The methods of Lie algebras are used to construct and solve a generalization of the XY model

Montgomery, W.

Solomon, Allan I.

English

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DIAS-TP-76-25 -1--1. IntroductionGeneralized XY ModelIn this paper we apply the methods of Lie Algebras to solve and generalize theXV Model (Lieb, Schultz end Tiattis 1901, Ketsura 1962). The methods employedoriginate in the Spectrum Generating Algebras of particle physics - whh areW. Montgomery end A. I. Solomon non-symmetry algebras of the Hemiltonian, end provide elegant solutions forquantum statistical problems [Solomon 1971, 1974). We briefly outline theapproach we adopt in the present context.Open University, Milon Keynes, U.K.In equilibrium statistical mechanics the thermodynamic behavieur of a system,andwhose Hamiltonian is H, follows from evaluation of the partition functionSchool of Theoretical PhysicsQ = trace fexp (-6H)),Dublin Institute for Advanced StudiesDublin 4, Ireland where is the inverse of the absolute temperature times Boltzmann’s constant.Classically, the trace may be interpreted as the sum over all the allowedconfigurations of the systems in quantum mechanics, as the usual Hubert spacetrace.In our aigebraic treatment we shall consider H to LJC an elcrnent of a suitab]eLie algebra of rank i. This means that one can find a Cartan basis for thealgebra which includes the 9. mutually commuting elements, h1, h2 h9..The solution of the problem is obtained by finding an automorphism of the algebra,Abstractimplemented by U say, such thatThe methods of Lie Algebras are used to construct and solve9.generalizetior ‘f the XV model. H-U H U1 A hmm= 1where the A are known scalars (elements of the underlying field). Since inprinciple the spectra of the h are known, such an autornorphism effects diagonalisation end clearly leaves the partition function Q unchanged.Physics Abstracts Classification Number: 1.660-2-Therefcre the strategy to be adopted is in three parts:A. Determine a suitable Lie algebra which is to generate the spectrum ofH. The Haniltonian H will be an element of the algebra in some(usually large) representation.B. Choose a small-dimensional, faithfui representation in which to implementthe eutomorthIsm (1.1).Now return to the original repressntation, in which (1.1) remains trueand in particular the values of the scalars Am are unchanged, to evaluatethe spectrum of H and the partition function.In the case of the XV model, on a cyclic lattice of N points, the application ofthe three—part strategy givesA. The Hamiltonian is an element of a x 2N dimensional representationof so(2N) sc(2N). This is rank 2N algebra.B. We implement the automorphism (1.1) in the faithful 4N x 4N dimensionalrenresentation, determining the values of the 2N constants Ac. We roturn to toe 2N x represortiition to evaluate the partition function.The reason that toe soluticn of the XV NodJ is so readily obtained, in spite ofthe seemingly cumbursome nature o’ thu machinery outlined alove (nobdy cancagnnalizi even a 4N x 4h matrix in general!) is that the translationalbn:: of physically ±ntsresting models moans toat in these cases the under—ulebra is a mu us aller one, and effectively reduces the computation inall ench casco to thc clagccalation of a small (in our Dase, 2 x 2) matrix.The generalizd XV Nobel we treat is in fact too most ttooral translationally—invariant model consistent with the so(2N) B so(2N) algebra of the original XVNobel.in C to that of a system of uncoupled spins, to which the model is tharefcreequivalent in an algebraic sense, we now briefly treat such a Free Spin Nodel.NH - ) A Z (2.is cm‘n 1The A are positive scalars aridmu1where the matrices occurring in the direct product are all 2 x 2, and a , whiroccurs at the mth position, is the third of the three Pauli spinors,11 ro -a[i oj Oj Q -tThe ce:all negative sign ifl (2.1) ensures tha 3lignment along the posItivez-direction lowers the energy.The partition functionNQ(N. ) — trace (exp BA Z).m1may be evaluated as a straightforward matrix trace, using the properties ofdirect products, asNL(N, B) 11 {2 cosh BA }.In1n I2. Free Spin NodelConsider the following Hemiltonlan, representing a system of N uncoupled spins:The free energy per particle f is given in thu thermodjnamic limit bySince the automorphism (1.1) reduces the computation of the partition function—4— -5-I I 3. Algebra of the XV Flodel-= urn N log Q(N,6)We consider a one-dimensional lattice of N sites, labelled 1, 2,..., N. The=.-_J log{2 cosh3A(Jd XY [lodel is given by the following Hamiltonian of neareot-neighbour type:0 Nwhere we replace the discrete valued A by the continuous function h(), where H1 = - {JXX X X + v Y } . (3.1)in m1 1 in m+1m m=1A( 3 = A at = 2nm/N Cm = 1, 2 N) The notation for X and Y is analogous to that of 2 in the previous section,m m in m m mso that, for example,All quantities of thermodynamid interest may be calculated from f.