^{Addendum B}

Quantum phase transitions: a wrong concept

(September 2003, revised November 2006)

Introduction of "quantum phase transitions" was not legitimate, insofar as it was based on misrepresentation of well established facts. To demonstrate this, the review article "Quantum Phase Transitions" by M. Vojta [cond-mat/0309604] is used here. It is helpful on two reasons. 1. It is very authoritative, for S. Sachdev, who had published the canonical book on quantum phase transitions, "contributed enormously to the writing of this article", and many other authorities also had "illuminating conversations and collaborations". 2. Its author tried to be logical in justifying the new class of phase transitions - and thus made it easier to find where he failed. Our analysis is basically structured as comments on the Vojta’s article.

Excerpt: The [non-quantum] phase transitions … occur at finite temperature; here macroscopic order … is destroyed by thermal fluctuations.

Comment: Those phase transitions are nucleation and growth of new phase in the solid medium of original phase, rather than "destroying" the original phase by thermal fluctuations as assumed by "second-order" (critical phenomena) theory. This fact is a major point of the book.

Excerpt: [Quantum phase transitions take] place at zero temperature. A non-thermal control parameter such as pressure, magnetic field, or chemical composition, is varied to access the transition point. There, order is destroyed solely by quantum fluctuations.
Comment: Since the previous assumption is wrong, the deduction from it hangs in the air. Vibration energy – including quantum – is only a part of crystal free energy. If changing of one of the above control parameters gives rise to a lower free energy, there is no reason for the phase transition not to proceed in the same way at low temperatures as well. In other words, quantum effects may contribute to the free energy of competing phases, but have nothing to do with mechanism of the phase transition: it is still nucleation-growth.

To the number of classifications of solid-state phase transitions, discussed in Chapter 1, one more was added by Vojta: "classical - quantum". How "quantum" phase transitions differ from "classical"? First, the latter need to be defined.

Excerpt: [Classical] phase transitions are traditionally classified into first-order and continuous transitions. At first-order transitions the two phases co-exist at the transition temperature – examples are ice and water at 0 C, or water and steam at 100 C.
Comment: It is not accidental that the chosen examples of first-order phase transitions are not solid-to-solid, even though "quantum" phase transitions are. The reason becomes evident since all "classical" solid-state transitions are declared "continuous" and a "critical phenomenon", conveniently omitting the fact that almost all of them have already been recognized first order in the experimental literature. Moreover, it is in direct contradiction with L. Landau, the "father" of "continuous" phase transitions. In Statistical Physics by Landau and Lifshitz we read: "Transition between different crystal modifications occurs usually by phase transition at which jump-like rearrangement of crystal lattice takes place and state of the matter changes abruptly. Along with such jump-like transitions, however, another type of transitions may also exist related to change in symmetry". Thus, according to Landau, phase transitions between crystal modifications are first order, but "continuous" phase transitions may (!) also exist (and, consequently, not necessarily exist). In reality, they don’t exist; the remaining few would be re-classified to first order upon careful reexamination. Landau used only two particular examples of "second-order" phase transitions - in NH₄Cl and BaTiO₃ - and both turned out first order.

Introduction of "quantum phase transitions" by Vojta can be briefly summarized as follows. All solid-solid "classical" phase transitions occur at critical points in which the previously existing order is destroyed by thermal fluctuations. Not far from 0 K the "classical" critical point becomes "quantum" and so does the phase transition.

Comment: introduction of "quantum phase transitions" as antithesis to first-order phase transitions and extension of the "critical" behavior of all other solid-state phase transitions (called "classical") is based on erroneous premise and, therefore, is not valid. While this already proves our point, it is useful to extend our dissection somewhat further.

Excerpt: In contrast, at continuous transitions the two phases do not co-exist. An important example is the ferromagnetic transition of iron at 770 C, above which the magnetic moment vanishes. This phase transition occurs at a point where thermal fluctuations destroy the regular ordering of magnetic moments – this happens continuously in the sense that the magnetization vanishes continuously when approaching the transition from below. The transition point of a continuous phase transition is also called critical point.
Comment: Ferromagnetic phase transitions have become the last resort for the conventional theory to exemplify continuous phase transitions and critical phenomena. The confusing Vojta’s explanation (magnetization changes continuously at critical point) illustrates that the theory reached impasse. At the present time this last resort has actually been eliminated by innumerous experimental examples of first order ferromagnetic phase transitions, including those in Fe, Co and Ni. But the task of introduction of "quantum" phase transitions required ignoring this and describing ferromagnetic phase transition in Fe in a quite erroneous way. It ignores evidence [Preston, Refs.192,193] that the transition is first order (therefore, nucleation-and-growth). Contrary to the Vojta’s description, the magnetization curve is not sharp when measured in either direction, leaving the "critical phenomenon" without its "critical point". The "continuous" change of magnetization is due to multiple nucleation over a temperature range of two-phase coexistence. See Chapter 4 for details.

To complete the picture, there were numerous publications, both theoretical and experimental, where certain "quantum" phase transitions were stated to be first order. Who are wrong: those arguing that "quantum" phase transitions are an antithesis to first-order ones and a "critical phenomenon", or those embracing "first-order quantum phase transitions"? The answer is: all of them are. The experimentalists, who concluded the "quantum" phase transitions they observed were first order, are less erroneous: the transitions are first order indeed. (In some such works the conclusions were based on the observed hysteresis). But they still believe that deal with "quantum" phase transitions rather than associating them with nucleation and growth.

Conclusion: the concept of "quantum" phase transitions is wrong, as is the description of "classical" solid-state phase transitions by theory of critical phenomena.