Berezinskii-Kosterlitz-Thouless Transition and the Haldane Conjecture: Highlights of the Physics Nobel Prize 2016
BBerezinski˘ı-Kosterlitz-Thouless Transition and the HaldaneConjecture: Highlights of the Physics Nobel Prize 2016
Wolfgang Bietenholz a , b and Urs Gerber a , ca Instituto de Ciencias NuclearesUniversidad Nacional Aut´onoma de M´exicoA.P. 70-543, C.P. 04510 Ciudad de M´exico, Mexico b Albert Einstein Center for Fundamental PhysicsInstitute for Theoretical PhysicsUniversity of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland c Instituto de F´ısica y Matem´aticasUniversidad Michoacana de San Nicol´as de Hidalgo, Edificio C-3Apdo. Postal 2-82, C.P. 58040, Morelia, Michoac´an, MexicoThe 2016 Physics Nobel Prize honors a variety of discoveries related to topologicalphases and phase transitions. Here we sketch two exciting facets: the groundbreak-ing works by John Kosterlitz and David Thouless on phase transitions of infiniteorder, and by Duncan Haldane on the energy gaps in quantum spin chains. Theseinsights came as surprises in the 1970s and 1980s, respectively, and they have bothinitiated new fields of research in theoretical and experimental physics.
PACS: General theory of phase transitions, 64.60.Bd; Statistical mechanics of model systems,64.60.De; General theory of critical region behavior, 64.60.fd; Equilibrium properties near criticalpoints, critical exponents, 64.60.F-
When we hear the word “spin” we usually think of Quantum Mechanics, whereparticles are endowed with an internal degree of freedom, which manifests itself likean angular momentum. So what does a “classical spin” mean?It is much simpler: it is just a vector (or multi-scalar) (cid:126)e , say with N components;here we assume them to be real, (cid:126)e = e (1) ·· e ( N ) ∈ R N . (1.1)1 a r X i v : . [ phy s i c s . pop - ph ] A p r odels which deal with such classical spin fields are often formulated on a lattice(or grid), such that a spin (cid:126)e x is attached to each lattice site x . In solid state physics, (cid:126)e x might represent a collective spin of some crystal cell. If it is composed of manyquantum spins, it appears classical [1].If the spin direction is fixed at each site x , we obtain a configuration, which wedenote as [ (cid:126)e ].In a number of very popular models, the length of each spin variable is normalizedto | (cid:126)e x | = 1 , ∀ x . Then the spin field maps the sites onto a unit sphere in the N -dimensional spin space, x → S N − . We are going to refer to this setting, and (forsimplicity) to a lattice of unit spacing, with sites x ∈ Z d in d dimensions.To define a model, we still need to specify a Hamilton function H [ (cid:126)e ] (no oper-ator), which fixes the energy of any possible spin configuration. Its standard formreads H [ (cid:126)e ] = J (cid:88) (cid:104) xy (cid:105) (1 − (cid:126)e x · (cid:126)e y ) − (cid:126)H · (cid:88) x (cid:126)e x , (1.2)where the symbol (cid:104) xy (cid:105) denotes nearest neighbor sites. J is a coupling constant, andwe see that J > ferromagnetic behavior: (approximately) parallel spinsare favored, since they minimize the energy. (cid:126)H is an external “magnetic field” (an“ordering field”, in a generalized sense), which may or may not be included; itspresence favors spin orientations in the direction of (cid:126)H .Thus we arrive at a set of highly prominent models in statistical mechanics, de-pending on the spin dimension N : N = 1 e x ∈ {− , +1 } Ising model N = 2 (cid:126)e T x = (cos ϕ x , sin ϕ x ) XY model N = 3 (cid:126)e T x = (sin θ x cos ϕ x , sin θ x sin ϕ x , cos θ x ) Heisenberg modelwhere ϕ x , θ x ∈ R . These models are discussed in numerous text books, such asRefs. [1–3].Although it might seem ridiculously simple, the Ising model is incredibly suc-cessful in describing a whole host of physical phenomena. The
XY model will beaddressed in Section 2; its best application is to model superfluid helium. The
Heisenberg model captures actual ferromagnets, like iron, cobalt and nickel. Section3.1 refers to its 2d version, which is also a toy model for Quantum Chromodynamics(QCD), since it shares fundamental properties like asymptotic freedom, topologicalsectors, and a dynamically generated mass gap. The large N limit also attractsattention, since it leads to simplifications, which enable analytical calculations, see e.g. Ref. [3]. Vice versa,
J < e.g.
