The triumphs of lattice gauge theory

When first proposed by Ken Wilson in 1974, lattice gauge theory was arguably a toy model, i.e., an oversimplification. He treated space-time as a discrete lattice purely to make analysis more tractable. Borrowing insights and techniques from lattice models in statistical mechanics, Wilson could then argue for quark confinement, showing that the confining potential was linear with distance.

Earlier, in 1971, Wegner had proposed a Z2 gauge theory in the context of generalised Ising models in statistical mechanics to show how a phase transition was possible without a local order parameter, i.e., without symmetry breaking. Later, it was shown that the phase transition is similar to the confinement-deconfinement phase transition that occurs in QCD. [A nice review from 2014 by Wegner is here]. This work also provided a toy model to illustrate the possibility of a quantum spin liquid.

Perhaps, what was not anticipated was that lattice QCD could be used to calculate accurately properties of elementary particles.

The discrete nature of lattice gauge theory means it is amenable to numerical simulation. It is not necessary to have the continuum limit of real spacetime because of universality. Due to increases in computational power over the past 50 years and innovations in algorithms, lattice QCD can be used to calculate properties of nucleons and mesons, such as mass and decay rates, with impressive accuracy. The figure below is taken from a 2008 article in Science. 

The mass of three mesons is typically used to fix the mass of the light and strange quarks and the length scale. The mass of nine other particles, including the nucleon, is calculated with an uncertainty of less than one per cent, and in agreement with experimental values.

An indication that this is a strong coupling problem is that about 95 per cent of the mass of nucleons comes from the interactions. Only about 5 per cent is from the rest mass of the constituent quarks.

For more background on computational lattice QCD, there is a helpful 2004 Physics Today article, which drew a critical response from Herbert Neuberger. A recent (somewhat) pedagogical review by Sasa Prelovsek just appeared on the arXiv.

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

Why is quantum matter so interesting?

Last year Ben Powell wrote a Perspective for Science, The Expanding Materials Multiverse. It begins with a nice statement about why quantum condensed matter is so interesting, exciting, and challenging.

High-energy physicists are limited to studying a single vacuum and its excitations, the particles of the standard model. For condensed-matter physicists, every new phase of matter brings a new “‘vacuum.” Remarkably, the low-energy excitations of these new vacua can be very different from the individual electrons, protons, and neutrons that constitute the material. The materials multiverse contains universes where the particle-like excitations carry only a fraction of the elementary electronic charge, are magnetic monopoles, or are their own antiparticles. None of these properties have ever been observed in the particles found in free space. Often, emergent gauge fields accompany these “fractionalized” particles, just as electromagnetic gauge fields accompany charged particles. On page 1101 of this issue, Hassan et al. provide a glimpse of the emergent behaviors of a putative new phase of matter, the dipole liquid. What particles live in this universe, and what new physics is found in this and neighboring parts of the multiverse?

There is also a nice figure which makes an everyday analogy to illustrate different states of matter.

Quantum spin liquid on the hyper-honeycomb lattice

Two of my UQ colleagues have a nice preprint that brings together many fascinating subjects including strong electron correlations and MOFs. Again it highlights an ongoing theme of this blog, how chemically complex materials can exhibit interesting physics. A great appeal of MOFs is the possibility of using chemical "tuneability" to design materials with specific physical properties.

A theory of the quantum spin liquid in the hyper-honeycomb metal-organic framework [(C2H5)3NH]2Cu2(C2O4)3 from first principles 
A. C. Jacko, B. J. Powell

What is a hyper-honeycomb lattice?
It is a three-dimensional version of the honeycomb lattice.
A simple tight-binding model on the lattice has Dirac cones, just like graphene.

The preprint is a nice example how one can start with a structure that is chemically and structurally complex and then use calculations based on Density Functional Theory (DFT) to derive a "simple" effective Hamiltonian (in this case an antiferrromagnetic Heisenberg model of coupled chains) to describe the low-energy physics of the material.

