A statistical mechanical approach to restricted integer partition functions

Zhou, Chi-Chun; Dai, Wu-Sheng

doi:10.1088/1742-5468/aabfc9

Cited by 11 publications

(11 citation statements)

References 50 publications

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“…[5], the energy eigenvalue spectrum of an interacting many-body system is calculated from the partition function. In statistical mechanics, there are many methods developed for calculating partition functions and grand partition functions, such as the cluster expansion method, the field theory method [63], and some mathematical methods [64,65]. The eigenvalue and the partition function are both spectral functions.…”

Section: Discussionmentioning

confidence: 99%

Scattering approach for calculating one-loop effective action and vacuum energy

Li¹,

Chen²,

Li³

et al. 2022

Preprint

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An approach for calculating one-loop effective actions and vacuum energies is suggested. Spectral functions are functions of the eigenvalue spectrum of an operator. The starting point is to regard one-loop effective actions and vacuum energies in quantum field theory and scattering phase shifts and amplitudes in quantum mechanics as spectral functions of an operator. Different spectral functions of the same operator may belong to different fields of physics. Methods for calculating various spectral functions can be interconverted through transform relations among spectral functions. In this paper, we convert the method for calculating scattering phase shifts and amplitudes to a method for calculating one-loop effective actions and vacuum energies. In principle, all methods for the calculation of scattering phase shifts and amplitudes can be converted to methods for the calculation of the one-loop effective actions and vacuum energies. As an example, we convert the Born approximation in quantum-mechanical scattering theory to a method for calculating one-loop effective actions and vacuum energies. In appendices, we provide integral representations of the Bessel function for calculating the sum encountered.

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Section: Discussionmentioning

confidence: 99%

Scattering approach for calculating one-loop effective action and vacuum energy

Li¹,

Chen²,

Li³

et al. 2022

Preprint

Self Cite

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show abstract

“…0 denotes an integer partition of integer M and a i is the element of this integer partition [27][28][29]. The length of (a) is l (a) = m. For example, (a) = (4).…”

Section: The Relation Between Permutation Group and Unitary Group In ...mentioning

confidence: 99%

“…The partitions for (4) are (4), (3, 1), (2, 2), 2, 1 2 , 1 4 . Each integer partition of the permutation group gives a conjugacy-class operator [27][28][29]. The conjugacyclass operator P (2, 1 ν−2 ) denotes the exchange of any two particles when the spin system includes ν particles, and (2, 1 ν−2 ) is the integer partition.…”

Section: The Relation Between Permutation Group and Unitary Group In ...mentioning

confidence: 99%

A group method solving many-body systems in intermediate statistical representation

Shen,

Zhou,

Dai

et al. 2021

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The exact solution of the interacting many-body system is important and is difficult to solve. In this paper, we introduce a group method to solve the interacting many-body problem using the relation between the permutation group and the unitary group. We prove a group theorem first, then using the theorem, we represent the Hamiltonian of the interacting many-body system by the Casimir operators of unitary group. The eigenvalues of Casimir operators could give the exact values of energy and thus solve those problems exactly. This method maps the interacting many-body system onto an intermediate statistical representation. We give the relation between the conjugacy-class operator of permutation group and the Casimir operator of unitary group in the intermediate statistical representation, called the Gentile representation. Bose and Fermi cases are two limitations of the Gentile representation. We also discuss the representation space of symmetric and unitary group in the Gentile representation and give an example of the Heisenberg model to demonstrate this method. It is shown that this method is effective to solve interacting many-body problems.

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“…For the generalized Heisenberg model consisting of N particles with the dimension of single-particle-Hilbert space m, by using Eq. (3.21), the number of distinct eigenvalues is P (N, m), where P (N, m) is the restrict integer partition number [19] that is the number of ways to express N as sum of other integers with the number of summands no larger than m. It is because one irreducible representation gives a distinct eigenvalue of Casimir operator and thus gives a distinct eigenvalue of the system. The irreducible representation is labeled by a set of number (a) = (a 1 , a 2 , ..., a m ) with a 1 a 2 ... a m 0 [17].…”

Section: The Highest Degeneracy Of Eigenstates Of a Lattice Modelmentioning

confidence: 99%

Converting Lattices into Networks: The Heisenberg Model and Its Generalizations with Long-Range Interactions

Zhou,

Shen,

Chen

et al. 2020

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In this paper, we convert the lattice configurations into networks with different modes of links and consider models on networks with arbitrary numbers of interacting particle-pairs. We solve the Heisenberg model by revealing the relation between the Casimir operator of the unitary group and the conjugacy-class operator of the permutation group. We generalize the Heisenberg model by this relation and give a series of exactly solvable models. Moreover, by numerically calculating the eigenvalue of Heisenberg models and random walks on network with different numbers of links, we show that a system on lattice configurations with interactions between more particle-pairs have higher degeneracy of eigenstates. The highest degeneracy of eigenstates of a lattice model is discussed.

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A statistical mechanical approach to restricted integer partition functions

Cited by 11 publications

References 50 publications

Scattering approach for calculating one-loop effective action and vacuum energy

Scattering approach for calculating one-loop effective action and vacuum energy

A group method solving many-body systems in intermediate statistical representation

Converting Lattices into Networks: The Heisenberg Model and Its Generalizations with Long-Range Interactions

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