The character of polarization correlations in six-vertex systems is discussed. With the aid of a connection between the 1-d Heisenberg--Ising chain and the six-vertex problem, existing results for the chain correlations are used to obtain information about long-wavelength polarization correlations in six-vertex models. These results are compared with a neutron scattering study of 2-d polarization correlations in the layered compound copper formate tetrahydrate. Because the six-vertex model is equivalent to a particular roughening model, these results also explicitly predict the critical behavior of that roughening model just above its roughening temperature. The results correspond to the predictions of Kosterlitz and Thouless for the phase transition in the 2-d Coulomb gas. 5 figures

Axe, J.D.

McCoy, B.M.

Youngblood, R.W.

UNT Digital Library

r BNL-26196Fluctuations in Two-Dimensional Six-Vertex SystemsR. W. YoungbloodJ . D. AxeB. H. McCoyMASTER- NOTICE-• prepared as iTills report wat r r  s in account ofsponsored by the United Stales Government. Neillicr theUnited States nor tfir United Slatet Department o(Energy, nor any of their employees, nor any of theircontractor!, subcontractors, or (heir employee, makeiany warranty, express or implied, or aisumes any legalliabilily or responsibility Tor the accuracy, completenessor usefulness of any informal fan, apparatus, product orprocess disclosed, or represents thai iti use would notinfringe privately owned rights.The submitted manuscript has been authored under contract EY-76-C-02-0016with the Oivision of Basic Energy Sciences, U.S. Department of Energy.Accordingly, the U.S. Government retains a nonexclusive, royalty-freelicense to publish or reproduce the published form of this contribution,or allow others to do so, for U.S. Government purposes.CHI" TIL'S BiiUJZUiliT IS UNLUilTElSFLUCTUATIONS IN TWO-DIMENSIONAL SIX-VERTEX SYSTEMS*R. W. Youngblood, J. D. Axe and B. M. McCoyPhysics DepartmentBrookhaven National LaboratoryOpton, H.Y. 11973Institute of Theoretical PhysicsState University of New York at Stony BrookStony Brook, N.Y. 11794ABSTRACTThe character of polarization correlations in six-vertexsystems will be discussed. Making use of a connection between the1-d Heisenberg-Ising chain and the six-vertex problem, we draw uponexisting results for the chain correlations to obtain informationabout long-wavelength polarization correlations in six-vertex models.These results are compared with a neutron scattering study of 2-dpolarization correlations in the layered compound copper formatetetrahydrate. Because the six-vertex model is equivalent to aparticular roughening model, these results also explicitly predictthe critical behavior of that roughening model just above itsroughening temperature. The results correspond to the predictionsof Kosterlitz and Thouless for the phase transition in the 2-dCoulomb gas.^Research has been performed under Contract EY-76-C-O2-OO16 with the Divisionof Basic Energy Sciences, U.S. Department of Energy.Although quite a bit is known about the statistical mechanicsof 2-d six-vertex systems [1], much remains to be said about thefluctuations in these models. Here, we will present a short dis-cussion of those aspects of the fluctuations that seem particularlygermane to the subject of this particular conference. In particu-lar, we will discuss analytic expressions for the long wavelengthpolarization correlation functions for these models which are asymp-totically exact at large distance [2]. We will apply these resultsto neutron scattering experiments on a quasi-2-d hydrogen-bondedsystem, copper formate tetrahydrate (CFT) [3]. We will also showthe relevance of these results to models of the solid-on-solid in-terfacial roughening transformation [4], and comment on the rela-tion of Kosterlitz-Thouless theory [5] to six-vertex