We consider the entanglement entropy for the 2D Ising model at the conformal fixed point in the presence of interfaces. More precisely, we investigate the situation where the two subsystems are separated by a defect line that preserves conformal invariance. Using the replica trick, we compute the entanglement entropy between the two subsystems. We observe that the entropy, just like in the case without defects, shows a logarithmic scaling behavior with respect to the size of the system. Here, the prefactor of the logarithm depends on the strength of the defect encoded in the transmission coefficient. We also comment on the supersymmetric case.We consider the entanglement entropy for the 2D Ising model at the conformal fixed point in the presence of interfaces. More precisely, we investigate the situation where the two subsystems are separated by a defect line that preserves conformal invariance. Using the replica trick, we compute the entanglement entropy between the two subsystems. We observe that the entropy, just like in the case without defects, shows a logarithmic scaling behavior with respect to the size of the system. Here, the prefactor of the logarithm depends on the strength of the defect encoded in the transmission coefficient. We also comment on the supersymmetric case.

arXiv:1505.02647May 11, 20150 citations

Entanglement entropy and negativity in the Ising model with defects

David Rogerson, Frank Pollmann, Ananda Roy

Abstract Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we analyze entanglement properties of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. In particular, we compute the entanglement entropy (EE) and the entanglement negativity (EN) in the presence of defects. For the EE, we consider the cases when the defect lies within the subsystem and at the edge of the subsystem. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial ‘finite-size correction’ in the EE obtained recently using free fermion computations [1]. These corrections arise when the subsystem size is appreciable compared to the total system size and lead to a deviation from the usual logarithmic scaling characteristic of one-dimensional quantum-critical systems. Using matrix product states with open and infinite boundary conditions, we numerically demonstrate the disappearance of the zero mode contribution for finite subsystem sizes in the thermodynamic limit. Our results provide further support to the recent free fermion computations, but clearly contradict earlier analytical field theory calculations based on twisted torus partition functions. Subsequently, we compute the logarithm of the EN (log-EN) between two disjoint subsystems separated by a defect. We show that the log-EN scales logarithmically with the separation of the subsystems. However, the coefficient of this logarithmic scaling yields a continuously-varying effective central charge that is different from that obtained from analogous computations of the EE. The defects leave their fingerprints in the subleading term of the scaling of the log-EN. Furthermore, the log-EN receives similar ‘finite size corrections’ like the EE which leads to deviations from its characteristic logarithmic scaling.Abstract Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we analyze entanglement properties of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. In particular, we compute the entanglement entropy (EE) and the entanglement negativity (EN) in the presence of defects. For the EE, we consider the cases when the defect lies within the subsystem and at the edge of the subsystem. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial ‘finite-size correction’ in the EE obtained recently using free fermion computations [1]. These corrections arise when the subsystem size is appreciable compared to the total system size and lead to a deviation from the usual logarithmic scaling characteristic of one-dimensional quantum-critical systems. Using matrix product states with open and infinite boundary conditions, we numerically demonstrate the disappearance of the zero mode contribution for finite subsystem sizes in the thermodynamic limit. Our results provide further support to the recent free fermion computations, but clearly contradict earlier analytical field theory calculations based on twisted torus partition functions. Subsequently, we compute the logarithm of the EN (log-EN) between two disjoint subsystems separated by a defect. We show that the log-EN scales logarithmically with the separation of the subsystems. However, the coefficient of this logarithmic scaling yields a continuously-varying effective central charge that is different from that obtained from analogous computations of the EE. The defects leave their fingerprints in the subleading term of the scaling of the log-EN. Furthermore, the log-EN receives similar ‘finite size corrections’ like the EE which leads to deviations from its characteristic logarithmic scaling.

arXiv:2204.03601Journal of High Energy PhysicsJun 29, 202233 citations

Entanglement Entropy in the Ising Model with Topological Defects

Ananda Roy, Hubert Saleur

Entanglement entropy (EE) contains signatures of many universal properties of conformal field theories (CFTs), especially in the presence of boundaries or defects. In particular, topological defects are interesting since they reflect internal symmetries of the CFT and have been extensively analyzed with field-theoretic techniques with striking predictions. So far, however, no lattice computation of EE has been available. Here, we present an ab initio analysis of EE for the Ising model in the presence of a topological defect. While the behavior of the EE depends, as expected, on the geometric arrangement of the subsystem with respect to the defect, we find that zero-energy modes give rise to crucial finite-size corrections. Importantly, contrary to the field-theory predictions, the universal subleading term in the EE when the defect lies at the edge of the subsystem arises entirely due to these zero-energy modes and is not directly related to the modular S matrix of the Ising CFT.Entanglement entropy (EE) contains signatures of many universal properties of conformal field theories (CFTs), especially in the presence of boundaries or defects. In particular, topological defects are interesting since they reflect internal symmetries of the CFT and have been extensively analyzed with field-theoretic techniques with striking predictions. So far, however, no lattice computation of EE has been available. Here, we present an ab initio analysis of EE for the Ising model in the presence of a topological defect. While the behavior of the EE depends, as expected, on the geometric arrangement of the subsystem with respect to the defect, we find that zero-energy modes give rise to crucial finite-size corrections. Importantly, contrary to the field-theory predictions, the universal subleading term in the EE when the defect lies at the edge of the subsystem arises entirely due to these zero-energy modes and is not directly related to the modular S matrix of the Ising CFT.

