There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS(3). We also compare the entropy computed in AdS(5)XS(5) with that of the free N=4 super Yang-Mills theory.

Holographic Derivation of Entanglement Entropy from the anti de Sitter Space/Conformal Field Theory Correspondence

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Table Of Integrals Series And Products

Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

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Entanglement in many-body systems

We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy SA = −Tr ρAlogρA corresponding to the reduced density matrix ρA 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.

https://iopscience.iop.org/article/10.1088/1742-5468/2004/06/P06002/pdf

Entanglement entropy and quantum field theory

We investigate the time evolution of the subsystem trace distance and Schatten distances after local operator quenches in two-dimensional conformal field theory (CFT) and in one-dimensional quantum spin chains. We focus on the case of a subsystem being an interval embedded in the infinite line. The initial state is prepared by inserting a local operator in the ground state of the theory. We only consider the cases in which the inserted local operator is a primary field or a sum of several primaries. While a nonchiral primary operator can excite both left-moving and right-moving quasiparticles, a holomorphic primary operator only excites a right-moving quasiparticle and an anti-holomorphic primary operator only excites a left-moving one. The reduced density matrix (RDM) of an interval hosting a quasiparticle is orthogonal to the RDM of the interval without any quasiparticles. Moreover, the RDMs of two intervals hosting quasiparticles at different positions are also orthogonal to each other. We calculate numerically the entanglement entropy, Renyi entropy, trace distance, and Schatten distances in time-dependent states excited by different local operators in the critical Ising and XX spin chains. These results match the CFT predictions in the proper limit.

/pdf/subsystem-distance-after-a-local-operator-quench-233ust36dx.pdf

Subsystem distance after a local operator quench

We study the symmetry resolved entanglement entropies in one-dimensional systems with boundaries. We provide some general results for conformal invariant theories and then move to a semi-infinite chain of free fermions. We consider both an interval starting from the boundary and away from it. We derive exact formulas for the charged and symmetry resolved entropies based on theorems and conjectures about the spectra of Toeplitz+Hankel matrices. En route to characterise the interval away from the boundary, we prove a general relation between the eigenvalues of Toeplitz+Hankel matrices and block Toeplitz ones. An important aspect is that the saddle-point approximation from charged to symmetry resolved entropies introduces algebraic corrections to the scaling that are much more severe than in systems without boundaries.

Boundary effects on symmetry resolved entanglement

The ground-state properties of a one-dimentional, attractively interacting Bose gas are studied at a fixed density. The exact solution obtained shows that an interesting phase transition can occur as the interaction strength is reduced to a vanishingly small value and the size of the system grows to the thermodynamic limit.

Local correlations in the attractive one-dimensional Bose gas: From Bethe ansatz to the Gross-Pitaevskii equation

We compute the two-point correlation functions of general quadratic operators
in the high-temperature phase of the three-dimensional O(N) vector model by
using field-theoretical methods. In particular, we study the small- and
large-momentum behavior of the corresponding scaling functions, and give
general interpolation formulae based on a dispersive approach. Moreover, we
determine the crossover exponent $\phi_T$ associated with the traceless
tensorial quadratic field, by computing and analyzing its six-loop perturbative
expansion in fixed dimension. We find: $\phi_T=1.184(12)$,
 $\phi_T=1.271(21)$, and $\phi_T=1.40(4)$ for $N=2,3,5$ respectively.

Critical structure factors of bilinear fields in O(N) vector models.

We consider the partial transpose of the spin reduced density matrix of two disjoint blocks in spin chains admitting a representation in terms of free fermions, such as XY chains. We exploit the solution of the model in terms of Majorana fermions and show that such partial transpose in the spin variables is a linear combination of four Gaussian fermionic operators. This representation allows to explicitly construct and evaluate the integer moments of the partial transpose. We numerically study critical XX and Ising chains and we show that the asymptotic results for large blocks agree with conformal field theory predictions if corrections to the scaling are properly taken into account.

Pasquale Calabrese

Papers

Subsystem distance after a local operator quench

Boundary effects on symmetry resolved entanglement

Local correlations in the attractive one-dimensional Bose gas: From Bethe ansatz to the Gross-Pitaevskii equation

Critical structure factors of bilinear fields in O(N) vector models.

Partial transpose of two disjoint blocks in XY spin chains