Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics B

The Phase Diagram of the Two-Dimensional t-J Model from High Temperature Expansions

W. O. Putikka, Marco U. Luchini, Thomas Maurice Rice

Published Mar 1, 1992• 0 citations
Publisher

Abstract

The phase diagram of the 2D t-J model has been investigated using high temperature expansions. Series for the Helmholtz free energy, the inverse compressibility, the chemical potential and the uniform spin susceptibility through tenth order on a square lattice have been calculated using the finite cluster method.1 The series are analytically continued beyond their radius of convergence by Padé and integral approximants. The most accurate extrapolations can be made for the Helmholtz free energy where for J/t≈0.3 and n≈0.9 we can reach T~t/5. We can test the accuracy of the continuation methods by comparing with the 1D results for the boundary of phase separation of Ogata et al.2 In 2D a region of phase separation was found at T=0 for J/t lying above a line extending from J/t=3.8 at zero filling to J/t=1.2 at half filling. No phase separation was found at very small J/t contrary to the earlier suggestion of Emery et al3 which was based on results from exact diagonalisation on 4×4 clusters but in agreement with Quantum Monte Carlo (QMC) calculations on the Hubbard model.4 For very small J/t near half filling where the Nagaoka effect is possible, we find a region of divergent uniform magnetic susceptibility at T=0. However the divergence is very weak when compared with the exponential behaviour expected from a 2D ferromagnet. This might imply a substantially reduced moment which is consistent with the recent QMC estimates of Zhang et al.5

DOI
10.1142/s0217979292000359

Related papers

Aspects of the phase diagram of the two-dimensionalt-Jmodel

W. O. Putikka, Marco U. Luchini, Thomas Maurice Rice

The phase diagram of the 2D t-J model is investigated using high-temperature expansions. Series for the Helmholtz free energy, the inverse compressibility, the chemical potential, and the uniform spin susceptibility through tenth order are calculated and analyzed. A region of phase separation is found at T=0 for J/t lying above a line extending from J/t=3.8 at zero filling to J/t=1.2 at half filling. For very small J/t near half filling where the Nagaoka effect is possible, we find a region of divergent uniform magnetic susceptibility at T=0.

Physical Review Letters•Jan 27, 1992•172 citations

Ferromagnetism in the two-dimensionalt-Jmodel

W. O. Putikka, Marco U. Luchini, Masao Ogata

The occurrance of ferromagnetism for the t-J model on a square lattice is investigated using high-temperature series for the Helmholtz free energy F and the uniform magnetic susceptibility ${\mathrm{\ensuremath{\chi}}}_{0}$. For J=0 fourteenth-order series have been calculated for F and ${\mathrm{\ensuremath{\chi}}}_{0}$ and we estimate F(T=0) to find the ground-state energy. A region of fully polarized ferromagnetism is found for J0. A ferrimagnetic region is observed for larger J/t where ${\mathrm{\ensuremath{\chi}}}_{0}$ is divergent, but F(T=0) is below that of spinless fermions. Our results do not support Nagaoka's theorem at a thermodynamic density of holes.

Physical Review Letters•Oct 12, 1992•97 citations

Magnetic phase diagram of the half-filled three-dimensional Hubbard model

Kok-Kwei Pan, Yung‐Li Wang

Magnetic properties of the half-filled Hubbard model on three-dimensional bipartite lattices have been studied by using the linked-cluster series expansion method. The series for the free energy and the staggered susceptibility have been calculated as a function of \ensuremath{\mathrm{B}}t for three-dimensional bipartite lattices for an arbitrary value of the Coulomb interaction U. We obtain a transition from a paramagnetic to an antiferromagnetic U/t-${\mathrm{T}}_{\mathrm{N}}$ /t phase diagram from the divergence of the staggered susceptibility series. Comparisons with Monte Carlo simulation, Onsager reaction field result of an effective spin Hamiltonian, and large-U Heisenberg high-temperature solutions are made.

Physical review. B, Condensed matter•Feb 1, 1997•5 citations

Numerical linked-cluster algorithms. II. t−J models on the square lattice

Marcos Rigol, Tyler Bryant, Rajiv R. P. Singh

We discuss the application of a recently introduced numerical linked-cluster (NLC) algorithm to strongly correlated itinerant models. In particular, we present a study of thermodynamic observables: chemical potential, entropy, specific heat, and uniform susceptibility for the t-J model on the square lattice, with Jt=0.5 and 0.3. Our NLC results are compared with those obtained from high-temperature expansions (HTE) and the finite-temperature Lanczos method (FTLM). We show that there is a sizeable window in temperature where NLC results converge without extrapolations whereas HTE diverges. Upon extrapolations, the overall agreement between NLC, HTE, and FTLM is excellent in some cases down to 0.25t . At intermediate temperatures NLC results are better controlled than other methods, making it easier to judge the convergence and numerical accuracy of the method.

arXiv:0706.3255•Physical Review E•Jun 21, 2007•61 citations

Ising expansion for the Hubbard model

Zhu-Pei Shi, Rajiv R. P. Singh

We develop series expansions for the ground state properties of the Hubbard model by introducing an Ising anisotropy into the Hamiltonian. For the two-dimensional square lattice half-filled Hubbard model, the ground state energy, local moment, sublattice magnetization, uniform magnetic susceptibility, and spin stiffness are calculated as a function of U/t, where U is the Coulomb constant and t is the hopping parameter. Magnetic susceptibility data indicate a crossover around U\ensuremath{\approxeq}4 between spin density wave antiferromagnetism and Heisenberg antiferromagnetism. Comparisons with Monte Carlo simulations, random phase approximation result, and mean-field solutions are also made.

arXiv:cond-mat/9507096•Physical review. B, Condensed matter•Oct 1, 1995•8 citations

Magnetic susceptibility of the two-dimensional Hubbard model

Adriana Moreo

Using quantum Monte Carlo techniques we study the magnetic susceptibility of the Hubbard model on a two-dimensional square lattice for values of the coupling U/t ranging from 4 to 20, and as a function of temperature and doping. For U/t=4 the magnetic susceptibility has a maximum at half filling and decreases with doping, in qualitative agreement with recent random-phase-approximation calculations. However, for U/t=10 the susceptibility increases with hole doping near half filling resembling the experimental results obtained for ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4}$. At low electronic density the temperature dependence of the susceptibility is compatible with that of a Fermi liquid.

