arXiv:2108.10626

Bargmann Representation of Spin Chains

M. W. AlMasri, Mohamed Ridza Wahiddin

Published Aug 24, 2021• 1 citation

Abstract

Spin chain Hamiltonians can be written in terms of complex differential operators using the Bargmann representation of the Jordan-Schwinger map. In this case, the eigenfunctions are expressed as the product of orthonormal monomials of the phase-space coordinates $z_{i}=Q_{i}+i P_{i}$ in the complex plane. Furthermore, the series constructed from each phase-space coordinate converges uniformly in any compact domain of the complex plane. Formulating spin chains with respect to the phase-space coordinates helps in discussing their classical limit and in the calculations of quasi-probability distributions.

DOI: 10.48550/arxiv.2108.10626

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arXiv:2108.10626

Bargmann Representation of Spin Chains

M. W. AlMasri, Mohamed Ridza Wahiddin

Published Aug 24, 2021• 1 citation

arXiv HTML PDF

Abstract

z_{i}=Q_{i}+i P_{i}

in the complex plane. Furthermore, the series constructed from each phase-space coordinate converges uniformly in any compact domain of the complex plane. Formulating spin chains with respect to the phase-space coordinates helps in discussing their classical limit and in the calculations of quasi-probability distributions.

DOI: 10.48550/arxiv.2108.10626

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Transport Theory and Statistical PhysicsJan 1, 197110 citations

z_{i}=Q_{i}+i P_{i}

When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation

T

of a real algebraic Lie group

G

. This requires defining an inner product on the Hilbert space

\\mathbb{H}

that carries the representation

T

. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of

T

, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet (

\\mathfrak{F}_{\\mathbb{H}_D}

\\mathbb{H}_D

\\mathfrak{F}^{\\mathfrak{H}_D}

) of Bargmann spaces. Coherent state wave functions -- the elements of

\\mathfrak{F}_{\\mathbb{H}_D}

and of

\\mathfrak{F}^{\\mathbb{H}_D}

-- are used to define the inner product on

\\mathbb{H}_D

in a way that simplifies the calculation of the overlaps. This inner product and the group action

\\Gamma

G

\\mathfrak{F}^{\\mathbb{H}_D}

are used to formulate expressions for the matrix elements of

T

with coefficients from the given subrepresentation.\n\nThe process of finding an explicit expression for

\\Gamma

relies on matrix factorizations in the complexification of

G

even though the representation

T

does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and

\\Gamma

action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in

\\mathbb{H}

. The scalar and vector coherent state methods will both be applied to Sp(

n

) and Sp(

n,\\mathbb{R}

z_{i}

and their partial derivatives. We finally discuss the computational complexity of the holomorphic representation and compare it with the computational complexity of the standard operator method and prove the efficiency of the holomorphic representation in computing some tasks. Our treatment is applicable in all quantum heat engines and refrigerators.