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Bipartite fidelity for models with periodic boundary conditions

Gilles Parez, Alexi Morin-Duchesne, Jean Liénardy

Published Jan 1, 2021• 10 citations
Publisher

Abstract

For a given statistical model, the bipartite fidelity \\({\\mathcal F}\\) is computed from the overlap between the groundstate of a system of size \\(N\\) and the tensor product of the groundstates of the same model defined on two subsystems \\(A\\) and \\(B\\), of respective sizes \\(N_A\\) and \\(N_B\\) with \\(N = N_A + N_B\\). In this paper, we study \\({\\mathcal F}\\) for critical lattice models in the case where the full system has periodic boundary conditions. We consider two possible choices of boundary conditions for the subsystems \\(A\\) and \\(B\\), namely periodic and open. For these two cases, we derive the conformal field theory prediction for the leading terms in the \\(1/N\\) expansion of \\({\\mathcal F}\\), in a most general case that corresponds to the insertion of four and five fields, respectively. We provide lattice calculations of \\({\\mathcal F}\\), both exact and numerical, for two free-fermionic lattice models: the XX spin chain and the model of critical dense polymers. We study the asymptotic behaviour of the lattice results for these two models and find an agreement with the predictions of conformal field theory.

DOI
10.1088/1742-5468/abc1eb

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For a given statistical model, the bipartite fidelity \({\mathcal F}\) is computed from the overlap between the groundstate of a system of size \(N\) and the tensor product of the groundstates of the same model defined on two subsystems \(A\) and \(B\), of respective sizes \(N_A\) and \(N_B\) with \(N = N_A + N_B\). In this paper, we study \({\mathcal F}\) for critical lattice models in the case where the full system has periodic boundary conditions. We consider two possible choices of boundary conditions for the subsystems \(A\) and \(B\), namely periodic and open. For these two cases, we derive the conformal field theory prediction for the leading terms in the \(1/N\) expansion of \({\mathcal F}\), in a most general case that corresponds to the insertion of four and five fields, respectively. We provide lattice calculations of \({\mathcal F}\), both exact and numerical, for two free-fermionic lattice models: the XX spin chain and the model of critical dense polymers. We study the asymptotic behaviour of the lattice results for these two models and find an agreement with the predictions of conformal field theory.
Abstract For a given statistical model, the bipartite fidelity F is computed from the overlap between the groundstate of a system of size N and the tensor product of the groundstates of the same model defined on two subsystems A and B , of respective sizes N A and N B with N = N A + N B . In this paper, we study F for critical lattice models in the case where the full system has periodic boundary conditions. We consider two possible choices of boundary conditions for the subsystems A and B , namely periodic and open. For these two cases, we derive the conformal field theory prediction for the leading terms in the 1/ N expansion of F , in a most general case that corresponds to the insertion of four and five fields, respectively. We provide lattice calculations of F , both exact and numerical, for two free-fermionic lattice models: the XX spin chain and the model of critical dense polymers. We study the asymptotic behaviour of the lattice results for these two models and find an agreement with the predictions of conformal field theory.
Abstract We investigate the bipartite fidelity for a lattice model described by a logarithmic CFT: the model of critical dense polymers. We define this observable in terms of a partition function on the pants geometry, where d defects enter at the top of the pants lattice and exit in one of the legs. Using the correspondence with the XX spin chain, we obtain an exact closed-form expression for and compute the leading terms in its 1/ N asymptotic expansion as a function of x = N A / N , where N is the lattice width at the top of the pants and N A is the width of the leg where the defects exit. We find an agreement with the results of Stéphan and Dubail for rational CFTs, with the central charge and conformal weights specialised to c = −2 and . We compute a second instance of the bipartite fidelity for d = 2 by imposing a different rule for the connection of the defects. In the conformal setting, this choice corresponds to inserting two boundary condition changing fields of weight that are logarithmic instead of primary. We compute the asymptotic expansion in this case as well and find a simple additive correction compared to , of the form . We confirm this lattice result with a CFT derivation and find that this correction term is identical for all logarithmic theories, independently of c and .
