Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1104.2544 · Physical Review B

Phase transition in the Rényi-Shannon entropy of Luttinger liquids

Jean-Marie Stéphan, Grégoire Misguich, Vincent Pasquier

Published Nov 28, 2011• 88 citations
arXivHTMLPDF

Abstract

The R\'enyi-Shannon entropy allows extraction of some universal information about many-body wave functions. For a critical spin chain with central charge $c=1$, we show that it exhibits a phase transition at some value ${n}_{c}$ of the R\'enyi parameter $n$ which depends on the Luttinger parameter $R$. A replica-free formulation establishes a connection to boundary entropies in conformal field theory and reveals that the transition is triggered by a vertex operator which becomes relevant at the boundary. Our numerical results (XXZ and ${J}_{1}\ensuremath{-}{J}_{2}$ spin chains) match the continuum limit prediction, confirming its universal character. The replica approach used in previous works turns out to be correct only for $n<{n}_{c}$. From the point of view of two-dimensional Rokhsar-Kivelson states, this transition reveals a singularity in the entanglement spectra.

DOI
10.1103/physrevb.84.195128

Related papers

Rényi entropy of a line in two-dimensional Ising models

Jean-Marie Stéphan, Grégoire Misguich, Vincent Pasquier

We consider the two-dimensional Ising model on an infinitely long cylinder and study the probabilities ${p}_{i}$ to observe a given spin configuration $i$ along a circular section of the cylinder. These probabilities also occur as eigenvalues of reduced density matrices in some Rokhsar-Kivelson wave functions. We analyze the subleading constant to the R\'enyi entropy ${R}_{n}=1/(1\ensuremath{-}n)\text{ln}({\ensuremath{\sum}}_{i}{p}_{i}^{n})$ and discuss its scaling properties at the critical point. Studying three different microscopic realizations, we provide numerical evidence that it is universal and behaves in a steplike fashion as a function of $n$ with a discontinuity at the Shannon point $n=1$. As a consequence, a field theoretical argument based on the replica trick would fail to give the correct value at this point. We nevertheless compute it numerically with high precision. Two other values of the R\'enyi parameter are of special interest: $n=1/2$ and $n=\ensuremath{\infty}$ are related in a simple way to the Affleck-Ludwig boundary entropies associated to free and fixed boundary conditions, respectively.

arXiv:1006.1605•Physical Review B•Sep 30, 2010•86 citations

Replica manipulation of the ground state in one-dimensional quantum spin systems

裕太 熊野

In the last decade it has become evident that information-theoretic quantities, which quantify a quantum entanglement encoded in a quantum state, are extremely useful to analyze a state of mane-body systems.Celebrated examples are the entanglement entropy and the Shannon entropy, which extract universal information of the underlying field theory of the system.Furthermore, a significant concept, the entanglement spectrum, has been proposed recently.It contains more complete information than the entanglement entropy.They have achieved a considerable success in characterizing exotic quantum phases that are beyond conventional descriptions.Inspired by these developments, we propose a new state, which we name a Rényified state, as follows: for a given quantum state |Ψ⟩ = ∑ i ψ i |i⟩ and the basis states |i⟩, the Rényified state is defined by raising the wave-function coefficients ψ i to the power of n (and normalized).The Shannon entropy can be obtained from the Rényified state.Moreover, the latter contains more information than the former.We study the Rényified state starting from a Tomonaga-Luttinger liquids (TLL), which is an important universality class of one-dimensional quantum systems.A TLL is characterized by a TLL parameter K, and described by the free boson field theory with a central charge c = 1 in the conformal field theory (CFT) context.We will show, using numerical calculations, that the Rényified state behaves as a TLL described by a modified TLL parameter K = K/n for small n.For larger n, the Rényified state behaves in a different way: the longitudinal correlation has long-range order while the transverse one remains algebraic.Based on the numerical results, we apply a replica field theory formulation to an analysis of the Rényified state.It suggests that the Rényified state is also a TLL described by a modified TLL parameter K = K/n, when n < 4K.The field theoretical analysis indicates that the TLL description breaks down at n = 4K, which is related to a phase transition in the Rényi-Shannon entropy.Beyond the transition, n > 4K, it suggests that the Rényified state is no longer a TLL since the longitudinal correlation has long-range order while the transverse one remains i algebraic.This exceptional behavior is unlikely to be realized in the ground state of a Hamiltonian with only short-range interactions.This indicates that the Rényified state beyond the transition belongs to a new class of exotic quantum phase.We explain an origin of the exotic behavior by constructing a particular conformal invariant boundary state of a two-component massless free boson.The relationship between the Rényified state and the Rényi-Shannon entropy is also elucidated in boundary CFT formalism.

