Phys. Rev. B 40, 4904 (1989)

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Paired-magnon analysis of quantum antiferromagnets

Efstratios Manousakis
Department of Physics, Center for Materials Research and Technology Supercomputer Computations Research Institute, Florida State University, Tallahassee, Florida 32306
(Received 8 March 1989)

We study the spin-(1/2 Heisenberg antiferromagnet on an infinite square lattice. The calculational scheme known as ``paired-phonon analysis'' developed for strongly correlated quantum fluids is extended to a ``paired-magnon analysis'' to study quantum antiferromagnets. We define a complete and orthonormal set of multimagnon states and calculate the matrix elements of the Hamiltonian using a separability approximation. Our results obtained by diagonalizing the Hamiltonian matrix analytically are very similar to those obtained in spin-wave theory. We obtain -0.3290, for the ground-state energy per bond in units of the antiferromagnetic coupling and 0.303 for the ground-state staggered magnetization. These results compare well with the best-known estimates -0.334±0.001 and 0.313, respectively. We derive the analytic form of the ground-state wave function in this approximation and find it to be of the same form as that assumed by Marshall in his variational studies.. The zero-point motion of long-wavelength excitations (spin waves) in the model, however, reflects a long-range tail in our wave function. We discuss the separability approximation by giving quantitative arguments which justify its validity.

©1989 The American Physical Society.

PACS: 75.10.Jm 74.20.-z

I. INTRODUCTION

The discovery of copper oxide superconductors has renewed the interest in certain quantum spin and fermion models. The examination of these materials by neutron scattering experiments^{1}shows long range antiferromagnetic (AF) correlations which may be understood^{2,3}in terms of the dynamics of a simple two dimensional (2D) spin cf12 AF Heisenberg model

H=J^sum from {langle R vec , R vec prime rangle}^S vec_{R}vec cdot S vec_{{R}vec prime} .   (1.1)

In this model S vec_{R}vec_ describes the Pauli spin cf12 operator of one electron of the ith CuO_{2_}cell being in a linear combination of the orbitals d_{{x^{2}-y^{2}_}of}the copper and p_{x_}and p_{y_}of the two oxygen atoms of the CuO_{2_}plane. This model can be obtained as the strong coupling limit of the Hubbard model, when the conduction band is half filled.

Contrary to its simplicity the model (1.1) lacks an exact solution in two or higher space dimensions and a growing number of numerical,^{3,4}analytical,^{5}or semianalytical^{6}techniques of an approximate nature have been employed. Even though there is no exact statement yet,^{7}based upon the above methods, it is believed that the spin cf12 AF Heisenberg model on the square lattice develops AF long range order at zero temperature. Moreover, the general picture emerging from these studies is that one may obtain a relatively good quantitative description of the ground state of the AF Heisenberg model by treating small spin fluctuations above the Negael state perturbatively.

The Hamiltonian (1.1) can be written as

H=H_{1}+H_{2 ,  }(1.2a)

H_{1}= case 1 over 4^sum from {langle R vec , R vec prime rangle}^sigma_{R}vec^{z}sigma_{{R}vec prime}^{z ,  }(1.2b)

H_{2}= cf12^sum from {langle R vec , R vec prime rangle}^ (S_{R}vec^{+}S_{{R}vec prime}^{-}+ S_{R}vec^{-}S_{{R}vec prime}^{+}) ,   (1.2c)

where sigma^{z}=2 S^{z,}S^{+}=S_{x}+iS_{y_}and S^{-}= S_{x}-iS_{y_,}and by letting J=1 we measure the energy in units of J.

We define the set of ``multimagnon'' states in the following way:

| ... n(k) ... ) == ^prod from k vec^ ( sigma_{k}vec^{z})^{n(k)}| phi rangle ,   (1.3a)

sigma_{k}vec^{z}= 1 over {sqrt N}^sum from R vec^ e^{{i}k vec cdot R vec} sigma_{R}vec^{z ,  }(1.3b)

where the sum runs over all N lattice vectors R vec and n(k) =0,1,2 ,..., N. The state | phi rangle is defined as follows:

| phi rangle == 1 over {(2^{N})^{1/2}^sum}from c^(-1)^{L(c)}|c rangle .   (1.4)

Here the sum is over all possible spin configurations c of the lattice and L(c) is the number of down spins in one sublattice contained in the configuration c. Therefore, the state | phi rangle is

| phi rangle =^prod from {R vec member A}^| R vec rangle_{+^prod}from {R vec member B}^| R vec rangle_{- ,  }(1.5a)

| R vec rangle_{+-}== 1 over {sqrt 2} (|+ rangle +- |- rangle ) ,   (1.5b)

where A and B represent the two sublattices and |+ rangle and |- rangle are eigenstates of sigma_{R}vec^{z_}with eigenvalues +1 and -1, respectively. The operator sigma_{R}vec^{z_}acting on | R vec rangle_{+-_}gives

sigma_{R}vec^{z}| R vec rangle_{+}= | R vec rangle_{- ,  }(1.6a)

sigma_{R}vec^{z}| R vec rangle_{-}= | R vec rangle_{+ .  }(1.6b)

Since the states | R vec rangle_{+_}and | R vec rangle_{-_}form a complete basis of the Hilbert space of the electron at R vec, all possible states of the Hilbert space for N spins can be obtained by acting on | phi rangle by all products of sigma_{{R}vec_{1}},..., sigma_{{R}vec_{l}^{z_}for}any l (0<l <= N) different sites. It can be easily verified that the state (1.4) [or (1.5)] has zero staggered magnetization in the z and y directions but has full staggered magnetization in the x direction. In fact, if we rotate the Negael state around the y axis by pi /2 we obtain the state (1.4) [or (1.5)].

