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Register a new userEntanglement hamiltonians in two-dimensional conformal field theory
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We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangling surface and the appearance of boundary entropies as universal O(1) terms in the entanglement entropy.
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We reconsider the moments of the reduced density matrix of two disjoint intervals and of its partial transpose with respect to one interval for critical free fermionic lattice models. It is known that these matrices are sums of either two or four Gaussian matrices and hence their moments can be reconstructed as computable sums of products of Gaussian operators. We find that, in the scaling limit, each term in these sums is in one-to-one correspondence with the partition function of the corresponding conformal field theory on the underlying Riemann surface with a given spin structure. The analytical findings have been checked against numerical results for the Ising chain and for the XX spin chain at the critical point.
The lectures provide a pedagogical introduction to the methods of CFT as applied to two-dimensional critical behaviour.
We review the imaginary time path integral approach to the quench dynamics of conformal field theories. We show how this technique can be applied to the determination of the time dependence of correlation functions and entanglement entropy for both global and local quenches. We also briefly review other quench protocols. We carefully discuss the limits of applicability of these results to realistic models of condensed matter and cold atoms.
We consider a quantum quench in a finite system of length $L$ described by a 1+1-dimensional CFT, of central charge $c$, from a state with finite energy density corresponding to an inverse temperature $betall L$. For times $t$ such that $ell/2<t<(L-ell)/2$ the reduced density matrix of a subsystem of length $ell$ is exponentially close to a thermal density matrix. We compute exactly the overlap $cal F$ of the state at time $t$ with the initial state and show that in general it is exponentially suppressed at large $L/beta$. However, for minimal models with $c<1$ (more generally, rational CFTs), at times which are integer multiples of $L/2$ (for periodic boundary conditions, $L$ for open boundary conditions) there are (in general, partial) revivals at which $cal F$ is $O(1)$, leading to an eventual complete revival with ${cal F}=1$. There is also interesting structure at all rational values of $t/L$, related to properties of the CFT under modular transformations. At early times $t!ll!(Lbeta)^{1/2}$ there is a universal decay ${cal F}simexpbig(!-!(pi c/3)Lt^2/beta(beta^2+4t^2)big)$. The effect of an irrelevant non-integrable perturbation of the CFT is to progressively broaden each revival at $t=nL/2$ by an amount $O(n^{1/2})$.
We study the logarithmic negativity and the moments of the partial transpose in the ground state of a two dimensional massless harmonic square lattice with nearest neighbour interactions for various configurations of adjacent domains. At leading order for large domains, the logarithmic negativity and the logarithm of the ratio between the generic moment of the partial transpose and the moment of the reduced density matrix at the same order satisfy an area law in terms of the length of the curve shared by the adjacent regions. We give numerical evidences that the coefficient of the area law term in these quantities is related to the coefficient of the area law term in the Renyi entropies. Whenever the curve shared by the adjacent domains contains vertices, a subleading logarithmic term occurs in these quantities and the numerical values of the corner function for some pairs of angles are obtained. In the special case of vertices corresponding to explementary angles, we provide numerical evidence that the corner function of the logarithmic negativity is given by the corner function of the Renyi entropy of order 1/2.
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