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.

Quantum states of matter and metrology

Two characteristics of states of matter are associated with them being referred to as quantum. One characteristic is the importance of quantum statistics of particles, i.e., that the system is composed of particles that obey Fermi-Dirac or Bose-Einstein statistics. The second characteristic is that a macroscopic property is quantized with values determined by Planck’s constant. I now discuss each of these with respect to emergence.

Quantum statistics. 

For a system of non-interacting  fermions and bosons at high temperatures the properties of the system are those of a classical ideal gas. As the temperature decreases there is a smooth crossover to low-temperature properties that are qualitatively different for fermions, bosons, and classical particles. This crossover occurs around a temperature, known as the degeneracy temperature, that is dependent on the particle density and Planck’s constant. 

Many of the properties resulting from quantum statistics also occur in systems of strongly interacting particles and this is central to the concept of Landau’s Fermi liquid and viewing liquid 4He as a boson liquid. If liquid 3He and the electron liquid in elemental metals are viewed as a gas of non-interacting fermions, the degeneracy temperature is about 1 K and 1000 K, respectively. Thermodynamic properties are qualitatively different above and below the degeneracy temperature. Low-temperature properties can have values that differ by orders of magnitude from classical values and have a different temperature dependence. In contrast to a classical ideal gas, a fermion gas has a non-zero pressure at zero temperature and its magnitude is determined by Planck’s constant. This degeneracy pressure is responsible for the gravitational stability of white dwarf and neutron stars.  

These properties of systems of particles can be viewed as emergent properties, in the sense of novelty, as they are qualitatively different from high-temperature properties. However, they involve a crossover as a function of temperature and so are not associated with discontinuity. They also are not associated with unpredictability as they are straightforward to calculate from a knowledge of microscopic properties.

Quantised macroscopic properties.

These provide a more dramatic illustration of emergence. Here I consider four specific systems: superconducting cylinders, rotating superfluids, Josephson junctions, and the integer Quantum Hall effect. All of these systems have a macroscopic property that is observed to have the following features.

i. As an external parameter is varied the quantity varies in a step-like manner with discrete values on the steps. This is contrast to the smooth linear variation seen when the material is not condensed into the quantum state of matter.

ii. The value on the steps is an integer multiple of some specific parameter.

iii. This parameter (unit of quantisation) only depends on Planck’s constant h and other fundamental constants. 

iv. The unit of quantisation does not depend on details of the material, such as chemical composition, or details of the device, such as its geometrical dimensions.

v. The quantisation has been observed in diverse materials and devices.

vi. Explanation of the quantisation involves topology.

Superconducting cylinders. A hollow cylinder of a metal is placed in a magnetic field parallel to the axis of the cylinder. In the metallic state the magnetic flux enclosed by the cylinder increases linearly with the magnitude of the external magnetic field. In the superconducting state, the flux is quantized in units of the magnetic flux quantum, Φ0 = h/2e where e is the charge on an electron. It is also found that in a type II superconductor the vortices that occur in the presence of an external magnetic field enclose a magnetic flux equal to Φ0.  

Rotating superfluids. When a cylinder containing a normal fluid is rotated about an axis passing down the centre of the cylinder the fluid rotates with a circulation proportional to the speed of rotation and the diameter of the cylinder. In contrast, in a superfluid, as the speed of rotation is varied the circulation is quantised in units of h/M where M is the mass of one atom in the fluid. This quantity is also the circulation around a single vortex in the superfluid. 

Josephson junctions. In the metallic state the current passing through a junction increases linearly with the voltage applied across the junction. In the superconducting state the AC Josephson effect occurs. If a beam of microwaves of constant frequency is incident on the junction, jumps occur in the current when the voltage is an integer multiple of h/2e. The quantisation is observed to better than one part in a million (ppm).

Integer Quantum Hall effect. In a normal conductor the Hall resistance increases linearly with the external magnetic field for small magnetic fields. In contrast, in a two-dimensional conductor at high magnetic fields the Hall resistance is quantized in units of h/2e^2. The quantisation is observed to better than one part in ten million. Reflecting universality, the observed value of the Hall resistance for each of the plateaus is independent of many details, including the temperature, the amount of disorder in the material, the chemical composition of system (silicon versus gallium arsenide), or whether the charge carriers are electrons or holes.

Other examples of macroscopic quantum effects are seen in SQUIDs (Superconducting Quantum Interference Devices). They exhibit quantum interference phenomena analogous to the double-slit experiment. The electrical current passing through the SQUID has a periodicity defined by the ratio of the magnetic flux inside the current loop of the SQUID and the quantum of magnetic flux.

