Bose condensation in quasi-one-dimensional antiferromagnets in strong fields
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PHYSICAL REVIEW B VOLUME 43, NUMBER 4 1 FEBRUARY 1991
Bose condensation in quasi-one-dimensional antiferromagnets in strong fields
Ian AleckCanadian Institute for Advanced Research and Physics Department, University ofBritish Columbia,
Vancouver, British Columbia, Canada V6T2A6(Received 10 July 1990j
Recent experiments show that axially symmetric integer-spin antiferromagnetic chains undergo aphase transition at a critical applied magnetic field. It was argued, using Landau-Ginzburg theory,that this is one-dimensional Bose condensation. The theory is further analyzed at the mean-fieldand Gaussian level. Then some exact results concerning the critical behavior are determined usingknown results on one-dimensional Bose fluids. These are shown to be consistent with recent numer-ical simulations on spin chains. Breaking of axial symmetry produces crossover to a two-dimensional Ising transition.
One-dimensional antiferromagnetic of integer spinhave an excitation gap, ' A. The lowest excited state is atriplet of massive bosons. As observed in recent experi-ments ' on NENP the application of a magnetic fieldcauses a Zeeman splitting of the triplet with one membercrossing the ground state at a critical field, h, =A. Aswas argued recently, using a Landau-Ginsburg theory,the ground state above h, may be regarded as a Bose con-densate of the low-energy boson. Varying the magneticfield is equivalent to varying the chemical potential forthis boson and the (uniform) magnetization correspondsto the boson number.
The microscopic Hamiltonian under consideration isthe integer spin antiferromagnetic chain in an appliedfield with crystal field anisotropy:
H=QIS S;+i+D(S;) +E[(S,") (S~) ]h S ] .
(Exchange anisotropy can also be included and does notchange any of the basic conclusions. We adsorb the Bohrmagneton and the g factors into the definition of the mag-netic field, h and set Pi= 1.) As discussed in Ref. 4, aLandau-Ginsburg formulation of the model, based on thelarge-s continuum limit, gives a Hamiltonian density:
3 Q2&=II + + g P;+A/ h(/XII) . 2 2 t)x i 2v
Here P is the staggered magnetization density and II isthe momentum canonically conjugate to P; v, b, , are phe-nomenological parameters representing the spin-wave ve-locity, and the gaps for staggered magnetization Auctua-tions with polarization i The P t.erm is included for sta-bility. In Ref. 4 only a partial mean-field and Gaussiananalysis of the Landau-Ginsburg model was reported,referring primarily to the region h &h, . Since then, a
different approximate theory, based on a fermionic ratherthan bosonic representation was presented. Further-more some numerical results were reported. ' The pur-pose of this paper is twofold. A more complete meanfield and Gaussian analysis of the Landau-Ginsburgtheory will be presented for both phases. Furthermore,some exact results concerning the critical behavior of thistheory will be derived.
The critical behavior is deduced from the following ob-servation. Since two elements of the triplet of excitedstates have an excitation gap at h they are irrelevantand may be integrated out. Furthermore, near hwemay approximate the dispersion relation for the low-energy magnon by E (k) =b. + v k /2b, . The e8'ectivelow-energy Landau-Ginsburg Lagrangian then becomesprecisely the standard one used to study Bose condensa-tion in a nonrelativistic Bose Quid with 6-function repul-sion. This model has been studied extensively in one di-mension. ' Although true off-diagonal long-range orderdoes not occur quasi-long-range phase coherence does.This corresponds to power-law decay of the staggeredmagnetization orthogonal to the applied field, with a con-tinuously varying exponent g. In the limit of zero density(h ~h, ) the Bose system becomes equivalent to a systemof free fermions and g + ,', in agreement with calcula-tions on finite spin-1 chains. The known result for theground-state energy as a function of density in the BoseQuid determines the magnetization near h, :
Q(h h, )2bM=
Note that this is different than the mean-field Landau-Ginsburg result M ~(h h, ) given in Ref. 4. It agreeswith the free fermion result suggested by a different typeof approximate theory in Ref. 5, but disagrees with theform suggested in Ref. 6.
