Detecting Goldstone modes with entanglement entropy

Kulchytskyy, Bohdan; Herdman, Chris M.; Inglis, Stephen; Melko, Roger G.

doi:10.1103/physrevb.92.115146

Cited by 31 publications

(38 citation statements)

References 44 publications

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“…in toroidal geometries of circumference 2L. The number of Nambu-Goldstone modes obtained from the prefactor of this term is in perfect agreement with field theoretical expectations [10,19,53,54], with accuracy at the percent level or lower. The fact that the VNE returns a value which is considerably closer to the field theoretical prediction when compared to the one extracted from Renyi entropies [17,19] may signal the fact that the latter are more affected by irrelevant operators, as observed in 1D [47,48,50], or may be due to the smoother continuity properties of the VNE.…”

Section: Two-dimensional Quantum Magnetssupporting

confidence: 83%

Measuring von Neumann entanglement entropies without wave functions

Mendes-Santos¹,

Giudici²,

Fazio³

et al. 2020

New J. Phys.

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We present a method to measure the von Neumann entanglement entropy of ground states of quantum many-body systems which does not require access to the system wave function. The technique is based on a direct thermodynamic study of lattice entanglement Hamiltonians-recently proposed in the paper [Dalmonte et al 2018 Nat. Phys. 14 827] via field theoretical insights-and can be performed by quantum Monte Carlo methods. We benchmark our technique on critical quantum spin chains, and apply it to several two-dimensional quantum magnets, where we are able to unambiguously determine the onset of area law in the entanglement entropy, the number of Goldstone bosons, and to check a recent conjecture on geometric entanglement contribution at critical points described by strongly coupled field theories. The protocol can also be adapted to measure entanglement in experiments via quantum quenches.

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Section: Two-dimensional Quantum Magnetssupporting

confidence: 83%

Measuring von Neumann entanglement entropies without wave functions

Mendes-Santos¹,

Giudici²,

Fazio³

et al. 2020

New J. Phys.

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“…We first recall in Section IV C 1 the origin of this logarithmic contribution in the presence of a broken symmetry, as stemming from the interplay between the tower-of-state entanglement spectrum of a subsystem and the low-lying Goldstone modes coupling two subsystems. In so doing we shall rephrase arguments which have been put forward in earlier works [57,59], but we will also generalize them to the case of long-range interactions, which add the new ingredient of a continuously varying dynamical exponent z ≤ 1. Furthermore we shall specialize in Section IV C 2 our discussion to the case of d = 1, where the analysis is somewhat more subtle.…”

Section: B Entanglement and Fluctuations In The Mediumand Short-rangmentioning

confidence: 99%

“…As discussed in Ref. [57,59], the Hamiltionan of the coupled A and B rotors can then be approximated as that of a harmonic oscillator of frequency ∆ G / [57,59]. Tracing out subsystem B leads to a densitymatrix description of subsystem A, in which ToS modes are populated up to an energy of the order of ∆ G , namely up to an angular momentum L max ∼ (I∆ G ) 1/2 ; hence, knowing that ToS levels L have a degeneracy of order L N −2 , the EE can be estimated by simple state counting as S A ∼ log Ω A , where…”

Section: Xy Phase: Logarithmic Term From the Tower-of-state Spectrummentioning

confidence: 99%

Entanglement and fluctuations in the XXZ model with power-law interactions

2017

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We investigate the ground-state properties of the spin-1/2 XXZ model with power-law-decaying (1/r α ) interactions, describing spins interacting with long-range transverse (XX) ferromagnetic interactions and longitudinal (Z) antiferromagnetic interactions, or hardcore bosons with long-range repulsion and hopping. The long-range nature of the couplings allows us to quantitatively study the spectral, correlation and entanglement properties of the system by making use of linear spin-wave theory, supplemented with density-matrix renormalization group in one-dimensional systems. Our most important prediction is the existence of three distinct coupling regimes, depending on the decay exponent α and number of dimensions d: 1) a short-range regime for α > d + σc (where σc = 1 in the gapped Néel antiferromagnetic phase exhibited by the XXZ model, and σc = 2 in the gapless XY ferromagnetic phase), sharing the same properties as those of finite-range interactions (α = ∞); 2) a long-range regime α < d, sharing the same properties as those of the infinite-range interactions (α = 0) in the thermodynamic limit; and 3) a most intriguing medium-range regime for d < α < d + σc, continuously interpolating between the finite-range and the infinite-range behavior. The latter regime is characterized by elementary excitations with a long-wavelength dispersion relation ω ≈ ∆g + ck z in the gapped phase, and ω ∼ k z in the gapless phase, exhibiting a continuously varying dynamical exponent z = (α − d)/σc. In the gapless phase of the model the z exponent is found to control the scaling of fluctuations, the decay of correlations, and a universal sub-dominant term in the entanglement entropy, leading to a very rich palette of behaviors for ground-state quantum correlations beyond what is known for finite-range interactions.

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“…These include Goldstone modes due to spontaneous breaking of a continuous symmetry [1][2][3][4]; topological order [5][6][7]; Fermi surfaces [8,9]; quantum criticality [10][11][12][13]; and more. If a system is divided into two spatial regions A and B, the entanglement in a pure state can be characterized by the von Neumann entropy associated with the reduced density matrix of either subsystem,…”

Section: Introductionmentioning

confidence: 99%

Unusual corrections to scaling and convergence of universal Renyi properties at quantum critical points

et al. 2016

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At a quantum critical point, bipartite entanglement entropies have universal quantities which are subleading to the ubiquitous area law. For Renyi entropies, these terms are known to be similar to the von Neumann entropy, while being much more amenable to numerical and even experimental measurement. We show here that when calculating universal properties of Renyi entropies, it is important to account for unusual corrections to scaling that arise from relevant local operators present at the conical singularity in the multi-sheeted Riemann surface. These corrections grow in importance with increasing Renyi index. We present studies of Renyi correlation functions in the 1+1 transverse-field Ising model (TFIM) using conformal field theory, mapping to free fermions, and series expansions, and the logarithmic entropy singularity at a corner in 2 + 1 for both free bosonic field theory and the TFIM, using numerical linked cluster expansions. In all numerical studies, accurate results are only obtained when unusual corrections to scaling are taken into account. In the worst case, an analysis ignoring these corrections can get qualitatively incorrect answers, such as predicting a decrease in critical exponents with the Renyi index, when they are actually increasing. We discuss a two-step extrapolation procedure that can be used to account for the unusual corrections to scaling.

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Detecting Goldstone modes with entanglement entropy

Cited by 31 publications

References 44 publications

Measuring von Neumann entanglement entropies without wave functions

Measuring von Neumann entanglement entropies without wave functions

Entanglement and fluctuations in the XXZ model with power-law interactions

Unusual corrections to scaling and convergence of universal Renyi properties at quantum critical points

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