# What determines a system to experience crossover or phase transition?

## Main Question or Discussion Point

More specifically, consider a Hamiltonian with a changeable parameter a. When changing a, the ground state of the system will change. In some cases, one phase crossovers to another, like in other cases, there's a phase transition. What factor determines the difference? Actually I am now working on a specific model, the Dicke model. And I want to consider the above question in this and related model. So before that, I want some general guidances about the question to get to the point quicker. Thx.

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Redbelly98
Staff Emeritus
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Are you talking about where there is an avoided energy level crossing, due to off-diagonal terms in the Hamiltonian?

The theory is covered in the first 4 pages of a paper by Rubbmark et al.:
Phys. Rev. A 23, 3107–3117 (1981): "Dynamical effects at avoided level crossings: A study of the Landau-Zener effect using Rydberg atoms"

At the risk of oversimplifying things, it is determined by the rate at which a is changed. For a fast da/dt, the system will cross over; for a slow da/dt, it does not cross over.

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More specifically, consider a Hamiltonian with a changeable parameter a. When changing a, the ground state of the system will change. In some cases, one phase crossovers to another, like in other cases, there's a phase transition. What factor determines the difference? Actually I am now working on a specific model, the Dicke model. And I want to consider the above question in this and related model. So before that, I want some general guidances about the question to get to the point quicker. Thx.
A phase transition is characterised by non-analytic (smooth or continuous) changes. Crossovers will not have this non-analytic behaviour.

That, at least, is the definition. However, in practice, several things complicate matters.

1. Finite size. Truly singular behaviour can only occur in thermodynamic systems, with infinite volume/particle number, etc. Real systems, especially those used in experiments with cold atoms or (for you?) exciton/polariton systems, are finite and sometimes very finite (N is O(100)). This will prevent singularities from being directly observed, but sometimes some work based on finite-size scaling can recover the theoretical limit.

2. Suppose that the discontinuity is in a high order (high derivative). This will again usually end up being obscured by experimental noise, etc.

3. It's usually *very* hard to tune a system to be actually on the phase transition itself, which will tend again to round off singularities.

In real life these things tend to be the basis for some very fervent discussions (see BCS/BEC literature, or cuprates, for examples of where decades of arguments were essentially on whether the observed phenomenon were really phase transitions).

If you don't mind me asking, where are you working/studying?

What I am talking is about theories under thermodynamic limit, so no finite size problem or experimental difficulties. I got some new thoughts today, welcome commence.
I agree with Redbelly98 that it's about energy level crossing or avoiding. Crossing corresponds to a transition, avoiding to a crossover. The avoiding, in many cases, comes from the second-order perturbation, which mixes the original two “near-energy” state and opens a gap. When the mixing is forbidden, a transition happens. There're several ways to forbid it. The most common factor is that the original two states are of different symmetries. This kind of transition is continuous one. Another factor may be topological index, which I have only heard about and know little. Thanks a lot.

You are completely correct that level crossings tend to cause phase transitions. There is a very nice formalism which deals with this point of view, using the concept of quantum fidelity (basically, overlap between ground states with different parameters). Review paper: http://prl.aps.org/abstract/PRL/v99/i9/e095701