Elastic interactions and complex patterns in binary systems

One of the many beauties of condensed matter physics is that it can reveal and illuminate how two systems or phenomena that at first appear to be quite different actually involve similar physics. This is an example of universality: for emergent phenomena, many details don't really matter. One example is the similarities between superconductivity and superfluidity. A consequence of universality is that the same concepts, techniques, toy models, and effective theories can be used to describe a wide range of systems.

The complex organometallic molecules, known by the misnomer "spin crossover" compounds, exhibit a rich range of phase transitions and types of spatial order. Key aspects of the physics are the following.

• Each transition metal ion can be in one of two possible states: low-spin or high-spin. 
• The size of each molecular complex depends on the spin state.
• Consequently, the molecules interact with their neighbours via elastic interactions.

A toy model that can describe this is expanding balls connected by springs. Various versions of this type of model are reviewed here. The simplest version is the chain model below.

It turns out there are other classes of systems described by similar models. As far as I am aware, this was first pointed out in Consequences of Lattice Mismatch for Phase Equilibrium in Heterostructured Solids Layne B. Frechette, Christoph Dellago, Phillip L. Geissler

That paper is motivated by experiments on the growth of semiconductor quantum dots, by ion exchange, such as when CdSe is bathed in an Ag-rich solution and Ag2Se is produced with heterostructures (i.e., patterns of Ag and Se ions) that are different from the bulk crystal.

They consider the balls and springs model above on a triangular lattice.

They also point out how similar physics is relevant to binary metal alloys, e.g, AgCu, citing 

Ising model for phase separation in alloys with anisotropic elastic interaction—I. Theory, P. Fratzl and O. Penrose

Those authors consider a square lattice with elastic interactions associated with bond stretching along the edges and diagonals of the squares and bending of the square angles.

Frechette et al. also mention experiments on thin films of  DNA modified metallic nanoparticles. Compared to atomic systems these can tolerate larger lattice-mismatch before the formation of defects due to lattice strain.

Other systems (not mentioned) described by similar Ising models are metal-hydrogen systems, where the Ising pseudospin signifies whether a hydrogen atom is present at a particular site in the metallic crystal.

Frechette et al. start with the ball and springs model and "integrate out" the springs to obtain an effective Hamiltonian, which is an Ising model.

The spatial range of the interaction between Ising spins is shown in the colour-shaded plot below.

The interaction has two components.

One is an infinite range "ferromagnetic" part, seen as the light blue below.

The second is a short-range interaction which is mostly "antiferromagnetic" (i.e., red), but extends over several lattice sites. (Note, this interaction will be frustrated on the triangular lattice).

Using this toy model, Frechette et al. can obtain complex patterns (heterostructures) similar to those seen in quantum dots grown by ion exchange.

There is some subtle (and confusing) physics associated with deriving the Ising model from the ball and springs model. 

Due to the long-range nature of elastic interactions, the boundary conditions matter. 

The infinite range part of the Ising interaction arises from dealing with the lattice constant for the crystal, depending on the net "magnetisation" of the "spins". But that is a story for another day.

Elastic frustration in molecular crystals

Crystals of large molecules exhibit diverse structures. In other words, the geometric arrangements of the molecules relative to one another are complex. Given a specific molecule, theoretically predicting its crystal structure is a challenge and is the basis of a competition.

One of the reasons the structures are rich and the theoretical problem is so challenging is that there are typically many different interactions between different molecules, including electrostatic, hydrogen bonding, pi-pi,...

Another challenge is to understand the elastic and plastic properties of the crystals.

Some of my UQ colleagues recently published a paper that highlights some of the complexity.

Origins of elasticity in molecular materials

Amy J. Thompson, Bowie S. K. Chong, Elise P. Kenny, Jack D. Evans, Joshua A. Powell, Mark A. Spackman, John C. McMurtrie, Benjamin J. Powell, and Jack K. Clegg

They used calculations based on Density Functional Theory (DFT) to separate the contributions to the elasticity from the different interactions between the molecules. The figure below shows the three dominant interactions in the family of crystals that they consider.

The figure below shows the energy of interaction between a pair of molecules for the different interactions.

Note the purple vertical bar, which is the value of the coordinate in the equilibrium geometry of the whole crystal. The width of the bar represents variations in both lengths that occur in typical elastic experiments.

What is striking to me is the large difference between the positions of the potential minima for the individual interactions and the minima for the combined interactions.

This is an example of frustration: it is not possible to simultaneously minimise the energy of all the individual pairwise interactions. They are competing with one another.

A toy model illustrates the essential physics. I came up with this model partly motivated by similar physics that occurs in "spin-crossover" materials.

The upper (lower) spring has equilibrium length a (b) and spring constant k (k'). In the harmonic approximation, the total elastic energy is

The equilibrium separation of the two molecules is given by

which is intermediate between a + 2R and b. This illustrates the elastic frustration. Neither of the springs (bonds) is at its optimum length.

The system is stable provided that k + k' is positive. Thus, it is not necessary that both k and k' be positive. The possibility that one of the k's is negative is relevant to reality. Thompson et al. showed that the individual molecular interaction energies are described by Morse potentials. If one is far enough from the minimum of the potential, the local curvature can be negative. 

