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:0801.4111 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Entanglement entropy in d+1 SU(N) gauge theory

Alexander Velytsky

Published Apr 24, 2008• 39 citations
arXivHTMLPDF

Abstract

We consider the entanglement entropy for a subsystem in $d+1$ dimensional $SU(N)$ lattice gauge theory. The $1+1$ gauge theory is treated exactly and shows trivial behavior. Gauge theories in higher dimensions are treated within Migdal-Kadanoff approximation. We consider the gauge theory in the confinement phase. We demonstrate the existence of a nonanalytical change from the short distance to long distance form in the entanglement entropy in such systems ($d>2$) reminiscent of phase transition. The transition is manifested in nontrivial change in the renormalization group flow of character expansion coefficients defining the partition function.

DOI
10.1103/physrevd.77.085021

Related papers

Entanglement entropy in SU(N) gauge theory

Alexander Velytsky

The entanglement entropy of SU(N) lattice gauge theory is studied exactly in 1+1 space-time dimensions and in Migdal-Kadanoff approximation in higher dimensional space. The existence of a non-analytical behavior reminiscent of a phase transition for a characteristic size of the entangled region is demonstrated for higher dimensional theories.

arXiv:0809.4502•International Symposium on Lattice Field Theory (LATTICE)•Jun 30, 2009•6 citations

Entanglement entropy in $SU(2)$ lattice gauge theory

P. V. Buividovich, M.I. Polikarpov

The entropy of entanglement between a three-dimensional slab of thickness l and its complement is studied numerically for four-dimensional SU (2) lattice gauge theory. We find a signature of a nonanalytic behavior of the entanglement entropy, which was predicted recently for large Nc confining gauge theories in the framework of AdS/CFT correspondence. The derivative of the entanglement entropy over l is likely to have a discontinuity at some l = lc ≈ 0.5 fm. The application of the concepts of quantum information theory to quantum field theories in continuous space-times and on the lattices has led recently to many important advances in our understanding of their quantum behavior [1–3]. One such concept is the quantum entanglement of states of systems with many degrees of freedom, which turned out to be a very useful modelindependent characteristic of the structure of the ground state of quantum fields. In particular, for quantum fields on the lattice in the vicinity of a quantum phase transition (i.e. a phase transition which occurs at zero temperature when some parameters of the system, such as the coupling constants, are varied) the ground state is a strongly entangled superposition of the states of all elementary lattice degrees of freedom (such as spins in Heisenberg model, or link variables in lattice gauge theory), and different phases of lattice theories can be characterized by different patterns of entanglement [1–3]. Quantum entanglement thus appears to be an adequate concept for the description of the emergence of collective degrees of freedom in quantum field theories [1–3]. A commonly used measure of quantum entanglement of the ground state of quantum fields in (D − 1) + 1-dimensional space-time is the entropy of entanglement S [A] between some (D − 1)-dimensional region A and its (D − 1)-dimensional complement B, which characterizes the amount of information shared between A and B [1, 2, 4]. Entanglement entropy is defined as the usual von Neumann entropy for the reduced density matrix ρA associated with the region A: S [A] = −Tr A (ρA ln ρA) (1) The reduced density matrix is obtained from the density matrix of the ground state of the theory, ρAB = |0〉〈0| , by tracing over all degrees of freedom which are localized outside of A, i.e. within B [2, 4]: ρA = Tr B ρAB = Tr B |0〉〈0| (2) This density matrix describes the state of quantum fields as seen by an observer who can only perform measurements within A. ∗ e-mail: buividovich@tut.by † e-mail: polykarp@itep.ru

•Jan 1, 2008•0 citations

Entanglement entropy for 2D gauge theories with matters

Sinya Aoki A, Norihiro Iizuka, Kotaro Tamaoka +1 more

We investigate the entanglement entropy in $1+1$-dimensional $SU(N)$ gauge theories with various matter fields using the lattice regularization. Here we use extended Hilbert space definition for entanglement entropy, which contains three contributions; (1) classical Shannon entropy associated with superselection sector distribution, where sectors are labeled by irreducible representations of boundary penetrating fluxes, (2) logarithm of the dimensions of their representations, which is associated with ``color entanglement,'' and (3) EPR Bell pairs, which give ``genuine'' entanglement. We explicitly show that entanglement entropies (1) and (2) above indeed appear for various multiple ``meson'' states in gauge theories with matter fields. Furthermore, we employ transfer matrix formalism for gauge theory with fundamental matter field and analyze its ground state using hopping parameter expansion (HPE), where the hopping parameter $K$ is roughly the inverse square of the mass for the matter. We evaluate the entanglement entropy for the ground state and show that all (1), (2), (3) above appear in the HPE, though the Bell pair part (3) appears in higher order than (1) and (2) do. With these results, we discuss how the ground state entanglement entropy in the continuum limit can be understood from the lattice ground state obtained in the HPE.

