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arXiv:0807.0104 · Physical Review B

Universal subleading terms in ground-state fidelity from boundary conformal field theory

Lorenzo Campos Venuti, Hubert Saleur, Paolo Zanardi A

Published Mar 13, 2009• 22 citations
arXivHTMLPDF

Abstract

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 $O({L}^{\ensuremath{\alpha}})$ ($L$ system size), where $\ensuremath{\alpha}$ is equal to the spatial dimension $d$ for gapped systems, and otherwise depends on the critical exponents. Here we show that when parameters are changed along a critical manifold, a subleading $O(1)$ 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 $g$ factors and partition functions of (boundary) conformal field theory in $d=1$ and $d=2$ dimensions. Numerical checks are presented on the simple example of the $XXZ$ chain.

DOI
10.1103/physrevb.79.092405

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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\ensuremathα)O({L}^{\ensuremath{\alpha}})O(L\ensuremathα) (LLL system size), where \ensuremathα\ensuremath{\alpha}\ensuremathα 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.
In quantum field theories defined on a space-time with boundaries, the entanglement entropy exhibits subleading, boundary-induced corrections to the ubiquitous area law. At critical points described by conformal field theories (CFTs), and when the entangling surface intersects the physical boundary of the space, new universal terms appear in the entropy and encode valuable information about the boundary CFT. In 2+12+12+1 dimensions, the universal subleading boundary term is logarithmic with coefficient $b(\ensuremath{\theta})$ depending on the angle $\ensuremath{\theta}$ at which the entangling surface intersects the boundary, as well as on the boundary conditions (BCs). In this paper, we conduct a numerical study of $b(\ensuremath{\theta})$ for free bosons on finite-size square lattices. We find a surprisingly accurate fit between our lattice results and the corresponding holographic function available in the literature. We also comment on the ratio ${b}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\pi}/2)/{A}_{T}$, where AT{A}_{T}AT​ is the central charge in the near boundary expansion of the stress tensor, for which a holographic analysis suggests that it may be a universal quantity. Though we show evidence that this ratio is violated for the free boson with Dirichlet BCs, we conjecture its validity for free bosons evenly split between Dirichlet and Neumann BCs.
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 interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
We analyze universal terms that appear in the large system size scaling of the overlap between the Neel state and the ground state of the spin-1/2 XXZ chain in the antiferromagnetic regime. In a critical theory, the order one term of the asymptotics of such an overlap may be expressed in terms of g-factors, known in the context of boundary conformal field theory. In particular, for the XXZ model in its gapless phase, this term provides access to the Luttinger parameter. In its gapped phase, on the other hand, the order one term simply reflects the symmetry broken nature of the phase. In order to study the large system size scaling of this overlap analytically and to compute the order one term exactly, we use a recently derived finite-size determinant formula and perform an asymptotic expansion. Our analysis confirms the predictions of boundary conformal field theory and enables us to determine the exponent of the leading finite-size correction.
Universal finite-size amplitude and anomalous entangment entropy of z=2z=2z=2 quantum Lifshitz criticalities in topological chains
We consider Lifshitz criticalities with dynamical exponent z=2 z = 2 that emerge in a class of topological chains. There, such a criticality plays a fundamental role in describing transitions between symmetry-enriched conformal field theories (CFTs). We report that, at such critical points in one spatial dimension, the finite-size correction to the energy scales with system size, L L , as \sim L^{-2} ∼ L − 2 , with universal and anomalously large coefficient. The behavior originates from the specific dispersion around the Fermi surface, \epsilon \propto \pm k^2 ϵ ∝ ± k 2 . We also show that the entanglement entropy exhibits at the criticality a non-logarithmic dependence on l/L l / L , where l l is the length of the sub-system. In the limit of l\ll L l ≪ L , the maximally-entangled ground state has the entropy, S(l/L)=S_0+(l/L)\log(l/L) S ( l / L ) = S 0 + ( l / L ) log ( l / L ) . Here S_0 S 0 is some non-universal entropy originating from short-range correlations. We show that the novel entanglement originates from the long-range correlation mediated by a zero mode in the low energy sector. The work paves the way to study finite-size effects and entanglement entropy around Lifshitz criticalities and offers an insight into transitions between symmetry-enriched criticalities.
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.