The experimentally relevant double occupancy presents an interesting doubly nonmonotonic temperature dependence at intermediate values of the interaction strength and also at large repulsion and magnetic fields close to the critical value. The susceptibility in zero magnetic field has a logarithmic singularity at low temperatures for all filling factors similar to the behavior of the same quantity in the spin-1/2 isotropic Heisenberg model. We determine the density profiles for a harmonically trapped system and show that while the total density profile seems to depend mainly on the value of chemical potential at the center of the trap the distribution of phases in the inhomogeneous system changes dramatically as we increase the magnetic field.
We show that the Hubbard model with infinite-range Coulomb coupling is equivalent to an ideal gas of three species of particles obeying fractional exclusion statistics. A full appreciation of this mapping requires an extension of the pertinent formalism. This very simple, but rather peculiar model is exactly solvable in any dimension and exhibits a Mott metal-insulator transition, whose universality class is shown to be that of a free spinless Fermi gas. A modified version of the Luttinger theorem is shown to apply in any dimension. We also characterize the metallic and insulating phases by obtaining the electronic band structure as well as the interacting density of states. The fractional statistics manifests itself on the amplitudes of several thermodynamic quantities and, in particular, the Pauli spin susceptibility is subdominant in all metallic phases, and a Curie-type of response appears.