Cr, Mn, Fe O and NiS . In field theory one usually deals with “source fields”, which correspond to a space dependentexternal field of this kind, i.e. to a term (cid:80) x (cid:126)H x · (cid:126)e x . σ -models, or O( N ) models, since — with theHamilton function (1.2) at (cid:126)H = (cid:126) N ) symmetry (or Z (2)symmetry in case of the Ising model): the energy remains invariant if we performthe same rotation on all spins, (cid:126)e x → Ω (cid:126)e x , Ω ∈ O( N ). If the system has temperature T , the probability for a configuration [ (cid:126)e ] is givenby p [ (cid:126)e ] = 1 Z e −H [ (cid:126)e ] /T , with Z = (cid:88) [ (cid:126)e ] e −H [ (cid:126)e ] /T = e − F/T . (1.3)The partition function Z is obtained by summing (or integrating) over all possibleconfigurations, and F = − T ln Z is the free energy . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 1: Examples for a uniform configuration of minimal energy (left) and for anon-uniform configuration of higher energy (right), in the 2d XY model.For
J > − V H (where H = | (cid:126)H | ). An example is shown in Fig. 1 (left), and inthe limit T → T ,fluctuating configurations — like the one in Fig. 1 (right) — gain more importance.They carry higher energy, so the exponential exp( −H [ (cid:126)e ] /T ) suppresses them (theyare less suppressed for increasing T ). On the other hand, there are many of them,and the combinatorial factor is relevant too. This is the entropy effect, which alsomatters for their impact, and which plays a key rˆole in Section 2. The O(4) model is of interest as well, in particular due to the local isomorphy O(4) ∼ SU(2) ⊗ SU(2). The latter is the flavor chiral symmetry of QCD with two massless flavors. Here themagnetic field corresponds to a small mass of the quark flavors u and d , which breaks the symmetrydown to O(3) ∼ SU(2). We express the temperature in units of the Boltzmann constant k B , which amounts to setting k B = 1 throughout this article. In N ≥ V , i.e. with V lattice sites, their number is 2 V . Even for a modest volume, say a 32 ×
32 lattice, this isa huge number of O (10 ), so straight summation is not feasible, not even with supercomputers.Hence to compute expectation values (see below) one resorts to importance sampling by means ofMonte Carlo simulations. For a text book and a recent introductory review, see Refs. [4]. .1 n -point functions and phase transitions What does it mean to have an “impact”? What physical quantities are affected?In exact analogy to field theory, the physical terms are expectation values of someproducts of spins; if they involve n factors, they are called n -point functions. The most important observable is the 2-point function, or correlation function, (cid:104) (cid:126)e x · (cid:126)e y (cid:105) = 1 Z (cid:88) [ (cid:126)e ] (cid:126)e x · (cid:126)e y e −H [ (cid:126)e ] /T . (1.4)One often focuses on its “connected part”, which — in most cases — decays expo-nentially in the distance | x − y | , (cid:104) (cid:126)e x · (cid:126)e y (cid:105) con = (cid:104) (cid:126)e x · (cid:126)e y (cid:105) − (cid:104) (cid:126)e x (cid:105) · (cid:104) (cid:126)e y (cid:105) = (cid:104) (cid:126)e x · (cid:126)e y (cid:105) − (cid:104) (cid:126)e (cid:105) ∝ e −| x − y | /ξ . (1.5)With the Hamilton function (1.2) the system is lattice translation invariant, so the1-point function (cid:104) (cid:126)e x (cid:105) does not depend on the site x , and we can just write (cid:104) (cid:126)e (cid:105) , (cid:104) (cid:126)e x (cid:105) = 1 Z (cid:88) [ (cid:126)e ] (cid:126)e x e −H [ (cid:126)e ] /T = (cid:104) (cid:126)e (cid:105) . (1.6)The decay rate of (cid:104) (cid:126)e x · (cid:126)e y (cid:105) con is given by the correlation length ξ , which serves as the scale of the system: any dimensional quantity is considered “large” or “small” basedon its comparison with (the suitable power of) ξ . Regarding the energy spectrum, ξ represents the inverse energy gap, /ξ = E − E . In quantum field theory, this isjust the mass of the particle, which emerges by the minimal (quantized) excitationsof the field under consideration.The phase transitions that we are interested in are of order 2 or higher, and theyare characterized by the property that ξ diverges. In a phase diagram, with axeslike T and H , this happens in a critical point, in particular at a critical temperature T c . The way how ξ diverges in the vicinity of a critical point defines the criticalexponent ν , lim T → T c ξ ∝ ( T − T c ) − ν , (1.7)where we assume the same power regardless whether T c is approached from above orfrom below (which usually holds). There are a number of critical exponents, whichcharacterize the system close to a critical point; we will see further examples below.In the limit ξ → ∞ , the spacing between the lattice points becomes insignificant(it is negligible compared to ξ ), so this is the continuum limit . This is why thevicinity of a critical point is so much of interest. Phase transitions of first order are more frequent, and they do not correspond to a criticalpoint, but we won’t discuss them. ξ . So does ξ diverge only in the limit T → d = 2 [6] or higher, and the same applies to N > condensate , given in eq. (1.6),which also defines the magnetization M (in some lattice volume V ), (cid:126)m [ (cid:126)e ] = (cid:88) x (cid:126)e x , M = |(cid:104) (cid:126)m (cid:105)| = V |(cid:104) (cid:126)e (cid:105)| . (1.8) M > N ) symmetry is broken. An external field H > infinite volume, V → ∞ )and gradually turn it off, the destiny of the system depends on the temperature: • At low T , the system keeps a dominant orientation in the direction of (cid:126)H , at T → N − • At high T , the system allows for wild fluctuations, and after turning off (cid:126)H it hardly “remembers” its direction. In this case, the O( N ) symmetry is re-stored, since the dominant contributions to an expectation value are due toconfigurations without such a preferred orientation.Thus the magnetization M discriminates the scenarios where the O( N ) symmetryis broken ( M > order ) or intact ( M (cid:39) disorder ). Therefore it is an orderparameter: it is finite (it vanishes) below (above) the critical temperature T c , whichis also called Curie temperature. The way how it converges to 0, as T approaches T c from below, defines another critical exponent β ,lim T (cid:37) T c M ∝ ( T c − T ) β . (1.9)The follow-up example is the critical exponent γ , which characterizes the divergenceof the magnetic susceptibility χ m , at a temperature T close to T c , χ m = 1 V (cid:16) (cid:104) (cid:126)m (cid:105) − (cid:104) (cid:126)m (cid:105) (cid:17) ∝ | T − T c | − γ . (1.10)As in the case of ν , also the exponent γ is usually the same for T > ∼ T c and for T < ∼ T c .There are classes of systems, which may look quite different, but which sharethe same critical behavior; we say that they belong to the same universality class. In particular the (dimensionless) critical exponents coincide within a universalityclass. The enormous success of the Ising model is due to the fact that there aremany models — and real systems — in the same universality class, so the Isingmodel captures their behavior next to a continuum limit.5
