*In response to “Spontaneous Symmetry Breaking” (Vol. 4, No. 2).*

*To the editors:*

It is impossible to disagree with the statement at the beginning of Édouard Brézin’s essay: “Contemporary physics is constantly confronted with questions of broken symmetry.” Already the short list of examples that Brézin gives at the end of his introduction is extremely impressive: from ferromagnetism, superfluidity, and superconductivity to modern quantum field theory. Indeed, a big part of Steven Weinberg’s fundamental three-volume treatise, *The Quantum Theory of Fields*, is devoted to the detailed description of how the ideas of symmetry and its breaking are shaping the worldview of not only physicists, but also philosophers interested in the natural sciences.

I would like to take a more modest point of view on the symmetry problem, including all its various manifestations and violations, from the practical position of a quantum theoretician working in low-energy physics, for example, nuclear or atomic theory. As we are teaching our students, the ideas of symmetry and related conservation laws penetrate the whole of physics. The closed elliptical Kepler orbits of planets reflect the special symmetry of Newtonian gravity—the existence, along with the usual orbital momentum, of another constant of motion, the Laplace–Runge–Lenz vector. The existing perturbations, including those of relativistic nature, lead to perihelion displacement, one of the earliest indications for the validity of general relativity.

In quantum mechanics, we look at such problems from a different, or rather complementary, viewpoint. The existence of alternative symmetries generated by non-commuting operators, as in the pure 1/*r* potential, leads to a large energetic degeneracy of stationary states. In the same way, the stationary states of a spherically symmetric quantum harmonic oscillator are strongly degenerate; the mean square radius of orbits with quite different shapes is the same. The simplest nuclear shell model is based on a harmonic oscillator potential plus a spin-orbit coupling that partly removes the oscillator degeneracy. An even more innocent example is any closed quantum system in a stationary state with a non-zero angular momentum. In the absence of external fields, the rotational symmetry implies the degeneracy of states with different angular momentum projections on an—arbitrary!—quantization axis. Of course, those examples are trivial.

Looking at such examples from a different perspective, we can argue that an atom or a nucleus in a stationary state with fixed quantum numbers, *J*, of the total angular momentum, *M*, of its projection on a randomly taken z-axis is a possible, albeit primitive, example of the spontaneously broken symmetry. The symmetry is restored by the possibility of rotation to the state with a different *M*-value. Brézin gives an example of spontaneous magnetization, where the broken symmetry reveals itself as a result of the double limiting transition, first to a very large particle number or volume, *N* → ∞, and then to the vanishing field that has selected the original orientation of magnetization, *h* → 0. Here, there is no first limit, we stay with our finite quantum sample. The analogy still persists: the original quantization axis was produced as a result of the previous manipulations with specific reactions or external fields, equivalent to the magnetic field, *h*. After the creation act, we can also switch off every source of asymmetry. As these sources can be just random perturbations, we have an analog of spontaneously broken symmetry in a small quantum system.

Our symmetry is of course restored by the existence of equivalent degenerate states with different orientations. They have the same *J* value but different *M* or superpositions of *M* corresponding to the rotation of our system as a whole without violating its intrinsic structure. We can think about this occurrence as an embryo of the Goldstone mode that starts with zero excitation energy.

Molecular and nuclear physics provide the next level of analogy. At some degree of occupancy of the highest nuclear shells, a deformed shape becomes energetically preferable. Then the nucleus acquires a natural internal asymmetry. Again, this does not require any transition to an infinite particle number. And here the real analog of the Goldstone mode emerges, the rotational branch of the nuclear excitation spectrum. In molecules, rotational levels are abundant as a consequence of the specific molecular shape. Due to the finite moment of inertia, the rotational spectrum is discrete, but it seems difficult to deny that symmetry-breaking physics is appearing already on the level of a small, autonomous quantum system.

In many classical examples, such as superfluidity or superconductivity, the phase transition actually does not occur absolutely spontaneously. It is governed by the external heat bath bringing the system to a critical temperature. This is also seen clearly in Rayleigh–Bénard convective instability. In a similar way, the nuclear shape phase transition is driven, in the absence of a thermostat, by increasing occupancy rates of upper shells, or, we can say, by the chemical potential. There are examples where this shape transformation is quite sharp as a function of the nucleon number. On the other hand, an analog of the superfluid phase transition in nuclei is indeed different from that in the bulk superfluid. The low-lying states have a condensate of pairs with the formally evaluated correlation length exceeding the size of the nucleus. But here the effect of the finiteness of the system is clearly evident: excitation of the nucleus, similar to the heating, does not lead to the sharp phase transition at some critical temperature. Instead, we see just gradual crossovers to normal nuclear matter.

I believe that such a broad understanding of phase transitions and symmetry breaking in applications to finite quantum systems can be traced back to the pioneering work of James Rainwater, Aage Bohr, and Ben Mottelson on nuclear deformation and Bohr, Mottelson, and David Pines on nuclear superfluidity.^{1} My late teacher Spartak Belyaev has shown how nuclear pairing changes all observable nuclear properties, including the moment of inertia, through coexistence of those two critical phenomena in atomic nuclei.^{2} When he received the Eugene Feenberg Medal in 2004, Belyaev gave a lecture titled “Many-Body Physics and Spontaneous Symmetry Breaking,” combining physics of macroscopic superfluidity and nuclear collective phenomena.^{3}

**Vladimir Zelevinsky**

**Édouard Brézin** *replies:*

I would like to thank our colleagues for their comments, which indeed point to other relevant, significant contributions to the topic of spontaneous symmetry breaking. In my short article, I wanted to illustrate how much this concept was ubiquitous; from condensed matter to cosmology and particle physics, I have tried to point out how our understanding of many phenomena is based on spontaneous symmetry breaking.

*Vladimir Zelevinsky is Professor of Physics at Michigan State University.*

*Édouard Brézin is Professor Emeritus at the Université Pierre et Marie Curie in Paris.*

- James Rainwater, “Nuclear Energy Level Argument for a Spheroidal Nuclear Model,”
*Physical Review*79 (1950): 432–34. Aage Bohr and Ben R. Mottelson,*Nuclear Structure: Volume II, Nuclear Deformations*(New York: Benjamin, 1974). Aage Bohr, Ben Mottelson, and David Pines, “Possible Analogy between the Excitation Spectra of Nuclei and Those of the Superconducting Metallic State,”*Physical Review*110 (1958): 936–38. ↩ - Spartak Belyaev, “Effect of Pairing Correlation on Nuclear Properties,”
*Matematisk-fysiske Meddelelser Kongelige Danske Videnskabernes Selskab*31, no. 11 (1959). Alexei Barabanov and Vladimir Zelevinsky, “Spartak Timofeevich Belyaev,”*Physics Today*70, no. 6 (2017): 72. ↩ - Spartak Belyaev, “Many-Body Physics and Spontaneous Symmetry Breaking,” in
*Recent Progress in Many-Body Theories: Proceedings of the 12th International Conference*, ed. Joseph A. Carlson and Gerardo Ortiz (Singapore: World Scientific, 2006), 13–24. ↩