Abstract We study the parallel transport of modular Hamiltonians encoding entanglement properties of a state. In the case of 2d CFT, we consider a change of state through action with a suitable diffeomorphism on the circle: one that diagonalizes the adjoint action of the modular Hamiltonian. These vector fields exhibit kinks at the interval boundary, thus together with their central extension they differ from usual elements of the Virasoro algebra. The Berry curvature associated to state-changing parallel transport is the Kirillov-Kostant symplectic form on an associated coadjoint orbit, one which differs appreciably from known Virasoro orbits. We find that the boundary parallel transport process computes a bulk symplectic form for a Euclidean geometry obtained from the backreaction of a cosmic brane, with Dirichlet boundary conditions at the location of the brane. We propose that this gives a reasonable definition for the symplectic form on an entanglement wedge.Abstract We study the parallel transport of modular Hamiltonians encoding entanglement properties of a state. In the case of 2d CFT, we consider a change of state through action with a suitable diffeomorphism on the circle: one that diagonalizes the adjoint action of the modular Hamiltonian. These vector fields exhibit kinks at the interval boundary, thus together with their central extension they differ from usual elements of the Virasoro algebra. The Berry curvature associated to state-changing parallel transport is the Kirillov-Kostant symplectic form on an associated coadjoint orbit, one which differs appreciably from known Virasoro orbits. We find that the boundary parallel transport process computes a bulk symplectic form for a Euclidean geometry obtained from the backreaction of a cosmic brane, with Dirichlet boundary conditions at the location of the brane. We propose that this gives a reasonable definition for the symplectic form on an entanglement wedge.

arXiv:2111.05345Journal of High Energy PhysicsMar 1, 202214 citations

Projective Approach to Dynamical Transport

Jochen Gemmer B, Mathias Michel A, G. Mahler

Lecture notes in physicsJan 1, 20090 citations

Virasoro algebra from modular Hamiltonians

Suchetan Das, Bobby Ezhuthachan, Somnath Porey +1 more

We construct an infinite class of eigen modes with integer eigenvalues for the Vacuum Modular Hamiltonian of a single interval in 2d CFT. An interesting feature of this construction is that these eigen modes together with the Modular Hamiltonian form a Virasoro algebra. The construction is geometric in the sense that it is intimately related to the partition of the interval ( $N$ ) into subintervals ( $N'$ ) and ( $N''$ ). In particular, we show that given a set consisting of a causal diamond associated to an interval ( $N$ ) and its sub-intervals ( $N'$ , $N''$ ) on a constant time slice in a 2d CFT, one can construct modular eigen modes, satisfying the Virasoro algebra, which we call the 'modular Virasoro algebra'. For ( $n=\pm 1, 0$ ), the corresponding generators are related to the modular Hamiltonians of the interval and its sub intervals. Another interesting feature of our construction is that OPE blocks associated with the interval $N$ transform as the modes of the primary highest weight field under transformations generated by these 'modular Virasoro generators'. We end with some comments on the bulk dual of the modular Virasoro algebra.We construct an infinite class of eigen modes with integer eigenvalues for the Vacuum Modular Hamiltonian of a single interval in 2d CFT. An interesting feature of this construction is that these eigen modes together with the Modular Hamiltonian form a Virasoro algebra. The construction is geometric in the sense that it is intimately related to the partition of the interval ($N$) into subintervals ($N'$) and ($N''$). In particular, we show that given a set consisting of a causal diamond associated to an interval ($N$) and its sub-intervals ($N'$, $N''$) on a constant time slice in a 2d CFT, one can construct modular eigen modes, satisfying the Virasoro algebra, which we call the 'modular Virasoro algebra'. For ($n=\pm 1, 0$), the corresponding generators are related to the modular Hamiltonians of the interval and its sub intervals. Another interesting feature of our construction is that OPE blocks associated with the interval $N$ transform as the modes of the primary highest weight field under transformations generated by these 'modular Virasoro generators'. We end with some comments on the bulk dual of the modular Virasoro algebra.

Jan 25, 20210 citations

A nonlinear quantum transport theory

Leonardo Lauck, Aurea Rosas Vasconcellos, Roberto Luzzi

Physica A Statistical Mechanics and its ApplicationsOct 1, 1990133 citations

Lattice Virasoro algebra and corner transfer matrices in the Baxter eight-vertex model

H. Itoyama, H. B. Thacker

A lattice Virasoro algebra is constructed for the Baxter eight-vertex model. The operator ${\mathrm{L}}_{0}$ is obtained from the logarithm of the corner transfer matrix and is given by the first moment of the XYZ spin-chain Hamiltonian. The algebra is valid even when the Hamiltonian includes a mass term, in which case it represents lattice coordinate transformations which distinguish between even and odd sublattices. We apply the quantum inverse scattering method to demonstrate that the Virasoro algebra follows from the Yang-Baxter relations.A lattice Virasoro algebra is constructed for the Baxter eight-vertex model. The operator ${\mathrm{L}}_{0}$ is obtained from the logarithm of the corner transfer matrix and is given by the first moment of the XYZ spin-chain Hamiltonian. The algebra is valid even when the Hamiltonian includes a mass term, in which case it represents lattice coordinate transformations which distinguish between even and odd sublattices. We apply the quantum inverse scattering method to demonstrate that the Virasoro algebra follows from the Yang-Baxter relations.

Physical Review LettersApr 6, 198772 citations

Quantum transport theory

Antti‐Pekka Jauho

Springer eBooksJan 1, 199849 citations

N

) into subintervals (

N'

) and (

N''

). In particular, we show that given a set consisting of a causal diamond associated to an interval (

N

) and its sub-intervals (

N'

N''

) on a constant time slice in a 2d CFT, one can construct modular eigen modes, satisfying the Virasoro algebra, which we call the 'modular Virasoro algebra'. For (

n=\pm 1, 0

), the corresponding generators are related to the modular Hamiltonians of the interval and its sub intervals. Another interesting feature of our construction is that OPE blocks associated with the interval

N

transform as the modes of the primary highest weight field under transformations generated by these 'modular Virasoro generators'. We end with some comments on the bulk dual of the modular Virasoro algebra.

A lattice Virasoro algebra is constructed for the Baxter eight-vertex model. The operator

{\mathrm{L}}_{0}

is obtained from the logarithm of the corner transfer matrix and is given by the first moment of the XYZ spin-chain Hamiltonian. The algebra is valid even when the Hamiltonian includes a mass term, in which case it represents lattice coordinate transformations which distinguish between even and odd sublattices. We apply the quantum inverse scattering method to demonstrate that the Virasoro algebra follows from the Yang-Baxter relations.