The precision of the quantisation provides a means to accurately determine fundamental constants. Indeed, the title of the paper announcing the discovery of the integer quantum Hall effect was, “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance.” It is astonishing that a macroscopic measurement of a property of a macroscopic system, such as the electrical resistance, can determine fundamental constants that are normally associated with the microscale and properties of atomic systems. 

Laughlin and Pines claimed that the quantisation phenomena described above reflect organizing principles associated with emergent phenomena, and their universality supports their claim of the unpredictability of emergent properties. 

Quantum states of matter and metrology

The universality of these macroscopic quantum effects has practical applications in metrology, the study of measurement and the associated units and standards. In 1990 new international standards were defined for the units of voltage and electrical resistance, based on the quantum Hall effect and the AC Josephson effect, respectively.

Prior to 1990 the standard used to define one volt was based on a particular type of electrical battery, known as a Weston cell. The new standard using the AC Josephson effect allowed voltages to be defined with a precision of better than one part per billion. This change was motivated not only by improved precision, but also improved portability, reproducibility, and flexibility. The old voltage standard involved a specific material and device and required making duplicate copies of the standard Weston cell. In contrast, the Josephson voltage standard is independent of the specific materials used and the details of the device. 

Prior to 1990 the international standard for the ohm was defined by the electrical resistance of a column of liquid mercury with constant cross-sectional area, 106.3 cm long, a mass of 14.4521 grams and a temperature 0 °C. Like the Josephson voltage standard, the quantum Hall resistance standard has the advantage of precision, portability, reliability, reproducibility, and independence of platform. The independence of the new voltage and resistance standards from the platform used reflects the fact that the Josephson and quantum Hall effects have the universality characteristic of emergent phenomena.

This post is an adaptation of material in Condensed Matter Physics: A Very Short Introduction

Quantifying the obscurity of academic writing

 Occasionally The Economist publishes nice graphs capturing social and economic trends. Here is one.

It is part of a nice article

Academic writing is getting harder to read—the humanities most of all

The downward trend in the humanities and social sciences is dramatic and perhaps not surprising.

I was surprised that in the natural sciences the trend wasn't worse. The Flesch reading ease score goes from about 26 around 1960 to about 18 today. 

However, this metric misses a lot. It only measures the number of words per sentence and the number of syllables per word. Thus, it is not sensitive to the amount of technical jargon or to overall clarity.

Self-organised criticality and emergence in economics

A nice preprint illustrates how emergence is central to some of the biggest questions in economics and finance. Emergent phenomena occur as many economic agents interact resulting in a system with properties that the individual agents do not have.

The Self-Organized Criticality Paradigm in Economics & Finance

Jean-Philippe Bouchaud

The paper illustrates several key characteristics of emergence (novel properties, universality, unpredictability, ...) and the value of toy models in elucidating it. Furthermore, it illustrates the elusive nature of the "holy grail" of controlling emergent properties. 

The basic idea of self-organised criticality

"The seminal idea of Per Bak is to think of model parameters themselves as dynamical variables, in such a way that the system spontaneously evolves towards the critical point, or at least visits its neighbourhood frequently enough"

A key property of systems exhibiting criticality is power laws in the probability distribution of a property. This means that there are "fat tails" in the probability distribution and extreme events are much more likely than in a system with a Gaussian probability distribution.

Big questions

The two questions below are similar in that they concern the puzzle of how markets produce fluctuations that are much larger than expected when one tries to explain their behaviour in terms of the choices of individual agents.

A big question in economics

"A longstanding puzzle in business cycle analysis is that large fluctuations in aggregate economic activity sometimes arise from what appear to be relatively small impulses. For example, large swings in investment spending and output have been attributed to changes in monetary policy that had very modest effects on long-term real interest rates."

This is the "small shocks, large business cycle puzzle", a term coined by Ben Bernanke, Mark Gertler and Simon Gilchrist in a 1996 paper. It begins with the paragraph above. [Bernanke shared the 2022 Nobel Prize in Economics for his work on business cycles].

A big question in finance

The excess volatility puzzle in financial markets was identified by Robert Shiller: The volatility "is at least five times larger than it "should" be in the absence of feedback". In the views of some, this puzzle highlights the failings of the efficient market hypothesis and the rationality of investors, two foundations of neoclassical economics. [Shiller shared the 2013 Nobel Prize in Economics for this work]. 