•Mar 24, 2015•1 citation

Shannon and entanglement entropies of one- and two-dimensional critical wave functions

Jean-Marie Stéphan, Shunsuke Furukawa, Grégoire Misguich +1 more

We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant ${S}_{0}$. In $c=1$ critical systems (Tomonaga-Luttinger liquids), we find that ${S}_{0}$ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 $XXZ$ chain. In a massive (crystal) phase, ${S}_{0}$ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a $c=1/2$ critical point.

arXiv:0906.1153•Physical Review B•Nov 20, 2009•187 citations

Logarithmic Terms in Entanglement Entropies of 2D Quantum Critical Points and Shannon Entropies of Spin Chains

Michael P. Zaletel, Jens Hjorleifur Bardarson, Joel E. Moore

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.

arXiv:1103.5452•Physical Review Letters•Jul 5, 2011•84 citations

Oddities in the entanglement scaling of the quantum six-vertex model

Sunny Pradhan, Jesús Cobos, E. Rico +1 more

We investigate the entanglement properties of the Quantum Six-Vertex Model on a cylinder, focusing on the Shannon-Renyi entropy in the limit of Renyi order $n = \infty$. This entropy, calculated from the ground state amplitudes of the equivalent XXZ spin-1/2 chain, allows us to determine the Renyi entanglement entropy of the corresponding Rokhsar-Kivelson wavefunctions, which describe the ground states of certain conformal quantum critical points. Our analysis reveals a novel logarithmic correction to the expected entanglement scaling when the system size is odd. This anomaly arises from the geometric frustration of spin configurations imposed by periodic boundary conditions on odd-sized chains. We demonstrate that the scaling prefactor of this logarithmic term is directly related to the compactification radius of the low-energy bosonic field theory description, or equivalently, the Luttinger parameter. Thus, this correction provides a direct probe of the underlying Conformal Field Theory (CFT) describing the critical point. Our findings highlight the crucial role of system size parity in determining the entanglement properties of this model and offer insights into the interplay between geometry, frustration, and criticality.

arXiv:2502.19152•Physical review. B./Physical review. B•Feb 23, 2026•0 citations

Universal behavior of the Shannon and Rényi mutual information of quantum critical chains

Francisco Castilho Alcaraz, M. A. Rajabpour

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 $c$ of the underlying conformal field theory of the model. In particular, we verified that for large index $n$, the MI of a subsystem of size $\ensuremath{\ell}$ in a periodic chain with $L$ 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 ${b}_{n}$ and the central charge of the system. We will support our findings with detailed numerical calculations for the transverse field Ising model, $Q=3,4$ quantum Potts chain, quantum Ashkin-Teller chain, and the XXZ quantum chain. We will also present some additional results of the Shannon mutual information ($n=1$), for the parafermionic ${Z}_{Q}$ quantum chains with $Q=5,6,7$, and 8.

