Fidelity mechanics: analogues of the four thermodynamic laws and Landauer's principle

Fidelity mechanics is formalized as a framework to investigate quantum critical phenomena in quantum many-body systems. This is achieved by introducing fidelity temperature to properly quantify quantum fluctuations, which, together with fidelity entropy and fidelity internal energy, constitute three basic state functions in fidelity mechanics, thus enabling us to formulate analogues of the four thermodynamic laws and Landauer's principle at zero temperature. Fidelity flows are defined and may be interpreted as an alternative form of renormalization group flows. Thus, both stable and unstable fixed points are characterized in terms of fidelity temperature and fidelity entropy: divergent fidelity temperature for unstable fixed points and zero fidelity temperature and (locally) maximal fidelity entropy for stable fixed points. In addition, an inherently fundamental role of duality is clarified, resulting in a canonical form of the Hamiltonian in fidelity mechanics. Dualities, together with symmetry groups and factorizing fields, impose the constraints on a fidelity mechanical system, thus shaping fidelity flows from an unstable fixed point to a stable fixed point. A detailed analysis of fidelity mechanical state functions is presented for the quantum XY model, the transverse field quantum Ising chain in a longitudinal field, the spin-$1/2$ XYZ model and the XXZ model in a magnetic field.