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Topological Transition of Condensed Matter Physics
MASAKI OSHIKAWA
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Nobel Prize in Physics 2016
"Theoretical Discoveries of Topological Phase Transitions and Topological Phases of Matter"

Topological Transition of Condensed Matter Physics

MASAKI OSHIKAWA
INSTITUTE FOR SOLID STATE PHYSICS
KAVLI INSTITUTE FOR THE PHYSICS AND MATHEMATICS OF THE UNIVERSE
UNIVERSITY OF TOKYO

INTRODUCTION

On October 4, 2016, the Nobel Foundation announced that the Nobel Prize in Physics was awarded to David J. Thouless, Michael Kosterlitz, and F. Duncan M. Haldane for "Theoretical Discoveries of Topological Phase Transitions and Topological Phases of Matter". These subjects are now central issues in condensed matter physics, to even in physics as a whole. The three laureates have indeed made pioneering contributions, which have profoundly affected later developments.

There is an infinite variety of matter, including artificially engineered ones. Even for a single material, we can change its properties by controlling various parameters such as the temperature, the pressure, and the external magnetic field. In most cases, changing these parameters slightly does not alter the properties immensely. On the other hand, sometimes changing a parameter leads to a drastic change of the properties such as a discontinuous jump or a singular behavior. Thus, it is convenient to introduce the notion of the phases of matter. A "drastic change" is identified with a phase transition, separating different phases. Two states belonging to the same phase may exhibit quantitatively different properties, but are qualitatively similar. Classic examples of phase transitions include the melting of a crystal (between solid and liquid phases), and the Curie transition of a ferromagnet (between ferromagnetic and paramagnetic phases). These classic examples can be understood in terms of spontaneous symmetry breaking (SSB): solid crystals are characterized by a spontaneous breaking of the spatial translation symmetry by the atomic configuration, and ferromagnets are characterized by a spontaneous breaking of the spatial rotation symmetry by the macroscopic magnetization. Liquids and paramagnets are "disordered" phases that do not break those symmetries. The understanding and description of phases of matter and transitions among them based on SSB, pioneered by Landau, have been quite successful and even applicable to more "quantum" phase transitions such as superfluid transitions in three spatial dimensions. The superfluid phase in three dimensions is characterized by a spontaneous breaking of the U(1) symmetry corresponding to the phase of a quantum mechanical wave function. Although the quantum mechanical phase ("phase of the wave function") is certainly of quantum mechanical nature, once the relevant U(1) symmetry is identified, the phase transition belongs to the same universality class as those in classical systems where the spontaneous breaking of the same U(1) symmetry occurs in three dimensions. In particular, it belongs to the same universality class as the "Curie" transition of the classical ferromagnetic XY model (planar spin model).

Although Landau's framework provides a unified understanding of many phases of matter and transitions among them, there are interesting, more exotic phases and phase transitions that are beyond description with Landau's framework. This year's Nobel Prize in Physics was indeed awarded for the discovery of novel kinds of phases and phase transitions, where topology plays a crucial role. Topology is a branch of mathematics concerning geometrical properties that are robust against continuous deformation. Applications of topology to condensed matter physics started as early as in the 1970s, in the classification of defects. One can introduce a disturbance to a crystal structure by moving atoms around. Many of the disturbances can be "healed" and can be regarded just as fluctuations. On the other hand, if the disturbance cannot be removed by a continuous, local deformation of the atomic configuration, it is robust and stable. Such stable "defects" can be naturally classified by topology. However, the topological approach was initially limited to the classification of a single static defect, given a conventionally ordered phase with spontaneous symmetry breaking. On the other hand, this year's Nobel Prize in Physics recognizes the importance of topology in governing phases and phase transitions, namely in many-body collective phenomena. While there are now many interesting examples demonstrating this, the following three major contributions were specifically recognized as the subjects of the Nobel Prize:
a) (Berezinskii-)Kosterlitz-Thouless ((B)KT) transition in two dimensions,
b) Quantized Hall conductivity in two dimensions as a topological invariant, by Thouless-Kohmoto-Nightingale-den Nijs (TKNN), and
c) Haldane gap in integer-spin quantum antiferromagnetic chains.

In the following, I will give a brief overview on each of these three subjects. If the reader is interested in more details, the "Advanced Information" [N2016] provided by the Nobel Foundation is a good start.