We construct a tight-binding model of [(C2H5)3NH]2Cu2(C2O4)3 from Wannier orbital overlaps. Including interactions within the Jahn-Teller distorted Cu-centered eg Wannier orbitals leads to an effective Heisenberg model. The hyper-honeycomb lattice contains two symmetry distinct sublattices of Cu atoms arranged in coupled chains. One sublattice is strongly dimerized, the other forms isotropic antiferromagnetic chains. Integrating out the strongest (intradimer) exchange interactions leaves extremely weakly coupled Heisenberg chains, consistent with the observed low temperature physics.

There is some rather subtle physics involved in the superexchange processes that determine the magnitude of the antiferromagnetic interactions J between neighbouring spins. In particular, there are destructive quantum interference effects that reduce one of the J's by an order of magnitude and increases another by an order of magnitude. To illustrate this effect, the authors also evaluate the J's when one flips the sign of some of the matrix elements in the tight-binding model. Similar subtle physics has also been observed in different families of organic charge transfer salts.

As an aside, there is some similarity (albeit many differences) with the basic chemistry of the insulating phase of cuprates: the parent compound involves a lattice of copper ions (d9) where there are three electrons in eg orbitals that are split by a Jahn-Teller distortion. The differences here are first, that the interactions between the frontier orbitals on the Cu sites is not via virtual processes involving oxygen p-orbitals but rather via pi-orbitals on the oxalate bridging orbitals. Second, the lattice of Cu orbitals is not a square but the hyper-honeycomb lattice.

The preprint is motivated by a recent experimental paper in JACS
Quantum Spin Liquid from a Three-Dimensional Copper-Oxalate Framework 
Bin Zhang, Peter J. Baker, Yan Zhang, Dongwei Wang, Zheming Wang, Shaokui Su, Daoben Zhu, and Francis L. Pratt

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.

Subtle paths to effective Hamiltonians in complex materials

Many of the most interesting materials involve significant chemical and structural complexity. Indeed, it is not unusual for a unit cell for a crystal to contain the order of one hundred atoms.
Yet, for a given class of materials, one would like to find an effective Hamiltonian involving as few degrees of freedom and parameters as possible.

Following Kino and Fukuyama, twenty years ago I argued that the simplest possible effective Hamiltonian for a large class of superconducting organic charge transfer salts was a one-band Hubbard model on an anisotropic triangular lattice at half filling.
It seemed natural to then argue that the relevant model for the spin degrees of freedom in the Mott insulating phase is the corresponding frustrated Heisenberg model with spatial anisotropy determined by the anisotropy in the tight-binding model.

However, it turns out this is not the case.
There are some subtle quantum interference effects that I overlooked in the "derivation"  of these effective models, leading to a different spatial anisotropy.
This is shown by some of my colleagues in a nice recent paper, accepted for PRL.

Dynamical reduction of the dimensionality of exchange interactions and the "spin-liquid" phase of κ-(BEDT-TTF)2X 
B. J. Powell, E. P. Kenny, J. Merino

This raises questions about what the relevant effective Hamiltonian is for the metallic, superconducting, and (possibly) ferroelectric phases.

The paper's significance goes beyond organic charge transfer salts to the general problem of finding effective Hamiltonians in complex materials.

Similar interference effects have been found to arise in quite a different class of materials.

Heisenberg and Dzyaloshinskii-Moriya interactions controlled by molecular packing in trinuclear organometallic clusters 
B. J. Powell, J. Merino, A. L. Khosla, and A. C. Jacko

This also reminds me of subtleties (and debates) that occur in the Zhang-Rice "derivation" of the t-J model from a three-band Hubbard model.

A molecular material and a model Hamiltonian with rich physics

Some of my UQ colleagues and Jaime Merino have written a series of nice papers inspired by an organometallic molecular material Mo3S7(dmit)3. They have considered possible model effective Hamiltonians to describe it and the different ground states that arise depending on the model parameters.
There is a rich interplay of strong correlations, Hund's rule coupling, spin frustration, spin-orbit coupling, flat bands, and Dirac cone physics.
Possible ground states include some sort of Mott insulator, a Haldane phase, semi-metal, ...

