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Physical review. B, Condensed matter

Spin-Peierls ground states of the quantum dimer model: A finite-size study

Subir Sachdev A

Published Sep 1, 1989• 79 citations
Publisher

Abstract

Exact diagonalization results on the quantum dimer Hamiltonian of Rokshar and Kivelson on lattice sizes up to 6\ifmmode\times\else\texttimes\fi{}6 are presented. Correlation functions of a new order parameter characterizing the crystalline states of the model are calculated. All of the results are consistent with the existence of a direct transition between a columnar crystalline state and a staggered crystalline state with no intermediate spin-fluid state. However, the existence of a spin-fluid state over a small, but finite, range of parameters has not been definitively ruled out.

DOI
10.1103/physrevb.40.5204

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Exact diagonalization results on the quantum dimer Hamiltonian of Rokshar and Kivelson on lattice sizes up to 6\ifmmode\times\else\texttimes\fi{}6 are presented. Correlation functions of a new order parameter characterizing the crystalline states of the model are calculated. All of the results are consistent with the existence of a direct transition between a columnar crystalline state and a staggered crystalline state with no intermediate spin-fluid state. However, the existence of a spin-fluid state over a small, but finite, range of parameters has not been definitively ruled out.
We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size L\ifmmode\times\else\texttimes\fi{}L, where L\ensuremath{\leqslant}8, with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of V/J. This state is consistent with the plaquette resonating-valence-bond state. The energy gap of this state vanishes as a power law of L. It exhibits columnar dimer order, but has disorder within the dimer columns. This state persists up to V/J1, and the system changes to a dimer liquid state only at V/J=1. \textcopyright{} 1996 The American Physical Society.
We study the nature of the ground state of the quantum dimer model proposed by Rokhsar and Kivelson by diagonalizing the Hamiltonian of the model on square lattices of size L×LL\times LL×L, where L≤8L\leq 8L≤8, with periodic boundary conditions. Finite-size scaling studies of the columnar order parameter and the low lying excitation spectrum show no evidence of a dimer liquid state in any finite region of the zero temperature phase diagram. In addition, we find evidence of a transition from the columnar dimer state to an intermediate state at a negative value of V/JV/JV/J. This state is identified to be the plaquette resonating valence bond (RVB) state. The energy gap of this state vanishes as a power law of LLL. It exhibits columnar dimer order, but has disorder {\it within} the dimer columns. This state persists up to V/J<1V/J<1V/J<1, and the system changes to a dimer liquid state only at V/J=1V/J=1V/J=1.
The plaquette phase of the square-lattice quantum dimer model is studied using a continuous-time reptation quantum Monte Carlo method for lattices of sizes up to $48\ifmmode\times\else\texttimes\fi{}48$ sites. We determine the location of the phase transition between the columnar and plaquette phases to occur at ${V}_{c}∕J=0.60\ifmmode\pm\else\textpm\fi{}0.05$ which is significantly larger than inferred from previous exact diagonalization studies on smaller lattices. Off diagonal correlation functions are obtained. They exhibit long-range order in the plaquette phase but not at the Rokhsar-Kivelson point. We also observe significant finite-size corrections to scaling for the transition between the plaquette phase and the critical resonating valence bond liquid. This study demonstrates the importance of understanding finite-size effects when considering critical properties of the square-lattice quantum dimer model.
We study the doped and undoped quantum dimer model using exact diagonalization. In the undoped case, previous studies are consistent with the picture that in the region V/J < 0 the ground state is a spin-Peierls state which is four-fold degenerate. We extend the previous calculation and show convincing evidence that a gap exists between the degenerate ground state and the other excitation states. This is further evidence for the spin-Peierls phase. In the doped model, we search for the existence of a stripe phase where the hole-rich region is anisotropic. We introduce a new dimer-hole correlation function, which can indicate the shape of the hole-rich region. Our results show evidence that the holes tend to cluster together with an anisotropic spatial distribution when the holes are not very mobile. But we do not see strong evidence for the existence of a stripe phase.
We consider the spin-1/21/21/2 Hamiltonians proposed by Cano and Fendley [Phys. Rev. Lett. 105, 067205 (2010)], which were built to promote the well-known Rokshar-Kivelson (RK) point of quantum dimer models to spin-1/21/21/2 wave functions. We first show that these models, besides the exact degeneracy of RK point, support gapless spinless excitations as well as a spin gap in the thermodynamic limit, signatures of an unusual spin liquid. We then extend the original construction to create a continuous family of SU(2) invariant spin models that reproduces the phase diagram of the quantum dimer model and, in particular, show explicit evidences for existence of columnar and staggered phases. The original models thus appear as multicritical points in an extended phase diagram. Our results are based on the use of a combination of numerical exact simulations and analytical mapping to effective generalized quantum dimer models.
A two-leg spin-1/2 ladder with diagonal interactions is investigated numerically. We focus our attention on the possibility of columnar dimer phase, which was recently predicted based on a reformulated bosonization theory. By using density matrix renormalization group technique and exact diagonalization method, we calculate columnar dimer order parameter, spin correlation on a rung, string order parameters, and scaled excitation gaps. Carefully using various finite-size scaling techniques, our results show no support for the existence of columnar dimer phase in the spin ladder under consideration.