A field theory representation of sum of powers of principal minors and physical applications
arxiv(2024)
摘要
We introduce a novel field theory representation for the Sum of Powers of
Principal Minors (SPPM), a mathematical construct with profound implications in
quantum mechanics and statistical physics. We begin by establishing a Berezin
integral formulation of the SPPM problem, showcasing its versatility through
various symmetries including SU(n), its subgroups, and particle-hole
symmetry. This representation not only facilitates new analytical approaches
but also offers deeper insights into the symmetries of complex quantum systems.
For instance, it enables the representation of the Hubbard model's partition
function in terms of the SPPM problem. We further develop three mean field
techniques to approximate SPPM, each providing unique perspectives and
utilities: the first method focuses on the evolution of symmetries post-mean
field approximation, the second, based on the bosonic representation, enhances
our understanding of the stability of mean field results, and the third employs
a variational approach to establish a lower bound for SPPM. These methods
converge to identical consistency relations and values for SPPM, illustrating
their robustness. The practical applications of our theoretical advancements
are demonstrated through two compelling case studies. First, we exactly solve
the SPPM problem for the Laplacian matrix of a chain, a symmetric tridiagonal
matrix, allowing for precise benchmarking of mean-field theory results. Second,
we present the first analytical calculation of the Shannon-Rényi entropy for
the transverse field Ising chain, revealing critical insights into phase
transitions and symmetry breaking in the ferromagnetic phase. This work not
only bridges theoretical gaps in understanding principal minors within quantum
systems but also sets the stage for future explorations in more complex quantum
and statistical physics models.
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