Physical review. B, Condensed matter•Aug 1, 1993•39 citations
The phase diagram of the 2D t-J model has been investigated using high temperature expansions. Series for the Helmholtz free energy, the inverse compressibility, the chemical potential and the uniform spin susceptibility through tenth order on a square lattice have been calculated using the finite cluster method.1 The series are analytically continued beyond their radius of convergence by Padé and integral approximants. The most accurate extrapolations can be made for the Helmholtz free energy where for J/t≈0.3 and n≈0.9 we can reach T~t/5. We can test the accuracy of the continuation methods by comparing with the 1D results for the boundary of phase separation of Ogata et al.2 In 2D a region of phase separation was found at T=0 for J/t lying above a line extending from J/t=3.8 at zero filling to J/t=1.2 at half filling. No phase separation was found at very small J/t contrary to the earlier suggestion of Emery et al3 which was based on results from exact diagonalisation on 4×4 clusters but in agreement with Quantum Monte Carlo (QMC) calculations on the Hubbard model.4 For very small J/t near half filling where the Nagaoka effect is possible, we find a region of divergent uniform magnetic susceptibility at T=0. However the divergence is very weak when compared with the exponential behaviour expected from a 2D ferromagnet. This might imply a substantially reduced moment which is consistent with the recent QMC estimates of Zhang et al.5
The phase diagram of the 2D t-J model is investigated using high-temperature expansions. Series for the Helmholtz free energy, the inverse compressibility, the chemical potential, and the uniform spin susceptibility through tenth order are calculated and analyzed. A region of phase separation is found at T=0 for J/t lying above a line extending from J/t=3.8 at zero filling to J/t=1.2 at half filling. For very small J/t near half filling where the Nagaoka effect is possible, we find a region of divergent uniform magnetic susceptibility at T=0.
The occurrance of ferromagnetism for the t-J model on a square lattice is investigated using high-temperature series for the Helmholtz free energy F and the uniform magnetic susceptibility ${\mathrm{\ensuremath{\chi}}}_{0}$. For J=0 fourteenth-order series have been calculated for F and ${\mathrm{\ensuremath{\chi}}}_{0}$ and we estimate F(T=0) to find the ground-state energy. A region of fully polarized ferromagnetism is found for J0. A ferrimagnetic region is observed for larger J/t where ${\mathrm{\ensuremath{\chi}}}_{0}$ is divergent, but F(T=0) is below that of spinless fermions. Our results do not support Nagaoka's theorem at a thermodynamic density of holes.
Magnetic properties of the half-filled Hubbard model on three-dimensional bipartite lattices have been studied by using the linked-cluster series expansion method. The series for the free energy and the staggered susceptibility have been calculated as a function of \ensuremath{\mathrm{B}}t for three-dimensional bipartite lattices for an arbitrary value of the Coulomb interaction U. We obtain a transition from a paramagnetic to an antiferromagnetic U/t-TN{\mathrm{T}}_{\mathrm{N}}TN​ /t phase diagram from the divergence of the staggered susceptibility series. Comparisons with Monte Carlo simulation, Onsager reaction field result of an effective spin Hamiltonian, and large-U Heisenberg high-temperature solutions are made.
We discuss the application of a recently introduced numerical linked-cluster (NLC) algorithm to strongly correlated itinerant models. In particular, we present a study of thermodynamic observables: chemical potential, entropy, specific heat, and uniform susceptibility for the t-J model on the square lattice, with Jt=0.5 and 0.3. Our NLC results are compared with those obtained from high-temperature expansions (HTE) and the finite-temperature Lanczos method (FTLM). We show that there is a sizeable window in temperature where NLC results converge without extrapolations whereas HTE diverges. Upon extrapolations, the overall agreement between NLC, HTE, and FTLM is excellent in some cases down to 0.25t . At intermediate temperatures NLC results are better controlled than other methods, making it easier to judge the convergence and numerical accuracy of the method.
We develop series expansions for the ground state properties of the Hubbard model by introducing an Ising anisotropy into the Hamiltonian. For the two-dimensional square lattice half-filled Hubbard model, the ground state energy, local moment, sublattice magnetization, uniform magnetic susceptibility, and spin stiffness are calculated as a function of U/t, where U is the Coulomb constant and t is the hopping parameter. Magnetic susceptibility data indicate a crossover around U\ensuremath{\approxeq}4 between spin density wave antiferromagnetism and Heisenberg antiferromagnetism. Comparisons with Monte Carlo simulations, random phase approximation result, and mean-field solutions are also made.
Using quantum Monte Carlo techniques we study the magnetic susceptibility of the Hubbard model on a two-dimensional square lattice for values of the coupling U/t ranging from 4 to 20, and as a function of temperature and doping. For U/t=4 the magnetic susceptibility has a maximum at half filling and decreases with doping, in qualitative agreement with recent random-phase-approximation calculations. However, for U/t=10 the susceptibility increases with hole doping near half filling resembling the experimental results obtained for ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​. At low electronic density the temperature dependence of the susceptibility is compatible with that of a Fermi liquid.