Multi-step spin-state transitions in organometallics and frustrated antiferromagnetic Ising models

In previous posts, I discussed how "spin-crossover" material is a misnomer because many of these materials do not undergo crossovers but phase transitions due to collective effects. Furthermore, they exhibit rich behaviours, including hysteresis, incomplete transitions, and multiple-step transitions. Ising models can capture some of these effects.

Here, I discuss how an antiferromagnetic Ising model with frustrated interactions can give multi-step transitions. This has been studied previously by Paez-Espejo, Sy and Boukheddaden, and my UQ colleagues Jace Cruddas and Ben Powell. In their case, they start with a lattice "balls and spring" model and derive Ising models with an infinite-range ferromagnetic interaction and short-range antiferromagnetic interactions. They show that when the range of these interactions (and thus the frustration) is increased, more and more steps are observed.

Here, I do something simpler to illustrate some key physics and some subtleties and cautions.

fcc lattice

Consider the antiferromagnetic Ising model on the face-centred-cubic lattice in a magnetic field. 

[Historical trivia: the model was studied by William Shockley back in 1938, in the context of understanding alloys of gold and copper.]

The picture below shows a tetrahedron of four nearest neighbours in the fcc lattice.

Even with just nearest-neighbour interactions, the lattice is frustrated. On a tetrahedron, you cannot satisfy all six AFM interactions. Four bonds are satisfied, and two are unsatisfied.

The phase diagram of the model was studied using Monte Carlo by Kammerer et al. in 1996. It is shown above as a function of temperature and field. All the transition lines are (weakly) first-order.

The AB phase has AFM order within the [100] planes. It has an equal number of up and down spins.

The A3B phase has alternating FM and AFM order between neighbouring planes. Thus, 3/4 of the spins have the same direction as the magnetic field.

The stability of these ordered states is subtle. At zero temperature, both the AB and A3B states are massively degenerate. For a system of 4 x L^3 spins, there are 3 x 2^2L AB states, and 6 x 2^L   A3B states. At finite temperature, the system exhibits “order by disorder”.

On the phase diagram, I have shown three straight lines (blue, red, and dashed-black) representing a temperature sweep for three different spin-crossover systems. The "field" is given by h=1/2(Delta H - T Delta S). In the lower panel, I have shown the temperature dependence of the High Spin (HS) population for the three different systems. For clarity, I have not shown the effects of the hysteresis associated with the first-order transitions.

If Delta H is smaller than the values shown in the figure, then at low temperatures, the spin-crossover system will never reach the complete low-spin state.

Main points.

Multiple steps are possible even in a simple model. This is because frustration stabilises new phases in a magnetic field. Similar phenomena occur in other frustrated models, such as the triangular lattice, the J1-J2 model on a chain or a square lattice.

The number of steps may change depending on Delta S. This is because a temperature sweep traverses the field-temperature phase diagram asymmetrically.

Caution.

Fluctuations matter.

The mean-field theory phase diagram was studied by Beath and Ryan. Their phase diagram is below. Clearly, there are significant qualitative differences, particularly in the stability of the A3B phase.

The transition temperature at zero field is 3.5 J, compared to the value of 1.4J from Monte Carlo.

Monte Carlo simulations may be fraught.

Because of the many competing ordered states associated with frustration, Kammerer et al. note that “in a Monte Carlo simulation one needs unusually large systems in order observe the correct asymptotic behaviour, and that the effect gets worse with decreasing temperature because of the proximity of the phase transition to the less ordered phase at T=0”. 

Open questions.

The example above hints at what the essential physics may be how frustrated Ising models may capture it. However, to definitively establish the connection with real materials, several issues need to be resolved.

1. Show definitively how elastic interactions can produce the necessary Ising interactions. In particular, derive a formula for the interactions in terms of elastic properties of the high-spin and low-spin states. How do their structural differences, and the associated bond stretches or compressions, affect the elastic energy? What is the magnitude, range, and direction of the interactions?

[n.b. Different authors have different expressions for the Ising interactions for a range of toy models, using a range of approximations. It also needs to be done for a general atomic "force field".]

2. For specific materials, calculate the Ising interactions from a DFT-based method. Then show that the relevant Ising model does produce the steps and hysteresis observed experimentally.

Emergent gauge fields in spin ices

Spin ices are magnetic materials in which geometrically frustrated magnetic interactions between the spins prevent long-range magnetic order and lead to a residual entropy similar to in ice (solid water).

Spin ices provide a beautiful example of many aspects of emergence, including how surprising new entities can emerge at the mesoscale. I think the combined experimental and theoretical work on spin ice was one of the major achievements of condensed matter physics in the first decade of this century.

Novelty

Spin ices are composed of individual spins on a lattice. The system exhibits properties that the individual spins and the high-temperature state do not have. The novel properties can be understood in terms of an emergent gauge field. Novel entities include spin defects reminiscent of magnetic monopoles and Dirac strings.

State of matter

Spin ices exhibit a novel state of matter, the magnetic Coulomb phase. There is no long-range spin order, but there are power-law (dipolar) correlations that fall off as the inverse cube of distance.

Toy models

Classical models such as the Ising or Heisenberg models with antiferromagnetic nearest-neighbour interactions on the pyrochlore lattice exhibit the emergent physics associated with spin ices: absence of long-range order, residual entropy, ice type rules for local order, and long-range dipolar spin correlations exhibiting pinch points. These toy models can be used to derive the gauge theories that describe emergent properties such as monopoles and Dirac strings.