A good place to start is the following paper
Low-energy effective theories of the two-thirds filled Hubbard model on the triangular necklace lattice 
C. Janani, J. Merino, Ian P. McCulloch, and B. J. Powell

The figure below (taken from this paper) shows some of the molecular structure and some of the hopping integrals that are associated with an underlying decorated honeycomb lattice.

This model could be called kagomene, because it interpolates between the kagome lattice and the honeycomb lattice (graphene). The figure below is taken from this paper, which uses DFT and Wannier orbitals to estimate the tight-binding parameters and the spin-orbit coupling. Interaction driven topological insulator states are possible on this lattice.

There are a few things that are not "normal" about the physics, arising from the 4/3 band filling and the molecular orbitals that are delocalised over the triangles. Specifically, the orbital degeneracy does not arise from atomic orbital degeneracy (cf. d orbitals, or t2g and eg), but rather the E representation associated with C3 symmetry of the triangles.

Hund's rule coupling. 
This involves the E orbitals and arises purely from the Hubbard U on the non-degenerate orbital on a single lattice site.

Spin-orbital coupling.
This is Spin Molecular Orbital Coupling, where the electron spin couples to the angular momentum associated with motion around the triangle, not the angular momentum of degenerate atomic orbitals.

Haldane phase.
The associated spin-1's arise from the triplet ground state of four electrons on a triangle.
A DMRG study shows that this is the ground state of a three leg-ladder Hubbard model at 2/3 filling.

Many interesting and important open questions remain about the general phase diagram of the Hubbard model on the kagomene lattice. For example, the nature of the Mott insulator, different types of topological order, the possibility of superconductivity.....

Hopefully, these studies will stimulate new experimental studies and synthesis of new chemical compounds in this fascinating class of materials.

The challenge of multiferroism in organic Mott insulators

A theoretical picture of the Mott insulating phase of organic charge transfer salts [such as (BEDT-TTF)2X] is that they can be described by a single-band Hubbard model on an anisotropic triangular lattice at half filling. The spin excitations can then be described by the corresponding Heisenberg model. In these models, each lattice site corresponds to a single anti-bonding orbital on a pair (dimer) of BEDT-TTF molecules. Thus the internal structure of the dimer and the corresponding two-band Hubbard model at three-quarters filling is "integrated out" leaving a one-band picture.

However, there are some dielectric relaxation experiments that can be interpreted as inconsistent with the picture above.
The key question is whether there is charge order within the dimer, in particular, does it have a net dipole moment?
A 2010 theory paper by Hotta proposed this and an effective Hamiltonian for the Mott insulating phase where the spin on the dimer and the dipoles are coupled together. She suggested that a spin liquid phase could be driven by the dipoles, rather than spin frustration.
This picture also leads to the possibility of a multiferroic phase: coexisting ferromagnetic and ferroelectric phases.

There are two helpful recent reviews, presenting alternative views of the experiments.

Dielectric spectroscopy on organic charge-transfer salts
P Lunkenheimer and A Loidl

Ferroelectricity in molecular solids: a review of electrodynamic properties 
S Tomić and M Dressel

The figure below shows experimental measurements from this paper. (The authors of the first review above and Hotta are co-authors.) The figure shows the temperature dependence of the real part of the dielectric constant at different frequencies. Note how it becomes very large at low frequencies (almost static) near about 25 K, which coincidentally is the temperature at which this organic charge transfer salt becomes antiferromagnetic (with weak ferromagnetism due to spin canting).

The above dielectric behaviour is similar to what one sees at a ferroelectric transition.

However, one should be cautious about this interpretation for multiple reasons. These are tricky experiments.

Dielectric dispersion spectroscopy is a bulk probe, not a microscopic one. One is not measuring the electric dipole moment of a single unit cell but rather the electric polarisation of a bulk crystal that has surfaces and contains defects, and impurities. For example, charge accumulation on the sample surface can enhance the measured dielectric constant and lead to significant frequency dependence, even when the actual material has no intrinsic frequency dependence (This is known as Maxwell-Wagner polarisation or the space-charge effect).