Berezinski˘ı-Kosterlitz-Thouless Transitionin the 2d XY Model
This section deals with the 2d XY model, which is among the classical spin modelsintroduced in Section 1. We can imagine a 2d square lattice, where each site x =( x , x ) , x µ ∈ Z , carries a “watch hand” (cid:126)e x , like an arrow from the origin to somepoint on a unit circle. These arrows are parameterizable by an angle ϕ x , (cid:126)e x =(cos ϕ x , sin ϕ x ), as we mentioned before.We formulate the angular difference between two spins as∆ ϕ x,y = ( ϕ y − ϕ x ) mod 2 π ∈ ( − π, π ] , (2.1) i.e. the modulo operation acts such that it picks the minimal absolute value.Now let us consider one plaquette, i.e. one elementary square of the lattice withcorners x , x + ˆ1, x + ˆ2, x + ˆ1 + ˆ2, where ˆ µ is a unit vector in µ -direction. For a givenconfiguration, each plaquette has a vortex number v x , v x = 12 π (cid:16) ∆ ϕ x,x +ˆ1 + ∆ ϕ x +ˆ1 ,x +ˆ1+ˆ2 + ∆ ϕ x +ˆ1+ˆ2 ,x +ˆ2 + ∆ ϕ x +ˆ2 ,x (cid:17) ∈ {− , , +1 } . (2.2)If the configuration is smooth (close to uniform) in the range of this plaquette, weexpect v x = 0. In case of sizable angular differences | ∆ ϕ x,x ± ˆ µ | , however, we mightencounter a topological defect: this could be a vortex , which we denote as V, or an anti-vortex , AV. They correspond to vortex number +1 and −
1, respectively,vortex V v x = +1anti-vortex AV v x = − . (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Figure 2: Examples for configurations with one vortex V (left), and with one anti-vortex AV (right), in the 2d XY model.Examples for a configuration with one V or one AV are shown in Fig. 2. On theother hand, the configurations in Fig. 1 do not contain any topological defects.6n numerical studies, we have to deal with a finite lattice volume V , and weusually implement periodic boundary conditions in both directions; this provideslattice translation invariance. Then the volume represents a torus, and the totalvorticity vanishes, (cid:80) x v x = 0, due to Stokes’ Theorem. So the number of vorticesmust be equal to the number of anti-vortices, n V = n AV , and the configurations ofFig. 2 are actually incompatible with periodic boundaries.In fact, the global system does not have topological sectors, since its homotopygroup is trivial, Π ( S ) = { } . Nevertheless, the local topological defects V and AVare the crucial degrees of freedom for its phase transition. A first look suggests the following picture: • The presence of many V and AV, i.e. a high vorticity density ρ = (cid:104) n V + n AV (cid:105) /V = 2 (cid:104) n V (cid:105) /V , means that strong fluctuations are powerful, and they destroy the long-rangecorrelations. Hence the corresponding smooth configurations are suppressed,the correlation function (cid:104) (cid:126)e x · (cid:126)e y (cid:105) decays rapidly, as in relation (1.5), and weobtain a correlation length ξ of a few lattice spacings. Due to the interpretationof 1 /ξ as a mass, this is called the massive phase. • On the other hand, for a low vorticity density , ρ (cid:28)
1, long-range correlationdominates. It is not disturbed significantly by the few V and AV that arefloating around, and we are in the massless phase , where ξ = ∞ . Here thecorrelation function (cid:104) (cid:126)e x · (cid:126)e y (cid:105) does not decay exponentially, but only with somenegative power of | x − y | . The system is conformal , i.e. scale-invariant.If we start from low temperature and increase T gradually, this gives more im-portance to “rough” rather than smooth configurations — they are far from uniform,with strong fluctuations. This increases the vorticity density ρ , and at the criticaltemperature ρ is large enough to mess up the long-range correlations, so the systementers its massive phase.To make this point more explicit, we estimate the energy that it takes to im-plement one V or one AV in an otherwise smooth configuration. We do so in asimplified scheme of a quasi-continuous plane: close to the transition this can bejustified, since ξ (the relevant scale) is much larger than the lattice spacing. Thenthe angular field ϕ ( x ) of the simplest (rotationally symmetric) V or AV, with itscore at x = 0, obeys | (cid:126) ∇ ϕ ( x ) | = 1 r , r = | x | , (2.3)7ith opposite gradient directions for a V or an AV, see Fig. 2. In this continuumpicture, the vorticity v is given by a curl integral, anti-clockwise around the core, v = 12 π (cid:73) d(cid:126)x · (cid:126) ∇ ϕ ( x ) = 12 π (cid:90) π r dϕ (cid:16) ± r (cid:17) = ± (cid:26) a vortexan anti-vortex . (2.4)Regarding the energy, we note that the Hamilton function (1.2) (at (cid:126)H = (cid:126)
0) can beconsidered as a kinetic term, made of discrete derivatives, J (cid:88) µ =1 (1 − (cid:126)e x · (cid:126)e x +ˆ µ ) (cid:39) J (cid:88) µ =1 ∆ ϕ x,x +ˆ µ (cid:39) J (cid:126) ∇ ϕ ( x ) · (cid:126) ∇ ϕ ( x ) . (2.5)Here we switched from lattice to continuum notation, and we neglect O (∆ ϕ x,x +ˆ µ ).If we insert relations (2.3) and (2.5) into the Hamilton function, we obtain anestimate for the energy requirement for inserting one V or AV into a smooth “back-ground”, E V = J (cid:90) d x (cid:126) ∇ ϕ ( x ) · (cid:126) ∇ ϕ ( x ) ≈ J π (cid:90) L dr r = J π ln L . (2.6)Note that (despite the continuum notation) L expresses the system size in latticeunits, so it is dimensionless (and taking its logarithm makes sense). The integralover the plane is a bit sloppy regarding the shape of the volume; it is approximatedby a circle of radius L , except for a small inner disc with the radius of one latticespacing (which we have set to 1). The latter matches the illustrations in Fig. 2, andsuch an UV cutoff is needed to obtain a finite result.Even this simplified consideration captures relevant properties. The energy for asingle V or AV is considerable: it is enhanced ∝ ln L , so it takes a high temperatureto make such vortex excitations frequent. In the thermodynamic limit, L → ∞ , theyseem to be excluded, but we will see in Section 2.2 why the topological defects areso important nevertheless. Vadim L. Berezinski˘ı (1935-80) explored these propertiesin 1970/1 [7]. He was working in Moscow, where he pioneered the vortex picture [8],which was later an inspiration in the search for a confinement mechanism in QCD [9].
