Tony Leggett (1938-2026): condensed matter theorist

Tony Leggett died last week. The New York Times has a nice obituary. One measure of his influence on me is that more than 20 posts on this blog feature his work. He received the Nobel Prize in 2003 for developing the theory of superfluid 3He.

In 1972, a graduate student at Cornell, Doug Osheroff, discovered a phase transition around a temperature of 2 mK in liquid 3He. In the 1960s liquid 3He was established to be a Fermi liquid that was beautifully described by Landau's theory. Osheroff and his advisors, David Lee and Robert Richardson, incorrectly identified the phase transition as arising from antiferromagnetic order in the solid phase of 3He.

However, Leggett argued that it was actually due to superfluidity that there were two distinct superfluid phases, A and B, with different order parameters. 

Lee, Osheroff, and Richardson shared the Nobel Prize in 1996 for their discovery.

Leggett was primed to make rapid progress, as in 1965 and 1966 he had written three papers about superfluidity in liquid 3He, albeit assuming s-wave pairing. Indeed, by 1975 he wrote a comprehensive review article on the two superfluid phases.

For many reasons superfluid 3He was significant for the broader field of condensed matter. BCS showed that in elemental metals, superconductivity resulted from Cooper pairing of electrons due to an attractive electron-phonon interaction.  The order parameter (Cooper pair wave function) had s-wave spin singlet symmetry.

In contrast, superfluid 3He showed that Cooper pairing could also occur in a neutral Fermi liquid, and have non-trivial symmetry, i.e., p-wave symmetry and spin triplet. The order parameter has 18 components, compared to only 2 for elemental superconductors. There is spontaneous symmetry breaking of the local gauge symmetry, and spin or orbital rotational symmetries. 

The Cooper pairing in superfluid 3He is not due to a fermion-phonon interaction but due to spin fluctuations.

The fact that Cooper pairing was possible for different symmetries and mechanisms than for elemental superconductors was significant in that it meant it was reasonable to consider this possibility for superfluidity in neutron stars, and superconductivity in cuprates, strontium ruthenate, heavy fermions, and organic charge transfer salts.

There is rich physics associated with the symmetry breaking: 18 collective modes of the order parameter, textures such as boojums, and exotic vortex cores. For vortices, there is also some (controversial) connection to cosmic strings, including experiments that test the Kibble-Zurek mechanism and the electro-weak phase transition in the early universe.

Aside: My Ph.D. thesis was on the theory of the non-linear interaction of zero sound with the order parameter collective modes in the B-phase.

Leggett's development of the theory of superfluid 3He was amazing and certainly worthy of a Nobel. However, I think he made an even greater contribution to physics through his work on the theory of macroscopic quantum effects in Josephson junctions. This work was the basis for the experimental work that was honoured with the Nobel Prize last year.

With his student Amir Caldeira, Leggett performed concrete calculations of the effects of decoherence on quantum tunnelling in Josephson junctions.

[The NY Times obituary mistakenly says this work began after Leggett moved to Urbana. It was done while he was still at Sussex].

The formalism they developed involving the spectral density is the basis for most theoretical treatments of decoherence in superconducting qubits. A relevant toy model is the spin-boson model, and in 1987 Leggett published a seminal (but rather dense) review on the subject.

Leggett aided our understanding of cuprate superconductors. He contributed to the theoretical ideas that were the basis of the phase-sensitive measurements that established the d-wave nature of the order parameter. He also showed that experiments with inconsistent with  Anderson's interlayer tunneling theory.

I recommend reading Leggett's own scientific autobiography, Matchmaking Between Condensed Matter and Quantum Foundations, and Other Stories: My Six Decades in Physics and his book, The Problems of Physics

2025 Nobel Prize in Physics: Macroscopic quantum effects

John Clarke, Michel H. Devoret, and John M. Martinis received the prize  “for the discovery of macroscopic quantum mechanical tunnelling and energy quantisation in an electric circuit.”

The work was published in three papers in PRL in 1984 and 1985. The New York Times has a nice discussion of the award, including comments from Clarke, Martinis, Tony Leggett, and Steve Girvin.

There is some rich, subtle, and beautiful physics here. As a theorist, I comment on the conceptual and theoretical side, but don't want to minimise that doing the experiments was a technical breakthrough.

The experiments were directly stimulated by Tony Leggett, who, beginning in the late 70s, championed the idea that Josephson junctions and SQIDs could be used to test whether quantum mechanics was valid at the macroscopic level. Many in the quantum foundations community were sceptical. Leggett and Amir Caldeira, performed some beautiful, concrete, realistic calculations of the effect of decoherence and dissipation on quantum tunneling in SQUIDs. The results suggested that macroscopic tunneling should be observable.

Aside: Leggett rightly received a Nobel in 2003 for his work on the theory of superfluid 3He. Nevertheless, I believe his work on quantum foundations is even more significant.

Subtle point 1. What do we mean by a macroscopic quantum state?

It is commonly said that superconductors and superfluids are in a macroscopic quantum state. Signatures are the quantisation of magnetic flux in a superconducting cylinder and how the current through a Josephson junction oscillates as a function of the magnetic flux through the junction. I discuss this in the chapter on Quantum Matter in my Very Short Introduction.

Leggett argued that these experiments are explained by the Josephson equations, which treat the phase of the superconducting order parameter as a classical variable. For example, in a SQUID, it satisfies a classical dynamical equation. 

If the state is truly quantum, then the phase variable should be quantised.

Aside: a nice microscopic derivation, starting from BCS theory and using path integrals, of the effective action to describe the quantum dynamics was given in 1982 by Vinay Ambegaokar, Ulrich Eckern, Gerd Schön

Subtle point 2. There are different signatures of quantum theory: energy level quantisation, tunnelling, coherence (interference), and entanglement.

In 1984-5, Clarke, DeVoret, and Martinis observed the first two. Macroscopic quantum coherence is harder to detect and was only observed in 2000. 

In a nice autobiographical article

Matchmaking Between Condensed Matter and Quantum Foundations, and Other Stories: My Six Decades in Physics

Leggett commented in 2020,

Because of the strong prejudice in the quantum foundations community that it would never be possible to demonstrate characteristically quantum-mechanical effects at the macroscopic level, this assertion made us [Leggett and Garg, 1985] the target of repeated critical comments over the next few years. Fortunately, our experimental colleagues were more open-minded, and several groups started working toward a meaningful experiment along the lines we had suggested, resulting in the first demonstrations (29, 30) of MQC [Macroscopic Quantum Coherence] in rf SQUIDs (by then rechristened flux qubits) at the turn of the century. However, it would not be until 2016 that an experiment along the lines we had suggested (actually using a rather simpler protocol than our original one) was carried out (31) and, to my mind, definitively refuted macrorealism at that level.  

