THE ISING MODEL IN A TRANSVERSE MAGNETIC FIELD: A TEST ...

THE ISING MODEL IN A TRANSVERSE MAGNETIC FIELD: A TEST-BED FOR MANY-BODY THEORIES R.F. Bishop

Department of Physics, UMIST (University of Manchester Institute of Science and Technology) P. O. Box 88, Manchester M60 1QD, United Kingdom. D.J.J. Farnell and M.L. Ristig

Institut fur Theoretische Physik, Universitat zu Koln Zulpicher Str., 50937 Koln, Germany. 1. INTRODUCTION The coupled cluster method (CCM) [1-4] and the correlated basis function (CBF) method [5-7] are well-known techniques of quantum many-body theory. The CCM has been extensively applied to various T = 0 lattice quantum spin systems recently [3,4] with great success. Similarly, the CBF method has been recently applied [6,7] to the speci c case of the Ising model in a transverse magnetic eld (also called the transverse Ising model) considered here, with much success also. In this work we seek to apply for the rst time the CCM to the spin-half, T = 0 transverse Ising model, and also to extend previous CBF calculations for this model to the case of the triangular lattice. The spin-half transverse Ising Hamiltonian is given by,

 Jz

 X X H = 2 +  N J2 iz jz  ix ; i;j i

(1)

where the index i runs over all N lattice sites on the square, triangular, and cubic lattices; the index j runs over all z possible lattice nearest-neighbour sites to i; and periodic boundary conditions are assumed.

2. THE COUPLED CLUSTER METHOD (CCM) 2.1 The Paramagnetic Regime The starting point of any CCM calculation is to choose a normalized model or reference state, denoted ji. A complete set of mutually commuting many-body

creation operators fCI+ g with respect to ji may now be de ned such that any state jI i, from a complete set of basis states, denoted fjI ig, may be obtained by the application of the operators fCI+ g to ji. In the paramagnetic regime we choose a model state in which all of the spins point in the positive x-direction, which is an exact ground-eigenstate of Eq. (1) in the limit  ! 1. The ground-state wave function may now be written within the normal coupled cluster method (NCCM) parametrizations of the ground ket- and bra-state wave functions given, respectively, by, X (2) j i = eS ji ; S = SI CI+ ; ~ h ~ j = hjSe

I 6=0

S

;

X S~ = 1 + S~I CI ; I 6=0

(3)

in a notation in which C0+  1, and CI  (CI+ )y. The operators CI+ and CI are formed from products of Pauli spin-raising and spin-lowering operators, respectively, and the coecients SI and S~I are referred to as the CCM ket-state and bra-state correlation coecients, respectively. The parametrization of Eqs. (2) and (3) naturally leads to the following expression for the CCM ground-state energy, Eg = hje S HeS ji : (4) Equation (4) illustrates a key feature of the CCM, which is the similarity transform, the importance of which in any such CCM calculation is considered in detail in Refs. [1-4]. One now wishes to obtain numerical values for the CCM ket-state and bra-state correlation coecients in order to determine ground-state expectation values, such as the energy. In order to do this one must evaluate, analytically or computationally, the following equations, hjCI e S HeS ji = 0 ; 8I 6= 0 ; (5) ~ S [H; CI+]eS ji = 0 ; 8I 6= 0 ; (6) hjSe which are easily derived [1-4] from Eqs. (2) and (3) and the ket and bra ground-state Schrodinger equations. In order to perform any realistic CCM calculation, one needs to truncate the otherwise in nite number of cluster correlations, for N ! 1, in S and S~ to some tractable subset. There are three approximation schemes, which by now are well understood [3,4], namely: the SUBn scheme, in which all correlations involving only n or fewer spins are retained, but no further restriction is made concerning their spatial separation on the lattice; the SUBn-m sub-approximation, in which all SUBn correlations spanning a range of no more than m adjacent lattice sites are retained; and the localized LSUBm scheme, which retains all multi-spin correlations over distinct locales on the lattice de ned by m or fewer contiguous sites. In order to treat this problem in a manner analogous to previous CCM calculations for such spin problems [3,4], a rotation of the local axes of the spins which point along the positive x-axis (such that all spins now point in the downwards z-direction) is performed, de ned by: x ! z , y ! y , z ! x . Hence, the Hamiltonian of Eq. (1), with J = 1 and   12 (x  iy ), is now given by,  X   X + + 1 z + + i j + i j + i j + i j +  iz : (7) H = 2 + N 2 i i;j

