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arXiv:1812.03758 · Physical review. B./Physical review. B

Quantum phase transitions in the spin-boson model without the counterrotating terms

Yanzhi Wang, Shu He, Liwei Duan, Qing‐Hu Chen

Published Sep 3, 2019• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.100.115106

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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.
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.
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.
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.
By employing a variational approach, the density matrix renormalization group (DMRG), the exact diagonalization, and symmetry and mean-field analyses, the ground-state properties of the two-bath spin-boson model with simultaneous diagonal and off-diagonal coupling are systematically studied in the sub-Ohmic regime. A quantum phase transition from a doubly degenerate ``localized phase'' to the other doubly degenerate ``delocalized phase'' is uncovered. Via the multi-D1{\mathrm{D}}_{1}D1​ Ansatz as the variational wave function, transition points are determined accurately, consistent with the results from DMRG and exact diagonalization. An effective spatial dimension deff=2.37(6){d}_{\mathrm{eff}}=2.37(6)deff​=2.37(6) is then estimated, which is found to be compatible with the mean-field prediction. Furthermore, the quantum phase transition is inferred to be of first order for the baths described by a continuous spectral density function. In the single-mode case, however, the transition is softened.
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 spin-boson model (SBM) with two spins in staggered biases by a numerically exact method based on variational matrix product states. Several observables such as the magnetization, the entanglement entropy between the two spins and the bosonic environment, the ground-state energy, as well as the correlation function for two spins are calculated exactly. The characteristics of these observables suggest that the staggered biases can drive the second-order quantum phase transition (QPT) to the first-order QPT in the sub-Ohmic SBM, while the Kosterlitz-Thouless QPT in the Ohmic SBM goes directly to the first-order one. A quantum tricritical point, where the continuous QPT meets the first-order one, can then be detected. It is found that the staggered biases would not change the universality of the phase transition in this model below the quantum tricritical point.