The set of states defined by (1.3) form a nonorthogonal basis. In the next section of this paper we orthonormalize them and calculate the matrix elements of H in a separability approximation. This approximation was introduced in the theory of quantum fluids^{8}in the context of ``paired phonon analysis'' of strongly correlated Bose liquids. Following Ref. 8 we extend the method of ``paired phonon analysis'' to a ``paired magnon analysis'' to study the spin cf12 Heisenberg antiferromagnet. The separability approximation neglects the coupling of paired multimagnon states. In Sec. III we diagonalize the Hamiltonian matrix and find the ground state and elementary excitations. We obtain -0.3290 J for the ground state energy per bond of the infinite square lattice, which is in <2% agreement with the most accurate estimates.^{6}In Sec. IV we derive the analytic form of the ground state wave function and show that it has the form assumed by Marshall in his variational studies. Our wave function, however, has long distance behavior consistent with the existence of low lying long wavelength excitations (spin waves) in the model and their zero point motion. In Sec. V, we calculate the staggered magnetization and find 0.303 the same value with the results of spin wave theory.^{5}In the same section, we give quantitative justification of our separability approximation. Since the ground state properties of the spin cf12 Heisenberg antiferromagnetic can be analytically and accurately calculated with this technique, it will be interesting to study the presence of one, two, or more holes with the effective Hamiltonian obtained from the Hubbard model in the strong coupling limit.

II. MULTIMAGNON STATES AND MATRIX ELEMENTS

Let us start from the following states:

|m,n) == ( sigma_{k}vec^{z})^{m}( sigma_{{-}k vec}^{z})^{n}| phi rangle .   (2.1)

These states, however, do not form an orthogonal set. We modify the definition as follows:

|m,n rangle == ( [N-(m+n)]! over m!n!N! right )^{1/2^sum}from {lcurl R vec_{i}, r vec_{j}rcurl_{C}^}e^{{i}k vec cdot ( R vec_{1}+ ... + R vec_{m})} e^{{-}k vec cdot ( r vec_{1}+ ... + r vec_{n})} sigma_{{R}vec_{1}^{z}...}sigma_{{R}vec_{m}^{z}sigma}sub {r vec_{1}^{z}...}sigma_{{r}vec_{n}^{z}|}phi rangle ,   (2.2)

with k vec != 0. Here, lcurl R vec_{i}, r vec_{j}rcurl_{C_}means that the sum is over all lcurl R vec_{1},..., R vec_{m}, r vec_{1},..., r vec_{n}rcurl_ with the constraint R vec_{i}!= R vec_{j}!= r vec_{k}!= r vec_{l}!= R vec_{1_.}The overlap between such states is given by

langle m prime , n prime |m,n rangle = mark 1 over N! ( {[N-(m+n)]![N-(m prime + n prime )]!} over {m!m prime !n!n prime !} right )^{1/2  

}

lineup times^sum from {lcurl R vec_{i}, r vec_{j}rcurl_{C}, lcurl R vec_{i^{sprime},}r vec_{j^{sprime}rcurl}sub C} ^mark e^{{i}k vec cdot ( R vec_{1}+ ... + R vec_{m})} e^{{-}i k vec cdot ( R vec_{1^{sprime}+}... + R vec_{{m}prime}^{sprime})} e^{{-i}k vec cdot ( r vec_{1}+ ... + r vec_{n})} e^{{i}k vec cdot ( r vec_{1^{sprime}+}... + r vec_{{n}prime} )}  

lineup times langle phi | sigma_{{R}vec_{1^{sprime}}sup}z ... sigma_{{R}vec_{{m}prime}^{sprime}}sup z sigma_{{r}vec_{1^{sprime}}sup}z ... sigma_{{r}vec prime _{{n}prime}}^{z}sigma_{{R}vec_{1}^{z}...}sigma_{{R}vec_{m}^{z}sigma}sub {r vec_{1}^{z}...}sigma_{{r}vec_{n}^{z}|}phi rangle  

= delta_{{n,n}prime} delta_{{m,m}prime} .   (2.3)

In order to obtain nonzero contribution, the sigma's must occur in pairs such that ( sigma^{z})^{2}=1 (because langle phi | sigma^{z}| phi rangle =0). However, all the sites lcurl R vec_{1},..., R vec_{m}, r vec_{1},..., r vec_{n}rcurl_ are different and no two sites in the set lcurl R vec_{1^{sprime},...,}R vec_{{m}prime}^{sprime}, r vec_{1^{sprime},...,}r vec_{{n}prime}^{sprime}rcurl_ are the same. Moreover, R vec_{i}!= r vec_{j^{sprime_}and}R vec_{i^{sprime}!=}r vec_{j_,}because if R vec_{i}= r vec_{j^{sprime_}or}R vec_{i}= r vec_{j^{sprime_}we}obtain k vec = 0 vec by summing over R vec_{i_}or R vec_{i^{sprime_}respectively.}Hence m^{sprime}= m, n prime =n and the sites R vec_{1},..., R vec_{m_}must be identical to any permutation of R vec_{1^{sprime},...,}R vec_{m^{sprime_}and}the sites r vec_{1},..., r vec_{n_}must be identical to any permutation of r vec_{1^{sprime},...,}r vec_{n^{sprime_.}There}are m!n! such permutations. The matrix element of sigma's is unity for each such term and the summation over all different R vec_{1},..., R vec_{m_}and r vec_{1},..., r vec_{n_}gives a factor of N(N-1)(N-2) ... [N-(m+n-1)]. Therefore, the states |m,n rangle defined by (2.2) form an orthonormal set.

The states defined by Eq. (2.2), however, do not form a complete set. The entire Hilbert space is spanned by

| ... m_{k}vec , m_{{-}k vec} ... rangle == ^prod from {k vec , k_{x}>0}^ [ ( {[N-(m_{k}vec +m_{{-}k vec} )]!} over {m_{k}vec !m_{{-}k vec } !N!} right )^{1/2^sum}from {lcurl R vec_{i}, r vec_{j}rcurl_{C}^mark}e^{{i}k vec cdot ( R vec_{1}+ ... + R vec_{{m}sub k vec})} e^{{-i}k vec cdot ( r vec_{1}+ ... + r vec_{{m}sub {- k vec} )}}  

left lineup times sigma_{{R}vec_{1}^{z}...}sigma_{{R}sub m_{k}vec}^{z}sigma_{{r}vec_{1}^{z}...}sigma_{{r}vec_{{m}sub {- k vec}}} right ] | phi rangle ,   (2.4)

where m_{k}vec_ and m_{{-k}vec}_ are the number of magnons in the momentum states k vec and - k vec, respectively. These paired multimagnon states are nonorthogonal. We can proceed further by introducing a separability approximation^{8}in the calculation of the matrix elements of the unit operator and the Hamiltonian. Namely

langle ... m_{k}vec^{sprime}, m_{{-}k vec}^{sprime}... | ... m_{k}vec , m_{{-}k vec} ... rangle  L

