Geometric Critical Exponents in Classical and Quantum Phase Transitions

Abstract

We define geometric critical exponents for systems that undergo continuous second order classical and quantum phase transitions. These relate scalar quantities on the information theoretic parameter manifolds of such systems, near criticality. We calculate these exponents by approximating the metric and thereby solving geodesic equations analytically, near curvature singularities of two dimensional parameter manifolds. The critical exponents are seen to be the same for both classical and quantum systems that we consider, and we provide evidence about the possible universality of our results.

1 + 15 pages, LaTeX. Expanded version with several new examples added. Main results and conclusions are unchanged. Version to appear in Phys. Rev. E

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• Complexity and information geometry in spin chains
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• Classical analog of the quantum metric tensor
• The geometry of RN-AdS fluids
• Thermodynamic curvature of the Schwarzschild-AdS black hole and Bose condensation
• Measure of the density of quantum states in information geometry and its application in the quantum multi-parameter estimation
• Demystifying the Holographic Mystique
• Determination of the Critical Manifold Tangent Space and Curvature with Monte Carlo Renormalization Group
• Hidden Symmetries, the Bianchi Classification and Geodesics of the Quantum Geometric Ground-State Manifolds
• Geometry of AdS black hole thermodynamics in extended phase space
• Classical description of the parameter space geometry in the Dicke and Lipkin-Meshkov-Glick models
• Quantum geometry in many-body systems with precursors of criticality
• Conformal Fisher information metric with torsion
• Generalized quantum geometric tensor for excited states using the path integral approach
• Geometrical aspect of susceptibility critical exponent