There are reports of significant sample dependence; the dielectric constant can vary by up to two orders of magnitude!

The origin of the dependence of the results on the direction of the electric field is not clear (at least to me). One usually finds the largest effects when the electric field is parallel to the least conducting direction (i.e. perpendicular the layers) in the crystal.

The magnitude of the electric dipole moment that one deduces from the magnitude of the dielectric constant (by fitting the temperature dependence to a Curie form, as in the dashed line in the figure above) is an order of magnitude larger than the moment on single dimers that is deduced from infrared (IR) measurements. This last discrepancy is emphasized by the authors of the second review above.

(IR measures the vibrational frequencies of the BEDT-TTF molecules; spectral shifts are correlated with the charge density on the molecule. Splitting of spectral lines corresponds to the presence of charge order, as discussed here.)

If one does accept that charge order occurs, further questions that arise include:

How do we know that the charge order is occurring within the dimers not between dimers?

Are these dielectric properties necessary or relevant for the insulating, superconducting, and magnetic properties (antiferromagnetism or spin liquid) or is it just a second-order effect (causality or correlation)?

What is the relevant effective Hamiltonian in the Mott insulating phase?

How is this similar and different to multiferroic behaviour in inorganic materials?

What role does spin-orbit coupling [and specifically the Dzyaloshinskii-Moriya interaction] play?

What experimental signatures could be considered a "smoking gun" for the presence of electric dipoles on single dimers?

How does one understand the different experiments which probe the system on very different time scales?

Desperately seeking organic spin liquids

A spin liquid is a state of matter where there is no magnetic order (spontaneous breaking of spin rotational symmetry) at zero temperature. The past few decades has seen a desperate search for both real materials and Heisenberg spin models in two spatial dimensions that have this property. I have written many posts on the subject. An important question is what is a definitive experimental signature of such a system.

Strong candidate materials are the Mott insulating phase of several organic charge transfer salts, which was reviewed in detail in 2011 by Ben Powell and I.

One experimental signature is the temperature dependence of the specific heat. In particular, some theories predict spin liquid states with spinon excitations with a Fermi surface. This would lead to a linear term in the temperature dependence of the specific heat, as one sees in a simple metal that is a Landau Fermi liquid. This paper is one of several that claims to observe this signature.

However, it is important to bear in mind two subtle issues with interpreting these experiments. 

First, one always have to subtract off the large contribution to the specific heat from lattice vibrations. There are two main ways to do this. One is to fit the data, including a cubic term, T^3 in the temperature dependence. The second method is to subtract the data from a different compound (e.g. a deuterated one) which has a different electronic (magnetic) ground state but a similar crystal structure. Due to subtle isotope effects and hydrogen bonding, sometimes deuterated compounds are argued to meet this requirement for the magnetic contributions to be different and the phonon contributions to be the same.

However, there are problems with both of these subtraction methods. 

First, curve fitting with many parameters can be getting the tail of the elephant to wiggle, as discussed more below. Second, changing the chemistry does change the phonon spectrum and so also changes the lattice contribution.

Finally, what about the linear in T term? 

In a News and Views about a 2008 Nature Physics paper claiming to observe this linear in T term, Art Ramirez showed one could take the same experimental data and fit it to an alternative expression involving T^(2/3) which was proposed by an alternative theory. 

This is shown in the Figure below.

I also worry about how the low T specific heat is dominated by the 1/T^2 term associated with the Schottky anomaly from two level systems.

Unfortunately, Ramirez's concerns seem to have been ignored in following papers.

We really need more direct experimental probes of spin liquid behaviour. Unfortunately, there is a paucity of realistic ones.

The many scales of emergence in the Haldane spin chain

The spin-1 antiferromagnetic Heisenberg chain provides a nice example of emergence in a quantum many-body system. Specifically, there are three distinct phenomena that emerge that were difficult to anticipate: the energy gap conjectured by Haldane, topological order, and the edge excitations with spin-1/2.

An interesting question is whether anyone could have ever predicted these from just knowing the atomic and crystal structure of a specific material. I suspect Laughlin and Pines would say no.

