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arXiv:cond-mat/9301030 · Physical review. B, Condensed matter

Phase separation in the large-spin t−J model

Antimo Angelucci, Sandro Sorella

Published Apr 1, 1993• 13 citations
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Abstract

We investigate the phase diagram of the two-dimensional $t\ensuremath{-}J$ model using a recently developed technique that allows one to solve the mean-field model Hamiltonian with a variational calculation. The accuracy of our estimate is controlled by means of a small parameter $\frac{1}{q}$, analogous to the inverse spin magnitude $\frac{1}{s}$ employed in studying quantum spin systems. The mathematical aspects of the method and its connection with other large-spin approaches are discussed in detail. In the large-$q$ limit the problem of strongly correlated electron systems turns into the minimization of a total-energy functional. We have performed this optimization numerically on a finite but large $L\ifmmode\times\else\texttimes\fi{}L$ lattice. For a single hole the static small-polaron solution is stable except for small values of $J$, where polarons of increasing sizes have lower energy. At finite doping we recover phase separation above a critical $J$ and for any electron density, showing that the Emery et al. picture represents the semiclassical behavior of the $t\ensuremath{-}J$ model. Quantum fluctuations are expected to be very important, especially in the small-$J$-small-doping region, where phase separation may also be suppressed.

DOI
10.1103/physrevb.47.8858

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We investigate the phase diagram of the two-dimensional t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model using a recently developed technique that allows one to solve the mean-field model Hamiltonian with a variational calculation. The accuracy of our estimate is controlled by means of a small parameter 1q\frac{1}{q}q1​, analogous to the inverse spin magnitude 1s\frac{1}{s}s1​ employed in studying quantum spin systems. The mathematical aspects of the method and its connection with other large-spin approaches are discussed in detail. In the large-qqq limit the problem of strongly correlated electron systems turns into the minimization of a total-energy functional. We have performed this optimization numerically on a finite but large L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL lattice. For a single hole the static small-polaron solution is stable except for small values of JJJ, where polarons of increasing sizes have lower energy. At finite doping we recover phase separation above a critical JJJ and for any electron density, showing that the Emery et al. picture represents the semiclassical behavior of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model. Quantum fluctuations are expected to be very important, especially in the small-JJJ-small-doping region, where phase separation may also be suppressed.
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.
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Using the variational cluster approach, we study the transition from the antiferromagnetic to the superconducting phase of the two-dimensional Hubbard model at zero temperature. Our calculations are based on a method to evaluate the VCA grand potential which employs a modified Lanczos algorithm and avoids integrations over the real or imaginary frequency axis. Thereby, very accurate results are possible for cluster sizes not accessible to full diagonalization. This is important for an improved treatment of short-range correlations, including correlations between Cooper pairs in particular. We apply this improved method in order to investigate the cluster-size dependence of the phase-separation tendency that has been proposed recently on the basis of calculations for smaller clusters. While the energy barrier associated with phase separation rapidly decreases with increasing cluster size for both hole and electron doping, the extension of the phase-separation region behaves differently in the two cases. More specifically, our results suggest that phase separation remains persistent in the hole-doped case and disappears in the electron-doped case. We also study the evolution of the single-particle spectrum as a function of doping and point out the relevance of our results for experimental findings in electron and hole-doped materials.
Solid state physics, a very broad field in natural science, aims to explain physical properties of materials modeled as a many-body ensemble of the order of ∼ 10^23 interacting particles. For interacting metallic systems, one of the most powerful tools for unraveling the consequences of the collective interaction is the Fermi liquid theory. While still being an approximation, it excellently models many materials especially in the low-temperature and low-energy regime by treating themany-body problem as being equivalent to effectively non-interacting ”particles”, which is done by introducing the concept of quasiparticles. During the last few decades, several exotic phases of matter have been discovered, which originate from strong Coulomb interactions between the electrons, whose description is beyond the Fermi liquid theory. This thesis addresses the question of the validity of Fermi liquid theory once a single hole is introduced into an antiferromagnet with completely localized spins due to the strong interactions. Here, the most-studied case concerns the so-called 2D t − J model, which hosts a well-defined quasiparticle, the spin polaron. Focusing on a more tractable case, the t − Jz model without quantum spin fluctuations, we study the stability of the spin polaron solution once the nextnearest neighbor hopping t′ is introduced into the system. We focus on observations at the Γ-point and the X-point of systems with fixed Jz = 0.4t and varying t′ ∈ [−0.5t, 0.5t]. The model is numerically solved using exact diagonalization on finite-size lattices of up to 32 sites. A study of the quasiparticle weight under finite-size scaling reveals an “anomalous” regime at the Γ-point fort′ ≤ −0.3t, indicated by a vanishing quasiparticle weight. In order to characterize the low-energy dynamics within this “anomalous” regime, we investigated the spin correlation centered around the hole, as well as the magnon number distribution of a single-doped 20-site system. Based on the results, we present a preliminary picture of the antiferromagnetic ground state doped with one hole within that “anomalous” regime (t′ ≤ −0.3t) that deviates from the standard spin polaron as the cloud of magnetic fluctuations, typically localized around the hole, delocalizes over the whole finite-size lattice. Consequently, such systems show a weakly suppressed antiferromagnetic configuration under single-hole doping.
Based on the U(1) slave-boson functional integral approach to the $t\ensuremath{-}J$ model Hamiltonian, we investigate the phase diagram of phase separation for the hole-doped two-dimensional system of antiferromagnets. We discuss the phase separation (into a hole-rich and a hole-free phase) as a function of J/tJ/tJ/t where J is the Heisenberg (antiferromagnetic) coupling constant and t the hopping energy. For the physically interesting region of relatively small Heisenberg coupling, 0<J/t<0.2,0<J/t<0.2,0<J/t<0.2, we find that the phase separation occurs in a region of hole doping concentration below ${x}_{c}\ensuremath{\sim}0.1$ and that in the same region the critical hole doping concentrations remain robust with nearest-neighbor hole-hole repulsive interactions. On the other hand, in the region of larger J/tJ/tJ/t the phase separation boundary is found to be highly sensitive to the variation of the repulsive hole-hole interactions. In agreement with other studies, we find that the phase separation occurs at a critical value of ${J}_{c}\ensuremath{\approx}3.4t$ corresponding to the critical hole concentration xc=1.{x}_{c}=1.xc​=1.