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arXiv:2506.14818

Non-degenerate Ground State of the Spin-Boson Model under Abelian Diagonalization

Liu, Tao, Juhao Wu, Ying, Mark

Published Jun 9, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2506.14818

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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.
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.
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.
We explore the spin-boson model in a special case, i.e., with zero local field. In contrast to previous studies, we find no possibility for quantum phase transition (QPT) happening between the localized and delocalized phases, and the behavior of the model can be fully characterized by the even or odd parity as well as the parity breaking, instead of the QPT, owned by the ground state of the system. The parity breaking mentioned in our case is completely different from the spontaneously broken symmetry relevant to the conventionally defined QPT in previous studies. Our analytical treatment about the eigensolution of the ground state of the model presents for the first time a rigorous proof of no-degeneracy for the ground state of the model, which is independent of the bath type, the degrees of freedom of the bath and the calculation precision. We argue that the QPT mentioned previously appears due to incorrect employment of the ground state of the model and/or unreasonable treatment of the infrared divergence existing in the spectral functions for Ohmic and sub-Ohmic dissipations.
We study the phase diagram of a one-dimensional version of the Kitaev spin-1/2 model with an extra "Γ term," using analytical, density matrix renormalization group and exact diagonalization methods. Two intriguing phases are found. In the gapless phase, although the exact symmetry group of the system is discrete, the low energy theory is described by an emergent SU(2)_{1} Wess-Zumino-Witten (WZW) model. On the other hand, the spin-spin correlation functions exhibit SU(2) breaking prefactors, even though the exponents and the logarithmic corrections are consistent with the SU(2)_{1} predictions. A modified non-Abelian bosonization formula is proposed to capture such exotic emergent "partial" SU(2) symmetry. In the ordered phase, there is numerical evidence for an O_{h}→D_{4} spontaneous symmetry breaking.
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).
We study the anisotropic spin-boson model (SBM) with the subohmic bath by a numerically exact method based on variational matrix product states. A rich phase diagram is found in the anisotropy-coupling strength plane by calculating several observables. There are three distinct quantum phases: a delocalized phase with even parity (phase I), a delocalized phase with odd parity (phase II), and a localized phase with broken Z2{Z}_{2}Z2​ symmetry (phase III), which intersect at a quantum tricritical point. The competition between those phases gives an overall picture of the phase diagram. For small power of the spectral function of the bosonic bath, the quantum phase transition (QPT) from phase I to III with mean-field critical behavior is present, similar to the isotropic SBM. The phase diagram with three different phases can be found at large power of the spectral function: For the highly anisotropic case, the system experiences the QPTs from phase I to II via first order, and then to phase III via second order with the increase of the coupling strength. For the low anisotropic case, the system only experiences the continuous QPT from phase I to phase III with the non-mean-field critical exponents. Very interestingly, at the moderate anisotropy, the system would display the continuous QPTs for several times but with the same critical exponents. This unusual reentrance to the same localized phase is discovered in light-matter interacting systems. The present study on the anisotropic SBM could open an avenue to the rich quantum criticality.