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arXiv:cond-mat/9912294 · Physical review. B, Condensed matter

Limits on phase separation for two-dimensional strongly correlated electrons

W. O. Putikka, Marco U. Luchini

Published Jul 15, 2000• 23 citations
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Abstract

From calculations of the high-temperature series for the free energy of the two-dimensional t-$J$ model we construct a series for ratios of the free energy per hole. The ratios can be extrapolated very accurately to low temperatures and used to investigate phase separation. Our results confirm that phase separation occurs only for $J/t\ensuremath{\gtrsim}1.2.$ Also, the phase transition into the phase-separated state has ${T}_{c}\ensuremath{\approx}0.25J$ for large $J/t.$

DOI
10.1103/physrevb.62.1684

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Phase separation in the two-dimensional Hubbard model: A fixed-node quantum Monte Carlo study

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Fixed-node Green's-function Monte Carlo calculations have been performed for very large 16x16 two-dimensional Hubbard lattices, large interaction strengths U=10,20, and 40, and many (15 similar to 20) densities between empty and half-filling. The nodes were fixed by a simple Slater-Gutzwiller trial wave function. For each value of U we obtained a sequence of ground-state energies which is consistent with the possibility of a phase separation close to half-filling, with a hole density in the hole-rich phase which is a decreasing function of U. The energies suffer, however, from a fixed-node bias: more accurate nodes are needed to confirm this picture. Our extensive numerical results and their test against size, shell, shape, and boundary-condition effects also suggest that phase separation is quite a delicate issue, on which simulations based on smaller lattices than considered here are unlikely to give reliable predictions. RI Capone, Massimo/A-7762-2008; Guidoni, Leonardo/G-3759-2011

•Jan 1, 1998•0 citations
From calculations of the high-temperature series for the free energy of the two-dimensional t-JJJ model we construct a series for ratios of the free energy per hole. The ratios can be extrapolated very accurately to low temperatures and used to investigate phase separation. Our results confirm that phase separation occurs only for J/t\ensuremath≳1.2.J/t\ensuremath{\gtrsim}1.2.J/t\ensuremath≳1.2. Also, the phase transition into the phase-separated state has Tc\ensuremath≈0.25J{T}_{c}\ensuremath{\approx}0.25JTc​\ensuremath≈0.25J for large J/t.J/t.J/t.
The ground state of the two-dimensional (2D) t-J model is studied by exact diagonalization for seven different unit cells with N=10, 16, 18, 20, 26, 32, and 36 sites on a square lattice. In the limit of low electron densities (up to six electrons) we calculate the lowest eigenvalues and classify the ground state according to wave vector K, total spin S, and point-group symmetry both in the limit J=0 (U=\ensuremath{\infty} Hubbard model) and for finite J>0. The transition to a completely phase-separated state for J>JPS{\mathit{J}}_{\mathrm{PS}}JPS​(N) is examined by calculating the free energy as a function of the electron concentration. We find that JPS{\mathit{J}}_{\mathrm{PS}}JPS​(N) increases with the lattice size N. An extrapolation of our finite-size results to the thermodynamic limit gives JPS{\mathit{J}}_{\mathrm{PS}}JPS​(\ensuremath{\infty})/t=4.1 (\ifmmode\pm\else\textpm\fi{}0.1).
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
We develop a general numerical method to study the zero-temperature properties of strongly correlated electron models on large lattices. The technique, which resembles Green's-function Monte Carlo, projects the ground-state component from a trial wave function with no approximations. We use this method to determine the phase diagram of the two-dimensional $t\ensuremath{-}J$ model, using the Maxwell construction to investigate electronic phase separation. The shell effects of fermions on finite-sized periodic lattices are minimized by keeping the number of electrons fixed at a closed-shell configuration and varying the size of the lattice. Results obtained for various electron numbers corresponding to different closed shells indicate that the finite-size effects in our calculation are small. For any value of interaction strength, we find that there is always a value of the electron density above which the system can lower its energy by forming a two-component phase separated state. Our results are compared with other calculations on the $t\ensuremath{-}J$ model. We find that the most accurate results are consistent with phase separation at all interaction strengths.
Fixed-node Green's-function Monte Carlo calculations have been performed for very large 16\ifmmode\times\else\texttimes\fi{}16 two-dimensional Hubbard lattices, large interaction strengths U=10,20,U=10,20,U=10,20, and 404040, and many $(15\ensuremath{\sim}20)$ densities between empty and half-filling. The nodes were fixed by a simple Slater-Gutzwiller trial wave function. For each value of U we obtained a sequence of ground-state energies which is consistent with the possibility of a phase separation close to half-filling, with a hole density in the hole-rich phase which is a decreasing function of UUU. The energies suffer, however, from a fixed-node bias: more accurate nodes are needed to confirm this picture. Our extensive numerical results and their test against size, shell, shape, and boundary-condition effects also suggest that phase separation is quite a delicate issue, on which simulations based on smaller lattices than considered here are unlikely to give reliable predictions.
The phase separation boundary of isotropic $t\ensuremath{-}J$ ladders is analyzed using density matrix renormalization group techniques. The complete boundary to phase separation as a function of J/tJ/tJ/t and doping is determined for a chain and for ladders with two, three, and four legs. Six-chain ladders have been analyzed at low hole doping. We use a direct approach in which the phase separation boundary is determined by measuring the hole density in the part of the system which contains both electrons and holes. In addition, we examine the binding energy of multihole clusters. An extrapolation in the number of legs suggests that the lowest J/tJ/tJ/t for phase separation to occur in the two-dimensional $t\ensuremath{-}J$ model is $J/t\ensuremath{\sim}1.$
Fixed-node Green's-function Monte Carlo calculations have been performed for very large 16x16 two-dimensional Hubbard lattices, large interaction strengths U=10,20, and 40, and many (15 similar to 20) densities between empty and half-filling. The nodes were fixed by a simple Slater-Gutzwiller trial wave function. For each value of U we obtained a sequence of ground-state energies which is consistent with the possibility of a phase separation close to half-filling, with a hole density in the hole-rich phase which is a decreasing function of U. The energies suffer, however, from a fixed-node bias: more accurate nodes are needed to confirm this picture. Our extensive numerical results and their test against size, shell, shape, and boundary-condition effects also suggest that phase separation is quite a delicate issue, on which simulations based on smaller lattices than considered here are unlikely to give reliable predictions. RI Capone, Massimo/A-7762-2008; Guidoni, Leonardo/G-3759-2011