The string tension at low T and the free energy of domain walls at high T can
be computed from one and the same observable. We show by explicit calculation
that domain walls in hot Z(2) gauge theory have good thermodynamical behaviour.
This is due to roughening of the wall, which expresses the restoration of
translational symmetry.Comment: Contributed paper to the proceedings of the second workshop on
  Continuous Advances in QCD, ITP, University of Minnesota, Minneapolis,
  Minnesota, USA, March 28-31, 1996. 11 pages, figures and style file are
  appended in uuencoded gzip.tar.fil

Altes, C. P. Korthals

Bronoff, S.

English

arXiv

The string tension at low T and the free energy of domain walls at high T can be computed from one and the same observable. We show by explicit calculation that domain walls in hot Z(2) gauge theory have good thermodynamical behaviour. This is due to roughening of the wall, which expresses the restoration of translational symmetry.The string tension at low T and the free energy of domain walls at high T can be computed from one and the same observable. We show by explicit calculation that domain walls in hot Z(2) gauge theory have good thermodynamical behaviour. This is due to roughening of the wall, which expresses the restoration of translational symmetry

Korthals Altes, C.P.

CERN Document Server

arXiv:hep-lat/9607016v1  4 Jul 1996CERN-TH/96-164INTERFACES IN HOT GAUGE THEORYS. BRONOFFCentre Physique The´orique au C.N.R.S., B.P. 907,Campus de Luminy, F13288 Marseille CEDEX, FranceC.P. KORTHALS ALTESTheory Division, CERN, CH-1211 Geneva 23 Switzerland,and Centre Physique The´orique au C.N.R.S., B.P. 907,Campus de Luminy, F13288 Marseille CEDEX, FranceConsider Hamiltonian Z(2) gauge theory in a heat bath with temperature T. Thestringtension at low T and the free energy of domainwalls at high T can be com-puted from one and the same observable. We show by explicit calculation thatdomain walls in hot Z(2) gauge theory have good thermodynamical behaviour.This is due to roughening of the wall, which expresses the restoration of transla-tional symmetry.Contribution to the Proceedings of the Second Workshop on“Continuous Advances in QCD”28–31 March 1996, Minnesota, U.S.A.CERN-TH/96-164June 199611 IntroductionTraditionally, spontaneously broken symmetries are restored at high enoughtemperature. High enough means the scale of the energy gap in the theory.Nature provides us with such systems abundantly, such as superconductivity,ferromagnetism and the electroweak sector of the standard model of particlephysics.Much less known is the fact that not only nature, but also the laboratory ofLagrangian field theory, provides us with systems where the opposite happens.Weinberg 1 quotes the example of Seignette salt, a ferroelectric compound;below the critical temperature it orthorombhic and above only isoclinic. Also,during the last two decades examples in field theory have been given mainlyin connection with particle physics and cosmology.Independently of these developments, the study of SU(N) gauge theorieshas led to the notion of spontaneously broken centergroup (Z(N)) symmetry 4taking place above the deconfinement transition. This notion has been usefulin understanding certain universal properties of the transition, like criticalexponents 5.Other consequences of the spontaneous breaking are domain walls betweencoexisting phases. These questions have been addressed at very high tempera-tures where semi-classical techniques are possible9. It was soon discovered thatthese walls showed thermodynamic anomalies10; and other criticism sprang up,going as far as to deny the concept of broken Z(N) symmetry.In this paper we want to restore the balance and point out that, in sim-ple gauge systems, a fully controlled calculation gives perfectly sensible an-swers to thermodynamical behaviour of the walls. The model in question isZ(2) gauge theory, formulated with a lattice Hamiltonian in two space dimen-sions. The reason one has control is that the model can be reformulated as atwo-dimensional Ising model. Its three-dimensional version