systems.The models under discussion have been introduced by Dr. Weeksin his lectures at this conference [4]. The allowed vertex con-figurations are prescribed by the ice rules. The vertex weightingscheme is shown in Figure 1. (In adopting these weights, we aretacitly ruling out applied fields.) If vertices 1 and 2 are favoredenergetically, the ground state is ferroelectric along ± y; if 3 and4 are favored, the ground state is ferroelectric along ± x; if 5 and6 are favored, the ground state is antiferroelectric. The parameterA, defined in Fig. 1, is a measure of how close the system is tobeing ferro- or antiferroelectric. n, also defined in Fig. 1,measures the anisotropy in the polar configurations. Baxter [5]showed that singularities in the free energy corresponding to ferro-electric and antiferroelectric transformations occur at A • +1 and-1, respectively. The relation between the three phases (antiferro-electric, disordered, and ferroelectric) is summarized in Fig. 1.For the weighting scheme we employ, no single physical system dis-plays the full range of behavior shown in Fig. 1; the point A * 1/2corresponds to T • °°. But the polarization correlations we discussare a property of the entire disordered regime, and it is natural todiscuss them as a function of A rather than T. We will also haveoccasion to mention the so-called FSOS roughening model [4] and the1-d Heisenberg-Ising chain; certain relevant attributes which arisefrom equivalences within these models are summarized in Fig. 1.Our primary interest is in long-wavelength polarization fluctu-ations. For this purpose, it is convenient to define a coarse-grained polarization. Consider the correlation function betweentwo parallel arrows for the isotropic case, a • b. Introduce thevariable a* (m,n) to denote the sense of the arrow at site (m,n)(the subscript j • 1,2 denotes the arrow direction — right or leftsloping in Fig. 1). For A • 0, Sutherland used an equivalence to afree fermion system to show that for large r2<o\(0)o. (m,n)> -v -*-=•ir r(-1)nt+n (m —n )(m2+n2)(1)V V V V V VA A A A A AC AANTIFERROELECTRICSMOOTH INTERFACEGAP IN EIGENVALUESPECTRUMu —TR-12 , ,2 2a + b - c2ab2TR -0DISORDEREDROUGHNO GAP INEIGENVALUESPECTRUMb «-*aTFerro Jv A+1FERROELECTRICTILTED INTERFACEGAP IN FIGENVALUESPECTRUMK - - ASA S iFigure 1.where m,n are integer coordinates and r • /m +n . We see thatthere is an antiferroelectric contribution which oscillates rapidly,and a ferroelectric component which does not. We therefore formthe "coarse-grained" ferroelectric polarizationPj(r) • a.(m,n) + a.(m+l,n).It is easy to see that the leading contribution to <Pj(O)Pj(r)> isthe smooth contribution to <o^(0)aj(m,n)> ; the other contributionscancel, to leading order in 1/r. In fact, making use of a connec-tion between this model and nhe dimer model [6], we have shown [2]that leading contributions t.. the coarse-grained polarization cor-relations at A • 0 have the form<Py(0)Py(r)> -A<Py(0)Px(r)> A(x2-y2)<xV)22yx<P (0)Px2 2(2)(3)(4)2vhere A * 2/ir . These quantities also depend on n» for simplicity*we have temporarily set n • 1. In these expressions and In thefollowing, the symbol * means "is asymptotically equal to in thelimit of large r."The simple fora of these functions reflects ea underlyingsimplicity in the system. The slat-vertex condition (Fig. 1) is thecondition that polarization be locally "divergenceless" at eachlattice site. This condition is largely responsible for the foraof Eq. (2-4). The ice rules (Fig. 1) state that the polarization ±a"divergenceless" at each lattice site, which in turn means that?(r) is a solenoidal vector field, V-?C?) - 0. This guaranteesthat ?(r) is the curl of a mathematically simpler vector field,n*(r). Fortunately, there Is a physical as well as a mathematicalmotivation for introducing h(r). We know from the work of vanBeijeren [3] that 1i(r) is simply related *to the height variable ofa surface roughening model. _'Dr Weeks' lecture in this volume [4j discusses the relationof a particular roughening model (which .