The legacy of 40 years of cuprate superconductivity

In February 1986, Bednorz and Müller made a stunning discovery: superconductivity at a temperature of 35 K in a doped copper oxide (cuprate). Arguably, this discovery changed condensed matter physics. In April 1986, they submitted their results to Z. Phys. B. Only nineteen months later, they were awarded the Nobel Prize in Physics, the shortest time ever between a discovery and the award. A nice and short review of the history is here.

One measure of my estimate of the influence of this discovery is that it received about 5 pages of coverage in my Condensed Matter Physics: A Very Short Introduction. (See Chapter 5, Adventures in Flatland).

How things have developed over the past forty years, for better and worse, may be representative of how science advances: discovery by serendipity, hype about applications, unexpected secondary benefits, foundational questions, new concepts, unification, and incremental advances.

Hype about technological applications

On March 20, 1987, The New York Times had a front-page article, DISCOVERIES BRING A 'WOODSTOCK' FOR PHYSICS, by James Gleick. This followed the 1987 APS March meeting. It began 

"Physicists from three continents converged on the New York Hilton for a hastily scheduled special conference on a string of discoveries that seem certain to produce a rapid cascade of commercial applications in electricity, magnetism and electronics.There are many things we know and understand that we did not when they were first discovered."

This has largely been unfulfilled. There are a few niche applications, but cuprates are not used in electricity distribution or even in the superconducting magnets in hospital MRI machines, which are probably the main commercial application of superconductors. One of the significant obstacles is that it is hard to make wires from these materials, as they are ceramics. This is an example of the common gap between research laboratory science and commercially viable technology.

After 40 years, do we have a successful theory?

It depends on who you ask. But I would say there is a lot we do understand.

We have a phenomenological theory for all the macroscopic phenomena associated with the superconducting state: Ginzburg-Landau theory!

Properties of the superconducting state are well-described by a BCS wavefunction with a d-wave order parameter and the associated Bogoliubov quasiparticles. [This is somewhat puzzling, as in the metallic state quasi-particles are not well defined].

Although not everyone agrees, I think it is fair to say that the essential physics is in a one-band Hubbard model, and the key physics is:

strong electronic correlations,

a doped antiferromagnetic Mott insulator,

d-wave pairing that is "mediated"/caused from some mixture/variant of antiferromagnetic spin fluctuations or RVB spin singlets,.....

We certainly don't understand the cuprates at the same level as elemental superconductors. But we do understand the essential physics.

What is harder to describe and understand are the states adjacent to the superconducting state in the phase diagram: the pseudogap state and the strange metal.

Strongly correlated electron materials became a large, vibrant and unified field

Before 1986, there were small, disconnected communities intermittently interested in transition metal oxides, rare earths, Kondo impurities, Mott metal-insulator transitions, organic superconductors, heavy fermions, and quantum antiferromagnets.

The discovery of the cuprates brought together these communities as they found common interests, challenges, questions, concepts, and techniques.

The discovery of superconductivity in strontium ruthenate, alkali fullerides, iron pnictides and chalcogenides, twisted bilayer graphene and more cuprates, organic charge-transfer salts, and heavy fermions has shown how rich these systems are. The challenge is to understand the similarities and differences between these chemically and structurally diverse systems. In many of them, superconductivity is proximate to a Mott insulating state.

The unity and excitement were probably stimulated and enhanced by the activities and ideas of high-profile theorists such as Anderson, Schrieffer, Scalapino, Pines, Rice, and Varma. On the other hand, their acrimonious disagreements probably did not help.

Secondary theoretical benefits

The things I list below were not new ideas when the cuprate discovery happened. However, interest in the cuprates led them to become major research themes and ideas.

Importance of phase diagrams, including as a function of interaction parameters in toy models

Highlighting the limitations of electronic structure methods based on Density Functional Theory with approximate Exchange-Correlation functionals (i.e., anything computational). In the presence of strong correlations, DFT methods have spectacular failures. For example, predicting a metallic state instead of the Mott insulator.

Low dimensionality leads to qualitatively different behaviour, including the possibility of new types of order and quasiparticles. This is most dramatic in one dimension, where one has Luttinger liquids and spin-charge separation.

Spin liquids. Landau was wrong. Spontaneous symmetry breaking does not always occur in antiferromagnets.

Non-Fermi liquids. Landau was wrong. Not all metals are Fermi liquids.