arXiv:2111.04534Physical Review LettersMar 2, 202251 citations

Entanglement entropy with localized and extended interface defects

Ferenc Iglói, Zsolt Szatmári, Yu‐Cheng Lin

The quantum Ising chain of length, $L$ , which is separated into two parts by localized or extended defects is considered at the critical point where scaling of the interface magnetization is nonuniversal. We measure the entanglement entropy between the two halves of the system in equilibrium, as well as after a quench, when the interaction at the interface is changed for time $t>0$ . For the localized defect the increase in the entropy with $\text{log}\text{ }L$ or with $\text{log}\text{ }t$ involves the same effective central charge, which is a continuous function of the strength of the defect. On the contrary for the extended defect the equilibrium entropy is saturated but the nonequilibrium entropy has a logarithmic time dependence the prefactor of which depends on the strength of the defect.The quantum Ising chain of length, $L$, which is separated into two parts by localized or extended defects is considered at the critical point where scaling of the interface magnetization is nonuniversal. We measure the entanglement entropy between the two halves of the system in equilibrium, as well as after a quench, when the interaction at the interface is changed for time $t>0$. For the localized defect the increase in the entropy with $\text{log}\text{ }L$ or with $\text{log}\text{ }t$ involves the same effective central charge, which is a continuous function of the strength of the defect. On the contrary for the extended defect the equilibrium entropy is saturated but the nonequilibrium entropy has a logarithmic time dependence the prefactor of which depends on the strength of the defect.

arXiv:0903.3740Physical Review BJul 6, 200975 citations

Entanglement Requires Fluctuations at Conformal Interfaces

Yuya Kusuki, Kosei Suzuyama, Yin Tang

We conjecture that the universal logarithmic coefficient for capacity of entanglement $c_{\mathrm{cap}}$ is bounded below by the corresponding entropy coefficient $c_{\mathrm{eff}}$ at conformal interfaces. We prove the bound in free boson and fermion CFTs and in holographic models with interface branes, and establish it using defect perturbation theory. Numerical calculations at interacting interfaces between the tricritical Ising and Ising CFTs further support the bound. Despite this constraint, capacity provides independent interface data. We construct interfaces with identical entropy and energy transmission coefficients but different capacity coefficients, and show that capacity can even increase as entropy decreases.We conjecture that the universal logarithmic coefficient for capacity of entanglement $c_{\mathrm{cap}}$ is bounded below by the corresponding entropy coefficient $c_{\mathrm{eff}}$ at conformal interfaces. We prove the bound in free boson and fermion CFTs and in holographic models with interface branes, and establish it using defect perturbation theory. Numerical calculations at interacting interfaces between the tricritical Ising and Ising CFTs further support the bound. Despite this constraint, capacity provides independent interface data. We construct interfaces with identical entropy and energy transmission coefficients but different capacity coefficients, and show that capacity can even increase as entropy decreases.

hep-thCross-listedarXiv:2610.02309Oct 5, 20260 citations

Entanglement and topological interfaces

E. Brehm, Ilka Brunner A, D. Jaud +1 more

In this paper we consider entanglement entropies in two-dimensional conformal field theories in the presence of topological interfaces. Tracing over one side of the interface, the leading term of the entropy remains unchanged. The interface however adds a subleading contribution, which can be interpreted as a relative (Kullback-Leibler) entropy with respect to the situation with no defect inserted. Reinterpreting boundaries as topological interfaces of a chiral half of the full theory, we rederive the left/right entanglement entropy in analogy with the interface case. We discuss WZW models and toroidal bosonic theories as examples.In this paper we consider entanglement entropies in two-dimensional conformal field theories in the presence of topological interfaces. Tracing over one side of the interface, the leading term of the entropy remains unchanged. The interface however adds a subleading contribution, which can be interpreted as a relative (Kullback-Leibler) entropy with respect to the situation with no defect inserted. Reinterpreting boundaries as topological interfaces of a chiral half of the full theory, we rederive the left/right entanglement entropy in analogy with the interface case. We discuss WZW models and toroidal bosonic theories as examples.

arXiv:1512.05945Fortschritte der PhysikApr 13, 201631 citations

The quantum Ising chain of length,

L

, which is separated into two parts by localized or extended defects is considered at the critical point where scaling of the interface magnetization is nonuniversal. We measure the entanglement entropy between the two halves of the system in equilibrium, as well as after a quench, when the interaction at the interface is changed for time

t>0

. For the localized defect the increase in the entropy with

\text{log}\text{ }L

or with

\text{log}\text{ }t

involves the same effective central charge, which is a continuous function of the strength of the defect. On the contrary for the extended defect the equilibrium entropy is saturated but the nonequilibrium entropy has a logarithmic time dependence the prefactor of which depends on the strength of the defect.

We conjecture that the universal logarithmic coefficient for capacity of entanglement

c_{\mathrm{cap}}

is bounded below by the corresponding entropy coefficient

c_{\mathrm{eff}}

at conformal interfaces. We prove the bound in free boson and fermion CFTs and in holographic models with interface branes, and establish it using defect perturbation theory. Numerical calculations at interacting interfaces between the tricritical Ising and Ising CFTs further support the bound. Despite this constraint, capacity provides independent interface data. We construct interfaces with identical entropy and energy transmission coefficients but different capacity coefficients, and show that capacity can even increase as entropy decreases.