Imbalance entanglement: Symmetry decomposition of negativity

Cornfeld, Eyal; Goldstein, Moshe; Sela, Eran

doi:10.1103/physreva.98.032302

Cited by 121 publications

(159 citation statements)

References 51 publications

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“…Note that some works normalize each block by each trace [21,22,23] before calculating the entropies, which thus quantify the entanglement after a projective charge measurement. We prefer not to do so and instead use (6), following [19,20], because the resulting entropies are not only more accessible to calculations, but are also directly experimentally measurable, using either the replica trick [20,24,25], or random time evolution which conserves the charge [26,27]. Let us also note that S 1 (Q A ) is simply the distribution P (Q A ) of charge in subsystem A. WhenQ can assume any integer value (e.g., when particle number or total S z are conserved), we define the flux-resolved REE as S n (α) = Tr ρ n A e iαQ A .…”

Section: Introductionmentioning

confidence: 99%

Symmetry resolved entanglement: exact results in 1D and beyond

Fraenkel¹,

Goldstein²

2020

J. Stat. Mech.

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113

160

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In a quantum many-body system that possesses an additive conserved quantity, the entanglement entropy of a subsystem can be resolved into a sum of contributions from different sectors of the subsystem's reduced density matrix, each sector corresponding to a possible value of the conserved quantity. Recent studies have discussed the basic properties of these symmetry-resolved contributions, and calculated them using conformal field theory and numerical methods. In this work we employ the generalized Fisher-Hartwig conjecture to obtain exact results for the characteristic function of the symmetry-resolved entanglement ("flux-resolved entanglement") for certain 1D spin chains, or, equivalently, the 1D fermionic tight binding and the Kitaev chain models. These results are true up to corrections of order o L −1 where L is the subsystem size. We confirm that this calculation is in good agreement with numerical results. For the gapless tight binding chain we report an intriguing periodic structure of the characteristic functions, which nicely extends the structure predicted by conformal field theory. For the Kitaev chain in the topological phase we demonstrate the degeneracy between the even and odd fermion parity sectors of the entanglement spectrum due to virtual Majoranas at the entanglement cut. We also employ the Widom conjecture to obtain the leading behavior of the symmetry-resolved entanglement entropy in higher dimensions for an ungapped free Fermi gas in its ground state.

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Section: Introductionmentioning

confidence: 99%

Symmetry resolved entanglement: exact results in 1D and beyond

Fraenkel¹,

Goldstein²

2020

J. Stat. Mech.

Self Cite

113

160

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“…We obtain the flux resolved RNs by adding vertex operators at the boundaries between the subsystems 43 . The additivity of the scaling dimensions, Eq.…”

Section: B the Entanglement Negativitymentioning

confidence: 99%

“…The numerical method for the exactly solvable XX model is developed in Ref. 43 and based on the fact that the partially transposed RDM is a sum of two Gaussian matrices 81,82 :…”

Section: B Entanglement Negativitymentioning

confidence: 99%

“…which could be analytically continued from an even integer n to yield the negativity, using ρ T2 A1∪A2 = lim n→1/2 N (2n) A1,A2 . While the RNs are not even entanglement monotones (as opposed to the REs), they are still useful indicators of entanglement since (like the REs), they are experimentally measurable for both bosons 43,44 and fermions 40 , and are easier to calculate.…”

Section: Introductionmentioning

confidence: 99%

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Dynamics of charge-resolved entanglement after a local quench

2019

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Quantum entanglement and its main quantitative measures, the entanglement entropy and entanglement negativity, play a central role in many body physics. An interesting twist arises when the system considered has symmetries leading to conserved quantities: Recent studies introduced a way to define, represent in field theory, calculate for 1+1D conformal systems, and measure, the contribution of individual charge sectors to the entanglement measures between different parts of a system in its ground state. In this paper, we apply these ideas to the time evolution of the chargeresolved contributions to the entanglement entropy and negativity after a local quantum quench. We employ conformal field theory techniques, the time-dependent density matrix renormalization group algorithm, and exact solution in the noninteracting limit, finding good agreement between all these methods.

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“…For any given state, clearly E = log(2N + 1). The (logarithmic) negativity has been studied in several contexts, ranging from harmonic chains and lattices [44][45][46][47][48][49][50][51][52][53] to quantum spin models [54][55][56][57][58][59][60][61][62][63][64][65] , from conformal and integrable field theories [66][67][68][69][70][71][72][73][74][75] to non-equilibrium situations [75][76][77][78][79][80][81] and intrinsic and symmetry-protected topological orders [82][83][84][85][86][87][88][89][90] . For fermionic models, it has been shown that the partial timereversal transpose is a more appropriate object to characterise t...…”

Section: Introductionmentioning

confidence: 99%

Negativity spectrum in the random singlet phase

2020

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Entanglement features of the ground state of disordered quantum matter are often captured by an infinite randomness fixed point that, for a variety of models, is the random singlet phase. Although a copious number of studies covers bipartite entanglement in pure states, at present, less is known for mixed states and tripartite settings. Our goal is to gain insights in this direction by studying the negativity spectrum in the random singlet phase. Through the strong disorder renormalization group technique, we derive analytic formulas for the universal scaling of the disorder averaged moments of the partially transposed reduced density matrix. Our analytic predictions are checked against a numerical implementation of the strong disorder renormalization group and against exact computations for the XX spin chain (a model in which free fermion techniques apply). Importantly, our results show that the negativity and logarithmic negativity are not trivially related after the average over the disorder. arXiv:1910.09571v1 [cond-mat.stat-mech]

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Imbalance entanglement: Symmetry decomposition of negativity

Cited by 121 publications

References 51 publications

Symmetry resolved entanglement: exact results in 1D and beyond

Symmetry resolved entanglement: exact results in 1D and beyond

Dynamics of charge-resolved entanglement after a local quench

Negativity spectrum in the random singlet phase

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