Classification of excited-state quantum phase transitions for arbitrary number of degrees of freedom

Abstract

Classical stationary points of an analytic Hamiltonian induce singularities of the density of quantum energy levels and their flow with a control parameter in the system's infinite-size limit. We show that for a system with f degrees of freedom, a non-degenerate stationary point with index r causes a discontinuity (for r even) or divergence (r odd) of the (f −1) th derivative of both density and flow of the spectrum. An increase of flatness for a degenerate stationary point shifts the singularity to lower derivatives. The findings are verified in an f = 3 toy model.

Monodromy in Dicke superradiance

“…(19). The largest time derivative takes place at p = 1, which corresponds to the equator of the atomic Bloch sphere (z = 0), and a similar conclusion, based on Eqs.…”

Excited-state quantum phase transitions and their manifestations in an extended Dicke model

“…, 1, and a local quadratic minimum in V would yield r = 0. The Hamiltonian (6) can be integrated analytically [8], leading to the formulaē…”

“…A natural extension of the qpt into excited spectra is the Excited-State Quantum Phase Transition (esqpt) [3,4,5,6,7,8]. It shows up as a singularity that begins in the qpt critical point λ c at ground-state energy and continues into the (λ × energy) plane along the curve called critical borderline.…”

“…(the constant C 4 > 0 can be expressed explicitly [8]). Formula (9) suggests that the smooth level density is discontinuous in the ⌈g − 1⌉ derivative.…”

“…From this expression it follows that the types of nonanalyticity inρ(E, λ) andφ(E, λ) are the same, providedρ(E, λ) 0 φ (E, λ). More detailed discussion can be found in [8]. The situation may be slightly different for systems with f = 1, where the level density itself exhibits discontinuities at esqpt, therefore formula (13) cannot be used.…”