->^prod from {k vec , k_{x}> 0}^langle m_{k}vec^{sprime}, m_{{-}k vec}^{sprime}|m_{k}vec , m_{{-}k vec} rangle  

=^prod from {k vec , k_{x}> 0}^delta_{{m}sub k vec^{sprime}, m_{k}vec} , delta_{{m}sub {- k vec}^{sprime}, m_{{-}k vec}} ,   (2.5)

and

langle ... m_{k}vec^{sprime}, m_{{-}k vec}^{sprime}... | H-E_{phi}| ... m_{k}vec , m_{{-}k vec} ... rangle  L

->^sum from {q vec , q_{x}> 0}^mark langle m_{q}vec^{sprime}, m_{{-}q vec}^{sprime}| H-E_{phi}|m_{q}vec , m_{{-}q vec} rangle  

lineup times^prod from {k vec != q vec , k_{x}> 0}^langle m_{k}vec^{sprime}, m_{{-}k vec}^{sprime}|m_{k}vec , m_{{-}k vec} rangle .   (2.6)

where E_{phi}= langle phi |H| phi rangle_. Since we have orthogonalized the states (1.2) for all k vec, within the separability approximation the states (2.4) are orthogonal. This approximation neglects the matrix elements which couple a subspace defined by (2.2) for a definite value of m_{k}vec , m_{{-}k vec}_ with another subspace defined by (2.2) and characterized by different values. This approximation makes sense only in a limited function space characterized by

sum from k vec^m_{k}vec << N .   (2.7)

Our results are subject to the validity of the approximation (2.5) and (2.6). In the rest of the treatment we do not introduce any further approximation. In Sec. V, we will come back to this point and show that our solution satisfies the condition (2.7) to a reasonable degree.

The expectation value of the Hamiltonian (1.1) with the state | phi rangle is given by

E_{phi}mark == langle phi |H| phi rangle  

lineup = cf12^sum from {langle R vec , R vec prime rangle}^langle phi |S_{R}vec^{+}S_{{R}vec prime}^{-}+S_{{R}vec_{prime}^{+}S}sub R vec^{-}| phi rangle =- dN over 4 .  

Here, it requires the same effort to work on a generalization of the square lattice to a hypercubic one in d dimensions.

In the Appendix we calculate the matrix elements of H in the separability approximation (2.6). We find that the nonzero matrix elements are .EQL (2.9) langle m,n|H-E_{phi}|m,n rangle =d(m+n) ( 1- 3 over 4 m+n over N right ) ,   (2.8)

langle m-1,n-1|H-E_{phi}|m,n rangle =d sqrt mn ( 1 - m+n over N right ) gamma (k) ,  L

.EQL langle m+1,n+1|H-E_{phi}|m,n rangle   (2.10)

=d sqrt (m+1)(n+1) ( 1- m+n over N right ) gamma (k) ,  R (2.11)

where

gamma (k)= 1 over d^sum from {mu =1} to d^ cos (k_{mu}) .   (2.12)

III. DIAGONALIZATION. GROUND STATE AND ELEMENTARY EXCITATIONS

In the approximation (2.5) and (2.6), the ground state wave function can be written as

| psi_{0}rangle =^prod from {k vec , k_{x}> 0}^ F_{k}vec | phi rangle .   (3.1)

The state F_{k}vec | phi rangle_ can be written as a linear superposition of states (2.2) with m=n, namely

F_{k}vec | phi rangle =^sum from m=0 to inf^C_{m}|m,m rangle .   (3.2)

This function is the eigenstate of H hat with the lowest eigenvalue in the subspace spanned by (2.2) for a specific value of k vec and all m=n. The matrix elements of the Hamiltonian in this subspace are

langle m,m |H-E_{phi}|m,m rangle =2dm ( 1- 3 over 2 m over N right ) ,  L (3.3)

langle m-1,m-1|H-E_{phi}|m,n rangle =dm ( 1-2 m over N right ) gamma (k) ,  L (3.4)

langle m+1,m+1|H-E_{phi}|m,m rangle =d(m+1) ( 1-2 m over N right ) gamma (k) .  L

We need to diagonalize H-E_{phi_}in this subspace. First we neglect the terms of order m/N. We will show that their contribution to the ground state energy vanishes in the limit N -> inf. The eigenvalue problem (3.6) langle n,n|(H-E_{phi}) F_{k}vec | phi rangle =E langle n,n|F_{k}vec | phi rangle   (3.5)

reduces to the following recursion relation:

2dnC_{n}+d gamma (k)(n+1)C_{n+1}+d gamma (k)nC_{n-1}=EC_{n .  

}

It can be verified that the normalized solution of Eq. (3.7) is (3.8) C_{n}=C_{0}D (k)^{n ,  }(3.7)

C_{0}= [1-D(k)^{2}]^{1/2  }(3.9)

with

D(k) = {-1 +- [1- gamma (k)^{2}]^{1/2}}over {gamma (k)} ,   (3.10)

E= langle phi |F_{k}vec^{dag}HF_{k}vec | phi rangle =d lcurl -1 +- [1- gamma (k)^{2}]^{1/2}rcurl   (3.11)

and we must choose the solution with the plus sign because the other leads to an unstable ground state against creation of excitations (see the discussion of the excitation spectrum below).

Using (3.1) and the approximation (2.5) and (2.6), the ground state energy is given by

E_{0}=E_{phi}+d^sum from {k vec , k_{x}>0}^lcurl -1+ [1- gamma (k)^{2}]^{1/2}rcurl .   (3.12)

Evaluation of this expression for a large enough square lattice gives E_{0}/ dN =- 0.3290_. This value is in <2% agreement with the best estimates^{6}of -0.334 +- 0.001. For the one dimensional lattice we find E_{0}/N=-0.4317_, which is within 3% of its exact value, even though we do not except the separability approximation to be accurate in 1D for arguments given in Sec. IV. For an infinite three dimensional cubic lattice we find E_{0}/dN =- 0.2986_. Our expression (3.12) is the same as that obtained with linear spin wave theory.^{5}

Next, we examine the contribution of the terms neglected in (3.3)\(en(3.5). First, let us calculate the expectation value with F_{k}vec | phi rangle_ of the neglected parts H hat_{r_}of H hat:

langle phi | F_{k^{dag}H}sub r F_{k}vec | phi rangle = mark - 3d 1 over N^sum from m^C_{m^{2}m^{2  