To understand the emergent properties one needs to derive effective Hamiltonians at several different length and energy scales. I have tried to capture this in the diagram below. In the vertical direction, the length scales get longer and the energy scales get smaller.

It is interesting that one can get the Haldane gap from the non-linear sigma model. However, it coarse grains too much and won't give the topological order or the edge excitations.

It seems to me that the profundity of the emergence that occurs at the different strata (length scales) is different. At the lower levels, the emergence is perhaps more "straightforward" and less surprising or less singular (in the sense of Berry).

Aside. I spend too much time making this figure in PowerPoint. Any suggestions on a quick and easy way to make such figures?

Any comments on the diagram would be appreciated.

Visit to a state university in India

Like everything in India, higher education is incredibly diverse, both in quality, resources, and culture. These statistics give some of the flavour. There are about 800 universities. A significant distinction is between state and central universities. The former are funded and controlled by state governments. The latter (and IITs, IISERs, IISc, TIFR...)  are funded and controlled by the central (i.e. national/federal) government. Broadly, the quality, resources, and autonomy (i.e. freedom from political interference) of the latter is much greater. On my many trips to India I have only visited these centrally funded institutes and universities.

This afternoon I looking forward to visiting the Physics Department of Vidyasagar University. It is funded by the West Bengal state government, and was started in 1981. It is named in honour of Ishwar Chandra Vidyasagar, a significant social reformer from the 19th century.

I am giving my talk on "Emergent Quantum Matter".
Here are the slides.

Update. I enjoyed my visit and interacting with the faculty and students. On the positive side, people were enthusiastic and there were some excellent questions from the students. I want to write a blog post about one question. On the negative side, it is sad to see how poorly places like this are resourced: whether infrastructure, lab equipment, lab supplies, library, faculty, or salaries. For example, there are 5 physics faculty members and they teach a full M.Sc. [2 years course work] to about 100 students. This is 2 courses per faculty per semester and obviously, their expertise is stretched to cover all courses. The Ph.D. students mostly have full-time jobs elsewhere and come in the afternoons and evenings to work on their projects. One travels 2 hours each way on public transport.

Organic spin liquid talk at IIT-Kgp

Today I am giving a seminar in the Physics Department at Indian Institute of Technology (IIT) Kharagpur,

"Frustrated organic Mott insulators: from quantum spin liquids to superconductors."
Slides are here.

Due to the recent Nobel Prize to Haldane, I included one slide about quantum spin liquids in one dimension.

The talk material is covered in great detail in a review article, written together with Ben Powell.

2016 Nobel Prize in Physics: Topology matters in condensed matter

I was delighted to see this year's Nobel Prize in Physics awarded to Thouless, Haldane, and Kosterlitz 
”for theoretical discoveries of topological phase transitions and topological phases of matter”.

A few years ago I predicted Thouless and Haldane, but was not sure they would ever get it. I am particularly glad they were not bypassed, but rather pushed forward, by topological insulators.

There is a very nice review of the scientific history on the Nobel site.

Here are a few random observations, roughly in order of decreasing importance.

First, it is important to appreciate that there are two distinct scientific discoveries here. They do both involve Thouless and topology, but they really are distinct and so Thouless’ contribution in both is all the more impressive.
The “topological phase transition” concerns the Kosterlitz-Thouless transition which is a classical phase transition (i.e. driven by thermal fluctuations) which is driven by vortices (topological objects,
which can also be viewed as non-linear excitations).
The KT transition and the low temperature phase is remarkably different from other phase transitions and phases of matter. It is a truly continuous transition in that all the derivatives of the free energy are continuous and a Taylor expansion about the critical temperature is not defined.
Yet the superfluid density undergoes a jump at the KT transition temperature.
The low temperature phase has power law correlations with an exponent which is not only irrational but non-universal (i.e. it depends on the coupling constant and temperature).
There are deep connections to quantum phase transitions in one-dimensional systems, e.g. in a spin-1/2 XXZ spin chain, but that is another story.