arXiv:1705.01549•Physical review. D/Physical review. D.•Aug 30, 2017•10 citations

Decomposition of entanglement entropy in lattice gauge theory

William Donnelly

We consider entanglement entropy between regions of space in lattice gauge theory. The Hilbert space corresponding to a region of space includes ``edge states'' that transform nontrivially under gauge transformations. By decomposing the edge states in irreducible representations of the gauge group, the entropy of an arbitrary state is expressed as the sum of three positive terms: a term associated with the classical Shannon entropy of the distribution of boundary representations, a term that appears only for non-Abelian gauge theories and depends on the dimension of the boundary representations, and a term representing nonlocal correlations. The first two terms are the entropy of the edge states, and depend only on observables measurable at the boundary. These results are applied to several examples of lattice gauge theory states, including the ground state in the strong coupling expansion of Kogut and Susskind. In all these examples we find that the entropy of the edge states is the dominant contribution to the entanglement entropy.

arXiv:1109.0036•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 4, 2012•238 citations

Entanglement entropy of ( 2 + 1 )-dimensional SU(2) lattice gauge theory on plaquette chains

Lukas Ebner, Andreas Schäfer, Clemens Seidl +2 more

We study the entanglement entropy of Hamiltonian SU(2) lattice gauge theory in 2 + 1 dimensions on linear plaquette chains and show that the entanglement entropies of both ground and excited states follow Page curves. The transition of the subsystem size dependence of the entanglement entropy from the area law for the ground state to the volume law for highly excited states is found to be described by a universal crossover function. Quantum many-body scars in the middle of the spectrum, which are present in the electric flux truncated Hilbert space, where the gauge theory can be mapped onto an Ising model, disappear when higher electric field representations are included in the Hilbert space basis. This suggests the continuum ( 2 + 1 )-dimensional SU(2) gauge theory does not have such scarred states. Published by the American Physical Society 2024

arXiv:2401.15184•Physical review. D/Physical review. D.•Jul 9, 2024•21 citations

Topological entanglement entropy in d-dimensions for Abelian higher gauge theories

J.P. Ibieta-Jimenez, M. Petrucci, L.N. Queiroz Xavier +1 more

Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log $$ \left({GSD}_{\tilde{A}}\right) $$ GSD A ˜ , where $$ {GSD}_{\tilde{A}} $$ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity $$ {GSD}_{\tilde{A}} $$ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.