I find it rather amusing that nowadays the younger generation of experimentalists in the superconducting qubit area blithely writes papers with words like “artificial atom” in their titles, apparently unconscious of how controversial that claim once was.

Two final comments on the sociology side.

Superconductivity and superfluidity have now been the basis for Nobel Prizes in six years and four years, respectively.

The most widely cited of the three PRLs that were the basis of the Prize is the one on quantum tunnelling with about 500 citations on Google Scholar. (In contrast, Devoret has more than 20 other papers that are more widely cited). From 1986 to 1992 it was cited about a dozen times per year. Between 1993 and 2001 is was only cited a total of 30 times. Since, 2001 is has been cited about 20 times per year.

This is just one more example of how citation rates are a poor measure of the significance of work and a predictor of future success.

Reviewing emergent computational abilities in Large Language Models

Two years ago, I wrote a post about a paper by Wei et al, Emergent Abilities of Large Language Models

Then last year, I posted about a paper Are Emergent Abilities of Large Language Models a Mirage? that criticised the first paper.

There is more to the story. The first paper has now been cited over 3,600 times. There is a helpful review of the state of the field.

Emergent Abilities in Large Language Models: A Survey

Leonardo Berti, Flavio Giorgi, Gjergji Kasneci

It begins with a discussion of what emergence is, quoting from Phil Anderson's More is Different article [which emphasised how new properties may appear when a system becomes large] and John Hopfield's Neural networks and physical systems with emergent collective computational abilities, which was the basis of his recent Nobel Prize. Hopfield stated

"Computational properties of use to biological organisms or the construction of computers can emerge as collective properties of systems having a large number of simple equivalent components (or neurons)."

Berti et al. observe, "Fast forward to the LLM era, notice how Hopfield's observations encompass all the computational tasks that LLMs can perform."

They discuss emergent abilities as in-context learning, defined as the "capability to generalise from a few examples to new tasks and concepts on which they have not been directly trained."

Here, I put this review in the broader context of the role of emergence in other areas of science.

Scales. 

Simple scales that describe how large an LLM is include the amount of computation, the number of model parameters, and the size of the training dataset. More complicated measures of scale include the number of layers in a deep neural network and the complexity of the training tasks.

Berti et al. note that the emergence of new computational abilities does not just follow from increases in the simple scales but can be tied to the training process. I note that this subtlety is consistent with experience in biology. Simple scales would be the length of an amino acid chain in a protein or base pairs in a DNA molecule, the number of proteins in a cell or the number of cells in an organism. More subtle scales include the number of protein interactions in a proteome or gene networks in a cell. Deducing what the relevant scales are is non-trivial. Furthermore, as emphasised by Denis Noble and Robert Bishop, context matters, e.g., a protein may only have a specific function if it is located in a specific cell.

Novelty. 

When they become sufficiently "large", LLMs have computational abilities that they were not explicitly designed for and that "small" versions do not have. 

The emergent abilities range "from advanced reasoning and in-context learning to coding and problem-solving."

The original paper by Wei et al. listed 137 emergent abilities in an Appendix!

Berti et al. give another example.

"Chen et al. [15] introduced a novel framework called AgentVerse, designed to enable and study collaboration among multiple AI agents. Through these interactions, the framework reveals emergent behaviors such as spontaneous cooperation, competition, negotiation, and the development of innovative strategies that were not explicitly programmed."

An alternative to defining novelty in terms of a comparison of the whole to the parts is to compare properties of the whole to those of a random configuration of the system. The performance of some LLMs is near-random (e.g., random guessing) until a critical threshold is reached (e.g., in size) when the emergent ability appears.

Discontinuities.

Are there quantitative objective measures that can be used to identify the emergence of a new computational ability? Researchers are struggling to find agreed-upon metrics that show clear discontinuities. That was the essential point of Are Emergent Abilities of Large Language Models a Mirage? 

In condensed matter physics, the emergence of a new state of matter is (usually) associated with symmetry breaking and an order parameter. Figuring out what the relevant broken symmetry and the order parameter often requires brilliant insight and may even lead to a Nobel Prize (Neel, Josephson, Ginzburg, Leggett,...) A similar argument can be made with respect to the development of the Standard Model of elementary particles and gauge fields. Furthermore, the discontinuities only exist in the thermodynamic limit (i.e., in the limit of an infinite system), and there are many subtleties associated with how the data from finite-size computer simulations should be plotted to show that the system really does exhibit a phase transition.

Unpredictability.

The observation of new computational abilities in LLMs was unanticipated and surprised many people, including the designers of the specific LLMs involved. This is similar to what happens in condensed matter physics, where new states of matter have mostly been discovered by serendipity.

Some authors seem surprised that it is difficult to predict emergent abilities. "While early scaling laws provided some insight, they often fail to anticipate discontinuous leaps in performance."

Given the largely "black box" nature of LLMs, I don't find it the unpredictability surprising. It is hard for condensed matter systems, and they are much better characterised and understood.

Modular structures at the mesoscale.

Modularity is a common characteristic of emergence. In a wide range of systems, from physics to biology to economics, a key step in the development of the theory of a specific emergent phenomenon has been the identification of a mesoscale (intermediate between the micro- and macro-scales) at which modular structures emerge. These modules interact weakly with one another, and the whole system can be understood in these terms. Identification of these structures and the effective theories describing them has usually required brilliant insight. An example is the concepts of quasiparticles in quantum many-body physics, pioneered by Landau.

Berti et al. do not mention the importance of this issue. However, they do mention that "functional modules emerge naturally during training" [Ref. 7,43,81,84] and that "specialised circuits activate at certain scaling thresholds [24]".

Modularity may be related to an earlier post, Why do deep learning algorithms work so well? In the training process, a neural network rids noisy input data of extraneous details...There is a connection between the deep learning algorithm, known as the "deep belief net" of Geoffrey Hinton, and renormalisation group methods (which can be key to identifying modularity and effective interactions).

Is emergence good or bad?

Undesirable and dangerous capabilities can emerge. Those observed include deception, manipulation, exploitation, and sycophancy.

These concerns parallel discussions in economics. Libertarians, the Austrian school, and Federich Hayek tend to see the emergence as only producing socially desirable outcomes, such as the efficiency of free markets [the invisible hand of Adam Smith]. However, emergence also produces bubbles and crashes and recessions.

Resistance to control

A holy grail is the design, manipulation, and control of emergent properties. This ambitious goal is promoted in materials science, medicine, engineering, economics, public policy, business management, and social activism. However, it largely remains elusive, arguably due to the complexity and unpredictability of the systems of interest. Emergent properties of LLMs may turn out to offer similar hopes, frustrations, and disappointments. We should try, but have realistic expectations.

Toy models.