Topological states of matter are strictly quantum.
Having done the KT transition there is no reason why Thouless would have been led to the formulation of the quantum Hall effect in terms of topological invariants.
That is really an independent discovery. Furthermore, the topology and maths is much more abstract because it is not in real space but involves fibre bundles, Chern numbers, and Berry connections.

All of this phenomena are striking examples of emergence in physics: surprising new phenomena, entities, and concepts.
But, here there is a profound issue about theory preceding experiment.
Almost always emergent phenomena are discovered experimentally and later theory scrambles to explain what is going on.
But, here it seems to be different. KT was predicted and then observed.
The Haldane phase was predicted and then observed in real materials.
When I give my emergent quantum matter talk, I sometimes say: “I can’t think of an example of where a new quantum state of matter was predicted and then observed. Sometimes people give the example of BEC in ultracold atomic gases and of topological insulators but they are essentially non-interacting systems."

On the other hand, it is important to acknowledge that all of this was done with effective Hamiltonians (XY models and Heisenberg spin chains). No one started with a specific material (chemical composition) and then predicted what quantum state it would have without any input from experiment.

The background article helped me better appreciate the unique contributions of Kosterlitz. I was in error not to suggest him before. By himself he worked out the renormalisation group (RG) equations for the transition. Also with Nelson he predicted the universal jump in the superfluid density.
As an aside, it is fascinating that the same RG equations appear in the anisotropic Kondo model and were discovered earlier by Phil Anderson, which was also before Wilson did RG.

The background article also notes how it took a while for Haldane’s 1983 conjecture (that integer spin chains had an energy gap to the lowest excited triplet state) to be accepted, and suggests experiment decided.  It should be pointed out that on the theory side that the numerics was not clear (see e.g., this 1989 review by Ian Affleck) until Steve White developed the DMRG (Density Matrix Renormalisation Group) for one-dimensional quantum many-body systems and laid the matter to rest in 1994 by calculating the energy gap and correlation length to five significant figures!

Later I have some minor sociology comments, but don’t want to spoil all the lovely science in this post.

Spin liquid state in the spin-1 Kagome antiferromagnet

There is a nice paper
Plaquette-triplon analysis of a trimerized spin-1 Kagomé Heisenberg antiferromagnet 
Pratyay Ghosh, Akhilesh Kumar Verma, Brijesh Kumar

They consider an antiferromagnetic Heisenberg model on the Kagome lattice with three different interactions, J, J', and J'', shown below.

The case J'=J and J''=0 is the regular Kagome lattice model.

For J'=J''=0 one has isolated triangles for which the ground state is a singlet with an energy gap to three low-lying triplet states.
[Aside: for a nice general treatment of such triangles (and tetrahedrons) see this paper].
This state is the starting point for an analysis using bosonic excitations corresponding the triplet excitations on the triangular plaquette.
The calculated phase diagram is below.

 

One sees that turning on the inter triangle interaction J' has no effect on the quantum numbers or symmetry of the ground state. It remains a singlet, with no spontaneously broken symmetry, with an energy gap to the lowest lying triplet, i.e. a spin liquid. 

I found this surprising and interesting. But, it is also consistent with some numerical work, such as a recent DMRG study by Changlani and Lauchli. 

Only when one turns on the further frustrating next nearest neighbour interaction (J'') does one obtain a different ground state. Furthermore, a relatively small value of J''/J less than 0.2 is sufficient.

For a discussion about a possible spin liquid grounds state in the spin-1/2 Kagome model, see the suggested reading in an earlier post.

I thank the authors for helping me understand the paper.

From a spin liquid to a correlated Dirac metal

There is an interesting paper
Theoretical prediction of a strongly correlated Dirac metal 
 I. I. Mazin, Harald O. Jeschke, Frank Lechermann, Hunpyo Lee, Mario Fink, Ronny Thomale, Roser Valentí

The compound Herbertsmithite ZnCu3(OH)6Cl2 has attracted a lot of interest because it is a Mott insulator with a layered crystal structure where the Cu2+ ions (spin 1/2) are arranged in a kagome lattice.
There is some evidence both experimentally and theoretically that the ground state is a spin liquid.
[However, inevitably there are complications such as the role of impurities and the Dzyaloshinskii-Moriya interaction].

