Heat destroys everything, but physicists find an exception

In everyday life, physics tells us that heat destroys the structure of matter, such as when ice melts into water. But physicists have discovered a new model that can theoretically maintain this order at any temperature.

Written by | Ant

In the American TV series "The Big Bang Theory", Leonard returned from working in the Arctic for three months and gave Penny a snowflake that would never melt. Penny said with emotion that this was the most romantic gift she had ever received.

The Big Bang Theory Season 3 Episode 1

Strictly speaking, Leonard's gift is actually a "fossilized imprint" of snowflakes, just like the fossilized dinosaur footprints in Dinosaur Valley State Park in Texas. Just as the dinosaurs have long since become extinct, the snowflakes have long since melted, leaving behind an imprint that will never fade.

Left: Snowflake "imprint fossils". Image source: snowcrystal.com; Right: Dinosaur footprint fossils in Dinosaur Valley State Park. Image source: ksat.com

Recently, the famous popular science magazine Quanta Magzine published an article titled “Heat Destroys All Order. Except for in This One Special Case.”[1] The article discussed a magical phenomenon in quantum field theory, a substance that exists in an ordered state that can never be destroyed by heat. The picture accompanying the article is a snowflake in fire.

Image credit: Kristina Armitage/Quanta Magazine

In 1937, Landau proposed the famous theory of spontaneous symmetry breaking[2]. According to this theory, the freezing of water into ice is actually a process of spontaneous symmetry breaking. Compared with ice, water has a higher symmetry. Let's take rotational symmetry as an example. If you rotate water by an angle, it will look the same as before. Ice is different. Ice has a lattice structure. Only when you rotate it at a certain angle can the ice crystal structure remain unchanged. When water freezes into ice, the original symmetry is destroyed, which is called symmetry breaking. Compared with ice, the molecules in water can move freely, so it is in a more disordered state.

The square lattice remains unchanged only when it is rotated 90 degrees.

Due to the constraints of the second law of thermodynamics, when a phase transition occurs, the high-temperature phase needs to have a higher entropy. In general, systems with high symmetry are more disordered and therefore have higher entropy. This also explains why ice turns into water when heated. So are there exceptions? Indeed, there are. In 1950, Soviet physicist Isaak Pomeranchuk predicted a peculiar behavior of liquid He-3, that when we heat liquid He-3, He-3 will turn into a solid. In 1969, the Pomeranchuk effect was experimentally confirmed. This discovery was later used to continue refrigeration at extremely low temperatures by pressurizing, which also led to the discovery of He-3 superfluidity. [3]

In 1976, Stephen Weinberg began to pay attention to this phenomenon. He proposed a quantum field theory composed of scalar fields and successfully realized inverse symmetry breaking [4]. We can understand this model constructed by Weinberg as two coupled spins, one of which can only take the spin-up or spin-down state. The spin of the other subsystem can take values ​​on a high-dimensional sphere. As shown in the figure below, the red spin is in an ordered state, while the blue spin is in a disordered state. The coupling between them means that when the temperature is increased, the ordered state of the red spin will not be destroyed. The additional entropy generated by the increase in temperature is absorbed by the blue spin. This is very similar to the Pomeranchuk effect. Since He-3 has an additional spin degree of freedom (He-3 is a fermion), entropy is absorbed by the spin, thus realizing an anomalous phase transition.

In the model constructed by Weinberg, the phase transition temperature of symmetry breaking can be increased by adjusting some parameters. The ordered state of symmetry breaking can sometimes even exist at infinite temperature, thus achieving unmeltable order. This is like a snowflake that never melts, which is very counterintuitive. However, there is a flaw in this model. In 4-dimensional space-time, the scalar field is ultraviolet incomplete (as is the standard model of particle physics), and the coupling constant of the system will diverge at a certain high energy scale. This means that when the temperature exceeds a certain value, we can no longer trust the predictions of the theory.

In fact, the efforts of Weinberg and many subsequent physicists are related to cosmology. At the beginning of the universe, it was at extremely high temperatures, and the energy per unit volume was very high. As the universe expanded at an accelerated rate, 0.01 ns after the birth of the universe, when the temperature gradually dropped to 1015 Kelvin, the characteristic energy of particles in the universe dropped to about 100 GeV. The standard model of particle physics combined with cosmology predicts that an electroweak symmetry breaking phase transition will occur at this time. Similar to the phase transition of water freezing into ice that we mentioned earlier, the universe changes from a disordered state at high temperature to an ordered state at low temperature. The electroweak symmetry breaking phase transition has a name that may be more familiar to people - the Higgs mechanism. Through this mechanism, many elementary particles, including electrons, gain mass, and the universe changes to a state that is more similar to our world today. It can be said that the basic physics theories known to mankind can already explain the universe from 0.01 ns at the beginning of the Big Bang to today.