Quantum criticality. Although this is a robust concept for certain toy models, whether it is relevant to the cuprates remains contentious.

Systematic improvements in approximation schemes and numerical techniques - exact diagonalisation, DMRG, DMFT, quantum Monte Carlo,...

Emergence. Chemical complexity and strong interactions can lead to new states of matter.

Secondary experimental benefits

Better probes. The desire to characterise the cuprates helped drive significant improvements in the resolution of ARPES (Angle-Resolved PhotoEmission Spectroscopy), STM (Scanning Tunnelling Microscopy), and inelastic neutron scattering. These advances have born fruit in the study of a wide range of other materials, beyond the cuprates.

Growth of single crystals. The early days of the cuprates produced a lot of junk experimental results because of the poor quality of the samples produced by "shake and bake". However, the involvement of solid-state chemists has improved things. The techniques have also led to the production of single crystals for a wide range of strongly correlated materials.

Why is there so little research on cuprates today?

Today, there is little research directly on cuprates, both theoretically and experimentally. It is hard to get funding to work on them, even though there is a lot we don't understand really well.

This is because of the problem of fashion in science. The low-lying fruit has been picked. There is a continuous new stream of materials being discovered with exotic properties, the latest being twisted bilayer van der Waals compounds.

Angle-Dependent Magnetoresistance as a probe of Fermi surface properties in cuprates

About twenty-five years ago I became interested in how the Fermi surface of the metallic state of organic charge-transfer salts could be mapped out by measuring the interlayer resistance as a function of the direction of a large applied magnetic field. [A nice review from 2004 is by Mark Kartsovnik]. Later this technique was used for a range of other metals including strontium ruthenate, iron pnictides, semiconductor heterostructures, and finally cuprates, mostly in the overdoped region.

For the cuprates, it was discovered that one could not only map out the shape of the intralayer Fermi surface, but also anisotropies in the scattering rate and the interlayer hopping integral. Of particular interest was the finding that the overdoped cuprates were not simple Fermi liquids, as usually claimed, but more like anisotropic marginal Fermi liquids.

It should be stressed that the Fermi surface information is extracted indirectly by comparing experimental curves of angle-dependence to calculations based on different models for the shape of the Fermi surface, anisotropies in the scattering rate, and interlayer hopping. Thus, there is a fair bit of curve fitting to determine the parameters of the model. However, when one has observations at several magnetic fields, temperatures, and curves for the angle dependence in all directions, there are a lot of constraints, and specific anisotropies tend to produce some specific qualitative features in the shapes of the curves. Examples are shown below, taken from the Nature paper referenced below.

Recently, measurements have been reported on samples of the cuprate Nd-LSCO 

[La1.6−xNd0.4SrxCuO4] at dopings of p=0.21 and p=0.24, lying on both sides of the putative quantum critical point at p=0.23. 

The differences between the ADMR at these two dopings are analysed quantitatively in a preprint, which claims to show that at p=0.21 the Fermi surface is reconstructed due to (pi,pi) ordering. This is important as it relates to the fundamental question as to the origin of the pseudogap state.

Fermi surface transformation at the pseudogap critical point of a cuprate superconductor

Yawen Fang, Gael Grissonnanche, Anaelle Legros, Simon Verret, Francis Laliberte, Clement Collignon, Amirreza Ataei, Maxime Dion, Jianshi Zhou, David Graf, M. J. Lawler, Paul Goddard, Louis Taillefer, B. J. Ramshaw

Submitted on 3 Apr 2020 (v1), last revised 26 Nov 2020 (v2)

Aside: There is also a Nature paper, Linear-in temperature resistivity from an isotropic Planckian scattering rate, by the same group that compares the p=0.24 observations to those on the overdoped cuprate Tl2201 [p=0..29]. The arxiv notes "substantial text overlap" between the preprint above and the preprint for the Nature paper. [Figure 2 in v1 of the preprint above is in the Nature paper].

Here I focus on the first preprint as it stimulated a nice theory preprint

Interpreting Angle Dependent Magnetoresistance in Layered Materials: Application to Cuprates

Seth Musser, Debanjan Chowdhury, Patrick A. Lee, T. Senthil

They present a strong case against the main claim of Fang et al. that their ADMR data supports a reconstructed Fermi surface for the p=0.21 system.

There are several nice things about this preprint.

