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.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.

arXiv:1212.2703Physical Review BSep 25, 20136 citations

Entanglement entropy of U (1) quantum spin liquids

Michael Pretko, T. Senthil

Entanglement entropy is a powerful tool for characterizing spin liquids and other systems with long-range entanglement. For gapped spin liquids, this problem has been well-studied, finding a universal subleading constant term in the entanglement entropy, known as the topological entanglement entropy. In this paper, the idea of topological entanglement entropy is generalized to the gapless $U$ (1) spin liquid. First, it is shown that the entanglement entropy separates into two pieces, one due to emergent charges of the theory (assumed to be gapped) and one due to the emergent gapless gauge mode. Both give universal subheading contributions to the entanglement entropy, which are logarithmic in the size of the region considered. The photon contribution is due to the local physics of the gapless mode. The particle contribution, on the other hand, is a topological term, generalizing the notion of topological entanglement entropy. A construction is presented which can isolate this topological piece by eliminating the photon contribution and other local terms, such as the area law. This topological term in the entanglement entropy can serve as a powerful diagnostic tool for $U$ (1) spin liquids.Entanglement entropy is a powerful tool for characterizing spin liquids and other systems with long-range entanglement. For gapped spin liquids, this problem has been well-studied, finding a universal subleading constant term in the entanglement entropy, known as the topological entanglement entropy. In this paper, the idea of topological entanglement entropy is generalized to the gapless $U$(1) spin liquid. First, it is shown that the entanglement entropy separates into two pieces, one due to emergent charges of the theory (assumed to be gapped) and one due to the emergent gapless gauge mode. Both give universal subheading contributions to the entanglement entropy, which are logarithmic in the size of the region considered. The photon contribution is due to the local physics of the gapless mode. The particle contribution, on the other hand, is a topological term, generalizing the notion of topological entanglement entropy. A construction is presented which can isolate this topological piece by eliminating the photon contribution and other local terms, such as the area law. This topological term in the entanglement entropy can serve as a powerful diagnostic tool for $U$(1) spin liquids.

arXiv:1510.03863Physical review. B./Physical review. BSep 8, 201642 citations

Entanglement Entropy for Quantum Phases of Extended Hubbard Model

Jon Spalding, Shan-Wen Tsai

Bulletin of the American Physical SocietyMar 13, 20170 citations

Investigating topological order using recurrent neural networks

Mohamed Hibat-Allah, Roger G. Melko, Juan Carrasquilla

Recurrent neural networks (RNNs), originally developed for natural language processing, hold great promise for accurately describing strongly correlated quantum many-body systems. Here, we employ two-dimensional RNNs to investigate two prototypical quantum many-body Hamiltonians exhibiting topological order. Specifically, we demonstrate that RNN wave functions can effectively capture the topological order of the toric code and a Bose-Hubbard spin liquid on the kagome lattice by estimating their topological entanglement entropies. We also find that RNNs favor coherent superpositions of minimally entangled states over minimally entangled states themselves. Overall, our findings demonstrate that RNN wave functions constitute a powerful tool to study phases of matter beyond Landau's symmetry-breaking paradigm.Recurrent neural networks (RNNs), originally developed for natural language processing, hold great promise for accurately describing strongly correlated quantum many-body systems. Here, we employ two-dimensional RNNs to investigate two prototypical quantum many-body Hamiltonians exhibiting topological order. Specifically, we demonstrate that RNN wave functions can effectively capture the topological order of the toric code and a Bose-Hubbard spin liquid on the kagome lattice by estimating their topological entanglement entropies. We also find that RNNs favor coherent superpositions of minimally entangled states over minimally entangled states themselves. Overall, our findings demonstrate that RNN wave functions constitute a powerful tool to study phases of matter beyond Landau's symmetry-breaking paradigm.

arXiv:2303.11207Physical review. B./Physical review. BAug 22, 202327 citations

{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.

U

(1) spin liquid. First, it is shown that the entanglement entropy separates into two pieces, one due to emergent charges of the theory (assumed to be gapped) and one due to the emergent gapless gauge mode. Both give universal subheading contributions to the entanglement entropy, which are logarithmic in the size of the region considered. The photon contribution is due to the local physics of the gapless mode. The particle contribution, on the other hand, is a topological term, generalizing the notion of topological entanglement entropy. A construction is presented which can isolate this topological piece by eliminating the photon contribution and other local terms, such as the area law. This topological term in the entanglement entropy can serve as a powerful diagnostic tool for

U

(1) spin liquids.