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arXiv:1211.0321 · Physical Review B

Entanglement entropy at generalized Rokhsar-Kivelson points of quantum dimer models

Alexander Selem, C. M. Herdman, K. Birgitta Whaley

Published Mar 6, 2013• 11 citations
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Abstract

We study the $n=2$ R\'enyi entanglement entropy of the triangular quantum dimer model via Monte Carlo sampling of Rokhsar-Kivelson- (RK-) like ground-state wave functions. Using the construction proposed by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and an adaptation of the Monte Carlo algorithm described by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)], we compute the topological entanglement entropy (TEE) at the RK point $\ensuremath{\gamma}=(1.001\ifmmode\pm\else\textpm\fi{}0.003)\mathrm{ln}2$, confirming earlier results. Additionally, we compute the TEE of the ground state of a generalized RK-like Hamiltonian and demonstrate the universality of TEE over a wide range of parameter values within a topologically ordered phase approaching a quantum phase transition. For system sizes that are accessible numerically, we find that the quantization of TEE depends sensitively on correlations. We characterize corner contributions to the entanglement entropy and show that these are well described by shifts proportional to the number and types of corners in the bipartition.

DOI
10.1103/physrevb.87.125105

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We study the n=2n=2n=2 R'enyi entanglement entropy of the triangular quantum dimer model via Monte Carlo sampling of Rokhsar-Kivelson- (RK-) like ground-state wave functions. Using the construction proposed by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and an adaptation of the Monte Carlo algorithm described by Hastings et al. [Phys. Rev. Lett. 104, 157201 (2010)], we compute the topological entanglement entropy (TEE) at the RK point \ensuremathγ=(1.001\ifmmode±\else\textpm\fi0.003)ln2\ensuremath{\gamma}=(1.001\ifmmode\pm\else\textpm\fi{}0.003)\mathrm{ln}2\ensuremathγ=(1.001\ifmmode±\else\textpm\fi0.003)ln2, confirming earlier results. Additionally, we compute the TEE of the ground state of a generalized RK-like Hamiltonian and demonstrate the universality of TEE over a wide range of parameter values within a topologically ordered phase approaching a quantum phase transition. For system sizes that are accessible numerically, we find that the quantization of TEE depends sensitively on correlations. We characterize corner contributions to the entanglement entropy and show that these are well described by shifts proportional to the number and types of corners in the bipartition.
A highly efficient and simple-to-implement Monte Carlo algorithm is proposed for the evaluation of the Rényi entanglement entropy (REE) of the quantum dimer model (QDM) at the Rokhsar-Kivelson (RK) point. It makes possible the evaluation of REE at the RK point to the thermodynamic limit for a general QDM. We apply the algorithm to a QDM defined on the triangular and the square lattice in two dimensions and the simple and the face centered cubic (fcc) lattice in three dimensions. We find the REE on all these lattices follows perfect linear scaling in the thermodynamic limit, apart from an even-odd oscillation in the case of the square lattice. We also evaluate the topological entanglement entropy (TEE) with both a subtraction and an extrapolation procedure. We find the QDMs on both the triangular and the fcc lattice exhibit robust Z2 topological order. The expected TEE of ln2 is clearly demonstrated in both cases. Our large scale simulation also proves the recently proposed extrapolation procedure in cylindrical geometry to be a highly reliable way to extract the TEE of a topologically ordered system.
Thanks to Pfaffian techniques, we study the Rényi entanglement entropies and the entanglement spectrum of large subsystems for two-dimensional Rokhsar–Kivelson wavefunctions constructed from a dimer model on the triangular lattice. By including a fugacity t on some suitable bonds, one interpolates between the triangular lattice ( t = 1) and the square lattice ( t = 0). The wavefunction is known to be a massive topological liquid for t > 0 whereas it is a gapless critical state at t = 0. We mainly consider two geometries for the subsystem: that of a semi-infinite cylinder and the disc-like set-up proposed by Kitaev and Preskill (2006 Phys. Rev. Lett. 96 110404). In the cylinder case, the entropies contain an extensive term—proportional to the length of the boundary—and a universal subleading constant s n ( t ). Fitting these cylinder data (up to a perimeter of L = 32 sites) provides s n with a very high numerical accuracy (10 − 9 at t = 1 and 10 − 6 at t = 0.5). In the topological liquid phase we find s n ( t > 0) = − ln2, independent of the fugacity t and the Rényi parameter n . At t = 0 we recover a previously known result, for n < 1 and s n ( t = 0) = − ln(2)/( n − 1) for n > 1. In the disc-like geometry—designed to get rid of the boundary contributions—we find an entropy s n KP ( t 0) = − ln2 in the whole massive phase whatever n > 0, in agreement with the result of Flammia et al (2009 Phys. Rev. Lett. 103 261601). Some results for the gapless limit are discussed.
