Category Archives: paper

What is condensed matter physics?

Below is a lightly-edited excerpt from Ch. 1 of my dissertation in which I describe my field in the broadest possible terms. My dissertation is currently in production for publication in the “Springer Theses” series.

This dissertation is in the field of condensed matter physics, which in the most informal sense possible, could be described as ‘the study of stuff that is not especially hot nor moving especially fast’  [1]. A more formal (but no less vague) definition is ‘the study of the behavior of large collections of interacting particles’ [2]. The haziness of this definition is appropriate since condensed matter is a very broad field encompassing the study of almost all everyday matter including liquids, solids and gels as well as exotic matter like superconductors. Condensed matter physics is a tool for answering questions like: Why are some materials liquids? Why are others magnetic? What sorts of materials make good conductors of electricity? Why are ceramics brittle? Our understanding of condensed matter physics underlies much of modern technology; some prominent examples include ultra-precise atomic clocks, transistors [3], lasers, and both the superconducting magnets and the superconducting magnetometers used for magnetic resonance imaging (MRI). Condensed matter physics overlaps with the fields of magnetism, optics, materials science and solid state physics.

Condensed matter physics is concerned with the behavior of large collections of particles. These particles are easy to define: they will sometimes be atoms or molecules and occasionally electrons and nuclei; condensed matter is almost never concerned with any behavior at higher energy scales (i.e. no need to worry about quarks). The key word in the definition is large. Atoms are very small, so any macroscopic amount of matter has a huge number of them, somewhere around Avogadro’s number: 1023. Large ensembles of particles display emergent phenomena that are not obvious consequences of underlying laws that govern the behavior of their microscopic components. In the words of P.W. Anderson:

The ability to reduce everything to simple fundamental laws does not imply the ability to start from those laws and reconstruct the universe. …hierarchy does not imply that science X is “just applied Y.” At each stage entirely new laws, concepts, and generalizations are necessary, requiring inspiration and creativity to just as great a degree as in the previous one. Psychology is not applied biology, nor is biology applied chemistry. [4]

Emergent phenomena are not merely difficult to predict from the underlying microscopic laws, but they are effectively unrelated. At the most extreme scale, no one would argue that consciousness is somehow a property of standard model particles, or that democracy is a state that could ever be described in terms of quantum field theory. Here I will focus on two such emergent phenomena: phase transitions, where symmetries of the underlying laws are spontaneously violated and behavior is independent of microscopic details, and quasiparticles, an almost infinite variety of excitations of many-body states of matter that bear no resemblance to the ‘real’ particles that make up the matter itself [5].

The most interesting problems tend to involve systems with interactions. To highlight the importance of interactions, let us first consider the case of noninteracting particles. The canonical example here is the ideal gas, which is composed of classical point-like particles that do not interact with each other. Because they do not interact, the motion of the particles is independent; if we want to know the energy of any particle, it is easy to calculate from its speed (E = mv22). The behavior of the whole system can be described by an ensemble of independent single particles. When the particles are interacting things are very different. Instead of an ideal gas, let us consider a gas of classical electrons interacting via the Coulomb force 1∕r. For two electrons the equations of motion can be solved analytically, but in a solid there are 1023 electrons (for all practical purposes, we can round 1023 up to infinity). To write down the energy of of one of them, we must account for the position of every single other electron. Thus the energy of just one electron is a function of 3N variables. Even with just three particles, analytic (pen and paper) solutions are impossible in most cases. An analytic solution for the motion of 1023 electrons is impossible, and “it’s not clear that such a solution, if it existed, would be useful” [6]. This is many-body physics. Instead of following individual particles, we describe their collective motion and the resulting emergent phenomena such as quasiparticles and phase transitions. Consider waves crashing on the beach. It would be foolish to try to understand this phenomenon by following the motion of all the individual water molecules. Instead, we can treat the water as a continuous substance with some emergent properties like density and viscosity. We can then study the waves as excitations the ground state of the water (the state without waves).

