Quantum phase transition in XXZ central spin model

The gist: The central spin model exhibits a normal-to-superradiant phase transition in the limit where the ratio of the transition frequency of the central spin to that of the bath spins and the number of bath spins tend to infinity

Model Description

The central spin model can be described as a single spin- 1/2 particle coupled to N spin- 1/2 particles (bath spins), which is also called qubit-big spin model in Ref. [26] <ref:2207.04191#pg3> For different strengths of the longitudinal and the transverse interactions, the Hamiltonian of this model can be written as H = ω0 squared σ(0)z + ω squared N ∑ k=1 σ(k)z + N ∑ k=1 ∆k squared σ(k)z σ(0)z + N ∑ k=1 Ak squared σ(k)x σ(0)x +σ(k)y σ(0)y <ref:2207.04191#pg3> For simplicity, the case that the central spin is uniformly coupled to bath spins (Ak = A, ∆k = ∆), the Hamiltonian becomes H = ω0Sz +ωJz +A(J+S− +J−S+) +2∆JzSz <ref:2207.04191#pg3> The Hamiltonian of the XXZ central spin model exhibits a U(1) symmetry <ref:2207.04191#pg3>

Phase Transition Conditions

The paper investigates the superradiant quantum phase transition in the XXZ central spin model, focusing on the limit where N → ∞ and the ratio of frequencies tends to infinity <ref:2207.04191#pg2> It shows that for ∆ < ω, a similar critical point with the JC model exists when ∆ = 0 <ref:2207.04191#pg2> However, for ∆ < ω, the critical point is different from the one before and an analytical solution is provided using the theory of low-energy effective Hamiltonian <ref:2207.04191#pg2> The necessary condition for the superradiance QPT in the XXZ central spin model is ∆

Influence of Longitudinal Interaction

The longitudinal interaction ∆ significantly influences the excitation number and coherence of the ground state, being remarkably enhanced for negative longitudinal interaction while suppressed for positive longitudinal interaction <ref:2207.04191#pg2> For ∆ = 0, the excitation number of the ground state is ng = 0 for g However, for g > 1, the excitation number of the ground state is given by ng = η / (4g squared - g-2) <ref:2207.04191#pg3> In Fig. 3, we see that the macroscopic excitations of bath spins are significantly enhanced in the case of the negative longitudinal interaction (−ω < ∆ < 0), but for the situation of the positive longitudinal interaction (0

Characterization and Measurement

The quantum Fisher information (QFI) is used to characterize the QPT of the XXZ central spin model, and a measurement scheme can be illustrated <ref:2207.04191#pg2> The QFI is given by Fq = -4 ∂ squared f(g˜,δg˜) / ∂ (δg˜) squared δg˜→0 <ref:2207.04191#pg6> In Fig. 4(a), the QFI has an abrupt change at the critical point g c = 1, which is due to a significant change in the ground state of the system <ref:2207.04191#pg6> The inverse variance of the parameter ˜g is given by Fg˜ = (∂g˜hσxi) squared / (1-hσxi 2) <ref:2207.04191#pg6>

Low-Energy Effective Hamiltonian

The low-energy effective Hamiltonian in the normal phase is obtained by applying a Holstein-Primakoff transformation and a Schrieffer-Wolff transformation e S with the anti-Hermitian operator S = λa†S− −aS+ / ωe0 <ref:2207.04191#pg8> This leads to Henp = -ωe0 squared − ω j +ωe 1−g˜2 a†a, where a (a†) is the bosonic annihilation (creation) operator <ref:2207.04191#pg8> For the superradiant phase, a displacement operator D(α) is applied to Hhp, leading to the low-energy effective Hamiltonian H 0sp = ωα squared − ω0 squared +κ0x squared −ω j <ref:2207.04191#pg9> The second derivative of the ground state energy with respect to ˜g is discontinuous at the critical point, which means the QPT occurs in the XXZ central spin model is a second-order phase transition like the JC model <ref:2207.04191#pg9>

Practical Application

Quantum criticality can be viewed as a quantum resource for quantum sensing, and quantum Fisher information (QFI) is a key concept in quantum metrology The observable of the central spin hσxi is able to measured, which is given by hσx(t)i = 2Ren b∗ ↑b↓ hϕ e it[2ωe(ge 2+ ∆ωe)n] ϕi o The precision of the coupling strength can be calculated using the error propagation formula [46] It is worth noting that choosing different longitudinal interaction ∆ has a significant influence on the positions of the maximum value and minimum values of Fg, meaning we can change the coupling strength ˜g at which maximum precision is achieved by controlling the value of the longitudinal interaction ∆