KOSTERLITZ-THOULESS AND TOPOLOGICAL PHASE TRANSITION

In two dimensions, it is proved in general that a continuous symmetry such as U(1) cannot be broken spontaneously at finite temperatures [39, 53]. This excludes the possibility of a conventional phase transition in, say, the two-dimensional XY model or "two-dimensional superfluid". Nevertheless, following the pioneering work by Berezinskii [8,9], Kosterlitz and Thouless theoretically predicted that there is a well-defined phase transition in a two-dimensional system with U(1) symmetry at a finite temperature [35,36]. This novel type of phase transition is in fact related to vortices, a kind of topologically stable defect. Suppose you allow a point defect at the origin, and an arbitrary configuration of the XY spins away from the origin. Let us assume that the spin configuration is "continuous", namely that the neighboring spins are nearly parallel, except in the very neighborhood of the defect. Then the point defect can be classified by the configuration of spins on a circle C encircling the defect on the two-dimensional plane. This configuration corresponds to a mapping from the circle C to another circle (configuration space of the single planer spin). In the spirit of topology, we identify different configurations that can be continuously deformed into each other. In mathematical terminology, we consider the equivalence class of the mapping from a circle to a circle, with the multiplication of two elements defined by concatenation of the mappings. This is nothing but the "fundamental group" appearing in homotopy theory: π1(S1)= meaning that such a defect (vortex) can be characterized by an integer corresponding to "winding number" (Fig. 1).

Fig. 1: An example of the configuration of XY (planar) spins in two dimensions. Along the circle C encircling the origin O, the spins rotate once, namely by 360 degrees. In fact, this number of rotations is unchanged even if the spin configuration is deformed, as long as the deformation is local and the defect does not cross C. This means that this configuration has a winding number 1, which is a topological invariant. Thus we can identify a vortex located at O, as a topological defect.

In practice, a vortex with a winding number n can be regarded as a bound state of n elementary vortices with a winding number 1, so it usually suffices to consider elementary vortices and antivortices. The creation of a vortex distorts spin configurations infinitely far away from the vortex. As a consequence, the total energy of a vortex in fact diverges logarithmically as a function of the system size. This seems to imply that a vortex is unphysical after all. However, this is not the case.

If we consider a pair consisting of a vortex and an antivortex, the winding number defined at a distance is zero, and the pair can have a finite energy. There is a logarithmic attractive potential between the vortex and the antivortex, much like the Coulomb interaction in two-dimensional electromagnetism. At zero temperature, the system is in the ground state with a long-range order, and there is no vortex or antivortex. As the temperature is increased from zero, vortex-antivortex pairs appear, but they do not affect the long-distance properties of the system when the temperature is sufficiently low. The system loses the long-range order even without vortices at any temperature above zero, however, due to strong fluctuations in the local distortions of the spin orientations. Still, the spin-spin correlation function exhibits a power-law decay as a function of the distance, rather than an exponential decay that is ubiquitous at sufficiently high temperatures. This is sometimes called a "quasi long-range order". In the context of two-dimensional superfluids, this also implies that the "helicity modulus", which is a rigidity against a twist of the quantum mechanical phase, is indeed non-vanishing in this low-temperature phase. Since the helicity modulus is related to the superfluid density, this means that superfluidity does exist at a sufficiently low but non-zero temperature in two dimensions even though the long-range order is absent. As the temperature is increased further, more and more vortex-antivortex pairs appear, and the distance between nearest pairs becomes comparable to the size of each pair. Then each vortex would not be bound to a particular antivortex and move freely, resulting in a vortex-antivortex "plasma". This corresponds to the high-temperature disordered phase where the spin-spin correlation function decays exponentially, or the superfluid density vanishes. Thus, we can expect a phase transition driven by the topological defects — vortices. Moreover, Kosterlitz [34] applied the renormalization-group method (which was still in infancy at that time) to the transition, leading to several universal scaling relations near the transition. The celebrated "Kosterlitz Renormalization Group Flow" still remains one of the most remarkable applications of the renormalization group, and also appears in many different contexts. One of the most striking consequences of the renormalization-group analysis is that the helicity modulus vanishes discontinuously at the transition point. This implies that the superfluid density drops to zero at the transition temperature, with a "universal jump" in the superfluid density [43]. This has been indeed confirmed in experiments on thin films of liquid helium-4 [10]. This remains one of the most intriguing triumphs of theoretical physics to this day.