\e denotes FSOS) with thesix-vertex model which we are discussing? In particular. Fig. 3 ofhis third lecture shows on a discrete lattice how to associate aspin variable with the local gradient of the column height h. Withthe appropriate choice or coordinate syitram, this is justo^fcn.n) - -fh(m,n + §) - h(m,n - |>] (5)o2(m,n) - [h(m + ̂ ,n) - h(m - |,n)'j (6)Thus, for example (Fig. 2a), an ordered polar 3ix-vertex configura-tion pointing along (say) y, is equivalent to a surface with a mono-tonically increasing Jtieight, 3h/3x - coast ant. The local polariza-tion conservation 7«P • 0 completely suppresses longitudinal polari-zation fluctuations, which in the language of the height variabletranslate Into a discontinuous "tear" on the crystal surface, asshown in Fig. 2b. Such configurations correspond to unacceptablylarge step sizes In the roughening modal.It is clear from jche foregoing paragraph that the proper choiceof vector potential n(r) - h(r)z, so thatThen, i f we define a height-height correlation function<h(r)h(rQ)> ^ -Fig. 2a. The completelypolarized state of the six-vertex lattice correspondsto a monotonically slopingsurface. (Compare vertex 3of Fig. 3 of Weeks' thirdlecture.) The numbers givethe heights of the latticesites above which theyappear.1 2 3 4 5 6 7 8Fig. 2b. A longitudinalpolarization fluctuation(violating 7-P* * 0) corres-ponds to a tear in thesurface (violating thestep size constraint inthe roughening model).Eqs. (2-4) can be restated very^concisely in terms of the height-height correlation function, V(T-TQ).(8)<Py(r0)Px(r)><px(?o)px(?)>-^y-l7o(9)(10)Thus far, all of our results have been restricted to 4 » 0,where the problem is especially simple. However, we can now proceedto extend the results throughout the regime - 1<A<1, by making con-tact with previous work on the 1-d Heisenberg-Ising chain. McCoyand Wu [9] showed that the transfer matrix of the six-vertex problemcommutes with the Heisenberg-Ising Hamlltonian, shown in Fig. 1.Therefore, there is a connection between the correlation functionsin the two problems. Recent work by Luther and Peschel [10] andFogedby [11] gives the leading asymptotic contribution to<Sz(0,0)Sz(x,t»0)>, which (by virtue of the above-mentioned commu-tation relation) has the same general form as <0j(O,Q)Oj(m,n»O)>(equation 1), including a prefactor A which is a known function ofA. This result effectively prescribes the coarse-grained correlationfunction along one axis. Since V-? * 0, there is still a generatingvector potential which can be analytically continued from that axisto cover the entire x-y plane. Thus we arrive at the followingresult, valid for - 1<A<1.f(r) -v -Ain r + constant (11)2 -1where A - (ir 9) (Luther and Peschel, [10], Fogedby [11])and e " 7 3^n'' * (Johnson, Krinsky and McCoy [12])The formulation of Che correlation function ?(r) given inEq. (11), together with Eq. (8-10) (which generate the asymptoticpolarization correlation functions), constitute the principal re-sults of this paper. We now turn to a discussion of their signifi-cance in two different areas.Since Chui and Weeks [13] have shown that the discrete Gaussianroughening model maps onto the 2-d Coulomb gas problem, and sincevan Beijeren [8] has explicitly demonstrated that a similar rougen-ing model maps exactly onto the F model, it is natural to suppose(along with Shugard et al. [14] and others [4]),that there is aclose connection between the critical behavior of the Kosterlitz-Thouless transition and the critical behavior of the six-vertex tran-sition at A » -1. In particular, the prefactor K,, appearing [14] inG(r) - 2(<h(O)2>- < h(0)h(r)>) % ^ 2 — In r + c (12)is to be