Quenches on thermofield double states and time reversal symmetry

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.


II. TIME REVERSAL SYMMETRY
A system with time reversal operation has an anti-unitary operation T such that T 2 = 1.
For our arguments, we are assuming a system where any state has an even number of fermions excited to begin with, this way we can assume that there are no degeneracies for the eigenstates of the Hamiltonian 1 .
Notice that if we have an eigenstate of T with eigenvalue 1, T |ψ = |ψ , then it is true that T (i|ψ ) = −iT |ψ = −i|ψ changes the sign of the eigenvalue. We will call the eigenstates of T with eigenvalue 1 real, and the ones with eigenvalue −1 imaginary. Any state can be decomposed into a real and imaginary part as follows The Hilbert space can be spanned by complex combinations of real states. We say that the

THERMOFIELD-DOUBLE STATES
A thermofield double state is a state on a product Hilbert space of two copies of a physical system (left and right, or L, R for short), which is of the form where |n are eigenstates of the energy operator H L acting on the left copy of the system.
We can use a shorthand notation where |T F D ≃ exp(−βH L /2) as a matrix. Here, for T -symmetry, we require that the basis |n is real: this unambiguously determines the phase of the different amplitudes of the thermofield double state. In [5] the authors use the antiunitary CP T operator to define the thermofield double state and study how to engineer the thermofield double state in simple systems. For us, the presence of additional symmetries like C, P lead to additional degeneracies that we want to avoid, so we will only study systems with T -symmetry under the assumption that there are no degeneracies.
The total Hamiltonian for the system is H tot = H L + H R , where H R has the same matrix elements of H L , but on the right copy. If we evolve the system in time, we get that the amplitudes of the state in the energy basis develop phases but the reduced density matrix ρ L = exp(−βE n )|n n| stays unchanged. The condition of T invariance requires that the phases are aligned, but moreover, we want the coefficients to be positive. This will be important later on. A thermofield double state is also annihilated by H L − H R .
We will say a generalized thermofield double state is a state that is of a similar form, where it is a sum of products of (repeated) real states with positive real coefficients. That is, it is of the schematic form exp(−βH ef f /2) for some effective Hamiltonian that is time symmetric. This is the modular Hamiltonian for the density matrix ρ L . We can also require that it is annihilated by H L − H R . For example a thermofield double for a microcanonical ensemble in some window would be of this form. We will not make a distinction between generalized and standard thermofield double states in what follows.
Notice the following fact. If |T F D is a thermofield double state and φ is a T-even operator on a single copy of the system, then the matrix elements of φ are real. In particular, one checks that For the purposes of this paper we are interested on the sign of the response, so we do not have to normalize the states, otherwise, we should use an additional factor of the inverse of the partition function on the right. Similarly, if φ L = φ R are T-odd, they are purely imaginary and we find that so the correlation has the opposite sign. The T-odd observables are anticorrelated.
A double sided quenchá la Gao-Jafferis-Wall will be a unitary operator of the form acting on the thermofield double state. Here We will now study the linearized response in ǫ of the (generalized) thermofield double state to the quench. For this, consider a T -even operator W L and consider the response to the quench from Kubo's formula.
It is easy to argue that for a time symmetric ǫ, the operator as argued in the previous section, the basis that diagonalizes the thermofield Hamiltonian H ef f state is real and the thermofield double state itself is real. As such, the matrix elements of both ǫ(t)φ L (t)φ R (t) and W are real in this basis. It immediately follows that δ ǫ W L = 0, as the right hand side has no imaginary part. The condition of shortness of the pulse is that W should be measured after the quench, but at a small enough time that it can be replaced by W (0). This depends on W itself.
For example, this result pertains to the Hamiltonian H L and any time averaged (equilibrium) expectation value. Time averaging removes the off-diagonal elements of an operator (so long as the Hamiltonian is non-degenerate) and gives rise to a T-even operator that is time independent. Because of time translation invariance of these quantities, they can be measured after the quench and they will be T -even at any time, not only at t = 0. Indeed, one can say based on this result that the double sided quench does not change the thermodynamic properties of the final state, to linearized order, with respect to the initial state after tracing over the right system degrees of freedom. Now, for any T -odd W observable, we have that W T F D = 0 by a very slight variant of the same argument as above: the expectation value is real as W is hermitian, but the right hand side is purely imaginary. In general, for these T-odd W , to linearized order the quench changes the expectation value. That is because the commutator in Kubo's formula is now purely imaginary and there is no constraint on its expectation value.
We will now argue that there is also no change in the entanglement entropy between both sides either, to linearized order, for this type of quench. To do this, we need to consider the thermofield double as a matrix M. The quench will change M as follows which is appropriate for global coordinates of the double sided black holes that results in an expression of the form O L (t)O R (−t) rather than as above.
where φ T is the transpose matrix of φ. The density matrix for the left system is given by To linearized order in ǫ we get that The idea is to show that the right hand side is purely imaginary. This is straightforward to prove for an instantaneous quench: we use the reality properties of M, φ to do so. In particular in a real basis we have that M † = M is real, and φ is either purely real or purely imaginary. Thus the right hand side is purely imaginary. As such δρ is off-diagonal, after all, the initial ρ is characterized by a hermitian operator, and it is real in a real basis. It follows that the eigenvalues of ρ do not change to linearized order, by a standard perturbation theory calculation. This is where we require that the spectrum is non-degenerate. If the eigenvalues of ρ only change to quadratic order in ǫ, then the entanglement entropy or any of the corresponding Renyi entropies are fixed at this order.
We now need to show that this imaginary property is true in the general case. We concentrate our attention on a fixed t and its time reversed mirror time −t. Look at the first term on the right, summed over these two times The result is clearly invariant under t → −t. Now, decompose φ L (t) = φ e (t) + φ o (t) under even and odd parts with respect to t. Both of these are hermitian. We do a similar decomposition for φ R . It is clear that φ E will have the same time reversal properties as φ(0), so it inherits its reality properties (either purely real or purely imaginary) while the other piece will have the opposite time reversal property. Because of the time symmetry of the full expression, only the even-even and odd-odd pieces contribute in the sum. They also have definite reality properties: either purely real or purely imaginary. For each of these, we get a net reality property of the expression that is the same as in the instantaneous quench problem. We see that δρ is a sum (integral) of purely imaginary terms. As such, it is imaginary and follows that it must be off-diagonal. By our argument for the instantaneous quench this implies that the entanglement entropy is not modified to linearized order.

