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arXiv:1209.2424 · Physical Review B

Establishing non-Abelian topological order in Gutzwiller-projected Chern insulators via entanglement entropy and modular S -matrix

Yi Zhang, Ashvin Vishwanath

Published Apr 18, 2013• 36 citations
arXivHTMLPDF

Abstract

We use entanglement entropy signatures to establish non-Abelian topological order in projected Chern-insulator wave functions. The simplest instance is obtained by Gutzwiller projecting a filled band with Chern number $C=2$, whose wave function may also be viewed as the square of the Slater determinant of a band insulator. We demonstrate that this wave function is captured by the $\text{SU}{(2)}_{2}$ Chern-Simons theory coupled to fermions. This is established most persuasively by calculating the modular $\mathcal{S}$-matrix from the candidate ground-state wave functions, following a recent entanglement-entropy-based approach. This directly demonstrates the peculiar non-Abelian braiding statistics of Majorana fermion quasiparticles in this state. We also provide microscopic evidence for the field theoretic generalization, that the $N$th power of a Chern number $C$ Slater determinant realizes the topological order of the $\text{SU}{(N)}_{C}$ Chern-Simons theory coupled to fermions, by studying the $\text{SU}{(2)}_{3}$ (Read-Rezayi--type state) and the $\text{SU}{(3)}_{2}$ wave functions. An advantage of our projected Chern-insulator wave functions is the relative ease with which physical properties, such as entanglement entropy and modular $\mathcal{S}$-matrix, can be numerically calculated using Monte Carlo techniques.

DOI
10.1103/physrevb.87.161113

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We use entanglement entropy signatures to establish non-Abelian topological order in projected Chern-insulator wave functions. The simplest instance is obtained by Gutzwiller projecting a filled band with Chern number C=2C=2C=2, whose wave function may also be viewed as the square of the Slater determinant of a band insulator. We demonstrate that this wave function is captured by the SU(2)2\text{SU}{(2)}_{2}SU(2)2​ Chern-Simons theory coupled to fermions. This is established most persuasively by calculating the modular S\mathcal{S}S-matrix from the candidate ground-state wave functions, following a recent entanglement-entropy-based approach. This directly demonstrates the peculiar non-Abelian braiding statistics of Majorana fermion quasiparticles in this state. We also provide microscopic evidence for the field theoretic generalization, that the NNNth power of a Chern number CCC Slater determinant realizes the topological order of the SU(N)C\text{SU}{(N)}_{C}SU(N)C​ Chern-Simons theory coupled to fermions, by studying the SU(2)3\text{SU}{(2)}_{3}SU(2)3​ (Read-Rezayi--type state) and the SU(3)2\text{SU}{(3)}_{2}SU(3)2​ wave functions. An advantage of our projected Chern-insulator wave functions is the relative ease with which physical properties, such as entanglement entropy and modular S\mathcal{S}S-matrix, can be numerically calculated using Monte Carlo techniques.
The topological order is encoded in the pattern of long-range quantum entanglements, which cannot be measured by any local observable. Here we perform an exact diagonalization study to establish the non-Abelian topological order for topological band models through entanglement entropy measurement. We focus on the quasiparticle statistics of the non-Abelian Moore-Read and Read-Rezayi states on the lattice models with bosonic particles. We identify multiple independent minimal entangled states (MESs) in the ground state manifold on a torus. The extracted modular S matrix from MESs faithfully demonstrates the Ising anyon or Fibonacci quasiparticle statistics, including the quasiparticle quantum dimensions and the fusion rules for such systems. These findings unambiguously demonstrate the topological nature of the quantum states for these flatband models without using the knowledge of model wave functions.
Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the topological entanglement entropy (TEE) upon partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasiparticle excitations of the topological phase. The transformation of these states allows for the determination of the modular S\mathcal{S}S and U\mathcal{U}U matrices which encode quasiparticle properties. We demonstrate our method by extracting the modular S\mathcal{S}S matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate the TEE and prove that the quasiparticles obey semionic statistics. This method offers a route to nearly complete determination of the topological order in certain cases.
We develop an approach based on edge theories to calculate the entanglement entropy and related quantities in (2+1)-dimensional topologically ordered phases. Our approach is complementary to, e.g., the existing methods using replica trick and Witten's method of surgery, and applies to a generic spatial manifold of genus ggg, which can be bipartitioned in an arbitrary way. The effects of fusion and braiding of Wilson lines can be also straightforwardly studied within our framework. By considering a generic superposition of states with different Wilson line configurations, through an interference effect, we can detect, by the entanglement entropy, the topological data of Chern-Simons theories, e.g., the RRR symbols, monodromy, and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement/correlation measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent noncontractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders.
We implement the universal wave-function overlap (UWFO) method to extract modular SSS and TTT matrices for topological orders in Gutzwiller-projected parton wave functions (GPWFs). The modular SSS and TTT matrices generate a projective representation of SL(2,Z)SL(2,\mathbb{Z})SL(2,Z) on the degenerate-ground-state Hilbert space on a torus and may fully characterize the 2+1D topological orders, i.e., the quasiparticle statistics and chiral central charge (up to E8{E}_{8}E8​ bosonic quantum Hall states). We use the variational Monte Carlo method to computed the SSS and TTT matrices of the chiral spin liquid (CSL) constructed by the GPWF on the square lattice, and we confirm that the CSL carries the same topological order as the $\ensuremath{\nu}=\frac{1}{2}$ bosonic Laughlin state. We find that the nonuniversal exponents in the UWFO can be small, and direct numerical computation can be applied on relatively large systems. The UWFO may be a powerful method to calculate the topological order in GPWFs.
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 $\ensuremath{\gamma}=\mathrm{log}\sqrt{3}$.
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.