This is not discussed in the review. As I have argued before, a key to understanding a specific emergent phenomenon is the development of toy models that illustrate the phenomenon and the possible essential ingredients for it to occur. The following paper may be a step in that direction.

An exactly solvable model for emergence and scaling laws in the multitask sparse parity problem

Yoonsoo Nam, Nayara Fonseca, Seok Hyeong Lee, Chris Mingard, Ard A. Louis

In a similar vein, another possibly relevant paper is the review

Statistical Mechanics of Deep Learning

Yasaman Bahri, Jonathan Kadmon, Jeffrey Pennington1, Sam S. Schoenholz, Jascha Sohl-Dickstein and Surya Ganguli

They considered a toy model for the error landscape for a neural network, and show that the error function for a deep neural net of depth D corresponds to the energy function for a D-spin spherical spin glass. [Section 3.2 in their paper].

Emergence of the arrow of time

Time has a direction. Microscopic equations of motion in classical and quantum mechanics have time-reversible symmetry. But this symmetry is broken for many macroscopic phenomena. This observation is encoded in the second law of thermodynamics. We experience the flow of time and distinguish past, present, and future. The arrow of time is manifest in phenomena that occur at scales covering many orders of magnitude. Here are some of these different arrows of time, listed in order of increasing time scales. These are discussed by Tony Leggett in chapter 5 of The Problems of Physics.

Elementary particle physics. CP violation is observed in certain phenomena associated with the weak nuclear interaction, such as the decay of neutral kaons observed in 1964. The CPT symmetry theorem shows that any local quantum field theory that is invariant under the “proper” Lorentz transformations must also be invariant under combined CPT transformations. This means that CP violation means that time-reversal symmetry is broken. In 1989, the direction violation of T symmetry was observed.

Electromagnetism. When an electric charge is accelerated an electromagnetic wave propagates out from the charge towards infinity. Energy is transferred from the charge to its environment. We do not observe a wave that propagates from infinity into the accelerating charge, i.e., energy being transferred from the environment to the charge. Yet this possibility is allowed by the equations of motion for electromagnetism. There is an absence of the “advanced” solution to the equations of motion. 

Thermodynamics. Irreversibility happens in isolated systems. Heat never travels from a cold body to a hotter one. Fluids spontaneously mix. There is a time ordering of the thermodynamic states of isolated macroscopic systems. The thermodynamic entropy encodes this ordering.

Psychological experience. We remember the past and think we can affect the future. We don’t think we can affect the past or know the future.

Biological evolution. Over time species adapt to their environment and become more complex and more diverse.

Cosmology. There was a beginning to the universe. The universe is expanding not contracting. Density perturbations grow independent of cosmic time (Hawking and Laflamme).

It is debatable to what extent these arrows of time are related to one another. 

The problem of how statistical mechanics connects time-reversible microscopic dynamics with macroscopic irreversibility is subtle and contentious. Joel Lebowitz claimed this problem was solved by Boltzmann, provided the distinction between typical and average behaviour are accepted, along with the Past Hypothesis. This states that the universe was initially in a state of extremely low entropy. David Wallace discussed the need to accept the idea of probabilities in law of physics and that the competing interpretations of probability as frequency or ignorance matter. In contrast, David Deutsch claims that the second law of thermodynamics is an “emergent law”: it cannot be derived from microscopic laws, like the principle of testability.

I find the Past Hypothesis fascinating because it connects the arrow of time seen in the laboratory and everyday life (time scales of microseconds to years) to cosmology, covering timescales of the lifetime of the universe (10^10 years) and the “initial” state of the universe, perhaps at the end of the inflationary epoch (10^-33 seconds). This also raises questions about how to formulate the Second Law and the concept of entropy in the presence of gravity and on cosmological length and time scales. 

Can emergent properties be explained?

An important question about emergent properties is whether they can be explained solely in terms of the properties of the components of the system. Here I explore the question from the point of view of Hempel's covering law of scientific explanation, discussed in my last post.

According to Hempel, a scientific explanation E of a specific phenomena P is a logical argument that starts with some premises, at least one of which is a scientific law L, and which logically implies P.

I now give a version of this that describes a microscopic scientific explanation of some emergent property.

Suppose that a macroscopic system S has property X. S is composed of many interacting microscopic components whose properties, including their interactions, have a finite enumeration x1, x2, x3,...xn. None of these properties is X. Hence, in the sense of novelty, X is an emergent property of S. Let l1, l2, l3,.., lm be a finite number of microscopic laws. Then X has a microscopic scientific explanation if it can be deduced from the x's and l's.

A possible problem with most microscopic "explanations" of emergent properties may be whether they at some point implicitly assume some "emergent" scientific law, such as spontaneous symmetry breaking, or the existence of X. Let me illustrate this possible problem with some examples.

Irreversibility. Microscopic laws are invariant under time-reversal. But macroscopic systems exhibit irreversible behaviour such as the mixing of two distinct fluids. This is encoded in the second law of thermodynamics. This problem of the "arrow of time" is nicely discussed by Tony Leggett in The Problems of Physics, in a chapter entitled "Skeletons in the Cupboard." An alternative perspective is that of Joel Lebowitz, who claims Boltzmann solved the problem.

Superconductivity. One could claim that BCS theory provides a microscopic explanation of superconductivity. We start with the properties of electrons, ions, Coulomb's law, quantum mechanics, and statistical mechanics. These properties and microscopic laws can be used to show that there is an effective attractive interaction between electrons. One then considers the BCS variational wavefunction and calculates the properties of the macroscopic system. They are consistent with experimental observations of superconductivity. It is explained!

However, there are several problems on the way, which all in some sense involve assuming that superconductivity does occur. First, investigating the variational wave function only shows that the superconducting state has lower energy than the normal metallic state. This does not prove it is the true ground state. In fact, in one dimension it is not.

But potentially more fatal to the claimed microscopic explanation is that it assumes that spontaneous symmetry breaking is allowed, including (in some subtle sense that people still argue about) the breaking of the gauge symmetry of electromagnetism. One of the major points that Phil Anderson was trying to make in More is Different is that spontaneous symmetry breaking is a law of nature that should be viewed as of similar status to microscopic laws such as Schrodinger's equation. 

Mean-field theory of antiferromagnetism. One might claim that one can start with a classical Heisenberg or Ising model, and classical statistical mechanics, crank the mathematical handle and get antiferromagnetic. If one does mean-field theory, then one is not really doing statistical mechanics as one is considering a weird ensemble and a Hamiltonian that is no longer microscopic. Suppose instead one does the exact solution of the Ising model. That can give the magnetic state and all the critical exponents. But, it is not clear to me that when one takes the thermodynamic limit, one assumes that the broken symmetry state is allowed. Similar questions arise for me if one does a computer simulation on large lattices and uses clever finite-size scaling techniques to deduce physical properties of the emergent state. Does the assumption of the validity of these techniques amount to some extra (macroscopic) law of nature?  

