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arXiv:cond-mat/0612227 · Physical Review B

Topological entanglement entropy in the quantum dimer model on the triangular lattice

Shunsuke Furukawa, Grégoire Misguich

Published Jun 5, 2007• 146 citations
arXivHTMLPDF

Abstract

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 ${\mathbb{Z}}_{2}$ 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 ${\mathbb{Z}}_{2}$ 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.

DOI
10.1103/physrevb.75.214407

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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.
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 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 $\ensuremath{\gamma}=(1.001\ifmmode\pm\else\textpm\fi{}0.003)\mathrm{ln}2$, 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.
We elucidate the topological features of the entanglement entropy of a region in two-dimensional quantum systems in a topological phase with a finite correlation length $\ensuremath{\xi}$. First, we suggest that simpler reduced quantities, related to the von Neumann entropy, could be defined to compute the topological entropy. We use our methods to compute the entanglement entropy for the ground-state wave function of a quantum eight-vertex model in its topological phase and show that a finite correlation length adds corrections of the same order as the topological entropy which come from sharp features of the boundary of the region under study. We also calculate the topological entropy for the ground state of the quantum dimer model on a triangular lattice by using a mapping to a loop model. The topological entropy of the state is determined by loop configurations with a nontrivial winding number around the region under study. Finally, we consider extensions of the Kitaev wave function, which incorporate the effects of electric and magnetic charge fluctuations, and use it to investigate the stability of the topological phase by calculating the topological entropy.
It is known that for a topologically ordered state the area law for the entanglement entropy shows a negative universal additive constant contribution, $\ensuremath{-}\ensuremath{\gamma}$, called the topological entanglement entropy. We theoretically study the entanglement entropy of the two-dimensional Shastry-Sutherland quantum antiferromagnet using exact diagonalization on clusters of 16 and 24 spins. By utilizing the Kitaev-Preskill construction [A. Kitaev and J. Preskill, Phys. Rev. Lett. 96, 110404 (2006)] we extract a finite topological term, $\ensuremath{-}\ensuremath{\gamma}$, in the region of bond-strength parameter space corresponding to high geometrical frustration. Thus, we provide strong evidence for the existence of an exotic topologically ordered state and shed light on the nature of this model's strongly frustrated, and long controversial, intermediate phase.
Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log (GSDA~) \left({GSD}_{\tilde{A}}\right) (GSDA~​) GSD A ˜ , where GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.
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.