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arXiv:1106.0015 · Physical Review B

Topological entanglement entropy of Z2 spin liquids and lattice Laughlin states

Yi Zhang, Tarun Grover, Ashvin Vishwanath

Published Aug 9, 2011• 102 citations
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Abstract

We study entanglement properties of candidate wave functions for SU(2) symmetric gapped spin liquids and Laughlin states. These wave functions are obtained by the Gutzwiller projection technique. Using topological entanglement entropy $\ensuremath{\gamma}$ as a tool, we establish topological order in chiral spin liquid and ${\mathbb{Z}}_{2}$ spin liquid wave functions, as well as a lattice version of the Laughlin state. Our results agree very well with the field theoretic result $\ensuremath{\gamma}=\mathrm{log}D$ where $D$ is the total quantum dimension of the phase. All calculations are done using a Monte Carlo technique on a $12\ifmmode\times\else\texttimes\fi{}12$ lattice enabling us to extract $\ensuremath{\gamma}$ with small finite-size effects. For a chiral spin liquid wave function, the calculated value is within $4%$ of the ideal value. We also find good agreement for a lattice version of the Laughlin $\ensuremath{\nu}=1/3$ phase with the expected $\ensuremath{\gamma}=\mathrm{log}\sqrt{3}$.

DOI
10.1103/physrevb.84.075128

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We study entanglement properties of candidate wave functions for SU(2) symmetric gapped spin liquids and Laughlin states. These wave functions are obtained by the Gutzwiller projection technique. Using topological entanglement entropy \ensuremathγ\ensuremath{\gamma}\ensuremathγ as a tool, we establish topological order in chiral spin liquid and Z2{\mathbb{Z}}_{2}Z2​ spin liquid wave functions, as well as a lattice version of the Laughlin state. Our results agree very well with the field theoretic result \ensuremathγ=logD\ensuremath{\gamma}=\mathrm{log}D\ensuremathγ=logD where DDD is the total quantum dimension of the phase. All calculations are done using a Monte Carlo technique on a 12\ifmmode×\else\texttimes\fi1212\ifmmode\times\else\texttimes\fi{}1212\ifmmode×\else\texttimes\fi12 lattice enabling us to extract \ensuremathγ\ensuremath{\gamma}\ensuremathγ with small finite-size effects. For a chiral spin liquid wave function, the calculated value is within 44%4 of the ideal value. We also find good agreement for a lattice version of the Laughlin \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 phase with the expected \ensuremathγ=log3\ensuremath{\gamma}=\mathrm{log}\sqrt{3}\ensuremathγ=log3​.
The topological entanglement entropy (TEE) of Gutzwiller projected RVB state is studied with Monte Carlo simulation. With tricks to speed up the convergence of TEE, we are able to show that the spin liquid state studied in Zhang et al. [Phys. Rev. B 84, 075128 (2011)] actually does not support Z2{Z}_{2}Z2​ topological order. We find that both an approximate Marshall sign structure and a long-ranged RVB amplitude are responsible for the suppression of the vison gap in this spin liquid state. On the other hand, a TEE of ln2\mathrm{ln}2ln2 is clearly demonstrated for a short range RVB state on triangular lattice proposed in Yunoki et al. [Phys. Rev. B 74, 014408 (2006)]. We find the TEE of the RVB wave function is mainly determined by its phase rather than its amplitude. Our results indicate that the effective field theory misses some key information for the topological property of a RVB state.
Quantum spin liquids are phases of matter whose internal structure is not captured by a local order parameter. Particularly intriguing are critical spin liquids, where strongly interacting excitations control low energy properties. Here we calculate their bipartite entanglement entropy that characterizes their quantum structure. In particular we calculate the Renyi entropy S(2) on model wave functions obtained by Gutzwiller projection of a Fermi sea. Although the wave functions are not sign positive, S(2) can be calculated on relatively large systems (>324 spins) using the variational Monte Carlo technique. On the triangular lattice we find that entanglement entropy of the projected Fermi sea state violates the boundary law, with S(2) enhanced by a logarithmic factor. This is an unusual result for a bosonic wave function reflecting the presence of emergent fermions. These techniques can be extended to study a wide class of other phases.
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Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin-12\frac{1}{2}21​ quantum spin liquids with Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)1{}_{1}1​), as is the case for the $\ensuremath{\nu}$=12\frac{1}{2}21​ (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.
On the triangular and kagome lattices, short-ranged resonating valence-bond wave functions can be sampled without the sign problem using a recently developed Pfaffian Monte Carlo scheme. In this Rapid Communication, we study the Renyi entanglement entropy in these wave functions using a replica-trick method. Using various spatial bipartitions, including the Levin-Wen construction, our finite-size scaled Renyi entropy gives a topological contribution consistent with $\ensuremath{\gamma}=ln(2)$, as expected for a gapped Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquid. We prove that the mutual statistics is consistent with the toric code anyon model and rule out any other quasiparticle statistics such as the double semion model.
The Kalmeyer-Laughlin chiral spin liquids (CSL) and the Z2 spin liquids are two of the simplest topologically ordered states. Here I develop a theory of a direct quantum phase transition between them. Each CSL is characterized by an integer n and is topologically equivalent to the 1/2n Laughlin fractional quantum Hall (FQH) state. Depending on the parity of n, the transition from the CSL is either to a "twisted" version of the Z2 spin liquid, the "doubled semion" model, or the conventional Z_2 spin liquid ("toric code"). In the presence of SU(2) spin symmetry, the triplet gap remains open through the transition and only singlet operators acquire algebraic correlations. An essential observation is that the CSL/Laughlin FQH states can be understood in terms of bosonic integer quantum Hall (BIQH) states of Schwinger bosons or vortices, respectively. I propose several novel many-body wave functions that can interpolate through the transition.