[1] This definition distinguishes condensed matter from particle physics (the other broad subdiscipline of physics), which is the ‘study of really hot and really fast-moving objects.’
[2] In practice, condensed matter tends to be the term used to describe physics that does not fit into one of the smaller, more well-defined subdisciplines like high-energy physics or cosmology.
[3] Both transistors and atomic clocks are essential to cellular telephones and satellite navigation systems like GPS.
[4]This quote is taken from “More is different” Science 177, 393 (1972) by P.W. Anderson , an excellent refutation of reductionism and discussion of emergent phenomena written in a manner that should be accessible to non-physicists.
[5] I hope to post non-technical descriptions of phase transitions and quasiparticles at some point in the future.
[6] Chaikin and Lubensky, 1998, p. 1

figure 4 from the paper

New paper: “Metamagnetism and zero-scale-factor universality in the two-dimensional J-Q model”

This week my paper “Metamagnetism and zero-scale-factor universality in the two-dimensional J-Q model” came out in Physical Review B. This paper builds on my previous work studying the 1D J-Q model, focusing on the saturation transition at high magnetic field using quantum Monte Carlo and exact techniques.

Below a critical coupling value (Q/J)min, this saturation transition is continuous, and is an example of “zero-scale-factor” (ZSF) universality. In ZSF, the order parameter (in this case the magnetization) is described by a universal function of the bare coupling constants and no non-universal numbers (Sachdev, 1994). 2D is the upper critical dimension of ZSF, so we expect to see logarithmic violations of the scaling at low temperature. The form of these violations is predicted by Sachdev et al. This paper is the first numerical test of this form, and we find to our surprise that the logarithmic violations do not match the form proposed by Sachdev et al. The reasons for this are currently unclear and merit further investigation.

Above the coupling ratio (Q/J)min, the saturation transition take the form of a sudden magnetization jump an example of metamagnetism, a kind of first-order phase transition. This metamagnetic transition is broadly similar to the one discovered in our previous work on the 1D J-Q model (Iaizzi, 2017). It is caused by the onset of bound states of magnons (spin flips against a polarized background). Using an exact method, we extract the value of (Q/J)min.

This paper is available directly from PRB (paywall); it can also be found on arxiv, or as a free PDF here.

Thanks to my coauthors Kedar Damle and Anders Sandvik.

New preprint: Metamagnetism and zero-scale-factor universality in the 2D J-Q model

I just submitted a new manuscript to PRB and posted it on arXiv (arXiv:1804.06045). This paper is a collaboration with Anders Sandvik and Kedar Damle. This is paper builds on our previous work on the field-driven saturation transition in the 1D J-Q model to study the saturation transition in 2D.

figure 4 from the paper

Fig 4. Magnetization density as a function of field for various values of s=Q with J+Q=1.

Metamagnetism: Using QMC, we find magnetization jumps to the saturated state (metamagnetism) above a critical coupling ratio (Q/J)min. We then use an exact method based on a high magnetization expansion to determine (Q/J)min. Above (Q/J)min the saturation transition is discontinuous (featuring a magnetization jump) and below (Q/J)min the transition is continuous.

Zero-scale-factor universality: When the saturation transition is continuous it is governed by zero-scale-factor universality. First proposed by Sachdev, Senthil and Shankar in 1994, zero-scale-factor universality is characterized by response functions that depend only on the bare parameters and no non-universal numbers. Remarkably, there had be no previous numerical or experimental verification of the scaling forms predicted in the 1994 Sachdev paper. Our QMC results confirm that the leading order scaling forms work and find logarithmic divergences at low temperature. These divergences are to be expected, since two spatial dimensions is the upper critical dimension of the zero-scale-factor universality, and the Sachdev paper even proposes a form for this divergence. When comparing to our QMC results, we find that the Sachdev form for the logarithmic divergences does not appear to be correct.

If you would like to know more all the details are on arxiv.