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arXiv:1212.2703 · Physical Review B

Topological entanglement entropy in Gutzwiller projected spin liquids

Jiquan Pei, Steve Han, Haijun Liao, Tao Li

Published Sep 25, 2013• 6 citations
arXivHTMLPDF

Abstract

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 ${Z}_{2}$ 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 $\mathrm{ln}2$ 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.

DOI
10.1103/physrevb.88.125135

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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.
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 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 $\ensuremath{\gamma}=\mathrm{log}D$ where DDD 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 44%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 $\ensuremath{\gamma}=\mathrm{log}\sqrt{3}$.
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.
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.
The Gutzwiller projection of fermionic wave functions is a well-established method for generating variational wave functions describing exotic states of matter, such as quantum spin liquids. We investigate the conditions under which a projected wave function constructed from fermionic partons can be rigorously shown to possess topological order. We demonstrate that these conditions can be precisely determined in the case of projected Majorana stabilizer codes. We then use matrix product states to study states that interpolate between two distinct Majorana fermion codes, one yielding a Z2{\mathbb{Z}}_{2}Z2​ spin liquid and the other a trivial polarized state upon projection. While the free-fermion states are adiabatically connected, we find that the projected states undergo a phase transition detected by the topological entanglement entropy. Our work underscores the profound impact of the Gutzwiller projection and cautions against inferring properties of quantum spin liquids solely from their unprojected counterparts.
We analyze the properties of a non-Abelian spin-one chiral spin liquid state proposed by Greiter and Thomale [Phys. Rev. Lett. 102, 207203 (2009)] using Monte Carlo. In this state the bosonic $\ensuremath{\nu}=1$ Moore-Read Pfaffian wave function is used to describe a gas of bosonic spin flips on a square lattice with one flux quantum per plaquette. For toroidal geometries there is a three-dimensional space of these states corresponding to the topological degeneracy of the bosonic Moore-Read state on the torus. We show that spin correlations for different states in this space become indistinguishable for large system size. We also calculate the Renyi entanglement entropy for different system partitions to extract the topological entanglement entropy and provide evidence that the topological order of the lattice spin-liquid state is the same as that of the continuum Moore-Read state from which it is constructed.
We construct a short-range resonating-valence-bond (RVB) state on the ruby lattice, using projected entangled-pair states with bond dimension D=3D=3D=3. By introducing nonlocal moves to the dimer patterns on the torus, we distinguish four distinct sectors in the space of dimer coverings, which is a signature of the topological nature of the RVB wave function. Furthermore, by calculating the reduced density matrix of a bipartition of the RVB state on an infinite cylinder and exploring its entanglement entropy, we confirm the topological nature of the RVB wave function by obtaining a nonzero topological contribution, $\ensuremath{\gamma}=\ensuremath{-}\mathrm{ln}\phantom{\rule{4pt}{0ex}}2$, consistent with that of a Z2{\mathbb{Z}}_{2}Z2​ topological quantum spin liquid. We also calculate the ground-state energy of the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model on the ruby lattice and compare it with the RVB energy. Finally, we construct a quantum dimer model for the ruby lattice and discuss it as a possible parent Hamiltonian for the RVB wave function.