These results should be exact in the limit of axial sym-metry. This symmetry can be broken either by crystalfield anisotropy [the E term of Eq. (1)] or by the applica-tion of the external field along a direction other than thesymmetry axis. Such anisotropy can be studied using the
43 3215 1991 The American Physical Society
3216 IAN AFFLECK 43
Haldane formalism for the one-dimensional Bose Auid.It is a relevant perturbation, producing long-range Isingorder above h, (it also leads to nontrivial renormalizationof the ualue of h, ). The correlation length thus becomesfinite above h' it scales as a power v of the inverse an-isotropy with a density dependent exponent, which ap-proaches 1 as the density goes to zero. The Ising transi-tion itself is in the two-dimensional universality class(since time plays the role of a second dimension here). Aswas pointed out in Ref. 7 three-dimensional couplingstend to produce true long-range off-diagonal order (i.e.,Bose condensation) even without anisotropy.
A rough picture of the various phases is as follows. Inthe axially symmetric case, for h (h there is no magnet-ic moment (uniform or alternating) on length scales longcompared to the correlation length (about seven latticesites for the isotropic spin-1 case). Above h there is auniform magnetic moment in the direction of the appliedfield (taken to be the 3-axis), corresponding to the cantedspin structure shown in Fig. 1(a). However, the spins un-dergo long wavelength precession about the 3-axis, sothat there is only quasi-long-range order (i.e., power-lawdecay) of the alternating transverse spin components, i.e.,
canted spins (for example, lying in the 1-3 plane) asshown in Fig. 1(b). Below h fluctuations produce afinite correlation length for alternating order of threetransverse spin components, but allow a finite uniformmagnetization in the field direction. Above hboth theuniform 3-component and the staggered transverse com-ponent have long-range order, i.e.,
(S, ) =const
(S ) =constX( 1)'In Sec. II we analyze the Landau-Ginsburg model in
the mean-field and Gaussian approximations, with andwithout axial symmetry. Section III then takes into ac-count the one-dimensional fluctuation eA'ects usingHaldane's method, deriving various exact results. Sec-tion IV contains conclusions and a comparison of thesepredictions with numerical simulations and experiment.Some of the details of the Gaussian approximation calcu-lations of Sec. II are relegated to the Appendix.
II. MEAN-FIELD AND GAUSSIANAPPROXIMATIONS
(S, ) =constgab
(S,'S~ ) ~( 1)' ~ (a, b=1,2) .li j "
(4)A simple way of performing a mean-field analysis of
the Landau-Ginsburg model, is to first make a canonicaltransformation to the Lagrangian density:
Weak three-dimensional couplings will make this intotrue long-range order. Axial symmetry breaking terms inthe Hamiltonian pick out two preferred orientations of
2. +hxPa~ U ay ' ~' , 42v Bt 2 Bx
We see that the magnetic field adds a quadratic term tothe eA'ective potential:
V= y ' y,' (hxy)'+ay'.(a) For small fields the minimum of V is at /=0 but above a
critical field the minimum occurs at nonzero P.A. Axially symmetric case
FICs. 1. (a) Spin configuration in axially symmetric case:there is a static uniform component in the 3 direction and a pre-cessing alternating component in the 1-2 plane. (b) Spinconfiguration with axial symmetry breaking: there is a staticuniform component in the 3 direction and a static alternatingcomponent in the 1 direction.
2h b.M= I/XII=Lh =Lh
v 4v A.(10)
Let us first consider the simplest case where the field isapplied along a symmetry axis, corresponding approxi-mately to a field applied along the b axis in NENP. Weassume that A, =62=6 and that the field points alongthe 3-axis. We see from V that the phase transitionoccurs at h, =A. Beyond the critical field the symmetryof rotation about the z axis is spontaneously broken (inmean-field theory) and P has a nonzero (constant) expec-tation value lying in the xy plane of magnitude:
(h 6 )4v A.
We may immediately calculate the magnetization, M,from Eq. (6):
43 BOSE CONDENSATION IN QUASI-ONE-DIMENSIONAL. . . 3217
v ap'2 Bx
2 0o a8,v v at
Solving the resulting classical equations of motion givesfrequencies
to2 (3h 2 Q2)+ V 2k 2++(3h 2 Q2)2+4v2h 2k 2 (13)
The lower frequency solution is the sound or Goldstonemode. At k ~0, this frequency approaches
vk(h b, )3h b, (14)
Thus the speed of sound is v, =v+(h b, )/(3h b, ),which goes to zero as h ~h, .
where L, is the length of the system. Note that, in mean-field theory, M is linear in h has announced in Ref. 4.