compared with A(A) appearing in Eq. (11). If we can iden-tify K Ĵ T) with 2TTA(A), we can expand A(A) to obtainK (T -•• T_+) - § + (nonuniversal constant). (T-Tw)1/2+.. (13)09 R IT R2Both the value Km(Tn) a ^ and the leading square root behavior arepredicted by Kosterlitz-Thouless theory for the unbinding of vor-tices. Thus, the present results for ¥ explicitly support the ideathat the six-vertex model (together with its various equivalents)is in the same universality class as the 2-d Coulomb gas (togetherwith its equivalents).Now we turn to polarization fluctuations in CFT. A full de-scription of this work is contained in Refs. [2] and [3]. Crystalsof CFT contain 2-d layers of water molecules interleaved with 2-dlayers of copper formate. Above T • T o • 248 K (in the deuteratedcompound), there is icelike disorder in the hydrogen-bond network.Below To, the layers become ferroelectric, the direction of polari-zation alternating between +b and -b from one layer to the next.The in-plane longitudinal momentum coordinate is K; the transversemomentum coordinate is H. Figure 3 shows some results of a diffuseneutron scattering study of the long-wavelength polarization corre-lations. ("Polarization" here means that of the hydrogen atompositions, measured from the centers of their respective bonds.)The quantity plotted is the measured intensity, !(($).(14)($) is a geometrical structure factor, which is a rather slow-ly varying function of ($ • <» + "?t where 5 is a reciprocal latticeINTENSITYKFig. 3. Shown here are results of a neutron scattering study ofpolarization fluctuations in the 2-d hydrogen-bond network of copperformate tetrahydrate (Ref. [3]). The origin of the indicated co-ordinate system corresponds to a reduced intralayer momentum trans-fer q of zero. Data are shown for essentially an entire (intralayer)Brillouin zone.vector. Syy(q) is the Fourier transform of <Py(0)Py(r)>, and thebar denotes an average over instrumental resolution. The importantfeature in this plot is the notch at q * 0. Typically Cfor examplein Ising-type systems), pair correlations near To give rise toscattering which peaks at q =• 0; as T c is approached, the diffusepeak associated with order parameter fluctuations is seen to grow.Here, instead of peaking, the intensity dips to near zero, inspite of the incipient transition to an ordered state in whichthere is a Bragg peak at q » 0. (In calling the origin of Fig. 3q • 0, we have suppressed the L momentum coordinate, upon which thescattering is only weakly dependent. This is the expected quasi-2-d behavior. Hcwever, it is important to note that the data ofFig. 3 lie in a plane in which L is half-integral, in which theantiferroelectric Bragg peaks occur below T ,)In the CFT problem,n 5* 1, and the hitherto suppressed n de-pendence of the correlations must be taken into account. It isshown in Ref. [2] that this can be done by replacing r in Eq. (11)withp(r) - /x2 + X2y2 (15)For & =• 0, we have X = n- Polarization correlations are still givenby Eqs. (8-10)_. The Fourier transform of <Py(0)Py(r)> is given by2at small q. (16)S w(q)This function is plotted in Fig. 4 in units of irA, with X chosen tocorrespond roughly to the experimental observations. Note thatthere is a strong formal resemblance between the singularity in thisfunction and that occurring in dipolar-coupled systems (see Dr.Als-Nielsen's notes on LiTbF4). However, we stress that the calcu-lated effect is due entirely to six-vertex ineractions. The absenceof longitudinal fluctuations (Syy(rh,k*0)*0) is a direct consequenceof the ice rule restrictions, 7>p • 0. In real scattering experi-ments, the cross section one observes is somewhat smeared by finiteinstrumental resolution. , This effect is partially taken into accountin Fig. 5. There is a strong resemblance between Figs. 3 and 5;thus, the small-q regime (the notch) of