Abstract

We derive an explicit matrix representation for the Hamiltonian of the Ising model in mutually orthogonal external magnetic fields, using as basis the eigenstates of a system of non-interacting spin 1/2 particles in external magnetic fields. We subsequently apply our results to obtain an analytical expression for the ground state energy per spin, to the fourth order in the exchange integral, for the Ising model in perpendicular external fields.

1. Introduction

Field-induced effects in low-dimensional quantum spin systems have been studied for a long time ^{1, 2}. Hamiltonian models incorporating external magnetic fields are gaining popularity among experimentalists as well as theoreticians (see references ^{3, 4, 5, 6}). A longitudinal field is often introduced mainly to facilitate the calculation of order parameter and associated susceptibility as can be seen for example in references ^{7, 8, 9}, and a transverse field to introduce quantum fluctuations ^{10, 11}.

Our main objective in this paper is to give an explicit matrix representation for the Hamiltonian of a system of spin-1/2 particles on a cyclic one dimensional lattice chain, interacting via nearest neighbour exchange, in the presence of transverse and longitudinal external magnetic fields.

The Hamiltonian is

(1)

where and are the uniform external transverse magnetic fields, is the uniform longitudinal field, is the nearest neighbour exchange interaction, are the usual spin- operators and the fields , and are measured in units where the splitting factor and Bohr magneton are equal to unity. Periodic boundary condition is assumed so that , and so on. The parameters , , and are all assumed to be non-negative.

It is convenient to write , where

and

describes a system of non-interacting spin particles in mutually orthogonal external magnetic fields.

The model (1) has been widely studied for various combinations of the parameters and especially for phase transitions (see ^{3, 5, 12} and the references therein). Our aim is to give an explicit matrix representation for using the eigenstates of as basis.

Throughout this paper we will make use of the following identities which hold for

(2)

2. Quantization of a System of Non-interacting Spin 1/2 Particles in External Magnetic Fields

A system of non-interacting spin particles in mutually orthogonal external magnetic fields , and is described by the Kronecker sum Hamiltonian

where, for , each single particle Hamiltonian , at the site, has the matrix elements, in unit of ,

with respect to the eigenstates of the spin operator , whose elements, in unit of , are

The remaining two spin operators and have matrix elements given by

Parameters , and are the external magnetic fields and .

Explicitly,

Solving the eigenvalue equation , the normalized eigenstates , , are found to be

with corresponding eigenvalues

(3)

where

(4)

Note that

The diagonalizing matrix has elements for Thus, is similar to the diagonal matrix having elements , that is

With respect to the new basis, , and for the Pauli spin matrices have the representation

and

Since is a Hermitian operator that lives in a dimensional Hilbert space, its eigenstates form a complete orthonormal basis, suitable for giving matrix representations for operators living in and with the same conditions at the boundary. The eigenvalue equation for is

For each the eigenstate is a direct product of the eigenstates of while the eigenvalue is the sum of the respective eigenvalues , that is

where

where the floor of is the smallest integer not greater than Thus each state is uniquely represented by a binary vector .

Thus, any operator in has the matrix representation with elements given by

Using (3) we get

(5)

Note that counts the number of states in the direct product state . The degeneracy of the state is therefore Thus only the ground state and the most excited state are non-degenerate.

3 Quantization of the One Dimensional Spin 1/2 Ising Model in External Magnetic Fields

Explicit matrix representation

Since is diagonal in the basis , the only task is to find the matrix elements of and then add them to those of . We have

(6)

where and where we have introduced an dimensional vector whose components are symmetric binary matrices defined by

(7)

Thus if either the two vectors and are one and the same vector, that is , or they differ only at the consecutive and entries, otherwise .

Note that

(8)

where we have introduced another dimensional vector whose components are symmetric binary matrices with elements given by

(9)

Thus if either the two vectors and are one and the same vector, , or they differ only at the component, otherwise .

Motivated by the definitions in (7), (8) and (9) we introduce two more dimensional vectors, and whose components are symmetric binary matrices, in terms of which the and matrices may also be expressed. The and matrices are defined through their elements by

It is straightforward to verify the following properties for the and matrices:

(10)

where

is the all-ones matrix. The and matrices are singular and have trace equal to . The eigenvalues of are repeated twice and repeated times while those of are repeated four times and repeated times. Finally using multinomial expansion theorem and (10), it is readily established that the matrices and satisfy

It is now obvious that

(11)

(12)

From (11) and (12) we find

and

Explicitly

and

From the definitions of the and matrices the following additional properties are evident:

1. , , for .

