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arXiv:1010.3716 · Journal of Statistical Mechanics Theory and Experiment

Universal behavior of a bipartite fidelity at quantum criticality

Jérôme Dubail, Jean-Marie Stéphan

Published Mar 28, 2011• 52 citations
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Abstract

We introduce the (logarithmic) bipartite fidelity of a quantum system as the (logarithm of the) overlap between its ground state wavefunction and the ground state that one would obtain if the interactions between two complementary subsystems A and B were switched off. We argue that it should typically satisfy an area law in dimension d > 1. In the case of one-dimensional quantum critical points (QCP) we find that it admits a universal scaling form, , where is the typical size of the smaller subsystem. The prefactor is proportional to the central charge c and depends on the geometry. We also argue that this quantity can be useful for locating quantum phase transitions, allows for a reliable determination of the central charge, and in general exhibits various properties that are similar to the entanglement entropy. Like the entanglement entropy, it contains subleading universal terms in the case of a 2D conformal QCP.

DOI
10.1088/1742-5468/2011/03/l03002

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We introduce the (logarithmic) bipartite fidelity of a quantum system as the (logarithm of the) overlap between its ground state wavefunction and the ground state that one would obtain if the interactions between two complementary subsystems A and B were switched off. We argue that it should typically satisfy an area law in dimension d > 1. In the case of one-dimensional quantum critical points (QCP) we find that it admits a universal scaling form, , where is the typical size of the smaller subsystem. The prefactor is proportional to the central charge c and depends on the geometry. We also argue that this quantity can be useful for locating quantum phase transitions, allows for a reliable determination of the central charge, and in general exhibits various properties that are similar to the entanglement entropy. Like the entanglement entropy, it contains subleading universal terms in the case of a 2D conformal QCP.
The entanglement entropy of a pure quantum state of a bipartite system A union or logical sumB is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. In one dimension, the entanglement of critical ground states diverges logarithmically in the subsystem size, with a universal coefficient that for conformally invariant critical points is related to the central charge of the conformal field theory. We find that the entanglement entropy of a standard class of z=2 conformal quantum critical points in two spatial dimensions, in addition to a nonuniversal "area law" contribution linear in the size of the AB boundary, generically has a universal logarithmically divergent correction, which is completely determined by the geometry of the partition and by the central charge of the field theory that describes the critical wave function.
The entanglement entropy of a pure quantum state of a bipartite system is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have an entanglement entropy that diverges logarithmically in the subsystem size, with a universal coefficient that is is related to the central charge of the associated conformal field theory. Here I will discuss recent extensions of these ideas to a class of quantum critical points with dynamic critical exponent z = 2 in two space dimensions and to 2D systems in a topological phase. The application of these ideas to quantum dimer models and fractional quantum Hall states will be discussed.
Much has been learned about universal properties of entanglement entropies in ground states of quantum many-body lattice systems. Here we unveil universal properties of the average bipartite entanglement entropy of eigenstates of the paradigmatic quantum Ising model in one dimension. The leading term exhibits a volume-law scaling that we argue is universal for translationally invariant quadratic models. The subleading term is constant at the critical field for the quantum phase transition and vanishes otherwise (in the thermodynamic limit); i.e., the critical field can be identified from subleading corrections to the average (over all eigenstates) entanglement entropy.
Universal logarithmic terms in the entanglement entropy appear at quantum critical points (QCPs) in one dimension (1D) and have been predicted in 2D at QCPs described by 2D conformal field theories. The entanglement entropy in a strip geometry at such QCPs can be obtained via the "Shannon entropy" of a 1D spin chain with open boundary conditions. The Shannon entropy of the XXZ chain is found to have a logarithmic term that implies, for the QCP of the square-lattice quantum dimer model, a logarithm with universal coefficient ±0.25. However, the logarithm in the Shannon entropy of the transverse-field Ising model, which corresponds to entanglement in the 2D Ising conformal QCP, is found to have a singular dependence on the replica or Rényi index resulting from flows to different boundary conditions at the entanglement cut.
The study of the (logarithm of the) fidelity, i.e., of the overlap amplitude, between ground states of Hamiltonians corresponding to different coupling constants provides a valuable insight on critical phenomena. When the parameters are infinitesimally close, it is known that the leading term behaves as $O({L}^{\ensuremath{\alpha}})$ (LLL system size), where $\ensuremath{\alpha}$ is equal to the spatial dimension ddd for gapped systems, and otherwise depends on the critical exponents. Here we show that when parameters are changed along a critical manifold, a subleading O(1)O(1)O(1) term can appear. This term, somewhat similar to the topological entanglement entropy, depends only on the system's universality class and encodes nontrivial information about the topology of the system. We relate it to universal ggg factors and partition functions of (boundary) conformal field theory in d=1d=1d=1 and d=2d=2d=2 dimensions. Numerical checks are presented on the simple example of the XXZXXZXXZ chain.
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.