The picture of Section 2.1 can be criticized for assuming either a single V or a singleAV in the entire configuration, although we stressed before that their number mustbe equal (with periodic boundaries). So the minimal excitation of topological defectsleads to one V plus one AV, as illustrated in Fig. 3, and the above calculation hasto be revised. In fact, the result is not E isolatedV , AV = 2 E V = 2 πJ ln L , but instead E V , AV = 2 πJ ln r V , AV , (2.7)8here r V , AV is the distance between the V and AV core. This can be understoodqualitatively: if the V–AV pair is tightly bound, its long-distance impact cancels; faraway, the configuration can be practically uniform, as in the absence of any vortices.If we observe the system with a low resolution (corresponding to a large ξ ), we mightnot see this pair at all.Figure 3: Profile of a configuration with a V–AV pair, with zero total vorticity: theV (AV) core is indicated by a red dot (blue square). Its energy is estimated in eq.(2.7).Only pulling them far apart leads to “free” V and AV, which are visible to suchan observer. When r V , AV reaches the magnitude of L , the energy requirement is ofthe order of E isolatedV , AV .From eq. (2.7) we see that the trend towards minimal energy implies an attractiveforce ∝ /r V , AV between the V and AV cores. In d = 2 this is a Coulomb force, soa few V and AV spread over the plane can be considered as a Coulomb gas . Its freeenergy F consists of the total energy E , plus an entropy term (cf. Section 1).In the period 1972-4, John M. Kosterlitz (born 1942 in Aberdeen), and
David J.Thouless (born 1934 in Bearsden), both from Scotland, worked on this issue at theUniversity of Birmingham. They concluded that the driving force of the transitionbetween the massive and the massless phase is not exactly the density ρ (referredto in Section 2.1), but the density of “free vortices and anti-vortices”, i.e. V or AVwithout any opposite partner nearby. So the phase transition is actually driven bythe (un-)binding of V–AV pairs [10].To make this picture more explicit, we consider the free energy F , say in asub-volume which is large enough to accommodate one free V. It is convenient tocall its size L , and to recycle formula (2.6). The entropy S is the logarithm of themultiplicity of such configurations, here this is just the number of L plaquetteswhere the vortex could be located. This yields F = E V − T S = J π ln L − T ln L = ( J π − T ) ln L , (2.8)9nd the phase of the system depends on the question which of these two termsdominates. • At low T there are hardly any free V or AV (they are suppressed when L becomes large), though there might be some tight V–AV pairs. • At high T these pairs unbind: due to the dominance of the second term, alarge size L makes it easy to spread free V and AV all over the system.In this setting, eq. (2.8) suggests that the critical temperature, where the tran-sition happens, amounts to T c = J π/
Kosterlitz and Thouless predicted a type of phase transition, which had been un-known before the 1970s. The correlation length diverges when T decreases downto T c , as in the well-known phase transitions of second order, but in contrast tothem ξ = ∞ persists at T < T c , and no symmetry breaking is involved. This meansa step beyond Landau’s Theory, which successfully describes second order phasetransitions with the concept of spontaneous symmetry breaking. In low dimensions( d ≤ M >
0. This has been demonstrated generally by theMermin-Wagner Theorem [11], and specifically for the 2d O( N ) models in Ref. [12].The characteristics of the BKT transition were also confirmed experimentally, inparticular in thin films of superfluid He [13] and of superconductors [14].With respect to the critical exponents, this transition was discussed comprehen-sively by Kosterlitz in 1974 [15], based on Renormalization Group techniques. Hepointed out that this is a phase transition of infinite order, an essential phase tran-sition . The correlation length ξ is not described by a power divergence as in relation(1.7), but by an essential singularity, ξ ∝ exp (cid:16) const . ( T − T c ) ν e (cid:17) , T > ∼ T c . (2.9)Thus one defines a critical exponent ν e for the exponential growth of ξ ; Kosterlitzderived its value ν e = 1 / β , and the susceptibility χ m does not follow relation(1.10) either. The critical exponents of Section 1.1 all refer to infinite volume, but inthe 2d XY model at V = L × L → ∞ , χ m diverges throughout the massless phase.Kosterlitz predicted how it diverges as a function of L (the scale which is left) [15], χ m ∝ L − η e (ln L ) − r e , η e = 1 / , r e = − / . (2.10)10his prediction is hard to verify numerically: studying the logarithmic term (andfurther sub-leading logarithms) requires huge volumes. A particularly extensiveinvestigation with the standard Hamilton function (1.2) in Ref. [16] (see also Ref.[17]) is based on simulations up to size L = 2048, and the outcome is consistentwith the predicted exponents η e and r e , though r e comes with a large error.At this point we mention an alternative and entirely different Hamilton functionfor the O( N ) spin models. Unlike the term (1.2), it does not include any (discrete)derivative term, but just a cutoff δ for the angular difference between any two nearestneighbor spins, H [ (cid:126)e ] = (cid:26) | ∆ ϕ x,x +ˆ µ | < δ ∀ x, µ ∞ otherwise . (2.11)Such a constraint Hamilton function is topologically invariant, which means thatmost small modifications of a configuration leave the energy exactly constant. Thisis highly unusual: part of the configurations are excluded (those that violate theconstraint), while all others have energy 0. Still, it has the same symmetries as thestandard action, and it belongs to the same universality class [18–20].There is no temperature in this formulation, but the constraint angle δ plays arˆole, which bears some analogy. In fact, there is a critical δ c , and the system is in itsmassive (massless) phase for δ > δ c ( δ < δ c ). When δ approaches its critical valuewithin the massive phase, the correlation length exhibits an exponential divergenceas in relation (2.9) [19], ξ ∝ exp (cid:16) const . ( δ − δ c ) ν e (cid:17) , δ > ∼ δ c . (2.12)This observation singles out the critical constraint angle δ c = 1 .
