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Physical review. B, Condensed matter

Supersymmetrict-Jmodel: An exact result for two dimensions

Ken Cappon

Published Apr 1, 1991• 5 citations
Publisher

Abstract

The t-J model exhibits an additional symmetry at J=2t. The irreducible representations of this symmetry include states with different quantities of holes and total spin. This leads to degeneracies between systems with differing number of holes and gives an upper bound on the ground-state energy for one- and two-hole systems of arbitrary dimension. Also, it leads to an exact result for the difference in the ground-state energy for momenta k=(0,0) and k=(\ensuremath{\pi},\ensuremath{\pi}) between the single-hole t-J model and the Heisenberg antiferromagnet in two dimensions.

DOI
10.1103/physrevb.43.8698

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arXiv:supr-con/9502003•Physical review. B, Condensed matter•Nov 1, 1995•6 citations
The t-J model exhibits an additional symmetry at J=2t. The irreducible representations of this symmetry include states with different quantities of holes and total spin. This leads to degeneracies between systems with differing number of holes and gives an upper bound on the ground-state energy for one- and two-hole systems of arbitrary dimension. Also, it leads to an exact result for the difference in the ground-state energy for momenta k=(0,0) and k=(\ensuremath{\pi},\ensuremath{\pi}) between the single-hole t-J model and the Heisenberg antiferromagnet in two dimensions.
We calculated the binding energy of two holes with antiparallel spin in the t-J model using a variational wave function based on the string or spin-bag picture. The symmetry of the bound state is dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2. Although the size of matrices to be diagonalized is only 3\ifmmode\times\else\texttimes\fi{}3 we could reproduce the results of various extensive numerical calculations with reasonable accuracy. We show that a sizeable contribution of the binding energy is due to the quantum fluctuations in the ground state of the Heisenberg antiferromagnet. In contrast to exact diagonalization results on small clusters in the thermodynamical limit the binding energy decreases rapidly with increasing t/J. A detailed discussion of the various finite-size effects in a small cluster is also included.
Exact many-body states of square clusters up to 18 sites (with periodic boundary conditions) are investigated in order to study the dynamics of one or two holes in the t-J model. Our method takes full advantage of the translation, rotation, and reflection symmetries of the cluster. Using a modified Lanczos algorithm, we calculate the lowest state of all the different space-group representations and its total spin as well as its spin and hole correlations. For a single hole, at intermediate J/t values, the magnetic, kinetic energies, and staggered magnetization follow J/t power-law behaviors with size-dependent exponents. A binding energy of order J between two holes appears for sufficiently large J/t(\ensuremath{\ge}0.25). One-hole and two-hole calculations give different results regarding the stability of the Nagaoka state (against the singlet state) when J/t\ensuremath{\rightarrow}0. Energy expectation values of variational resonating-valence-bond states are calculated exactly and optimized. Significant overlaps with the exact states are found for any symmetry.
The one dimensional t-J Hamiltonian is diagonalized exactly for the supersymmetric case 2t=\ifmmode\pm\else\textpm\fi{}J using the Bethe ansatz. In this limit it is identical with models previously considered by Lai, Sutherland, and Schlottmann. In the present paper we discuss the ground-state properties and excitation spectrum in zero magnetic field. The ground state is a liquid of singlet bonds with varying spatial separation. Its most remarkable feature is the presence of bonds connecting particles at arbitrarily large distances. The ground-state energy is an analytic function of the band filling. There is no difference in the chemical potential for adding one or two particles and no evidence for the binding of holes. The low-lying part of the spectrum consists of two types of gapless excitations (charge and spin) with effective Fermi surfaces at 2kF{\mathit{k}}_{\mathit{F}}kF​ and kF{\mathit{k}}_{\mathit{F}}kF​, respectively. An interpretation of the energy spectrum in terms of spinons and holons is appropriate at low energies.
Th t-t'-t''-J model is investigated by a rigorous diagonalization in a two-dimensional square lattice as large as \ensuremath{\surd}20 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}20 , which has no additional degeneracy. The ground state of one hole in the t-J model is at a momentum k near (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2) with a total spin S=1/2 and that of two holes is at k=(0,0) with S=0. The hole pairing and nonclustering may be preferable for the ground state of the t-t'-t''-J model in the region of t'>0 and t''\ensuremath{\lesssim}0.
The ground state of a single hole in the one- and two-dimensional t-J model is studied for finite systems as a function of the phase in the kinetic energy term. The coherent effective hopping tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​ and the incoherent hopping ti∗{\mathit{t}}_{\mathit{i}}^{\mathrm{*}}ti∗​, evaluated in this way, are connected via the conductivity sum rule. It is shown that for even and odd chains tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​\ensuremath{\sim}ti∗{\mathit{t}}_{\mathit{i}}^{\mathrm{*}}ti∗​, while their reduction from the free value t remains small even for large J/t. The Bethe ansatz solution, obtained for J=2t, is consistent with the latter result, yielding tc∗{\mathit{t}}_{\mathit{c}}^{\mathrm{*}}tc∗​/t\ensuremath{\sim}0.938. In the 4\ifmmode\times\else\texttimes\fi{}4 system, the phase variation yields a nontrivial energy manifold. The minimum appears to be near the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},0) line in the Brillouin zone for J/t1, while moving towards the (\ensuremath{\pi}/2,\ensuremath{\pi}/2)-(\ensuremath{\pi},\ensuremath{\pi}) line for J/t\ensuremath{\gg}1.
We consider a modified t-J model with the minima of single-hole dispersion at the points (0,\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}), (\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi},0). It is shown that two holes in the antiferromagnetic background produce a bound state whose properties strongly differ from the states known in the unmodified t-J model. The bound state is d wave; it has four nodes on the face of the magnetic Brillouin zone. However, in the coordinate representation it looks like the usual s wave.