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Communications in Theoretical Physics

Quantum tricritical point induced by staggered qubit biases in a two-qubit quantum Rabi model

Yanzhi Wang, Tian Ye, Qing‐Hu Chen

Published Jun 24, 2025• 1 citation
Publisher

Abstract

Abstract We investigate the quantum phase transitions (QPTs) of the two-qubit quantum Rabi model with staggered qubit biases. In the limit of an infinite qubit-to-cavity frequency ratio, we analytically derive the mean-field Hamiltonian and the order-parameter-dependent energy density functional, which yields the ground-state energy and order parameter. The rich superradiant phase transitions (SRPTs), including both second- and first-order QPTs and a tricritical point (TCP), are analytically derived. Specifically, we derive the analytical expressions for all phase transition points, including the nonperturbative point of the first-order SRPT. The analytical findings are further corroborated by numerical finite-size scaling analysis. It is found that both the critical correlation-length and order-parameter exponents at the TCP differ from those of the original second-order SRPTs, implying that the TCP belongs to a new universality class. This work provides a reliable theoretical framework for designing new, simple experimental platforms to explore the rich QPTs.

DOI
10.1088/1572-9494/ade761

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Although the superradiant phase transition (SRPT) is prohibited in the paradigmatic quantum Rabi model due to the no-go theorem caused by the A2{\mathbf{A}}^{2}A2 term, we demonstrate two distinct types of SRPTs emerging from the normal phase in the anisotropic quantum Rabi model. A discontinuous phase transition between the two types of superradiant phases also emerges in the presence of a strong A2{\mathbf{A}}^{2}A2 term. Additionally, a rich phase diagram featuring a triple point, which connects first- and second-order phase transitions, is derived analytically and confirmed through numerical diagonalization at large effective system sizes. Finally, distinct critical behavior at the triple point is revealed and contrasted with that of a single continuous SRPT. This work may open a new avenue for observing SRPTs in their intrinsic form without altering the A2{\mathbf{A}}^{2}A2 term, while also offering a practical platform for exploring rich quantum phenomena.
Quantum tricriticality, a unique form of high-order criticality, is expected to exhibit fascinating features including unconventional critical exponents and universal scaling laws. However, a quantum tricritical point (QTCP) is much harder to access, and the corresponding phenomena at tricriticality have rarely been investigated. In this study, we explore a tricritical quantum Rabi model, which incorporates a nontrivial parameter for adjusting the coupling ratio between a cavity and a three-level atom. The QTCP emerges at the intersection of a first- and second-order superradiant phase transitions according to Landau theory. By using finite-frequency scaling analyses for quantum fluctuations and the mean photon number, universal critical exponents differentiate the QTCP from the second-order critical point. We find that the phase transition at the tricritical point goes beyond the conventional second-order phase transition. Our work explores an interesting direction in the generalization of the well-known Rabi model for the study of higher-order critical points due to its high control and tunability.
In this paper, we analyze the quantum criticality of the Rabi-Stark model at finite ratios of the qubit and cavity frequencies in terms of the energy gap and the order parameter, as well as the fidelity, if the Stark coupling strength is the same as the cavity frequency. The critical exponents are derived analytically. The energy gap and the length critical exponents are different from those in the quantum Rabi model and the Dicke model. A finite-size scaling analysis for the order parameter and the fidelity susceptibility is also performed. The universal scaling behaviors are demonstrated, and several finite-size exponents then can be extracted. Furthermore, universal critical behavior also can be established in terms of the bosonic Hilbert space truncation number, and the corresponding critical scaling exponents are found. Interestingly, the critical correlation length exponents in terms of the photonic truncation number as well as the equivalently effective length scales are different in the Rabi-Stark model and the quantum Rabi model, suggesting they belong to different universality classes. The second-order quantum phase transition is convincingly corroborated in the Rabi-Stark model at finite frequency ratios; by contrast, it emerges only at the infinite frequency ratio in the original quantum Rabi model without the Stark coupling.
Abstract We explore an extended quantum Rabi model describing the interaction between a two-mode bosonic field and a three-level atom. Quantum phase transitions of this few degree of freedom model is found when the ratio η of the atom energy scale to the bosonic field frequency approaches infinity. An analytical solution is provided when the two lowest-energy levels are degenerate. According to it, we recognize that the phase diagram of the model consists of three regions: one normal phase and two superradiant phases. The quantum phase transitions between the normal phase and the two superradiant phases are of second order relating to the spontaneous breaking of the discrete Z 2 symmetry. On the other hand, the quantum phase transition between the two different superradiant phases is discontinuous with a phase boundary line relating to the continuous U (1) symmetry. For a large enough but finite η , the scaling function and critical exponents are derived analytically and verified numerically, from which the universality class of the model is identified.
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.