1. It shows how one should be careful about interpreting ADMR

2. It highlights the possible role of an anisotropic quasi-particle weight, Z(phi), where phi denotes the position on the intralayer Fermi surface, not the direction of the field. Anisotropy can arise from correlation effects and or "coherence factors" associated with Fermi surface reconstruction due to an ordered state. 

2. In their modeling, Fang et al. did not include the effects of Z(phi) and Musser et al. show that when it is included the qualitative differences in the ADMR that they claim arise due to the ordered state do not appear.

3. The authors consider a "toy" model for which some analytical results can be obtained. 

4. This provides some physical insight into the origins of the different features in the data, such as the peak around theta=40 degrees [It is just the magic angle associated with the average radius of the Fermi surface] and how the behaviour near theta=90 degrees depends on the relative size of different parameters [see especially equation (16)].

5. What is happening in this material may not be generic to the cuprates. "The van Hove filling in Nd-LSCO is located between the two dopings, p = 0.21 and p = 0.24, respectively. Thus what was a large Fermi surface centered at the Γ-point on the overdoped side will become a Fermi surface centered at (π, π) on the underdoped side, assuming no reconstruction occurs"

6. The most important insight is at the beginning of Section V. When the value of of the interlayer hopping integral t_perp(phi) averaged over the Fermi surface, changes from non-zero to zero an upturn in the ADMR at low angles (i.e. fields almost parallel to the layers) to a downturn. This suggests an alternative explanation for the transition seen in the preprint.

7. It highlights the often overlooked fact that observation of ADMR is not conclusive evidence of a three-dimensional Fermi surface. Using the parameters from the experimental preprint gives typical values of t_perp * tau ~ 0.1, and so the materials are far from the regime of a coherent three-dimensional Fermi surface.

I have a few minor comments

a. Like many others, the authors incorrectly credit with Yamaji explaining the magic angles associated with ADMR. However, Yamaji's explanation is not the correct one because it involves quantised orbits, whereas the effect is semi-classical, as explained by Kartsovnik, Laukhin, Pesotskii, Schegolev, and Yakovenko. 

b. Investigation of the role of small closed orbits when the magnetic field is almost parallel to the layers is credited to Schofield and Cooper. However, there was earlier and more detailed work by Hanasaki et al. Albeit, both of these papers consider the clean high field limit and so are of debatable relevance.

c. It would be nice to know the status of Fang et al., preprint on which this paper is based, particularly as the first authors of both are in the same department.

What do we really understand about cuprate superconductors?

 At a recent meeting of the condensed matter theory group at UQ we watched the first half of a Harvard (online) seminar that Steve Kivelson gave (at the end of 2020), What do we know about the essential physics of high temperature superconductivity after one third of a century?

As a springboard he takes Phil Anderson's final posting of the arXiv, Last Word's on the Cuprates, from the end of 2016. He was 93 years old then!

Kivelson considers that there are two things we really understand about the cuprates. The first, is that the d-wave superconductivity is intimately connected with the antiferromagnetism of the undoped materials.

The second, is that Tc, the superconducting transition temperature, is determined by thermal disordering of the phase of the order parameter for the superconducting state. This is in contrast to conventional superconductors, where Tc is determined by the amplitude of the order parameter vanishing. 

Kivelson's argument for the first point is based on nice work done a decade ago with Sri Raghu and Doug Scalapino, and that led to other work I have blogged about. It should be stressed that this work is a weak-coupling renormalisation group treatment and so the question remains as to whether the phase diagram for weak-coupling is adiabatically connected to that for strong coupling, which is the regime of the actual cuprate materials. In different works, as U/t increases from very small values to large values there are no phase transitions. Cluster Dynamical Mean-Field Theory (DMFT) studies give some confidence that this is the case. However, not everyone will be convinced by that.

The talk is worth watching, even if at times it gets a bit too technical. It is very important that we have such honest and open reflections about how much progress is (not) being made in a field. I largely agree with Kivelson, but do find the lack of progress rather discouraging and cannot see that this will be inspiring bright young graduate students to enter the field or for funding agencies to put more money into it.

The significance of the discovery of cuprate superconductivity

This is some draft text for Condensed Matter Physics: A Very Short Introduction. 

Feedback appreciated.

A big change in condensed matter physics occurred in the late 1980s due to an unexpected discovery that led to whole new areas of research. Why was this discovery so significant?

