Work statistics and generalized Loschmidt echo for the Hatano-Nelson model

We focus on the biorthogonal work statistics of the interacting many-body Hatano-Nelson model after switching on the imaginary vector potential. We introduce a generalized Loschmidt echo $G(t)$ utilizing the biorthogonal metric operator. It is well suited for numerical analysis and its Fourier transform yields the probability distribution of work done. The statistics of work displays several universal features, including an exponential decay with the square of both the system size and imaginary vector potential for the probability to stay in the ground state. Additionally, its high energy tail follows a universal power law with exponent $-3$. This originates from the peculiar temporal power law decay of $G(t)$ with a time dependent exponent. The mean and the variance of work scale linearly and logarithmically with system size while all higher cumulants are non-extensive. Our results are relevant for non-unitary field theories as well.