is not analyticallysolvable, but similar properties hold.Although the reader may think Z(2) theory a little outlandish, we believethe results will carry over to an SU(2) theory, because the main actors areelectric fluxes and are present in both theories.In the next section we will formulate the problem in a general way. Thesubsequent section deals with the Z(2) case. Most of Sections 2 and 3 has beenknown for 15 years. Only from Sec. 4 on are there developments which to thebest of our knowledge have not appeared in the litterature: a more preciseformulation of boundary conditions, and the roughening of the surface.22 Formulation of the ProblemLet us start with some simple observations on a domain wall in between twoordered phases.If we have a system that breaks the symmetry at low temperature, the wallreflects the properties of the high temperature phase. This implies that entropyand internal energy of the wall are higher than those in the surroundings.Needless to say, the free energy of the wall is higher because otherwise thesystem would be unstable against wall formation.Now the case under study: the symmetry breaks at high temperature.Then the wall has the properties of the low T phase- hence a lower entropyand internal energy! Of course stability requires that the resulting free energybe higher, just as in the previous case.Local free energy and entropy are not unambiguously defined inside thewall. Only the internal energy is locally defined as the expectation value ofthe local Hamiltonian.The free energy F of a gauge theory is related to the Gibbs sum Z overphysical states:exp−FT≡ Z =∑phys〈phys| exp −HT|phys〉. (1)A physical state in a gauge theory can be written as the average over allgauge transformations Ω(~x):|phys〉 =∫DΩ| ~AΩ〉. (2)We suppose that boundary conditions on Ω have been imposed, such thatany product of two of them again obeys the boundary conditions.With this definition a physical state is gauge-invariant. A gauge transfor-mation in Hilbert space is generated by the Gauss operator G(~x) ≡ ~D · ~E(~x)in the following way:| ~AΩ〉 = exp iω ·G| ~A〉 (3)The dot means summation over all indices, such as space and colour and Ω ≡exp iω. The Gauss operator commutes with the Hamiltonian, and the gaugeinvariance of a physical state leads to:〈 ~AΩ| exp −HT|phys〉 = 〈 ~A| exp −HT|phys〉. (4)In other words, the presence of the physical state renders the gauge trans-formation on other | ~A〉 states redundant. This is an important property and3corresponds, in the Euclidean path integral version of the free energy, to thefreedom in dropping integrations over A0 variables in the Euclidean time di-rection without changing the free energy.Here we want to avoid Euclidean path integrals altogether. Let it sufficeto mention that in the path integral one can define the Polyakov loop as theordered product of all the A0 potentials along the line in the Euclidean timedirection from 0 to 1T. This formulation is useful numerically, but is obscur-ing the issues at hand, i.e. the formation of surfaces in space. Due to theabove invariance the Wilson line can be reduced to the sole gauge transformΩ. Nowhere will we introduce Euclidean time.Instead, we will combine Eqs. (1), (2) and (4) and do the integrationsover ~A to get an effective action in the same dimensionality as the originalHamiltonian:∫D ~A〈 ~A| exp −HT| ~AΩ〉 ≡ exp−Seff (5)so thatexp−FT=∫DΩexp−Seff (Ω). (6)The gauge transformation lives in the same space as that of the quantum statesand the Hamiltonian. The effective action describes the interactions betweenthe gauge group variables. There is no Euclidean time in Eq. (6).What purpose does the effective action serve?It defines averages such as 〈Ω(~0〉 (the average of the Wilson line) andmore generally correlations. These correlations are, in the language of thequantum statistical Gibbs sum (8), simply the correlations of heavy charges 3.These correlations teach us about the forces present in the system for varioustemperatures. The import of the effective action is now clear, and in Sec. 3these correlations are worked out for the simple Z(2) model.3 