What symmetries distinguish liquids, crystals, glasses, and isotropic solids?

 One of the most important ideas in condensed matter physics is that different states of matter are associated with different symmetries. These different symmetries result in different types of elementary excitations such as the Goldstone bosons associated with continuous symmetry breaking. The symmetries of the low-lying excited states reflect the symmetries of the ground state.

For example, consider the transition from a liquid to a cubic crystal. The continuous rotational and translational symmetry of the liquid is broken to the discrete rotational and translational symmetry of the crystal. Long-wavelength sound waves reflect these changes in symmetry. In the crystal, there are three distinct sound waves: one longitudinal and two shear modes. In contrast, in the liquid, there are only longitudinal modes. 

An isotropic solid, such as studied in elasticity theory, supports two types of distortions: compression and shear. Consequently, there are three types of sound waves (longitudinal and transverse phonons. The latter can have two different polarisations). The isotropic solid has continuous, not discrete, rotational and translational symmetries. A glass is an example.

This leads to a fundamental question:

What is the difference between liquids and solids at the level of fundamental symmetries?

In different words, what is the order parameter for the liquid-solid transition? A possible answer is the shear modulus G, which vanishes in the liquid state.

A related question is: What is the fate of the transverse phonons upon transitioning from the solid state to the liquid state?

I would have thought that these questions would have been settled decades ago. However, they have not. Just two years ago, Physical Review E published a 22 page article that aims to address the questions above.

Deformations, relaxation, and broken symmetries in liquids, solids, and glasses: A unified topological field theory

Matteo Baggioli, Michael Landry, and Alessio Zaccone

The paper immediately drew a Comment claiming the paper

"contradicts the known hydrodynamic theory of classical liquids." The authors have a Reply.

I do not have the expertise to give insight on the subtle technical issues in this debate. My only comment is that it is amazing how we are struggling to answer such basic questions.

I thank Jean-Noel Fuchs for getting me interested in these subtle questions. This happened when he kindly pointed out an error in Condensed Matter Physics: A Very Short Introduction. On page 40, I erroneously stated that shear sound waves exist in a liquid. This was part of a confused discussion about how sound waves can be used to distinguish different states of matter.  I have drafted a corrected paragraph and inserted it in my post listing the errors in my book.

I welcome any comments about the issues discussed above.

My mental health update

I have struggled with my mental health on and off since the time of my Ph.D. studies. Several readers have commented that has been helpful for them to hear my story. Here I give a small update on both my health and some recent reading.

I have been thinking about the issue more because I have been invited to give a talk in October for a research centre at UQ, as part of Mental Health Week. I may adapt a talk that I gave for a school colloquium at UQ six years ago. I welcome suggestions for things people think I should talk about.

My mental health is the best it has been for almost a decade. There are probably many reasons for this: retirement, managing stress, no international travel, being connected to a church community, and practising the basics (diet, exercise, less screen time, less caffeine, ...), ...

Until a year ago, I believed I would be on antidepressants for the rest of my life. But my doctor told me we should explore my getting off antidepressants. It is now the view of the medical establishment that there are too many people on them who do not need to be, there can be long-term complications, and that the longer a patient is on them the harder it is to get off them. Over the past 2 years, The Economist had helpful articles along these lines (see below).

In April we agreed that we would start the experiment of reducing my dose, following the now standard practice of slowly reducing the dose every three weeks. He warned me to look for side effects, such as random brain zaps. There were no side effects. I got to zero dosage a month ago.

Unfortunately, I am now experiencing one side effect which I have now learned is not uncommon: uncontrollable sobbing. The first instance was July 21 when I learned that Biden was not going to run again for President. The fact that this triggered ten minutes of sobbing shows there is something not quite right with my brain chemistry!

I had several other incidents with my family. The tears are out of proportion to the significance of the event that triggers them. Sometimes I choke up when talking to people I care about or on an issue that concerns me.

I had an appointment with my doctor this week and we agreed that for now, we would stay the course, not resume the medication, and monitor the situation.

How to make better use of antidepressants: Identify those who really need them, and wean other people off them

(The Economist, October 19, 2022)

Antidepressants are over-prescribed, but genuinely help some patients: In around 15% of cases, they offer large benefits

(The Economist, January 20, 2023)

The graphs above are amazing. They show several striking things.

1. There is a massive placebo effect for antidepressants. This is shown by the two coloured curves being almost identical.

2. There is a massive variation between patients with regard to how effective the drugs are. This is shown by the very broad distribution. It reminds me of journal impact factors: the distribution is so broad that discussions about the mean are meaningless.

Antidepressants can cause withdrawal symptoms – here’s what you need to know

(The Conversation, June 23, 2023)

Psychiatry’s Incurable Hubris: The biology of mental illness is still a mystery, but practitioners don’t want to admit it.

(The Atlantic, April 2019).

Disclaimer. I am not a mental health professional. Mental health is an incredibly complex issue. Everyone is different. Do not take any action based on what you read about my experience. I do not present my experience as an example others should follow. Rather I present my experience so others may know that mental health struggles are not unusual, including among those who may appear to have "successful" lives.

Do arrows of explanation point down or up?

The figure above shows the stratification of objects that interest physicists. As one goes down the chain length and time scales get smaller and energy scales get larger.

A reductionist seeks to explain the objects at each strata in terms of the objects that occur at the next lower strata.

In 1987 Steven Weinberg gave a talk at the University of Cambridge at the Tercentenary Celebration of Newton's Principia.

The talk was published in Nature. Newtonianism, Reductionism, and the Art of Congressional Testimony

Part of the talk is about Weinberg's testimony to a US Congressional Committee making the case for the construction of the SSC (Superconducting Super Collider). Phil Anderson spoke against the SSC.

Weinberg argued that the SSC should be built because particle physics is "in some sense more fundamental than other areas of physics." He claims that this is because "the arrows of explanation point down", as in the diagram shown above.

A contrasting perspective is that of Andrew Steane. His book, Science and Humanity, contains the figure below.

In his picture of the explanatory relationship between physics, chemistry, and biology, Steane draws arrows pointing in both directions. The up arrow is denoted “supports [allows and physically embodies the expression of]” and the down arrow is denoted “enarches [exhibits the structures and behaviours that make sense in their own terms and are possible within the framework of].”

Weinberg's article is worth reading in full. It has many insights about science and physics worth considering, including the relationship between emergence and reductionism.

Aside: It is also reproduced in his book of essays, Facing Up: Science and Its Cultural Adversaries, published in 2001.