Actual materials that exhibit spin ice physics such as dysprosium titanate (Dy₂Ti₂O₇) and holmium titanate (Ho₂Ti₂O₇) are more complicated. They involve quantum spins, ferromagnetic interactions, spin-orbit coupling, crystal fields, complex crystal structure and dipolar magnetic interactions. Chris Henley says these materials

"are well approximated as having nothing but (long-ranged) dipolar spin interactions, rather than nearest-neighbor ones. Although this model is clearly related to the “Coulomb phase,” I feel it is largely an independent paradigm with its own concepts that are different from the (entropic) Coulomb phase..."

Effective theory

Gauge fields described by equations analogous to electrostatics and magnetostatics in Maxwell’s electromagnetism are emergent in coarse-grained descriptions of spin ices. 

Consider a bipartite lattice where on each site we locate a tetrahedron. The "ice rules" require that two spins on each tetrahedron point in and two out. We can define a field L(i) on each lattice site i which is the sum of all the spins on the tetrahedron. The magnetic field B(r) is a coarse-graining of the field L(i). The ice rules and local conservation of flux require that 

The classical ground state of this model is infinitely degenerate. The emergent “magnetic” field [which it should be stressed is not a physical magnetic field] allows the presence of monopoles [magnetic charges]. These correspond to defects that do not satisfy the local ice rules in the spin system.

It can be shown that the total free energy of the system is

K is the "stiffness" or "magnetic permeability" associated with the gauge field. It is entirely of entropic origin, just like the elasticity of rubber.

[Aside: I would be curious to see a calculation of K from a microscopic model and an estimate from experiment. I have not stumbled upon one yet. Do you know of one? Henley points out that in water ice the entropic elasticity makes a contribution to the dielectric constant and this "has been long known."]

  A local spin flip produces a pair of oppositely charged monopoles. The monopoles are deconfined in that they can move freely through the lattice. They are joined together by a Dirac string.

This contrasts with real magnetism where there are no magnetic charges, only magnetic dipoles; one can view magnetic charges as confined within dipoles.

There is an effective interaction between the two monopoles [charges] that has the same form as Coulomb’s law.  There are only short-range (nearest neighbour) direct interactions between the spins. However, these act together to produce a long-range interaction between the monopoles (which are deviations from local spin order).

Universality

The novel properties of spin ice occur for both quantum and classical systems, Ising and Heisenberg spins, and for a range of lattices. The same physics occurs with water ice, magnetism, and charge order.

Modularity at the mesoscale

The system can be understood as a set of weakly interacting modular units. These include the tetrahedra of spins, the magnetic monopoles, and the Dirac strings. The measured temperature dependence of the specific heat of Dy₂Ti₂O₇  is consistent with that calculated from Debye-Huckel theory for deconfined charges interacting by Coulomb's law, and shown as the blue curve below. The figure is taken from here.

Pinch points.

The gauge theory predicts that the spin correlation function (in momentum space) has a particular singular form exhibiting pinch points [also known as bow ties], which are seen experimentally.

Unpredictability

Most new states of matter are not predicted theoretically. They are discovered by experimentalists, often by serendipity. Spin ice and the magnetic Coulomb phase seems to be an exception. Please correct me if I am wrong.

Sexy magnetic monopoles or boring old electrical charges?

I am hoping a reader than clarify this issue. What is wrong with the following point of view. In the discussion above the "magnetic field" B(r) could equally well be replaced with an "electric field" E(r). Then the spin defects are just analogous to electrical charges and the "Dirac strings" become like a polymer chain with opposite electrical charges at its two ends. This is not as sexy. 

Note that Chris Henley says Dirac strings are "a nebulous and not very helpful notion when applied to the Coulomb phase proper (with its smallish polarisation), for the string's path is not well defined... It is only in an ordered phase... that the Dirac string has a clear meaning."

Or is the emergent field actually "magnetic"? It describes spin defects and these are associated with a local magnetic moment. Furthermore, the long-range dipolar correlations (with associated pinch points) of the gauge field are detected by magnetic neutron scattering and so the gauge field should be viewed as "magnetic" and not "electric".

Emergent gauge fields in quantum many-body systems?

In spin ice, the emergent gauge field is classical and arises in a spin system that can be described classically. This does raise two questions that have been investigated extensively by Xiao-Gang Wen. First, he has shown how certain mean-field treatments of frustrated antiferromagnetic (with quantum spin liquid ground states) and doped Mott insulators lead to emergent gauge fields. As fascinating as his work is, it needs to be stressed that there is no definitive evidence for these emergent gauge fields. They just provide appealing theoretical descriptions. This is in contrast to the emergent gauge fields for spin ice.

Second, based on Wen's success at constructing these emergent gauge fields he has pushed provocative (and highly creative) ideas that the gauge fields and fermions that are considered "fundamental" in the standard model of particle physics may be emergent entities. This is the origin of the subtitle of his 2004 book, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Electrons.

To prepare this post I found the articles below helpful.

