# Is Bose-Einstein condensation a quantum phase transition?

Classical phase transitions are driven by thermal fluctuations which die out at $$T = 0$$, hence classical phase transitions don't happen at zero temperature. Quantum phase transitions are driven by quantum fluctuations which prevail at $$T = 0$$ and thus can lead to macroscopic change in the system (quantum phase transition). This is the case in the superfluid-Mott insulator transition in the Bose-Hubbard model, for example, which is due to interactions.

Bose-Einstein condensation (BEC) is a consequence of quantum statistics, it is not directly related to any microscopic parameter in the hamiltonian. A real phase transition (not a crossover) from classical thermal gas to a condensate happens at non-zero temperature. This suggests that BEC is a classical phase transition. Contrary, in Subir Sachdev's book I find a graph (Fig 16.4) that suggests BEC is a quantum phase transition like any other, with a quantum critical point at $$T = 0$$ and $$\mu<0$$. As a specific example, we could consider a weakly-interacting Bose-gas in three dimensions.

In other words, what is the defining feature of a quantum phase transition: Either it happens only at $$T = 0$$ or it happens not only at $$T > 0$$ but also at $$T = 0$$?

The terminology is an absolute mess. The way I think about it, the defining feature of a quantum phase transition is:

A quantum phase transition is a phase transition that takes place in a system that lives in $$d$$-dimensional space, but whose universal properties are described by a $$(d+1)$$-dimensional field theory.

(If your immediate reaction is "The terminology 'quantum phase transition' does not at all convey that idea!" ... then you're right.)

Consider an infinitely large quantum system (at any temperature, zero or otherwise) that is near, but not necessarily at, a phase transition. Its equilibrium statistical properties are described by fields on a $$(d+1)$$-dimensional space that is given by the Cartesian product of the (infinitely large) actual spatial manifold $$M$$ times a compactified (i.e. periodic, and so finite) dimension called the "imaginary time" dimension, whose "circumference" (periodicity) is the inverse temperature $$\beta$$. Away from the transition, the quantum effects are certainly important even at finite temperature, as the corresponding classical system is described by a completely different $$d$$-dimensional field theory with no imaginary time direction.

However, very close the transition, the correlation length $$\xi$$ diverges. In particular, if the temperature is greater than zero and so $$\beta$$ is finite, then as you get very close to the transition, the correlation length inevitably eventually becomes much larger than $$\beta$$. In this case, the field value gets "locked" to a single value as you circle around the imaginary time direction, and that degree of freedom effectively drops out. So the dynamically fluctuating fields only live on the infinitely large spatial manifold $$M$$, just as in the classical case. So (only when you are extremely close to the transition), the imaginary time dimension becomes unimportant and the quantum statistical properties become identical to that of the corresponding classical critical system. That's why phase transitions of quantum systems at finite temperature are called "classical"; they are "actually" quantum, but the quantum effects do not affect the universal properties that are captured by field theory.

But for a quantum phase transition that occurs at zero temperature, the imaginary time period $$\beta$$ is infinite, so it actually isn't periodic at all; the imaginary time direction is simply another infinitely large dimension. If the dynamical exponent $$z = 1$$, which is often the case, then the imaginary time dimension is more or less on the same footing as the $$d$$ spatial dimensions, up to unimportant constants.

That's why the Mermin-Wagner theorem sometimes allows spontaneous symmetry breaking at one lower spatial dimension than usual at zero temperature (e.g. spontaneous breaking of continuous symmetries in $$d = 2$$ spatial dimensions); the infinitely large quantum imaginary time dimension effectively gives you a third dimension to play with. In a completely classical system, nothing unusual would happen at zero temperature, and no new types of spontaneous symmetry breaking would become allowed as they do in the quantum case.

What's really going on here under the hood is that there are three relevant length scales that all formally go to infinity at a zero-temperature quantum phase transition: the spatial system size $$L$$, the inverse temperature $$\beta$$, and the correlation length $$\xi$$. In any real system, none of these three will be literally infinite, and which field theory accurately describes the statistical physics depends on their relative ratios. In the regime $$L, \beta \gg \xi$$, it's a $$(d+1)$$-dimensional quantum field theory and a quantum phase transition; in the regime $$L \gg \xi \gg \beta$$ it's a $$d$$-dimensional classical field theory and a classical phase transition; and in the regime $$\xi \geq L$$ you can't really use field theory at all, and you're screwed.