I wonder whether some of these issues would be clarified (or just muddied) by considering the Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics by David Ruelle. In particular, does he make clear how an equilibrium broken symmetry magnetic state is fundamentally different from the microscopic equilibrium state associated with a finite number of spins.

I welcome ideas on how to clarify these issues.

A Myth about Condensed Matter Physics?

What is condensed matter physics about? 

In his beautiful book, The Problems of Physics (originally published in 1987), Leggett has a nice chapter about condensed matter physics, Physics on a human scale. The abstract begins:

This chapter argues that the widespread notion that the discipline of condensed matter physics is devoted to deriving the properties of complex many-body systems from that of their atomic-level components is a myth, and that the analogy of map-making is much more appropriate.

Here are some quotes that clarify Leggett's argument.

a number of cases, particularly in the traditional areas of the physics of gases and crystalline solids, in which a model which treats the behaviour of the whole as essentially just the sum of that of its parts (atoms or electrons) has been quite successful; and a few more in which, even if a ‘one- particle’ picture fails, a description in terms of pairs of particles interacting in a way which is not particularly sensitive to the environment gives good results. But these cases, despite the fact that they totally dominate the presentation of the subject in most elementary textbooks, are actually the exception rather than the rule. 

In virtually all the frontier areas of modern condensed-matter physics, the relationship between our understanding of the behaviour of matter at the microscopic level of single atoms and electrons, and at the macroscopic level of (say) liquids and solids, is actually a good deal more complicated than this.

If the activity just described is not what condensed-matter physics is all about, then what is it about? I would claim that the most important advances in this area come about by the emergence of qualitatively new concepts at the intermediate or macroscopic levels—concepts which, one hopes, will be compatible with one's information about the microscopic constituents, but which are in no sense logically dependent on it. 

... [these new concepts] provide a new way of classifying a seemingly intractable mass of information, of selecting the important variables from the innumerable possible variables which one can identify in a macroscopic system;

All this is not to deny that an important role is played in condensed-matter physics by attempts to relate the macroscopic behaviour of bulk matter to our knowledge concerning its constituent atoms and electrons. Indeed, the theoretical literature on the subject is full of papers which at first sight seem to be claiming to ‘derive’ the former from the latter—that is, to do exactly what I have just said condensed-matter physicists do not do. 

It is precisely this compelling need to isolate, from a vast and initially undifferentiated mass of information, the features which are relevant to the questions one wishes to ask, which distinguishes condensed-matter physics qualitatively from areas such as atomic or particle physics...

In this situation I believe that it is sensible to reorient our view of the kinds of questions that we are really asking in condensed-matter physics. Rather than chasing after the almost certainly chimerical goal of deducing the behaviour of macroscopic bodies rigorously from postulates regarding the microscopic level, it may be better to view the main point of the discipline as, first, the building of autonomous concepts or models at various levels, ranging all the way from the level of atomic and subatomic physics to that of thermodynamics; and, second, the demonstration that the relation between these models at various levels is one not of deducibility but of consistency—that is, that there are indeed ‘physical approximations’ we can make which make the models at various levels mutually compatible.

In different words, condensed matter physics is all about emergence! [Although, I know Leggett does not like the way the word is used]. 

The centrality of intermediate scales was also emphasised by Tom McLeish in Soft Matter: A Very Short Introduction.

When I recently read Leggett's chapter I was concerned that this might be in conflict with my draft manuscript of Condensed Matter Physics: A Very Short Introduction.  In the first chapter, I wrote the following.

The central question of Condensed Matter Physics

Generally, condensed matter physicists grapple with one question. Because it is so important I state the question in three different ways.

• How do macroscopic properties emerge from microscopic properties? 

• How do the properties of a state of matter emerge from the properties of the atoms in the material and the interactions between the atoms?

• How do the many atoms in a material interact with one another to collectively produce a particular property of the material? 

I think this is consistent with Leggett's perspective, particularly because I do later emphasise emergence and intermediate scales. On the other hand, I may not have the same emphasis (or strong language) that Leggett does. 

Leggett's view is particularly pertinent today because a quarter of a century later there are probably a lot more people who would say that they are condensed matter physicists but would subscribe to the "myth". This is because of the rise of computational materials science due to massive increases in computational power and better computational methods such as those based on Density Functional Theory (DFT), using "better" functionals and DMFT (Dynamical Mean-Field Theory).

What do you think?

Fifty years ago: three big discoveries in condensed matter

For the marketing plan for my Very Short Introduction, I was recently asked whether there were any significant anniversaries happening in condensed matter physics (and associated conferences). This is not something I normally think about.

I realised that fifty years ago there were three big discoveries. All eventually led to Nobel Prizes. Each discovery had a profound effect on the formation of condensed matter as a distinct discipline built around a few unifying concepts. At the time the discoveries and ideas appeared quite independent, but there are deep connections between them.

Renormalisation group and critical phenomena

In 1971 Ken Wilson published two papers  [PRB 4, 3174, and PRB 4, 3184] laying the foundations, followed by two PRLs in 1972, including one with the provocative title, Critical Exponents in 3.99 Dimensions. 

Wilson received the Nobel Prize in 1982. This work had many implications and applications. 

Explained universality in critical phenomena.

Highlighted how spatial dimensionality changes physics.

Illustrates why effective Hamiltonians work (so well).

Showed the power of quantum field theory techniques.

Defined concepts of scaling and fixed points.

Superfluidity in liquid 3He

In 1972,  Osheroff, Richardson, and Lee reported new phase transitions in liquid/solid 3He. Tony Leggett identified these transitions as due a superfluid phases and also identified the order parameters. The experimentalists shared the Nobel Prize in 1996 and Leggett in 2003. The discovery was significant for many reasons, beyond just being a new state of matter.

It provided a rich example of a state of matter with multiple broken symmetries. The order parameter has eighteen components, which can be viewed as a combined superfluid, ferromagnet, and liquid crystal.

The rich order parameter led to an exploration of diverse topological defects, from superfluid vortices with magnetic cores to boojums. This highlighted the concepts of broken symmetry, rigidity, and topological defects.

This was the first example of an unconventional fermionic superfluid. Specifically, it could be described by BCS theory, but not with s-wave pairing nor with the pairing mechanism of the electron-phonon interaction in elemental superconductors. This showed the adaptability of BCS theory. It laid the groundwork for understanding unconventional superconductivity in heavy fermions, organics, and cuprates.

Berezinskii-Kosterlitz-Thouless phase transitions

In Berezinskii published papers in 1970 and 1971, and Kosterlitz and Thouless published papers in 1972 and 1973. This work was significant for reasons including the following.