Emergent particles and gauge fields in quantum matter

Ben J. Powell

Maxwell electromagnetism as an emergent phenomenon in condensed matter

J. Rehn and R. Moessner

The “Coulomb Phase” in Frustrated Systems

Chris Henley

Anthony Jacko (1985-2022): condensed matter theorist

I was very sad when last week I learned of the death of Anthony Jacko, a former member of the Condensed Matter Theory group at UQ. He was only 36 years old, having been diagnosed with stage 4 cancer at the end of last year.

Jacko's funeral was this week. Family and friends spoke warmly of his intelligence, humour, faithfulness, passion for life, and endearing quirkiness. There were both tears and laughs.

I will say something here about his scientific contributions, though at times like this what we achieve professionally does not really seem that important.

I first met Jacko as an undergraduate at UQ when he took an advanced undergraduate condensed physics course with me in 2006. That year he did an undergraduate honours (fourth year) project with Ben Powell and John Fjaerestad, on the Kadowaki-Woods ratio. This work eventually led to a Nature Physics paper, that I discussed in this blog post.

In 2007 I was quite happy when Jacko decided to do a Ph.D. with me and Ben Powell. We tried to come up with simple effective Hamiltonians for organometallic complexes that are used in organic LEDs and solar cells. Although we made some progress, I think the questions we tried to address have still not been answered definitively. The most progress has subsequently been made by Ben Powell.

For a postdoc, Jacko moved to Frankfurt to work with Roser Valenti and Harald Jeschke (now at Okayama University). I was really impressed how Jacko learned how to do reliable DFT-based electronic structure calculations and to use Wannier orbitals to extract tight-binding model parameters. Jacko brought this expertise back to Ben Powell's group at UQ, where he worked from 2013 to 2018.

During that time Jacko co-authored a string of really nice papers that inspired me to write multiple blog posts, such as those below. Looking back over that work I see how careful, solid, and systematic it is. Basically, good science, that we do not see enough of these days.

The broad issue is as follows. Understanding strong electron correlations in complex molecular materials requires effective Hamiltonians that are a realistic representation of the essential physics and chemistry. Sometimes next-nearest-neighbour interactions and subtleties in crystal structure really do matter. Other times they do not. The methods used by Jacko provided a robust way of doing this.

How robust are your tight-binding model parameters?

Common challenges with constructing diabatic states and tight-binding models

Quantum spin liquid on the hyper-honeycomb lattice

Springy stringy molecular crystals

Faculty hope that former students will come to their funeral. We also hope that we won't have to attend the funeral of any of our students. It is very sad.

An endowment is being created at The University of Queensland, to fund an undergraduate physics prize that will be awarded each year in honour of Jacko.

My condolences to Jacko's partner, Alana, and to family and friends.

Unusual metal-insulator transitions arising from interplay of frustration, flat bands, and strong correlations

My colleagues and I recently posted a preprint

C3 symmetry breaking metal-insulator transitions near a flat band in the half-filled Hubbard model on the decorated honeycomb lattice

H. L. Nourse, Ross H. McKenzie, B. J. Powell

We study the single-orbital Hubbard model on the half-filled decorated honeycomb lattice. In the non-interacting theory at half-filling, the Fermi energy lies within a flat band where strong correlations are enhanced and the lattice exhibits frustration. We find a correlation driven first-order metal-insulator transition to two different insulating ground states - a dimer valence bond solid Mott insulator when inter-triangle correlations dominate, and a broken C3 symmetry antiferromagnet that arises from frustration when intra-triangle correlations dominate.

The metal-insulator transitions into these two phases have very different characters. 

The metal-broken C3 antiferromagnetic transition is driven by spontaneous C3 symmetry breaking that lifts the topologically required degeneracy at the Fermi energy and opens an energy gap in the quasiparticle spectrum. 

The metal-dimer valence bond solid transition breaks no symmetries of the Hamiltonian. It is caused by strong correlations renormalizing the electronic structure into a phase that is adiabatically connected to both the trivial band insulator and the ground state of the spin-1/2 Heisenberg model in the relevant parameter regime. 

Therefore, neither of these metal-insulator transitions can be understood in either the Brinkmann-Rice or Slater paradigms.

We welcome comments.

2021 Nobel Prize in Physics: from spin glasses to complexity theory

I was delighted to hear of the award of the Nobel Prize in Physics for 2021. The committee continues to surprise us. I did not make any predictions this year, because I had nothing new to predict. I am still surprised that experimental tests of Bell inequalities (Aspect, Clauser, Zeilinger) have still not got a prize. Maybe next year.

Here I will just write about the award to Giorgio Parisi “for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales” as it involves condensed matter theory, beginning with spin glasses, and like many things with Phil Anderson!

The popular science background and the scientific background to the prize are worth reading, as always. It notes that in a Physics Today column Anderson wrote in 1988,

“The history of spin glass may be the best example I know of the dictum that a real scientific mystery is worth pursuing to the ends of the Earth for its own sake, independently of any obvious practical importance or intellectual glamour.” 

This was in the first of a series of seven Reference Frame columns he wrote on spin glasses. The fifth column described the work of Parisi.

Here I will describe the basic ideas, particularly as they show that sometimes obscure basic science questions, very abstract ideas and mathematical formulations can be useful for very practical scientific questions, across a wide range of disciplines.

