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arXiv:cond-mat/9411042 · Physical review. B, Condensed matter

Two-dimensionalt-Jmodel at low electron density

C. Stephen Hellberg, Efstratios Manousakis

Published Aug 15, 1995• 28 citations
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Abstract

The phase diagram of the two-dimensional t-J model is determined accurately at low electron density by a combination of analytic and numerical techniques. The ground state exhibits three phases in the limit of zero electron density: an unpaired state at small J/t, a gas of s-wave pairs for 2J/t\ensuremath{\lesssim}3.4367, and a phase-separated state at larger interaction strengths. Bound states of larger clusters are never realized, and the instabilities present at small densities are discussed.

DOI
10.1103/physrevb.52.4639

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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>${\mathit{J}}_{\mathrm{PS}}$(N) is examined by calculating the free energy as a function of the electron concentration. We find that ${\mathit{J}}_{\mathrm{PS}}$(N) increases with the lattice size N. An extrapolation of our finite-size results to the thermodynamic limit gives ${\mathit{J}}_{\mathrm{PS}}$(\ensuremath{\infty})/t=4.1 (\ifmmode\pm\else\textpm\fi{}0.1).

Physical review. B, Condensed matter•Sep 15, 1993•10 citations
The phase diagram of the two-dimensional t-J model is determined accurately at low electron density by a combination of analytic and numerical techniques. The ground state exhibits three phases in the limit of zero electron density: an unpaired state at small J/t, a gas of s-wave pairs for 2J/t\ensuremath{\lesssim}3.4367, and a phase-separated state at larger interaction strengths. Bound states of larger clusters are never realized, and the instabilities present at small densities are discussed.
The phase diagram of the two-dimensional t-J model in the limit of infinitesimal electron density is determined exactly with analytic approaches. Numerical work is only required to show other potential ground states are never stabilized. Significant differences with previous work on the model are found. The ground state exhibits pairing for all interaction strengths insufficient to cause phase separation. The pairing is p-wave for small interaction strengths, d{sub x{sup 2}{minus}y{sup 2}} for intermediate J/t, and s-wave for larger J/t.
A new phase of a gas of pairs of electrons bounded in a singlet state is found in the one-dimensional t-J model for J>2t and the density of electrons less than 0.2. This phase was conjectured in the study of the diagonalization of small lattices [Phys. Rev. Lett. 66, 2388 (1991)]. The existence of this new phase for much larger lattice sizes is demonstrated by a combination of two numerical methods, the variational Monte Carlo and the power method. A trial wave function for this phase is proposed and shown to be in good agreement with the ground state obtained by the power method.
The phase diagram of the one-dimensional t-J model is investigated by analyzing the results of exact diagonalization and the exact solutions at J/t=0 and 2. Phase separation takes place above a critical value of J around Jc{\mathit{J}}_{\mathit{c}}Jc​/t=2.5--3.5 depending on the electron density. In the small-J region, Tomonaga-Luttinger liquid theory holds and its correlation exponents are calculated as a function of J/t and the electron density. Superconducting correlations become dominant in a region between the solvable case (J/t=2) and phase separation. A spin-gap region is also found at low density.
The two-dimensional t-J model at low electron densities is unstable against various forms of electron pairing at low enough temperatures. At parameter values J/t<<1, the leading instability, although only at very low temperature, is to p-wave pairing similar to the Hubbard model. At values of J/t>1, an instability against d-wave pairing sets in at a higher temperature as found numerically by Dagotto et al (1993). In addition, at values of J/t>2, which are beyond the threshold for a bound state in the low-electron-density limit, a region of predominantly s-wave pairing is found.
We report results of large-scale ground-state density matrix renormalization group (DMRG) calculations on t-[Formula: see text]-J cylinders with circumferences 6 and 8. We determine a rough phase diagram that appears to approximate the two-dimensional (2D) system. While for many properties, positive and negative [Formula: see text] values ([Formula: see text]) appear to correspond to electron- and hole-doped cuprate systems, respectively, the behavior of superconductivity itself shows an inconsistency between the model and the materials. The [Formula: see text] (hole-doped) region shows antiferromagnetism limited to very low doping, stripes more generally, and the familiar Fermi surface of the hole-doped cuprates. However, we find [Formula: see text] strongly suppresses superconductivity. The [Formula: see text] (electron-doped) region shows the expected circular Fermi pocket of holes around the [Formula: see text] point and a broad low-doped region of coexisting antiferromagnetism and d-wave pairing with a triplet p component at wavevector [Formula: see text] induced by the antiferromagnetism and d-wave pairing. The pairing for the electron low-doped system with [Formula: see text] is strong and unambiguous in the DMRG simulations. At larger doping another broad region with stripes in addition to weaker d-wave pairing and striped p-wave pairing appears. In a small doping region near [Formula: see text] for [Formula: see text], we find an unconventional type of stripe involving unpaired holes located predominantly on chains spaced three lattice spacings apart. The undoped two-leg ladder regions in between mimic the short-ranged spin correlations seen in two-leg Heisenberg ladders.
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).