Superconductivity is amazing. At extremely low temperatures, many metals can conduct electricity without generating any heat. A ``holy grail’’ of physics is to discover a material that is a superconductor at room temperature. This could revolutionise the transmission of electricity. Until 1986, the highest temperature at which superconductivity was possible was about 23 K (-250 °C), in Nb₃Ge, a combination of the elements Niobium and Germanium. This requires cooling the material with liquid helium, which is expensive, and consequently limits commercial applications, such as MRI machines in hospitals.

In 1986, Alex Bednorz and Karl Muller, working at an IBM laboratory in Switzerland, investigated whether a chemical compound composed of the elements lanthanum, barium, copper, and oxygen (La, Ba, Cu, O) would be superconductor at a higher temperature. They were motivated by theoretical arguments that strong interactions between the electrons and the vibrations of the atoms could enhance the superconducting transition temperature, Tc. They found superconductivity below a Tc of 36 K, a new record. The material consisted of layers of copper and oxygen atoms and this lead to many other groups investigating similar classes of material. Within a year, materials were discovered with a Tc of 120 K (-150 °C). This was significant because superconductivity could be achieved by cooling with liquid nitrogen, which is about the same price as beer. Unfortunately, over the past thirty years, there has been little progress at increasing Tc to higher temperatures. Room temperature superconductivity remains a holy grail.

The discovery of Bednorz and Muller generated considerable excitement in the physics community, attracting many new researchers to superconductivity research. In March 1987 a special session was held during a regular meeting of the American Physical Society in New York City. A large ballroom was overflowing with more than one thousand physicists. I was one of thousands more outside watching on a TV monitor. Each speaker was only allowed three minutes to present their work and the session went long past midnight. This event was described on the front page of The New York Times as the ``Woodstock of Physics’’, in honour of the famous rock music festival held in New York state, and considered as emblematic of the 1960s.

Scientific history is full of serendipity. Some discoveries are accidental. It is now known that the reason that Bednorz and Muller chose to focus on this class of materials (strong interaction of electrons with atomic vibrations) was actually wrong. The high Tc does not arise from this interaction, but rather from a strong magnetic interaction between the electrons and from the two-dimensionality of the materials. The latter is a result of the layered crystal structure shown in Figure 5.2. Although the discovery was arguably somewhat serendipitous, its profound significance was shown in 1987, when Bednorz and Muller were awarded the Nobel Prize. In contrast, most scientists receive the prize decades after their ground-breaking discovery.

Figure 1. The repeat unit for the crystal structure of a superconducting copper oxide, BiSrCaCuO. A key ingredient is the layers of copper and oxygen atoms, which are isolated from each other by a large number of other atoms. The chemical complexity is reflected in the unit cell containing more than 50 atoms, including 5 different chemical elements.

From a fundamental physics point of view the superconductivity of these materials is not the only interesting and theoretically challenging property. The materials exhibit two new states of matter, known as the pseudogap state and the strange metal (see the phase diagram in Figure 5.3.). These conducting states have properties distinctly different from those found in common metals such as copper and gold.

Figure 2. Phase diagram of copper oxide superconductors. Temperature (T) versus doping. Doping describes the chemical composition of the material, particularly the density of charge carriers. The different states are antiferromagnet (AF), superconductor (SC), regular metal (FL), strange metal, and pseudogap.

Since 1986 more than ten thousand scientific papers have been published concerning possible theories to describe the different states in the phase diagram (Figure 5.3). Although some progress has been made, there is still no single accepted theory, and particularly no theory that has the simplicity and predictive power of the BCS theory of superconductivity in simple metals such as lead and tin. Developing a comprehensive theory remains one of the outstanding problems in Condensed Matter Physics.

Both experimental and theoretical studies suggest that all this rich new physics requires the low-dimensionality of these materials, i.e., they are almost living in Flatland.

But, why is low-dimensionality crucial? There is no simple explanation for this, but generally as the dimension of a system gets lower, the constituents fluctuate more (e.g. the atoms move around more), conventional orders become less stable, and new states of matter become possible.

Any comments?

Particularly how to make this more accessible and interesting to a general audience.

Signatures of strong vs. weak coupling in the superconducting phase?

Superconductivity in strongly correlated systems such as cuprates, organic charge transfer salts, and the Hubbard model presents the following interesting puzzle or challenge.

On the experimental side the superconducting phase can extend from a region of strong correlation (close proximity to the Mott insulator) to one of weak correlation (a Fermi liquid metal with a small mass enhancement).

