Scaling of Tripartite Entanglement at Impurity Quantum Phase Transitions

Bayat, Abolfazl

doi:10.1103/physrevlett.118.036102

Cited by 47 publications

(39 citation statements)

References 60 publications

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“…In Eq. (16), N (ρ T ) has two parts, n N (R n ) and n δ n . The first part n N (R n ) is the sum of the entanglement in each density matrix R n , and the second n δ n counts contribution from mixtures of different R n 's.…”

Section: A Nrg Approximation Of Negativitymentioning

confidence: 99%

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Numerical renormalization group method for entanglement negativity at finite temperature

2018

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We develop a numerical method to compute the negativity, an entanglement measure for mixed states, between the impurity and the bath in quantum impurity systems at finite temperature. We construct a thermal density matrix by using the numerical renormalization group (NRG), and evaluate the negativity by implementing the NRG approximation that reduces computational cost exponentially. We apply the method to the single-impurity Kondo model and the single-impurity Anderson model. In the Kondo model, the negativity exhibits a power-law scaling at temperature much lower than the Kondo temperature and a sudden death at high temperature. In the Anderson model, the charge fluctuation of the impurity contribute to the negativity even at zero temperature when the on-site Coulomb repulsion of the impurity is finite, while at low temperature the negativity between the impurity spin and the bath exhibits the same power-law scaling behavior as in the Kondo

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Section: A Nrg Approximation Of Negativitymentioning

confidence: 99%

“…Due to the convexity of the negativity [22,23], δ n ≥ 0 is guaranteed. Equations (16) and (17) are exact, given construction of density matrix ρ T .…”

Section: A Nrg Approximation Of Negativitymentioning

confidence: 99%

Numerical renormalization group method for entanglement negativity at finite temperature

2018

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“…Therefore, quantum entanglement may be an alternative way to detect a quantum phase transition [15,25], other than the thermodynamic quantities. Besides interest from quantum information and quantum phase transitions [26][27][28][29][30][31], quantum entanglement is not only a powerful theoretical concept but also has been measured in several recent experiments [32][33][34][35]. In particular, (cluster) spin models can be implemented in experiments and simulated in quantum information processors [36][37][38][39][40][41][42][43][44].…”

Section: Introductionmentioning

confidence: 99%

Geometric entanglement and quantum phase transition in generalized cluster-XY models

Deger

Wei

2019

Quantum Inf Process

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In this work, we investigate quantum phase transition (QPT) in a generic family of spin chains using the ground-state energy, the energy gap, and the geometric measure of entanglement (GE). In many of prior works, GE per site was used. Here, we also consider GE per block with each block size being two. This can be regarded as a coarse grain of GE per site. We introduce a useful parameterization for the family of spin chains that includes the XY models with n-site interaction, the GHZ-cluster model and a cluster-antiferromagnetic model, the last of which exhibits QPT between a symmetry-protected topological phase and a symmetry-breaking antiferromagnetic phase. As the models are exactly solvable, their ground-state wavefunctions can be obtained and thus their GE can be studied. It turns out that the overlap of the ground states with translationally invariant product states can be exactly calculated and hence the GE can be obtained via further parameter optimization. The QPTs exhibited in these models are detected by the energy gap and singular behavior of geometric entanglement. In particular, the XzY model exhibits transitions from the nontrivial SPT phase to a trivial paramagnetic phase. Moreover, the halfway XY model exhibits a first-order transition across the Barouch-McCoy circle, on which it was only a crossover in the standard XY model.

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“…There are studies of multipartite entanglement in spin chains that involve figures of merit for three spins [15]. This can be done using the tangle, which corresponds to a hyperdeterminant of a tensor of three two-valued indices.…”

Section: Introductionmentioning

confidence: 99%

Multipartite entanglement in spin chains and the hyperdeterminant

Cervera-Lierta¹,

Celades²,

Latorre³

et al. 2018

J. Phys. A: Math. Theor.

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A way to characterize multipartite entanglement in pure states of a spin chain with n sites and local dimension d is by means of the Cayley hyperdeterminant. The latter quantity is a polynomial constructed with the components of the wave function ψ i1,...,in which is invariant under local unitary transformation. For spin 1/2 chains (i.e. d = 2) with n = 2 and n = 3 sites, the hyperdeterminant coincides with the concurrence and the tangle respectively. In this paper we consider spin chains with n = 4 sites where the hyperdeterminant is a polynomial of degree 24 containing around 2.8 × 10 6 terms. This huge object can be written in terms of more simple polynomials S and T of degrees 8 and 12 respectively. Correspondingly we compute S, T and the hyperdeterminant for eigenstates of the following spin chain Hamiltonians: the transverse Ising model, the XXZ Heisenberg model and the Haldane-Shastry model. Those invariants are also computed for random states, the ground states of random matrix Hamiltonians in the Wigner-Dyson Gaussian ensembles and the quadripartite entangled states defined by Verstraete et al. in 2002. Finally, we propose a generalization of the hyperdeterminant to thermal density matrices.We observe how these polynomials are able to capture the phase transitions present in the models studied as well as a subclass of quadripartite entanglement present in the eigenstates.

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Scaling of Tripartite Entanglement at Impurity Quantum Phase Transitions

Cited by 47 publications

References 60 publications

Numerical renormalization group method for entanglement negativity at finite temperature

Numerical renormalization group method for entanglement negativity at finite temperature

Geometric entanglement and quantum phase transition in generalized cluster-XY models

Multipartite entanglement in spin chains and the hyperdeterminant

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