https://doi.org/10.1140/epjp/s13360-025-06795-y
Regular Article
Variational principle for the time evolution operator, its usefulness in effective theories of condensed matter systems, and insight into the role played by the quantum geometry of unitary transformations
1
Physics Department, King Fahd University of Petroleum and Minerals, 31261, Dhahran, Saudi Arabia
2
Interdisciplinary Research Center (IRC) for Intelligent Secure Systems, KFUPM, Dhahran, Saudi Arabia
Received:
13
April
2025
Accepted:
27
August
2025
Published online:
7
September
2025
We discuss a variational approach to determining the time evolution operator with the aim to improve on perturbation theory while preserving its simple structure. An immediate insight of our work is the appearance of a generalization of the quantum geometric tensor for unitary operators in the variational equations of motion. Quantum geometry, we then conclude, plays an important role in the evolution of variational parameters. We employ 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 
, time t is not forced to appear at 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 2
2 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). Our 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. We suggest a possible remedy to issues with the thermodynamic limit and demonstrate that that the approach in the limit of small perturbative parameter t agrees with typical results—even in the thermodynamic large N limit. Our work is an example of how the appearance of mathematically beautiful concepts like quantum geometry can indicate an opportunity to dig for improved approximations beyond typical perturbation theory.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