[X,Y] 2i ZIn the model we shall be considering, the Hamiltonian can in fact be rotated to in n mn nthe somewhat more general formWe may also include additionally a contribution from an external magnetic field hN+H =— {(i - y)A + (1 + y)A }Z (2.2)m m m Nm= IH = - h 2 . (3.2)0 inm=1where A are positive scalars andiny = Z1 22 .... z The XV flodel described by H = H H is exactly solvable, and although it does0 1not exhibit a phase transition in the thermodynamic limit for finite 6. itsThe partition funLtlor. corresponding to this Hamiltonien may be equally readilythermodynamic behaviour has been extensively studied. urther, the XY flodel ivaluated,intimately re]nted to the solution of the two-dmensioral Icing fiodel in trensferN-I. +— matrix form (Suzuki lS7)j this connection is evDn more expllcit in the casD of= 2 ii cosh 6A + II cosh SA + H sinh 5A — H sinh SA )in Ifl m inthe generalized model we shall describe in the next section. of which the X? odelThe free energy per particle determined from this partition function will depend is only a special case.+on the relative magnitudes of A and A for example, when A > A the firstin rn m rnterm will dominate. We now implement Part A of our strategy by determining the Spectrum GeneratingAlgebra for the XY flodel. Define the following matrices yrr 1 2 r-1 rr=2,3 N.Y ZZ ...ZN+r I 2 r-1 rYl — X1 V1The rnt:ices y Ir ,2,..., 2N) then generate a Clifford algebra with Then the operatorsanti-commutation relations given by L(8)- (I- a-r)L a — ±r’y} 2 • close on the algebra so(2N) so(2N)The transformation from {X Y • 2 } to (y } is sometimes called the Jordanm m m r [L LJ - (6 L(8) - 6 + 6 L(a) - 6 L)Wignor transformation. Using the y we may construct the N(2N — 1) matrices rs pq ab rs sq sp z-q rq ps sq prLrs This enlarged algebra now contains ll the previously required quantitiesL — i/4Cy,y ] (r, s 1, 2,..., 2N) (3.3)rs r sL -L +L•rs rs rswhiuh close under the commutation relations of the Lie algebra so(2N)as well as the cyclic terms[L , L ] i(6 L - 6 L . + 6 L - 6 L ) . (3.4)rs pq rp sq ep rq rq ps sq prX X - 2(L- )N I 1,2N 1,2N(We retain the i in expressions such s (3.3) and (3.4) only when we wish toY Y 2(L.- L’ ) imaintain the hermiticity of the operators concerned. The algebras, such as N I N,NI N,N•Isa(2., that we are interested in ae of course real Lie algebras whose definingend so we may write the XY Hamiltunian H H0 + H • equations (3.1) and (3.2)relaticns do not in”clve i.)I explicitly as an element of so(2N) so(2N). This’completes Part A of ourFran the fullcwrig exp:eusions. which hold for m 1. 2 N-I, strategy.XX -2L YY 2L 2 “2Lrn m Nrn, m÷1 m m÷1 N+m+l, m m m. Nm4. Translational InvarianceWe see that in the case of the XY flodel with free ends, where the summation inThe most general element of our so(2N) 8 so(2N) algebra may be written(3.1) goes from 1 to N-I, we may immediately express H as an element of so(2N).in the cyciic case, however, we require the additional terms XNXI and N1 H r (a) L(a) V (4ma innend so must enlarge the algebra. This is readily done as follows, a ± m,n 1Introduce the matrix y - Z12... ZN . This obeysor, more compactly,{y, 0 - ‘1 Cy, L] - 0• H - - 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(Al) and (A2), may be chosen to have particularly simple, -(a)closed forms; for in the infinite range limit the coupling constants in (A2) We determine the elements A of the canonical forn ofmcan be taken as the Fourier coefficients of (fairly arbitrary) functions. N—I (cz’. 5A) (Al)The Hamiltonian considered, though of doubtul direct physical interest, hasa useful interpretation as the associated Hamiltonian of tho two-dimensional Assume that the eigenvalue ec4uation for is of the formIsing problem; that is, an operator commuting with the transfer matrix.—- (A2)Although we h.ve only considered translationally invariant models in this note,it is a straigotforward matter to proceed to the non-invariant case. For (a) (a)with eigenvaluewhere e is en eigenvector of Aexample, if the Harniltonian is invariant under v-translations, where v is somepositive integer dividing N, 4t4) E’1T’HT’ = HBy substituting the expression (Al) for in (A2) we obtainwe obtain the general solution by solving the modification of (4.3) ,j / ()z A —r— 0so that i is the eigenvelue corresponding to the eigenvector’Iof the 2.x..This modification will be the subject of another note. (a)Hamiltonian matrix HN-IReferences. .(),r_(A3)atsura S 1962 Phys. Rev. 127 1508Lieb E H, Schultz T 0 and flattis 0 C 1961 Ann. Phys., NY 16 941 Rewriting (a) . (a)__________(1 in terms of the four real numbers mIiS: skens Ch 3, Lapel H W and Gaemers K 3 F 1974 Phys.Lett. 50A 261Solomon A I 1971 3. Math. Rhyc. 12 390 (A)1974 Proceedings of the 3rd Internatio,a Colloquium on GroupTheoretical Methods in Physics (Marseille) Vol I p. 318.C)(UoCtI-..C)’—’0)p.CU‘UCUClp:3:r:3<(UCt(1)—(UC)4,XCtc:(UC‘C)Ci):iC)>‘,-‘.C)U)3,,,,_(p.(UU)c):30(U-.‘1Z3:3‘%_C—.(U<(UU’Ci)C)rt-Ci)C—’—h(U.—C.I(U::•‘-a3C(UC-—’3(C(+‘i—i‘i:,.-‘-‘;;‘—-,cL,:3C))p.Z)‘00Ci)0-C,-0Ctp.0)•U)C,,,,zp.C-,:-<Cr•>(UC-’i.(flcC-“p.•<CtC-’.:3Cra)0))(UaC))C)(UcC-).U‘1“),,C-”CC-(CU)-CtC-”—‘C’.CDC,C).3“(UIIC’,,,u-‘‘4-•C;,,‘CIC’(U-<c)jl”

Generalized XY Model

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Generalized XY Model

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