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Physical review. B, Condensed matter

Finite-temperature phase diagram of thet-Jmodel: Renormalization-group theory

Alexis Falicov, A. Nihat Berker

Published May 1, 1995• 16 citations
Publisher

Abstract

The finite-temperature phase diagram of the t-J model of electronic conduction is calculated in d dimensions, using the Migdal-Kadanoff renormalization-group procedure. No finite-temperature phase transition in d=1 and a finite-temperature first-order boundary ending at a critical point in d=2 are found. In d=3, a remarkably complex multicritical phase diagram is found, with a new phase, between hole dopings of 0.3 and 0.4 and temperatures 1/J below 0.4, in which the hopping strength t renormalizes to infinity under rescaling. Our results are confirmed by comparison of calculated electron densities, kinetic energies, and nearest-neighbor density-density and spin-spin correlation functions with exact finite-cluster results.

DOI
10.1103/physrevb.51.12458

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Physical review. B./Physical review. B•Dec 9, 2024•2 citations
The finite-temperature phase diagram of the t-J model of electronic conduction is calculated in d dimensions, using the Migdal-Kadanoff renormalization-group procedure. No finite-temperature phase transition in d=1 and a finite-temperature first-order boundary ending at a critical point in d=2 are found. In d=3, a remarkably complex multicritical phase diagram is found, with a new phase, between hole dopings of 0.3 and 0.4 and temperatures 1/J below 0.4, in which the hopping strength t renormalizes to infinity under rescaling. Our results are confirmed by comparison of calculated electron densities, kinetic energies, and nearest-neighbor density-density and spin-spin correlation functions with exact finite-cluster results.
A generalized t-J Hamiltonian is analyzed within the framework of a quantum real-space renormalization group. Besides the usual hopping (t), exchange (J), and chemical-potential (\ensuremath{\mu}) terms, the Hamiltonian contains a nearest-neighbor charge interaction term and is invariant under the renormalization. The finite-temperature phase diagram of the d=2 and d=3 model is calculated in the full range of the parameters. Our results show that many of the critical properties of the La- and Y-based high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors can be explained by the present model.
Recently, robust d-wave superconductive (SC) order has been unveiled in the ground state of the 2D t-t^{'}-J model-with both nearest-neighbor (t) and next-nearest-neighbor (t^{'}) hoppings-by density matrix renormalization group studies. However, there is currently a debate on whether the d-wave SC holds up strong on both t^{'}/t>0 and t^{'}/t<0 cases for the t-t^{'}-J model, which correspond to the electron- and hole-doped sides of the cuprate phase diagram, respectively. Here, we exploit state-of-the-art thermal tensor network approach to accurately obtain the phase diagram of the t-t^{'}-J model on cylinders with widths up to W=6 and down to low temperature as T/J≃0.06, pushing the boundaries of contemporary finite-T calculations. For t^{'}/t>0, we find a domelike SC regime with a diverging d-wave pairing susceptibility, χ_{SC}∝1/T^{α} below a characteristic temperature T_{c}^{*}. Near optimal doping, T_{c}^{*} reaches its highest value of about 0.15J. Above T_{c}^{*} yet below a higher crossover temperature T^{*}, the magnetic susceptibility becomes suppressed, which can be related to the onset of pseudogap (PG) behaviors. On the other hand, for t^{'}/t<0, we find the pairing correlations are much weaker, although there exhibits a node-antinode structure in the PG regime as observed in the hole-doped cuprates. The thermal tensor network calculations of the t-t^{'}-J model underscore both the similarities and differences in the finite-temperature phase diagram between the fundamental model and cuprates, yielding unique insights into their intricate behaviors.
The global phase diagram of the spinless Falicov-Kimball model in d=3d=3d=3 spatial dimensions is obtained by renormalization-group theory. This global phase diagram exhibits five distinct phases. Four of these phases are charge-ordered (CO) phases, in which the system forms two sublattices with different electron densities. The CO phases occur at and near half filling of the conduction electrons for the entire range of localized electron densities. The phase boundaries are second order, except for the intermediate and large interaction regimes, where a first-order phase boundary occurs in the central region of the phase diagram, resulting in phase coexistence at and near half filling of both localized and conduction electrons. These two-phase or three-phase coexistence regions are between different charge-ordered phases, between charge-ordered and disordered phases, and between dense and dilute disordered phases. The second-order phase boundaries terminate on the first-order phase transitions via critical endpoints and double critical endpoints. The first-order phase boundary is delimited by critical points. The cross-sections of the global phase diagram with respect to the chemical potentials and densities of the localized and conduction electrons, at all representative interactions strengths, hopping strengths, and temperatures, are calculated and exhibit ten distinct topologies.
The field-free quantum-spin-1/2 XXZ model is studied in three-dimensions using renormalization group theory. We obtain global phase diagrams and critical properties. We identify zero-temperature quantum phase transitions at isotropic points, as well as long-range order below critical temperatures at all anisotropies. We show that the excitation spectrum is gapless in XY-like, and gapped in Ising-like anisotropy regimes. Besides, internal energy density, specific heat, nearest-neighbor entanglement measures and spin-spin correlations are obtained globally at all temperatures. Our numerical results cover the global range of anisotropy parameter, for both ferromagnetic and antiferromagnetic cases.
We examine the ground-state phase diagram of the ttt-JJJ model in one dimension by means of the density matrix renormalization group. This model is characterized by a rich phase diagram as a function of the exchange interaction JJJ and the density nnn, displaying Luttinger-liquid (LL) behavior both of repulsive and attractive (i.e., superconducting) natures, a spin-gap phase, and phase separation. The phase boundaries separating the repulsive from the attractive LL phase as JJJ is increased, and also the boundaries of the spin-gap region at low densities, and phase separation at even larger JJJ, are determined on the basis of correlation functions and energy gaps. In particular, we shed light on a contradiction between variational and renormalization-group (RG) results about the extent of the spin-gap phase, which results larger than the variational but smaller than the RG one. Furthermore, we show that the spin gap can reach a sizable value ( 0.1t~0.1t 0.1t) at low enough filling, such that preformed pairs should be observable at temperatures below these energy scales. No evidence for a phase with clustering of more than two particles is found on approaching phase separation.
We study the phase diagram of the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the pyrochlore lattice with S=1S=1S=1 spins at zero and finite temperatures. We use a combination of complementary state-of-the-art quantum many-body approaches such as density matrix renormalization group (DMRG), density-matrix purification, and pseudo-Majorana functional renormalization group (PMFRG). We present an efficient approach to preserve the applicability of the PMFRG for spin-1 systems at finite temperatures despite the inevitable presence of unphysical spin states. The good performance of our methods is first demonstrated for the nearest-neighbor pyrochlore Heisenberg model where the finite-temperature behavior of the specific heat and uniform susceptibility show excellent agreement within PMFRG and density-matrix purification. Including an antiferromagnetic second-neighbor coupling we find that the nonmagnetic ground-state phase of the nearest-neighbor model extents up to ${J}_{2}/{J}_{1}\ensuremath{\sim}0.02$ within DMRG, beyond which magnetic $\mathbit{k}=0$ long-range order sets in. Our PMFRG calculations find the phase transition in a similar regime ${J}_{2}/{J}_{1}\ensuremath{\sim}0.035(8)$ which, together with the DMRG result, provides a strong argument for the existence of a small but finite nonmagnetic ground-state phase in the spin-1 pyrochlore Heisenberg model. We also discuss the origin of discrepancies between different versions of the functional renormalization group concerning the location of this phase transition.