Renyi entanglement entropies of descendant states in critical systems with boundaries: conformal field theory and spin chains

Taddia, Luca; Ortolani, F.; Palmai, Tamas

doi:10.1088/1742-5468/2016/09/093104

Cited by 35 publications

(56 citation statements)

References 67 publications

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“…For homogeneous systems this generalisation was considered in Refs. [84,85]. However, as soon as descendent fields are involved, the calculations become much more cumbersome.…”

Section: Discussion and Outlookmentioning

confidence: 99%

Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems

2019

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Conformal field theories in curved backgrounds have been used to describe inhomogeneous one-dimensional systems, such as quantum gases in trapping potentials and non-equilibrium spin chains. This approach provided, in a elegant and simple fashion, non-trivial analytic predictions for quantities, such as the entanglement entropy, that are not accessible through other methods. Here, we generalise this approach to low-lying excited states, focusing on the entanglement and relative entropies in an inhomogeneous free-fermionic system. Our most important finding is that the universal scaling function characterising these entanglement measurements is the same as the one for homogeneous systems, but expressed in terms of a different variable. This new scaling variable is a nontrivial function of the subsystem length and system's inhomogeneity that is easily written in terms of the curved metric. We test our predictions against exact numerical calculations in the free Fermi gas trapped by a harmonic potential, finding perfect agreement. arXiv:1810.02287v1 [cond-mat.stat-mech]

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“…For homogeneous systems this generalisation was considered in Refs. [84,85]. However, as soon as descendent fields are involved, the calculations become much more cumbersome.…”

Section: Discussion and Outlookmentioning

confidence: 99%

Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems

2019

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“…[60,61] which extended the method of Chatterjee and Zamolodchikov 55 by changing geometry to a semi-infinite cylinder; similarly, one may tackle finite size systems. The method can also be tested in excited states 62,63 . The method is fully analytic and allows the treatment of other measures of entanglement such as negativity [32][33][34][35][36] corresponding to an analytic continuation to n → 1/2.…”

Section: Conclusion and Lookoutmentioning

confidence: 99%

Entanglement entropy and boundary renormalization group flow: Exact results in the Ising universality class

Cornfeld

Sela

2017

Phys. Rev. B

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The entanglement entropy in one dimensional critical systems with boundaries has been associated with the noninteger ground state degeneracy. This quantity, being a characteristic of boundary fixed points, decreases under renormalization group flow, as predicted by the g-theorem. Here, using conformal field theory methods, we exactly calculate the entanglement entropy in the boundary Ising universality class. Our expression can be separated into the well known bulk term and a boundary entanglement term, displaying a universal flow between two boundary conditions, in accordance with the g-theorem. These results are obtained within the replica trick approach, where we show that the associated twist field, a central object generating the geometry of an n-sheeted Riemann surface, can be bosonized, giving simple analytic access to multiple quantities of interest. We argue that our result applies to other models falling into the same universality class. This includes the vicinity of the quantum critical point of the two-channel Kondo model, allowing to track in real space the presence of a region containing one half of a qubit with entropy 1 2 log(2), associated with a free local Majorana fermion.

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“…However, there are many examples of eigenstates with sub-volume (logarithmic) scaling of the entanglement entropy (see, e.g., Refs. [25,27]), in particular when the low-energy part of the spectrum is described by a CFT for which exact analytic predictions are obtainable [28][29][30][31][32][33][34][35][36]. Motivated by this evidence, there is strong common belief that ground states of "physically reasonable local hamiltonians" fulfil the area law, and that violations are at most logarithmic (see, however, Ref.…”

Section: Introductionmentioning

confidence: 99%

Unusual area-law violation in random inhomogeneous systems

Alba¹,

Santalla²,

Ruggiero³

et al. 2019

J. Stat. Mech.

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The discovery of novel entanglement patterns in quantum many-body systems is a prominent research direction in contemporary physics. Here we provide the example of a spin chain with random and inhomogeneous couplings that in the ground state exhibits a very unusual area law violation. In the clean limit, i.e., without disorder, the model is the rainbow chain and has volume law entanglement. We show that, in the presence of disorder, the entanglement entropy exhibits a power-law growth with the subsystem size, with an exponent 1/2. By employing the Strong Disorder Renormalization Group (SDRG) framework, we show that this exponent is related to the survival probability of certain random walks. The ground state of the model exhibits extended regions of short-range singlets (that we term "bubble" regions) as well as rare long range singlet ("rainbow" regions). Crucially, while the probability of extended rainbow regions decays exponentially with their size, that of the bubble regions is power law. We provide strong numerical evidence for the correctness of SDRG results by exploiting the free-fermion solution of the model. Finally, we investigate the role of interactions by considering the random inhomogeneous XXZ spin chain. Within the SDRG framework and in the strong inhomogeneous limit, we show that the above area-law violation takes place only at the free-fermion point of phase diagram. This point divides two extended regions, which exhibit volume-law and area-law entanglement, respectively. arXiv:1807.04179v1 [cond-mat.str-el] A FIG. 1: Setup used in this work. (top) Definition of the random inhomogeneous XX chain.The chain couplings are denoted as Jn = e −|n|h Kn, with n half integer numbers, h a real inhomogeneity parameter, and Kn independent random variables distributed with (4). In this work we focus on the entanglement entropy of a subregion A of length (shaded area in the Figure). Subsystem A starts from the chain center. local, translational invariant models that exhibit area-law violations has been constructed by Movassagh and Shor in Ref. [44]. Their ground-state entanglement entropy is ∝ 1/2 , thus exhibiting a polynomial violation of the area law. Importantly, the exponent of the entanglement growth originates from universal properties of the random walk. This is due to the fact that the ground state of the model is written in terms of a special class of combinatorial objects, called Motzkin paths [51]. A similar result can be obtained [45] using the Fredkin gates [52]. An interesting generalization of Ref. 44 obtained by deforming a colored version of the Motzkin paths has been presented in Ref. 46. The ground-state phase diagram of the model exhibits two phases with area-law and volume-law entanglement, respectively. These are separated by a "special" point, where the ground state displays square-root entanglement scaling.In this paper, we show that unusual area-law violations can be obtained in a one-dimensional inhomogeneous local system in the presence of disorder. Specifically, here we investiga...

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Renyi entanglement entropies of descendant states in critical systems with boundaries: conformal field theory and spin chains

Cited by 35 publications

References 67 publications

Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems

Entanglement and relative entropies for low-lying excited states in inhomogeneous one-dimensional quantum systems

Entanglement entropy and boundary renormalization group flow: Exact results in the Ising universality class

Unusual area-law violation in random inhomogeneous systems

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