10 100 1000 1.86 1.88 1.9 1.92 1.94 1.96 1.98 2 ξ δ datafit with δ c = 1.775, ν e = 0.503 Figure 4: The exponential divergence of the correlation length ξ , as δ decreasestowards δ c (cid:39) . ν e .11his divergence as δ > δ c decreases, and the fit yields ν e = 0 . χ m is consistent with the relation (2.10), and η e is confirmed to two digits [19],whereas the value for r e is plagued by large uncertainties, as in Ref. [16].Another prediction for the BKT transition in the 2d XY model refers to the helicity modulus (or spin stiffness ). In its dimensionless form, it is defined asΥ = 1 T ∂ ∂α F | α =0 , (2.13)where α is a twist angle in the boundary conditions. The free energy F is minimalat α = 0 (periodic boundaries), and Υ is the curvature in this minimum.The qualitative picture is illustrated in Fig. 5 (left): in the large volume limit,one expects Υ to perform a universal jump at T c [21]. Soon after the BKT transitionhad been put forward, the height of this jump was predicted as 2 /π [22]. Later asmall correction was subtracted [23] to arrive at the theoretical valueΥ c , theory = 2 π (cid:16) − e − π (cid:17) (cid:39) . . (2.14)Regarding the constraint Hamilton function, we can interpret Z exp( − F ( α ) /T ) moderate volumelarge volumeinfinite volume ϒ ϒ c T T c ϒ c ϒ c, theory Figure 5: A qualitative picture of the helicity modulus Υ depending on the tem-perature (left), and an overview over numerical results for its helicity jump at thecritical point, Υ c (right).generally as the probability for a (dynamical) twist angle α , so the helicity moduluscan be studied without needing the concept of temperature.Fig. 5 (right) summarizes simulation results for Υ c obtained with various latticeHamilton functions. The standard formulation (1.2) is very tedious in this regard:even simulations at L = 2048 yielded Υ c = 0 . | ∆ ϕ x,x +ˆ µ | exceeds δ = π/
2, the energy contributionsof this pair of neighboring spins jumps from zero to some finite value, which is varied(instead of varying δ ). Here L = 256 led to Υ c = 0 . L extrapolation with the theoreticalvalue in eq. (2.14).This compatibility was finally demonstrated beyond doubt with the constraintHamilton function (2.11). As a function of δ (replacing T ), Υ behaves exactly asdepicted in Fig. 5 (left): a jump is observed around δ c , and it becomes more markedas the volume increases. At δ c the value Υ c = 0 . L = 64, and larger volumes confirmed the agreement with eq. (2.14) [20]. This isone of the clearest pieces of numerical evidence that the BKT transition does occur,and that the corresponding quantitative predictions are valid. The height of this jump, Υ c , is related to the essential critical exponent η e . Itcan be translated into the jump of the superfluid density [22], which has been ob-served in films of He [13], and recently also in an optically trapped 2d Bose gas [26].All this seems nicely consistent, but in some sense it is puzzling: in Section 2.2we reviewed the consideration of energy vs. entropy in the vortex picture, whichpredicts the BKT transition. This picture is standard, and it has been brought intofurther prominence by the Nobel Prize Committee. However, in the formulationwith the constraint Hamilton function the energy cost for any V or AV is zero, butstill the BKT transition is beautifully observed [19, 20]. Is this a contradiction?
Figure 6: Maps of typical configurations of the XY model on a 128 ×
128 lattice,with the constraint Hamilton function (2.11) and δ = 1 .
8, 1 . . δ ≤ . Alternative numerical evidence is obtained from the Step Scaling Function [19,25]. It expressesthe change of the ratio
L/ξ when the size L is altered, at fixed (nearly critical) T > ∼ T c or δ > ∼ δ c .
13 first hint is given by Fig. 6, which shows “maps” of the V and AV found intypical configurations at δ = 1 . , . .
5. For small δ , when only few V andAV show up, the trend to a V–AV pair formation is obvious. At δ = 2 . ρ free r of “freeV” plus “free AV”, defined by the property that there is no opposite partner withindistance r , with r = 1 , L = 128). We see an onset around δ (cid:39) . δ exceeds 1 .
9. Hence ρ free r behaves indeed like an (inverse)“order parameter” for the BKT transition (although, strictly speaking, there is noordering). ρ r f r ee δ r = 1r = 2r = 4 Figure 7: Left: The density of “free vortices”, ρ free r , i.e. of V or AV without anopposite partner within distance r = 1 , δ > ∼ δ c , so ρ free r issimilar to an (inverse) order parameter. Right: The mean distance squared betweenV–AV pairs, D , for optimal pairing (black line). For small n V (few vortices), D is much shorter than the corresponding term for random distributed V and AV (redline). Around n V (cid:38)
50 (typical for δ ≈ .
9) this striking discrepancy fades away.This confirms the V–AV (un-)binding mechanism in the BKT transition.Fig. 7 (left) shows the mean distance squared between nearby V and AV cores, d , AV , in configurations with n V vortices (and n V anti-vortices), also at L = 128, D = 1 n V n V (cid:88) i =1 d , AV ,i . (2.15)The V and AV pairs are formed such that D is minimal. This is compared to D for artificial configurations, where the same number of V and AV are randomdistributed over the volume. For small n V — which corresponds to small δ — we The finite volume shifts the apparent critical angle somewhat up. δ , when n V increases ( δ = 1 . n V = 50).We conclude that the V–AV (un-)binding mechanism is at work, which confirmsonce more the elegant picture by Kosterlitz and Thouless for the BKT phase transi-tion. This observation holds even when topological defects do not cost any energy;then it is a pure entropy effect. Therefore the standard argument for this picture —outlined in Section 2.2 — should be extended.Figure 8: From left to right: Vadim L’vovich Berezinski˘ı (1935-1980) was bornin Kiev (USSR) and graduated 1959 at Moscow State University. After workingat the Textile Institute and the Research Institute for Heat Instrumentation, hejoined 1977 the Landau Institute of Theoretical Physics in Moscow.
John MichaelKosterlitz was born 1942 in Aberdeen (Scotland), studied at Cambridge Univer-sity, and graduated 1969 in Oxford. In 1974 he become Lecturer at BirminghamUniversity, and in 1982 Professor at Brown University in Rhode Island, USA.
