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arXiv:1511.00881 · Physical review. B./Physical review. B

Finite-volume energy spectrum, fractionalized strings, and low-energy effective field theory for the quantum dimer model on the square lattice

Debasish Banerjee, M. Bögli, Christoph P. Hofmann, Fu-Jiun Jiang, P. Widmer, U.‐J. Wiese

Published Sep 8, 2016• 24 citations
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Abstract

We present detailed analytic calculations of finite-volume energy spectra, mean-field theory, as well as a systematic low-energy effective field theory for the square lattice quantum dimer model. An emergent approximate spontaneously broken $\text{SO}(2)$ symmetry gives rise to a pseudo-Goldstone boson. Remarkably, this soft phononlike excitation, which is massless at the Rokhsar-Kivelson (RK) point, exists far beyond this point. The Goldstone physics is captured by a systematic low-energy effective field theory. We determine its low-energy parameters by matching the analytic effective field theory with exact diagonalization results. This confirms that the model exists in the columnar (and not in a plaquette or mixed) phase all the way to the RK point.

DOI
10.1103/physrevb.94.115120

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We present detailed analytic calculations of finite-volume energy spectra, mean-field theory, as well as a systematic low-energy effective field theory for the square lattice quantum dimer model. An emergent approximate spontaneously broken SO(2)\text{SO}(2)SO(2) symmetry gives rise to a pseudo-Goldstone boson. Remarkably, this soft phononlike excitation, which is massless at the Rokhsar-Kivelson (RK) point, exists far beyond this point. The Goldstone physics is captured by a systematic low-energy effective field theory. We determine its low-energy parameters by matching the analytic effective field theory with exact diagonalization results. This confirms that the model exists in the columnar (and not in a plaquette or mixed) phase all the way to the RK point.
We revisit the phase diagram of Rokhsar-Kivelson models, which are used in fields such as superconductivity, frustrated magnetism, cold bosons, and the physics of Josephson junction arrays. From an extended height effective theory, we show that one of two simple generic phase diagrams contains a mixed phase that interpolates continuously between columnar and plaquette states. For the square lattice quantum dimer model we present evidence from exact diagonalization and Green's function Monte Carlo techniques that this scenario is realized, by combining an analysis of the excitation gaps of different symmetry sectors with information on plaquette structure factors. This presents a natural framework for resolving the disagreement between previous studies.
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.
Aiming to describe frustrated quantum magnets with nonmagnetic singlet ground states, we have extended the Rokhsar-Kivelson (RK) loop expansion to derive a generalized quantum dimer model containing only connected terms up to arbitrary order. For the square-lattice frustrated Heisenberg antiferromagnet (${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ model), an expansion up to eighth order shows that the leading correction to the original RK model comes from dimer moves on length-6 loops. This model free of the original sign problem is treated by advanced numerical techniques. The results suggest that a rotationally anisotropic plaquette phase [A. Ralko, D. Poilblanc, and R. Moessner, Phys. Rev. Lett. 100, 037201 (2008)] is the ground state of the Heisenberg model in the parameter region of maximum frustration.
We determine the dynamical dimer correlation functions of quantum dimer models at the Rokhsar–Kivelson point on the bipartite square and cubic lattices and the non-bipartite triangular lattice. On the basis of an algorithmic idea by Henley, we simulate a stochastic process of classical dimer configurations in continuous time and perform a stochastic analytical continuation to obtain the dynamical correlations in momentum space and the frequency domain. This approach allows us to observe directly the dispersion relations and the evolution of the spectral intensity within the Brillouin zone beyond the single-mode approximation. On the square lattice, we confirm analytical predictions related to soft modes close to the wavevectors (π,π) and (π,0) and further reveal the existence of shadow bands close to the wavevector (0,0). On the cubic lattice the spectrum is also gapless but here only a single soft mode at (π,π,π) is found, as predicted by the single-mode approximation. The soft mode has a quadratic dispersion at very long wavelength, but crosses over to a linear behavior very rapidly. We believe this to be the remnant of the linearly dispersing 'photon' of the Coulomb phase. Finally the triangular lattice is in a fully gapped liquid phase where the bottom of the dimer spectrum exhibits a rich structure. At the M point the gap is minimal and the spectral response is dominated by a sharp quasiparticle peak. On the other hand, at the X point the spectral function is much broader. We sketch a possible explanation based on the crossing of the coherent dimer excitations into the two-vison continuum.
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.