It showed states of matter and phase transitions were qualitatively different in two and three dimensions.

New concepts such as topological order, quasi-long-range order, essential singularities, and defect-mediated phase transitions were introduced.

Like that of Wilson, this work highlighted universality. There were connections between superfluids, superconductors, and XY magnets.

Scaling equations provided insight.

Kosterlitz and Thouless were awarded the Nobel Prize in 2016

We should celebrate!

Wow! Quite the Golden Jubilee!

Does anyone know of any conferences, events, or books that are planned to mark these anniversaries?

Will there be big new discoveries in condensed matter physics?

 There are two aspects to this question concerning the future of condensed matter physics. First, are there big things to be discovered? If yes, will they be discovered?

I believe the first answer is yes for two reasons. First, the past hundred years have given us a continual stream of discoveries, many of them unexpected. Every time that things get a little boring, pretty soon there is something exciting and new. Second, condensed matter physics is all about emergent phenomena in materials. Emergent phenomena are extremely hard to anticipate or predict. Because of the combinatorics of chemistry, the list of possible materials to study is endless. CMP presents an endless frontier to explore. However, just because such a frontier exists does not mean that it will be explored. Successful explorers require courage, creativity, resources, time, and freedom.

I am concerned that the wild frontiers of condensed matter may not be explored. It is worth reflecting on who were some of the pioneers of CMP and the character of their institutional environments.  Consider Kammerlingh Onnes, Landau, Kapitsa, Anderson, de Gennes, and Leggett. Some common elements of the context (institutional, historical, political) in which they made their discoveries were time, stability, job security, mental space, and intellectual freedom. For example, Anderson spent almost three decades at Bell Labs in its heyday. Thanks to the monopoly of Bell in providing telephone services in the USA, the parent company had a very secure and stable income, providing it the ability to provide substantial financial and institutional support for basic research.

These pioneers played a long game. They had the freedom to fail, to choose research topics, and to change directions. They did not follow fashion and were fiercely independent thinkers. Andrew Zangwill highlights this about Anderson in his biography. They largely had the resources they needed and did not have to worry or fight for funding. Their daily life was very different from that of a researcher today. Their mental space was not filled with an endless stream of distractions such as emails, grant proposals, conferences, reporting, reviewing, committees, metrics, ... Most of their time and mental energy was simply focused on curiosity-driven research. 

Today, there is intense competition for funding, institutional status, and career benefits associated with obtaining it, and a pressure to produce in the short term "outputs" (papers) and "impact" (citations) and "national benefit" (technological, commercial, security, and social). This naturally leads to researchers working on "safe" projects in fashionable areas that they are confident will produce results in the short term.

I hope that I am wrong. But, I fear that great discoveries may be missed.

Why Josephson matters

Reflecting on macroscopic quantum effects in condensed matter I have come to the view that Brian Josephson is a key figure. But, the observation of magnetic flux quantisation in superconducting cylinders is also a landmark.

The significance of Josephson is nicely laid out in a fascinating article published by in Physics Today in 2001 by Donald G. McDonald

The Nobel Laureate Versus the Graduate Student 

John Bardeen, the leading condensed matter theorist of his day, was quite wrong when he dismissed a startling prediction by the unknown Brian Josephson. 

The article nicely lays out several important precursors to Josephson's work that all occurred after BCS theory in 1957.

1. The experimental (unanticipated) discovery by Ivar Giaever in 1960 of single-particle tunneling in SIS junctions [superconductor-insulator-superconductor sandwiches]. I-V curves clearly showed the structure of the BCS energy gap.

[Aside. This discovery was also laid the foundation for John Rowell's tunneling experiments that allowed a quantitative (strong-coupling BCS) analysis of the electron-phonon interaction responsible for superconductivity.]

2. The discovery by Hans Meissner [not the discoverer of the Meissner effect!] in 1960 of the proximity effect, where superconductivity is induced in a non-superconducting metal, by close proximity to a superconductor.

3. The discovery in 1961 by two independent experimental groups that the magnetic flux inside a cylinder was quantised in units of h/2e where h is Planck's constant and e is the electronic charge. This effect had been predicted by Fritz London in 1948, albeit without the factor of 2.

These experiments provided ``the first direct demonstration of a macroscopic quantum effect.''

The data above is from a 1971 paper, observing flux quantisation within one-half of a per cent.

A nice article on the history of the discovery is

50 Years of Fluxoid Quantization: 2e or Not 2e by Dietrich Einzel 

 Aside. These experiments also clearly showed the physical nature of the magnetic vector potential, A, and illustrated the Aharonov-Bohm effect.

4. Josephson's attendance at a series of lectures ``Concepts in Solids" that Phil Anderson gave to graduate students at Cambridge in 1961-1962. In particular, at the end, Anderson introduced the concept of broken symmetry as an organising principle to describe ``condensed systems" such as antiferromagnets, superfluid 4He, ferroelectrics, and superconductors.

Distinctly quantum phenomena are tunneling, superposition (and the associated coherence and interference), and entanglement. Josephson junctions can be used to illustrate all of these at the macroscopic scale.

This is explored in a nice autobiographical article by Tony Leggett.

Matchmaking Between Condensed Matter and Quantum Foundations, and Other Stories: My Six Decades in Physics

Because of the strong prejudice in the quantum foundations community that it would never be possible to demonstrate characteristically quantum-mechanical effects at the macroscopic level, this assertion made us [Leggett and Garg] the target of repeated critical comments over the next few years. Fortunately, our experimental colleagues were more open-minded, and several groups started working toward a meaningful experiment along the lines we had suggested, resulting in the first demonstrations (29, 30) of MQC [Macroscopic Quantum Coherence] in rf SQUIDs (by then rechristened flux qubits) at the turn of the century. However, it would not be until 2016 that an experiment along the lines we had suggested (actually using a rather simpler protocol than our original one) was carried out (31) and, to my mind, definitively refuted macrorealism at that level. I find it rather amusing that nowadays the younger generation of experimentalists in the superconducting qubit area blithely writes papers with words like “artificial atom” in their titles, apparently unconscious of how controversial that claim once was.

The 90% University

A helpful starting point for me when considering universities after the pandemic is the cover story of The Economist, from two weeks ago, The 90% Economy. They were reflecting on the world economy will change in the coming years, suggesting three key characteristics.

 1. the economy will be more fragile 
 2. there will be less innovation 
 3. there will be even greater inequality 

The reason that it is called the 90% economy is because in the next few years rather than growing a few percent each year it will decrease in size by about 10%. 
Now on one level that doesn't sound too bad, but the problem is that it is not a uniform decrease across every sector, company, and individual. The changes will be quite heterogeneous. Rather, there will be significant gaps, that because of the interconnectedness of everything there will be problems. 