THOULESS-KOHMOTO-NIGHTINGALE-DEN NIJS AND TOPOLOGICAL QUANTIZATION

In the quantum Hall effect, the Hall conductivity is quantized to integer or fractional multiples of e2 /h, which is the quantity solely determined by the fundamental constants. In their groundbreaking paper, Thouless, Kohmoto, Nightingale, and den Nijs (TKNN) brought about a new understanding regarding the quantum Hall effect [51]. It in fact goes beyond the single phenomenon of the quantum Hall effect, and led to the concept of topological phases of matter. Although the actual experiments on quantum Hall effect are performed on two-dimensional electron gas with a constant electron density by varying the magnetic field, it helps to consider a fictitious situation in which the magnetic field is constant and the chemical potential is changed. The textbook solution for free electrons in a magnetic field is the discretized Landau levels. When we increase the chemical potential, each Landau level is completely filled once the chemical potential surpasses that level, resulting in a step-like increase of the electron density. The Hall conductivity is then also given as a step-like function of the chemical potential: the Hall conductivity jumps by e2 /h each time a Landau level is filled. Thus the Hall conductivity is quantized to integral multiples of e2 /h on the plateaus. This looks very much like the actual quantum Hall effect, but we need to remember that this simple "quantization" only applies to the fictitious situation where the chemical potential is varied, and that impurities are essential for the actual quantum Hall effect. Nevertheless, this simple picture is suggestive and captures some essence of physics. TKNN studied what happens to this simple picture if electrons are placed on a periodic potential (or periodic lattice). Electrons in a periodic potential under a magnetic field is an intriguing problem by itself, with a fractal structure of energy levels as a function of the magnetic field [H1976]. If we regard the periodic potential as a perturbation, it generally splits each Landau level into several subbands. How can the Hall conductivity then behave as a function of the chemical potential? Now, generically each (sub)band has some width, so the Hall conductivity would change as the chemical potential is increased when it is within a (sub)band. However, when the chemical potential is in a band gap, (sub)bands below the chemical potential are completely filled while those above are empty. Since each (filled) Landau level carries e2 /h of Hall conductivity, it might appear that each (filled) subband, which arise from splitting of a single Landau level, should carry a fraction of e2 /h. However, Laughlin's gauge argument [37] (which was in fact a kind of topological argument) implies that, whenever the chemical potential is placed within a band gap of free electrons, the Hall conductivity must be an integral multiple of e2 /h. By an explicit calculation based on the linear response theory, TKNN found that indeed each filled (sub)band contributes to the Hall conductivity by an integral multiple of e2 /h, and that integer can have either sign depending on the band structure. At the time of the TKNN paper, however, the meaning of this intriguing "TKNN integer" was not quite clear. The deep meaning of the TKNN integer was later elucidated by Avron, Seiler, and Simon [5, S1983], and by Kohmoto [33]. Applying the Bloch theorem, each single-electron eigenstate is represented by the Bloch wavefunction labelled by a two-dimensional crystal momentum. There is a freedom in the choice of the phase factor of the Bloch wavefunction, which is very much like the gauge symmetry in the real space. One can define a "Berry connection" in the momentum space, which is an analog of the electromagnetic U(1) vector potential in the real space. The TKNN integer turns out to be nothing but the total number of fictitious magnetic flux (in unit flux quantum) throughout the entire momentum space (Brillouin zone, BZ):


Since the momentum space (Brillouin zone) obeys periodic boundary conditions, it is topologically equivalent to a two-dimensional torus. Naively, the total (fictitious) magnetic flux through a closed surface without a boundary like the torus should be zero. This would immediately follow from Stokes' theorem


for a two-dimensional region Ω and its boundary ∂Ω, once the fictitious vector potential is well-defined over the entire Brillouin zone. However, the subtlety is that sometimes the fictitious vector potential cannot be defined over the entire Brillouin zone. In other words, even though there is no singularity and the Bloch wave function can be well-defined everywhere in the Brillouin zone, sometimes there is no single gauge choice that makes the Bloch wave function and the Berry connection smoothly defined over the entire Brillouin zone. In mathematical terminology, such a Bloch wave function is related to a nontrivial fiber bundle that cannot be written as a simple product. A simpler example of a nontrivial fiber bundle is the Möbius strip, which is locally a product of a circle (base space) and a line segment (fiber), but is globally not a simple product of them. A simple product of the same base space and the fiber gives a cylinder, which is topologically distinct from the Möbius strip.