arXiv:1405.1074•Physical Review B•Aug 19, 2014•43 citations
The R'enyi-Shannon entropy allows extraction of some universal information about many-body wave functions. For a critical spin chain with central charge c=1c=1c=1, we show that it exhibits a phase transition at some value nc{n}_{c}nc​ of the R'enyi parameter nnn which depends on the Luttinger parameter RRR. A replica-free formulation establishes a connection to boundary entropies in conformal field theory and reveals that the transition is triggered by a vertex operator which becomes relevant at the boundary. Our numerical results (XXZ and J1\ensuremath−J2{J}_{1}\ensuremath{-}{J}_{2}J1​\ensuremath−J2​ spin chains) match the continuum limit prediction, confirming its universal character. The replica approach used in previous works turns out to be correct only for n<ncn<{n}_{c}n<nc​. From the point of view of two-dimensional Rokhsar-Kivelson states, this transition reveals a singularity in the entanglement spectra.
We consider the two-dimensional Ising model on an infinitely long cylinder and study the probabilities pi{p}_{i}pi​ to observe a given spin configuration iii along a circular section of the cylinder. These probabilities also occur as eigenvalues of reduced density matrices in some Rokhsar-Kivelson wave functions. We analyze the subleading constant to the R\'enyi entropy ${R}_{n}=1/(1\ensuremath{-}n)\text{ln}({\ensuremath{\sum}}_{i}{p}_{i}^{n})$ and discuss its scaling properties at the critical point. Studying three different microscopic realizations, we provide numerical evidence that it is universal and behaves in a steplike fashion as a function of nnn with a discontinuity at the Shannon point n=1n=1n=1. As a consequence, a field theoretical argument based on the replica trick would fail to give the correct value at this point. We nevertheless compute it numerically with high precision. Two other values of the R\'enyi parameter are of special interest: n=1/2n=1/2n=1/2 and $n=\ensuremath{\infty}$ are related in a simple way to the Affleck-Ludwig boundary entropies associated to free and fixed boundary conditions, respectively.
In the last decade it has become evident that information-theoretic quantities, which quantify a quantum entanglement encoded in a quantum state, are extremely useful to analyze a state of mane-body systems.Celebrated examples are the entanglement entropy and the Shannon entropy, which extract universal information of the underlying field theory of the system.Furthermore, a significant concept, the entanglement spectrum, has been proposed recently.It contains more complete information than the entanglement entropy.They have achieved a considerable success in characterizing exotic quantum phases that are beyond conventional descriptions.Inspired by these developments, we propose a new state, which we name a Rényified state, as follows: for a given quantum state |Ψ⟩ = ∑ i ψ i |i⟩ and the basis states |i⟩, the Rényified state is defined by raising the wave-function coefficients ψ i to the power of n (and normalized).The Shannon entropy can be obtained from the Rényified state.Moreover, the latter contains more information than the former.We study the Rényified state starting from a Tomonaga-Luttinger liquids (TLL), which is an important universality class of one-dimensional quantum systems.A TLL is characterized by a TLL parameter K, and described by the free boson field theory with a central charge c = 1 in the conformal field theory (CFT) context.We will show, using numerical calculations, that the Rényified state behaves as a TLL described by a modified TLL parameter K = K/n for small n.For larger n, the Rényified state behaves in a different way: the longitudinal correlation has long-range order while the transverse one remains algebraic.Based on the numerical results, we apply a replica field theory formulation to an analysis of the Rényified state.It suggests that the Rényified state is also a TLL described by a modified TLL parameter K = K/n, when n < 4K.The field theoretical analysis indicates that the TLL description breaks down at n = 4K, which is related to a phase transition in the Rényi-Shannon entropy.Beyond the transition, n > 4K, it suggests that the Rényified state is no longer a TLL since the longitudinal correlation has long-range order while the transverse one remains i algebraic.This exceptional behavior is unlikely to be realized in the ground state of a Hamiltonian with only short-range interactions.This indicates that the Rényified state beyond the transition belongs to a new class of exotic quantum phase.We explain an origin of the exotic behavior by constructing a particular conformal invariant boundary state of a two-component massless free boson.The relationship between the Rényified state and the Rényi-Shannon entropy is also elucidated in boundary CFT formalism.
We study the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum model. This entropy is related to the entanglement entropy of a Rokhsar-Kivelson-type wave function built from the corresponding two-dimensional classical model. In both critical and massive cases, we observe that it is composed of an extensive part proportional to the length of the system and a subleading universal constant S0{S}_{0}S0​. In c=1c=1c=1 critical systems (Tomonaga-Luttinger liquids), we find that S0{S}_{0}S0​ is a simple function of the boson compactification radius. This finding is based on a field-theoretical analysis of the Dyson-Gaudin gas related to dimer and Calogero-Sutherland models. We also performed numerical demonstrations in the dimer models and the spin-1/2 XXZXXZXXZ chain. In a massive (crystal) phase, S0{S}_{0}S0​ is related to the ground-state degeneracy. We also examine this entropy in the Ising chain in a transverse field as an example showing a c=1/2c=1/2c=1/2 critical point.
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.
We investigate the entanglement properties of the Quantum Six-Vertex Model on a cylinder, focusing on the Shannon-Renyi entropy in the limit of Renyi order n=∞n = \inftyn=∞. This entropy, calculated from the ground state amplitudes of the equivalent XXZ spin-1/2 chain, allows us to determine the Renyi entanglement entropy of the corresponding Rokhsar-Kivelson wavefunctions, which describe the ground states of certain conformal quantum critical points. Our analysis reveals a novel logarithmic correction to the expected entanglement scaling when the system size is odd. This anomaly arises from the geometric frustration of spin configurations imposed by periodic boundary conditions on odd-sized chains. We demonstrate that the scaling prefactor of this logarithmic term is directly related to the compactification radius of the low-energy bosonic field theory description, or equivalently, the Luttinger parameter. Thus, this correction provides a direct probe of the underlying Conformal Field Theory (CFT) describing the critical point. Our findings highlight the crucial role of system size parity in determining the entanglement properties of this model and offer insights into the interplay between geometry, frustration, and criticality.
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.