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arXiv:1510.03863 · Physical review. B./Physical review. B

Entanglement entropy of U (1) quantum spin liquids

Michael Pretko, T. Senthil

Published Sep 8, 2016• 42 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.94.125112

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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 UUU(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 UUU(1) spin liquids.
Abstract An outstanding challenge involves understanding the many-particle entanglement of liquid states of quantum matter that arise in systems of interacting electrons. The Fermi liquid (FL) shows a violation of the area-law in real-space entanglement entropy of a subsystem, believed to be a signature of the ground state of a gapless quantum critical system of interacting fermions. Here, we apply a T = 0 renormalization group approach to the FL, unveiling the long-wavelength quantum fluctuations from which long-range entanglement arises. A similar analysis of non-Fermi liquids such as the 2D marginal Fermi liquid (MFL) and the 1D Tomonaga–Luttinger liquid reveals a universal logarithmic violation of the area-law in gapless electronic liquids, with a proportionality constant that depends on the nature of the underlying Fermi surface. We extend this analysis to classify the gapped quantum liquids emergent from the destabilisation of the Fermi surface by renormalisation group relevant quantum fluctuations arising from backscattering processes.
Quantum spin liquids (QSLs) are phases of interacting spins that do not order even at the absolute zero temperature, making it impossible to characterize them by a local order parameter. In this paper, we review the unique view provided by the quantum entanglement on QSLs. We illustrate the crucial role of topological entanglement entropy in diagnosing the non-local order in QSLs, using specific examples such as the chiral spin liquid. We also demonstrate the detection of anyonic quasi-particles and their braiding statistics using quantum entanglement. In the context of gapless QSLs, we discuss the detection of emergent fermionic spinons in a bosonic wavefunction, by studying the size dependence of entanglement entropy.
A bstract Despite the seeming simplicity of the theory, calculating (and even defining) entanglement entropy for the Maxwell theory of a U(1) gauge field in (3+1) dimensions has been the subject of controversy. It is generally accepted that the ground state entanglement entropy for a region of linear size L behaves as an area law with a subleading logarithm, S = αL 2 − γ log L . While the logarithmic coefficient γ is believed to be universal, there has been disagreement about its precise value. After carefully accounting for subtle boundary corrections, multiple analyses in the high energy literature have converged on an answer related to the conformal trace anomaly, which is only sensitive to the local curvature of the partition. In contrast, a condensed matter treatment of the problem yielded a topological contribution which is not captured by the conformal field theory calculation. In this perspective piece, we review aspects of the various calculations and discuss the resolution of the discrepancy, emphasizing the important role played by charged states (the “extended Hilbert space”) in defining entanglement for a gauge theory. While the trace anomaly result is sufficient for a strictly pure gauge field, coupling the gauge field to dynamical charges of mass m gives a topological contribution to γ which survives even in the m → ∞ limit. For many situations, the topological contribution from dynamical charges is physically meaningful and should be taken into account. We also comment on other common issues of entanglement in gauge theories, such as entanglement distillation, algebraic definitions of entanglement, and gauge-fixing procedures.
Quantum spin liquids (QSLs) are long-range entangled phases of matter and a natural setting for exploring how quantum correlations generate quantum complexity. Motivated by the emergence of magic, or nonstabilizerness, as a diagnostic of many-body complexity beyond entanglement, we study magic of QSLs by computing the stabilizer Rényi entropy (SRE) of the Kitaev honeycomb model. We derive a correspondence between Pauli strings and products of itinerant Majorana operators in the model's free-fermion description; this enables the sampling of SRE using an optimized algorithm for Gaussian states in systems with thousands of spins. Our results show that magic is largest in the gapless phase, where subleading volume-law corrections indicate nonlocal contributions, while in the gapped phases, SRE decreases in quantitative agreement with a perturbative expansion that we develop in the anisotropic limit. At the topological phase transition, SRE exhibits universal scaling dictated by critical exponents.
Topological order, reflected in long-range patterns of entanglement, is quantified by the topological entanglement entropy $\ensuremath{\gamma}$. We show that for gapped quantum spin liquids it is possible to extract $\ensuremath{\gamma}$ using two-spin local correlators. We demonstrate our method for the gapped Z2{\mathbb{Z}}_{2}Z2​ Kitaev spin liquid on a honeycomb lattice with anisotropic interactions. We show that the $\ensuremath{\gamma}=ln2$ for Z2{\mathbb{Z}}_{2}Z2​ topological order can be simply extracted from local two-spin correlators across two different bonds that involve only Majorana fermions, with an accuracy comparable to or higher than the Kitaev-Preskill construction. This implies that even though the ground state can be factorized into the product of Majorana and gauge sectors, the different superselection sectors of Z2{\mathbb{Z}}_{2}Z2​ gauge theory determined by global Wilson loop operators can be reflected locally in the Majorana sector.
We analyze in detail the effect of nontrivial band topology on the area-law behavior of the entanglement entropy in Kitaev's honeycomb model. By mapping the translationally invariant 2D spin model onto 1D fermionic subsystems, we identify those subsystems responsible for universal entanglement contributions in the gapped phases and those responsible for critical entanglement scaling in the gapless phases. For the gapped phases, we analytically show how the topological edge states contribute to the entanglement entropy and provide a universal lower bound for it. For the gapless semimetallic phases and topological phase transitions, the identification of the critical subsystems shows that they fall always into the Ising or the XY universality classes. As our study concerns the fermionic degrees of freedom in the honeycomb model, qualitatively similar results are expected to apply also to generic topological insulators and superconductors.