In this paper the authors replace the Zn2+ ions with (isoelectronic) Ga3+ ions. This means that in non-interacting electron picture the bands go from half filling (n=1) to two-third filling (n=4/3). This is of particular interest because for a tight-binding model on the kagome lattice there are symmetry protected Dirac points, just like in graphene, at this band filling.

There are subtle interlayer effects because the kagome layers order ABCABC....

This changes the three-dimensional Bravais lattice from hexagonal to rhombohedral and a doped system will have a Fermi surface like that below.

However, one needs to take into account the strong interactions associated with the localised Cu orbitals that lead to a Mott insulator at half filling. The authors use a range of theoretical techniques (rotationally invariant slave bosons, functional RG, Dynamical Cluster Approximation (DMFT)), to investigate instabilities in the associated Hubbard model.
They find a subtle competition between metallicity, charge ordering, ferromagnetism, and f-wave superconductivity.

Hopefully, someone will make this compound soon!

I thank Ben Powell for bringing the paper to my attention. He and Anthony Jacko recently considered an organometallic material with a rich band structure that interpolates between honeycomb and kagome.

Classification of topological orders

Quantum many-body states such as quantum Hall states, spin liquids, and topological insulators differ from superconductors, superfluids, and anti-ferromagnets in that they do not exhibit spontaneously broken symmetries. The latter is a major organising principle of quantum condensed matter: the broken symmetry can be used to distinguish different states and leads to new low-energy collective excitations (Goldstone bosons).

So, how does one characterise and categorise different states without broken symmetries?

Topological order has been proposed by Xiao-Gang Wen to be the relevant organising principle.
An earlier post considered the role of edge states in such a classification.

How does topology enter?
1. Consider a fractional quantum Hall system on different surfaces with different genus (sphere, torus, connected donuts, ...). Then the ground state is degenerate (in the thermodynamic limit) and the degeneracy depends on the genus of the surface.
In contrast, if one considers a two dimensional non-interacting gas of fermions, there is a unique ground state on both a sphere and a donut (plane with periodic boundary conditions).
2. For a system with an energy gap to the lowest excited state one can have edge states (low energy excitations that are spatially confined to the edge of the sample) and these are described by a topological field theory. [I am hazy on what this means; something like that the coupling constant in the action can have only integer values; these depend only on the topology of the space time.]

At the cake [weekly UQ condensed matter] meeting we are struggling with the paper
Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order

Here are a few of the things I have learnt.

I think this is only about gapped states, i.e, where there is a non-zero energy gap to the lowest excited state.

Broadly the topologically ordered states are divided into two classes, depending on whether they have short- or long-range quantum entanglement. The former means that one can perform a set of spatially localised unitary transformations that map the state into a product (i.e. non-entangled) state.

Class I. Long-range entanglement
Topological order is "stable" (i.e. adiabatically connected) to any perturbation of the Hamiltonian.
Examples: fractional quantum Hall states, chiral spin liquids, Kitaev's toric code, topological Mott insulators. Topological superconductors (e.g., a p+ip state) are in this class but also spontaneously break symmetry too.

Class II. Short-range entanglement
Symmetry-protected topological order. This means only "stable" (i.e. adiabatically connected) to perturbations of the Hamiltonian that preserve a specific symmetry.
Examples: Haldane and AKLT phases of Heisenberg spin-1 antiferromagnetic chains, topological insulators.

The paper goes on to consider how for tensor product states one can define renormalisation group flows that will lead to a fixed point which will reveal a "simpler" wave function that can be classified in terms of the several tensors with many indices [provided the relevant symmetry groups are finite dimensional].

I welcome corrections and clarifications.

Tata Colloquium on organic Mott insulators

Tomorrow I am giving the theory colloquium at the Tata Institute for Fundamental Research in Mumbai. My host is Kedar Damle.

Here are the slides for "Frustrated Mott insulators: from quantum spin liquids to superconductors".
A related review article was co-authored with Ben Powell.