Fig. 3 is evidence that thepair correlations in CFT are qualitatively obeying Eq. (2). Itwould be of some interest to perform further measurements with muchhigher resolution to test Eq. (8) quantitatively.In summary, we see that in the disordered phase of a particularclass of six-vertex systems, long-wavelength polarization correla-tions are governed by a logarithmic (2d-Coulomb-like) potential.The scattering cross section of hydrogen-bonded systems of thistype depends on particular spatial derivatives of this potential;Fig. 4. This is a contour plot of SyyCq) (see Eq. (16)). Thecontours are at integer multiples of Arr, for the case X = A . Thearrow indicates a longitudinal resolution width (see Fig. 5).X= .250 = .0333 = .125 CT= .0333These are plots of I(Q) calculated from Eq. (14). 5yygiven by Eq. (16), with A held constant. F(Q) corresponding tothe data shown in Fig. 3 is included, a is the halfwidth along kof a Gaussian instrumental resolution function (the resolution alongh is assumed to be perfect), a, h, and k are given in reciprocallattice units.the observed cross section from such correlations is quite distinc- 10tive. Given certain rigorous connections between the Heisenberg-Ising problem and the six-vertex problem, we can draw upon existingresults to obtain new information about the six-vertex problem awayfrom the free fermion limit, £ » 0. In particular, we can inferthe temperature dependence of the prefactor of the logarithmicpotential. In addition to making an explicit statement about aparticular roughening model, the results correspond gratifyingly topredictions of Kbsterlitz-Thouless theory for the vortex-unbindingtransformation, thereby reinforcing the idea that roughening modelsare in the same universality class.ACKNOWLEDGMENTTwo of us (RUT and JDA) have profited from conversations heldat this Institute with J. D. Weeks, on the connection betweenroughening models and the six-vertex problem, and from severalconversations with V. J. Emery.REFERENCES1. For a review, see the article by E. H. Lieb and F. Y, Wu in"Phase Transitions and Critical Phenomena," ed. C. Domb andM. S. Green, Academic, New York (.1977).2. R. Youngblood, J. D. Axe. and B. McCoy, submitted to Phys. Rev.3. R. Youngblood and J. D. Axe, Phys. Rev. B17, 3639 (1978).4. For reviews, see J, D. Weeks, this volume; J. D, Weeks andG. H. Gilmer in "Advances in Chemical Physics," ed. I.Prigogine and S. A. Rice, Vol. 40 (1979); H. Muller-Kvumbhaarin "1976 Crystal Growth and Materials," North Holland, NewYork (1977).5. J. M. Kosterlitz and D. J. Thouless, J. Phys. C6, 1181 (1973);J. M. Kosterlitz, J. Phys. C7., 1046 (1974). A review isgiven by A. P. Young in this volume.6. R. J. Baxter, Ann. Phys. (N.Y.) 70, 193 (1972).7. B. Sutherland, Phys. Lett. 26A, 532 (1968).8. H. van Beijeren, Phys. Rev. Lett. _38_, 993 (1977).9. B. M. McCoy and T. T. Wu, Nuovo Cimento 56B, 311 (1968).10. A. Luther and I. Peschel, Phys. Rev. B12_, 3908 (1975).11. H. C.Fogedby, J. Phys. CU, 4767 (1978).12. J. D. Johnson, S. Krinsky, and B. M. McCoy, Phys. Rev. A8, 2526(1973).13. S. T. Chui and J. D. Weeks, Phys. Rev. B14, 4978 (1978).14. W. J. Shugard, J. 0. Weeks and G. H. Gilmer, Phys. Rev. Lett.41, 1399 (1978).BNL-26196Fluctuations in Two-Dimensional Six-Vertex SystemsR. W. YoungbloodJ. D. AxeB. M. McCoyThe submitted manuscript has been authored under contract EY-76-C-02-0016with the Division of Basic Energy Sciences, U.S. Department of Energy.Accordingly, the U.S. Government retains a nonexclusive, royalty-freelicense to publish or reproduce the published form of this contribution,or allow others to do so, for U.S. Government purposes.

Fluctuations in two-dimensional six-vertex systems

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Fluctuations in two-dimensional six-vertex systems

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