"Asset prices frequently undergo large jumps for no particular reason, when financial economics asserts that only unexpected news can move prices. Volatility is an intermittent, scale-invariant process that resembles the velocity field in turbulent flows..." (page 2)

Emergent properties

Close to a critical point, the system is characterised by fat-tailed fluctuations and long memory correlations.

Avalanches. They allow very small perturbations to generate large disruptions.

Dragon Kings

Minsky moment

The holy grail: control of emergent properties

It would be nice to understand superconductivity well enough  to design a room-temperature superconductor. But, this pales in significance compared to the "holy grail" of being about to manage economic markets to prevent bubbles, crashes, and recessions.

Bouchaud argues that  the quest for efficiency and the necessity of resilience may be mutually incompatible. This is because markets may tend towards self-organised criticality which is characterised by fragility and unpredictability (Black swans).

The paper has the following conclusion

"the main policy consequence of fragility in socio-economic systems is that any welfare function that system operators, policy makers of regulators seek to optimize should contain a measure of the robustness of the solution to small perturbations, or to the uncertainty about parameters value.

Adding such a resilience penalty will for sure increase costs and degrade strict economic performance, but will keep the solution at a safe distance away from the cliff edge. As argued by Taleb [159], and also using a different language in Ref. [160], good policies should ideally lead to “anti-fragile” systems, i.e., systems that spontaneously improve when buffeted by large shocks."

Toy models

Toy models are key to understanding emergent phenomena. They ignore almost all details to the point that critics claim that the models are oversimplified. The modest goal of their proponents is simply to identify what ingredients may be essential for a phenomenon to occur. Bouchaud reviews several such models. All provide significant insight.

A trivial example (Section 2.1)

He considers an Ornstein-Uhlenbeck process for a system relaxing to equilibrium. As the damping rate tends to zero [κ⋆ → 0], the relaxation time and the variance of fluctuations diverge at the same rate. In other words, "in the limit of marginal stability κ⋆ →0, the system both amplifies exogenous shocks [i.e., those originating outside the system] and becomes auto-correlated over very long time scales."

The critical branching transition (Section 2.2)

The model describes diverse systems: "sand pile avalanches, brain activity, epidemic propagation, default/bankruptcy waves, word of mouth,..."

The model involves the parameter R0 which became famous during the COVID-19 pandemic. R0 is the average number of uninfected people who become infected due to contact with an infected individual. For sand piles R0 is the average number of grains that start rolling in response to a single rolling grain.

when R0 = 1 the distribution of avalanche sizes is a scale-free, power-law distribution 1/S^3/2, with infinite mean.

"most avalanches are of small size, although some can be very large. In other words, the system looks stable, but occasionally goes haywire with no apparent cause."

A generalised Lotka-Volterra model (Sections 3.3 and 4.2) 

This provides an analogue between economic production networks and ecology. Last year I reviewed recent work on this model, concerning how to understand the interplay of evolution and ecology.

A key result is how in the large N limit (i.e., a large number of interacting species/agents) qualitatively different behaviour occurs. Ecosystems and economies can collapse. 

 "any small change in the fitness of one species can have dramatic consequences on the whole system – in the present case, mass extinctions...

"most complex optimisation systems are, in a sense, fragile, as the solution to the optimisation problem is highly sensitive to the precise value of the parameters of the specific instance one wants to solve, like the Aij entries in the Lotka-Volterra model. Small changes of these parameters can completely upend the structure of the optimal state, and trigger large-scale rearrangements,..." 

Balancing stick problem (Section 3.4)

 The better one is able to stabilize the system, the more difficult it becomes to predict its future evolution! 

Propagation of production delays along the supply chain (Section 4.1)

An agent-based firm network model (Section 4.3)

This has the phase diagram shown below. The horizontal axis is the strength of forces counteracting supply/demand and profit imbalances. The vertical axis is the perishability of goods.

There are four distinct phases.

Leftmost region (a, violet): the economy collapses; 

Middle region (b, blue): the economy reaches equilibrium relatively quickly;

Right region (c, yellow): the economy is in perpetual disequilibrium, with purely endogenous fluctuations. 

The green vertical sliver (d) corresponds to a deflationary equilibrium

Phase diagrams illustrate how quantitative changes can produce qualitative differences.

Universality

The toy models considered describe emergent phenomena in diverse systems, including in fields other than economics and finance. 

Here are a few other recent papers by Bouchaud that are relevant to this discussion.

Navigating through Economic Complexity: Phase Diagrams & Parameter Sloppiness

From statistical physics to social sciences: the pitfalls of multi-disciplinarity

This includes the opening address from a workshop on "More is Different" at the College de France in 2022.