Vortex mass in the three-dimensional O(2) scalar theory
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Vortex mass in the three-dimensional O (2) scalar theory Gesualdo Delfino , , Walter Selke and Alessio Squarcini , SISSA – Via Bonomea 265, 34136 Trieste, Italy INFN sezione di Trieste, 34100 Trieste, Italy Institute for Theoretical Physics,RWTH Aachen University, 52056 Aachen, Germany Max-Planck-Institut f¨ur Intelligente Systeme,Heisenbergstr. 3, D-70569, Stuttgart, Germany IV. Institut f¨ur Theoretische Physik,Universit¨at Stuttgart, Pfaffenwaldring 57,D-70569 Stuttgart, Germany
We study the spontaneously broken phase of the XY model in three dimensions, with boundaryconditions enforcing the presence of a vortex line. Comparing Monte Carlo and field theoreticdeterminations of the magnetization and energy density profiles, we numerically determine themass of the vortex particle in the underlying O (2)-invariant quantum field theory. The result shows,in particular, that the obstruction posed by Derrick’s theorem to the existence of stable topologicalparticles in scalar theories in more than two dimensions does not in general persist beyond theclassical level. Topological excitations are among the most fascinatingobjects in quantum field theory (QFT) [1, 2]. Being as-sociated to extended configurations of the fields appear-ing in the action, they are intrinsically non-perturbativeand difficult to characterize as quantum particles. A wellknown exception is provided by two-dimensional space-time, where sine-Gordon solitons correspond, throughfermionization, to the fundamental fields of the massiveThirring model [3]; in addition, integrability allows a fulland exact quantum description [4]. No similar methods,however, are available in higher dimensions.In three dimensions the simplest theory with symme-try properties allowing for topological excitations – vor-tices – is the O (2)-invariant scalar theory. This describesthe universality class of the XY lattice model, whichincludes the superfluid transition of He as a particu-larly interesting representative (see [5]). While a directresponsibility of vortices in the phase transition of thethree-dimensional O (2) theory has been debated [6, 7],it is a fact that the transition is of the type associatedto spontaneous symmetry breaking, which occurs also inabsence of non-trivial topology. In field theory, vorticesin the scalar theory have usually been considered only topoint a problem, i.e. that the energy (mass) of the staticclassical solution diverges logarithmically [8], a particu-lar case of Derrick’s theorem [1, 2, 9]. This divergenceat the classical level suggested the absence of a vortexparticle in the scalar QFT.In this paper we consider the three-dimensional XY model in its spontaneously broken phase, slightly belowthe critical temperature T c , with boundary conditionsenforcing the presence of a vortex line. As the otherproperties of the near-critical system, the correspondingenergy density and order parameter profiles have to beaccounted for by the O (2) scalar QFT describing the con-tinuum limit. Remarkably, these profiles are calculable in the field-theoretical framework, and we compare the an-alytic results with the numerical determination obtainedby Monte Carlo simulations, finding excellent agreementas we vary the temperature and the end-to-end distanceof the vortex line. In the process we numerically deter-mine the mass m V of the vortex particle, which for small | T − T c | can be expressed as m V ≈ . m + , (1)where m + is the mass of the fundamental particles inthe phase with unbroken symmetry ( T > T c ). This re-sult provides the first direct verification that Derrick’stheorem, as a statement concerning classical field config-urations, does not prevent the existence of stable topo-logical particles in quantum theories of self-interactingscalar fields in more than two dimensions.We consider the XY model with reduced Hamiltonian H = − T X s i · s j , (2)where s i is a two-component unit vector (spin) located atthe site i of a cubic lattice, and the sum is performed overall pairs of nearest neighboring sites. We focus on thecase T < T c in which the O (2) symmetry of the Hamilto-nian is spontaneously broken, i.e. h s i i 6 = 0; h· · ·i denotesthe average over spin configurations weighted by e −H .Close to T c the intrinsic length scale of the system be-comes much larger than lattice spacing and the system isdescribed by a O (2)-invariant Euclidean scalar field the-ory, which in turn is the continuation to imaginary timeof a QFT in (2 + 1) dimensions. Switching to notationsof the continuum, we denote by x = ( x , x , τ ) ≡ ( x , τ ) apoint in Euclidean space, τ being the imaginary time di-rection, and by s ( x ) = ( s ( x ) , s ( x )) the two-componentspin field. The field theory is the usual one defined by x x R/ L R/ L FIG. 1. Geometry considered for the XY model