}

}

lineup -2d 1 over N gamma (k)^sum from m^C_{m}C_{m-1}m^{2  

}

lineup - 2d 1 over N gamma (k)^sum from m^C_{m}C_{m+1}m(m+1) ,  

and using Eqs. (3.8) and (3.9) we obtain langle phi | F_{k}vec^{dag}H_{r}F_{k}vec | phi rangle = d 1 over N langle m^{2}rangle -2d 1 over N gamma (k) D langle m rangle ,   (3.13)

(3.14b) langle m^{p}rangle ==^sum from m^|C_{m}|^{2}m^{p}= (1-D^{2})^sum from m^D^{2m}m^{p .  }(3.14a)

We find

langle m rangle = {D^{2}}over {1-D^{2}}= 1 over {2(1- gamma^{2})^{1/2}}- 1 over 2 ,  L (3.15a)

langle m^{2}rangle = {D^{2}(1+D^{2})} over {(1-D^{2})^{2}}= 1 over {2(1- gamma^{2})} - 1 over {2(1- gamma^{2})^{1/2} .  L}(3.15b)

The contribution of H_{r_}to the ground state expectation value per bond is obtained as

1 over dN^sum from {k vec , k_{x}> 0}^langle phi |F_{k}vec^{dag}H_{r}F_{k}vec | phi rangle  L

= 1 over {N^{2}^sum}from {k vec , k_{x}>0}^langle m^{2}rangle -2 1 over {N^{2}^sum}from {k vec , k_{x}>0}^gamma D langle m rangle .  R (3.16)

The sum (1/N) size -2 sum gamma D langle m rangle converges for a square lattice and therefore the second term vanishes in the limit N -> inf. For large N, the sum of (1/N) size -2 sum langle m^{2}rangle app ln (N) (i.e., logarithmically divergent with the size of the lattice). Hence, the first term vanishes in the limit N -> inf as (1/N) ln (N).

A single magnon excitation of momentum q vec can be defined as

| psi_{q}vec rangle =G_{q}vec^prod from {k vec != q vec ,k_{x}>0}^F_{k}vec | phi rangle ,   (3.17)

where F_{k}vec_ for k vec != q vec is identical to the ground state operator defined by (3.2) and G_{q}vec_ is defined as

G_{q}vec | phi rangle =^sum from m=0 to inf^B_{m}|m+1,m rangle .   (3.18)

The excitation energy e(q) in the separability approximation (2.5) and (2.6) is given by

e(q) mark == langle psi_{q}vec |H| psi_{q}vec rangle - langle psi_{0}|H| psi_{0}rangle  

lineup = langle phi | G_{q}vec^{dag}HG_{q}vec | phi rangle - langle phi |F_{q}vec^{dag}HF_{q}vec | phi rangle ,   (3.19)

because the expectation values in the separability approximation are sums over all k vec. Therefore, we need to determine G_{q}vec_ in the subspace of |m+1,m rangle. Neglecting the 1/N terms in (2.9)\(en(2.11), the eigenvalue problem

(H-E_{phi}) G_{q}vec | phi rangle =E G_{q}vec | phi rangle   (3.20)

reduces to the following recursion relation:

d(2n+1)B_{n}+d gamma (q) sqrt (n+1)(n+2) B_{n+1  L

}

lineup +d gamma (q) sqrt n(n+1) B_{n-1}=EB_{n .  R}(3.21)

It can be verified that the solution to this recursion relation is

B_{n}=B_{0}sqrt n+1 D(q)^{n ,  }(3.22)

B_{0}=1-D(q)^{2 .  }(3.23)

D(q) is the same as that found for the ground state and given by (3.10). The eigenvalue is given by

E= sqrt phi |G_{q}vec^{dag}HG_{q}vec | phi rangle =- d +- 2d [1- gamma (q)^{2}]^{1/2 ,  }(3.24)

and choosing the plus sign (3.19) takes the form

e(q)=d [1- gamma (q)^{2}]^{1/2 .  }(3.25)

We need to choose the solution with the positive sign for the stability of the ground state.

IV. THE FORM OF THE GROUND STATE WAVE FUNCTION

Next, we determine the form of the ground state wave function. The normalized states defined by (2.2) for m=n can be expressed as

|m,m rangle =^sum from n=0 to m^C_{n^{m}|n,n) ,  }(4.1)

where |n,n) are nonorthonormal states given by Eq. (2.1). We can determine C_{n^{m_}by}projecting both sides to langle m prime , m prime |. We obtain

delta_{{m,m}prime} =^sum from {n=m prime} to m^C_{n^{m}langle}m prime , m prime |n,n) .   (4.2)

For N >> n and N >> m prime we obtain

langle m prime , m prime |n,n) mark = {(n!)^{2}}over {m prime !(n-m prime )!} , n >= m prime  

lineup =0, n<m prime .   (4.3)

It can be verified that the solution to (4.2) is

C_{n^{m}=}{(-1)^{m-n}}over n! ( pile {m above n} right ) .   (4.4)

Here

( pile {m above n} right ) == m! over (m-n)!n! .  

Therefore, the ground state wave function (3.1) is given by

| psi_{0}rangle =^prod from {k vec , k_{x}>0}^ (1-D^{2})^{1/2^sum}from m=0 to inf ^sum from n=0 to m^mark D^{m}{(-1)^{m-n}}over n! ( pile {m above n} right )  L

lineup times (| sigma_{k}vec^{z}|^{2})^{n}| phi rangle .   (4.5)

The size -2 sum_{m=0^{inf}size}-2 sum_{n=0^{m_}can}be changed to size -2 sum_{n=0^{inf}size}-2 sum_{{m}>= n}^{inf_,}and by changing the summation variable m to l=m-n, we obtain

| psi_{0}rangle =^prod from {k vec , k_{x}>0}^(1-D^{2})^{1/2^sum}from n=0 to inf^mark {D^{n}}over n! (| sigma_{k}vec^{z}|^{2})^{n  L

}

lineup times sum from l=0 to inf^(-D)^{l}( pile {l+n above n} right ) | phi rangle .  

