Abstract
In this paper we study a quench protocol on thermofield double states in the presence of time-reversal symmetry that is inspired by the work of Gao, Jafferis and Wall. The deformation is a product of hermitian operators on the left and right systems that are identical to each other and that lasts for a small amount of time. We study the linear dependence on the quench to the properties of the deformation under time reversal. If the quench is time symmetric, then the linear response after the quench of all T-even operators vanishes. This includes the response of the energy on the left system and all the thermodynamic expectation values (the time averaged expectation values of the operators). Also, we show under an assumption of non-degeneracy of the Hamiltonian that the entanglement entropy between left and right is not affected to this order. We also study a variation of the quench where an instantaneous deformation is given by an operator of fixed T-parity and it's time derivative. It is shown that the sign of the response of the Hamiltonian is correlated with the T-parity of the operator. We can then choose the sign of the amplitude of the quench to result in a reduction in the energy. This implies a reduction of the entanglement entropy between both sides.
Highlights
The eternal black hole has been argued by Maldacena to be given by a thermofield double state [1] on two copies of a conformal field theory
In this paper we study this problem in general quantum systems under the assumption of having a time-reversal symmetry T, where the Hamiltonian is time symmetric
We find that under mild assumptions of nondegeneracy of the spectrum and the thermofield double state there is no change of the expectation value of either the energy or the entanglement entropy, so long as ε is time symmetric
Summary
The eternal black hole has been argued by Maldacena to be given by a thermofield double state [1] on two copies of a conformal field theory. We prove that this is true for any T-even Hermitian operator, as is the case of time averaged operators that would correspond to thermodynamic properties of the system This shows that any such effect to linearized order requires us to break T-invariance, either in the pulse ε, or in the choice of operators in the protocol. We find that there is a nonzero response for the left Hamiltonian to leading order This follows from positivity properties of the amplitudes for states in the thermofield double state. If an initial generalized thermofield double state is time symmetric and diagonal, we show that the response on the left and the right Hamiltonian are the same. If O is Hermitian, the matrix elements of O are antisymmetric in any real basis
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