Exact scaling of geometric phase and fidelity susceptibility and their breakdown across the critical points

Abstract

It was shown via numerical simulations that geometric phase (GP) and fidelity susceptibility (FS) in some quantum models exhibit universal scaling laws across phase transition points. Here we propose a singular function expansion method to determine their exact form across the critical points as well as their corresponding constants. For the models such as anisotropic XY model where the energy gap is closed and reopened at the special points ($k_0=0,\pi$), scaling laws can be found as a function of system length $N$ and parameter deviation $\lambda -{\lambda }_c$ (where ${\lambda }_c$ is the critical parameter). Intimate relations for the coefficients in GP and FS have also been determined. However in the extended models where the gap is not closed and reopened at these special points, the scaling as a function of system length $N$ breaks down. We also show that the second order derivative of GP also exhibits some intriguing scaling laws across the critical points. These exact results can greatly enrich our understanding of GP and FS in the characterization of quantum phase transitions.

arXiv comment6 pages, 2 figures

References in corpus (56)

• Topological Insulators
• Topological insulators and superconductors
• Berry Phase Effects on Electronic Properties
• Classification of topological insulators and superconductors in three spatial dimensions
• Entanglement in quantum critical phenomena
• Topological insulators and superconductors: ten-fold way and dimensional hierarchy
• Scaling of Entanglement close to a Quantum Phase Transitions
• Entanglement in a simple quantum phase transition
• Direct Measurement of the Zak phase in Topological Bloch Bands
• Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms
• Ground state overlap and quantum phase transitions
• Geometric quantum computation with NMR
• Geometric Manipulation of Trapped Ions for Quantum Computation
• Decay of Loschmidt Echo Enhanced by Quantum Criticality
• Fidelity approach to quantum phase transitions
• Classical Correlation and Quantum Discord in Critical Systems
• Fidelity, dynamic structure factor, and susceptibility in critical phenomena
• Non-Abelian gauge potentials for ultra-cold atoms with degenerate dark states
• Quantum critical scaling of the geometric tensors
• Quantum Phase Transitions and Bipartite Entanglement
• Entanglement and the Phase Transition in Single Mode Superradiance
• Entanglement and quantum phase transition in the extended Hubbard model
• Local entanglement and quantum phase transition in the Hubbard model
• Quantum discord and quantum phase transition in spin chains
• Geometric Quantum Mechanics
• Observation of Berry's Phase in a Solid State Qubit
• Detection of Geometric Phases in Superconducting Nanocircuits
• Unconventional geometric quantum computation
• Finite-Size Scaling Exponents of the Lipkin-Meshkov-Glick Model
• Scaling of geometric phases close to quantum phase transition in the XY chain
• Geometric Quantum Computation
• Three-spin interactions in optical lattices and criticality in cluster Hamiltonians
• Entanglement, quantum phase transition and scaling in XXZ chain
• Quantum and classical thermal correlations in the XY spin-1/2 chain
• Geometric phases and criticality in spin chain systems
• Ground-State Fidelity and Bipartite Entanglement in the Bose-Hubbard Model
• Mixed-state fidelity and quantum criticality at finite temperature
• Majorana zero modes in a quantum Ising chain with longer-ranged interactions
• Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model
• Intrinsic relation between ground-state fidelity and the characterization of a quantum phase transition
• Quantum fidelity and quantum phase transitions in matrix product states
• Fidelity and Quantum phase transition for the Heisenberg chain with the next-nearest-neighbor interaction
• Quantum Monte Carlo simulations of fidelity at magnetic quantum phase transitions
• Large N Scaling Behavior of the Lipkin-Meshkov-Glick Model
• Topological Characterization of Extended Quantum Ising Models
• Quantum correlations in topological quantum phase transitions
• Critical Property of Geometric Phase in the Dicke Model
• Fidelity analysis of topological quantum phase transitions
• Effect of disorder on quantum phase transitions in anisotropic XY spin chains in a transverse field
• Measurement of geometric phase for mixed states using single photon interferometry
• Fidelity susceptibility made simple: A unified quantum Monte Carlo approach
• The Euler Number of Bloch States Manifold and the Quantum Phases in Gapped Fermionic Systems
• Scaling of Berry's Phase Close to the Dicke Quantum Phase Transition
• Geometric phases and quantum phase transitions
• Quench dynamics and ground state fidelity of the one-dimensional extended quantum compass model in a transverse field
• Fidelity in topological superconductors with end Majorana fermions

Cited by in corpus (5)

• Quantum enhanced measurements without entanglement
• Fidelity susceptibility of the anisotropic $XY$ model: The exact solution
• Anomalous relaxation and multiply time scales in the quantum $XY$ model with boundary dissipation
• Scaling of Quantum Geometry Near the Non-Hermitian Topological Phase Transitions
• Geometric phase and topological phase diagram of the one-dimensional $XXZ$ Heisenberg spin chain in a longitudinal field