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Numerical studies of the doped and undoped quantum dimer model

Kam-Fung Yu

Published Jan 1, 2001• 0 citations
Publisher

Abstract

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.

DOI
10.14711/thesis-b716410

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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.
The hole-doped two-dimensional(2D) $t\ensuremath{-}J$ model is studied by exact diagonalization on a $5\ifmmode\times\else\texttimes\fi{}4$ cluster which, unlike the standard tilted-square clusters, can in principle accomodate an antiphase domain wall. For hole concentration 1010%10 and J/t> 0.5J/t>~0.5J/t> 0.5 the ground-state energy/site is lower than the conventional tilted-square $\sqrt{20}\ifmmode\times\else\texttimes\fi{}\sqrt{20}$ cluster. In the ground state two holes form a loosely bound pair pinned to an antiphase domain wall. The dynamical density correlation function shows sharp quasiparticlelike peaks, reminiscent of the ``holons'' in 1D chains, which suggest the existence of solitonlike propagating domain walls. The dynamical correlation function of the bond-singlet operator has a low-energy peak structure characteristic of columnar spin-Peierls order, the dynamical spin-correlation function shows an intense and isolated ``resonance peak'' near $(\ensuremath{\pi},\ensuremath{\pi}).$
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.
A family of models is proposed to describe the motion of holes in a fluctuating quantum dimer background on the square lattice. Following Castelnovo et al. [Ann. Phys. (N.Y.) 318, 316 (2005)], a generalized Rokhsar-Kivelson Hamiltonian at finite doping which can be mapped on a doped interacting classical dimer model is constructed. A simple physical extension of this model is also considered. Using numerical computations and simple considerations based on the above exact mapping, we determine the phase diagram of the model showing a number of quantum phases typical of a doped Mott insulator. The two-hole correlation function generically exhibits short-range or long-range algebraic correlations in the solid (columnar) and liquid (critical) phases of the model, respectively. Evidence for an extended region of a doped valence bond solid phase exhibiting holon pairing but no phase separation is given. In contrast, we show that hole deconfinement occurs in the staggered dimer phase.
"We have studied the extended Hubbard and the Peierls-extended Hubbard models in one dimension using primarily computational methods. We discuss the derivation of the models and the computational methods used to study them, focusing primarily on our extensions over existing numerical methods. In particular, we present the construction of ""operator-loop"" updates for spin-1/2 fermions and the inclusion of electron-phonon interactions in the Stochastic Series Expansion Quantum Monte Carlo method and the use of Quantum Parallel Tempering to improve the quality of the numerical data. For the extended Hubbard model at half-filling, we have confirmed the existence of a recently reported bond-order wave phase for the ground state in a small region of parameter space between the well-known charge- and spin-density wave phases. We have studied in detail the nature of the associated quantum phase transitions between all the phases. Upon doping away from half-filling, the doubly-degenerate, spin and charge gapped long-range-ordered ground states of the model are shown to evolve to a Luther-Emery liquid with no charge gap but a finite spin gap. For the Peierls-extended Hubbard model, which includes both electron-phonon and electron-electron interactions, we have studied the dimerization transition of the half-filled system in the presence of fully quantum mechanical phonons. We show that the transition to a dimerized state occurs only when the electron-phonon coupling exceeds a finite critical value that depends on the bare phonon frequency. This is in contrast to previous studies in similar models with adiabatic phonons that had predicted a dimerized state for arbitrarily small values of the electron-phonon coupling."
The quantum dimer model is a low-energy effective model for many magnetic systems (materials) that are candidates for quantum spin liquids. It has strict local constraints which is described by gauge field theory. However, since constraints hinder the application of numerical algorithms, the phase diagrams of quantum dimer models are still controversial, even on a square lattice. The core controversy is whether the mixed state exists due to the restriction. In this paper, we give strong evidence to solve this dispute. With our sweeping cluster quantum Monte Carlo method, we studied the phase diagram of a large parameter region by introducing the definition of the pair correlation function and other supporting evidence to distinguish the mixed phase from the columnar phase with high precision. In particular, we find that the ground state belongs to the mixed phase for a vast parameter region.
We establish exact results for the one-dimensional spin-orbital model for special values of the four-spin interaction VVV and dimerization parameter δδδ. The first exact result is at δ=1/2δ=1/2δ=1/2 and V=−2V=-2V=−2. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approximate analytical techniques. The second result is for the homogeneous chain with V=−4V=-4V=−4 and gives evidence that the system has a spontaneously dimerized ground state. Numerical results indicate that the interplay between dimerization and interaction could result in gapless phases in the regime $0\leq V