Just like the economy going back to ``normal'' universities will continue to have students, continue to do teaching, continue to graduate people but things, won't be quite the same, in some quite significant ways. It is not just a matter of possible 10-20 percent budget cuts.

1. Universities will be more fragile

Less stability and predictability is particularly bad for universities. Key ingredients for universities to achieve their real purpose are time and stability. Significant research, teaching, and learning all require time to build up knowledge, explore different possibilities, make mistakes, and not be distracted by crises, whether personal, institutional, or societal. Economic uncertainty is just one of many dimensions to the forthcoming fragility.

2. There will be less innovation in universities

Research is all about innovation, discovering new things, and exploring new ideas.  Drawing on The Economist article. More virtual meetings, less international travel, and fewer face-to-face meetings mean less brainstorming. Less random meetings and informal interactions will mean fewer new ideas. There will also be less money available for discretionary funding for new initiatives and fewer startup funds for new faculty, even if hiring freezes end.
After any crisis, people are more cautious and more risk-averse. A problem before the pandemic was that science was increasingly being done in a very risk-averse manner. People, particularly those without tenure, focus on low-lying fruit, working on problems that they are pretty sure they can solve in a year or less. Even senior people can only get funding if they have a ``track record'' in an area. This means they will just keep doing the same thing and not move into new areas. 
In reviewing his scientific life, Tony Leggett recently made the following comments.

Indeed, when I look back on ... a 60-year career in physics, I think I have been fortunate in many ways. I have had a marvelous constellation of graduate students and postdocs, from all corners of the globe... But if I had to pick out one thing that made all the difference, particularly in the early stages, it would be the tolerant and relaxed environment that I experienced at Sussex when starting there in the late 1960s. When I recall this and then look around at the current environment for people at the postgraduate, postdoc or junior faculty level, I feel quite concerned ... I get the impression that many of them feel that there will be no hope of obtaining the kind of postdoctoral/faculty/tenured position .. unless they have not only published three or four papers but published them in high-impact journals... I fear that one almost ineluctable outcome is that there is a strong temptation to focus all one’s energy on problems that can be reasonably guaranteed to yield results within the relevant time frame, typically two or three years. And almost by definition, these are not the really worthwhile problems! ...  the best advice I can give to any younger colleagues who seek my opinion is deliberately to put aside some fraction (30%, 25%, even 20%) of their research time for problems that they not only are not sure they can solve within the two- or three-year deadline but are not even sure that they (or anyone) can solve at all.

A similar conclusion can be drawn from a brilliant podcast, The Obscure Virus Club, by Malcolm Gladwell.

3. There will be more inequality, both within universities and between universities

This is part of the social tragedy that the rich get richer and the poor get poorer, particularly during and after a crisis, such as a pandemic. The poor do not have the resources to adapt and survive. If you live in a slum you can't practice social distancing. If you don't have access to clean running water or disinfectant it is hard to practice personal hygiene.  If you lose your job you can't use your savings to go and get training for a new job. In contrast, if you are wealthy and have significant cash reserves you can wait things out, see your competitors fail, and snap up cheap investments. Similarly, with universities, it's hard for me to believe that institutions such as Harvard with massive endowments won't come out of the pandemic relatively stronger. Weak institutions will fold. Others will really struggle to survive and go through periods of stagnation.

There will also be inequality within institutions related to access, gender, and seniority. Because of the background of an economic downturn it will be harder for students from poorer backgrounds to afford tuition or access scholarships. Furthermore, poor job prospects will make the financial cost and risks of student loans not seem worthwhile. There will also be a push within universities to increase the number of adjunct faculty (i.e. people on short-term teaching contracts with no benefits). People with tenure who are well established will do fine because they also have a good strong social and professional networks. In a more online environment, it's harder to build those professional networks and so disparities may increase. The Economist article mentions a study that found the productivity of female economics faculty, as measured by the production of research papers, fell relatively to male ones, since the pandemic. That's arguably because women are more likely to have to take care of homeschooling and entertainment of bored children during the lockdown.

All this is a bit depressing. However, with a crisis, there are always opportunities. There will be plenty of opportunism (where people exploit a situation without regard to moral considerations and the impact on others). But, there will also be opportunities for the wealthy, powerful, and privileged to do good and facilitate much-needed changes in universities. For example, large philanthropies and wealthy universities can make long-term investments that others won't or can't. Well-established faculty can provide support/cover to junior faculty, students, and postdocs. In the longer term of decades, these will likely be the institutions and individuals at the forefront of what universities really should be about: thinking, writing, teaching, and learning, at the deepest level.

Mean-field theories: helpful or misleading? From Hubbard to COVID-19 models

Mean-field theory (self-consistent field theory) is incredibly valuable. It gives significant insights into what is possible with a particular model.
What kind of phases and broken symmetries may be possible?
How does the phase diagram depend on different parameters in a model?
Indeed, mean-field theory is the basis of the whole Landau paradigm for spontaneous symmetry breaking and phase transitions.
Implementations of Density Functional Theory (DFT) in computational materials science are basically mean-field theories. Most of computational quantum chemistry involves some sort of mean-field theory.

Mean-field theories do not take into account fluctuations, dynamic or spatial.
Basically, a many-body problem is reduced to a one-body problem.

A good mean-field theory can win you a Nobel Prize. That's what Anderson, BCS, Ginsberg, Abrikosov, and Leggett all did!
Can you think of others?

However, mean-field theory does have its limitations.
It is usually quantitatively wrong. It often gives unreliable values for transition temperatures. In spatial dimensions less than four, mean-field theory gives the wrong values for the critical exponents near a phase transition.

An even bigger problem is that mean-field can be qualitatively wrong.
For many models (e.g. the Ising model or Heisenberg model) mean-field theory always gives a transition from a disordered to an ordered phase at a non-zero temperature.
However, in one dimension the Ising model has no phase transition in one dimension. For a Heisenberg ferromagnet or antiferromagnet, there is no transition at finite temperature in two dimensions.
The Mermin-Wagner theorem states that in two dimensions a superconductor or superfluid never has long-range order at finite temperature. Instead, there is a Kosterlitz-Thouless transition, to a distinct state of matter, with power-law correlations.

Mean-field theory can also fail to predict the existence of states of matter. For example, for Hubbard models, mean-field theory can produce several states: a Fermi liquid metal, a ferromagnetic metal, an antiferromagnetic metal, and a spin-density-wave insulator. But it is quite possible the model also can have non-magnetic Mott insulating phases, superconductivity, non-Fermi liquid metals, and pseudogap states.

In the next post, I will discuss some issues that arise in mean-field theories used in modeling the COVID-19 epidemic.

