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The entanglement entropy of solvable lattice models

Robert Weston

Published Jan 1, 2006• 27 citations
OpenAlex

Abstract

We consider the spin κ/2 analogue of the XXZ quantum spin chain. We compute the entanglement entropy S associated with splitting the infinite chain into two semi-infinite pieces. In the scaling limit, we find S ≃ cκ 6 ln(ξ) plus a term that depends upon the boundary conditions. Here ξ is the correlation length and cκ = 3κ κ+2 is the central charge associated with the sl2 WZW model at level κ. Our result generalises previous observations and suggests that this is a simple and perhaps rather general way of extracting the central charge of the ultraviolet CFT associated with the scaling limit of a solvable lattice model. In this letter, we consider the spin κ/2 analogue of the XXZ quantum spin chain (which corresponds to the choice κ = 1). We derive an exact formula for the entanglement entropy S (i,κ) associated with splitting the infinite chain into two semi-infinite pieces. We consider the scaling limit and find that S (i,κ) ≃ cκ

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We consider the spin κ/2 analogue of the XXZ quantum spin chain. We compute the entanglement entropy S associated with splitting the infinite chain into two semi-infinite pieces. In the scaling limit, we find . Here ξ is the correlation length and c κ = 3κ/(κ+2) is the central charge associated with the Wess–Zumino–Witten (WZW) model at level κ. ln( g ) is the boundary entropy of the WZW model. Our result extends previous observations and suggests that this is a simple and perhaps rather general way both of extracting the central charge of the ultraviolet conformal field theory (CFT) associated with the scaling limit of a solvable lattice model, and of matching lattice and CFT boundary conditions.

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Entanglement entropy fluctuations in quantum Ising chains

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The behavior of Ising chains with the spin-spin interaction value λ in a transverse magnetic field of constant intensity ( h = 1) is considered. For a chain of infinite length, exact analytical formulas are obtained for the second central moment (dispersion) of the entropy operator Ŝ = -lnρ with reduced density matrix ρ, which corresponds to a semi-infinite part of the model chain occurring in the ground state. In the vicinity of a critical point λ c = 1, the entanglement entropy fluctuation Δ S (defined as the square root of dispersion) diverges as Δ S ∼ [ln(1/|1 − λ|)] 1/2 . For the known behavior of the entanglement entropy S , this divergence results in that the relative fluctuation δ S = Δ S / S vanishes at the critical point, that is, a state with almost nonfluctuating entanglement is attained.

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We consider the spin κ/2 analogue of the XXZ quantum spin chain. We compute the entanglement entropy S associated with splitting the infinite chain into two semi-infinite pieces. In the scaling limit, we find S ≃ cκ 6 ln(ξ) plus a term that depends upon the boundary conditions. Here ξ is the correlation length and cκ = 3κ κ+2 is the central charge associated with the sl2 WZW model at level κ. Our result generalises previous observations and suggests that this is a simple and perhaps rather general way of extracting the central charge of the ultraviolet CFT associated with the scaling limit of a solvable lattice model. In this letter, we consider the spin κ/2 analogue of the XXZ quantum spin chain (which corresponds to the choice κ = 1). We derive an exact formula for the entanglement entropy S (i,κ) associated with splitting the infinite chain into two semi-infinite pieces. We consider the scaling limit and find that S (i,κ) ≃ cκ
We consider the spin κ/2 analogue of the XXZ quantum spin chain. We compute the entanglement entropy S associated with splitting the infinite chain into two semi-infinite pieces. In the scaling limit, we find . Here ξ is the correlation length and c κ = 3κ/(κ+2) is the central charge associated with the Wess–Zumino–Witten (WZW) model at level κ. ln( g ) is the boundary entropy of the WZW model. Our result extends previous observations and suggests that this is a simple and perhaps rather general way both of extracting the central charge of the ultraviolet conformal field theory (CFT) associated with the scaling limit of a solvable lattice model, and of matching lattice and CFT boundary conditions.
We study the entanglement entropy scaling of the XXZ chain. While in the critical XY phase of the XXZ chain the entanglement entropy scales logarithmically with a coefficient that is determined by the associated conformal field theory, at the ferromagnetic point, however, the system is not conformally invariant yet the entanglement entropy still scales logarithmically albeit with a different coefficient. We investigate how such an nontrivial scaling at the ferromagnetic point influences the estimation of the central charge ccc in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of ccc. In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.
We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A logρ A corresponding to the reduced density matrix ρ A 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.
Random spin chains at quantum critical points exhibit an entanglement entropy between a segment of length LLL and the rest of the chain that scales as log2L{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}Llog2​L with a universal coefficient. Since for pure quantum critical spin chains this coefficient is fixed by the central charge of the associated conformal field theory, the universal coefficient in the random case can be understood as an effective central charge. In this paper we calculate the entanglement entropy and effective central charge of the spin-1 random Heisenberg model in its random-singlet phase and also at the critical point at which the Haldane phase breaks down. The latter is the first entanglement calculation for an infinite-randomness fixed point that is not in the random-singlet universality class. Our results are consistent with a ccc-theorem for flow between infinite-randomness fixed points. The formalism we use can be generally applied to calculation of quantities that depend on the RG history in $s\ensuremath{\geqslant}1$ random Heisenberg chains.
We present a study of the scaling behavior of the R\'enyi entanglement entropy (REE) in SU(N)\mathrm{SU}(N)SU(N) spin chain Hamiltonians, in which all of the spins transform under the fundamental representation. These SU(N)\mathrm{SU}(N)SU(N) spin chains are known to be quantum critical and described by a well known Wess-Zumino-Witten (WZW) nonlinear sigma model in the continuum limit. Numerical results from our lattice Hamiltonian are obtained using stochastic series expansion quantum Monte Carlo for both closed and open boundary conditions. As expected for this 1D critical system, the REE shows a logarithmic dependence on the subsystem size with a prefactor given by the central charge of the SU(N)\mathrm{SU}(N)SU(N) WZW model. We study in detail the subleading oscillatory terms in the REE under both periodic and open boundaries. Each oscillatory term is associated with a WZW field and decays as a power law with an exponent proportional to the scaling dimension of the corresponding field. We find that the use of periodic boundaries (where oscillations are less prominent) allows for a better estimate of the central charge, while using open boundaries allows for a better estimate of the scaling dimensions. We also present numerical data on the thermal R\'enyi entropy which equally allows for extraction of the central charge.
The behavior of Ising chains with the spin-spin interaction value λ in a transverse magnetic field of constant intensity ( h = 1) is considered. For a chain of infinite length, exact analytical formulas are obtained for the second central moment (dispersion) of the entropy operator Ŝ = -lnρ with reduced density matrix ρ, which corresponds to a semi-infinite part of the model chain occurring in the ground state. In the vicinity of a critical point λ c = 1, the entanglement entropy fluctuation Δ S (defined as the square root of dispersion) diverges as Δ S ∼ [ln(1/|1 − λ|)] 1/2 . For the known behavior of the entanglement entropy S , this divergence results in that the relative fluctuation δ S = Δ S / S vanishes at the critical point, that is, a state with almost nonfluctuating entanglement is attained.