We analyze the topological deformations of the ground state manifold of a quantum spin-1/2 in a magnetic field $\mathbit{H}=H(sin\ensuremath{\theta}cos\ensuremath{\phi},sin\ensuremath{\theta}sin\ensuremath{\phi},cos\ensuremath{\theta})$ induced by a coupling to an ohmic quantum dissipative environment at zero temperature. From Bethe ansatz results and a variational approach, we confirm that the Chern number associated with the geometry of the reduced spin ground state manifold is preserved in the delocalized phase for $\ensuremath{\alpha}<1$. We report a divergence of the Berry curvature at ${\ensuremath{\alpha}}_{c}=1$ for magnetic fields aligned along the equator $\ensuremath{\theta}=\ensuremath{\pi}/2$. This divergence is caused by the complete quenching of the transverse magnetic field by the bath associated with a gap closing that occurs at the localization Kosterlitz-Thouless quantum phase transition in this model. Recent experiments in quantum circuits have engineered nonequilibrium protocols to access topological properties from a measurement of a dynamical Chern number defined via the out-of-equilibrium spin expectation values. Applying a numerically exact stochastic Schr\"odinger approach we find that, for a fixed field sweep velocity $\ensuremath{\theta}(t)=vt$, the bath induces a crossover from (quasi)adiabatic to nonadiabatic dynamical behavior when the spin bath coupling $\ensuremath{\alpha}$ increases. We also investigate the particular regime $H/{\ensuremath{\omega}}_{c}\ensuremath{\ll}v/H\ensuremath{\ll}1$ with large bath cutoff frequency ${\ensuremath{\omega}}_{c}$, where the dynamical Chern number vanishes already at $\ensuremath{\alpha}=1/2$. In this regime, the mapping to an interacting resonance level model enables us to analytically describe the behavior of the dynamical Chern number in the vicinity of $\ensuremath{\alpha}=1/2$. We further provide an intuitive physical explanation of the bath-induced breakdown of adiabaticity in analogy to the Faraday effect in electromagnetism. We demonstrate that the driving of the spin leads to the production of a large number of bosonic excitations in the bath, which strongly affect the spin dynamics. Finally, we quantify the spin-bath entanglement and formulate an analogy with an effective model at thermal equilibrium.
We investigate the topological phase transition of Kitaev spin liquid in an external magnetic field by calculating the Berry curvature and the Fubini-Study metric. Employing Jordan-Wigner transformation and effective perturbative theory to transform the Hamiltonian into fermionic quadratic form, the Berry curvature is calculated by choosing the effective magnetic field as the parameter, and we find that the
xy component of the Berry curvature has the same behavior around the critical lines with the phase diagram. Especially, we relate the Berry curvature with the derivative of effective magnetic susceptibility which can be regarded as the signature of topological phase transition. Moreover, we related the second nonlinear susceptibility with the interband Berry connection. Then we analytically calculate the generalized Berry curvature in the mixed state called the mean Uhlmann curvature, which can be related with the spectral function. The curves where the Uhlmann curvature changes temperature with different coupling constants reveal that it will have a peak at
T=0.2K when adjusting
Jx from 0.2 to 0.3 with
Jy=0.25, in which the first-order derivative is discontinuous, revealing the characteristic of thermodynamic phase transition from the
B phase quantum spin liquid state to the magnetic ordered state. Last, we analytically calculate the quantum geometric tensor in the effective magnetic-field space whose imaginary part is the Berry curvature and the real part is the Fubini-Study metric, and we find that the
zz component of the Fubini-Study metric and the phase diagram are highly correlated, which will peak at the cross point of three phases, and then the Fubini-Study metric is extended to the finite temperature, called the Bures metric, which has the same behavior of the Uhlmann curvature at
T=0.2K in
B phase.
We study the magnetic susceptibility of 1D quantum XY model, and show that when the temperature approaches zero, the magnetic susceptibility exhibits the finite-temperature scaling behavior. This scaling behavior of the magnetic susceptibility in 1D quantum XY model, due to the quantum-classical mapping, can be easily experimentally tested. Furthermore, the universality in the critical properties of the magnetic susceptibility in quantum XY model is verified. Our study also reveals the close relation between the magnetic susceptibility and the geometric phase in some spin systems, where the quantum phase transitions are driven by an external magnetic field.
Consider a d-dimensional antiferromagnet with a quantum disordered ground state and a gap to bosonic excitations with nonzero spin. In a finite external magnetic field, this antiferromagnet will undergo a phase transition to a ground state with nonzero magnetization, describable as the condensation of a dilute gas of bosons. The finite-temperature properties of the Bose gas in the vicinity of this transition are argued to obey a hypothesis of zero scale-factor universality for d2, with logarithmic violations in d=2. Scaling properties of various experimental observables are computed in an expansion in \ensuremath{\epsilon}=2-d, and exactly in d=1.
We analyze the Bures metric over the manifold of thermal density matrices for systems featuring a zero temperature quantum phase transition. We show that the quantum critical region can be characterized in terms of the temperature scaling behavior of the metric tensor itself. Furthermore, the analysis of the metric tensor when both temperature and an external field are varied, allows one to complement the understanding of the phase diagram including crossover regions which are not characterized by any singular behavior. These results provide a further extension of the scope of the metric approach to quantum criticality.
From the Aharonov-Bohm effect to general relativity, geometry plays a central role in modern physics. In quantum mechanics, many physical processes depend on the Berry curvature. However, recent advances in quantum information theory have highlighted the role of its symmetric counterpart, the quantum metric tensor. In this paper, we perform a detailed analysis of the ground state Riemannian geometry induced by the metric tensor, using the quantum
XY chain in a transverse field as our primary example. We focus on a particular geometric invariant, the Gaussian curvature, and show how both integrals of the curvature within a given phase and singularities of the curvature near phase transitions are protected by critical scaling theory. For cases where the curvature is integrable, we show that the integrated curvature provides a new geometric invariant, which like the Chern number characterizes individual phases of matter. For cases where the curvature is singular, we classify three types (integrable, conical, and curvature singularities) and detail situations where each type of singularity should arise. Finally, to connect this abstract geometry to experiment, we discuss three different methods for measuring the metric tensor: via integrating a properly weighted noise spectral function or by using leading-order responses of the work distribution to ramps and quenches in quantum many-body systems.