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arXiv:1511.06401 · Physical review. B./Physical review. B

Formation probabilities and Shannon information and their time evolution after quantum quench in the transverse-field XY chain

Khadijeh Najafi, M. A. Rajabpour

Published Mar 28, 2016• 26 citations
arXivHTMLPDF

Abstract

We first provide a formula to calculate the probability of occurrence of different configurations (formation probabilities) in a generic free fermion system. We then study the scaling of these probabilities with respect to the size in the case of the critical transverse-field XY chain in the ${\ensuremath{\sigma}}^{z}$ bases. In the case of the transverse field Ising model, we show that all the ``crystal'' configurations follow the formulas expected from conformal field theory (CFT). In the case of the critical XX chain, we show that the only configurations that follow the formulas of the CFT are the ones which respect the filling factor of the system. By repeating all the calculations in the presence of open and periodic boundary conditions we find further support for our classification of different configurations. Using the developed technique, we also study Shannon information of a subregion in our system. In this respect we distinguish particular configurations that are more important in the study of the scaling limit of the Shannon information of the subsystem. Finally, we study the evolution of formation probabilities, Shannon information, and Shannon mutual information after a quantum quench in free fermion system. In particular, for the initial state considered in this paper, we demonstrate that the Shannon information after quantum quench first increases with the time and then saturates at time ${t}^{*}=\frac{l}{2}$, where $l$ is the size of the subsystem.

DOI
10.1103/physrevb.93.125139

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We first provide a formula to calculate the probability of occurrence of different configurations (formation probabilities) in a generic free fermion system. We then study the scaling of these probabilities with respect to the size in the case of the critical transverse-field XY chain in the \ensuremathσz{\ensuremath{\sigma}}^{z}\ensuremathσz bases. In the case of the transverse field Ising model, we show that all the ``crystal'' configurations follow the formulas expected from conformal field theory (CFT). In the case of the critical XX chain, we show that the only configurations that follow the formulas of the CFT are the ones which respect the filling factor of the system. By repeating all the calculations in the presence of open and periodic boundary conditions we find further support for our classification of different configurations. Using the developed technique, we also study Shannon information of a subregion in our system. In this respect we distinguish particular configurations that are more important in the study of the scaling limit of the Shannon information of the subsystem. Finally, we study the evolution of formation probabilities, Shannon information, and Shannon mutual information after a quantum quench in free fermion system. In particular, for the initial state considered in this paper, we demonstrate that the Shannon information after quantum quench first increases with the time and then saturates at time t∗=l2{t}^{*}=\frac{l}{2}t∗=2l​, where lll is the size of the subsystem.
Abstract We calculate exactly the probability to find the ground state of the XY chain in a given spin configuration in the transverse σ z -basis. By determining finite-volume corrections to the probabilities for a wide variety of configurations, we obtain the universal boundary entropy at the critical point. The latter is a benchmark of the underlying boundary conformal field theory characterizing each quantum state. To determine the scaling of the probabilities, we prove a theorem that expresses, in a factorized form, the eigenvalues of a sub-matrix of a circulant matrix as functions of the eigenvalues of the original matrix. Finally, the boundary entropies are computed by exploiting a generalization of the Euler–MacLaurin formula to non-differentiable functions. It is shown that, in some cases, the spin configuration can flow to a linear superposition of Cardy states. Our methods and tools are rather generic and can be applied to all the periodic quantum chains which map to free-fermionic Hamiltonians.
We study the Shannon and R\'enyi mutual information (MI) in the ground state (GS) of different critical quantum spin chains. Despite the apparent basis dependence of these quantities we show the existence of some particular basis (we will call them conformal basis) whose finite-size scaling function is related to the central charge ccc of the underlying conformal field theory of the model. In particular, we verified that for large index nnn, the MI of a subsystem of size $\ensuremath{\ell}$ in a periodic chain with LLL sites behaves as $\frac{c}{4}\frac{n}{n\ensuremath{-}1}ln[\frac{L}{\ensuremath{\pi}}sin(\frac{\ensuremath{\pi}\ensuremath{\ell}}{L})]$, when the ground-state wave function is expressed in these special conformal basis. This is in agreement with recent predictions. For generic local basis, we will show that, although in some cases ${b}_{n}ln[\frac{L}{\ensuremath{\pi}}sin(\frac{\ensuremath{\pi}\ensuremath{\ell}}{L})]$ is a good fit to our numerical data, in general, there is no direct relation between bn{b}_{n}bn​ and the central charge of the system. We will support our findings with detailed numerical calculations for the transverse field Ising model, Q=3,4Q=3,4Q=3,4 quantum Potts chain, quantum Ashkin-Teller chain, and the XXZ quantum chain. We will also present some additional results of the Shannon mutual information (n=1n=1n=1), for the parafermionic ZQ{Z}_{Q}ZQ​ quantum chains with Q=5,6,7Q=5,6,7Q=5,6,7, and 8.
Shannon-Rényi and stabilizer entropies are key diagnostics of structure, "nonstabilizerness," phase transitions, and universality in quantum many-body states. We establish an exact correspondence for quadratic fermions: for any Gaussian eigenstate, the stabilizer Rényi entropy equals the Shannon-Rényi entropy of a number-conserving free-fermion eigenstate on a doubled system, evaluated in the computational basis. Specializing to the transverse-field Ising (TFI) chain, the TFI ground-state stabilizer entropies map to the Shannon-Rényi entropies of the XX-chain ground state of length 2L. Building on this correspondence, together with other exact identities we prove, we derive closed expressions for the stabilizer entropy at indices α=1/2,2, 4 for a broad class of critical closed free-fermion systems. Each of these can be written with respect to the universal functions of the TFI chain. We further derive conformal-field-theory scaling laws for the stabilizer entropy at arbitrary Rényi index under both periodic and open boundary conditions. At α=4, these scaling forms display a discontinuity for both open and periodic boundary conditions.
Abstract We calculate formation probabilities of the ground state of the finite size quantum critical chains using conformal field theory (CFT) techniques. In particular, we calculate the formation probability of one interval in the finite open chain and also formation probability of two disjoint intervals in a finite periodic system. The presented formulas can be also interpreted as the Casimir energy of needles in particular geometries. We numerically check the validity of the exact CFT results in the case of the transverse field Ising chain.
Abstract We study the Rényi entropy and subsystem distances on one interval for the finite size and thermal states in the critical XY chains, focusing on the critical Ising chain and XX chain with zero transverse field. We construct numerically the reduced density matrices and calculate the von Neumann entropy, Rényi entropy, subsystem trace distance, Schatten two-distance, and relative entropy. As the continuum limit of the critical Ising chain and XX chain with zero field are, respectively, the two-dimensional free massless Majorana and Dirac fermion theories, which are conformal field theories, we compare the spin chain numerical results with the analytical results in conformal field theories and find perfect matches in the continuum limit.
We consider the Shannon mutual information of subsystems of critical quantum chains in their ground states. Our results indicate a universal leading behavior for large subsystem sizes. Moreover, as happens with the entanglement entropy, its finite-size behavior yields the conformal anomaly c of the underlying conformal field theory governing the long-distance physics of the quantum chain. We study analytically a chain of coupled harmonic oscillators and numerically the Q-state Potts models (Q=2, 3, and 4), the XXZ quantum chain, and the spin-1 Fateev-Zamolodchikov model. The Shannon mutual information is a quantity easily computed, and our results indicate that for relatively small lattice sizes, its finite-size behavior already detects the universality class of quantum critical behavior.