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arXiv:2201.06677 · Physical Review A

Finite-temperature strong-coupling expansions for the SU(N) Hubbard model

Rajiv R. P. Singh, J. Oitmaa

Published Mar 24, 2022• 7 citations
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Abstract

We develop finite-temperature strong-coupling expansions for the $\mathrm{SU}(N)$ Hubbard model in powers of $\ensuremath{\beta}t, w=exp(\ensuremath{-}\ensuremath{\beta}U)$ and $\frac{1}{\ensuremath{\beta}U}$ for arbitrary filling. The expansions are done in the grand canonical ensemble and are most useful at a density of one particle per site, where for $U$ larger than or of order of the bandwidth, the expansions converge over a wide temperature range ${t}^{2}/U\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}T\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}10U$. By taking the limit $w\ensuremath{\rightarrow}0$, valid at temperatures much less than $U$, the expansions turn into a high-temperature expansion for a dressed $\mathrm{SU}(N)$ Heisenberg model that includes nearest-neighbor exchange, further neighbor exchanges, and ring exchanges known from the $T=0$ perturbation theory of the SU(2) Hubbard model. Below a filling of one particle per site, the $w\ensuremath{\rightarrow}0$ limit corresponds to an effective $t\text{\ensuremath{-}}J$ model. The onset of strong correlations can be identified by a plateaulike behavior in the entropy as a function of temperature. At small deviations from one particle per site, the expansions can be arranged in powers of a small parameter $\ensuremath{\delta}=1\ensuremath{-}n$, the deviation from one particle per site, where the leading $\ensuremath{\beta}t$-dependent terms correspond to holes sloshing around in a disordered $\mathrm{SU}(N)$ background. We use these expansions to calculate the thermodynamic properties of the model at moderate and high temperatures over a wide parameter range.