The Z(2) Gauge Model3.1 The ModelWe study a system with gauge group Z(2) on a spatial lattice with continoustime at equilibrium with a thermostat at temperature T. We will partly followKiskis 7 The Hamiltonian readsH =∑lK(1− σx(l)), (7)4K is the coupling and σx(l) is the first Pauli matrix defined on the link l. Itacts as such on the two-component wave function in the link l and acts asunity on wave functions on the other links. The system is represented by theproduct of those two-component ψ’s:Ψ = Πlψ(l) .The Hamiltonian contains only a kinetic term analogous to an electricterm in an SU(N) gauge theory. We omit the magnetic plaquette term forsimplicity. It is not essential to the argument of this paper.The ensuing theory is not trivial! The partition function Z in Eq. (1) hasphysical states satisfying Gauss’s lawG(~x)|phys〉 ≡ Π{l←l~x}σx(l)|phys〉 = |phys〉 (8)We can diagonalize all the σx and obtain on any link an eigenvalue 2K or 0.We then haveZ(T/K) =′∑{φ(l)}Πl{exp−2KTφ(l)}, (9)where the prime denotes the Gauss’s constraint and φ(l) takes the values 0 or1. If the latter we say there is “flux” on link l.It is easy to see that a physical state can be represented by a configurationof fluxes such that an even number enters into every vertex of the lattice. Onecan define the chargeQ(~x) =∑{l←l~x}φ(l) ≡ Φ(~x),m¯od2 (10)so Gauss’s law in Eq. (8) does not allow charge.3.2 Order ParameterWe will now establish the relation between correlations of Wilson lines andthe correlations between heavy test charges. First we need states with a testcharge at ~y. To this end we introduce the two projectors :Pe(~x) ≡ 12∑ω(~x)=0,1G(~x)ω(~x) (11a)andPo(~x) ≡ 12∑ω(~x)=0,1(exp iπω(~x))G(~x)ω(~x) (11b)5We introduced here the notationΩ(~x) ≡ exp iπω(~x) = ±1, (12)using G(~x) as defined in Eq. (8).In the basis where the Hamiltonian and hence the Gauss operator arediagonal, the projectors read:Pe(~x) =12∑Ω(~x)=±1Ω(~x)Φ(~x) (13a)andPo(~x) =12∑Ω(~x)=±1Ω(~x)Φ(~x)+1. (13b)The first projector renders the number of fluxes ending in ~x even , the secondone admits only an odd number, so creates a charge mod 2. The projectorsobviously commute with the Hamiltonian.Consider a quantum state of the gauge field, which is physical everywhereexcept at ~y, where an odd number of fluxes ends (so where the charge sits). Itcan be written with the help of the projectors as:Po(~y)Π~x 6=~yPe(~x)| ~A〉 ≡ |Q(~y), ~A〉. (14)The free energy FQ of the system with a charge is defined by:exp−FQT=∑{ ~A}〈 ~A,Q(~y)| exp−HT| ~A,Q(~y)〉. (15)The projectors on the bra can be commuted through the Boltzmann factor andgive, with Eqs. (11):exp−FQT=∑{Ω}〈 ~A| exp−HTΩ(~y)| ~AΩ〉Together with the definition of Seff in Eq. (5) we find:exp−FQT=∑{Ω}Ω(~y) exp−Seff (16)The reader will have noticed that an important point has been glossedover. It is the fact that a state with only one charge in some point ~y is hard6to realise, with Gauss’ law satified everywhere else. In terms of the fluxes:imagine one flux leaving ~y ( never to return there anymore) it has necessarilyan end point (or rather on odd number of them). These end points must be onthe boundary, if we want Gauss‘s law everywhere inside our volume. Thus weneed at least one charge on the boundary, on which the flux can land. Moregenerally we need an odd number of charges on the boundary. So Gauss’s lawforces us in computing correlations of an even number of Wilson lines, and thefree energy FQ is obtained exponentially fast with distance.4 Effective Action and the Ising ModelWe start from Eq. (9). We can undo the Gauss constraint on the sum byintroducing the projectors of Eqs. (13) and find for the partition function:Z(T/K) = 2−V∑{Ω}Πl exp−2φ(l)TΠ~x(Ω(~x))Φ(~x). (17)V and L stand in this section for number of points and number of links on thelattice.The summation over all gauge field configurations {φ(l)} is now easily doneand leaves us with:exp−Seff = Πl(1 + (ΩΩ′)l exp−2KT). (18)In this equation only products of nearest-neighbour gauge transforms appearbecause a given flux φ(l) in Eq. (10) is present as power in both, in Eqn. (17).Clearly the Ising model has