The last summation gives 1/(1+D)^{n+1}and therefore | psi_{0}rangle mark =^prod from {k vec , k_{x}> 0}^ ( 1-D over 1+D right )^{1/2^sum}from n=0 to inf^1 over n! ( D over 1+D | sigma_{k}vec^{z}|^{2}right )^{n}| phi rangle   (4.6)

lineup = LAMBDA^exp ( sum from {k vec , k_{x}>0}^ D(k) over 1+D(k) | sigma_{k}vec^{z}|^{2}right ) | phi rangle  

lineup = LAMBDA^exp ( - cf12^sum from i<j^u_{ij}sigma_{i^{z}sigma}sub j^{z}right ) | phi rangle ,   (4.7)

where

u_{ij}== 1 over N^sum from k vec^ [ ( {1+ gamma (k)} over {1- gamma (k)} right )^{1/2}-1 right ] e^{{i}k vec cdot ( R vec_{i}- R vec_{j})} .   (4.8)

Variational wave functions of similar form were introduced and studied by Hulthen^{9}and Kastelijn^{10}for one dimension and Marshall^{11}for one, two, and three dimensions. Starting from perturbation theory, similar variational studies were also performed by Bartkowski.^{12}More recently, the same form was studied by Huse and Elser^{13}using the variational Monte Carlo (VMC) approach. They took u(1)=u_{1_}and u(r)=c/r^{p}for r>1, where r=| R vec_{i}- R vec_{j}|_, and treated u_{1_,}c, and p as variational parameters. The best energy obtained in this approach^{14}is -0.3319 J for u_{1}app 0.65_, c app 0.475, and p app 0.7. Similar VMC studies were carried out by Horsch and Linden^{14}where using only u(1) as a variational parameter [and u(r >1)=0] they found -0.322 J for the ground state energy in our units. Notice that our u is not a function of the distance r between two points on the lattice but rather a function of the two components x and y of the vector R vec_{ij_.}In Fig. 1 we plot our u(x,y=0) (open circles) and compare it with the results of VMC (solid line). The form (4.7) and (4.8) has long distance behavior consistent with the existence of long wavelength spin wave excitations. From Eq (4.8) we find that

u(r -> inf )= {sqrt 2} over {pi r} .   (4.9)

This form [Eq. (4.9)] is shown by the dashed line in Fig. 1 and we see that the onset of the asymptotic form starts from essentially r=2. The question of the long range tails is well known in liquid ^{4He}where the existence of long wavelength excitations (zero sound) influence the long range behavior of the Jastrow correlation factor.^{15}In the helium case the long range behavior of the wave function does not give significant contribution to the ground state energy. However, it has important consequencies to the spectrum of elementary excitations when the same wave function is used to define the Feynman Cohen states or to construct a correlated basis.^{16}We notice that the tails of the wave function of Ref. 13 and that of Eqs. (4.7) and (4.8) are quite different. The reason for that may be that the ground state energy is not sensitive to the exact tail of the wave function. The numerical results of Ref. 14 does not seem to support the significance of the tail of the wave function for the long wavelength excitations in the system since they find that the numerically calculated structure factor and the excitation spectrum are linear at low momenta. While this work was reviewed for publication, however, we have received the Green's-function Monte Carlo work of Ref. 17, where the conclusions of the authors confirm the results of our analytical calculations. It will be interesting to perform a variational calculation using a wave function where one treats u for the first few neighbors as variational parameters and a tail proportional to that of (4.7) and (4.8) to account for low lying spin wave excitations. The proportionality constant can be found by requiring consistency of the calculation and sum rules.^{18}

V. STAGGERED MAGNETIZATION. VALIDITY OF THE SEPARABILITY APPROXIMATION

We define the z component of the staggered magnetization operator as

M hat^sub st^{z}==1 over N^sum from x,y^(-1)^{x+y}S_{R}vec^{z ,  }(5.1)

where x and y are the two components of the vector R vec in units of the lattice spacing. The expectation value of M hat^sub st^{z}withthe wave function (4.7) and (4.8) vanishes.

Let us consider the x component of the staggered magnetization

M hat^sub st^{x}==1 over N^sum from x,y^(-1)^{x+y}{S_{R}vec^{+}+ S_{R}vec^{-}}over 2 ,   (5.2)

where we have used the identity S^{x}=(S^{+}+S^{-})/2. We would like to calculate its expectation value with the ground wave function (4.7) and (4.8). Since (4.7) and (4.8) is a superposition of states with the same number of magnons in states having k vec and - k vec, let us consider the matrix elements of (5.2) with such states

langle m prime , m prime |(-1)^{x+y}S_{R^{+}|m,m}rangle = mark {(-1)^{x+y}}over {N!m!m prime !} sqrt {(N-2m)!(N-2m prime )!}  L

lineup times^sum from {lcurl R vec_{i}, r vec_{j}rcurl_{C}, lcurl R vec_{i^{sprime},}r vec_{j^{sprime}rcurl}sub C}^ mark e^{{i}k vec cdot ( R vec_{1}+ ... + R vec_{m})} e^{{-i}k vec cdot ( R vec_{1^{sprime}+}... + R vec_{{m}prime}^{sprime})} e^{{-}i k vec cdot ( r vec_{1}+ ... + r vec_{m})} e^{{i}k vec cdot ( r vec_{1^{sprime}+}... + r_{{m}prime }^{sprime})}  

lineup times langle phi | sigma_{{R}vec_{1^{sprime}}sup}z ... sigma_{{R}vec_{{m}prime}^{sprime}}sup z sigma_{{r}vec_{1^{sprime}}sup}z ... sigma_{{r}vec_{{m}prime}^{sprime}}sup z S_{R}vec^{+}sigma_{{R}vec_{1}^{z}...}sigma_{{R}vec_{m}^{z}sigma}sub {r vec_{1}^{z}...}sigma_{{r}vec_{m}^{z}|}phi rangle .   (5.3)

There are terms in which R vec is none of the lcurl R vec_{i}rcurl_ and lcurl r vec_{i}rcurl_. These matrix elements are nonzero if m=m prime and the set lcurl R vec_{i}rcurl_ is identical to any permutation of the set lcurl R vec_{i^{sprime}rcurl_}and the set lcurl r vec_{i}rcurl_ is identical to any permutation of the set lcurl r vec_{i^{sprime}rcurl_.}The expectation value (-1)^{x+y}langle phi |S_{R}vec^{+}| phi rangle = cf12_ and summing over all R vec we obtain

cf12 (1-2m/N) delta_{{m,m}prime}  

There are also nonzero terms in which one of the lcurl R vec_{i}rcurl_ and one of the lcurl R vec_{i^{sprime}rcurl_}are equal to R vec. In this case we obtain a nonzero contribution if m=m prime and the remaining m-1 elements of lcurl R vec_{i}rcurl_ are identical to any permutation of the remaining m-1 elements of lcurl R vec_{i^{sprime}rcurl_}and the lcurl r vec_{i}rcurl_ are identical to any permutation of lcurl r vec_{i^{sprime}rcurl_.}The expectation value langle phi |(-1)^{x+y}sigma_{R}vec S_{R}vec^{+}sigma_{R}vec | phi rangle =- cf12_, hence summing over all R vec we obtain

- 1 over 2 m over N delta_{{m,m}prime} .  

