Criticality, phase transitions and closing bandgap

[thephystudent]

At critical points, when correlation lengths diverge near a phase transition, people often say that the spectral gap closes, which is to my understanding just the energy difference between the ground state and the first excited state (the first two eigenvalues of the hamiltonian).

How should we interpret this? The typical toy models for phase transitions are spin chains as the Ising model. But I would associate 'gap' primarily with the gap between the valence and the conduction band for electrons in a solid. Does this bandgap for electrons close at some kind of phase transition? If so, between which phases? And how can level repulsion between the bands vanish?

Thanks

 

[thephystudent]

Aha, the gap is used as an order parameter for e.g. superconductivity, it does not have to close for all phase transitions?

 

[ZapperZ]

At critical points, when correlation lengths diverge near a phase transition, people often say that the spectral gap closes, which is to my understanding just the energy difference between the ground state and the first excited state (the first two eigenvalues of the hamiltonian).

How should we interpret this? The typical toy models for phase transitions are spin chains as the Ising model. But I would associate 'gap' primarily with the gap between the valence and the conduction band for electrons in a solid. Does this bandgap for electrons close at some kind of phase transition? If so, between which phases? And how can level repulsion between the bands vanish?

Thanks

Aha, the gap is used as an order parameter for e.g. superconductivity, it does not have to close for all phase transitions?

What phase transition are you talking about here?

When I read your first post, it appears that you're talking about the transition between the insulating phase and the metallic phase ("... the spectral gap closes ... ", "... bandgap for electrons... "). But then you bring out order parameter for a superconductor.

You're all over the place here and I'm confused.

Zz.

 

[thephystudent]

What phase transition are you talking about here?

When I read your first post, it appears that you're talking about the transition between the insulating phase and the metallic phase ("... the spectral gap closes ... ", "... bandgap for electrons... "). But then you bring out order parameter for a superconductor.

You're all over the place here and I'm confused.

Zz.

Yes, that's probably because was a bit confused myself. I was kind of trying to reconcile some things that I sometime picked up at different occasions. So at first I thought that phase transitions are always associated with some closing of a gap in the spectrum. But than I realized that I was mistaken and the gap is specifically used as an order parameter at superfluid-normal phase transitions: a gap in the spectrum for a superfluid means that energy cannot dissipate trough small fluctuations?

So what is the precise difference between this 'superfluid gap' and the band-gap of materials?
Since you bring up the transition between insulating and metallic phase, what is the order parameter there and does it have to do anything with the band structure? And what parameter do you tune trough that transition, temperature? I do remember hearing about cold atom experiments about the transition between mott-insulating and superfluid phase by tuning the optical lattice or so.

Thanks for your input.

 

[ZapperZ]

Yes, that's probably because was a bit confused myself. I was kind of trying to reconcile some things that I sometime picked up at different occasions. So at first I thought that phase transitions are always associated with some closing of a gap in the spectrum. But than I realized that I was mistaken and the gap is specifically used as an order parameter at superfluid-normal phase transitions: a gap in the spectrum for a superfluid means that energy cannot dissipate trough small fluctuations?

Again, WHAT type of phase transition?

You should know that there are SEVERAL species of "phase transition". I already mentioned the metal-insulator transition, but there is the thermodynamic phase transition, the structural phase transition, the superconducting phase transition, etc... etc. And at quantum critical points, there are even quantum phase transition.

So which one are you specifically referring to here?

So what is the precise difference between this 'superfluid gap' and the band-gap of materials?

This is whole chapters of a book!

Since you bring up the transition between insulating and metallic phase, what is the order parameter there and does it have to do anything with the band structure? And what parameter do you tune trough that transition, temperature? I do remember hearing about cold atom experiments about the transition between mott-insulating and superfluid phase by tuning the optical lattice or so.

Thanks for your input.

There are at least half a dozen different issues here in this one paragraph alone!

If you want to learn about metal-insulator transition, ranging from the Mott-Hubbard model all the way to the Zhang-Rice singlet state, read this review:

http://webhome.phy.duke.edu/~baranger/articles/strong_cor/MItransitions_rmp.pdf

Otherwise, I don't think I can handle this many different things all at once, at least, not on here.

Zz.

 

[thephystudent]

You should know that there are SEVERAL species of "phase transition". I already mentioned the metal-insulator transition, but there is the thermodynamic phase transition, the structural phase transition, the superconducting phase transition, etc... etc. And at quantum critical points, there are even quantum phase transition.

Well, that was not the answer I was expecting, but a useful one for sure. I knew there was a difference between first and second (or higher) order phase transitions (leaving aside the topological ones), but thought criticality referred to all of them. I should probably review my statistical mechanics course :) . Skimming trough it, the things I was mainly referring to with phase transitions were Landau theory and the corresponding Landau-wilson model. I'm a bit surprised this is all so complex, especially this metal-insulator transition; because of the existence of universality classes I thought there was kind of a clear classification and way of handling all of them.
Regarding my original question on the gap, I picked up that in dissipative systems it is the liouvillian gap that replaces the hamiltonian gap, but that was probably in a very specific context then. I will look at that in more detail myself.

 

[thephystudent]

Aha, I found the solution. The specific part on superconductors and so on was a bit confusing I admit, the closing of a spectral gap is generic for phase transitions.

That is, for a first order phase transition, the gap closes only exactly at the critical value, and opens at both sides. For a (symmetry breaking) second order transition, the gap is open on one side, and closed on the other.

The point is that this spectral gap is meant only for the whole many-body system, in fact after taking the thermodynamic limit first.
This is fundamentally different to a 'band gap', which is a gap in the spectrum of individual electrons.