A functional integral approach to quantum spin systems

Journal of Magnetism North-Holland

and Magnetic

A functional

Materials

104-107

integral

(1992) 855-856

approach

to quantum

spin systems

G.J. Mata a and G.B. Arnold b a Departamento de Fisica, UniLlersidad Sim6n

Boli’uar, Apartado 89000, Caracas 108OA, Venezuela ’ Department of Physics, UnilJersity of Notre Dame, Notre Dame, IN 46556, USA We use a cluster-decomposition technique to derive a functional integral representation of the partition function of quantum spin systems on a lattice. The free energy is decomposed into the sum of the free energy of the isolated clusters and that of a system of interacting vector fields. Our formalism allows a constraint-free description of the thermodynamics and excitations in the rotationally invariant phase. The transition to the broken-symmetry phases appears to be linked to the softening of spin wave modes. We briefly discuss applications to one- and two-dimensional spin systems.

The physics of spin systems has attracted considerable attention in recent years. Interest in low-dimensional systems, in particular, has led to efforts to understand the excitations in the rotationally invariant phases [l-4], where no long-range order exists. In this case standard methods, based upon expansion about some ordered state, do not apply. We have previously developed an effective cluster method [5] for spin systems in which the computation of ground-state properties (and finite-temperature equilibrium properties away from critical points) can be carried out with a minimal amount of computational effort. In this method, the system is broken up into identical clusters that are small enough so that the many-body Hamiltonian can be exactly diagonalized for each cluster in isolation. The interaction between clusters is approximately taken into account through the effect of an additional term in the isolated cluster Hamiltonian. Fluctuations in different clusters are assumed to be uncorrelated. Here we report an extension of this method to include the effect of dynamic correlation between different clusters and thus develop a theory of the spin wave spectrum. We consider a spin system divided into clusters. The FLth component of the spin at the (Y site within the cluster i is denoted by Sk. The exchange interaction between spins Sk and Sk, is denoted by J,j,,,,. The Hamiltonian is written as

i

(1)

ij

where summation over repeated greek indices is implied. The first term in this equation, 2$Y0,represents interactions of spins within the same cluster. The second term, 2Y1, represents interactions between different clusters. In the interaction representation the partition function may be expressed as Z = Tr{ emPP”T[exp(

0312-8853/92/$05.00

sdTz,(T))]}.

0 1992 - Elsevier

Science

Publishers

We perform a Hubbard-Stratonovitch tion in 3, only. We get the following gral representation for Z:

z = z,

Jb4vtt(~)l

exp(4[@1

jQ4%(~)1

transformafunctional inte-

+Al[41)

exp(4,[ILl)

’

(3)

where Z, = Tr(eFP*l)

(4)

is the partition function of their interactions,

of the clusters

in the absence

,

exp/oirdTx$:(r)S,@(r)

i

(6)

Zfl

The brackets denote thermal averages and the indices LY have been omitted for clarity. (In two-sublattice antiferromagnets the integral is made convergent by redefining the spins in the down sublattice so that the interaction changes sign.) The free energy is thus recast into the sum of the free energy of a system of noninteracting clusters and that of interacting vector fields. A ,[$I describes the response of a cluster to arbitrary time-dependent fields and it is the generator of the connected correlations for the isolated cluster. Up to this point the transformation is exact. However, A,[$] cannot be computed exactly and even if it were known we still would be left with a nonlinear field theory. Here we consider the simplest level of approximation and we retain terms that lead to a free field action. That is, we keep only second-order connected correlations. In addition we assume that the system is in the rotationally invariant phase, in which spin corre-

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G.J. Mata, G.B. Arnold / Integral approach to quantum spin systems

856

lations

are scalar. The action now becomes

A,[$]

= f zj’drL’dTI&!+)G;(r i

original found:

- r’)+,?(r),

w(k)

O (7)

lattice.

The

following

dispersion

= \/4 ’ e ~PJ/(1+3e~P’)+2sin2k.

relation

is

(13)

At finite temperatures a gap opens at k = 0 and at temperature the gap becomes exactly zero. The bandwidth at T = 0 is equal to &J, which is within 10% of the exact value. We have also examined a spin S = f antiferromagnet on a square lattice. Using a pair decomposition we find a spectrum which shows a temperature-dependent gap at the F and W points in the two-dimensional Brillouin zone. The gap is found to close at a temperature T, = 0.91/. (T, coincides with the transition predicted by the effective cluster method.) Below this temperature the rotationally invariant phase becomes unstable (in this approximation) and the system becomes antiferromagnetic. We see that the transition can be associated to the softening of an antiferromagnetic spin wave. (The k = 0 spin wave has zero spectral weight and does not give rise to long-range correlations.) On the basis of the Mermin-Wagner theorem, we know that this system has no phase transition at any finite temperature. We get a finite phase transition because, having neglected nonlinear terms in the action, we cannot properly account for critical fluctuations. We expect nonlinear terms to renormalizc the gap and force T, down to zero. Nonetheless, the qualitative picture described here should hold. In summary, we have presented a method to calculate the spin wave spectrum of quantum spin systems, without reference to an ordered state, and without the introduction of constraints. The ordered state itself appears as an instability of the ‘normal’, rotationally invariant phase. Extensions of this work to incorporate the effect of long-range fluctuations and to study the spin waves in the ordered phase, are presently under way.

k = IT. At zero

where we note that G,(r - 7’) = (T[S/Yr)S:(r’)]

>;r,,,

(8)

is the generalized dynamic susceptibility of an isolated cluster. We recall that G;(T - 7’) is a matrix of rank equal to the cluster size. Upon Fourier transformations we get the full action

441= -i c c [eJn)1*

where, for identical clusters, the index in G is dropped, the 0,‘s are Matsubara frequencies, and k runs over the Brillouin zone of the lattice of clusters. From eq. (9) we can read the propagator DJw,) for the fields 4: ok(mn)

= [l -J(k)G(w,jm’J(k).

(10)

One can show that the spin propagator

is given by

lm

(11)

This is in turn related by analytic generalized dynamic susceptibility G,:(t)=

-ifi(t)([s:(t),

SJ‘]).

continuation

to the

(12)

Therefore the poles of Dk(w,) give the spin wave energies. We see that in order to compute the spin wave spectrum we need to compute the cluster susceptibility and the cluster interaction matrix Jjj. We have applied these results to an antiferromagnetic linear chain with spin S = i and nearest neighbour coupling J. We divide the chain into pairs of spins. The pair susceptibility and interaction matrix JLj are readily computed. Eq. (10) is used to construct the propagator. The spectrum is found to be doubly degenerate in the Brillouin zone of the lattice of pairs. The spectrum can also be ‘unfolded’ and viewed as a function of k vectors in the full Brillouin zone of the

This work was supported by CONICIT, Fundacion Polar, and Decanato de Investigaciones de la Universidad Simon Bolivar, all of Caracas, Venezuela. References [l] J.E. Hirsch and S. Tang, Phys. Rev. B 40 [2] D.P. Arovas and A. Auerbach, Phys. Rev. [3] M. Takahashi, Prog. Theor. Phys. Suppl. Phys. Rev. Lett. 58 (1987) 168. [4] E. Rastelli and A. Tassi, Phys. Lett. A 48 [5] G.J. Mata and G.B. Arnold, Phys. Rev. B

(1989) 4769. B 38 (1988) 316. 87 (1986) 233; (1974) 119. 38 (1988) 11582.