We obtain the same contribution if one of the lcurl r vec_{i}rcurl_ and one of the lcurl r vec_{i^{sprime}rcurl_}are the same with R vec. The operator (-1)^{x+y}S_{R^{-_}has}the same matrix elements and therefore we conclude that

langle m^{prime}, m prime | M hat_{x}| m,m rangle = cf12 ( 1- 4m over N right ) delta_{{m,m}prime} .   (5.4)

Hence, the ground state expectation value of the operator M hat_{x_}is given by

langle psi_{0}| M hat_{x}| psi_{0}rangle = cf12 - 1 over N ^sum from {k vec , k_{x}>0}^( 1 over {[1- gamma (k)^{2}]^{1/2}}-1 right ) .  

Evaluation of this expression for a square lattice and sufficiently large N gives langle psi_{0}| M hat_{x}| psi_{0}rangle =0.303_ which is app 61% of its classical value. In 1D the correction diverges, and in 3D we obtain 0.422, which is closer to its classical value. The expectation value of the y component of the staggered magnetization is zero. Note that the expression (5.5) is identical to that obtained in spin wave theory.^{5}

Next, we check the criterion for the validity of our separability approximation. The average number of virtually excited magnons in the interacting ground state for an infinite square lattice is small as compared to the number of sites. We find that their ratio is [see Eq. (3.15a)] (5.6) R= 1 over N^sum from {k vec , k_{x}> 0}^( 1 over {[1- gamma (k)^{2}]^{1/2}}-1 right ) =0.197 ,   (5.5)

which is a rather small number. In 1D this integral diverges and therefore the separability approximation can not be justified. Our estimate, however, for the ground state energy per bond for 1D is within 3% of its exact value. For an infinite 3D lattice the ratio R=0.078. Therefore, we expect the corrections due to our separability approximation in 3D to be even smaller than those in two dimensions.

In analogy with other interacting Bose systems m_{k}vec_, given by (3.15a), is the momentum distribution of the spin degrees of freedom in the interacting ground state. We demonstrate this as follows. We define the following operators:

a_{k}vec^{dag}|m,n rangle_{k}vec == sqrt (m+1) |m+1,n rangle_{k}vec ,   (5.7)

a_{k}vec |m,n rangle_{k}vec == sqrt m |m-1,n rangle_{k}vec ,   (5.8)

a_{{-}k vec}^{dag}|m,n rangle_{k}vec == sqrt (n+1) |m,n+1 rangle_{k}vec ,   (5.9)

a_{{-}k vec} |m,n rangle_{k}vec == sqrt n |m,n-1 rangle_{k}vec .   (5.10)

It is straightforward to show that these operators satisfy

[a_{k}vec , a_{k}vec^{dag}] = [a_{{-}k vec} , a_{{-}k vec}^{dag}] =1 ,   (5.11)

[a_{k}vec^{dag}, a_{{-}k vec}^{dag}]=[a_{k}vec , a_{{-}k vec} ]=0 .   (5.12)

Neglecting the terms of order (m+m)/N the Hamiltonian (2.9)\(en(2.11) is given by

H_{k}vec =d(a_{k}vec^{dag}a_{k}vec + a_{{-}k vec}^{dag}a_{{-}k vec} ) +d gamma (k)(a_{k}vec a_{{-}k vec} +a_{k}vec^{dag}a_{{-}k vec}^{dag}) .  L

This Hamiltonian can be exactly diagonalized by means of a canonical transformation: A_{k}vec == lambda (k) a_{k}vec - mu (k) a_{{-}k vec}^{dag} ,   (5.13)

A_{k}vec^{dag}== lambda (k) a_{k}vec^{dag}- mu (k)a_{{-}k vec} ,  N (5.14)

with A_{k}vec^{dag_}and A_{k}vec_ satisfying

[A_{k}vec , A_{k}vec^{dag}] =[A_{{-}k vec} , A_{{-}k vec}^{dag}] =1 ,   (5.15)

[A_{k}vec^{dag}, A_{{-}k vec}^{dag}] =[A_{k}vec , A_{{-}k vec} ] =0   (5.16)

by choosing

lambda (k)= 1 over {[1-D(k)^{2}]^{1/2} ,  }(5.17)

mu (k) = D(k) over {[1-D(k)^{2}]^{1/2} .  }(5.18)

The full Hamiltonian (1.2) in the separability approximation takes the following diagonal form:

H= mark E_{phi}+^sum from {k vec , k_{x}> 0}^[-d+e(k)]  

lineup +^sum from {k vec , k_{x}>0}^e(k)(A_{k}vec^{dag}A_{k}vec + A_{{-}k vec}^{dag}A_{{-}k vec} ) ,   (5.19)

where e(k) is given by (3.25). Therefore, these results are the same as those obtained in Sec. III.

Hence the momentum distribution of ``bare'' magnons is

langle psi_{0}| a_{k}vec^{dag}a_{k}vec | psi_{0}rangle = m_{k}vec .   (5.20)

The ``condensate fraction,'' i.e., fraction of degrees of freedom occupying the zero momentum state for the square lattice, is

m_{0}=1 - 1 over N^sum from {k vec != 0}^m_{k}=0.803 .   (5.21)

Namely, there is a significant fraction of degrees of freedom at k vec =0, even in the interacting ground state. We can conclude that the spin cf12 Heisenberg antiferromagnet relative to helium is not as strongly interacting a system. In the former the ``condensate'' fraction is app 80%, whereas in helium the strong interactions leave only app 9% of the atoms in the condensate.^{19}

It will be interesting to attempt extending this approach to the case of the effective Hamiltonian obtained from the Hubbard model in the strong coupling limit.^{20}Below half filling and at half filling, this Hamiltonian operates in a subspace of the Hilbert space having states with singly occupied sites. At half filling, it is equivalent to the Hamiltonian (1.1). The next obvious step is to study the case of one, two, or more holes with this Hamiltonian.