emerged up to some factors. We find with therelationexp−2KT= tanhJ (19)that the r.h.s. of Eq. (17) can be rewritten asexpJL1coshJLΠl exp−J(1− (ΩΩ′)l). (20)The relation between the parameters can be extended to a local one: Eq. (19)is true linkwise. Note that the dual coupling J∗ in the Ising model equals KT.This is the main result of this section. As expected from the previous sec-tion we have at high T (large J) spontaneous breakdown of the Z(2) symmetry.We note in Eq. (17) that the Boltzmann weight of the gauge fields is ac-companied by a factor depending on Ω(~x) which is negative when there is a7+++++++––––––+ + + + + – – – – – –– – – – – –+ + + + + +Figure 1: The system with boundary spins frozen, such that a domainwall develops in themiddle. Multiplying all spins on one side of the wall with −1 will generate a dislocationalong the centerline, and render the boundary conditions homogeneous.charge in ~x (only after integrating over the gauge transform do we recover theprojectors, which give of course non negative factors). The thermodynamicalbehaviour of the gauge fields in terms of Seff can therefore be quite unex-pected. On the other hand the density matrix in terms of the variables Ω,Eq. (20) is positive.Note that the behaviour is dual; it is obvious that at high T many quantumstates contribute to the partition sum, whereas at large J few classical spinconfigurations contribute, So the quantum entropy of the gauge system is highwhilst the classical entropy of the Ising model is low. We know that the Isingmodel allows coexistence of different phases, separated by domain walls. Onecan therefore ask how this translates into the gauge model at high temperature.5 Coexisting PhasesThere is a well-known device to create a domain wall in spin models: considerthe boundary spins. Fix the spins on one side to be up, on the other side tobe down (see Fig. 1).In the Ising model we have to introduce an external field hb at the boundarypoints b, which freezes the spins in question. Its counterpart in the gauge modelis a chemical potential µb that couples to the charge Q(b) of Eq. (10). Therelation between the two is like that between the link couplings in Eq. (19):exp−µbT= tanhhb (21).Freezing spins with a positive (negative) field means µb = 0+(0+ + iπ). Itis clear that at large J (high T in the gauge model) the spins on both sides ofthis wall will have opposite sign. We have created a domain wall. To detect8the wall one can put a charge on one side, and measure the correlation withanother charge. When going through the wall the correlation will go throughzero and change sign when emerging on the other side.Of course, the configuration of chemical potentials implies that any fluxline passing through the dislocation an odd number of times will contributewith a minus sign to the partition function Z+−! This is so because such a fluxwill necessarily end with an odd number of end points on the boundary on bothsides of the wall, picking up an odd number of minus signs. This is perhapseasier to see by transforming all the spin variables on one side of the wall witha minus sign. Then the border spins are all frozen the same way, and there isno minus sign from the border. But now it comes from the antiferromagneticdislocation (J → −J) we created by the transformation on the spins and byusing the relation (19) between J and T ! Below the critical Tc there is no wall.Correspondingly, the correlation drops exponentially fast with distance and iscontrolled by the string tension ρ(J).The ratio of the partition functions with and without mixed boundaryconditions signals in what phase we are:Z+−Z++= 1− exp−ρ(J)L if J < Jc= exp−LtrJα(J) if J > JcThe string tension has good thermodynamical properties 6. We now analysethose of the domain wall.6 Domain Wall and RougheningIn the Ising model for d = 2 the total free energy α(J) of the wall is knownand equals α(J) = 2(1− J∗J) for J ≥ J∗. When the two are equal, we have theCurie point, where the surface stops to exist.The location of the wall is subject to large fluctuations, of the order of√Ltr for d = 2. This is known as