below T c .Boundary spins point outwards orthogonally to the verticalexternal surfaces. On the top and bottom surfaces they arefixed as indicated, so that a vortex line (one configuration isshown) runs between the central points of these surfaces. the action A = Z d x (cid:8) [ ∂ µ s ( x )] + g s ( x ) + g [ s ( x )] (cid:9) , (3)with the XY critical point reachable tuning the couplings(see e.g. [10]).We consider the system as defined in the volume x ∈ ( − L/ , L/ x ∈ ( − L/ , L/ τ ∈ ( − R/ , R/ L → ∞ and R large but finite. The boundary conditionsare chosen in such a way that all spins on the external sur-faces parallel to the τ -axis point outwards orthogonallyto the surface. This implies the formation of a vortexon each section with constant τ , with the vortex centerforming a vortex (or defect) line as τ varies. Boundaryconditions on the surfaces τ = ± R/ x = 0, τ = ± R/
2. The vortex line corresponds tothe trajectory in imaginary time of a topological particle(the vortex V ) in the (2 + 1)-dimensional QFT [11].The boundary conditions at τ = ± R/ | B ( ± R/ i = e ± R H | B (0) i of the Euclideantime evolution; here H denotes the Hamiltonian of thequantum system. These boundary states can be ex-panded on the basis of asymptotic particle states of the QFT, and will contain the vortex as the contribution withminimal energy, i.e. | B ( ± R/ i = Z d p (2 π ) E p a p e ± R E p | V ( p ) i + . . . , (4)where p is the two-component momentum of the particle, E p = p p + m V its energy, a p an amplitude, and wenormalize the states by h V ( p ′ ) | V ( p ) i = (2 π ) E p δ ( p − p ′ ). In the calculations performed with the boundaryconditions we have chosen (which we indicate with a sub-script B ) the one-vortex contribution in (4) determinesthe asymptotics for R ≫ /m V (indicated below by thesymbol ∼ ). Then we have Z B ≡ h B ( R/ | B ( − R/ i = h B (0) | e − RH | B (0) i (5) ∼ | a | Z d p (2 π ) m V e − ( m V + p mV ) R = | a | πR e − m V R , while for the expectation value of a field Φ we obtain h Φ( x , i B = Z B h B ( R/ | Φ( x , | B ( − R/ i (6) ∼ R (2 π ) m V R d p d p F Φ ( p | p ) e − R mV ( p + p )+ i x · ( p − p ) , where F Φ ( p | p ) = h V ( p ) | Φ(0 , | V ( p ) i , p , p → h s ( x , i B (magnetization) hasto interpolate between zero at x = 0 (where the symme-try is unbroken) and the asymptotic valuelim | x |→∞ h s ( x , i B = v ˆ x , (8)where ˆ x = x / | x | , and v = |h s ( x , τ ) i| is the modulus of thebulk magnetization for free boundary conditions. It wasargued in [11] that such a behavior requires F s ( p | p )proportional to p − p | p − p | ; (9)it was shown in the same paper that, when inserted in(6), this expression produces the result h s ( x , i B ∼ v √ π F (cid:18) , − η (cid:19) η ˆ x , (10)= v √ π η (cid:2) I ( η /
2) + I ( η / (cid:3) e − η / ˆ x , where F ( α, γ ; z ) is the confluent hypergeometric func-tion, I k ( z ) are Bessel functions, and η ≡ r m V R | x | . (11)The generalization of (10) to h s ( x , τ ) i B is straightforwardand results in replacing η by η/ p − τ /R .We now compare the theoretical prediction (10) withMonte Carlo simulations of the XY model on the cubiclattice. Obviously, in the simulations L is finite, but wealways take it sufficiently large to exhibit the approachto the theoretical asymptotic values; lattices with L upto 161 and R up to 91 are considered (lengths enteringsimulations are expressed in units of the lattice spacing).Technical details are quite similar to those of our recentstudy of two-dimensional Potts models [12]. In particu-lar, the standard Metropolis algorithm [13] is used. Ther-mal averages are computed by averaging over several (atleast six) realizations, with each run being of length 10 Monte Carlo steps per site. The resulting error bars arenot depicted in figures 2 and 3 below; usually their sizedoes not exceed that of the symbols in those figures.The relations we write below are intended in the limitin which the temperature approaches the critical value T c , which is known very accurately; we quote here theresult 1 /T c = 0 . T c ≃ . T < T c , sufficiently close to T c to make cor-rections to scaling inessential, at least within the level ofaccuracy relevant for the purposes of this paper. Since for L large and sufficiently away from the boundaries all ra-dial directions in the plane τ = 0 are equivalent, we focuson the cases x = 0 or x = 0. We measured h s i i B alongthese axes and verified that, within error bars, only theradial component is non-zero. This component shouldthen be compared with (10), taking into account thatnear T c m V ≃ m V ( T c − T ) ν , (12) v ≃ v ( T c − T ) β , (13)with ν = 0 . β = 0 . v = 0 . m V is the only unknown quantity inthe comparison between theory and data. Our MonteCarlo results for different values of T and R are shown infigure 2, and seen to be in remarkable agreement with thetheoretical curves corresponding to m V = 2 .