Fig. 2: Möbius strip. It is a two-dimensional object, which is locally a product of a circle (thick curve) and a line section. However, globally it cannot be regarded as a simple product of the circle (base space) and the line segment (fiber), and is an example of a nontrivial fiber bundle.

The fiber bundle corresponding to the Bloch wave function of a two-dimensional system is characterized by an integer topological invariant called Chern number, which is physically the total fictitious flux discussed earlier. Thus a TKNN integer is nothing but the Chern number. Here, a direct connection between an observable physical quantity and a topological invariant was established. This is a very intriguing connection which is appealing in several respects. First, by the very definition of the topological invariant which is robust against continuous deformations (perturbations), it gives a natural understanding of the quantization of the physical quantity. While TKNN in fact does not give a direct explanation for the observed quantum Hall effect in standard settings where magnetic field is varied at a constant electron density, it provided a new perspective to the quantization of the Hall conductivity. In addition, the new understanding based on the fictitious magnetic field in the Brillouin zone stimulated several developments including anomalous Hall effect [NSOMO2010] and Weyl semimetals [NN1983, M2007, WTVS2011, YLR2011]. Moreover, TKNN and its later elucidations led to the concept of topological phases of matter which are characterized by a topological invariant.

Topological insulators, which look very much like trivial band insulators in the bulk but have topologically protected gapless edge/surface states, have been one of the "hottest" topics in condensed matter physics in recent years [B2013]. While TKNN does not directly apply to topological insulators, the concept of topological insulators was an extension of TKNN in its philosophy. In fact, the entire opening paragraph of Ref. [KM2005], which was the first clear theoretical proposal of topological insulators, was devoted to a review of TKNN.

HALDANE GAP AND TOPOLOGICAL PHASES

The antiferromagnetic Heisenberg model is a fundamental model in magnetism. The quantum version of the model exhibits quantum fluctuations; they are indeed so strong in one dimension that the long-range magnetic order is destroyed even in the ground state. This was recognized for the S=1/2 Heisenberg antiferromagnetic chain, based on the Bethe Ansatz exact solution. More specifically, the energy spectrum is gapless, namely there is a continuum of excited states just above the ground state, and the ground state exhibits a power-law decay of the spin-spin correlation function. In fact, this critical ground state of the quantum spin chain is in many ways similar to, and in a certain sense can be identified with, the low-temperature BKT phase of the two-dimensional classical XY model. On the other hand, apparently very little attention was paid to the same model with higher spin quantum numbers S >1/2 before the 1980s. Haldane then put forward a prediction, very bold at the time, that the ground-state properties of the Heisenberg antiferromagnetic chain is qualitatively different, depending on whether the spin quantum number S is an integer or a half-odd-integer [21, 23]. According to his prediction, for a half-odd-integer S, the system is gapless and spin-spin correlation function shows a power-law decay, as in the BKT phase of two-dimensional classical XY model. This was just a generalization of the known case of S=1/2 (but is rather nontrivial from a field-theory perspective). However, he also predicted that, for an integer S, the system has a non-vanishing excitation gap ("Haldane gap") above the ground state, and the spin-spin correlation decays exponentially, much like in the high-temperature disordered phase of a classical spin model. This certainly sounds strange, and was indeed against the "common sense" among physicists at that time. Moreover, his prediction was based on a topological term of a relativistic field theory, O(3) non-linear sigma model [HOO2017]. The topological term appears in the action of the field theory, and is given by a topological invariant of the space-time configuration of the field. Because of the topological nature, it remains invariant under an infinitesimal change of the field variable, and thus does not affect equation of motion at all. However, the topological term can drastically alter the nature of the field theory by its quantum effect. The departure from conventional methodologies in condensed matter physics was perhaps also a reason why few people initially believed in his prediction. Nevertheless, his striking prediction was subsequently confirmed by many analytical, numerical, and experimental studies [A1989] (Fig. 3).