On the theoretical side, one can obtain the d-wave superconducting state from a weak coupling approach (renormalisation group or random phase approximation) or a strong coupling approach such as an RVB variational wave function.
Aside: This also relates to the challenge/curse of intermediate coupling.

Given that in the two extremes the superconducting state emerges as an instability from two very different metallic states, the questions are:
What signatures or properties does the superconducting state (or "mechanism") have of these two distinct regimes (strong vs. weak coupling)?
Is it even possible that there is actually a phase transition (or at least a crossover) between different superconducting states?

Here is a partial answer, following this paper
Energetics of superconductivity in the two-dimensional Hubbard model 
E. Gull and A. J. Millis

In the weak coupling regime (smaller U, higher doping) the superconducting state becomes stable (as for traditional BCS theory) due to the fact that the potential energy decreases by more than the increase in kinetic energy.
In contrast, in the strong coupling regime (large U, lower doping, in the pseudogap region) the opposite occurs. The superconducting state becomes stable because the kinetic energy decreases by more than the increase in potential energy.
This is summarised in the figure below.

Aside: note how the condensation energy (the energy difference) is much less than the absolute values of the kinetic and potential energy. This highlights how, as often the case in strongly correlated systems, there is a very subtle energy competition. This is one reason why theory is so hard and why one can observe many competing phases.

I thank Andre-Marie Tremblay, Peter Hirschfeld and other Aspen participants for stimulating this post.

Deducing broken rotational symmetry from angle-dependent magnetoresistance

There is an interesting preprint
Broken rotational symmetry on the Fermi surface of a high-Tc superconductor 
B. J. Ramshaw, N. Harrison, S. E. Sebastian, S. Ghannadzadeh, K. A. Modic, D. A. Bonn, W. N. Hardy, Ruixing Liang, P. A. Goddard

They measure the interlayer magnetoresistance as function of magnetic field direction (see below) and from this deduce that the C4 symmetry of the crystal is broken to C2 in the charge density wave phase that occurs in the pseudogap region.

They then compare their experimental results to a calculation that uses a Fermi surface (that is reconstructed due to the CDW), a coherent three-dimensional Fermi surface, and a Boltzmann equation.

One might be concerned about the use of a three-dimensional Fermi surface because
a. the CDW correlation length between the layers is small
b. the interlayer charge transport is not necessarily coherent.

However, based on work I did long ago with Perez Moses and Malcolm Kennett [see for example this paper]. I think the theoretical results are robust to these concerns. What we showed is that for two contrasting situations shown below, the angle-dependent magnetoresistance is identical.

The top shows a coherent three-dimensional Fermi surface.
The bottom shows two layers that are coherently coupled together. The interlayer momentum is conserved in hopping between the layers.
One does not need coherence over more than two layers.

Another minor comment is that the authors did most of their calculations numerically. However, I think a lot can be done analytically using the expression below (from the Kennett paper) and simplifying for the case an isotropic scattering time (tau) and the low field limit (omega_c tau much less than one).

I thank Sam Lederer and Steve Hayden for bringing this work to my attention and asking about these issues.

Superconductivity in Aspen

For the next two weeks I am at the Aspen Center for Physics participating in a workshop on Superconductivity. A blog for the meeting captures its flavour, spanning a diverse range of systems and debates. I was not here for the first two weeks. Here are two related experimental results for the underdoped cuprates that have generated a lot of discussion.

1. A charge density wave (CDW) phase.

This has been observed directly with X-rays. The figure below is taken from this paper.

2. A jump in the charge carrier density versus doping.

Hall resistance measurements at high magnetic fields imply that for small doping the charge density scales with the doping p [p=0 corresponds to the Mott insulator that occurs at half filling] and at higher dopings, 1+p. This is summarised in the figure below from this paper.

A few comments.

1. Is the CDW phase relevant to understanding the pseudogap, superconductivity, and the strange metal phase?
There is debate about this. On the one hand it does compete with superconductivity. It could provide the much sought after quantum critical point below the superconducting dome in the phase diagram. CDW fluctuations could produce the pseudogap and/or strange metal properties. On the other hand, it may just be an "artefact" due to residual interactions that only become important when the magnetic field suppresses the interactions that determine the zero field phase diagram.
Why does it generate so much interest?
Well it is a concrete result and we are desperate for them and some clue to these long standing puzzles.

