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

Emergence and protein folding

Proteins are a distinct state of matter. Globular proteins are tightly packed with a density comparable to a crystal but without the spatial regularity found in crystals. The native state is thermodynamically stable, in contrast to the globule state of synthetic polymers which is often glassy and metastable, with a structure that depends on the preparation history.

For a given amino acid sequence the native folded state of the protein is emergent. It has a structure, properties, and function that the individual amino acids do not, nor does the unfolded polymer chain. For example, the enzyme catalase has an active site whose function is as a catalyst to make hydrogen peroxide (which is toxic) decay rapidly.

Protein folding is an example of self-organisation. A key question is how the order of the folded state arises from the disorder (random configuration) of the unfolded state.

There are hierarchies of structures, length scales, and time scales associated with the folding.

The hierarchy of structures are primary, secondary, tertiary, and ternary structures. The primary structure is the amino acid sequence in the heteropolymer. Secondary structures include alpha-helices and beta-sheets, shown in the figure above in orange and blue, respectively. The tertiary structure is the native folded state. An example of a ternary structure is in hemoglobin which consists of four myoglobin units in a particular geometric arrangement.

The hierarchy of time scales varies over more than fourteen orders of magnitude, including folding (msec to sec), helix-coil transitions (microsec), hinge motion (nanosec), and bond vibrations (10 fsecs).

Folding exhibits a hierarchy of processes, summarised in the figure below which is taken from

Cooperativity and modularity in protein folding

Masaki Sasai, George Chikenji, Tomoki P. Terada

Modularity 

"Protein foldons are segments of a protein that can fold into stable structures independently. They are a key part of the protein folding process, which is the stepwise assembly of a protein's native structure." (from Google AI)

See for example.

Discontinuities

The folding-unfolding transition [denaturation] is a sharp transition, similar to a first-order phase transition. This sharpness reflects the cooperative nature of the transition. There is a well-defined enthalpy and entropy change associated with this transition.

Universality

Proteins exhibit "mutational plasticity", i.e., native structures tolerant to many mutations (changes in individual amino acids). Aspects of the folding process such as its speed, reliability, reversibility, and modularity appear to be universal, i.e., hold for all proteins.

Diversity with limitations

On the one hand, there are a multitude of distinct native structures and associated biological functions. On the other hand, this diversity is much smaller than the configuration space, presumably because thermodynamic stability vasts reduces the options.

Effective interactions

These are subtle. Some of the weak ones matter as the stabilisation energy of the native state is of order 40 kJ per mole, which is quite small as there are about 1000 amino acids in the polymer chain. Important interactions include hydrogen bonding, hydrophobic, and volume exclusion. In the folded state monomers interact with other monomers that are far apart on the chain. The subtle interplay of these competing interactions produces complex structures with small energy differences, as is often the case with emergent phenomena.

Toy models

1. Wako-Saito-Munoz-Eaton model

This is an Ising-like model on a chain. A short and helpful review is

A simple theoretical model goes a long way in explaining complex behavior in protein folding

Victor Muñoz 

Note that the interactions are not pairwise but involves strings of "spins" between native contacts.

2. Dill's HP polymer on a lattice

This consists of a polymer which has only two types of monomer units and undergoes a self-avoiding walk on a lattice. H and P denote hydrophobic and polar amino acid units, denoted by red and blue circles, respectively, in the figure below. The relative simplicity of the model allows complete enumeration of all possible confirmations for short chains. The model is simpler in two dimensions, yet still captures essential features of the folding problem.  

As the H-H attraction increases the chain undergoes a relatively sharp transition to just a few conformations that are compact and have hydrophobic cores. The model exhibits much of the universality of protein folding. Although there are 20 different amino acids in real proteins, the model divides them into two classes and still captures much of the phenomena of folding, including mutational plasticity.

co-operativity - helical order-disorder transition is sharp

Organising principles

Certain novel concepts such as the rugged energy landscape and the folding funnel apply at a particular scale.

This post drew on several nice papers written by Ken Dill and collaborators including

The Protein Folding Problem, H.S. Chan and K.A. Dill, Physics Today, 1993

The Protein Folding Problem: The Role of Theory

Roy Nassar, Gregory L. Dignon, Rostam M. Razban, Ken A. Dill, Journal of Molecular Biology, 2021

Interestingly, in the 2021 article, Dill claims that the protein folding problem [which is not the prediction problem] has now essentially been solved.