A spin glass is a distinct state of matter. This means, that in terms of the Landau paradigm, there must be an order parameter and an associated broken symmetry. What are they? Parisi found the answers.

First, as one usually does in theoretical condensed matter, one needs to write down a minimal model Hamiltonian that is complex enough to capture the essential physics but is simple enough to be amenable to analytical and/or computational analysis. 

Sam Edwards and Anderson proposed the following model for a spin glass, and Ising model where the spins are on a regular lattice but the interaction between any pair of spins, J_ik, is a random Gaussian variable with zero mean and non-zero variance.

This means that the interspin interactions are equally likely to be ferromagnetic or antiferromagnetic, leading to significant frustration.

To solve such a model one needs to calculate the partition function Z for each realisation of the J's (disorder), calculate F=- T ln Z, and average over all the configurations of disorder. 

Averaging Z over disorder is just a Gaussian integral, but averaging ln Z is analytically intractable.

Anderson's physical intuition was combined with the mathematical trickery of Edwards, that he had cultivated with his earlier work on quantum field theory and soft matter.

The replica trick is based on an identity that one learns in introductory calculus.

One considers not one system but rather n identical copies (replicas) of the physical system, calculates the average of the partition function for this fictional n-system, and then treats n as a continuous analytical variable and takes the limit that n goes to zero in the formula above. Wow, that is abstract! But, it is tractable.

Personal aside: more than twenty years ago I learned and used the replica trick because (like supersymmetry) as it provides a powerful mathematical tool to treat disorder exactly in one-dimensional models. But, the spin-glass case is much richer and more subtle.

Strange things happen for the spin glass. Soon after Edwards and Anderson's work, Thouless, Anderson, Palmer, and others made the rather puzzling discovery that not all the replicas were the same below the temperature associated with transition to the spin-glass state. The replica symmetry was broken in the spin-glass state.

Parisi proposed the order parameter below for this broken symmetry state. i denotes a lattice site, and the indices alpha and beta denote replicas. When alpha and beta have different values, the order parameter only becomes non-zero when the replicas are different.

Parisi, Toulouse, Mezard, and others then showed that there is a hierarchical structure associated with the order parameter leading to the concept of ultrametricity which can be associated with the rugged energy landscape of not just the spin-glass problem, but also optimisation problems, simulated annealing, protein folding, neural networks, ...

A nice overview that puts the theory of spin glasses in a much broader scientific context is Physics and Complexity by David Sherrington.

On Doug Natelson's blog, nanoscale views, there is a nice discussion in the comments about Parisi's Nobel and the subtle issue of the connections between separation of time scales and ultrametricity, and the connections (or not) between Parisi and climate science.

Superconductivity in kagome metals

Condensed matter physics is driven by fashion (too much). Is it fair to say that the latest fashion is the vanadium-based kagome metals  AV3Sb3 (A=K,Rb,Cs)?

[The PRL reporting superconductivity was published less than six months ago and has already been cited 44 times.]

These are certainly fascinating materials and have probably attracted attention for the following reasons.

-Kagome lattices support rich physics such as flat bands, Dirac metals, massively degenerate ground states, and (possibly) spin liquids.

-unlike other Kagome metals these compounds have both inversion and time-reversal symmetries, there is a Z2 topological invariant associated with bands near the Fermi surface, and topologically non-trivial surface states

-they are superconducting; furthermore, there are two superconducting domes as a function of pressure

-an anomalous Hall effect has been observed, which may result from topological physics

-there may be several types of charge order, including chiral charge density wave order

-the materials may be a topological superconductor [which MAY mean that it can be used to construct qubits that are "topologically protected].

Here are a few papers that I have looked at to get a better feel for this topic. I add a few things I gleaned from the papers and some basic questions I have. I welcome suggestions of other papers, that may be more helpful introductions. 

CsV3Sb5: A Z2 Topological Kagome Metal with a Superconducting Ground State 

Brenden R. Ortiz, Samuel M. L. Teicher, Yong Hu, Julia L. Zuo, Paul M. Sarte, Emily C. Schueller, A. M. Milinda Abeykoon, Matthew J. Krogstad, Stephan Rosenkranz, Raymond Osborn, Ram Seshadri, Leon Balents, Junfeng He, and Stephen D. Wilson

The figure below shows a top-down view of a single layer. The V atoms (red) form a Kagome lattice. There are three V atoms per unit cell.

The authors present DFT-based band structure calculations, which are compared to ARPES data. The good agreement suggests to me that strong correlations are not important. 

The authors use their band structures to construct Wannier orbitals and a tight-binding model for the band structure. However, even in the Supplementary information, they provide no details of this. I would like to know answers to the following.

For bands near the Fermi energy what is the composition of the underlying atomic orbitals (especially, how much d on V and p on Sb)?

How much of the band structure is described by a simple tight-binding model on a Kagome lattice with only next-nearest neighbour hopping?

Is the hopping between V sites via the p orbitals on the intermediate Sb atoms (superexchange in chemistry language)?

What is the band filling? 

Simple charge counting suggests there is one electron per triangle (1/6 band filling).

At a temperature of 100 K the intralayer resistivity is about 10 microohm-cm, well below the Mott-Ioffe-Regel limit (where the mean-free path is comparable to the lattice spacing), also suggesting that strong correlations are not significant.

