arXiv:1607.07016 · Journal of Statistical Mechanics Theory and Experiment

Finite size corrections to scaling of the formation probabilities and the Casimir effect in the conformal field theories

M. A. Rajabpour

Published Dec 1, 2016• 10 citations

Abstract

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.

DOI: 10.1088/1742-5468/2016/12/123101

Related papers

Formation probabilities in quantum critical chains and Casimir effect

M. A. Rajabpour

arXiv:1512.01052Europhysics Letters (EPL)Dec 1, 201516 citations

Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

Yifan Liu, Haruki Shimizu, Atsushi Ueda +1 more

arXiv:2405.06891SciPost PhysicsOct 2, 20246 citations

Scaling of the formation probabilities and universal boundary entropies in the quantum XY spin chain

Filiberto Ares, M. A. Rajabpour, Jacopo Viti

arXiv:2004.10606Journal of Statistical Mechanics Theory and ExperimentAug 1, 20207 citations

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

Khadijeh Najafi, M. A. Rajabpour

arXiv:1511.06401Physical review. B./Physical review. BMar 28, 201626 citations

Finite-size corrections of the entanglement entropy of critical quantum chains

J. C. Xavier, Francisco Castilho Alcaraz

arXiv:1111.6577Physical Review BJan 24, 201243 citations

Finite-size scaling at quantum transitions

Massimo Campostrini B, Andrea Pelissetto C, Ettore Vicari

arXiv:1401.0788Physical Review BMar 21, 2014107 citations

arXiv:1607.07016 · Journal of Statistical Mechanics Theory and Experiment

Finite size corrections to scaling of the formation probabilities and the Casimir effect in the conformal field theories

M. A. Rajabpour

Published Dec 1, 2016• 10 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1742-5468/2016/12/123101

Related papers

Formation probabilities in quantum critical chains and Casimir effect

M. A. Rajabpour

We find a connection between logarithmic formation probabilities of two disjoint intervals of quantum critical spin chains and the Casimir energy of two aligned needles in two dimensional classical critical systems. Using this connection we provide a formula for the logarithmic formation probability of two disjoint intervals in generic $1+1$ dimensional critical systems. The quantity is depenedent on the full structure of the underlying conformal field theory and so useful to find the universality class of the critical system. The connection that we find also provides a very efficient numerical method to calculate the Casimir energy between needles using quantum critical chains. The agreement between numerical results performed on critical transverse field Ising model and XX chain with our exact results is very good. We also comment on the mutual Rényi information of two disjoint intervals.We find a connection between logarithmic formation probabilities of two disjoint intervals of quantum critical spin chains and the Casimir energy of two aligned needles in two dimensional classical critical systems. Using this connection we provide a formula for the logarithmic formation probability of two disjoint intervals in generic $1+1$ dimensional critical systems. The quantity is depenedent on the full structure of the underlying conformal field theory and so useful to find the universality class of the critical system. The connection that we find also provides a very efficient numerical method to calculate the Casimir energy between needles using quantum critical chains. The agreement between numerical results performed on critical transverse field Ising model and XX chain with our exact results is very good. We also comment on the mutual Rényi information of two disjoint intervals.

arXiv:1512.01052Europhysics Letters (EPL)Dec 1, 201516 citations

Finite-size corrections to the energy spectra of gapless one-dimensional systems in the presence of boundaries

Yifan Liu, Haruki Shimizu, Atsushi Ueda +1 more

We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.We present the finite-size scaling theory of one-dimensional quantum critical systems in the presence of boundaries. While the finite-size spectrum in the conformal limit, namely of a conformal field theory with conformally invariant boundary conditions, is related to the dimensions of boundary operators by Cardy, the actual spectra of lattice models are affected by both bulk and boundary perturbations and contain non-universal boundary energies. We obtain a general expression of the finite-size energy levels in the presence of bulk and boundary perturbations. In particular, a generic boundary perturbation related to the energy-momentum tensor gives rise to a renormalization of the effective system size. We verify our field-theory formulation by comparing the results with the exact solution of the critical transverse-field Ising chain and with accurate numerical results on the critical three-state Potts chain obtained by Density-Matrix Renormalization Group.

arXiv:2405.06891SciPost PhysicsOct 2, 20246 citations

Scaling of the formation probabilities and universal boundary entropies in the quantum XY spin chain

Filiberto Ares, M. A. Rajabpour, Jacopo Viti

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.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.

arXiv:2004.10606Journal of Statistical Mechanics Theory and ExperimentAug 1, 20207 citations

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

Khadijeh Najafi, M. A. Rajabpour

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.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.

arXiv:1511.06401Physical review. B./Physical review. BMar 28, 201626 citations

Finite-size corrections of the entanglement entropy of critical quantum chains

J. C. Xavier, Francisco Castilho Alcaraz

Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\ensuremath{\alpha}$-R\'enyi entropy of a single interval for several critical quantum chains. We considered models with $U(1)$ symmetry such as the spin-1/2 $XXZ$ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\ensuremath{\alpha}$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies is ${p}_{\ensuremath{\alpha}}=2{X}_{\ensuremath{\epsilon}}/\ensuremath{\alpha}$ for $\ensuremath{\alpha}>1$ and ${p}_{1}=\ensuremath{\nu}$, where ${X}_{\ensuremath{\epsilon}}$ denotes the dimensions of the energy operator of the model and $\ensuremath{\nu}=2$ for all the models.Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement $\ensuremath{\alpha}$-R\'enyi entropy of a single interval for several critical quantum chains. We considered models with $U(1)$ symmetry such as the spin-1/2 $XXZ$ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\ensuremath{\alpha}$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies is ${p}_{\ensuremath{\alpha}}=2{X}_{\ensuremath{\epsilon}}/\ensuremath{\alpha}$ for $\ensuremath{\alpha}>1$ and ${p}_{1}=\ensuremath{\nu}$, where ${X}_{\ensuremath{\epsilon}}$ denotes the dimensions of the energy operator of the model and $\ensuremath{\nu}=2$ for all the models.

arXiv:1111.6577Physical Review BJan 24, 201243 citations

Finite-size scaling at quantum transitions

Massimo Campostrini B, Andrea Pelissetto C, Ettore Vicari

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 $XY$ 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.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 $XY$ 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.

arXiv:1401.0788Physical Review BMar 21, 2014107 citations

1+1

dimensional critical systems. The quantity is depenedent on the full structure of the underlying conformal field theory and so useful to find the universality class of the critical system. The connection that we find also provides a very efficient numerical method to calculate the Casimir energy between needles using quantum critical chains. The agreement between numerical results performed on critical transverse field Ising model and XX chain with our exact results is very good. We also comment on the mutual Rényi information of two disjoint intervals.

{t}^{*}=\frac{l}{2}

, where

l

is the size of the subsystem.

U(1)

symmetry such as the spin-1/2

XXZ

and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of $\ensuremath{\alpha}$, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies. We conjecture that the exponent of the leading finite-size correction of the $\ensuremath{\alpha}$-R\'enyi entropies is ${p}_{\ensuremath{\alpha}}=2{X}_{\ensuremath{\epsilon}}/\ensuremath{\alpha}$ for $\ensuremath{\alpha}>1$ and ${p}_{1}=\ensuremath{\nu}$, where ${X}_{\ensuremath{\epsilon}}$ denotes the dimensions of the energy operator of the model and $\ensuremath{\nu}=2$ for all the models.

XY

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.