arXiv:1907.01608•Journal of High Energy Physics•Mar 1, 2020•12 citations
We consider the entanglement entropy for a subsystem in d+1d+1d+1 dimensional SU(N)SU(N)SU(N) lattice gauge theory. The 1+11+11+1 gauge theory is treated exactly and shows trivial behavior. Gauge theories in higher dimensions are treated within Migdal-Kadanoff approximation. We consider the gauge theory in the confinement phase. We demonstrate the existence of a nonanalytical change from the short distance to long distance form in the entanglement entropy in such systems (d>2d>2d>2) reminiscent of phase transition. The transition is manifested in nontrivial change in the renormalization group flow of character expansion coefficients defining the partition function.
Entanglement entropy in SU(2)SU(2)SU(2) lattice gauge theory
The entropy of entanglement between a three-dimensional slab of thickness l and its complement is studied numerically for four-dimensional SU (2) lattice gauge theory. We find a signature of a nonanalytic behavior of the entanglement entropy, which was predicted recently for large Nc confining gauge theories in the framework of AdS/CFT correspondence. The derivative of the entanglement entropy over l is likely to have a discontinuity at some l = lc ≈ 0.5 fm. The application of the concepts of quantum information theory to quantum field theories in continuous space-times and on the lattices has led recently to many important advances in our understanding of their quantum behavior [1–3]. One such concept is the quantum entanglement of states of systems with many degrees of freedom, which turned out to be a very useful modelindependent characteristic of the structure of the ground state of quantum fields. In particular, for quantum fields on the lattice in the vicinity of a quantum phase transition (i.e. a phase transition which occurs at zero temperature when some parameters of the system, such as the coupling constants, are varied) the ground state is a strongly entangled superposition of the states of all elementary lattice degrees of freedom (such as spins in Heisenberg model, or link variables in lattice gauge theory), and different phases of lattice theories can be characterized by different patterns of entanglement [1–3]. Quantum entanglement thus appears to be an adequate concept for the description of the emergence of collective degrees of freedom in quantum field theories [1–3]. A commonly used measure of quantum entanglement of the ground state of quantum fields in (D − 1) + 1-dimensional space-time is the entropy of entanglement S [A] between some (D − 1)-dimensional region A and its (D − 1)-dimensional complement B, which characterizes the amount of information shared between A and B [1, 2, 4]. Entanglement entropy is defined as the usual von Neumann entropy for the reduced density matrix ρA associated with the region A: S [A] = −Tr A (ρA ln ρA) (1) The reduced density matrix is obtained from the density matrix of the ground state of the theory, ρAB = |0〉〈0| , by tracing over all degrees of freedom which are localized outside of A, i.e. within B [2, 4]: ρA = Tr B ρAB = Tr B |0〉〈0| (2) This density matrix describes the state of quantum fields as seen by an observer who can only perform measurements within A. ∗ e-mail: buividovich@tut.by † e-mail: polykarp@itep.ru
We investigate the entanglement entropy in 1+11+11+1-dimensional SU(N)SU(N)SU(N) gauge theories with various matter fields using the lattice regularization. Here we use extended Hilbert space definition for entanglement entropy, which contains three contributions; (1) classical Shannon entropy associated with superselection sector distribution, where sectors are labeled by irreducible representations of boundary penetrating fluxes, (2) logarithm of the dimensions of their representations, which is associated with ``color entanglement,'' and (3) EPR Bell pairs, which give ``genuine'' entanglement. We explicitly show that entanglement entropies (1) and (2) above indeed appear for various multiple ``meson'' states in gauge theories with matter fields. Furthermore, we employ transfer matrix formalism for gauge theory with fundamental matter field and analyze its ground state using hopping parameter expansion (HPE), where the hopping parameter KKK is roughly the inverse square of the mass for the matter. We evaluate the entanglement entropy for the ground state and show that all (1), (2), (3) above appear in the HPE, though the Bell pair part (3) appears in higher order than (1) and (2) do. With these results, we discuss how the ground state entanglement entropy in the continuum limit can be understood from the lattice ground state obtained in the HPE.
We consider entanglement entropy between regions of space in lattice gauge theory. The Hilbert space corresponding to a region of space includes ``edge states'' that transform nontrivially under gauge transformations. By decomposing the edge states in irreducible representations of the gauge group, the entropy of an arbitrary state is expressed as the sum of three positive terms: a term associated with the classical Shannon entropy of the distribution of boundary representations, a term that appears only for non-Abelian gauge theories and depends on the dimension of the boundary representations, and a term representing nonlocal correlations. The first two terms are the entropy of the edge states, and depend only on observables measurable at the boundary. These results are applied to several examples of lattice gauge theory states, including the ground state in the strong coupling expansion of Kogut and Susskind. In all these examples we find that the entropy of the edge states is the dominant contribution to the entanglement entropy.
We study the entanglement entropy of Hamiltonian SU(2) lattice gauge theory in 2 + 1 dimensions on linear plaquette chains and show that the entanglement entropies of both ground and excited states follow Page curves. The transition of the subsystem size dependence of the entanglement entropy from the area law for the ground state to the volume law for highly excited states is found to be described by a universal crossover function. Quantum many-body scars in the middle of the spectrum, which are present in the electric flux truncated Hilbert space, where the gauge theory can be mapped onto an Ising model, disappear when higher electric field representations are included in the Hilbert space basis. This suggests the continuum ( 2 + 1 )-dimensional SU(2) gauge theory does not have such scarred states. Published by the American Physical Society 2024
Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log (GSDA~) \left({GSD}_{\tilde{A}}\right) (GSDA~​) GSD A ˜ , where GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.