Electronic instabilities of kagome metals: saddle points and Landau theory Takamori Park, Mengxing Ye, Leon Balents

Section V. A. discusses a tight-binding model. I think it is for the Kagome lattice with only nearest-neighbour hopping.

Double-dome superconductivity under pressure in the V-based Kagome metals AV3Sb5 (A = Rb and K)

C. C. Zhu, X. F. Yang, W. Xia, Q. W. Yin, L. S. Wang, C. C. Zhao, D. Z. Dai, C. P. Tu, B. Q. Song, Z. C. Tao, Z. J. Tu, C. S. Gong, H. C. Lei, Y. F. Guo, S. Y. Li

Answers to the following questions may determine whether interest in these materials is sustained.

Is the superconductivity topological?

Is the superconductivity unconventional? There are two independent parts to this question: does the superconductivity result from electron-phonon coupling or purely electronic interactions? Is the order parameter s-wave?

[On the related question of whether there are nodes in the superconducting energy gap there are already preprints with contradictory conclusions].

Is there any significant connection between any of the following: the topological character of the metal, the superconductivity, charge density orderings, and electron correlations?

A rich phase diagram for a Hubbard model on the decorated honeycomb lattice

 An important scientific idea is that simple rules can produce complex behaviour. In condensed matter theory, model Hamiltonians with just a few parameters can have rich phase diagrams with many competing ground states. My colleagues and I just completed a paper that is one more example of this.

Spin-0 Mott insulator to metal to spin-1 Mott insulator transition in the single-orbital Hubbard model on the decorated honeycomb lattice  H. L. Nourse, Ross H. McKenzie, B. J. Powell 

We study the interplay of strong electron correlations and intra-triangle spin exchange at two-thirds filling of the single-orbital Hubbard model on the decorated honeycomb lattice using rotationally invariant slave bosons (RISB). We find that the spin exchange tunes between a spin-1 Mott insulator, a metal, and a spin-0 Mott insulator when the exchange is antiferromagnetic. The Mott insulators occur from effective intra-triangle multi-orbital interactions and are adiabatically connected to the ground state of an isolated triangle. An antiferromagnetic spin exchange, as determined by the Goodenough-Kanamori rules, may occur in coordination polymers from kinetic exchange via the ligands. We characterize the magnetism in the regime where spin-triplets dominate. For small U a spin-1 Slater insulator occurs with antiferromagnetic order between triangles. Magnetism in the spin-1 Mott insulator is described by a spin-1 Heisenberg model on a honeycomb lattice, whose ground state is Néel ordered.

Comments are welcome.

The Devil is not in the details

Condensed matter physics aims to understand different and describe states of matter. Each state (phase) is associated with a particular type of order and phase diagrams encode the external parameters (temperature, pressure, chemical composition, magnetic field,...) that are necessary for each of the possible orderings to be stable.

Phase diagrams of even the simplest systems, such as binary alloys, can be quite rich, with many competing phases. Nevertheless, in many cases, simple microscopic models, with just a few degrees of freedom and a few parameters can describe these rich diagrams. Often a key is for the model to involve competing interactions, which can arise from different forces or from geometric frustration. Earlier, I wrote about how an Ising model on a hexagonal close-packed lattice could describe the plethora of distinct orderings that are observed in binary alloys. There the orderings are defined by the ordering wave vector and the composition of the unit cell. Another example occurs in spin-crossover materials and is described in recent work by my UQ colleagues.

Structure–property relationships and the mechanisms of multistep transitions in spin crossover materials and frameworks, Jace Cruddas and Ben J. Powell

A 2016 chemistry paper was First Step Towards a Devil's Staircase in Spin‐Crossover Materials

Another rich example is where there are two different spatial scales associated with interactions between the components of the system. In a lattice system, this can lead to ordering wavevectors that are incommensurate with the lattice. About forty years Per Bak wrote a nice series of papers that explored this situation.

Ising model with solitons, phasons, and "the devil's staircase", Per Bak and J. von Boehm 

This is an elegant and clear study of a simple Ising model in three dimensions with a frustrating interaction J_2 in the vertical direction. This is an example of an ANNNI model.

A mean-field theory for a state with a sinusoidally varying magnetisation (with wavevector 2 pi q) gives the phase diagram on the right above. The point P is a Lifshitz point, a tricritical point where the incommensurate (modulated) phase becomes stable. 

[Aside: in fermion models, a Lifshitz point is quite different: where the volume of the Fermi surface vanishes].

However, there is much more to the story. The authors then construct a mean-field theory where the magnetisation is commensurate, allowing for large unit cells. This leads to the phase diagram below.

The fractions p/q correspond to states with wavevector 2 pi p/q.

For example, the 1/4 state is below.

What does this have to do with a "devil's staircase"?

If for fixed J_2/J_1 the wavevector is plotted as a function of temperature it has steps of varying size and width.

A paper by Bruinsma and Bak considers an AFM Ising chain with 1/n^2 interaction, in a magnetic field, at zero temperature.

Note: In the figure below q is NOT the wavevector but rather the ratio of up to down spins. 

The magnetisation vs field curve has a fractal structure.

Bak also wrote a Physics Today article (that compares the phenomena to frequency mode locking) and a general review that includes examples of experimental realisations, ranging from magnets to atoms on surfaces.