We may calculate the excitation spectrum in the stan-dard harmonic approximation by expanding X to quadra-tic order in the small Auctuations around the classicalground state. Fluctuations of P in the 3-direction arecompletely unaffected by the applied field. On the otherhand planar Auctuations consist of a Goldstone phasemode together with a finite-gap longitudinal mode. Weintroduce amplitude and phase fiuctuations, P' and 8, re-spectively, writing P as
P=[(P o+P')c os8, (P o+P')sin8, $ ]3.The terms in X of quadratic order in 8, P' are
4o a8 vdo a8 1 ay2v Bt 2 Bx 2v Bt
Solving for the dispersion relations we find that the lowerenergy mode has a gap near h, of
(h b, , )(b2 b, , )to' (O)=2h 2+(Q2 +2)/7 (17)
We see that the gap rises as the square root of h h,times the square root of the anisotropy, in, the mean-fieldtheory.
Other effects of anisotropy immerge at the level ofone-loop corrections to mean-field theory (i.e., the Gauss-ian approximation). These results are derived in the Ap-pendix. In the axially syrnrnetric case the magnetizationis strictly zero for h (h, . Above h, it rises linearly in themean-field approximation but has a finite jump at theone-loop level (see Appendix). In the presence of anisot-ropy there is a nonzero magnetization for all nonzero h.In particular, as shown in Ref. 4 as h ~0, M oc h. Theslope ofIat h =0 (i.e., the susceptibility) is proportionalto the square of the anisotropy. M diverges logarithrnic-ally as h ~h, from below or above. Furthermore, in thepresence of anisotropy there is field-dependent renormal-ization of the gap parameters, so that higher-loop correc-tions can shift haway from 6&. Another striking effectof anisotropy is on the static correlation function. In theaxially symmetric case the ground state is completelyunaffected by the field below h, so the correlation lengthremains unchanged (and equals v/5). Above h, it isinfinite. On the other hand, with anisotropy, the correla-tion length diverges as h, is approached from below orabove: $~1/Qh, h, the standard mean-field result.(See Appendix. )
B. Axial symmetry breaking
2(h 6, )
'(b2 b f)
We now wish to discuss the effects of breaking of axialsymmetry. We begin again with the Lagrangian of Eq.(7) with the field in the 3-direction, but we now considerthe case 5, & h2.
At the classical level, the transition looks essentiallythe same as in the axially symmetric case. The potentialof Eq. (8) develops a minimum at nonzero P, for h )b,For h )h(P, ) is again given by Eq. (9) with b, ~b, Adifference appears, however, in the spectrum for h & h, .All modes now have a gap. We now pararnetrize thesmall fluctuations away from the ground state by
4'=(4o+0i 6 03) .As before P3 is unaffected by the magnetic field at treelevel. The terms in the Lagrangian quadratic in $, , $2 are
III. FLUCTUATION EFFECTS AND EXACT RESULTS
then the Lagrangian becomest' 2
v& ay ay' av at at ax
Q2 Q2I@I'4~1&1' .
This is precisely the standard nonrelativistic Lagrangianfor bosons with 5-function repulsion. The upper mode offrequency, co+, has disappeared and we are left with onlythe lower mode with a frequency which is now
A Axially symmetric case
In order to go beyond the mean-field approximation, itwill be useful to discuss two different low-energy approxi-mations to the Lagrangian of Eq. (7). The first approxi-rnation is valid only near h, . It consists of simply drop-ping the massive field, P', which is unaffected by theexternal field, as well as dropping the terms quadratic intime derivatives in the Lagrangian of Eq. (7). If we com-bine P and P into a complex field
3218 IAN AFFLECK 43
uk [2(h 6)+v k ]2h
This is the standard Bogliubov result; it reduces tov k /2b, at h ~h, . It can be checked that this agreeswith Eq. (13) up to O(k ) for h close to h, . The secondapproximation is valid for all h )hat low energies. Itconsists of integrating out the amplitude oscillation P inEq. (12) obtaining an efFective Lagrangian containing 8only
(h' a'), 2hkok' a6) h' ae 'v u Bt 4v g Bt
3h 4 BO h 4 808v 2g Bt 8A, Bx
The frequency obtained from the Lagrangian agrees withco to O(k ), the result of Eq. (14). Note that the firstapproximation, the Lagrangian of Eq. (19), was only validclose to h, but was correct to O(k ) and described bothsides of the critical point while the second approxima-tion, Eq. (22) is valid for all h )h, but only to O(k ).