roughening and its consequence is that thescale over which the profile is varying becomes√Ltr in lattice units. The sameis true for the energy density profile of the wall in the Ising model. The resultis known analytically 8 and reads:ǫI(l) =a√Ltrexp {− sinh(2(J − J∗)) z2Ltr}. (22)We will write z2Ltr= ζ2. The link l is supposed to run from z to z+1. Importantis to note that the distance over which the interface is varying is of the order of9-4 -2 0 2 4-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0ζε-4 -2 0 2 4-0.4-0.2 0.0 0.2 0.4 0.6 0.8 1.0ζεFigure 2: ǫ(ζ) = 〈1 − σx(l)〉(ζ) is the meanvalue of the local energy operator at physicaltemperature T with K/T = 0.3, Eqn. (23). ζis the rescaled variable z/√L). The length ofthe wall is Ltr = 5 in lattice units.Figure 3: As in Fig. 2 but now for Ltr = 25.√Ltr. As the integrated profile gives us the same total energy, the amplitudeof the energy profile decreases like√Ltr.This formula reflects the random walk our one dimensional surface is mak-ing. It is universal up to the typical Ising factor in front of the exponent. Theconstant a is such that upon integration we get the total energy 2(1 − ∂J∗∂J).To translate this result into the average ǫ(ζ) of the energy density 1− σx(l) inthe gauge model is straightforward. The result is:ǫ(ζ) = ǫ0(T )− (− ∂J∂J∗)ǫI(T ). (23)The first term is the link energy density of the ground state of the gauge model,and is simply the Ising ground state energy per link, which is analyticallyknown 8.The Jacobian- reflecting the change from the variable J to the variableKT−1 = J∗- is always negative so the energy density inside the wall is smallerthan outside. This is one of the hallmarks of a broken symmetry at high T, aswe argued in the introduction. In Figs. 2 and 3 we show the dependence onLtr.7 ConclusionsDomain walls and string tension in gauge theory can be described by theasymptotic properties of the same partition function, for different temperature10regimes. When the system is large enough both string tension and wall havereasonable thermodynamical properties, the latter thanks to roughening. Itsorigin is the restoration of the translation invariance broken by the interface.In d = 3 the√Ltr behaviour becomes logarithmic, and is only present for arange of temperatures not too far above Tc.If one considers bubbles of radius R, the width of the wall will grow as√Ror as logR. The total energy of the wall profile depends only on temperature.AcknowledgmentsWe are indebted to Per Elmfors, Keijo Kajantie, Misha Shaposhnikov, MishaStephanof, and Mike Teper for many useful discussions and criticism. One ofus (CPKA) thanks the organizers of this very inspiring meeting and the CERNTheory Division for hospitality, S.B. acknowledges an Allocation de RechercheMESR.References1. S. Weinberg, Phys. Rev. D9 (1974), 3357.2. K. Kajantie, L. Karkainen, and K. Rummukainen, Nucl. Phys. B357(1991), 693; S. Huang, J. Potvin, C. Rebbi and S. Sanielevici, Phys.Rev. D42 (1990), 2864; T. Trappenberg, U.-J. Wiese, Nucl. Phys. B372(1992), 703.3. L.D. MacLerran, and B. Svetitsky, Phys. Rev. D24 (1981), 450.4. B. Svetitsky, and L.G. Yaffe, Phys. Rev D26 (1982) 693; Nucl. Phys.B210 [FS6] (1982) 423.5. R.V. Gavai, F. Karsch, and B. Peterson, Nucl. Phys. B322 (1989), 738;M. Fukugita, M.‘Okawa, and U. Ukawa, Phys. Rev. Lett. 63 (1989)1768; Nucl. Phys. B337 (1990) 181.6. Y.Y. Goldschmidt and J. Shigemitsu, Nucl. Phys. B200[FS4] (1982) 149.7. J. Kiskis, Phys. Rev. D51:3781, 1995.8. D.B. Abraham, in Phase Transitions and Critical Phenomena, Vol. 10,C. Domb and J.L. Lebowitz eds., Academic Press, London, New York.9. T. Bhattacharya, A. Gocksch, C.P. Korthals Altes and R.D. Pisarski,Phys. Rev. Lett. 66 (1991), 998; T. Bhattacharya, A. Gocksch, C.P. Ko-rthals Altes and R.D. Pisarski, Nucl. Phys. B383 (1992), 497, T. Bhat-tacharya, A. Gocksch, C.P. Korthals Altes and R.D. Pisarski, Nucl. Phys.B (Proc. Suppl), 20, 305 (1991).10. V.M. Belyaev et al., Phys. Lett. B277 (1992), 331; A.V. Smilga, Ann.of Phys. 234 (1994), 1.11

Interfaces in hot gauge theory

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