DavidJames Thouless was born 1934 in Bearsden (Scotland). He studied at CambridgeUniversity as well, and graduated 1958 at Cornell University, his Ph.D. advisor wasHans Bethe. He worked in Birmingham with Rudolf Peierls, and later with JohnKosterlitz. In 1980 he became Professor at the University of Washington in Seattle.
We now proceed to quantum spin models, leaving behind the classical spins (albeitthey will be back in Section 3.1). Now the components of a spin vector are Hermitian operators, for spin 1 / s = 1 / , , / , , / . . . (in natural units, (cid:126) = 1), we write them as ˆ S ax , where x is15till a lattice site. These components obey the familiar relations[ ˆ S ax , ˆ S by ] = i δ xy (cid:15) abc ˆ S cx , (cid:88) a =1 ˆ S ax ˆ S ax = s ( s + 1) , (3.1)where (cid:15) is the Levi-Civita symbol. If we compare these terms at large s , we seethat the commutator is suppressed as O ( s ) (cid:28) O ( s ), and the spin appears nearlyclassical.For arbitrary spin we assemble the Hamilton operator ˆ H , and write down thepartition function, ˆ H = − J (cid:88) (cid:104) xy (cid:105) ,a ˆ S ax ˆ S ay , Z = Tr e − ˆ H/T . (3.2)It is analogous to the Hamilton function (1.2) and partition function (1.3), nowwith quantum spins. We recognize a global SU(2) symmetry, which transforms eachcomponent ˆ S ax .In addition, the current framework differs from the previous sections in the fol-lowing points: • We focus on spin chains, i.e. dimension d = 1, so now the sites are located ona line. • We consider anti-ferromagnets, with
J <
0, cf. footnote 1. • We skip the external magnetic field. • We drop the additive constant
J V d (“cosmological constant”) of the Hamiltonfunction (1.2). This change is irrelevant — what matters are solely energy differences.
For commutative spin components it would be trivial to write down a groundstate of such an anti-ferromagnetic spin chain: it consists of spins of opposite orien-tations, in alternating order (say | s, − s, s, − s, s, − s . . . (cid:105) ), known as a N´eel state.However, this is not an eigenstate of ˆ H . Quantum spins are far more complicated,and identifying a ground state is a formidable task, even in d = 1.The investigation of these systems has a history of almost 100 years. The ongoinginterest has been fueled by the fact that quantum spin chains exist experimentally;we will give examples below. A breakthrough was achieved by Hans Bethe in 1931,who constructed the ground state for spin s = 1 / s = E − E . We repeat that a fi-nite gap corresponds to a massive phase, with a correlation length ξ = 1 / ∆ s .In the 1950s and 1960s such systems were studied mostly with “spin wave the-ory”, an approach which was fashion at that time. It predicts a “quasi long-range16rder” (without Nambu-Goldstone bosons), which means a power decay of the cor-relation function, i.e. the massless case with ξ = ∞ . This was elaborated mostly inhigher dimensions, d ≥
2, doubts remained about the spin chain.For d = 1, the expected zero gap for s = 1 / s = 1 / , , / . . . .Therefore it came as a great surprise when F. Duncan M. Haldane (born 1951in London) contradicted in 1983 [29, 30]. According to the
Haldane Conjecture, the paradigm was correct only for the half-integer spins, but not for s ∈ N . Heconjectured s = 1 / , / , / . . . (half-integer) ∆ s = 0 gapless s = 1 , , . . . (integer) ∆ s > s = 1 [33]. Later the existence of a gap ∆ > − ( J/ (cid:80) (cid:104) xy (cid:105) ( (cid:80) a ˆ S ax ˆ S ay ) [34]. Forthe standard Hamiltonian (3.2), the value of ∆ was established numerically to highprecision since the early 1990s [35]. A study based on the diagonalization of an L = 22 spin chain, and a large L extrapolation, obtained ∆ = 0 . J [36].This is in agreement with experimental studies. In particular, the materialCs Ni Cl contains quasi-1d anti-ferromagnetic s = 1 spin chains. The scattering ofpolarized neutrons leads to a multi-peak structure, from which the value ∆ (cid:39) . J could be extracted [37]. Similar observations were made with Ni(C H N ) NO ClO [38], but no gap was found in materials with s = 1 / s ∈ N , it is difficult to observe such a gap: it has a conjectured extent∆ s ∼ exp( − πs ) [40], so it becomes tiny for increasing s . The case s = 2 is stilltractable numerically: a study up to L = 350 arrived at ∆ = 0 . J [41]. In summary, the Haldane Conjecture (3.3) has been proved rigorously for all half-integer spins. For the integer spins there are theoretical conjectures. In particular for ∆ they are supported by consistent numerical and experimental results. In additionthere is numerical evidence for ∆ > . .1 Mapping onto the 2d O(3) model A new perspective occurred by mapping such anti-ferromagnetic quantum spinchains onto the 2d O(3) model, or Heisenberg model. The latter emerged as alow energy effective theory, which was constructed by a large- s expansion, and itsvalidity was conjectured for all s [30, 42] (for a review, see Ref. [43]).Thus we are back with a classical spin model of Section 1. We write its Hamiltonfunction in continuum notation, T H [ (cid:126)e ] = (cid:90) d x (cid:104) g ∂ µ (cid:126)e · ∂ µ (cid:126)e − θ π i (cid:15) µν (cid:126)e · ( ∂ µ (cid:126)e × ∂ ν (cid:126)e ) (cid:105) = 1 T H − i θ Q [ (cid:126)e ] . (3.4)The 3-component classical spin field (cid:126)e ( x ) ∈ S has the form that we wrote down forthe Heisenberg model (below eq. (1.2)). The term H is just a continuum versionof the form (1.2) at (cid:126)H = (cid:126)
