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Bulletin of the American Physical Society

Entanglement spectroscopy of SU(2)-broken phases in two dimensions

Vincenzo Alba, Fabian Kolley, Stefan Depenbrock, Ian P. McCulloch, Ulrich Schollwoeck

Published Mar 6, 2014• 0 citations
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Abstract

(Received 11 August 2013; published 30 October 2013)In magnetically ordered systems, the breaking of SU(2) symmetry in the thermodynamic limit is associatedwith the appearance of a special type of low-lying excitations in finite-size energy spectra, the so-called towerof states (TOS). In the present work, we numerically demonstrate that there is a correspondence between theSU(2) tower of states and the lower part of the

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Entanglement spectroscopy of SU(2)-broken phases in two dimensions

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In magnetically ordered systems, the breaking of SU(2) symmetry in the thermodynamic limit is associated with the appearance of a special type of low-lying excitations in finite-size energy spectra, the so-called tower of states (TOS). In the present work, we numerically demonstrate that there is a correspondence between the SU(2) tower of states and the lower part of the ground-state entanglement spectrum (ES). Using state-of-the-art density matrix renormalization group (DMRG) calculations, we examine the ES of the 2D antiferromagnetic ${J}_{1}$-${J}_{2}$ Heisenberg model on both the triangular and kagome lattice. At large ferromagnetic ${J}_{2}$, the model exhibits a magnetically ordered ground state. Correspondingly, its ES contains a family of low-lying levels that are reminiscent of the energy tower of states. Their behavior (level counting, finite-size scaling in the thermodynamic limit) sharply reflects TOS features, and is characterized in terms of an effective entanglement Hamiltonian that we provide. At large system sizes, TOS levels are divided from the rest by an entanglement gap. Our analysis suggests that (TOS) entanglement spectroscopy provides an alternative tool for detecting and characterizing SU(2)-broken phases using DMRG.

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arXiv:1407.3830•Journal of Statistical Mechanics Theory and Experiment•Oct 15, 2014•11 citations
(Received 11 August 2013; published 30 October 2013)In magnetically ordered systems, the breaking of SU(2) symmetry in the thermodynamic limit is associatedwith the appearance of a special type of low-lying excitations in finite-size energy spectra, the so-called towerof states (TOS). In the present work, we numerically demonstrate that there is a correspondence between theSU(2) tower of states and the lower part of the
In magnetically ordered systems, the breaking of SU(2) symmetry in the thermodynamic limit is associated with the appearance of a special type of low-lying excitations in finite-size energy spectra, the so-called tower of states (TOS). In the present work, we numerically demonstrate that there is a correspondence between the SU(2) tower of states and the lower part of the ground-state entanglement spectrum (ES). Using state-of-the-art density matrix renormalization group (DMRG) calculations, we examine the ES of the 2D antiferromagnetic J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on both the triangular and kagome lattice. At large ferromagnetic J2{J}_{2}J2​, the model exhibits a magnetically ordered ground state. Correspondingly, its ES contains a family of low-lying levels that are reminiscent of the energy tower of states. Their behavior (level counting, finite-size scaling in the thermodynamic limit) sharply reflects TOS features, and is characterized in terms of an effective entanglement Hamiltonian that we provide. At large system sizes, TOS levels are divided from the rest by an entanglement gap. Our analysis suggests that (TOS) entanglement spectroscopy provides an alternative tool for detecting and characterizing SU(2)-broken phases using DMRG.
Li and Haldane observed that low-lying entanglement spectra of ground states of (2+1)D chiral topological phases display characteristic degeneracies of the (1+1)D conformal field theory (CFT) appearing at a physical boundary. This work explains the finite-size splittings of these degeneracies by irrelevant deformations of a (1+0)D boundary of the CFT, representing the ground state. These intriguingly arise from operators of fractional dimension in certain entanglement spectra. This approach has applications for diagnosing chirality in topological projected entangled pair states (PEPS).
We study entanglement properties of systems with spontaneously broken continuous symmetry. We find that in addition to the expected area law behavior, the entanglement entropy contains a subleading contribution which diverges logarithmically with the subsystem size in agreement with the Monte Carlo simulations of A. Kallin et. al. (Phys. Rev. B 84, 165134 (2011)). The coefficient of the logarithm is a universal number given simply by NG(d−1)/2N_G (d-1)/2NG​(d−1)/2, where NGN_GNG​ is the number of Goldstone modes and ddd is the spatial dimension. This term is present even when the subsystem boundary is straight and contains no corners, and its origin lies in the interplay of Goldstone modes and restoration of symmetry in a finite volume. We also compute the "low-energy" part of the entanglement spectrum and show that it has the same characteristic "tower of states" form as the physical low-energy spectrum obtained when a system with spontaneously broken continuous symmetry is placed in a finite volume.
The (emergent) symmetry of a critical point constitutes fundamental pieces of information for determining the universality class and effective field theory. However, the underlying symmetry thus far can be conjectured only indirectly from the dimension of the order parameters in symmetry-breaking phases, and its correctness requires further verification to avoid overlooking hidden order parameters, which by itself is also a difficult task. In this Letter, we introduce an unbiased numerical approach to identify the underlying (emergent) symmetry of a critical point in quantum many-body systems without prior knowledge about the corresponding low-energy effective field theory. By numerically calculating the reduced density matrix in a very small subsystem of the total system, the Anderson tower of states in the entanglement spectrum can be obtained, clearly reflecting the underlying (emergent) symmetry of criticality. This is attributed to the fact that the entanglement spectrum can reveal the broken symmetry of the ground state of entanglement Hamiltonian after cooling from the critical point along an extra entanglement-temperature axis.
2D projected entangled pair states (PEPS) provide a unique framework giving access to detailed entanglement features of correlated (spin or electronic) systems. For a bi-partitioned quantum system, it has been argued that the entanglement spectrum (ES) is in a one-to-one correspondence with the physical edge spectrum on the cut and that the structure of the corresponding entanglement Hamiltonian (EH) reflects closely bulk properties (finite correlation length, criticality, topological order, etc). However, entanglement properties of systems with spontaneously broken continuous symmetry are still not fully understood. The spin-1/2 square lattice Heisenberg antiferromagnet provides a simple example showing spontaneous breaking of SU(2) symmetry down to U(1). The ground state can be viewed as a 'quantum Néel state' where the classical (Néel) staggered magnetization is reduced by quantum fluctuations. Here I consider the (critical) resonating valence bond (RVB) state doped with spinons to describe such a state; this enables the use of the associated PEPS representation (with virtual bond dimension D = 3) to compute the EH and the ES for a partition of an (infinite) cylinder. In particular, I find that the EH is (almost exactly) a chain of a dilute mixture of heavy (↓ spins) and light (↑ spins) hardcore bosons, where light particles are subject to long-range hoppings. The corresponding ES shows drastic differences with the typical ES obtained previously for ground states with restored SU(2)-symmetry (on finite systems).