"Ferromagnetic" Ising models for spin-state transitions in organometallics

In recent posts, I discussed how "spin crossover" is a misnomer for the plethora of organometallic compounds that undergo spin-state phase transitions (abrupt, first-order, hysteretic, multi-step,...)

In theory development, it is best to start with the simplest possible model and then gradually add new features to the model until (hopefully) arriving at a minimal model that can describe (almost) everything. Hence, I described how the two-state model can describe spin crossover. An Ising "spin" has values of +1 or -1, corresponding to high spin (HS) and low spin (LS) states. The "magnetic" field is half of the difference in Gibbs free energy between the two states. 

The model predicts equal numbers of HS and LS at a temperature

The two-state model is modified by adding Ising-type interactions between the “spins” (molecules). The Hamiltonian is then of the form

 The temperature dependence in the field arises because this is an effective Hamiltonian.

The Ising-type interactions are due to elastic effects. The spin-state transition in the iron atom leads to changes in the Fe-N bond lengths (an increase of about 10 per cent in going from LS to HS), changing the size of the metal-ligand (ML6 ) complex. This affects the interactions (ionic, pi-pi, H-bond, van der Waals) between the complexes. The volume of the ML6 complex changes by about 30 per cent, but typically the volume of the crystal unit cell changes by only a few per cent. The associated relaxation energies are related to the J’s. Calculating them is non-trivial and will be discussed elsewhere. There are many competing and contradictory models for the elastic origin of the J’s.

In this post, I only consider nearest-neighbour ferromagnetic interactions. Later, I will consider antiferromagnetic interactions and further-neighbour interactions that lead to frustration. 

Slichter-Drickamer model

This model was introduced in 1972 is beloved by experimentalists, especially chemists, because it provides a simple analytic formula that can be fit to experimental data.

The system is assumed to be a thermodynamic mixture of HS and LS. x=n_HS(T) is the fraction of HS. The Gibbs free energy is given by

This is minimised as a function of x to give the temperature dependence of the HS population.

The model is a natural extension of the two-state model, by adding a single parameter, Gamma, which is sometimes referred to as the cooperativity parameter.

The model is equivalent to the mean-field treatment of a ferromagnetic Ising model, with Gamma=2zJ, where z is the number of nearest neighbours. Some chemists do not seem to be aware of this connection to Ising. The model is also identical to the theory of binary mixtures, such as discussed in Thermal Physics by Schroeder, Section 5.4.

Successes of the model.

good quantitative agreement with experiments on many materials.

a first-order transition with hysteresis for T_1/2 < Tc =z J.

a steep and continuous (abrupt) transition for T_1/2 slightly larger than Tc.

Values of Gamma are in the range 1-10 kJ/mol. Corresponding vaules of J are in the range 10-200 K, depending on what value of z is assumed.

Weaknesses of the model.

It cannot explain multi-step transitions.

Mean-field theory is quantitatively, and sometimes qualitatively, wrong, especially in one and two dimensions.

The description of hysteresis is an artefact of the mean-field theory, as discussed below.

Figure. Phase diagram of a ferromagnetic Ising model in a magnetic field. (Fig. 8.7.1, Chaikin and Lubensky). Vertical axis is the magnetic field, and the horizontal axis is temperature. Tc denotes the critical temperature, and the double-line denotes a first-order phase transition between paramagnetic phases where the magnetisation is parallel to the direction of the applied field.

Curves show the free energy as a function of the order parameter (magnetisation) in mean-field theory. The dashed lines are the lines of metastability deduced from these free-energy curves. Inside these lines, the free energy has two minima: the equilibrium one and a metastable one. The lines are sometimes referred to as spinodal curves.

The consequences of the metastability for a field sweep at constant temperature are shown in the Figure below, taken from Banerjee and Bar.

How does this relate to thermally induced spin-state transitions?

Consider the phase diagram shown above of a ferromagnetic Ising model in a magnetic field. The red and blue lines correspond to temperature scans for two SCO materials that have different values of the parameters Delta H and DeltaS.

The occurrence of qualitatively different behaviour is determined by where the lines intercept the temperature and field axes, i.e. the values of T_1/2 /J and Delta H/J. If the former is larger than Tc/J, as it is for the blue line, then no phase transition is observed. 

The parameter Delta H/J determines whether at low temperatures, the complete HS state is formed.

The figure below is a sketch of the temperature dependence of the population of HS for the red and blue cases.

Note that because of the non-zero slope of the red line, the temperature  T_1/2 is not the average of the temperatures at which the transition occurs on the up and down temperature sweeps.

Deconstructing hysteresis.

The physical picture above of metastability is an artefact (oversimplification) of mean-field theory. It predicts that an infinite system would take an infinite time to reach the equilibrium state from the metastable state.

(Aside: In the context of the corresponding discrete-choice models in economics, this has important and amusing consequences, as discussed by Bouchaud.)

In reality, the transition to the equilibrium state can occur via nucleation of finite domains or in some regimes via a perturbation with a non-zero wavevector. This is discussed in detail by Chaikin and Lubensky, chapter 4.