We now wish to go beyond the mean-field approxima-tion. In particular we will be able to state some exact re-sults in the limit of zero magnetization (i.e., density). Itcan be seen that the mean-field theory is certainly not val-id there since the dimensionless parameter which controlsperturbation theory is )(,v /(h b, ). In this limit, as dis-cussed in Ref. 8, the Bose Quid with a 6-function interac-tion, or for that matter, any short-range interaction be-comes equivalent to a Bose Auid with a hard-core repul-sion, which in turn is equivalent to a free fermion gas inone dimension. In this low-density limit the ground stateis obtained simply by arranging that the wave functionvanishes when any two particles are close to each other.Such a wave function is equal to a free fermion wavefunction times a sign function (i.e., a function which is+1 everywhere) in one dimension. Such a wave functionhas zero potential energy but has a kinetic energy of or-der Nu p /2b, where p is the density and N the numberof particles since the rnomenta of the bosons must be ofO(p). [A simple argument like this was constructed inRef. 6 but concluded that the ground-state energy wasproportional to Ne ' )'~ where g is the range of the in-teraction. The problem with that argument is that it ig-nores the quantum mechanical nature of the particles andassumes they can be localized on a length scale of O(g)without any cost in kinetic energy resulting from the un-certainty principle. ] Explicitly, the kinetic energy of anonrelativistic free fermion gas of mass 6, expressed interms of its density, p, is m Xv p /6A. Including the restenergy and the chemical potential, the energy as a func-tion of density is
2+ 2 2E= +(b, h)N .6A
Thus the magnetization per unit length is
A somewhat different free fermion approximation wasdiscussed in Ref. 5. The two models do not in generalagree in detail, but they do at the critical field, both giv-ing Eq. (24) near h, . The above argument shows that thisresult should be universal and exact, and not merely aconsequence of the approximations. Other critical prop-erties of the one-dimensional Bose superAuid have beencalculated using a scaling theory. The low-energy La-grangian for the phase field, 8, of Eq. (22) is qualitativelycorrect. However, long-range order does not occur dueto the logarithmic behavior of the propagator in two di-mensions. In general, with the complex fieldparametrized as
0-0 e"and the low-energy effective Lagrangian written
21 1 ()6)
277 v)v dt
The correlation function behaves as
( y( )ty(0) )(()(x)()(0)) l2 e
1'g = Q Vtv /V J (28)
The second term in this Lagrangian just comes from thekinetic energy and is unaffected by the interactions (atleast in the nonrelativistic limit), hence
27TV pVJ =277V p() = (29)
However, the first term, which arises from the interac-tions, in general has a coeKcient, which is much differentfrom the mean-field value of Eq. (22):
4u kir(3h b, ) (30)
(S (x)S (0)) =p + +, cos(2irpx),q(2~x )'
1 )xgab(S'(x)S (0) ) (a, b = 1,2),X
where p=M/I the magnetization per unit length. Theexponent g is given by
i)= ,' O(p) . (33)
B. Axial symmetry breaking
In particular, Haldane argues using a Jordan-Wignertransformation, that, at the critical point, g = ,' andhence u& =vJ +0. Near h, the original spin correlationfunctions have the behavior
&(h b, )2bM/L =p= (24)Finally we wish to consider fluctuation effects in the
presence of anisotropy. Since a Z2 symmetry is beingbroken in a (1+1)-dimensional quantum field theory, we
43 BOSE CONDENSATION IN QUASI-ONE-DIMENSIONAL. . . 3219
expect the transition to be in the two-dimensional Isinguniversality class. The external field plays the role of theeffective temperature, and the uniform magnetization ishence the effective energy operator. The Ising order pa-rameter is the staggered magnetization (in the x direc-tion). We may now invoke various known results aboutthe two-dimensional Ising model. At the critical pointthe staggered magnetization has a correlation exponentg= ,' and the uniform magnetization has g=2. Since theIsing model energy operator, corresponding to the uni-form magnetization, has scaling dimension 1, the singularpart of the magnetization near h is linear in h h, ; i.e.,the slope should be finite as h, is approached from belowor above, but should be discontinuous at this point. Thecorrelation length diverges as g~ ~h h, ~ ' as h, is ap-proached from below or above.