A brilliant insight about quantum decoherence in electronic circuits

Yesterday, Matthew Woolley gave an interesting Quantum science seminar at UQ about some of his recent work on Photon assisted tunnelling with non-classical light.

I just want to focus on one point that was deeply imbedded in the talk. It is a idea that is profound and central to the physics of quantum electronic circuits. The idea is so old now its profoundness and brilliance may be lost on a new generation.
The idea and result is easiest for me to explain in terms of the figure below which describes a superconducting (Josephson junction) qubit connected to an electrical circuit. It is taken from this review.

One can quantise the electromagnetic field and consider a spin-boson model to describe decoherence and dissipation of the qubit. This is associated with a spectral density that is proportional to frequency with a dimensionless pre factor alpha, which for this circuit is given by

where R_V is the electrical resistance of the circuit, R_K is the quantum of resistance, and the C's are capacitances.
Similar physics is at play in normal tunnel junctions (see for example this important paper, highlighted by Matthew in his talk).

Why do I find this profound?
First, this is a very simple formula that depends only on macroscopic parameters of the electrical circuit. One does not have to know anything about the microscopic details of  all the different electronic degrees of freedom in the circuit or how they individually couple to the qubit. I find this surprising.
Second, the underlying physics is the fluctuation-dissipation theorem. The quantum noise in the electronic circuit is related to fluctuations in the current. By Kubo and the fluctuation-dissipation relation tell us the fluctuations in the current are essentially the conductivity [the inverse of the resistivity].

Who was the first to have this insight and calculate this?
I feel it was Caldeira and Leggett, but I can't find the actual equation with the circuit resistance in their 1983 paper.
Or did someone else do this earlier?

Because of the above, whenever the spectral density depends linearly on the frequency, Leggett (and now everyone) calls it ohmic dissipation.

I first learnt this through the thesis work of my student Joel Gilmore, and described in this review. There we considered a more chemical problem, two excited electronic states of a molecule that are in a polar dielectric solvent. The coupling to the environment is completely specified in terms of the frequency dependent dielectric constant of the solvent (and some geometric factors).

Update: Caldeira answers the question in a comment below.

The uncertain status of career moves

An interesting question is: to what extent does the local institutional environment and the status of an institution affect the quality of the science done by an individual?
If I move to a more highly ranked institution will I do better science?
Or, if I move to a more lowly ranked institution will the quality of my work decline?

Some scientists are obsessed with "moving up", thinking that being at the "best" place is essential. They cannot fathom that one could do outstanding work at a mediocre institution.
However, consider the following. People at a high status university may get Nobel Prizes but that is not necessarily where they actually did the prize-winning work. Here are a few examples.

John Van Vleck: Wisconsin to Harvard
Joe Taylor: U. Mass to Princeton
Tony Leggett: Sussex to Urbana
William Lipscomb: Minnesota to Harvard

Can anyone think of other examples?

So can one actually measure how career moves affect the quality of science? One recent attempt is
Career on the Move: Geography, Stratification, and Scientific Impact
Pierre Deville, Dashun Wang, Roberta Sinatra, Chaoming Song, Vincent Blondel & Albert-László Barabási

The authors give an exhaustive analysis of the authors, affiliations, and citations of more than 400,000 papers from Physical Review journals, concluding

while going from elite to lower-rank institutions on average associates with modest decrease in scientific performance, transitioning into elite institutions does not result in subsequent performance gain. 

This made it into an article in the Economist magazine, entitled Why climb the greasy pole?
It is worth looking at the figure that this conclusion is based on, noting the size of the error bars.

The vertical axis is the change in citations and the horizontal axis the change in university ranking.

What role does reasoning by analogy have in science?

Two weeks ago I went to an interesting history seminar by Dalia Nassar that considered a debate between the philosopher Immanuel Kant and his former student Johann Gottfried von Herder.
Kant considered that thinking by analogy had no role in science whereas Herder considered it did. Apparently, for this reason Kant thought that biology [natural history] could never be a real science. Thinking objects were fundamentally different from non-thinking objects.

One of the reasons I like going to these seminars is that they stimulate my thinking in new directions. For example, a seminar last year helped me understand that one of my "problems" is that I view science as a vocation rather than a career, perhaps in the tradition of Robert Boyle and the Christian virtuoso.

After the seminar I had a brief discussion with some of my history colleagues about what scientists today think about analogy. I think it plays a very important role, because it can help us understand new systems and phenomena in terms of things we already understand. But where people sometimes come unstuck is when they start to assume that the analogy is reality or the complete picture. Here are a few important historical examples.

   * Electromagnetic radiation. The analogy of light waves with sound and water waves helped. But went array when people thought there must be a medium, i.e. the aether.

  * Quantum mechanics. Particles and waves. Again the analogy helped understand interference and quantisation of energy levels. But I also think that pushing to hard the partial analogies with classical mechanics and classical waves is the source of some of the confusion about quantum measurement and the quantum-classical crossover.

  * Quantum field theory and many-particle physics. Feynman diagrams, path integrals, renormalisation, symmetry breaking, Higgs boson,…. there is a lot of healthy cross-fertilisation.

 * Imaginary time quantum theory and classical statistical mechanics. Path integral = Partition function.

Coincidentally, yesterday when I was in the library [yes, the real physical library not the virtual one!] trying to track down Wigner's quote I stumbled across a 1993 Physics Today review by Tony Leggett  of Grigory Volovik's book Exotic properties of superfluid 3He. Leggett expresses his reservations about analogies.

As to the correspondences with particle physics, being the kind of philistine who does not feel that, for example, his understanding of the Bloch equations of nmr is particularly improved by being told that they are a consequence of Berry's phase, I have to confess to greeting the news that the "spin-orbit waves" of 3He-A are the analog of the W boson and the "clapping" modes the analog of the graviton with less than overwhelming excitement. These analogies no doubt display a certain virtuosity, but it is not clear that they actually help our concrete understanding of either the condensed matter or the particle-physics problems very much, especially when they have to be qualified as heavily as is done here.

What do you think? Does analogy have an important role to play? When does it cause problems?

Are quantum effects ever enhanced in condensed phases?

Previously, I asked the question: are there any condensed phase systems in chemistry where quantum effects [e.g. tunneling, interference, entanglement] are enhanced compared to the gas phase?

Let me clarify. Suppose we take a molecular system X and consider the magnitude of some quantum effect Y in the gas phase. We then put X in some condensed phase environment [e.g., solvent, protein, or glass] and measure or calculate Y.
It is quite possible that Y increases due to what I would call "physically trivial" effects, e.g. a change in the geometry of X which makes Y larger. For example, the polarity of a solvent can decrease the donor-acceptor distance for proton transfer in a molecule and thus increase quantum tunnelling.