• So is a BEC a quantum phase transition? – SuperCiocia Nov 27 '19 at 4:03
• @SuperCiocia I would say "no" according to the standard terminology, while simultaneously acknowledging that the standard terminology is really stupid, because any reasonable definition of the term "quantum phase transition" should include the BEC transition. – tparker Nov 27 '19 at 4:23
• This is a fascinating way of looking at it. If true, half the things I used to believe are wrong. – SuperCiocia Nov 27 '19 at 4:28
• @SuperCiocia Then feel free to upvote – tparker Nov 27 '19 at 4:30
• I strongly disagree with this definition. The simplest counter-example is a quantum phase transition with a dynamical exponent $z\neq 1$. And in the end, saying that the transition only happens for infinitely large $\beta$ goes back to the standard definition: a phase transition of a quantum system at $T=0$... – Adam Nov 27 '19 at 10:30

As I understand it, the terminology "quantum phase transition" is used for a change in the ordering/structure of a system at absolute zero temperature (therefore also zero entropy) taking place owing to a changing external parameter such as volume or applied field or something like that. BEC in three dimensions is not like that. It is a change taking place at some non-zero temperature, as a parameter such as density or temperature is changed. It has a critical point at a finite temperature. The order parameter starts to grow abruptly as the temperature passes through the critical point. Therefore it is not a quantum phase transition.

(In a preliminary treatment of BEC one may take as order parameter the ground state population. In more advanced treatments one uses other things such as measures of correlations.)

Finally, a remark on terminology. I think it is a pity the terminology got formed the way it did, because it suggests that there is something extra-quantum about systems at zero temperature, when this is not true. Systems are quantum no matter what their temperature, and quantum physics often has something useful to add to the discussion of phase transitions at any temperature (BEC being a particularly telling example). Also, when you read statements that phase transitions are "driven by" fluctuations I often wonder in what sense the word "driven" is being used. For a quantum phase transition, the change in the system's ground state is owing to the change in the Hamiltonian, so it is not "driven" in the sense of "caused" by anything other than that. So in what sense is the word "driven" being used? (Thus my answer finishes with a query back to the questioner, but I hope others will comment).

• Thanks! I used the word "driven" because i was used to read it in papers and hear it in conference talks but for no good other reason. – Quasilattice Nov 12 '19 at 12:40
• But you could cross the BEC transition by increasing just the density (at constant temperature). Would this count? You could even cook up a scenario in which the interaction strength takes you across the transition (resulting from a shift in Tc). – Quasilattice Nov 12 '19 at 12:52
• @Quasilattice Crossing as a function of density instead of temperature does count. If the system is at non-zero $T$ then this would be a phase transition but not called a quantum phase transition (according to the standard practice which I do not like). For BEC, interestingly, you don't need an interaction strength---except to make sure the system thermalises. I think the use of the word "driven" is to say that the fluctuations influence the details of the physics here quite a lot, especially for metastable case. – Andrew Steane Nov 12 '19 at 13:33
• @Quasilattice makes a very important point. The key distinction between classical and quantum phase transitions is not that the temperature is the quantity being varied, but instead that the transition occurs at finite temperature. In a plot of T vs. (any relevant non-temperature parameter), the phase transition line will generically be diagonal, and you'll get the same universal physics whether you pass through the transition "vertically" (i.e. by varying the temperature) or "horizontally" (i.e. by varying the other parameter at fixed temperature). – tparker Nov 27 '19 at 2:21
• @tparker Thanks; I have modified the opening paragraph to make this clearer. – Andrew Steane Nov 27 '19 at 8:32

When talking about the BEC transition, one usually thinks of the finite-temperature phase transition that happens in 3D for interacting bosons. This transition is in the universality class of the 3D classical XY model, and describes the transition between a normal gas and a BEC. More generally, in $$d\geq 2$$, it is a classical transition between a normal gas and a superfluid.

This, of course, does not qualify as a quantum phase transition, as defined by a transition between two phases of matter that happens at zero-temperature when a non-thermal parameter is varied. (Side note: a transition that happens at finite-temperature when changing a non-thermal parameter does NOT qualify as a quantum phase transition either.)