This illustrates an important point that is often made in complexity science. Simple rules (theoretical models) can produce complex behaviour. 

This also illustrates characteristics of emergent phenomena. A wide range of physical systems can exhibit the same phenomena. Many of the details do not matter.

The devil is not in the details.

Single orbital + multiple sites = Rich physics

Since the discovery of the iron-based superconductors, it has become clear that the combination of multiple orbitals and strong correlations can lead to rich physics, beyond what one sees in single orbital routes.
An alternative route to rich physics is a single orbital model on a lattice with multiple sites in a unit cell.

Henry Nourse, Ben Powell, and I just posted a preprint
Multiple insulating phases due to the interplay of strong correlations and lattice geometry in a single-orbital Hubbard model

We find ten distinct ground states for the single-orbital Hubbard model on the decorated honeycomb lattice, which interpolates between the honeycomb and kagome lattices and is the simplest two-dimensional net. The rich phase diagram includes a real-space Mott insulator, dimer, and trimer Mott insulators, a spin-triplet Mott insulator, flat band ferromagnets, and Dirac metals. It is determined as a function of interaction strength, band filling, and hopping anisotropy, using rotationally invariant slave boson mean-field theory.

We welcome comments.

Simple model Hamiltonians can describe complexity

An important idea in condensed matter physics, both soft and hard, is that the rich phenomena seen in materials that are chemically and/or structurally complex can often be described by relatively simple model Hamiltonians that involve only a few parameters. This is particularly true when the model and system have competing interactions. This often leads to two inter-related phenomena, that I have previously described for strongly interacting quantum many-body systems.

• the emergence of new energy scales that are much smaller than the energy parameters in the Hamiltonian
• subtle competition between different ground states that involve distinct types of ordering (or not)

These phenomena also occur in classical systems. A nice example is described in this 1993 paper.

hcp Ising model in the cluster-variation approximation 
R. McCormack, M. Asta, D. de Fontaine, G. Garbulsky, and G. Ceder

The authors studied the Ising model on the hexagonal close-packed (hcp) lattice in a magnetic field. The authors are all from materials science departments and are motivated by the fact that the problem of binary alloys AxB1_x can be mapped onto an Ising model.
Rich phase diagrams result by varying the relative concentration of the atoms A and B (e.g. gold and silver), or equivalently the difference in the chemical potential between A and B, or the relative size of the interatomic interactions, or the temperature. The phase diagram can contain many competing phases with well-defined stoichiometry: A, B, AB, A2B, A3B, A2B3, A3B5, ...
Furthermore, even for a single stoichiometry, there can be multiple possible distinct orderings (and crystal structures).

The hcp lattice can be viewed as layers of two-dimensional hexagonal lattices where each layer is displaced relative to others. A unit cell is shown below on the left, where V1, V2, and V3, denote nearest-neighbour (nn), next-nearest neighbour (nnn), and nnnn interactions.
For the case of perfect packing of hard spheres V1=V2.
Note, that even when only nn interactions are present, and they are antiferromagnetic, that the system is frustrated, and for a single layer the Ising model does not order at finite temperature and has a massively degenerate ground state (i.e. non-zero entropy).
The figure on the right shows a way to represent this unit cell and the interactions in terms of two hexagonal lattices superimposed on top of each other.

 The authors show that there is a plethora (menagerie) of possible ground states and stoichiometric orderings.

We predict 32 physically realizable ground states with stoichiornetries A, AB, A2B, A3B, A, B, and A4B3. Of these structures, six are stabilized by NN pairs and eight by NNN pairs; the remaining 18 structures require multiplet interactions for their stability. 

This is a nice example of how a simple model can describe complex and rich behaviour. It is also a nice example of emergence in that many of the details don't matter such as the identity of the atoms or the form of the interaction between them.

Relating frustrated spin models and flat bands in tight-binding models

What kind of theory paper to I enjoy?
Here are some personal tastes
- "simple" enough I can understand it
- physical insight
- some analytical results
- some pretty pictures that illuminate

This week I read the following paper which I consider nicely meets these criteria.

Band touching from real-space topology in frustrated hopping models
Doron L. Bergman, Congjun Wu, and Leon Balents

The quantum spin antiferromagnetic Heisenberg model on the kagome lattice attracts a lot of attention because it may have a spin liquid ground state, for spin-1/2 and spin 1. This is arguably driven by the large spin frustration. A reflection of this frustration is that the classical model has a non-zero entropy at zero temperature due to a manifold of degenerate states. For this reason, the kagome lattice is sometimes said to be "maximally frustrated". This is in contrast to the triangular lattice for which their is a unique classical ground state and the spin-1/2 model exhibits long-range order.

The kagome lattice is also of interest because of the band structure for the tight-binding model has a flat band, i.e. it is dispersionless. This means that in the presence of interactions the electrons in this band may be strongly correlated and susceptible to instability to new states of matter.

The question arises as to whether there is any connection between these two properties of models on a particular "frustrated" lattice: flat bands and a manifold of degenerate classical ground states.

The purpose of this paper is to show that for a whole class of lattices, in two and three dimensions, that there is an close relationship between these properties.
It turns out that a key feature is that the flat bands touch a dispersive band at one point in k-space.