We consider a family of generalized Rokhsar-Kivelson (RK) Hamiltonians, which are reverse-engineered to have an arbitrary edge-weighted superposition of dimer coverings as their exact ground state at the RK point. We focus on a quantum dimer model on the triangular lattice, with doubly periodic edge weights. For simplicity we consider a 2×12\times12×1 periodic model in which all weights are set to one except for a tunable horizontal edge weight labeled ααα. We analytically show that the model exhibits a continuous quantum phase transition at α=3α=3α=3, changing from a topological Z2\mathbb{Z}_2Z2​ quantum spin liquid (α<3α<3α<3) to a columnar ordered state (α>3α>3α>3). The dimer-dimer correlator decays exponentially on both sides of the transition with the correlation length ξ∝1/∣α−3∣ξ\propto1/|α-3|ξ∝1/∣α−3∣ and as a power-law at criticality. The vison correlator exhibits an exponential decay in the spin liquid phase, but becomes a constant in the ordered phase, which we explain in terms of loop statistics of the double-dimer model. Using finite-size scaling of the vison correlator, we extract critical exponents consistent with the 2D Ising universality class. Additionally, we analytically show that the topological Rényi entropy of order ∞\infty∞ (topological min-entropy) changes from log⁡2\log2log2 for the quantum spin liquid phase α<3α<3α<3, to 000 for the ordered phase α>3α>3α>3, thereby analytically confirming the topological nature of the phase transition.
An outstanding problem in the study of topologically ordered phases is to find methods to distinguish topologically ordered from unordered phases. The topological entanglement entropy, a negative shift to the entanglement entropy scaling, has emerged as one possible distinguishing probe. The n=2 Rényi topological entanglement entropy of the triangular quantum dimer model is studied by classical Monte Carlo sampling of Rokhsar-Kivelson-like ground state wave functions, and using quantum Monte Carlo sampling for Hamiltonian parameter values away from the Rokhsar-Kivelson point. Both results demonstrate the universality of topological entanglement entropy over a wide range of parameter values within a topologically ordered phase. For systems sizes that are accessible numerically, it is found that the quantization of topological entanglement entropy depends sensitively on correlations. At the Rokhsar-Kivelson point, competing corner contributions to the entanglement entropy are also characterized.Another class of topologically ordered materials existing in three dimensions are topological insulators, which exhibit Dirac modes on its surface. The anomalous currents arising from these modes are investigated. The Dirac energy eigenstates are computed on a flat torus (genus one topology) and closed cap cylinder (genus zero topology) for various mass-term geometries. From the resulting properties of the surface spectra, a potential application for a flux-charge qubit is presented.
A characterization of topological order in terms of bi-partite entanglement was proposed recently [A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006); M. Levin and X.-G. Wen, ibid. 96, 110405 (2006)]. It was argued that in a topological phase there is a universal additive constant in the entanglement entropy, called the topological entanglement entropy, which reflects the underlying gauge theory for the topological order. In the present paper, we evaluate numerically the topological entanglement entropy in the ground states of a quantum dimer model on the triangular lattice, which is known to have a dimer liquid phase with Z2{\mathbb{Z}}_{2}Z2​ topological order. We examine the two original constructions to measure the topological entropy by combining entropies on plural areas, and we observe that in the large-area limit they both approach the value expected for Z2{\mathbb{Z}}_{2}Z2​ topological order. We also consider the entanglement entropy on a topologically nontrivial ``zigzag'' area and propose to use it as another way to measure the topological entropy.
Using exact diagonalizations and Green's function Monte Carlo simulations, we have studied the zero-temperature properties of the quantum dimer model on the triangular lattice on clusters with up to 588 sites. A detailed comparison of the properties in different topological sectors as a function of the cluster size and for different cluster shapes has allowed us to identify different phases, to show explicitly the presence of topological degeneracy in a phase close to the Rokhsar-Kivelson point, and to understand finite-size effects inside this phase. The nature of the various phases has been further investigated by calculating dimer-dimer correlation functions. The present results confirm and complement the phase diagram proposed by Moessner and Sondhi on the basis of finite-temperature simulations [Phys. Rev. Lett. 86, 1881 (2001)].