04 Mar 2021-Vol. 10, Iss: 3, pp 054

Abstract: In the presence of a conserved quantity, symmetry-resolved entanglement entropies are a refinement of the usual notion of entanglement entropy of a subsystem. For critical 1d quantum systems, it was recently shown in various contexts that these quantities generally obey entropy equipartition in the scaling limit, i.e. they become independent of the symmetry sector. In this paper, we examine the finite-size corrections to the entropy equipartition phenomenon, and show that the nature of the symmetry group plays a crucial role. In the case of a discrete symmetry group, the corrections decay algebraically with system size, with exponents related to the operators' scaling dimensions. In contrast, in the case of a U(1) symmetry group, the corrections only decay logarithmically with system size, with model-dependent prefactors. We show that the determination of these prefactors boils down to the computation of twisted overlaps.

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Topics: Discrete symmetry (62%), Symmetry group (58%), Symmetry (physics) (57%) ... show more

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11 results found

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Sara Murciano^{1}, Riccarda Bonsignori^{1}, Pasquale Calabrese^{1}, Pasquale Calabrese^{2}•Institutions (2)

19 May 2021-

Abstract: We consider the problem of symmetry decomposition of the entanglement negativity in free fermionic systems. Rather than performing the standard partial transpose, we use the partial time-reversal transformation which naturally encodes the fermionic statistics. The negativity admits a resolution in terms of the charge imbalance between the two subsystems. We introduce a normalised version of the imbalance resolved negativity which has the advantage to be an entanglement proxy for each symmetry sector, but may diverge in the limit of pure states for some sectors. Our main focus is then the resolution of the negativity for a free Dirac field at finite temperature and size. We consider both bipartite and tripartite geometries and exploit conformal field theory to derive universal results for the charge imbalance resolved negativity. To this end, we use a geometrical construction in terms of an Aharonov-Bohm-like flux inserted in the Riemann surface defining the entanglement. We interestingly find that the entanglement negativity is always equally distributed among the different imbalance sectors at leading order. Our analytical findings are tested against exact numerical calculations for free fermions on a lattice.

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Topics: Quantum entanglement (52%), Charge (physics) (51%), Conformal field theory (51%)

16 Citations

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U(1) symmetry resolved entanglement in free 1+1 dimensional field theories via form factor bootstrap

David X. Horvath^{1}, Luca Capizzi^{1}, Pasquale Calabrese^{1}, Pasquale Calabrese^{2}•Institutions (2)

Abstract: We generalise the form factor bootstrap approach to integrable field theories with U(1) symmetry to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies. The bootstrap equations are solved for the free massive Dirac and complex boson theories, which are the simplest theories with U(1) symmetry. We present the exact and complete solution for the bootstrap, including vacuum expectation values and form factors involving any type and arbitrarily number of particles. The non-trivial solutions are carefully cross-checked by performing various limits and by the application of the ∆-theorem. An alternative and compact determination of the novel form factors is also presented. Based on the form factors of the U(1) composite branch-point twist fields, we re-derive earlier results showing entanglement equipartition for an interval in the ground state of the two models.

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Topics: Symmetry (physics) (55%), Field (physics) (51%), Form factor (quantum field theory) (51%) ... show more

14 Citations

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Abstract: Entanglement plays a prominent role in the study of condensed matter many-body systems: Entanglement measures not only quantify the possible use of these systems in quantum information protocols, but also shed light on their physics. However, exact analytical results remain scarce, especially for systems out of equilibrium. In this work we examine a paradigmatic one-dimensional fermionic system that consists of a uniform tight-binding chain with an arbitrary scattering region near its center, which is subject to a DC bias voltage at zero temperature. The system is thus held in a current-carrying nonequilibrium steady state, which can nevertheless be described by a pure quantum state. Using a generalization of the Fisher-Hartwig conjecture, we present an exact calculation of the bipartite entanglement entropy of a subsystem with its complement, and show that the scaling of entanglement with the length of the subsystem is highly unusual, containing both a volume-law linear term and a logarithmic term. The linear term is related to imperfect transmission due to scattering, and provides a generalization of the Levitov-Lesovik full counting statistics formula. The logarithmic term arises from the Fermi discontinuities in the distribution function. Our analysis also produces an exact expression for the particle-number-resolved entanglement. We find that although to leading order entanglement equipartition applies, the first term breaking it grows with the size of the subsystem, a novel behavior not observed in previously studied systems. We apply our general results to a concrete model of a tight-binding chain with a single impurity site, and show that the analytical expressions are in good agreement with numerical calculations. The analytical results are further generalized to accommodate the case of multiple scattering regions.

