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Entanglement Entropy of Quantum Wire Junctions

Pasquale Calabrese, Mihail Mintchev, Ettore Vicari

Published Jan 1, 2012• 56 citations
OpenAlex

Abstract

We consider a fermion gas on a star graph modeling a quantum wire junction and derive the entanglement entropy of one edge with respect to the rest of the junction. The gas is free in the bulk of the graph, the interaction being localized in its vertex and described by a non-trivial scattering matrix. We discuss all point-like interactions, which lead to unitary time evolution of the system. We show that for a finite number of particles N, the Rényi entanglement entropies of one edge grow as lnN with a calculable prefactor, which depends not only on the central charge, but also on the total transmission probability from the considered edge to the rest of the graph. This result is extended to the case with an harmonic potential in the bulk.

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Entanglement entropy of quantum wire junctions

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Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) S ( α ) = ln tr ( ρ α ) / ( 1 − α ) of the reduced density matrix \rho ρ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} S ( 2 ) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) S N ( 2 ) = − ln ∑ n p 2 ( n ) which is a function of the probability distribution p(n) p ( n ) of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.

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We consider a fermion gas on a star graph modeling a quantum wire junction and derive the entanglement entropy of one edge with respect to the rest of the junction. The gas is free in the bulk of the graph, the interaction being localized in its vertex and described by a non-trivial scattering matrix. We discuss all point-like interactions, which lead to unitary time evolution of the system. We show that for a finite number of particles N, the Rényi entanglement entropies of one edge grow as lnN with a calculable prefactor, which depends not only on the central charge, but also on the total transmission probability from the considered edge to the rest of the graph. This result is extended to the case with an harmonic potential in the bulk.
We consider a fermion gas on a star graph modeling a quantum wire junction and derive the entanglement entropy of one edge with respect to the rest of the junction. The gas is free in the bulk of the graph, the interaction being localized in its vertex and described by a non-trivial scattering matrix. We discuss all point-like interactions, which lead to a unitary time evolution of the system. We show that for a finite number of particles N , the Rényi entanglement entropies of one edge grow as ln N with a calculable prefactor, which depends not only on the central charge, but also on the total transmission probability from the considered edge to the rest of the graph. This result is extended to the case with a harmonic potential in the bulk.
We introduce a systematic framework to calculate the bipartite entanglement entropy of a compact spatial subsystem in a one-dimensional quantum gas which can be mapped into a noninteracting fermion system. We show that when working with a finite number of particles N, the Rényi entanglement entropies grow as lnN, with a prefactor that is given by the central charge. We apply this novel technique to the ground state and to excited states of periodic systems. We also consider systems with boundaries. We derive universal formulas for the leading behavior and for subleading corrections to the scaling. The universality of the results allows us to make predictions for the finite-size scaling forms of the corrections to the scaling. ar
Abstract We consider different disordered lattice models composed of M linear chains glued together in a star-like manner, and study the scaling of the entanglement between one arm and the rest of the system using a numerical strong-disorder renormalization group method. For all studied models, the random transverse-field Ising model (RTIM), the random XX spin model, and the free-fermion model with random nearest-neighbor hopping terms, the average entanglement entropy is found to increase with the length L of the arms according to the form . For the RTIM and the XX model, the effective central charge is universal with respect to the details of junction, and only depends on the number M of arms. Interestingly, for the RTIM, decreases with M , whereas for the XX model it increases. For the free-fermion model, depends also on the details of the junction, which is related to the sublattice symmetry of the model. In this case, both increasing and decreasing tendency with M can be realized with appropriate junction geometries.
We introduce a systematic framework for calculating the bipartite entanglement entropy of a compact spatial subsystem in a one-dimensional quantum gas which can be mapped into a non-interacting fermion system. We show that when working with a finite number of particles N , the Rényi entanglement entropies grow as ln N , with a prefactor that is given by the central charge. We apply this novel technique to the ground state and to excited states of periodic systems. We also consider systems with boundaries. We derive universal formulas for the leading behavior and for subleading corrections to the scaling. The universality of the results allows us to make predictions for the finite size scaling forms of the corrections to the scaling.
Entanglement in a pure state of a many-body system can be characterized by the Rényi entropies S^{(\alpha)}=\ln\textrm{tr}(\rho^\alpha)/(1-\alpha) S ( α ) = ln tr ( ρ α ) / ( 1 − α ) of the reduced density matrix \rho ρ of a subsystem. These entropies are, however, difficult to access experimentally and can typically be determined for small systems only. Here we show that for free fermionic systems in a Gaussian state and with particle number conservation, S^{(2)} S ( 2 ) can be tightly bound—from above and below—by the much easier accessible Rényi number entropy S^{(2)}_N=-\ln \sum_n p^2(n) S N ( 2 ) = − ln ∑ n p 2 ( n ) which is a function of the probability distribution p(n) p ( n ) of the total particle number in the considered subsystem only. A dynamical growth in entanglement, in particular, is therefore always accompanied by a growth—albeit logarithmically slower—of the number entropy. We illustrate this relation by presenting numerical results for quenches in non-interacting one-dimensional lattice models including disorder-free, Anderson-localized, and critical systems with off-diagonal (bond) disorder.