The consequence of this “metastability” for a first-order transition in an SCO system is that the width of the hysteresis region (in temperature) may depend on the rate at which the temperature is swept and whether the system is allowed to relax before the magnetisation (fraction of HS) is measured at any temperature. Emprically, this is observed and has been highlighted by Brooker, albeit without reference to the theoretical subtleties I am highlighting here. She points out that up to 2014, chemists seemed to have been oblivious to these issues and reported results without testing whether their observations depended on the sweep rate or whether they waited for relaxation.

(Aside. The dynamics are different for conserved and non-conserved order parameters. In a binary liquid mixture, the order parameter is conserved, i.e., the number of A and B atoms is fixed. In an SCO material, the number of HS and LS is not conserved.)

In the next post, I will discuss how an antiferromagnetic Ising model can give a two-step transition and models with frustrated interactions can give multi-step transitions.

Emergence and continuous phase transitions in flatland

In two dimensions the phase transition that occurs for superfluids, superconductors, and planar classical magnets is qualitatively different from those which occur in higher  dimensions. Known as the Berezinskii-Kosterlitz-Thouless (BKT) transition, it involves several unique emergent phenomena. 

Novelty

The low-temperature state does not exhibit long-range-order or spontaneous symmetry breaking. Instead, the order parameter has power-law correlations, below a temperature T_BKT. Hence, it is qualitatively different from the high-temperature disordered state, which has correlations that decay exponentially. It is a distinct state of matter, with properties that are intermediate between the low- and high-temperature states normally associated with phase transitions. The power law correlations are similar to those at a conventional critical point, which decay in powers of the critical exponent eta. However, the BKT phase diagram can be viewed as having a line of critical points, consisting of all the temperatures below TBKT. Along this line, the critical exponent eta varies continuously with a value that depends on interaction strength. In contrast, at conventional critical points, eta has a fixed value determined by the universality class.

Sometimes it is stated that the low-temperature state has topological order, but I am not really sure what that means. Has this been made precise somewhere? 

The mechanism of the phase transition is qualitatively different from that for conventional phase transitions. It is driven by the unbinding of vortex and anti-vortex pairs by thermal fluctuations. In contrast, conventional phase transitions are driven by thermal fluctuations in the magnitude of the order parameter.

Discontinuity

There is a discontinuity in the stiffness of the order parameter at this transition temperature.

Unlike for conventional phase transitions the specific heat capacity is a continuous function of temperature. This is why the BKT transition is sometimes referred to as a continuous transition.

Toy model

A classical Heisenberg model for a planar spin, also known as the XY model, captures the essential physics.

Modularity at the mesoscale

The quasiparticles of the system that are relevant to understanding the transition are not magnons (for magnets) or phonons (for superfluids), but vortices, i.e., topological defects.  

These entities are usually on the mesoscale, i.e, there size is much larger than the lattice spacing. The relevant effective theory is not a non-linear sigma model. Thermal excitation of vortex-antivortex pairs determines the temperature dependence of physical properties and the transition at T_BKT.  There is an effective interaction between a vortex and an anti-vortex that is attractive and a logarithmic function of their spatial separation, analogous to a two-dimensional Coulomb gas. 

Universality

The BKT transition occurs in diverse two-dimensional models and materials including superfluids, superconductors, ferromagnets, arrays of Josephson junctions, and the Coulomb gas. The discontinuity in the order parameter stiffness at T_BKT has a universal value. 

The renormalisation group (RG) equations associated with the transition are the same as those of a multitude of other systems. The classical two-dimensional systems include the Coulomb gas, Villain model, Z_n model for large n, solid-on-solid model, eight vertex model, and the Ashkin-Teller model. They also apply to classical Ising chain with 1/r^2 interactions. Aside: Phil Anderson discovered these RG equations for the Ising chain before BKT derived their own equations.

Quantum models with the same RG equations include the anisotropic Kondo model, spin boson model, XXZ antiferromagnetic Heisenberg spin chain, and the sine-Gordon quantum field theory in 1+1 dimensions. In other words, all these models are in the same universality class.

Singularity

The correlation length of the order parameter is a non-analytic function of the temperature. 

This is related to the non-perturbative nature of the corresponding quantum models at their critical point. 

Personal aside: I first encountered this singularity (long ago) when working on a spin-Peierls model with quantum phonons.

Two-dimensional crystals

Similar physics is relevant to the solidification of two-dimensional liquids. However, the relevant toy model is not the classical XY model as one needs to include the effect of the discrete rotational symmetry of the lattice of the solid. The low-temperature state exhibits discrete rotational, but not spatial, symmetry breaking, with power-law spatial correlations. This state does not directly melt into a liquid, but into a distinct state of matter, the hexatic phase. It has short-range spatial order and quasi-long-range orientational (sixfold) order. The phase transitions are driven by topological defects, disclinations and dislocations.

Predictability

The BKLT transition, the quasi-ordered low-temperature state, and the hexatic phase were all predicted theoretically before they were observed experimentally. This is unusual for emergent phenomena but shows that unpredictability is not equivalent to novelty.

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