The transition discussed by Sachdev is indeed a quantum phase transition, between the vacuum (empty system), and a BEC (more generally, in $$d\geq 1$$, between the vacuum and a superfluid). While this is indeed a transition from a non-condensed state to a BEC, it is of a completely different nature than the finite-temperature BEC. First, the critical point is somewhat trivial, since it corresponds to an empty state (with zero chemical potential), and all its properties can be computed somewhat easily. (For instance, the propagator is not renormalized, and is thus the one of a free non-relativistic particle, with dynamical exponent $$z=2$$.) Second, its universality class is a new one, sometimes called the dilute Bose gas universality class. However, it is interesting to consider, because it also describes other quantum phase transitions, both for strongly interacting bosons in the context of the Mott transition with non-conserved density (the so-called generic transition), and a transition in quantum XY spin models in presence of a transverse magnetic field. Finally, this quantum critical point influences the shape of the phase diagram at low enough temperature, and is the terminal point of the critical line of the finite-temperature BEC transition as one lowers the chemical potential.

Bose-Einstein condensation is not a quantum phase transition.

Quantum phase transitions occur only at zero temperature, and become crossovers for any $$T\neq 0$$. In the same way that the paramagnetic-ferromagnetic Ising transition only occurs at zero external field, and becomes a crossover for any $$H\neq 0$$.

Experimentally, of course, you'd only probe the $$T\rightarrow 0$$ limit but you'd still call it a phase transition. Because, why not. So you'd (practically) consider quantum phase transitions happening also at $$T\neq 0$$.

The reasoning behind all this is that there are two energy scales$$^\dagger$$:

• thermal: $$k_{\mathrm{B}} T$$
• quantum: $$\hbar \omega$$

Only when $$T=0$$ the thermal energy scale vanishes completely and the quantum one can dominate and control the dynamics. In the Ising analogy, these correspond to the external field interaction energy $$\mu H$$ and the inter-particle interaction $$J$$ respectively.

Hence, in quantum phase transitions, temperature drops out completely. The transition is then driven by something else. An order parameter. Magnetisation, disorder strength. Interaction strength. Both. Etc.

Classical (temperature-driven) phase transitions are characterised by power-law divergence of several parameters, for example the correlation length $$\xi_\ell$$ and the correlation time $$\xi_t$$:

$$\xi_\ell \propto |T-T_{\mathrm{c}}|^{-\nu},$$ $$\xi_t \propto |T-T_{\mathrm{c}}|^{-\nu'}.$$

We could come up with a characteristic frequency scale by taking $$\omega_c \sim 1/\xi_t$$: $$\omega_c \sim 1/\xi_t \propto |T-T_{\mathrm{c}}|^{\nu'} \rightarrow 0 \quad \mathrm{as} \quad T\rightarrow T_{\mathrm{c}}.$$

The above argument is sometimes used to justify why the quantum energy scale $$\hbar \omega_c$$ is always going to be shadowed by the thermal one in the vicinity of $$T_{\mathrm{c}}$$, thereby not playing any effect. And requiring $$T=0$$ to really be dominant.

Contrary, in Subir Sachdev's book I find a graph (Fig 16.4) that suggests BEC is a quantum phase transition like any other, with a quantum critical point at $$T=0$$ and $$\mu<0$$. As a specific example, we could consider a weakly-interacting Bose-gas in three dimensions.

Bose-Einstein condensation is a non-interacting effect. The fact that a macroscopically occupied ground state occurs despite interactions is more complicated and, I would say, another matter. As you said, BEC is driven by particle statistics only.

I don't have the book but I am assuming the figure you are referring to is the the same as fig. 3 here. In eq. 50 on the same page he has a dependence $$\mu/u_0$$ where $$u_0$$ depends on the interaction strength $$a$$. So in the non-interacting limit $$a\rightarrow 0$$, that goes away and the reasoning stops being applicable.

The chemical potential cannot be thought as the external parameter driving a quantum phase transition as you don't have independent control over it. It is determined by the system itself, as explained here.

$$^\dagger$$: Though I am not aware of situations where these two energy scales directly compete against each other, there are phase diagrams (e.g. unitary Bose gas) where you can go from BEC to thermal Bose gas by varying $$T$$, and from BEC to unitary BEC changing the interaction strength $$a$$. These are all crossovers but still, food for thoughts.

In three dimensions Bose-Einstein condensation happens at finite temperature but in one and two spatial dimensions it happens at $$T=0$$.