We may also specify a crossover exponent governingthe Aow from the unstable axially symmetric critical lineto the Ising ordered phase in the presence of anisotropy.For small anisotropy the critical behavior can be studiedusing the nonrelativistic model discussed above [see Eq.(19)]. The axial symmetry breaking modifies the La-grangian to
ih t BP Bgt BPU Bt dt Bx
h +0 with a coefticient proportional to the square of theanisotropy. It has a finite slope as h ~h, from below orabove, but the two slopes are not the same [Fig. 2(b)]. Inthe axially symmetric case, the correlation length for thealternating, transverse spin components is rigorously fieldindependent below h, and then is strictly infinite above h,[Fig. 3(a)] where power-law decay occurs with an ex-ponent g which is exactly , at h, and initially decreasesabove h, . With axial symmetry breaking, the transversealternating spin correlation length is finite everywhere ex-cept right at h, . It diverges linearly in I/~h h, ~ as h, isapproached from below or above [Fig. 3(b)]. It alsodiverges linearly in the inverse anisotropy, just above h, .At hg= 4 for the alternating transverse spin com-ponents and g=2 for the uniform spin component paral-lel to the applied field.
Finite-temperature effects should round off the discon-tinuous derivative of M, produce a finite-correlationlength in the axially invariant superfiuid phase (g ~ 1/T),and destroy the Ising order with axial symmetry break-ing. In the strictly one-dimensional system this occurs atT=O. Three-dimensional coupling will produce a finiteT'
Q2 Q2+ ' ' (y2+yt2) 4X~y~' . (34)
The axial symmetry breaking adds an additional term tothe phase-field Lagrangian of Eq. (26):
ae aevJ + p cos20 .2' U& Bt BX 2U(35)
The Ising symmetry breaking corresponds to the classicalground states with 0=0,~. The anisotropy term has ascaling dimension of 2g:QU~/Uz, where 21 is the stag-gered magnetization exponent in the absence of anisotro-py. In the zero-density limit, this scaling dimension goesto 1. Thus the anisotropy is indeed relevant along thecritical line.
Q(h h, )2hM= (36)
near h, [Fig. 2(a)]. With broken axial symmetry, themagnetization is nonzero at all h. It vanishes linearly as
Here we briefly summarize the expected results basedon a combination of the mean-field/Gaussian analysisand exact results in the critical region, and compare withnumerical simulations and experiments.
In the axially symmetric case, the magnetization isrigorously zero below h, =5 and then rises as
FIG. 2. (a) Qualitative sketch of the zero-temperature mag-netization in the axially symmetric case. (h) Qualitative sketchof the zero-temperature magnetization with axial symmetrybreaking.
3220 EAN AFFLECK 43
Numerical results on the magnetization were presentedin Ref. 6 for chains of length, L, up to 16, based on whichit was argued that the energy had an exponential depen-dence on magnetization:
E(M) Mb, ~Me (37)This disagrees with the cubic dependence of Eq. (23). Webelieve that the results of Ref. 6 are dominated by finite-size effects and do not reflect the true asymptotic behav-ior. While the (M = 1) energy gap can be determined toabout 10% accuracy from a chain of length 16 (and a fewpercent for length 32) much longer chains will be neces-sary to accurately determine the asymptotic form of themagnetization. The reason for this is that the finite-density corrections to Eq. (23) have the form of Eq. (37).This follows, essentially from the argument of Ref. 6; thetypical distance between bosons is 1/p and the interac-tion between the particles drops off with exponent g. Un-til these finite-density corrections become small theasymptotic behavior cannot be deduced. This occurswhen p((1/g, i.e., L/M))g. For spin 1 this means
v =110 K, 5=14 K, (38)we obtain the behavior of the susceptibility at T=O forNENP for fields applied along the b axis ignoring anybreaking of axial symmetry:
M~0.041+h h, (p~ per Ni +,hin T) . (39)Only a slight bump in dM/dh was observed in the experi-ment of Ref. 3 and none whatever in that of Ref. 2. Thisis presumably due to finite-temperature effects, axial sym-metry breaking, and crystal defects.
L ))7M. Since a reliable result for M = 1 requiredL = 1632, we might estimate that chains of length16M 32M will be necessary. Of' course the number ofstates decreases with increasing M for fixed L so this maynot be hopeless. Numerical results on the transversecorrelation function are completely consistent with thebehavior derived here, and, in fact, the value q= ,' at h,was guessed from extrapolation of finite-size results inRef. 7.
