# Time reversal symmetry of transverse field Ising model

Basically, the answer is yes: $H$ is TRI because it is real. Reality condition really means that the Hamiltonian obeys a certain anti-unitary symmetry. In this case, the time-reversal operation is simply $T=K$ where $K$ is the complex conjugation. It is not the usual one($T=K\prod_i i\sigma^y_i$), and in particular $T^2=1$, so there is no Kramers' theorem and the spectrum is not doubly degenerate. The fact that level statistics follows GOE of course is a consequence of the reality condition. In fact, I think if there was a $T^2=-1$ time-reversal symmetry, the statistics would follow a different ensemble (the symplectic one, I believe).

You asked what if one changes the transverse field $g\sum_i \sigma^x_i$ to the $y$ direction. In that case, the two Hamiltonians are unitarily related (i.e. a $\pi/2$ spin rotation $U_z$ around $z$ would bring it back). Let me call the Hamiltonian with $x$ transverse field $H_x(g)$ where $g$ is the transverse field, and with $y$ transverse field $H_y(g)$. Define $U_z=e^{i\pi \sum_i\sigma^z_i/4}$, then it is easy to check that $U_z H_y(g) U_z^{-1}= H_x(g)$. Since we know $H_x^*(g)=H_x(g)$, we can easily find $H_y^*=U_z^2 H_y U_z^{-2}$. Therefore one just has to redefine the time-reversal symmetry to be $T=K U_z^2$. If you really want to break the reality condition, in a way that can not be fixed by additional unitary transformations, then one needs to turn on transverse fields along all three dimensions.

Last comment on your "Arguments for yes": the first argument you gave, namely one also flips the external parameters, does not work. In this way, there would be no time-reversal symmetry breaking, except the CP violation in the fundamental processes! When we talk about the symmetry of a Hamiltonian, we should just treat the system on its own, not with all the external devices that generate the various terms -- unless you want to consider the dynamics of these devices, but then it is a different Hamiltonian.