4 keys concepts: Colloquium in Ljubjlana

For the next two weeks I am visiting the Stefan Institute in Ljubjlana, Slovenia. My hosts are Peter Prelovsek and Jure Kokalj.

On thursday I am giving a Physics Colloquium in the Faculty of the University, "Bad metals, good superconductors, and quantum spin liquids."

Here is the current version of my slides. I am concerned the talk is not at a basic enough level for the general audience. I highlight four key concepts stimulated by the discovery of cuprate superconductivity:

1. Mott insulator
2. Superconductivity resulting from purely repulsive electronic interactions
3. Quantum spin liquids
4. Bad metals 

These all find a nice realisation in organic charge transfer salts.

Picture is of Lake Bled, near Ljubljana, which we visited last saturday.

Copper sulphate is a spin liquid

It is amazing since a common science project for school children is to make blue crystals of copper sulphate [CuSO4.5H2O]!
[Although I was surprised and disappointed when my son just told me he never did it].

Perhaps, one may not have to look so hard for quantum materials.

The first X-ray crystallography experiment [by von Laue] was also performed on copper sulphate pentahydrate.

It turns out that the Cu2+ ions (spin-1/2) form chains that are very weakly coupled to one another and so are effectively one-dimensional antiferromagnetic Heisenberg chains above the three-dimensional Neel ordering temperature of about 100 mK.
[Caveat: strictly speaking half of the Cu2+ ions form chains; the other half are essentially isolated and non-interacting].

Minor caveat: the relevant intrachain exchange interaction J ~ 0.25 meV and so one only sees the spinons for temperatures of order a Kelvin.

I first learned all this in the introduction of this Nature Physics paper.

I dislike "arbitrary units" on graphs

It is not unusual in papers to see graphs in which the vertical scale is given in "arbitrary units".  The most common occurrence of this may be experimental measurements of some spectrum, for example, a graph of the absorbance versus frequency (photon energy) of a solution of a specific molecule.
However, some theoretical papers do this too.

There are several reasons why authors may do this.

Laziness. It can be hard work and confusing to work out the actual physical units for some theoretical calculations.

Complexity.  For experiments it can be extremely difficult to normalise and calibrate some detectors.

Uncertainty and embarrassment. Parameters such as detector efficiency, sample thickness,
geometric corrections, solution concentration can involve large uncertainties so the horizontal scale may be unknown by as much as an order of magnitude.
But I think these uncertainties should be reported because they present a challenge for improvement.

I realise that the measured "units" may be detector dependent and not particularly useful to experimenters in different labs. For example, the units may be "number of clicks per second in homemade photodetector 3 in Professor Smith's lab".
But I still think those details should be reported.

I strongly dislike the use of "arbitrary units" for the following reasons.

1. It belies the fundamental fact that any physical quantity does have actual units.

2. For experiments it means that the reported measurements are not reproducible.
i.e. they cannot be checked. The shape of the spectrum can be checked but not the magnitude.

3. This practice limits the comparison of theory and experiment.
The absolute intensity [spectral weight] of a spectrum will be predicted by theory.  It is important to test this experimentally. Just because a theoretical calculation gives the correct spectrum does not mean it is correct.
Absolute units allow one to test sum rules.

[Aside: I have a prejudice/suspicion that matrix elements may be generally more theory sensitive than energies. For example, in quantum many-body theory it is possible to get a very accurate ground state energy with a variational wave function that has a small overlap with the true ground state.]

Another example is the temperature dependence of transport properties.
Sometimes resistivity is reported in arbitrary units or the resistance [rather than resistivity is reported].
Simple theories can sometimes get the temperature dependence of the properties such as resistivity and thermopower correct but the absolute magnitude can be off by orders of magnitude.

For a nice example of people working very hard to normalise spectra, test sum rules, and get physical insight, see the paper
Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain.

A less impressive example [that I was involved in] is in the paper Transition dipole strength of eumelanin.

So, do the hard yards. Don't use "arbitrary units".
If you referee a paper that does, request the authors to do better.