A. A double sided quench that changes the energy
Now, we will concentrate on a slight variation of the double sided quench that leads to a change in the energy. The idea is that φ L and φ R should now have the opposite time reversal symmetry assignments, so that we can get a non-zero result. This is in lieu of considering cases where ǫ(t) = −ǫ(−t), which would produce a time asymmetry in the quench protocol.
We don't want the two sides to be unrelated. Instead, we will require that φ R =φ L and we will focus on an instantaneous quench for simplicity. More general results follow if we use the same type of arguments that came after equation (11). We will also focus on thermofield We get this way that If φ is T-even, then it is real, andφ is purely imaginary. The right hand side would therefore be positive. This follows from the arguments that lead to equation (5). If we choose the opposite assignment of T-parity, we change the sign. It is then clear that we can choose ǫ to lower the left energy depending on the T-parity of φ. Notice that so far we have not used the fact that the state is diagonal in the energy basis. This is used when we consider the response of the Hamiltonian on the right. We get that The matrix elements ofφ R (0) are given byφ nm = −(E n −E 2 m )φ nm . Also, the matrix elements of the time derivative of the field areφ nm = i(E n − E m )φ nm . We see that when we use the diagonal entries of the matrix M ≡ diag(s n ), with s n ≥ 0, we get that Notice that the result is symmetric with respect to both sides, only because the state is diagonal in the energy basis.
As an aside, if we do a quench with the symmetric operator φ LφR we double the response, by the argument above. The instantaneous quench we described is computing half the result for ǫ(t) = δ ′ (t), which is T-odd. That is, our result implies that the quench with very short time ǫ(t) = −ǫ(−t) has a definite sign for the response in the energy, and that this response changes sign if we change the sign of ǫ. The sign depends on if the operators φ are T-even or T-odd. The result has opposite signs if we change the T-parity of φ.
As a corollary, if we use the canonical thermofield double, it is a state that maximizes the entanglement entropy given the expectation value of the energy. Since we can choose the sign of δH to be negative by choosing the sign of ǫ, we reduce the energy to linearized order. This implies that the entanglement entropy between both sides is also reduced to linearized order, by an amount that is at least as large as the difference in entropy between the two corresponding thermofield double states at their different fixed energies.
That is, we get that for δH L < 0, the linearized response in the entropy is bounded below below by the thermodynamic relation dE = T dS, to be given by where β is the temperature. On the other hand, if we choose δH positive, we get the opposite inequality, δS ≤ βδH L . That is, we get that the change in entropy and the energy to first order are correlated exactly as in the thermodynamic relation. That is, we get that for the canonical thermofield double states δS = βδH L . this type of quench would also produce regenesis [6].
For some of the results, we depend on having a non-degenerate Hamiltonian and generalized thermofield double state. It would be interesting to generalize the calculations in this paper to more general cases involving degeneracies associated to additional symmetries of the system. In particular, if we have a charge that is odd under T-parity (as is common in angular momentum or electric charge), one can consider the problem of thermofield double states for systems with chemical potential. If the system is holographic, these setups would correspond to rotating black holes or charged black holes and one can expect that the response of the system can be classified further in terms of the charge assignments of the perturbations. The traversability of the corresponding black holes under double trace perturbations has been analyzed in examples in [7]. In general, we did not use the 'trace' counting that is appropriate to large N theories in our results. It is not clear to the author how this property of the perturbation affects the discussion of the quench.
In view of the vanishing results to linearized order in time symmetric quenches, it is also interesting to generalize the results in this paper to second order in the perturbation and check if it is possible to make statements of positivity of the response of various observables to second order.