My interest was stimulated by the work of some of my UQ colleagues on a class of organometallic compounds that exhibit a kagomene lattice (that interpolates between kagome and honeycomb (graphene). The associated band structure (taken from this paper) is shown below.

The abstract states:

We demonstrate that this band touching is related to states which exhibit nontrivial topology in real-space. Specifically, these states have support [i.e. non-zero values] on one-dimensional loops which wind around the entire system 􏰀with periodic boundary conditions􏰁. A counting argument is given that determines, in each case, whether there is band touching or none, in precise correspondence to the result of straightforward diagonalization. When they are present, the topological structure protects the band touchings in the sense that they can only be removed by perturbations, which also split the degeneracy of the flat band.

I know illustrate this with the kagome lattice.

It has a three site basis (mu=1,2,3) and so there are three bands. If q is the Bloch wave vector, the Bloch states for the flat band can be written

One of these plaquette states is shown on the left below. 

A key point is that there is constructive interference between these plaquette states. Thus, one can take superpositions of them. On the right is the superposition of three neighbouring plaquette states.

A whole line of plaquette states can lead to visualising something with nontrivial topology.

The authors then show how similar physics occurs in other two- and three-dimensional lattice models. The one below is the dice lattice.

Finally, they show that the corresponding Hubbard model leads to a Heisenberg model in the classical limit does have macroscopic degeneracy.

I thank Ben Powell for bringing the paper to my attention.

A new class of "spin ice" materials

Two of my UQ colleagues have just finished a nice paper:
Spin-state ice in geometrically frustrated spin-crossover materials 
Jace Cruddas, B. J. Powell

The paper brings together two fascinating topics I have written about before, spin crossover materials and spin ice. One thing that it is a little worrying and disappointing about spin ice materials is that there seem to be only two (?) of them!
This paper argues that some spin crossover materials may be a new class of materials that realise ice physics (residual entropy, emergent gauge fields, monopoles, ...) Here, the Ising spin variable is the two possible spin states (High Spin and Low Spin). These materials have the potential advantage that they may be tuneable due to the creativity of synthetic chemists.
The mechanism of the interaction between spins is rather unique and interesting. It is not an exchange interaction but rather and effective interaction mediated by the spin-lattice interaction, which in these compounds is arguably large.
It is also interesting that the sign of the frustrating interactions (which are key to the stability of the ice) is determined by the anharmonic potential associated with intermolecular interactions in the spin crossover compound.

Spin ice in a nutshell

What is spin ice? What its definitive and experimental signatures?

A good place to start is the lucid discussion by Roderich Moessner and Art Ramirez in a 2006 article on Geometrical Frustration. They emphasise two organising principles: local constraints on neigbouring spins and the emergence of new entities such as gauge fields.

First, let's discuss the "ice" bit since this involves some beautiful chemistry, physics, statistical mechanics, and history. In the solid phase of water at atmospheric pressure (ice Ih) the water molecules form a hexagonal lattice, with the oxygen atoms located a the vertices of the lattice. The molecules interact with one another via hydrogen bonds.

Now the key point is that there are many different ways of orienting the water molecules (arranging the protons). The only constraint is that one has to have two protons covalently bonded to the oxygen and two protons on next-nearest neighbour water molecules hydrogen bonded to the oxygen. This is known as the ice rule. Suppose we assign an Ising spin variable (+1,-1)=(in, out)  = (covalent, Hbond) to each "bond" on the lattice. Then the ice rule is that on each tetrahedron the sum of the four "spins" must be zero.

How much degeneracy is there?
There are 2^4= 16 possible spin states on a tetrahedron. But, only six (a fraction of 3/8) satisfy the ice rule. To see this, put +1 on site one, then one must put +1 on one of the other three sites, and -1 on the other two. This gives 6 = 2 x 3 options.
If one neglects the interaction between vertices, the thermodynamic entropy per tetrahedron (water molecule) is

S = k ln (3/2)

Historical asides.
This "residual" entropy in ice was observed experimentally by William Giauque in the chemistry department at Berkeley in the 1930s.
Linus Pauling explained this in 1935, even arguing it as evidence for a specific crystal structure of ice.
Pauling's picture led to the ice-type models that are very important  (from a mathematical and conceptual point of view) in classical statistical mechanics as they are exactly soluble in two dimensions.
In 1956 Phil Anderson (who else!) noted that Pauling's problem was equivalent to that of Ising spins on a pyrochlore lattice.
It was not until four decades later than an experimental realisation was observed in a magnetic material. The experimental data is shown below.

But there is much more to spin ice. The local constraints lead naturally to an emergent gauge field (a pseudo-magnetic field), analogues of "magnetic monopoles", and unusual spin correlations (algebraic correlations without criticality). I now discuss the latter as they can be viewed as a "smoking gun" of spin ice.

The "magnetic field" B satisfies the constraint Div B =0. As a result the spin correlations have a dipolar form, i.e. they have a distance and directional dependence similar to the magnetic field associated with a magnetic dipole. This means the spin correlations fall off algebraically. This is in contrast to conventional magnets where spin correlations decay exponentially, except at a critical point. Furthermore, if one plots or measures the static spin structure factor S(q) one finds "pinch points" occur in high symmetry planes. The figure below shows an experimental measurement for Holonium Titanate, taken from here.