ACKNOWLEDGMENTS

This work was supported in part by the Center for Materials Research and Technology of The Florida State University (FSU) by the U.S. Defense Advanced Research Projects Agency (DARPA) sponsored Florida Initiative in Advanced Microelectronics and Materials under Contract No. MDA972 88-J-1006 and by the Supercomputer Computations Research Institute of FSU which is partially funded by U.S. Department of Energy under Contract No. DE FC05 85ER 250000.

APPENDIX

In this appendix we calculate the matrix elements of the Hamiltonian with the set of states (2.2). First we consider the z term

langle m prime , n prime | sigma_{R}vec^{z}sigma_{{R}vec prime}^{z}|m,n rangle = mark 1 over N! ( {[N-(m+n)]![N-(m prime + n prime )]!} over {m!m prime !n!n prime !} right )^{1/2  

}

lineup times sum from {lcurl R vec_{i}, r vec_{j}rcurl_{C}, lcurl R vec_{i^{sprime},}r vec_{j^{sprime}rcurl}sub C}^mark e^{{i}k vec cdot ( R vec_{1}+ ... + R vec_{m})} e^{{-i}k vec cdot ( R vec_{1^{sprime}+}... + R vec_{{m}prime}^{sprime})} e^{{-i}k vec cdot ( r vec_{1}+ ... + r vec_{n})} e^{{i}k vec cdot ( r vec_{1^{sprime}+}... + r vec_{{n}prime}^{sprime})}  

lineup times langle phi | sigma_{{R}vec_{1^{sprime}}sup}z ... sigma_{{R}vec_{{m}prime}^{sprime}}sup z sigma_{{r}vec_{1^{sprime}}sup}z ... sigma_{{r}vec_{{n}prime}^{sprime}}sup z sigma_{R}vec^{z}sigma_{{R}vec prime}^{z}sigma_{{R}vec_{1}^{z}...}sigma_{{R}vec_{m}^{z}sigma}sub {r vec_{1}^{z}...}sigma_{{r}vec_{n}^{z}|}phi rangle .   (A1)

(1) We obtain nonzero diagonal matrix elements with m prime =m, n prime =n, in the following cases: (a) R vec is one of lcurl R vec_{i}rcurl_ and R vec prime one of lcurl R vec_{i^{sprime}rcurl_,}(b) R vec is one of lcurl R vec_{i^{sprime}rcurl_}and R vec prime one of lcurl R vec_{i}rcurl_, (c) R vec is one of lcurl r vec_{i}rcurl_ and R vec prime one of lcurl r vec_{i^{sprime}rcurl_,}and (d) R vec is one of lcurl r vec_{i^{sprime}rcurl_}and R vec prime one of lcurl r vec_{i}rcurl_. In the first case there are m^{2}terms because R vec can be any one of lcurl R vec_{i}rcurl_ and R vec prime any one of lcurl R vec_{i^{sprime}rcurl_.}Let us consider for example the first term where R vec = R vec_{1_}and R vec prime = R vec_{1^{sprime_.}In}order to obtain nonzero contribution the sigma's must occur in pairs and hence the sites R vec_{2},..., R vec_{m_}must be identical to any permutation of R vec_{2^{sprime},...,}R vec_{m^{sprime_}and}the sites r vec_{1},..., r vec_{n_}must be identical to any permutation of r vec_{1^{sprime},...,}r vec_{n^{sprime_.}There}are (m-1)!n! such permutation. The summation over all different R vec_{2},..., R vec_{m_}and r vec_{1},..., r vec_{n_}gives a factor of (N-2)(N-3) ... [N-(m+n)]. The contribution of the first case to (A1) is obtained as

delta_{{m}prime , m} delta_{{n}prime , n} m[N-(m+n)] over N(N-1) e^{{i}k vec cdot ( R vec - R vec prime )} .  

In the limit N >> 1 we obtain

delta_{{m}prime , m} delta_{{n}prime , n} m over N ( 1- m+n over N right ) e^{{i}k vec cdot ( R vec - R vec prime )} .  

The contribution of the other three cases can be obtained in a similar way. Summing up the contribution of the above four cases, we obtain

langle m , n | sigma_{R}vec^{z}sigma_{{R}vec prime}^{z}|m,n rangle = mark (m+n) over N ( 1- m+n over N right )  

lineup times (e^{{i}k vec cdot ( R vec - R vec prime )} +e^{{-}i k vec ( R vec - R vec prime )} ) .  

(2) We can also have nonzero off diagonal matrix elements with m prime =m-1, n prime =n-1, in the following cases: (a) R vec is one of lcurl R vec_{i}rcurl_ and R vec prime one of lcurl r vec_{i}rcurl_ and (b) R vec is one of lcurl r vec_{i}rcurl_ and R vec prime one of lcurl R vec_{i}rcurl_. It can be easily verified that the contribution of both cases for N >> 1 is .EQL langle m-1,n-1 | sigma_{R}vec^{z}sigma_{{R}vec prime}^{z}|m, n rangle = mark {sqrt mn} over N ( 1- m+n over N right )   (A2)

lineup times ( e^{{i}k vec cdot ( R vec - R vec prime )} + e^{{-i}k vec cdot ( R vec - R vec prime )} ) .  