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Topics: Quantum entanglement (58%), Quantum state (55%), Quantum information (54%) ... show more

9 Citations

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Abstract: We consider the symmetry resolution of relative entropies in the 1+1 dimensional free massless compact boson conformal field theory (CFT) which presents an internal U(1) symmetry. We calculate various symmetry resolved Renyi relative entropies between one interval reduced density matrices of CFT primary states using the replica method. By taking the replica limit, the symmetry resolved relative entropy can be obtained. We also take the XX spin chain model as a concrete lattice realization of this CFT to perform numerical computation. The CFT predictions are tested against exact numerical calculations finding perfect agreement.

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Topics: Conformal field theory (59%), Symmetry (physics) (57%)

5 Citations

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Abstract: We consider the symmetry resolution of relative entropies in the 1+1 dimensional free massless compact boson conformal field theory (CFT) which presents an internal $U(1)$ symmetry. We calculate various symmetry resolved Renyi relative entropies between one interval reduced density matrices of CFT primary states using the replica method. By taking the replica limit, the symmetry resolved relative entropy can be obtained. We also take the XX spin chain model as a concrete lattice realization of this CFT to perform numerical computation. The CFT predictions are tested against exact numerical calculations finding perfect agreement.

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Topics: Conformal field theory (59%), Symmetry (physics) (57%)

3 Citations

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50 results found

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Abstract: We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy SA = −Tr ρAlogρA corresponding to the reduced density matrix ρA of a subsystem A. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A. These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.

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Topics: Joint quantum entropy (65%), Quantum relative entropy (65%), Von Neumann entropy (64%) ... show more

2,488 Citations

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Abstract: We study the "entanglement spectrum" (a presentation of the Schmidt decomposition analogous to a set of "energy levels") of a many-body state, and compare the Moore-Read model wave function for the nu=5/2 fractional quantum Hall state with a generic 5/2 state obtained by finite-size diagonalization of the second-Landau-level-projected Coulomb interactions. Their spectra share a common "gapless" structure, related to conformal field theory. In the model state, these are the only levels, while in the "generic" case, they are separated from the rest of the spectrum by a clear "entanglement gap", which appears to remain finite in the thermodynamic limit. We propose that the low-lying entanglement spectrum can be used as a "fingerprint" to identify topological order.

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Topics: Symmetry protected topological order (64%), Topological order (63%), Schmidt decomposition (63%) ... show more

1,026 Citations

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Abstract: Form factors for local spin operators of the XXZ Heisenberg spin-z finite chain are computed. Representation theory of Drinfel'd twists in terms of F-matrices for the quantum affine algebra U q ( s l 2 ) in finite-dimensional modules is used to calculate scalar products of two states involving one Bethe state (leading to Gaudin formula) and to solve the quantum inverse problem for local spin operators in the finite chain. Hence, we obtain the representation of the n-spin correlation functions in terms of expectation values (in ferromagnetic reference state) of the operator entries of the quantum monodromy matrix satisfying Yang-Baxter algebra. This leads to the direct calculation of the form factors of the XXZ Heisenberg spin-! finite chain as determinants of usual functions of the parameters of the model. A two-point correlation function for adjacent sites is also derived using similar techniques.

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Topics: Quantum affine algebra (57%), Spin-½ (55%), Correlation function (53%) ... show more

518 Citations

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Abstract: We review the properties of reduced density matrices for free fermionic or bosonic many-particle systems in their ground state. Their basic feature is that they have a thermal form and thus lead to a quasi-thermodynamic problem with a certain free-particle Hamiltonian. We discuss the derivation of this result, the character of the Hamiltonian and its eigenstates, the single-particle spectra and the full spectra, the resulting entanglement and in particular the entanglement entropy. This is done for various one- and two-dimensional situations, including also the evolution after global or local quenches.

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Topics: Quantum entanglement (58%), Ground state (51%), Free lattice (51%) ... show more

496 Citations

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Abstract: Many two-dimensional spin models can be transformed into Coulomb-gas systems in which charges interact via logarithmic potentials. For some models, such as the eight-vertex model and the Ashkin-Teller model, the Coulomb-gas representation has added significantly to the insight in the phase transitions. For other models, notably theXY model and the clock models, the equivalence has been instrumental for almost our entire understanding of the critical behavior. Recently it was shown that theq-state Potts model and then-vector model are equivalent to a Coulomb gas with an asymmetry between positive and negative charges. Fieldlike operators in these spin models transform noninteger charges and magnetic monopoles. With the aid of exactly solved models the Coulombgas representation allows analytic calculation of some critical indices.

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Topics: Potts model (57%), Charge (physics) (53%), Spin-½ (53%) ... show more

467 Citations