0, up to the notation for the coupling constant. At largespin s , the (approximately classical) spin (cid:126)S can be written as (cid:126)S (cid:39) s (cid:126)e , still with theconvention | (cid:126)e | = 1, which leads to a weak coupling g (cid:39) T / ( J s ).The important novelty is the θ -term: its integrated form, Q [ (cid:126)e ], counts how manytimes the configuration [ (cid:126)e ] covers the sphere S in an oriented manner. Hence it isan integer, namely the topological charge , or winding number, Q [ (cid:126)e ] ∈ Z = Π ( S ). Therefore exp( −H /T ) is 2 π -periodic in θ , so it is sufficient to consider 0 ≤ θ < π .The Haldane-Affleck map of an anti-ferromagnetic quantum spin chain onto thismodel relates the quantum spin s to the vacuum angle θ as θ = 2 πs (within thelarge s construction) [30, 42, 43]. Taking into account the 2 π -periodicity in θ , weinfer the scheme Haldane Conjecture s integer θ = 0 gap s half-integer θ = π gaplessUnder this mapping, the Haldane Conjecture takes a new turn. It is remarkablethat the mysterious part flips to the other side: the gap for the 2d O(3) modelwithout a θ -term is well established, see e.g. Refs. [44]. On the other hand, it ishard to verify whether the limit θ = π is indeed gapless. If the mapping wererigorous, we could conclude (considering Ref. [32]) that everything is accomplished,but it is another conjecture. Hence the challenge is to investigate the case θ = π .Perturbation theory does not help (cf. footnote 10), so Ian Affleck (born 1952 inVancouver) suggested a non-perturbative topological picture [40], along the lines of The term in eq. (3.4) is usually interpreted as an Euclidean action. Here it is embedded intothe Hamiltonian formalism of statistical mechanics, which we are using throughout this article. Itis the negative exponent in the formula for the partition function, given in eq. (1.3). Small variations of a configuration (except for a subset of measure zero) do not change Q [ (cid:126)e ],so the θ -term is not visible in the field equations of motion, nor in perturbation theory (expansionin powers of g ). Still, it does affect the actual physics, which is non-perturbative (finite g ). H and adds an auxiliary potential term ∼ µ ( e (3) ( x )) ,which pushes the field (cid:126)e into the ( e (1) , e (2) )-plane; in the limit µ → ∞ we are backwith the 2d XY model. We call ϕ the angle within this preferred plane (as before),and α the (suppressed) angle out of it.Let us consider a sub-volume, where the configuration contains a V or an AVin the preferred plane. Its contribution to the topological charge Q is given by thevorticity computed in eq. (2.4), using assumption (2.3) and Stokes’ Theorem, butnow normalized by the area of S , q = 14 π (cid:73) d(cid:126)x · (cid:126) ∇ ϕ ( x ) = ± . (3.5)Local topological defects of this kind, with q = 1 / q = − /
2, are denoted as merons and anti-merons , respectively.The energy estimate is similar to eq. (2.6), in particular we still obtain the factorln
L/a (we now write explicitly a “lattice spacing” a ). The large- L limit only allowsfor configurations with total vorticity 0, as before, but it permits meron–anti-meronpairs (cf. eq. (2.7)). At the end we have to remove the auxiliary potential, µ → Q = ( n meron − n anti-meron) / ∈ Z . However, the (not so tight)meron–anti-meron pairs are mainly responsible for the energy gap.So far this is the picture for θ = 0. If we now include a vacuum angle θ , wesee from eqs. (3.4), (3.5) that this attaches to each region with a meron, or an anti-meron, a factor exp( ± i θ/ θ/ θ = π “neutralizes” all these pairs: theydo not appear in exp( −H /T ), so they cannot erase the long-range order anymore,and the gap vanishes.This picture refers to rather smooth configurations, which dominate at weakcoupling, i.e. at small g . This is the framework of the effective low energy theory[30,42], and we also mentioned that the mapping at large s leads to a small g ∝ /s .For the other extreme, g (cid:29)
1, Seiberg reported a cusp in the free energy, whichsignals a first order phase transition, at θ = π [45].Taking these conjectures together, we arrive at the expected phase diagramshown in Fig. 9. In particular, if we fix a small (or moderate) g , we should run intoa second order phase transition, and therefore into a continuum limit, for θ → π .A subtle study in Ref. [46] made this interesting feature quantitative. To thisend, it related the 2d O(3) model at θ ≈ π , at low energy, to a model of conformal We neglect higher topological defects, which are exceptional at weak coupling. ππ0 0 θ weakcouplingstrongcoupling Figure 9: The expected phase diagram of the 2d O(3) model with a topological θ -term. At weak coupling, the map from anti-ferromagnetic quantum spin chains,along with Haldane’s Conjecture, predicts a finite energy gap at θ = 0, but a gaplesssecond order phase transitions for θ → π .field theory, known as the k = 1 Wess-Zumino-Novikov-Witten model ( k is thecentral charge) [47], see also Ref. [48]. Assuming both to be in the same universalityclass (cf. Section 1), the asymptotic behavior of the correlation length was derivedas ξ ( θ ≈ π ) ∝ | ln( | θ − π | ) | / | θ − π | / . (3.6)In a finite volume L × L , this translates further into a finite size scaling of themagnetic susceptibility χ m (given in eq. (1.10)), and the topological susceptibility χ t = ( (cid:104) Q (cid:105) − (cid:104) Q (cid:105) ) /V : they are both predicted to exhibit a dominant scaling ∝ L ;for a (more abrupt) first order transition one would expect susceptibilities ∝ L (generally L d ). The conjectured form, refined by logarithmic corrections, reads χ m = L √ ln L g m ( L/ξ ) , χ t = L √ ln L g t ( L/ξ ) , (3.7)where g m and g t are “universal functions” with respect to variations of L and ξ .This is an explicit prediction, to be verified in order to check the above conjectureabout a second order phase transitions for θ → π . The way to study effects beyondperturbation theory, from first principle, are numerical Monte Carlo simulations ofthe lattice regularized model (we recall footnote 5 and Refs. [4]). Its idea is togenerate numerous random configurations with probability p [ (cid:126)e ] ∝ exp( −H [ (cid:126)e ] /T ),cf. eq. (1.3). A large set of such configurations enables the numerical measurementof expectation values of the physical terms.This is straightforward