Nonlinear Bogolyubov–Valatin transformations and quaternions

Holten, J.W. van; Scharnhorst, K.

doi:10.1088/0305-4470/38/47/012

Cited by 3 publications

(12 citation statements)

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“…(7), i.e., B(t) ≡ 0, the calculating method for quadratic fermionic systems is almost the same as that for quadratic bosonic systems mentioned above. However, when B(t) ≡ 0, the above method fails, because the Bogoliubov transformation does not apply in this situation [27,[32][33][34]. For this situation, we can employ a new method which is described as follows to bypass this difficulty.…”

Section: Pacs Numbersmentioning

confidence: 99%

Group-theoretical approach to the calculation of quantum work distribution

Fei

Quan

2019

Phys. Rev. Research

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Usually the calculation of work distributions in an arbitrary nonequilibrium process in a quantum system, especially in a quantum many-body system is extremely cumbersome. For all quantum systems described by quadratic Hamiltonians, we invent a universal method for solving the work distribution of quantum systems in an arbitrary driving process by utilizing the group-representation theory. This method enables us to efficiently calculate work distributions where previous methods fail. In some specific models, such as the time-dependent harmonic oscillator, the dynamical Casimir effect, and the transverse XY model, the exact and analytical solutions of work distributions in an arbitrary nonequilibrium process are obtained. Our work initiates the study of quantum stochastic thermodynamics based on group-representation theory. PACS numbers:Introduction.-In the past two decades, a great breakthrough in the field of nonequilibrium thermodynamics is the discovery of fluctuation theorems [1] and the establishment of stochastic thermodynamics [2, 3]. It extends the usual definition of work, heat and entropy production in a thermodynamic process from ensembleaveraged quantities to trajectory-dependent quantities. Based on these extensions, the second law is sharpened and rewritten into equalities (fluctuation theorems) for arbitrary nonequilibrium processes. For an isolated quantum system, a proper definition of the trajectory work is defined through the so-called two-point measurement [4, 5], which preserves the non-negativity of the probabilities of trajectories and Jarzynski equality at the same time [6, 7]. As is known, the work distribution of a system provides a great deal of information about a nonequilibrium process [8][9][10], which is an analogue to the partition function encoding essential information about an equilibrium state. However, very few analytical results of work distributions of quantum systems have been obtained so far. In literature, the few exceptions are associated with either specific models (e.g., a forced harmonic oscillator [11][12][13], a harmonic oscillator with a time-dependent frequency [14,15], quenched Luttinger liquid at zero temperature [8], a diving scalar field [16] or a transverse Ising model at zero temperature [15]) or very special protocols (e.g., the sudden quench protocol [17][18][19][20][21][22][23] or the quantum adiabatic protocol [24,25]). Hence, to develop a universal method to efficiently calculate the work distribution for various quantum systems in an arbitrary nonequilibrium process is one of the most challenging problems in this field.

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Section: Pacs Numbersmentioning

confidence: 99%

Group-theoretical approach to the calculation of quantum work distribution

Fei

Quan

2019

Phys. Rev. Research

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“…As in our earlier paper [49], the main focus lies on structural and methodical aspects of the problem of nonlinear Bogolyubov-Valatin transformations. We start in Sec.…”

Section: Introductionmentioning

confidence: 99%

“…2-6, p. 52, (We disregard here work done within the framework of the coupled-cluster method (CCM) [4] which is nonunitary.). 1 However, a systematic analytic study of general (nonlinear) Bogolyubov-Valatin transformations had not been undertaken until the publication of our article [49].…”

Section: Introductionmentioning

confidence: 99%

“…Certain special types of such nonlinear basis transformations in a Clifford algebras have been considered previously in[46,131,132] and in generality for the Clifford algebra C(0, 2) in[49]. For related aspects see[133].…”

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confidence: 99%

“…. For further references concerning linear Bogolyubov-Valatin transformations see[49], p. 10247, below from eq. (16) 34.…”

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Nonlinear Bogolyubov-Valatin transformations: Two modes

Scharnhorst

Holten

2011

Annals of Physics

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Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper we perform a thorough study of general (nonlinear) canonical transformations for two fermionic modes. We find that the Bogolyubov-Valatin group for n = 2 fermionic modes which can be implemented by means of unitary SU (2 n = 4) transformations is isomorphic to SO(6; R)/Z 2 . The investigation touches on a number of subjects. As a novelty from a mathematical point of view, we study the structure of nonlinear basis transformations in a Clifford algebra [specifically, in the Clifford algebra C(0, 4)] entailing (supersymmetric) transformations among multivectors of different grades. A prominent algebraic role in this context is being played by biparavectors (linear combinations of products of Dirac matrices, quadriquaternions, sedenions) and spin bivectors (antisymmetric complex matrices). The studied biparavectors are equivalent to Eddington's E-numbers and can be understood in terms of the tensor product of two commuting copies of the division algebra of quaternions H. From a physical point of view, we present a method to diagonalize any arbitrary two-fermion Hamiltonian. Relying on Jordan-Wigner transformations for two-spin-1 2 and single-spin-3 2 systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin-1 2 and single-spin-3 2 Hamiltonians. Finally, from a calculational point of view, we pay due attention to explicit parametrizations of SU (4) and SO(6; R) matrices (of respective sizes 4 × 4 and 6 × 6) and their mutual relation. † As a consequence of eq. (17) and the anti-Hermiticity of the operatorsĉ k , the operatorsĉ k [i.e., the basis elements of the paravector space associated with the Clifford algebra C(0, 5)] obey the relation (k, l = −1, . . . , 4)(304) 35 Below we have not indicated that the wedge product relates to a different vector space than in eq. (301), i.e.:

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Remarks on the mean-field theory based on the SO(2N + 1) Lie algebra of the fermion operators

Nishiyama

Providência

2019

Int. J. Geom. Methods Mod. Phys.

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Toward a unified algebraic theory for mean-field Hamiltonian (MFH) describing pairedand unpaired-mode effects, in this paper, we propose a generalized HB (GHB) MFH in terms of the SO(2N+1) Lie algebra of fermion pair and creation-annihilation operators. We diagonalize the GHB-MFH and throughout the diagonalization of which, we can first obtain the unpaired mode amplitudes which are given by the SCF parameters appeared in the HBT together with the additional SCF parameter in the GHB-MFH and by the parameter specifying the property of the SO(2N +1) group. Consequently, it turns out that the magnitudes of these amplitudes are governed by such parameters. Thus, it becomes possible to make clear a new aspect of such the results. We construct the Killing potential in the coset space SO(2N ) U (N ) on the Kähler symmetric space which is equivalent with the generalized density matrix (GDM). We show another approach to the fermion MFH based on such a GDM. We derive an SO(2N+1) GHB MF operator and a modified HB eigenvalue equation. We discuss on the MF theory related to the algebraic MF theory based on the GDM and the coadjoint orbit leading to the non-degenerate symplectic form.

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Nonlinear Bogolyubov–Valatin transformations and quaternions

Cited by 3 publications

References 57 publications

Group-theoretical approach to the calculation of quantum work distribution

Group-theoretical approach to the calculation of quantum work distribution

Nonlinear Bogolyubov-Valatin transformations: Two modes

Remarks on the mean-field theory based on the SO(2N + 1) Lie algebra of the fermion operators

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