(3) We can also have off diagonal matrix elements with m prime =m+1, m prime =n+1 when (a) R vec is one of lcurl R vec_{i^{sprime}rcurl_}and R vec prime one of lcurl r vec_{i^{sprime}rcurl_}and (b) R vec is one of lcurl r vec_{i^{sprime}rcurl_}and R vec prime one of lcurl R vec_{i^{sprime}rcurl_.}In these cases we obtain L langle m+1,n+1| sigma_{R}vec^{z}sigma_{{R}vec prime}^{z}|m,n rangle   (A3)

= mark {sqrt (m+1)(n+1)} over N ( 1- m+n over N right )  

lineup times ( e^{{i}k vec cdot ( R vec - R vec prime )} + e^{{-i}k vec cdot ( R vec - R vec prime )} ) .   (A4)

Summing over all nearest neighbors R vec, R vec prime we find

langle m,n|H_{1}|m,n rangle = d over 2 (m+n) ( 1 - m+n over N right ) gamma (k) ,  L (A5)

langle m-1,n-1|H_{1}|m,n rangle = d over 2 gamma (k) sqrt mn ( 1- m+n over n right ) ,  L

.EQL langle m+1,n+1|H_{1}|m,n rangle = mark d over 2 gamma (k) sqrt (m+1)(n+1)   (A6)

lineup times ( 1- m+n over n right ) ,   (A7)

where

gamma (k)= 1 over d^sum from {mu =1} to d^cos (k_{mu}) .   (A8)

The second term of the Hamiltonian (1.2) is

langle m prime , n prime | cf12 (S_{R}vec^{+}S_{{R}vec prime}^{-}+S_{R}vec^{-}S_{{R}vec prime}^{+}) | m,n rangle = mark 1 over N! ( {[N-(m+n)]![N-(m prime + n prime )]!} over {m!m prime !n!n prime !} right )^{1/2  L

}

lineup times^sum from {lcurl R vec_{i}, r vec_{j}rcurl_{C}, lcurl R vec_{i^{sprime},}r vec_{j^{sprime}rcurl}sub C}^ mark e^{{i}k vec cdot ( R vec_{1}+ ... + R vec_{m})} e^{{-i}k vec cdot ( R vec_{1^{sprime}+}... + R vec_{{m}prime}^{sprime})} e^{{-i}k vec cdot ( r vec_{1}+ ... + r vec_{n})} e^{{i}k vec cdot ( r vec_{1^{sprime}+}... + r vec_{{n}prime}^{sprime})}  

lineup times langle mark phi | sigma_{{R}vec_{1^{sprime}}sup}z ... sigma_{{R}vec_{{m}prime}^{sprime}}sup z sigma_{{r}vec_{1^{sprime}}sup}z ... sigma_{{r}vec_{{n}prime}^{sprime}}sup z cf12 (S_{R}vec^{+}S_{{R}vec prime}^{-}+S_{R}vec^{-}S_{{R}vec prime}^{+})  

lineup times sigma_{{R}vec_{1}^{z}...}sigma_{{R}vec_{m}^{z}sigma}sub {r vec_{1}^{z}...}sigma_{{r}vec_{n}^{z}|}phi rangle .   (A9)

(1) We can have nonzero diagonal contributions with m prime =m and n prime =n in one of the following cases. (a) None of R vec and R vec prime is one of the lcurl R_{i}rcurl , lcurl R_{i^{sprime}rcurl}, lcurl r_{i}rcurl , lcurl r_{i^{sprime}rcurl_.}In this case after summing over R vec and R vec prime, we obtain

- case 1 over 4 dN ( 1- m+n over N right )^{2 .  

}

(b) One of the R vec , R vec prime is a member of the sets lcurl R vec_{i}rcurl_,lcurl r vec_{i}rcurl_,lcurl R vec_{i^{sprime}rcurl}, lcurl r vec_{i}prime rcurl_, and the other is not. In this case summing over all R vec and R vec prime we obtain

d(m+n) over 2 ( 1- m+n over N right ) .  

(c) Also the following four cases give diagonal elements: (i) R vec is one of lcurl R vec_{i}rcurl_ and R vec prime one of lcurl R vec_{i^{sprime}rcurl_.}(ii) R vec is one of lcurl r vec_{i}rcurl_ and R vec prime one of lcurl r vec_{i^{sprime}rcurl_.}(iii) R vec is one of lcurl R vec_{i^{sprime}rcurl_}and R vec prime one of lcurl r vec_{i}rcurl_. (iv) R vec is one of lcurl r vec_{i^{sprime}rcurl_}and R vec prime one of lcurl r vec_{i}rcurl_.

Summing over all R vec and R vec prime the contribution of the above four cases combined is

- d over 2 gamma (k)(m+n) ( 1- m+n over N right ) .  

(2) We obtain off diagonal elements having m prime =m-1 and n prime =n-1 in the following cases. (a) R vec is one of lcurl R vec_{i}rcurl_ and R vec prime one of lcurl r vec_{i}rcurl_. (b) R vec is one of lcurl r vec_{i}rcurl_ and R vec prime one of lcurl R vec_{i}rcurl_. Summing over all R vec , R vec prime we obtain

d over 2 gamma (k) ( 1- m+n over N right ) sqrt mm delta_{{m}prime , m-1} delta_{{n}prime , n-1} .  

(3) We obtain nonzero off diagonal matrix elements having m prime =m+1 and n prime =n+1 when (a) R vec is one of lcurl R vec_{i^{sprime}rcurl_}and R vec prime one of lcurl r vec_{i^{sprime}rcurl_,}(b) R vec is one of lcurl r vec_{i}rcurl_ and R vec prime one of lcurl R vec_{i^{sprime}rcurl_.}We obtain

d over 2 gamma (k) ( 1- m+n over N right ) sqrt (m+1)(n+1) delta_{{m}prime , m +1} delta_{{n}prime , n+1} .  L

Collecting all the terms contributing to the matrix elements of the second terms of (1.2), we obtain

langle m,n|H_{2}|m,n rangle = mark - ( 1- m+n over N right )^{2  L

}

lineup + d(m+n) over 2 [1- gamma (k)] ( 1- m+n over N right ) ,  L

.EQL langle m-1,n-1|H_{2}|m,n rangle = d over 2 sqrt mn ( 1- m+n over N right ) gamma (k) ,   (A10)

.EQL langle m+1,n+1|H_{2}|m,n rangle = mark d over 2 sqrt (m-1)(n+1)   (A11)

lineup times ( 1 - m+n over N right ) gamma (k) .   (A12)

Adding the terms (A5)\(en(A7) and (A10)\(en(A12) we obtain

langle m,n|H-E_{phi}| m,n rangle =d(m+n) ( 1- 3 over 4 m+n over N right ) ,  

langle m-1,n-1|H|m,n rangle = d sqrt mn ( 1- m+n over N right ) gamma (k) ,   (A13)

langle m+1,n+1|H|m,n rangle = mark d sqrt (m+1)(n+1)   (A14)

lineup times ( 1- m+n over N right ) gamma (k) ,  

(A15) where E_{phi}== langle phi |H| phi rangle_ is given by Eq. (2.8).

FIGURE AND TABLE CAPTIONS




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