2. The large Fermi surface and carrier density equal to 1+ p at large doping is what one expects from a simple Fermi liquid and Luttinger's theorem. The fact that at small doping the charge density is proportional to p may have a boring explanation or an interesting one.
Boring: the antiferromagnetic (AFM) order that occurs for small p reconstructs the Fermi surface due to the periodicity associated with AFM.
Interesting: this is a strongly correlated effect associated with doping a Mott insulator.

3. In a strongly correlated metal deducing information about the charge carrier density and the Fermi surface from measurements of the Hall coefficient and/or the thermoelectric power is subtle, as emphasised by Shastry, and discussed here.

The challenge of the pseudogap in organic charge transfer salts

I am often sprucing [Aussie slang for promoting] Dynamical Mean Field Theory (DMFT), and particularly how it captures many quantitative details of charge transport and bad metals in organic charge transfer salts.
However, it is always good and important to be transparent about the limitations of any theory, particularly one that you are enthusiastic about.

There is a nice paper
Repulsive versus attractive Hubbard model: Transport properties and spin-lattice relaxation rate 
Rok Žitko, Žiga Osolin, and Peter Jeglič

The authors use DMFT to calculate various spectral functions using the numerical renormalisation group (NRG) as the impurity solver. This is probably, the most reliable method, at least for low temperatures.

There is a lot I found interesting in the paper. But for now I just want to focus on one result in the paper: the temperature dependence of the NMR relaxation rate, 1/T_1.
1/(T_1 T) is proportional to the slope of the local spin fluctuation spectral function

The authors find that in the metallic phase of the repulsive Hubbard model this slope monotonically decreases with increasing temperature. Similar results were found 20 years ago (with a more approximate treatment) by Jarrell and Pruschke.

Why is this interesting?
In the organic charge transfer salts 1/T1T versus temperature is not monotonic, but has a maximum at a temperature (T_NMR) around the coherence temperature, T_coh, marking the approximate crossover from a bad metal (at high temperatures) to a Fermi liquid (at low temperatures). Actual data for a wide range of materials is shown in the Figure below, (n.b. this is a plot of T_1 T vs. T not 1/T_1T) taken from a paper, with Ben Powell and Eddy Yusuf.

That paper also emphasised the discrepancy with single site DMFT.
But, it was good to be reminded of it again.
What is going on?
Basically, like in the cuprates a pseudogap must be opening up. A cluster DMFT calculation, such as this one by Jaime Merino and Olle Gunnarsson [or one by Emanuel Gull ] captures this.
But, it remains to be shown in detail that the NMR data can be quantitatively described and the relationship between the two temperature scales T_coh and T_NMR needs to be elucidated.

Not seeing the pseudogap in ultra cold 2D atoms

Two weeks ago it was nice to have Meera Parish visit UQ and give a colloquium Fermions in Flatland. She recently moved to Monash University from University College London.

One important point she made was the comparison of the two figures below, showing a colour intensity plot of the one fermion spectral function A(E,k) for a two-dimensional Fermi gas near the unitary limit (BCS-BEC crossover).

The bottom figure is experimental data from a Nature paper, 

Observation of a pairing pseudogap in a two-dimensional Fermi gas.

It makes much of the possible connection to the pseudogap seen in cuprate superconductors.

The top figure is from a theory paper

Pair Correlations in the Two-Dimensional Fermi Gas 

Vudtiwat Ngampruetikorn, Jesper Levinsen, and Meera M. Parish

Therefore, our results suggest that the observed pairing gap [Nature paper] effectively arises from two-body physics and does not correspond to a pseudogap regime. This view is further supported by the fact that the pairing gap in the spectrum persists to very high temperatures well above Tc, as shown in Fig. [above]. Moreover, we see that the “closure” of the gap with increasing temperature appears to be due to the thermal broadening of the two branches.

An earlier post discussed more recent measurements of the spectral function in three-dimensional ultra cold fermionic atoms near the unitary limit.

Quantum limits to the shear viscosity in the unitary Fermi gas

Previously, I posted about possible quantum limits to the shear viscosity in quantum many-body systems. This has attracted a lot of interest because of claims, based on string theory techniques [AdS-CFT correspondence] that there is a universal lower bound for the ratio of the shear viscosity to the entropy.