5. In partic-ular, the figure confirms that the magnetization profilesdepend on the scaling variable (11), which in turn orig-inates from the fact that the vortex is an asymptoticparticle of the underlying QFT. The data also implic-itly confirm the form (9) of F s ( p | p ). A singularity forequal momenta is known to appear also in the form fac-tor of the spin field on the soliton state in two dimensions[16, 17], where it accounts for new results in the theoryof phase separation and interfaces [18].As any critical amplitude, m V is non-universal (i.e.depends on lattice details), and it is relevant to obtain theuniversal ratio with another mass amplitude. Above T c ,the correlation function h s ( x ) · s (0) i decays exponentiallyat large distances as e − m + | x | , where m + ≃ m ( T − T c ) ν is the mass of the lightest particles and coincides (see e.g. FIG. 2. Analytic values of the magnetization (10) (continuouscurves) and corresponding Monte Carlo results (data points).In order of decreasing slope at x = 0, the curves refer to( T = 2 . R = 31), ( T = 2 . R = 61), and ( T = 2 . R = 61). The Monte Carlo curves were obtained for L = 61,101, 161, respectively. [19]) with the inverse of the correlation length determinednumerically in [20]. From the data reported in that paper(see Table 7 and Eq. (25)) we deduce the value m ≃ .
21, which then leads to the universal relation (1).Below T c , the presence of Goldstone bosons makesmore complicated extracting a mass scale from the decayof spin-spin correlations, and it has been common to con-sider instead the helicity modulus Υ, which measures thefree energy change under a twist of the spins [21]. On theother hand, we are seeing that a true mass, m V , emergesfrom the measurement of the magnetization (10). To-gether with (1), the result Υ /m + = 0 . /m V ≈ . ε i = P j ∼ i s i · s j , where the sum runs over the nearest neigh-bors of site i . The Monte Carlo data we obtainedalong radial directions at τ = 0 are shown in figure 3and clearly exhibit the localization of the vortex energyaround the center of the system. They also allow to seethat the depth of the minimum of the profiles scales as R − / . These features are accounted for by the choice F cε ( p | p ) ∝ ( | p || p | ) − / for the energy density field ε ( x ) ∼ s ( x ); the superscript c denotes the connectedpart. Upon insertion in (6) this yields h ε ( x , i B ∼ A √ R (cid:20) F (cid:18) , , − m V R x (cid:19)(cid:21) + E vac , (14)where the additive constant E vac = h | ε | i , correspond-ing to the vacuum energy density, comes from the dis-connected part (2 π ) m V δ ( p − p ) h | ε | i of F ε ( p | p ).The quantities appearing in (14) scale as E vac ≃ E ( T c − T ) νX ε , (15) A ≃ A ( T c − T ) ν ( X ε − / , (16) FIG. 3. Analytic values of the energy density profile (14)(continuous curves) and corresponding Monte Carlo results(data points). The top and bottom profiles refer to ( T = 2 . R = 31) and ( T = 2 . R = 91), respectively. The profilesin between are obtained for T = 2 . R = 31(deeper minimum) and R = 81. Simulations were performedwith L = 91 , ,
151 for T = 2 . , . , .
16, respectively. where X ε is the scaling dimension of the energy densityfield; since ν = 1 / (3 − X ε ), the value we already quotedfor this exponent yields X ε ≃ .
51. For | x | large andsufficiently away from the boundary, the Monte Carlodata for the energy density asymptotize to the bulk en-ergy density E , which differs from E vac by regular terms c n ( T c − T ) n , n = 0 , , . . . (see e.g. [20]); we obtain E fitting the data reported in [20] for a list of values of T . Having already determined m V , the only unknownparameter left in the comparison between theory anddata for the energy density is the amplitude A entering(16). The value A = − . F cε ( p | p ) = constant assumed [22]in [11], and to gain further insight into this previouslyunexplored sector of QFT.It follows from (10) that |h s ( x , i B | /v behaves as A | x | and 1 − ( B | x | ) − for small and large | x | , respectively, with A/B = √ π/ ≃ . .
412 for
A/B .In this form, the comparison overcomes the fact that thecharacteristic lengths are different in the two cases; inparticular, the length p R/ m V in (11) depends on thedistance R , which has no counterpart in the GP calcula-tion. In perspective, it will be interesting to see whetherour results can be relevant for the controversial prob-lem of defining an inertial mass per unit length of vortextubes in superfluids (see [24]).It must be noted that the result (10) for h s ( x ) i B reliesonly on the topological constraints, and does not require that the scalar field interacts only with itself. Hence, (10)should hold also if the scalar is coupled to the electromag-netic field [25]. This case, however, does not correspondto the XY universality class and, in particular, the massratio (1) will be different. Similarly, (10) should holdin the broken phase of XY models allowing for antiferro-magnetic bonds. Vortex lines in a model of this type havebeen considered [26] in connection with the paramagneticMeissner effect.Summarizing, we studied the spontaneously bro-ken phase of the three-dimensional XY model withboundary conditions enforcing the presence of a vortexline. Through comparison with analytic expressions, weshowed that the results of Monte Carlo simulations forthe order parameter and energy density profiles corre-spond to a field theory possessing the vortex as a stablequantum particle, and determined in the process the nu-merical value of its mass. The result also yields the firstdirect verification that Derrick’s theorem, as a statementfor classical field configurations, does not provide a fun-damental obstruction to the existence of topological par-ticles in purely scalar QFTs in more than two dimensions.The analytic form of the profiles for large end-to-end dis-tance of the vortex line relies on topological propertiesand should continue to hold when the scalar field is cou-pled to electromagnetism. Acknowledgments.
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