Quantum Phase Transitions and Fidelity Susceptibility

Quantum phase transitions occur at zero temperature when a many-body system undergoes a qualitative change in its ground state due to quantum fluctuations. Traditional order parameters and correlation functions capture symmetry breaking, but many models lack a clear local signature. Fidelity, defined as the overlap between ground states at neighbouring values of a control parameter, and its derivative measure, fidelity susceptibility, provide a universal probe of criticality without prior knowledge of an order parameter. Near a critical point, fidelity susceptibility diverges with system size according to universal exponents, revealing the scaling of the quantum metric tensor—a geometric structure that captures the infinitesimal distance in parameter space. Extensions to mixed states introduce Uhlmann curvature and finite-temperature susceptibility, enabling the study of non-equilibrium and thermal transitions. These geometric and information-theoretic tools have been applied across spin, fermion and boson systems, offering insights into universality classes, finite-size effects, and shortcuts to adiabatic dynamics. With growing computational methods and experimental platforms in ultracold atoms, trapped ions and engineered lattices, fidelity susceptibility has become a versatile diagnostic for novel quantum matter and critical phenomena.

Research from Nature Portfolio

Recent studies have introduced Uhlmann curvature as a geometric measure of mixed-state criticality in open quantum systems. By analysing Gaussian non-equilibrium steady states of lattice fermions coupled to local reservoirs, this approach differentiates quantum from classical fluctuations, correlating divergences of curvature with the closing dissipative gap and correlation length. In parallel, exact closed-form expressions for fidelity susceptibility and counterdiabatic Hamiltonian coefficients have been derived for finite-size quantum Ising chains. Summing hyperbolic functions yields analytic scaling laws and critical exponents, while providing explicit counterdiabatic terms to enable rapid adiabatic driving across the quantum critical point. Together, these works deepen the connection between parameter-space geometry and non-equilibrium critical phenomena, and supply practical formulas for finite systems.

Quantum Phase Transitions and Fidelity Susceptibility publication trend

The graph below shows the total number of articles in quantum phase transitions and fidelity susceptibility across all publications each year (not limited to Nature Index journals).

Technical terms

Quantum phase transition: A zero-temperature transformation of a ground state driven by quantum fluctuations rather than thermal energy.

Fidelity: The overlap magnitude between two quantum states as a function of variation in a control parameter.

Fidelity susceptibility: The second derivative of fidelity with respect to a control parameter; quantifies sensitivity to changes and signals criticality.

Quantum metric tensor: The symmetric component of the quantum geometric tensor, defining an infinitesimal distance in the parameter space of Hamiltonians.

Uhlmann curvature: A generalisation of the Berry curvature to mixed states, capturing geometric features of non-equilibrium steady-state transitions.

Counterdiabatic Hamiltonian: An auxiliary term added to drive a system rapidly and adiabatically through a quantum critical point.

Aubry–André–Harper (AAH) model: A one-dimensional lattice with an incommensurate potential displaying extended, localized and critical (multifractal) phases.

References

1. Fidelity Susceptibility Made Simple: A Unified Quantum Monte Carlo Approach. Physical Review X (2015).
2. Uhlmann curvature in dissipative phase transitions. Scientific Reports (2018).
3. The quantum Ising model: finite sums and hyperbolic functions. Scientific Reports (2015).
4. Uhlmann fidelity and fidelity susceptibility for integrable spin chains at finite temperature: exact results. New Journal of Physics (2021).
5. Application of metric space method in quantum information in quasi-periodic systems. Acta Physica Sinica (2024).