To me a physically "non-trivial" effect is where the environment enhances the quantum effect for the same reference system X [e.g., one uses the same geometry of the molecule in the gas and condensed phases]. I am not sure this ever happens. Generally, environments decohere quantum systems.

But I stress that such environmental effects can be far from "trivial" to chemists and biologists. e.g., they can make an enzyme work!

Historically, this distinction between "trivial" and "non-trivial" environmental effects was important and confusing for the Caldeira-Leggett model for quantum tunneling in the presence of an environment. To clarify this issue Caldeira and Leggett added a "counter-term" to the Hamiltonian to subtract off the renormalisation of the potential barrier by the environment. This is discussed in detail in the book by Weiss [page 19 in the second edition]. Chemists call this "counter-term" the solvation energy.

Aside: in nuclear physics these renormalisation effects are observable and calculable, as described in this PRL.

The wealth of a poor man's scaling

Chapter 3 of Hewson's The Kondo Problem to Heavy fermions reviews Anderson's poor man's scaling treatment of the Kondo model.

Starting with the anisotropic Kondo Hamiltonian

one rescales the electronic bandwidth D and see how the interactions J_z and J_ and J+ rescale.

To lowest order in perturbation theory this leads to the renormalisation group equations

Solving these gives the flow diagram below

A few important consequences

1. Antiferromagnetic (AFM) interactions flow to strong coupling.

2. The Kondo energy/temperature is invariant to the flow.

3. This is an example of asymptotic freedom [interactions can weaker at higher energies].

It is impressive that Anderson did this before Wilson and Fisher used renormalisation group ideas to describe critical phenomena in classical phase transitions.

It is fascinating that the same flow equations and flows describe the Kosterlitz-Thouless phase transition associated with topological order [vortex pair unbinding] in a classical two dimensional superfluid.

The spin boson model which describes the quantum decoherence of a single qubit in an ohmic environment can be mapped to the anisotropic Kondo model and so is also described by the same flow equations [See this famous (and rather dense) review by Leggett et al.]

Desperately seeking a way to find order parameters

Much of condensed matter physics is concerned with finding the relevant order parameter for new phases of matter. Indeed this is a good way to win a Nobel Prize! This is much of what was done by Ginzburg, Neel, Leggett, de Gennes,....

A fundamental and controversial question is whether one can a priori predict new order parameters. Historically, the progression has always been:

1. Experimental discovery of a new phase of matter.
2. Proposal of an order parameter and a phenomenological (Ginzburg-Landau) theory to explain a range of experiments.
3. Proposal of an effective Hamiltonian which has a ground state with the desired spontaneous symmetry breaking and associated order parameter.
4. Justification of the effective Hamiltonian from so-called "ab initio" electronic structure calculations starting with Schrodinger's equation and the actual chemical composition of specific materials. 

The grand challenge is to invert this process, even just one step.

Laughlin and Pines seem to claim that this is essentially impossible.

There are some interesting fundamental philosophical questions as to whether the obstacles are ones of practical difficulty versus fundamental physics.

There is really interesting 2006 PRL, Systematic Derivation of Order Parameters through Reduced Density Matrices, by Shunsuke Furukawa, Grégoire Misguich, and Masaki Oshikawa.
Essentially they claim to have found a way to go from 3. to 2. above. In particular, given the results of an exact diagonalisation calculation of the low lying states of a lattice model they give a procedure to find the order parameter from looking at two nearly degenerate ground states.
They then apply the method to two concrete examples: a Heisenberg spin model on a ladder with ring exchange, and a quantum dimer model on the Kagome lattice. The method gives the correct order parameters in the first case and for the second shows there is no order parameter. I found this quite impressive and promising.

The PRL also promises future work generalising the method to more than two degenerate ground states and suggests application to frustrated two-dimensional quantum antiferromagnets. Unfortunately, I have not been able to find such work.

Lecture on superfluids

Last week I gave half a lecture on the phase diagram of helium and superfluidity to a second year undergraduate class on Thermodynamics and Condensed Matter Physics. The lecture includes:

• a discussion of the differences between the phase diagrams of 3He and 4He
• a list of the 4 Nobel Prizes awarded for work in superfluidity [Landau, Kapitsa, Osheroff, Richardson, and Lee, Leggett] (I did not include BECs but should have]
• a video of the superfluid transition
• a discussion of an amazing experiment on the space shuttle which determined the critical exponent for the specific heat at the superfluid (lambda) transition to five significant figures.

Tunneling without instantons?

This attempts to answer questions raised in a previous post.
Here is one point of view.
Tunneling is always present and as one lowers the temperature (or increases the coupling to the environment) one just has a crossover from transitions dominated by activation over the barrier to tunneling under the barrier. Instantons [or the "bounce solution" which is a solution to the classical equations of motion in an inverted potential] are just a convenient calculational machinery which arises when evaluating a path integral approximately by finding saddle points. There is always a contribution from the trivial solution corresponding to the top of the barrier. Quadratic fluctuations about this saddle point give a "prefactor" which includes quantum corrections due to tunneling and reflection.

Below the crossover temperature T0 this first saddle point becomes unstable and there is  a second saddle point, which is the instanton solution.

I thank Eli Pollak for sharing his thoughts on this subject.

But all this seems against the spirit of the approach to tunneling in dissipative environments, pioneered by Leggett [and reviewed in detail here], which seems to assert that tunneling only exists when instanton solutions are present.
Perhaps, the key distinction is that the instanton captures coherent tunneling whereas the quadratic fluctuations only capture incoherent tunneling. Specifically, if one considers a double well system, the instanton can capture the level splitting associated with tunneling.

I am keen to hear others perspectives.

Do you really think the cat is really dead?

I just re-read most of Tony Leggett's review article, Testing the limits of quantum mechanics: motivation, state of play, prospects. He considers that different interpretations of quantum mechanics broadly fall into three classes.

1. The statistical interpretration.
This claims that quantum state amplitudes (i.e. wavefunctions) have no reality but are merely a calculational device to calculate the probabilities of the outcome of measurements. Questions about whether a cat is dead or alive before it is measured are
ruled to be not "meaningful." The most "trenchant" advocate is Leslie Ballentine.

2. The orthodox interpretation.
QM amplitudes are "real" at the microscopic level but effectively not real (or at least not relevant) at the macroscopic level. This is claimed to be because of decoherence one can "For All Practical Purposes" (FAPP) never observe quantum interference between macroscopically distinct states.

3. The many-worlds interpretation.

Quantum states do exist in "reality". But when I think I measure the cat is dead this is actually an "illusion" because the alive state is "equally real." I am just unaware of it.

Leggett's view is that 1. and 3. are largely hermeneutical gymnastics that allow one to avoid discussing the problem. His agenda is to propose experimental tests of 2.