Using the g factor g=2. 2 and experimentally deter-mined values' of the velocity and gap
I would like to thank J. Rossat-Mignod, M. Takahashi,and N. Weiss for helpful discussions and the authors ofRefs. 5 7 for sending me copies of their work prior topublication. This research was supported by NSERC.
Here we give more details of the one-loop correctionsto mean-field theory referred to in Sec. II. The one-loopcorrection to the ground-state energy from the field-dependent modes, co+ is simply the harmonic oscillatorzero point energy:
The one-loop contribution to the magnetization is ob-tained by differentiating Eo"..
M/L = f1 dk2 (2m)den (k) de+(k)
The field-dependent energies below h, were discussed inRef. 4. In the isotropic limit they are simply
co ='t/b, +U k +h (42)
Note that the co+ and m contributions to M cancel.With anisotropy, the energies are given by
2 2 2C02 CO i +2h (co2+co, )
FEG. 3. (a) Correlation length in the axially symmetric case.(b) Qualitative sketch of the correlation length with axial sym-metry breaking.
where we have defined
co;(k)=(/ b, +U k (i =1,2) . (44)
43 BOSE CONDENSATION IN QUASI-ONE-DIMENSIONAL. . . 3221
Now, in general, the two terms in M/L of Eq. (41) do notcancel. In particular, as h ~0, we have
d CO CO2 C012 2 2
dh2 3 2+ 2(47)
Q7+ ~Cg2, CO ~CO1
d co+ 3c02+ Q)12 2 2
dh 2 ~2 ~2(45)
d co 3c01+c02dh2 2 2
(COp CO~ )hf""217 2COiCOp( CO i + CO2 )
This agrees with the zero field susceptibility calculated inRef. 4, Eq. (3.26). On the other hand, when h ~b, &, M /Lblows up because co +0 while dec /dh remains finite:
co2~ [(v k +(h, h )j.3h +5
There is a logarithmic divergence at k +0, h ~hgiving
Iln(h h, )I . (4g)4~V 52+36,
Above hwe must use Eq. (14) for the two frequencies,in the axially symmetric case. cu +0 at k~0, as givenby Eq. (14), however, we see from this equation thatdco /dh also goes to zero, so consequently M/L has afinite limiting value as h, is approached from above.Hence at the one-loop level there is a finite jump in M/Lat h in the axially symmetric case. With axial asym-metry, co is only zero right at h, and d co /dh isnonzero there. Hence, there is again a logarithmic diver-gence in M/L as h, is approached from above.
We now consider the transverse Green's function inthe Gaussian approximation. The equal time Green'sfunction is given by
2hzK+M h2h~G( )
(i,j=1,2)K +co h2K2+~2 h 22
2h~ K +coi h1 dk dK e'"
(IC +CO+)(IC +CO ) (2')
Here ~ is the imaginary frequency. We may do the sc in-tegral by the contour method. The encircled poles are at~=ice+. In the axially invariant case, we find the h-independent result:
G;,,=6dk 'k" 1
2+6, +u k(50)
G,"~2 ++3 f
dk e'277 2Q(Q ~ h ~)+ v 2k 2
This gives the standard two-dimensional relativistic prop-agator, with asymptotic behavior:
The correlation length is g=u/b. , independent of h. Inthis case, the residue of the pole at ~=ice is finite ash~h, since the cu factor in the denominator is can-celled by a similar factor in the numerator. Such a can-cellation does not occur when the axial symmetry is bro-ken. Consequently the correlation length diverges as h, .Keeping only the dominant contribution as h ~hwefind
b, ,h ~2(h b, ), b, 2 h ~b, 2 b, . (54)
In the axially symmetric case, co ~0 as k~O for allh )h, . The residue of the pole at ~ diverges as k +0since no cancellation of the m in the denominatoroccurs. Hence the correlation length is infinite, the prop-agator behaving as
dk;k1G(x) ~ e'"" ~ const+in x2~
With broken axial symmetry, the correlation length isfinite above hbut diverges as h ~h
g~v /Qh h, , (56)due to the behavior of co and the noncancellation of theco factor in the denominator.
Thus the correlation length diverges at h, :
g~u /'1/ h, hAbove hwe see from Eq. (16) that the propagator isagain given by Eq. (49) except that now we must makethe replacements
3222 IAN AFFLECK 43
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