The entanglement spectra for a subsystem in a spin chain fine-tuned to a quantum-critical point contains signatures of the underlying quantum field theory that governs its low-energy properties. For an open chain with given boundary conditions described by a 2D conformal field theory~(CFT), the entanglement spectrum of the left/right half of the system coincides with a boundary CFT spectrum, where one of the boundary conditions arise due to the `entanglement cut'. The latter has been argued to be conformal and has been numerically found to be the `free' boundary condition for Ising, Potts and free boson theories. For these models, the `free' boundary condition for the lattice degree of freedom has a counterpart in the continuum theory. However, this is not true in general. Here, this question is analyzed for the unitary minimal models of 2D CFTs using the density matrix renormalization group technique. The entanglement spectra are computed for blocks of spins in open chains of A-type restricted solid-on-solid models with identical boundary conditions at the ends. The imposed boundary conditions are realized exactly for these lattice models due to their integrable nature. The obtained entanglement spectra are in good agreement with certain boundary CFT spectra. The boundary condition for the entanglement cut is found to be conformal and to coincide with the one with the highest boundary entropy. This identification enables determination of the exponents governing the unusual corrections to the entanglement entropy from the CFT partition functions. These are compared with numerical results.
On the logarithmic bipartite fidelity of the open XXZ spin chain at Δ=−1/2Δ=-1/2Δ=−1/2
The open XXZ spin chain with the anisotropy \Delta=-\frac12 Δ = − 1 2 and a one-parameter family of diagonal boundary fields is studied at finite length. A determinant formula for an overlap involving the spin chain’s ground-state vectors for different lengths is found. The overlap allows one to obtain an exact finite-size formula for the ground state’s logarithmic bipartite fidelity. The leading terms of its asymptotic series for large chain lengths are evaluated. Their expressions confirm the predictions of conformal field theory for the fidelity.
We develop the finite-size scaling (FSS) theory at quantum transitions. We consider various boundary conditions, such as open and periodic boundary conditions, and characterize the corrections to the leading FSS behavior. Using renormalization-group (RG) theory, we generalize the classical scaling ansatz to describe FSS in the quantum case, classifying the different sources of scaling corrections. We identify nonanalytic corrections due to irrelevant (bulk and boundary) RG perturbations and analytic contributions due to regular backgrounds and analytic expansions of the nonlinear scaling fields. To check the general predictions, we consider the quantum XYXYXY chain in a transverse field. For this model exact or numerically accurate results can be obtained by exploiting its fermionic quadratic representation. We study the FSS of several observables, such as the free energy, the energy differences between low-energy levels, correlation functions of the order parameter, etc., confirming the general predictions in all cases. Moreover, we consider bipartite entanglement entropies, which are characterized by the presence of additional scaling corrections, as predicted by conformal field theory.
Universal properties of a critical quantum spin chain are encoded in the underlying conformal field theory (CFT). This underlying CFT is fully characterized by its conformal data. We propose a method to extract the conformal data from a critical quantum spin chain with both periodic and antiperiodic boundary conditions (PBC and APBC) based on low-energy eigenstates, generalizing previous work on spin chains with only PBC. First, scaling dimensions and conformal spins are extracted from the energies and momenta of the eigenstates. Second, the Koo-Saleur formula of lattice Virasoro generators is generalized to APBC and used to identify conformal towers. Third, local and string operators on the lattice are identified with CFT operators with PBC and APBC, respectively. Finally, operator product expansion coefficients are extracted by computing matrix elements of lattice primary operators in the low-energy subspaces with PBC and APBC. To go beyond exact diagonalization, tensor network methods based on periodic uniform matrix product states are used. We illustrate our approach with critical and tricritical Ising quantum spin chains. In the latter case, we propose lattice operators that correspond to super-Virasoro generators and verify their action on low-energy eigenstates. In this way, we explore the emergence of superconformal symmetry in the quantum spin chain.