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arXiv:cond-mat/0503235 · Physical Review A

Entanglement used to identify critical systems

Stein Olav Skrøvseth, Kaare Olaussen

Published Aug 16, 2005• 10 citations
arXivHTMLPDF

Abstract

We promote use of the geometric entropy formula derived by Holzhey et al. from conformal field theory, ${S}_{\ensuremath{\ell}}\ensuremath{\sim}(c∕3)\mathrm{log}(\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\pi}\ensuremath{\ell}∕N)$, to identify critical regions in zero temperature 1D quantum systems. The method is demonstrated on a class of one-dimensional $XY$ and $XYZ$ spin-$\frac{1}{2}$ chains, where the critical regions and their correponding central charges can be reproduced with quite modest computational efforts.

DOI
10.1103/physreva.72.022318

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We promote use of the geometric entropy formula derived by Holzhey et al. from conformal field theory, S\ensuremathℓ\ensuremath∼(c∕3)log(sin\ensuremathπ\ensuremathℓ∕N){S}_{\ensuremath{\ell}}\ensuremath{\sim}(c∕3)\mathrm{log}(\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\pi}\ensuremath{\ell}∕N)S\ensuremathℓ​\ensuremath∼(c∕3)log(sin\ensuremathπ\ensuremathℓ∕N), to identify critical regions in zero temperature 1D quantum systems. The method is demonstrated on a class of one-dimensional XYXYXY and XYZXYZXYZ spin-12\frac{1}{2}21​ chains, where the critical regions and their correponding central charges can be reproduced with quite modest computational efforts.
We compute the entropy of entanglement between the first NNN spins and the rest of the system in the ground states of a general class of quantum spin chains. We show that under certain conditions the entropy can be expressed in terms of averages over ensembles of random matrices. These averages can be evaluated, allowing us to prove that at critical points the entropy grows like $\ensuremath{\kappa}{log}_{2}N+\stackrel{\texttildelow{}}{\ensuremath{\kappa}}$ as $N\ensuremath{\rightarrow}\ensuremath{\infty}$, where $\ensuremath{\kappa}$ and $\stackrel{\texttildelow{}}{\ensuremath{\kappa}}$ are determined explicitly. In an important class of systems, $\ensuremath{\kappa}$ is equal to one-third of the central charge of an associated Virasoro algebra. Our expression for $\ensuremath{\kappa}$ therefore provides an explicit formula for the central charge.
A practical use of the entanglement entropy in a 1D quantum system is to identify the conformal field theory describing its critical behavior. It is exactly (c/3)lnℓ for an interval of length ℓ in an infinite system, where c is the central charge of the conformal field theory. Here we define the geometric mutual information, an analogous quantity for classical critical points. We compute this for 2D conformal field theories in an arbitrary geometry, and show in particular that for a rectangle cut into two rectangles, it is proportional to c. This makes it possible to extract c in classical simulations, which we demonstrate for the critical Ising and three-state Potts models.
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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.
For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N>>1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory. We show that for a class of strongly random quantum spin chains, the same logarithmic scaling holds for mean entanglement at criticality and defines a critical entropy equivalent to central charge in the pure case. This effective central charge is obtained for Heisenberg, XX, and quantum Ising chains using an analytic real-space renormalization-group approach believed to be asymptotically exact. For these random chains, the effective universal central charge is characteristic of a universality class and is consistent with a c-theorem.
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.