Conclusion

The work reveals the superradiant QPT occurring in central spin systems and provides a new idea for the realization of criticality-enhanced quantum sensing The low-energy effective Hamiltonian given by the Schrieffer-Wolff transformation provides us with a new perspective to explain the quatum phase transition Unlike the JC model, the nonlinear term caused by the longitudinal interaction can greatly affect the excitation number and the coherence of the ground state For −ω < ∆ < 0, these two physical quantities increases significantly, while for the case of 0 < ∆ < ω they decreases remarkably The QFI is used to quantify the abrupt change of the ground state around the critical point, and we also utilize the error propagation formula to calculate the precision of the coupling strength This work reveals the superradiant QPT occurring in central spin systems and provides a new idea for the realization of criticality-enhanced quantum sensing

Acknowledgments

We acknowledge Dr. Chao Han for valuable suggestions <ref:2207.04191#pg7> This work was supported by the NSFC through Grant No. 11935012 and No. 11875231, the National Key Research and Development Program of China (Grants No. 2017YFA0304202 and No. 2017YFA0205700>

Appendix A: Derivation of Exact Energy Spectrum

Similar to the JC model, the dynamics of the XXZ central spin model is confined to the two-dimensional space spanned by ↑,n-1i and ↓,ni [44] <ref:2207.04191#pg10> For a given n, the matrix elements of H are h↑,n-1H ↑,n-1i = ω0 squared + (ω +∆)m/A√kn, h↓,nH ↓,ni = -ω0 squared + (ω −∆) (m+1), h↑,n-1H ↓,ni = h↓,n-1H ↑,n-1i = A p/kn <ref:2207.04191#pg10> The matrix representation is H = ω0 squared + (ω +∆)m A√kn / A√kn − ω0 squared + (ω −∆) (m+1) <ref:2207.04191#pg10> We set omega1 = 2A/√kn, omega2 = (2m + 1)∆ − ω + ω0, omega3 = ∆ − (2m+1)ω. Then we can obtain the energy eigenvalues given by E± = 1/2 (-omega3 ± q / Ω2(1 + Ωe 2), where m = n - 1 - j, kn = 2 j - n + 1 <ref:2207.04191#pg10>

Appendix B: Derivation of Low-Energy Effective Hamiltonian

The low-energy effective Hamiltonian is expressed as Henp = D↓˜H0↓˜E = −ωe0 squared − ωj +ωe 1−g˜2 a†a, where a (a†) is the bosonic annihilation (creation) operator <ref:2207.04191#pg8> The detailed derivation of Henp is presented in Appendix B <ref:2207.04191#pg8> The ground state energy is approximately equal to the mean-field energy in Eq. (9) <ref:2207.04191#pg9>

References

[1] S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, 2011) <ref:2207.04191#pg10>

[2] W. H. Zurek, U. Dorner, and P. Zoller, Dynamics of a quantum phase transition, Phys. Rev. Lett.

1. Use quantum Fisher information (QFI) to characterize Quantum Phase Transitions (QPTs). This allows AI systems to quantify the abrupt change of the ground state in the vicinity of a critical point, which serves as an indicator for QPTs [41, 51–53].

2. Implement a measurement scheme based on the observable given by equation (18) to obtain parameter information via the error propagation formula, specifically calculating [46] Fg˜ = (∂g˜hσxi)2 / (1-hσxi2) for precision estimation.

3. Design quantum sensing systems where the amplitude of oscillation increases as the coupling strength ˜g approaches the critical point [6] to maximize precision, by controlling the longitudinal interaction ∆ instead of being confined to its vicinity.

4. Develop AI models capable of distinguishing between different regimes based on the influence of ∆ on excitation number: "for-ω < ∆ < 0, ng can be described by Eq. (8), however, it is different for-ω < ∆ < 0 and 0 < ∆ < ω."

5. Create predictive models that forecast the behavior of macroscopic excitations in bath spins by analyzing how [14] becomes Eq. (6) under the condition of ∆ = 0 versus its behavior when "for 0 < ∆ < ω, the corresponding results are different."