The history of the evolution of the universe. Image source: Centre for Theoretical Cosmology, University of Cambridge

Electroweak symmetry breaking occurs when particles reach an energy scale of about 100 GeV, which is far from the energy scale (1019 GeV) when the universe was just born. It is hard to believe that there is no new physics between these two huge scale differences - this is the famous hierarchy problem. Popular models include Grand Unified theory, supersymmetry, large extra dimension, etc. Weinberg's article shows that the ordered state of symmetry breaking may not be melted even at infinitely high temperatures. This raises another possibility for cosmology: Is our universe in an order that cannot be melted? Is it possible that electroweak symmetry breaking occurs at a scale close to Planck? Or has it never occurred? Is there really no new physics? Sometimes, this hypothesis is also called symmetry non-restoration [5].

Although this hypothesis is very attractive, we still need to correct the flaws in Weinberg's theory. To do this, we can first try to answer the question: Is there an unmeltable order?

In fact, in the nearly 50 years since Weinberg, this question has not been well answered. On the one hand, this phenomenon is very counterintuitive and physicists have not really considered it seriously; on the other hand, it is difficult to solve the strong correlation quantum field theory itself, especially the gauge field theory. It was not until 2020 that Zohar Komargodski and his collaborators introduced a genius solution: using conformal field theory to construct a complete ultraviolet system [6]. Conformal field theory is similar to a fractal system and has self-similarity. As shown in the figure, if you take a part of the fractal system and zoom in, you will find that it is consistent with the original shape before zooming in.

The self-similarity of conformal field theory means that the behavior of conformal field theory at low energy scales is exactly the same as that at high energy scales. In other words, if the system is in an ordered state at low temperatures, it will also be in an ordered state at high temperatures. It is precisely by utilizing this feature of conformal field theory that Zohar Komargodski and his collaborators discovered a theory that is in an ordered state at any temperature.

Although an important step towards their goal, Komargodski's approach was still flawed because they used a dimensional renormalization perturbation calculation scheme introduced by Wilson and Fisher in 1972 [7], so strictly speaking it only works when the space-time dimension is 3.99. In September 2024, Michael Scherer, Junchen Rong, and Bilal Hawashin used the functional renormalization group in 3D to show that Zohar et al.'s model can indeed achieve unmeltable order [8]. When using the functional renormalization method, they had to ignore certain interactions, which made the proof imperfect. However, this inspired Komargodski and a new collaborator, Fedor Popov, to give a rigorous proof three months later [9]. This series of work completed the last piece of the puzzle of unmeltable order. (Readers interested in the technical details can refer to the above article.)

Although these works prove that quantum field theory can indeed achieve an order that never melts, this model still has some shortcomings. Scherer-Rong-Hawashin's article shows that the infusible order can only be achieved when the auxiliary spin system takes values ​​on the 14-dimensional sphere. In addition, the conformal field theory in this model is a multicritical point, which means that we need additional fine-tuning to achieve this system. It can be said that these works have solved the basic problem of whether infusible order exists or not from a theoretical level, but there is still a long way to go before it can be applied in cosmology. However, this may be the direction we should strive for in the future. As mentioned earlier, our universe is in an ordered state with broken electroweak symmetry. If we construct a similar conformal field theory by extending the standard model of particle physics, we may be able to prove that our universe is in an ordered state that cannot be melted. The road is long and difficult, but the future is promising.

(This article is dedicated to Fan Fan.)

References

[1] Charlie Wood, Heat Destroys All Order. Except for in This One Special Case, Quanta Magazine.

[2] Lev D. Landau (1937). On the Theory of Phase Transitions. Zh. Eksp. Teor. Fiz. 7: 19-32.

[3] David M. Lee, The extraordinary phases of liquid 3He, Rev. Mod. Phys. 69, 645

[4] Steven Weinberg. Gauge and Global Symmetries at High Temperature. Phys. Rev. D, 9:3357–3378, 1974.

[5] Patrick Meade, Harikrishnan Ramani, Unrestored Electroweak Symmetry, Phys. Rev. Lett.122.041802

[6] Noam Chai, Soumyadeep Chaudhuri, Changha Choi, Zohar Komargodski, Eliezer Rabinovici, Michael Smolkin, Thermal Order in Conformal Theories, Phys.Rev.D 102 (2020) 6, 065014, Symmetry Breaking at All Temperatures, Phys.Rev.Lett. 125 (2020) 13, 131603

[7] K. Wilson and M. Fisher, Critical Exponents in 3.99 Dimensions, Phys.Rev.Lett. 28 (1972) 240-243

[8] Bilal Hawashin, Junchen Rong, Michael M. Scherer, UV complete local field theory of persistent symmetry breaking in 2+1 dimensions, Phys.Rev.Lett. 134 (2025) 4, 041602

[9] Zohar Komargodski, Fedor K. Popov, Temperature-Resistant Order in 2+1 Dimensions, arXiv:1807.07578

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