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The Journal of Chemical Physics

Unitary transformation with double polaronic generator for spin-boson model: Ground state, thermodynamics and dynamical evolution

Hang Zheng, Zhiguo Lü

Published Jan 22, 2026• 1 citation
Publisher

Abstract

Unitary transformation with a double polaronic generator is proposed for the unbiased spin-boson Hamiltonian, which leads to a new ground state with lower ground state energy, as well as an effective Hamiltonian without the counter-rotating coupling. The infrared divergence associated with the nonadiabatic modes in Ohmic bath plays an important role in determining the quantum critical point αc of the ground state, which separates the non-degenerate ground state for α<αc from the degenerate one for α>αc and leads to logarithmic divergence of the bosonic numbers. The thermodynamic properties, the dynamical evolution, and time correlation of the SBM model are studied by the effective Hamiltonian obtained from the unitary transformation, which can be solved easily and transparently by the perturbation theory. As a check of our theoretical treatment and numerical results, we have calculated the Wilson ratio and shown that the spectral sum rule and Shiba's relation are exactly fulfilled.

DOI
10.1063/5.0308208

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Unitary transformation with a double polaronic generator is proposed for the unbiased spin-boson Hamiltonian, which leads to a new ground state with lower ground state energy, as well as an effective Hamiltonian without the counter-rotating coupling. The infrared divergence associated with the nonadiabatic modes in Ohmic bath plays an important role in determining the quantum critical point αc of the ground state, which separates the non-degenerate ground state for α<αc from the degenerate one for α>αc and leads to logarithmic divergence of the bosonic numbers. The thermodynamic properties, the dynamical evolution, and time correlation of the SBM model are studied by the effective Hamiltonian obtained from the unitary transformation, which can be solved easily and transparently by the perturbation theory. As a check of our theoretical treatment and numerical results, we have calculated the Wilson ratio and shown that the spectral sum rule and Shiba's relation are exactly fulfilled.
An analytic ground state is proposed for the unbiased spin-boson Hamiltonian, which is non-Gaussian and beyond the Silbey-Harris ground state with lower ground state energy. The infrared catastrophe in Ohmic and sub-Ohmic bosonic bath plays an important role in determining the degeneracy of the ground state. We show that the infrared divergence associated with the displacement of the nonadiabatic modes in bath may be removed from the proposed ground state for the coupling α < αc. Then αc is the quantum critical point of a transition from non-degenerate to degenerate ground state and our calculated αc agrees with previous numerical results.
By utilizing a unitary transformation, we derive the necessary and sufficient conditions for the degeneracy between the even- and odd-parity energy states of the spin-boson model (SBM). Employing the Rayleigh quotient of matrix algebra, we rigorously prove that the ground state energy of the SBM is lower than the systems lowest possible degenerate energy and possesses a definite parity. Based on the necessary and sufficient conditions for parity breaking, we provide an analytical expression for the parity-breaking critical value, which is closely related to the expansion order and computational accuracy. This expression reproduces the SBM phase diagram obtained by quantum Monte Carlo (QMC) and logarithmic discretization numerical renormalization group (NRG) methods. However, this phase diagram does not characterize the ground state of the system.
The structure of the ground state of the Hamiltonian of a two-level system linearly coupled to a single mode of a radiation Bose-like field is studied for arbitrary values of the coupling constant including the counter-rotating terms. First of all the Hamiltonian is canonically transformed, introducing a new set of unitary operators in such a way that the eigenstates of the transformed Hamiltonian are exactly factorizable into eigenstates of the new pseudospin (1/2) and of the new field. The ground state is then found by a variational procedure. The validity of the results obtained by this approach is shown introducing a suitable class of canonical transformations by which it is possible to see that this variational ground state differs from the exact one only for perturbative contributions for which we give explicit expressions. Furthermore, we present investigations on the properties of this system in its ground state based on the calculation of the covariance of suitable pairs of operators. In this way we succeed in obtaining, among other things, a physically transparent meaning for the mathematical variational condition which determines the ground state.
Quantum phase transition in the spin-boson model was claimed on the basis of various numerical studies, but not strictly proven. Here by using a unitary transformation to decompose the Hamiltonian into two branches of odd and even parity we obtained the necessary and sufficient condition for degeneracy to occur between states of opposite parity in the spin-boson model, and the analytical expression for such degenerate energies. It can be strictly proven that the ground state of spin-boson model with non-vanishing tunneling amplitude must have an energy lower than the lowest possible such degenerate energy, and have definite parity. Starting from the invariancy of the parity operator we show that finite expansion by numerical calculation induces the breaking of parity symmetry responsible for the phase transition. The critical dissipation parameter we obtained for parity-symmetry breaking, as a logarithmic function of summed diagonal matrix elements in the finite expansion for the bosonic part of the parity operator, can reproduce the phase diagram derived with quantum Monte Carlo method and logarithmically discretized numberical renormalization group approach. It reveals that the quantum phase transition in spin-boson model claimed by numerical procedures arises from symmetry breaking caused by finite expansion in practical calculation. The method we developed here may also be applicable to the discussion of quantum chaos and other similar problems.
A generalized trial wave function termed as the ``multi-D1{D}_{1}D1​ ansatz'' has been developed to study the ground state of the spin-boson model with simultaneous diagonal and off-diagonal coupling in the sub-Ohmic regime. Ground-state properties including energy and spin polarization are investigated, and the results are consistent with those from exact diagonalization and density matrix renormalization group approaches for the cases involving two oscillators and two baths described by a continuous spectral density function. Breakdown of the rotational and parity symmetries along the continuous quantum phase transition separating the localized phase from the critical phase has been uncovered. Moreover, the phase boundary is determined accurately with the corresponding rotational- and parity-symmetry parameters. A critical value of the spectral exponent s∗=0.49(1){s}^{*}=0.49(1)s∗=0.49(1) is predicted in the weak coupling limit, which is in agreement with the mean-field prediction of 1/21/21/2, but much smaller than the earlier literature estimate of 0.75(1)0.75(1)0.75(1).