Transfer matrices, non-Hermitian Hamiltonians and Resolvents: some spectral identities

Abstract

I consider the N-step transfer matrix T for a general block Hamiltonian, with eigenvalue equation L_n ψ_{n+1} + H_n ψ_n + L_{n-1}^\dagger ψ_{n-1} = E ψ_n where H_n and L_n are matrices, and provide its explicit representation in terms of blocks of the resolvent of the Hamiltonian matrix for the system of length N with boundary conditions ψ_0 =ψ_{N+1} =0. I then introduce the related Hamiltonian for the case ψ_0 = z^{-1} ψ_N and ψ_{N+1} = z ψ_1, and provide an exact relation between the trace of its resolvent and Tr(T-z)^{-1}, together with an identity of Thouless type connecting Tr(\log |T|) with the Hamiltonian eigenvalues for z=e^{iϕ}. The results are then extended to T^\dagger T by showing that it is itself a transfer matrix. Besides their own mathematical interest, the identities should be useful for an analytical approach in the study of spectral properties of a physically relevant class of transfer matrices. P.A.C.S.: 02.10.Sp (theory of matrices), 05.60 (theory of quantum transport), 71.23 (Anderson model), 72.17.Rn (Quantum localization)

plain TeX, 12 pages; to appear on Journal of Physics A: Math.Gen

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• Theory of directed localization in one dimension
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• The two-dimensional Anderson model of localization with random hopping
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Cited by in corpus (11)

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• Non-Hermitian spectra and Anderson localization
• "Single Ring Theorem" and the Disk-Annulus Phase Transition
• A Unified Perspective of Complex Band Structure: Interpretations, Formulations, and Applications
• Spectral duality and distribution of exponents for transfer matrices of block tridiagonal Hamiltonians
• Hybrid scale-free skin effect in non-Hermitian systems: A transfer matrix approach
• Universal subdiffusive behavior at band edges from transfer matrix exceptional points
• Identities and exponential bounds for transfer matrices
• On the Kolmogorov-Sinai entropy of many-body Hamiltonian systems
• Environment assisted superballistic scaling of conductance
• Lack of near-sightedness principle in non-Hermitian systems