This Hamiltonian now contains only products of even numbers of spin raising and lowering operators (together with the z operators). Hence, the ground-state wave function must also contain even numbers of spin ips with respect to ji. Thus, we restrict our calculations to contain only even numbers of Pauli-raising operators in S and even numbers of Pauli-lowering operators in S~. It is therefore Pseen P from the above de nitions that the LSUB2 approximation is de ned by S = b21 i  i+ i++ , where b1 is the nearest-neighbour CCM ket-state correlation coecient, and  runs over all nearest-neighbour lattice vectors. An exact expression for the ground-state energy of Eq. (4) in terms of b1 may now be obtained. Furthermore, the evaluation of the ket-state equation of Eq. (5) yields a quadratic equation which may be either solved directly or expanded as a power series in terms of  1 for the ket-state coecient b1. In this manner it is found that the LSUB2 approximation explicitly reproduces second-order perturbation theory in this case. Higher-order LSUBm calculations may also be performed computationally (see Refs. [3,4]). The SUB2 approximation now contains all possible two-body correlations, and may be determined and solved analytically [3,4]. It is found that the SUB2 approximation yields critical points, denoted c , at which the solution to the SUB2 equations becomes complex. This type of behaviour has been seen previously [3,4] and is associated with a phase transition of the real system. High-order LSUBm and also SUB2-n calculations are also found to contain critical points at which the second derivative of the ground-state energy diverges, and it is found that the SUB2-n series of critical points lies on a straight line when plotted against 1=n2. For the square lattice (with n  12) linear extrapolation of these points gives agreement with the position of the full SUB2 critical point to within 0:5%. In analogy with this extrapolation for the SUB2-n series of points, we perform the same extrapolation for the LSUBm (with m = 4; 6) critical points results and assume that the same 1=m2 extrapolation rule applies in this case. The results for the treatment presented in this section are discussed in Sec. 4.

2.2 The Ferromagnetic Regime In the ferromagnetic regime we choose a state in which all spins are aligned along the z-axis and point downwards. The Hamiltonian (with J = 1) is written in terms of the spin raising and lowering operators,   12 (x  iy ), as

z

 X X H = 2 +  N 12 iz jz  (i+ + i ) : i;j

i

(8)

The approximation is now the SUB1 approximation de ned by, S = P lowest-order + a i i , such that an exact expression (which is true for any choice of S ) for the CCM ground-state energy in terms of a may be obtained. The SUB1 equation is now analytically obtained and may be solved either directly or as a power series in terms of  for a. It is found that this series expansion again exactly reproduces second-order perturbation theory calculations for the ground-state energy in the ferromagnetic phase. Higher-order LSUBm calculations may be determined computationally, although no critical points are found in these calculations. Once the ket-state and bra-state equations have been solved at a particular value of  it is then possible

to determine other expectation values such as the lattice magnetization (i.e., the magnetization in the z-direction), M , de ned in the CCM framework by, N X 1 M = N h ~ j iz j i : i=1

(9)

The manner in which the bra-state coecients are determined has been discussed at length in previous articles [3,4], and results of CCM calculations for the transverse Ising model are presented in Sec. 4.

3. THE CORRELATED BASIS FUNCTION (CBF) METHOD One begins the treatment of the transverse Ising model via the CBF methodology by de ning, for a ground-state, trial wave function j i, the lattice magnetization (i.e., the magnetization in the z-direction), given in the CBF framework by, z M = h hj ji ji i : (10) One similarly de nes the transverse magnetization to be given by, x (11) A = h hj ji ji i : It is also found to be useful to de ne a spatial distribution function (which plays a crucial part in any CBF calculation) in the following manner, h j z z j i ; (12) g(n) = h i j ji where n is de ned by n = ri rj in this section. The ground-state energy per lattice site may now be de ned for the expectation value, Eg =N = h jH j i=(N h j i), (again with J = 1) as, Eg = z 1 X
(n)g(n) + (1 A) ; (13) N 2 2 i;j where the function
(n) is equal to unity when n is a nearest-neighbour lattice vector and is zero elsewhere. We note that the distribution function g(n) may be decomposed according to g(n) = n;0 + (1 n;0 )M 2 + (1 M 2)G(n). G(n) now contains the short-range part of the spatial distribution function and vanishes in the limit jnj ! 1. The magnetization M and the transverse magnetization A may now be expressed in a factorized form in terms of a spin-exchange strength, n12, such that, A = (1 M 2) 21 n12 : (14) The energy functional is now expressed in terms of G(n) and n12 as,     Eg = (1 M 2) z 1 X
(n)G(n) +  1 (1 M 2) 21 n (15) 12 : N 2 2 i;j

Note that in the standard, mean- eld approximation one sets G(n) in Eq. (15) to zero for all n. In order to determine the energy and the ground-state expectation values, the Hartree-Jastrow Ansatz for the ground-state wave function is now introduced. This is given by, j i = expfMUM + U gj0i ; (16) where the correlations in U and UM are written in terms of pseudopotentials u(rij ), u1 (ri ), and uM (rij ) as follows, N X 1 U = 2 u(rij )iz jz ; i