We introduce a subsystem generalization of the spectral form factor via pseudoentropy, the von-Neumann entropy for the reduced transition matrix. We consider a transition matrix between the thermofield double state and its time-evolved state in two-dimensional conformal field theories, and study the time dependence of the pseudoentropy for a single interval. We show that the real part of the pseudoentropy behaves similarly to the spectral form factor; it starts from the thermal entropy, initially drops to the minimum, then it starts increasing, and then finally it approaches the vacuum entanglement entropy. We also study the theory dependence of its behavior by considering theories on a compact space.
We introduce a subsystem generalization of the spectral form factor via pseudoentropy, the von-Neumann entropy for the reduced transition matrix. We consider a transition matrix between the thermofield double state and its time-evolved state in two-dimensional conformal field theories, and study the time dependence of the pseudoentropy for a single interval. We show that the real part of the pseudoentropy behaves similarly to the spectral form factor; it starts from the thermal entropy, initially drops to the minimum, then it starts increasing, and then finally it approaches the vacuum entanglement entropy. We also study the theory dependence of its behavior by considering theories on a compact space.
A bstract In this work, we develop a generalisation of the thermal entropy to complex inverse temperatures, which we call the thermal pseudo-entropy. We show that this quantity represents the pseudo-entropy of the transition matrix between Thermofield Double states at different times. We have studied its properties in various quantum mechanical setups, Schwarzian theory, Random Matrix Theories, and 2D CFTs, including symmetric orbifolds. Our findings indicate a close relationship between the averaged thermal pseudo-entropy and the spectral form factor, which is instrumental in distinguishing chaotic and integrable models. Moreover, we have observed a logarithmic scaling of this quantity in models with a continuous spectrum, with a universal coefficient that is sensitive to the scaling of the density of states near the edge of the spectrum. Lastly, we found the connection between the real and imaginary parts of the thermal pseudo-entropy through the Kramers-Kronig relations.
We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A logρ A corresponding to the reduced density matrix ρ A of a subsystem A . For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c , we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A . These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.
In this paper, we explore properties of pseudoentropy in quantum field theories and spin systems from several approaches. pseudoentropy is a generalization of entanglement entropy such that it depends on both an initial and final state and has a clear gravity dual via the AdS/CFT. We numerically analyze a class of free scalar field theories and the XY spin model. This reveals basic properties of pseudoentropy in quantum many-body systems, namely, the area-law behavior, the saturation behavior, and the nonpositivity of difference between the pseudoentropy and averaged entanglement entropy in the same quantum phase. In addition, our numerical analysis finds an example where the strong subadditivity of pseudoentropy gets violated. Interestingly, we find that the nonpositivity of the difference can be violated only if the initial and final state belong to different quantum phases. We also present analytical arguments, which support these properties by both conformal field theoretic and holographic calculations. When the initial and final state belong to different topological phases, we expect a gapless mode localized along an interface, which enhances the pseudoentropy, leading to the violation of the nonpositivity of the difference. Moreover, we also compute the time evolution of pseudoentropy after a global quantum quench, where we observe that the imaginary part of pseudoentropy shows an interesting characteristic behavior.
In this paper, we explore properties of pseudoentropy in quantum field theories and spin systems from several approaches. pseudoentropy is a generalization of entanglement entropy such that it depends on both an initial and final state and has a clear gravity dual via the AdS/CFT. We numerically analyze a class of free scalar field theories and the XY spin model. This reveals basic properties of pseudoentropy in quantum many-body systems, namely, the area-law behavior, the saturation behavior, and the nonpositivity of difference between the pseudoentropy and averaged entanglement entropy in the same quantum phase. In addition, our numerical analysis finds an example where the strong subadditivity of pseudoentropy gets violated. Interestingly, we find that the nonpositivity of the difference can be violated only if the initial and final state belong to different quantum phases. We also present analytical arguments, which support these properties by both conformal field theoretic and holographic calculations. When the initial and final state belong to different topological phases, we expect a gapless mode localized along an interface, which enhances the pseudoentropy, leading to the violation of the nonpositivity of the difference. Moreover, we also compute the time evolution of pseudoentropy after a global quantum quench, where we observe that the imaginary part of pseudoentropy shows an interesting characteristic behavior.
The generators of radial conformal symmetries in Minkowski space-time can be put in correspondence with generators of time evolution in conformal quantum mechanics. Within this correspondence we show that in conformal quantum mechanics the state corresponding to the inertial vacuum for a conformally invariant field in Minkowski spacetime has the structure of a thermofield double. The latter is built from a bipartite ”vacuum state” corresponding to the ground state of the generators of hyperbolic time evolution. These can evolve states only within a portion of the time domain. When such generators correspond to conformal Killing vectors mapping a causal diamond in itself and generators of dilations, the temperature of the thermofield double reproduces, respectively, the diamond temperature and the Milne temperature found for massless fields in Minkowski spacetime. Moreover we calculate the entanglement entropy associated to the thermofield double states and obtain a UV divergent logarithmic behaviour analogous to known results in two-dimensional conformal field theory in which the entangling boundary is point-like.
We study the late-time behaviors of pseudo-(Rényi) entropy of locally excited states in rational conformal field theories. To construct the transition matrix, we utilize two nonorthogonal locally excited states that are created by the application of different descendant operators to vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudoentropy and pseudo-Rényi entropy corresponds to the logarithm of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(Rényi) entropy compared to that for (Rényi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(Rényi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(Rényi) entropy of an effective transition matrix in a finite-dimensional Hilbert space.