2. The eigenvalues of are and , each repeated times while those of are , repeated times, and , repeated times.

3. The and matrices are singular and have trace .

Returning to (6) and substituting for the matrix elements we find, after some algebra,

where, (for ),

and

where

Explicitly,

and

Putting the results together we finally have the matrix elements for the Ising interaction Hamiltonian, , to be explicitly given by

where

Since we therefore have that the matrix elements of the Ising model in mutually orthogonal external magnetic fields are given by

with as defined above.

Defining

we have

(13)

and

(14)

where

4. Example Application: Ground State Energy of Weakly Interacting Spin 1/2 Particles in External Magnetic Fields

When the exchange integral is small, the Ising interaction term can be treated as a perturbation of . In this section, we employ (13) to find corrections, up to the fourth order in , to the energy of the ground state of weakly interacting spin particles in mutually orthogonal external magnetic fields. Since the ground state of , the unperturbed system, is non-degenerate, we will apply the non-degenerate Rayleigh-Schrödinger perturbation theory.

The following particular cases of (13) will often be useful.

(15)

In particular,

(16)

For

(17)

where

In particular,

(18)

Note also from (5) that

(19)

The first order correction to the energy of the ground state of is the expectation value of the perturbation in the ground state of .

Thus, quoting (16), we have

(20)

The second order correction to the energy of the ground state of is given by

According to (18),

We therefore see that contributions to come only from states with either (corresponding to ) or (corresponding to in the case when the two states of the direct product state are consecutive). A typical state with is the state

while a particular state with (and ) is the state

Therefore

and since there are vectors with and N vectors with , and using (19), we obtain

(21)

The results (20) and (21) were also obtained in ¹³.

The third order correction to the energy of the ground state of is obtainable from the formula

where

Note that in the above derivation we made use of the following summation identity

Evaluation of

• Contribution from states with ()

The contribution of the states with to the sum is therefore

• Contribution from states with (provided that )

The states with therefore contribute

to .

Putting these results together we have

(22)

Evaluation of

In each term of the sum, one of four different scenarios is possible, namely, or or or We look at each possible situation in turn.

• Contribution to when

In this case, for each vector, there are two possible vectors for which the matrix element does not vanish, as typified below:

In such a situation,

We also have

Since there are states, the contribution to the sum when is

• Contribution to when

As in the previous case, for each vector, there are only two possible vectors for which the matrix element does not vanish, as typified below:

In such a situation,

There is therefore zero contribution to when .

• Contribution to when

In this case, typical situations with an vector and the two vectors for which does not vanish are depicted below

From (17) we have

signifying a zero contribution to the sum.

• Contribution to when

Here as in the previous case we have , so that again there is zero contribution to the sum.

Adding all the contributions we have

(23)

Evaluation of

From (16), (18) and (19) we have immediately that

(24)

Finally combining (22), (23) and (24), we obtain the third order correction to the energy of the ground state of as

(25)

The fourth order correction to the energy of the ground state of is given by the standard Rayleigh-Schrödinger perturbation formula

Calculations completely analogous to those in the previous sections, but much more involved, give as

(26)

Adding the energy corrections (20), (21), (25) and (26) to the ground state energy (obtained by setting in (5)) of the non-interacting spin particles in external magnetic fields we therefore find, to the fourth order in the exhange integral, , that the energy of the ground state, of the one dimensional Ising model in mutually orthogonal external magnetic fields, for spin sites is given by

that is

where is the ground state energy per spin, and .

Since , we can also write

or, in a more compact form,

(27)

with

Note that when , then

in perfect agreement with the exact result for the ground state energy of the transverse field Ising model ¹⁰:

where is a complete elliptic integral of the second kind.

The form of (27) suggests an exact result for the ground state energy per spin of the Ising model in external magnetic fields:

where are positive rational numbers, and in particular, for .

The knowledge of allows the derivation of approximate analytic expressions for physical quantities such as the magnetization in each direction and the spin-spin correlation function for neighbouring spins.

Invoking the Hellmann-Feynman rule in (1) gives for the magnetization

and similar expressions for and , the and magnetizations.

According to (27),

so that for we obtain

(28)

Thus for ,

and for and , respectively,

and

Note that in the absence of interaction, (), .

The spin-spin correlation, , is given by

yielding

Note that in the absence of interaction, , we have while gives .

5. Conclusion

We have given an explicit matrix representation for the Hamiltonian of the Ising model in mutually orthogonal external magnetic fields, with basis the eigenstates of a system of non-interacting spin particles in external magnetic fields. We subsequently applied our results to obtain an analytical expression for the ground state energy per spin, to the fourth order in the exchange integral, for the Ising model in perpendicular external fields. Since the Hamiltonian of the non-interacting spin particles in external magnetic fields is a Hermitian operator that lives in a dimensional Hilbert space, its eigenstates form a complete orthonormal basis, suitable for giving matrix representations for any operator living in the same Hilbert space and with the same conditions at the boundary.

References