Question

I want to know that, for any spin-chains in condensed matter Physics like X-Y spin model, Kitaev model 1-D only in which degenerate point is critical point. Is it necessary that the critical points has to be degeneracy points (gapless)? It means that at degeneracy point system becomes gapless it also shows Quantum Phase Transition in the spin-chain. In the point of view of quantum information, at the critical point (degenerate point) concurrence is maximum shows quantum phase transition in the spin-chain. But my question is what if concurrence is maximum at not the degenerate points but at some other points. Then how will we find critical points and quantum phase transition, also if concurrence is maximum at non-degenerate. Does it means our Hamiltonian has critical points which is gapped.? My question is it necessary that Generate points has to be the critical points in any spin-chain condensed matter physics. Critical point means that Quantum phase transition occurred at that point and degenerate point means gapless system. Concurrence is a measure of entanglement Which also can be use to study QPT in spin-chains. Like for 1d Kitaev model at magnetic field is zero defined as critical point which also a degenerate point. So you will see maximum value of concurrence at that point. But what am saying in some cases other than Kitaev model at degenerate point concurrence value is zero but some other points which is non- degenerate point is shown to be maximum value. So can someone please clear that?

Answer 1

Critical points in a spin-chain may be identified by a gap closing, but they do not always coincide with energy degeneracy points. Concurrence, a measure of entanglement, can peak at non-degenerate points without contradicting the presence of a quantum phase transition. One must consider the system's entire ground state and excitation spectrum to identify critical points.

Step-by-step explanation:

You're asking whether critical points within a spin-chain model in condensed matter physics, which signal a quantum phase transition (QPT), must necessarily be points of energy degeneracy or 'gapless' points. Your question also relates to the concept of concurrence, a measure of quantum entanglement used to study QPTs, and its behavior at non-degenerate points.

It's important to clarify that while energy degeneracy at a critical point often correlates with a gap closing, this is not a strict necessity in all models. Critical points can in principle occur also at gapped phases, where the system remains 'gapful'. Concurrence can indeed be maximized at points that are not energy degenerate, but this does not invalidate them as critical points; it simply indicates that the system has critical behavior characterized by something other than energy degeneracy. Thus, identifying critical points and QPTs may involve looking at more than just energy gaps or concurrence—they might require a broader analysis of the system's ground state and its excitation spectrum.

It is important to use both analytical methods and numerical simulations, like exact diagonalization or tensor network methods, to study these non-trivial critical points. The Hamiltonian of the system can pass through various phases, and the identification of a phase transition could involve abrupt changes in physical observables, scaling behavior, or fidelity susceptibility, rather than just degeneracy or concurrence.