for H , but as soon as we include θ (cid:54) = 0, H and exp( −H /T )become complex, so they do not define a probability anymore. We could generate the20onfigurations using | exp( −H [ (cid:126)e ] /T ) | , and include a complex phase a posteriori byre-weighting the statistical entries. This is correct in principle, but the re-weightinginvolves lots of cancellations, hence a reliable measurement requires a huge statistics— the required number of configurations grows exponentially with the volume V .This is the notorious sign problem. In most cases where this problem occurs, in particular in QCD at high baryondensity, and also in QCD with a θ -term, it has prevented reliable numerical results.However, in the case of the 2d O(3) model, this problem was overcome thanks tothe exceptionally powerful meron cluster algorithm [49], applied to the constraintHamilton function (2.11) at δ = 2 π/ (cid:126)e x , the clusters, which are updated collectively [50]. This approach provides hugestatistics (including many configurations that do not need to be generated explicitly,“improved estimator”). Hence in this exceptional case, conclusive numerical resultswere obtained, and they clearly confirmed the predicted large- L scaling of eq. (3.7),including the ln L -refinement [49].In addition, the algorithm assigns an integer or half-integer topological charge q to each cluster (they sum up to the topological charge Q ∈ Z of the entire config-uration). At weak coupling, most clusters are neutral ( q = 0), and a few percentcarry charge q = ± / q = 1 / − /
2) as merons (anti-merons). Then the picture of pair neutralization appears in a new light: now it isstochastic, and it does not require any O(3) symmetry breaking (unlike the potential ∼ µ ( e (3) ) ). Hence it confirms the result for the second order phase transition, andit even endows the heuristic picture with a neat stochastic interpretation. We described the concept of classical and quantum spin models, the framework ofthe 2016 Physics Nobel Prize. We addressed aspects related to topology, i.e. toquantities which are invariant under (most) small deformations, and which can onlychange in discrete jumps. We referred to low dimensions, d = 1 and 2, where —at finite T — thermal fluctuations prevent the spontaneous breaking of continuoussymmetries [11], and therefore the dominance of ordered structures, but higher orderphase transitions happen nevertheless.In the classical 2d XY model we described the BKT phase transition [10], whichis essential (of infinite order), and driven by the (un-)binding of vortex–anti-vortexpairs. This transition has been observed experimentally, for instance in superfluids[13] and in superconductors [14], and recently also in systems of ultra-cold atomstrapped in optical lattices [26, 51].Then we summarized the history of anti-ferromagnetic quantum spin chain stud-21igure 10: Frederick Duncan Michael Haldane (left), was born 1951 in Londonand studied at Cambridge University, where he graduated in 1978. After workingat the University of Southern California, the Bell Laboratories and the Universityof California, San Diego, he became Eugene Higgins Professor at Princeton Univer-sity in 1990.
Ian Keith Affleck (right) was born 1952 in Vancouver, studied atTrent University (in Ontario, Canada), and graduated 1979 at Harvard University,his Ph.D. advisor was Sidney Coleman. He worked at Princeton University andBoston University, and since 2003 he is Killam Professor at the University of BritishColombia in Vancouver.ies, in particular the Haldane Conjecture [29, 30] about energy gaps for integer spinvs. gapless chains for half-integer spin. This insight agrees with experimental resultsas well [37, 38]. We further discussed the mapping onto a classical 2d O(3) modelwith a topological θ -term (the Haldane-Affleck map [30, 42]), and the manifestationof the Haldane Conjecture in that system.These are only selected topics of the works, which were awarded with the PhysicsNobel Prize 2016. For a review of aspects which have not been covered here — inparticular the quantum Hall effect and topological insulators — we refer to Ref. [52]. Acknowledgements:
We thank Michael B¨ogli, Ferenc Niedermayer, Michele Pepe,Andrei Pochinsky, Fernando G. Rej´on-Barrera and Uwe-Jens Wiese for their collab-oration in projects related to the BKT transition and the Haldane conjecture, andHosho Katsura for a helpful remark regarding Ref. [34].This work was supported by the Albert Einstein Center for Theoretical Physics,the European Research Council under the European Union’s Seventh FrameworkProgramme (FP7/2007-2013)/ERC grant agreement 339220, the Consejo Nacionalde Ciencia y Tecnolog´ıa (CONACYT) through project CB-2013/222812, and byDGAPA-UNAM, grant IN107915. 22 eferences [1] S.-K. Ma, Modern Theory of Critical Phenomena (Addison-Wesley Publishing,1976).[2] P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and toCritical Phenomena (John Wiley and Sons, 1977).D.J. Amit, Field Theory, the Renormalization Group and Critical Phenomena(McGraw-Hill, 1978).I. Herbut, A Modern Approach to Critical Phenomena (Cambridge UniversityPress, 2007).[3] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (ClarendonPress, Oxford, 2002).[4] I. Montvay and G. M¨unster, Quantum Fields on a Lattice (Cambridge Univer-sity Press, 1994).W. Bietenholz, Int. J. Mod. Phys. E 25, 1642008 (2016).[5] E. Ising, Zeitschrift f¨ur Physik 31, 253 (1925).[6] R. Peierls, Proc. Cambridge Phil. Soc. 32, 477 (1936).L. Onsager, Phys. Rev. 65, 117 (1944).[7] V.L. Berezinski˘ı, Zh. Eksp. Teor. Fiz. 59, 907 (1970) [Sov. Phys. JETP 32, 493(1971)]; Zh. Eksp. Teor. Fiz. 61, 1144 (1971) [Sov. Phys. JETP 34, 610 (1972)].[8] A.A. Abrikosov, L.P. Gor’kov, I.E. Dzyaloshinski˘ı, A.I. Larkin, A.B. Migdal,L.P. Pitaevski˘ı and I.M. Khalatnikov, Soviet Physics Uspekhi 24, 249 (1981).[9] A.M. Polyakov, in
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