Fig. 3: (A) Inelastic neutron scattering data of the S=1 Haldane gap compound, Y2BaNiO5. (Reprinted with permission from Fig. 3 of Ref. [X2000]). The absence of the dynamical structure factor (indicated by black color) clearly indicates the Haldane gap. (B) Inelastic neutron scattering data of the S=1/2 Heisenberg antiferromagnetic chain, KCuF3. (Reprinted with permission from Fig. 1 of Ref. [L2013]). In contrast to the S =1 case, the gapless continuum excitations down to zero energy are clearly seen. The agreement between the experimental data (panel (a)) and the theory (panel (b)) is remarkable. These are not the first experimental observations of the excitation spectra of S=1/2 and S=1 Heisenberg antiferromagnetic chains, but are examples of more recent data which offer convincing confirmations of Haldane's prediction, thanks to the remarkable progress in experimental techniques.

As much as TKNN, Haldane's triumph was very influential; it would be fair to say that the popular style in condensed matter theory has changed significantly after the prediction of the Haldane gap, as evidenced in the textbooks on field theory for condensed matter physics [AGD1963, 19]. Moreover, the "Haldane gap" has inspired several important developments beyond the quantum spin chains.

Important theoretical support for Haldane's prediction was given by Affleck, Kennedy, Lieb, and Tasaki (AKLT), who constructed a new class of exactly solvable quantum spin models [1, 2]. While the AKLT model differs from the standard Heisenberg antiferromagnetic chain, they proved the properties predicted by Haldane for a modified spin chain model with integer spins. The AKLT state, the ground state of the AKLT model, also uncovered interesting properties which were not foreseen in Haldane's original prediction [16, K1990]. These were identified as characteristics of the "Haldane phase", and were understood as a consequence of hidden symmetry breaking [30]. More recently, the "Haldane phase" (for an odd integer spin) is recognized as a canonical example of "symmetry-protected topological phases" [54,45]. On the other hand, the AKLT state was also the prototype of matrix-product states [FNW1992, KSZ1993] in one dimension, and more general tensor-network states. These are of increasing importance in classification of quantum phases and in developing efficient numerical algorithms for quantum many-body problems [O2014].

Haldane's prediction also stimulated generalizations of an old theorem by Lieb, Schultz, and Mattis (LSM) on S=1/2 spin chains [LSM1961]. First, the difference between an integer spin chain and a half-odd integer spin chain in terms of the LSM theorem was demonstrated by Affleck and Lieb [AL1986]. Later generalizations [OYA1997, O2000, H2004] revealed that the LSM theorem can be understood in more general context as a "filling-enforced" constraint, which is valid also in higher dimensions with applications in constructions of topological phases.

CONCLUSION

Although it is impossible to cover in this article the entire impact of the works which won the Nobel Prize in Physics this year, I hope that the readers are convinced of their fundamental importance and that the prize is very well deserved. These works are more than a theory on a single specific phenomenon, but instead laid out conceptual foundations that transformed condensed matter physics. After this "transition", it has become virtually impossible to discuss quantum condensed matter physics without referring to topology. It might actually be said that it is not so easy to pick this type of conceptual contribution for a Nobel Prize, and this could be a reason why it took so long for these works to be recognized by the Nobel Committee for Physics. Here, one may find a parallel with Yoichiro Nambu, who had laid out a conceptual foundation of modern high-energy theory stimulated by condensed matter physics since the 1960s, but was awarded a Nobel Prize rather recently in 2008. In any case, I am very happy that those fundamental contributions are now recognized in the form of the Nobel Prize, which is very encouraging for us — not only for those of us working on topological aspects of condensed matter physics, but for all of us who are working on scientific problems motivated by a fundamental interest.

Fig. 4: F. D. M. Haldane (left) and the author (right), on October 17, 2016 at the Kavli Institute for Theoretical Physics, UC Santa Barbara, for the Round-Table Talk [HOO2016].

References

The numbered references are taken from [N2016]. The other references are labelled by the initial of the author(s) and publication year.

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Masaki Oshikawa is a professor at the Institute for Solid State Physics (ISSP) and a senior scientist at the Kavli Institute for Physics and Mathematics of the Universe (IPMU), both at the University of Tokyo. His research field is condensed matter theory, in particular quantum many-body problems. His main virtue is the laziness that motivates him to find the simplest possible argument. After obtaining a PhD from the University of Tokyo in 1995, he worked at the University of British Columbia and Tokyo Institute of Technology before taking his current position at ISSP in 2006. He enjoys commuting and traveling on his bicycle in various places in the world.