There are two interesting papers

Hydrodynamic fluctuations and the minimum shear viscosity of the dilute Fermi gas at unitarity
Clifford Chafin and Thomas Schäfer

Temperature evolution of the shear viscosity in a unitary Fermi gas 
 Gabriel Wlazłowski, Piotr Magierski, Aurel Bulgac, and Kenneth J. Roche

The main result of the latter is shown in the Figure below. The error bars arise because the results are based on a Quantum Monte Carlo simulation with imaginary time data that must be analytically continued to real frequencies [a thorny problem]. Tc is the superfluid transition temperature and T* the pseudogap temperature.

A few reasons why this is interesting.

1. In 1936 the legendary theoretical chemist Henry Eyring proposed a lower bound for the viscosity of n hbar, where n is the particle density. Here, we see that bound is violated.

2. The inset shows that the string theory bound is respected.

3. At low temperatures, the "classical" bound of about 0.2 n hbar, proposed in the first paper is violated.

4. The temperature dependence shows this is a strongly correlated fermion fluid, a long way from a Fermi liquid. For the latter, such as liquid 3He, the viscosity at low temperatures goes like 1/T^2, i.e. increases as the temperature decreases, and is much larger than n hbar. The fermion fluid here is something like a "bad metal" since these small viscosities correspond to mean free paths comparable to the Fermi wavelength.

Is there a Fermi liquid associated with the pseudogap state of the cuprates?

To me this seems at first to be a strange idea. The phenomenology of the cuprates and doped Hubbard models is roughly that as the doping decreases one goes from a Fermi liquid (large overdoping with no superconductivity) to an anisotropic marginal Fermi liquid  (overdoped but superconducting) to strange metal (marginal Fermi liquid) (optimal doping) to pseudogap state (underdoping). Hence, I would have thought that everything was rather non-Fermi liquid like in the pseudogap state.

However, the observation in the pseudogap range of copings of quantum magnetic oscillations (that could be associated with a small Fermi surface) and Fermi arcs, raised the question of a Fermi liquid state.

Over the past few years Martin Greven and collaborators have performed a range of transport measurements on a relatively clean single layer cuprate material Hg1201. They find Fermi liquid type behaviour [e.g. resistivity quadratic in temperature, scattering rates quadratic in frequency] for a range of temperatures below the pseudogap temperature T*.

A recent preprint is
Validity of Kohler's rule in the pseudogap phase of the cuprate superconductors
M. K. Chan, M. J. Veit, C. J. Dorow, Y. Ge, Y. Li, W. Tabis, Y. Tang, X. Zhao, N. Barišić, M. Greven

What is Kohler's rule?

In simple metals the temperature and magnetic field dependence of the magnetoresistance is dominated by the orbital motion of the electrons and described by some function of the product of omega_c and tau.

omega_c is the cyclotron frequency which is proportional to the magnetic field B and independent of temperature.
tau is the scattering time, which is temperature dependent and field independent, and should have the same temperature dependence as 1/rho where rho(B=0) is the resistivity in zero field.

These observations lead to Kohler's rule which is obeyed by simple metals.
A plot of the ratio of the rho(B)/rho(B=0) versus B/rho(B=0) should be independent of temperature.

In 1995 Ong's group observed significant violations of Kohler's rule in the underdoped and overdoped cuprates. Similar results were found by a Japanese group.

In 1998 I pointed out that in one mysterious organic metal there were also significant violations. The paper also has an extensive discussion of reasons why Kohler's rule can fail.

It seems that in the overdoped cuprates a generalised version of Kohler's rule holds.

In the preprint, the authors find results consistent with Kohler's rule for Hg2201 samples with Tc=70 K and 81K, and temperatures between about 100 K and 200 K, and fields up to 30 Tesla.

The left plot is the bare data and the right plot is the data rescaled according to Kohler's rule.

Is the idea of Fermi liquid in the pseudogap region reasonable?

The scenario may be something like this.

There are Fermi liquid like quasi-particles near the nodes of the pseudogap. The non-Fermi liquid excitations occur towards the anti-nodal regions. This is the basic idea of the anisotropic marginal Fermi liquid developed for overdoped cuprates. Suppose one assumes something like that model actually applies for all doping. Then in the pseudogap region the non-Fermi liquid part will start to get gapped out and one will just be left with the Fermi liquid part. This can then undergo charge ordering instabilities to form Fermi surface pockets due to Fermi surface reconstruction.