DOI
10.1103/physreva.105.033317

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We develop finite-temperature strong-coupling expansions for the SU(N)\mathrm{SU}(N)SU(N) Hubbard model in powers of \ensuremathβt,w=exp(\ensuremath−\ensuremathβU)\ensuremath{\beta}t, w=exp(\ensuremath{-}\ensuremath{\beta}U)\ensuremathβt,w=exp(\ensuremath−\ensuremathβU) and 1\ensuremathβU\frac{1}{\ensuremath{\beta}U}\ensuremathβU1​ for arbitrary filling. The expansions are done in the grand canonical ensemble and are most useful at a density of one particle per site, where for UUU larger than or of order of the bandwidth, the expansions converge over a wide temperature range t2/U\ensuremath≲T\ensuremath≲10U{t}^{2}/U\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}T\phantom{\rule{4pt}{0ex}}\ensuremath{\lesssim}\phantom{\rule{4pt}{0ex}}10Ut2/U\ensuremath≲T\ensuremath≲10U. By taking the limit w\ensuremath→0w\ensuremath{\rightarrow}0w\ensuremath→0, valid at temperatures much less than UUU, the expansions turn into a high-temperature expansion for a dressed SU(N)\mathrm{SU}(N)SU(N) Heisenberg model that includes nearest-neighbor exchange, further neighbor exchanges, and ring exchanges known from the T=0T=0T=0 perturbation theory of the SU(2) Hubbard model. Below a filling of one particle per site, the w\ensuremath→0w\ensuremath{\rightarrow}0w\ensuremath→0 limit corresponds to an effective t\ensuremath-Jt\text{\ensuremath{-}}Jt\ensuremath-J model. The onset of strong correlations can be identified by a plateaulike behavior in the entropy as a function of temperature. At small deviations from one particle per site, the expansions can be arranged in powers of a small parameter \ensuremathδ=1\ensuremath−n\ensuremath{\delta}=1\ensuremath{-}n\ensuremathδ=1\ensuremath−n, the deviation from one particle per site, where the leading \ensuremathβt\ensuremath{\beta}t\ensuremathβt-dependent terms correspond to holes sloshing around in a disordered SU(N)\mathrm{SU}(N)SU(N) background. We use these expansions to calculate the thermodynamic properties of the model at moderate and high temperatures over a wide parameter range.
Thermodynamic properties of the SU(nnn) Heisenberg model in one dimension is studied by means of high-temperature expansion for arbitrary nnn. The specific heat up to O[(βJ)23]O[(βJ)^{23}]O[(βJ)23] and the correlation function up to O[(βJ)18]O[(βJ)^{18}]O[(βJ)18] are derived with βJβJβJ being the antiferromagnetic exchange in units of temperature. It is found for n>2n>2n>2 that the specific heat shows a shoulder in the high-temperature side of a peak. The origin of this structure is clarified by deriving the temperature dependence of the correlation function. With decreasing temperature, the short-range correlation with two-site periodicity develops first, and then another correlation with nnn-site periodicity at lower temperature. This behavior is in contrast to that of the inverse square interaction model, where the specific heat shows a single peak according to the exact solution. Our algorithm has an advantage that neither computational time nor memory depends on the multiplicity nnn per site; the series coefficients are obtained as explicit functions of nnn.
By combining conventional finite-temperature many-body perturbation theory with cluster expansions, we develop a systematic method to carry out high order arbitrary temperature perturbative calculations on a computer. The method is well suited to studying the thermodynamic properties of quantum disordered and quantum critical phases at finite temperatures. As an application, we calculate the magnetic susceptibility, internal energy, and specific heat of the bilayer Heisenberg model. It is shown that for a wide range of coupling constants these expansions show excellent convergence at all temperatures. Comparing the direct series (without extrapolations) for the bulk susceptibility to quantum Monte Carlo simulations we find an almost perfect agreement between the two methods even at the quantum critical coupling separating the dimerized and antiferromagnetic phases. The convergence fails only at very low temperatures, which are also difficult to reach by quantum Monte Carlo simulations.
We report on a nonperturbative approach to the one-dimensional (1D) and two-dimensional (2D) Hubbard models that is capable of recovering both strong- $(U\ensuremath{\gg}t)$ and weak-coupling $(U\ensuremath{\ll}t)$ limits, with U the on-site Coulomb repulsion and t the kinetic energy. Dynamical corrections to the electron self-energy in the single particle Green function are explicitly included by expanding in terms of the 16 eigenstates that characterize two nearest neighbor sites. We first show that even when U is much smaller than the bandwidth, the Mott-Hubbard gap never closes at half filling in both 1D and 2D. Consequently, the Hubbard model at half filling is always in the strong-coupling nonperturbative regime. For both large and small U, we find that the population of nearest-neighbor singlet states approaches a value of order unity as $\stackrel{\ensuremath{\rightarrow}}{T}0$ as would be expected for antiferromagnetic order. We also find that the double occupancy is a smooth monotonic function of U and approaches the anticipated noninteracting limit of 1/41/41/4 as $\stackrel{\ensuremath{\rightarrow}}{U}0$ and vanishes as $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}.$ Finally, we compute the heat capacity [C(T,U)][C(T,U)][C(T,U)] for both 1D and 2D. Our results for 1D at moderate to high temperatures are in quantitative agreement with those of the exact Bethe ansatz solution, differing by no more than 11%.1 In addition, we find that in 2D, the C(T,U)C(T,U)C(T,U) curves vs T for different values of U exhibit a universal crossing point at two characteristic temperatures $T\ensuremath{\approx}1.7t\ifmmode\pm\else\textpm\fi{}0.1t$ and $T\ensuremath{\approx}0.4\ifmmode\pm\else\textpm\fi{}0.1t$ as is seen universally in Hubbard models and experimentally in a wide range of strongly correlated systems such as 3He,{}^{3}\mathrm{He},3He, UBe3,{\mathrm{UBe}}_{3},UBe3​, and ${\mathrm{CeCu}}_{6\ensuremath{-}x}{\mathrm{Al}}_{x}.$ The success of this method in recovering well-established results that stem fundamentally from the Coulomb interaction suggests that local dynamics are at the heart of the physics of strongly correlated systems.
Using finite-temperature strong-coupling expansions for the SU(NNN) Hubbard model, we calculate the thermodynamic properties of the model in the infinite-UUU limit for arbitrary density $0\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1$ and all NNN. We express the ferromagnetic susceptibility of the model as a Curie term plus $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$, an excess susceptibility above the Curie behavior. We show that, on a bipartite lattice, graph by graph the contributions to $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$ are non-negative in the limit that the hole density $\ensuremath{\delta}=1\ensuremath{-}\ensuremath{\rho}$ goes to zero. By summing the contributions from all graphs consisting of closed loops we find that the low-hole-density ferromagnetic susceptibility diverges exponentially as $exp\mathrm{\ensuremath{\Delta}}/T$ as $T\ensuremath{\rightarrow}0$ in two and higher dimensions. This demonstrates that the Nagaoka-Thouless ferromagnetic state exists as a thermodynamic state of matter at low enough density of holes and sufficiently low temperatures. The constant $\mathrm{\ensuremath{\Delta}}$ scales with the SU(NNN) parameter NNN as 1/N1/N1/N implying that ferromagnetism is gradually weakened with increasing NNN as the characteristic temperature scale for ferromagnetic order goes down.
We present an approach to the magnetic properties of the half-filled Hubbard model, based on an approximate mapping of its low-energy transverse spin excitations on to those of an effective underlying Heisenberg model, but with effective spin interactions which are self-consistently determined and not confined solely to nearest-neighbor couplings. The mapping is exact in strong-coupling and is found to be accurate over a very wide range of interaction strengths, down to weak coupling. At zero temperature, it permits ready evaluation at finite U of the one-loop effects of zero-point spin fluctuations on, e.g., the sublattice magnetization. At finite temperatures, thermodynamic properties of the system in the thermal paramagnet are studied via a physically transparent Onsager reaction field approach, which amounts to a self-consistent treatment of paramagnetic spin correlations. This is central not only in recovering the correct dimensionality dependence of antiferromagnetic long-ranged order, but also for the d=3 case of primary interest here yields a N\'eel temperature in close agreement with known strong- and weak-coupling limits. Spin correlation functions and magnetic susceptibilities also show very good agreement with quantum Monte Carlo calculations over an appreciable temperature range in which the low-lying transverse spin excitations are thermally dominant. \textcopyright{} 1996 The American Physical Society.
The half-filled extended Hubbard model (including nearest-neighbor interaction) is considered in perturbation theory at weak coupling on a d-dimensional hypercubic lattice (d\ensuremath{\gtrsim}3). It is shown that the exact critical temperature and the exact order parameter in the weak-coupling limit differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion. The results depend sensitively on dimension and on the relative strength of the nearest-neighbor interaction. The renormalization factor in d=3, estimated from the 1/d expansion, is q\ensuremath{\simeq}0.282 for the pure Hubbard model, q\ensuremath{\simeq}0.179 near the spin-density-wave to charge-density-wave transition and q\ensuremath{\simeq}0.830 in the limit of pure nearest-neighbor interaction.