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arXiv:2603.07031 · Physical review. B./Physical review. B

Quantum criticality in sub-Ohmic systems with three competing terms: beyond conventional spin-boson physics

Nengji Zhou, Yulong Shen, Zhe Sun

quant-phcond-mat.stat-mechcond-mat.str-elSubmitted Mar 7, 2026• Announced Mar 10, 2026• 0 citations
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Abstract

Quantum phase transitions (QPTs) in the spin-boson model with/without the rotating-wave approximation (RWA) are systematically investigated through variational calculations using a sub-Ohmic bath with high spectral density. Four cases involving different system-environment interactions are examined, where transition points and critical exponents are accurately determined across varying tunneling strengths. Contrary to prior work, a rich phase diagram is revealed in the tunneling-coupling plane even at the low spectral exponent $s<1/2$, with a novel U(1)-symmetric phase being identified. As coupling increases, a multi-stage QPT sequence arises for the tunneling $0<Δ< Δ^*=0.074(1)$, whereas a single transition occurs beyond this range. Furthermore, an odd-parity phase is found to emerge even under the positive tunneling, exhibiting distinct characteristics relative to the prototype model.

Journal reference
Physical Review B 113, 125119 (2026)
DOI
10.1103/p7vp-74cd
Comments
18 pages, 13 figures

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Quantum phase transitions (QPTs) in the spin-boson model with/without the rotating-wave approximation (RWA) are systematically investigated through variational calculations using a sub-Ohmic bath with high spectral density. Four cases involving different system-environment interactions are examined, where transition points and critical exponents are accurately determined across varying tunneling strengths. Contrary to prior work, a rich phase diagram is revealed in the tunneling-coupling plane even at the low spectral exponent s<1/2s<1/2s<1/2, with a novel U(1)-symmetric phase being identified. As coupling increases, a multi-stage QPT sequence arises for the tunneling 0<Δ<Δ∗=0.074(1)0<Δ< Δ^*=0.074(1)0<Δ<Δ∗=0.074(1), whereas a single transition occurs beyond this range. Furthermore, an odd-parity phase is found to emerge even under the positive tunneling, exhibiting distinct characteristics relative to the prototype model.
We study the spin-boson model without the counterrotating terms by a numerically exact method based on variational matrix product states. Surprisingly, the second-order quantum phase transition (QPT) is observed for the sub-Ohmic bath in the rotating-wave approximations. Moreover, first-order QPTs can also appear before the critical points. With the decrease of the bath exponents, these first-order QPTs disappear successively, while the second-order QPT remains robust. The second-order QPT is further confirmed by multi-coherent-state variational studies, while the first-order QPT is corroborated with the exact diagonalization in the truncated Hilbert space. Extension to the Ohmic bath is also performed, and many first-order QPTs appear successively in a wide coupling regime, in contrast to previous findings. The previous pictures for many physical phenomena for the spin-boson model in the rotating-wave approximation have to be modified at least at strong coupling.
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.
The ground state properties and quantum phase transitions of sub-Ohmic spin-boson models are investigated using the multiple Davydov D2Ansatz in conjunction with the variational principle. Three variants of the model are studied: (i) a single bath with diagonal coupling, (ii) two independent baths with diagonal and off-diagonal couplings, and (iii) a single bath with simultaneous diagonal and off-diagonal couplings. For the purely diagonal model, the multiple Davydov D2Ansatz yields critical coupling strengths that are consistent with other methodologies, validating its accuracy and efficiency. In the two-bath model, the competition between diagonal and off-diagonal couplings drives a first-order transition for both symmetric and asymmetric spectral exponents, with the von Neumann exhibiting a continuous peak only under exact symmetry. Finally, for a single bath with simultaneous diagonal and off-diagonal couplings, we demonstrate that a rotational transformation maps the system to an equivalent purely diagonal model, enabling simpler and intuitive physical interpretation and reduced computational complexity.
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 propose a general extended coherent state approach to the qubit (or fermion) and multimode boson coupling systems. The application to the spin-boson model with the discretization of a bosonic bath with arbitrary continuous spectral density is described in detail, and very accurate solutions can be obtained. The quantum phase transition in the nontrivial sub-Ohmic case can be located by the fidelity and the order-parameter critical exponents for the bath exponents s<1/2s<1/2s<1/2 can be correctly given by the fidelity susceptibility, demonstrating the strength of the approach.
The influence of dissipation on quantum tunneling in the spin-boson model with a sub-Ohmic bath is studied by a variational calculation. By examining the evolution of solutions of the variational equation with the coupling strength near the phase boundary, we are able to present a scenario of discontinuous transition in sub-Ohmic dissipation case in accord with Ginzburg-Landau theory. Based on the constructed picture, it is shown that the critical point found in the general way is not thermodynamically the critical point, but the point where the second energy minimum begins to develop. The true cross-over point is calculated and the obtained phase diagram is in agreement with the result of numerical renormalization group calculation.