Constant of Motion Identifying Excited-State Quantum Phases

Abstract

We propose that a broad class of excited-state quantum phase transitions (ESQPTs) gives rise to two different excited-state quantum phases. These phases are identified by means of an operator Ĉ, which is a constant of motion in only one of them. Hence, the ESQPT critical energy splits the spectrum into one phase where the equilibrium expectation values of physical observables crucially depend on this constant of motion and another phase where the energy is the only relevant thermodynamic magnitude. The trademark feature of this operator is that it has two different eigenvalues AE1, and, therefore, it acts as a discrete symmetry in the first of these two phases. This scenario is observed in systems with and without an additional discrete symmetry; in the first case, Ĉ explains the change from degenerate doublets to nondegenerate eigenlevels upon crossing the critical line. We present stringent numerical evidence in the Rabi and Dicke models, suggesting that this result is exact in the thermodynamic limit, with finite-size corrections that decrease as a power law.

Dynamical and excited-state quantum phase transitions in collective systems

“…gs < < , at which a ESQPT takes place. For example, all the energy levels become two-fold degenerate below this critical energy, , in the TL [41,43], and therefore broken-symmetry equilibrium or steady states are allowed in this region [69][70][71]. Contrarily, for > all degeneracies are broken and thus broken-symmetry equilibrium states are no longer possible.…”

“…For the purposes of this paper, the trademark of the phase where > and < is the existence of an operator Ĉ that becomes a constant of motion in the TL, proposed in [43]. We have that Ĉ, ˆ = 0, where ˆ is the projector onto the eigenspace with energy , ∀ < , whereas Ĉ, ˆ ≠ 0, ∀ > .…”

“…This theory is proposed in [42], for which this work serves as a companion paper. There, it is shown that both kinds of DPTs are explained by the behavior of an operator, Ĉ, that is a constant of motion only in one of the two phases separated by the critical energy of the ESQPT [43]. Below this critical energy, Ĉ commutes with the energy projectors in the TL, the dynamical order parameter characteristic of DPTs-I can be different from zero, and the main mechanism leading to non-analytical points in the return probability is precluded.…”

Exceptional Spectral Phase in a Dissipative Collective Spin Model

“…Since the results in panel (b) only account for a pair of degenerate eigenvalues, we come back to the case with = 0.5 to determine the boundary of the ESP. Following [33], we perform a finite-size scaling consisting in: (i) select a given bound for the distance , such that if < we consider that the two eigenvectors | , and | + 1, have coalesced; (ii) identify a precursor of the critical eigenvalue, * , ( , ), as the doublet ( , + 1) with largest real parts fulfilling > , and (iii) study how this precursor changes with the system size. Hence, in panel (c) of Fig.…”

Exponential precision by reaching a quantum critical point

“…The frequency ratio acts as an effective system size, and finite-size critical exponents can also be defined in this case. These fully-connected systems admit a simple description in terms of an effective bosonic mode whose potential depends on a rescaled and dimensionless coupling strength g. These models constitute a suitable test-bed for the exploration of different aspects of quantum critical phenomena [42,43,45,46,[48][49][50][51][52][53][54][55][56][57][58][59][60][61][62]. Within the normal phase 0 ≤ g ≤ 1 and in the thermodynamic limit, the effective Hamiltonian describing these systems can be written as [1,35,60]…”