This work discusses a variational approach to determining the time evolution operator. We directly see a glimpse of how a generalization of the quantum geometric tensor for unitary operators plays a central role in parameter evolution. We try the method with the simplest ansatz (a power series in a time-independent Hamiltonian), which yields considerable improvements over a Taylor series. These improvements are because, unlike for a Taylor series of
exp(−iHt), time
t is not forced to appear in the same order as
H, giving more flexibility for the description. We demonstrate that our results can also be employed to improve degenerate perturbation theory in a non-perturbative fashion. We concede that our approach described here is most useful for finite-dimensional Hamiltonians. As a first example of applications to perturbation theory, we present AB bilayer graphene, which we downfolded to a 2x2 model; our energy results considerably improve typical second-order degenerate perturbation theory. We then demonstrate that the approach can also be used to derive a non-perturbatively valid Heisenberg Hamiltonian. Here, the approach for a finite-size lattice yields excellent results. However, the corrections are not ideal for the thermodynamic limit (they depend on the number of sites
N). Nevertheless, the approach adds almost no additional technical complications over typical perturbative expansions of unitary operators, making it ready for deployment in physics questions. One should expect considerably improved couplings for the degenerate perturbation theory of finite-size systems. More work is needed in the many-body case, and we suggest a possible remedy to issues with the thermodynamic limit. Our work hints at how the appearance of mathematically beautiful concepts like quantum geometry can indicate an opportunity to dig for approximations beyond typical perturbation theory
Based on a commonly used model for the time evolution of extensive variables and on the basis of the analogies between the nonlinear stochastic theory and imaginary time quantum mechanics in curved space, we set up a canonical operator formulation of non-equilibrium thermodynamics which emphasizes the fundamental role played by the Boltzmann constant k in fluctuations. The k to 0 of this theory yields the classical description of non-equilibrium thermodynamics. The new formulation is technically useful and provides new insights which have important consequences, namely the thermodynamic uncertainty principle and the quantization of the entropy production rate for stationary states.
Understanding the real-time evolution of many-electron quantum systems is essential for studying dynamical properties in condensed matter, quantum chemistry, and complex materials, yet it poses a significant theoretical and computational challenge. Our work introduces a variational approach for fermionic time-dependent wave functions, surpassing mean-field approximations by accurately capturing many-body correlations. We employ time-dependent Jastrow factors and backflow transformations, enhanced through neural networks parameterizations. To compute the optimal time-dependent parameters, we employ the time-dependent variational Monte Carlo technique and introduce a new method based on Taylor-root expansions of the propagator, enhancing the accuracy of our simulations. The approach is demonstrated in three distinct systems. In all cases, we show clear signatures of many-body correlations in the dynamics. The results showcase the ability of our variational approach to accurately describe the time evolution, providing insight into quantum dynamical effects in interacting electronic systems, beyond the capabilities of mean-field.