Resistance Inequalities for the IsotropiC Heisenberg Model

R. T. Powers

University of Pennsylvania

Philadelphia, PA 19174

The purpose of this talk is to point out the relation between the

isotropic Heisenberg model and electical resistance. It is our contention

that the problem of whether certain systems have a first order phase tran-

sition can be determined from the resistance properties of the system. We

begin with a discussion of the Heisenberg model, followed by a discussion

of the resistance and end with showing the connection between the two.

In this talk we will confine our attention to spin ~ although all the

results stated have generalizations to higher spin. To describe the spin

at a single spin ~ particle one specifies a state ~ of a (2 x 2)-

matrix algebra N. The algebra N is spanned as a linear space by the

identi ty I and the three Pauli spin matrices.

I

(We denote the triple

by the three numbers

(1 0) (j 1

(1)

.... (ux,uy,uz ) by u. A state w of N is determined

(w(u ),w(a- ),w(u )) x y z (a ,a ,a ) ~ a. x y z From the fact

161

that w is a state and therefore positive we have lal ~ 1.

To describe the spin of two ~articles of spin ~ one specifies a

state w of Nl ® N2 where

is the span of the identity

N2 are (2 x 2)-matrix algebras. Nl

and the Pau:Li spin matrices crj = Cf ® I

and N2 is spanned by the matrices u2

= I ® cr. Tb describe the spin of

a-countably infinite number of span ~ particles one specifies a state w

of ~ = ®i€L Ni where ~ is a set labelling the particle and Ni is a

(2 x 2)-matrix algebra. ~ is the infinite tensor product of (2 x 2)-matrix

algebras and is called a UHF algebra.

* We consider a particular dynamics of this C -algebra m. A dynamics is

a strongly continuous one parameter group of *-automorphisms of ~. We are

interested in the dynamic associated with the Heisenberg model. This dynamics

is specified by giving a Hamiltonian H of ~

H = 2::( .• ) G J(ij)(I - cr •. 0'.) ~,J € ~ J

(2)

where the sum is taken over all lines (i,j) of a graph G. The graph G

is simply a set of pairs of points (or vertices) (i,j) in £. The graphs of

greatest interest to phYSicists are where £ = Zl;n and G consists of all

lines connecting nearest neighbors of £. The number J(ij) are Positive.

The aseumption J(ij) > 0 corresponds to the statement; the interaction

is ferromagnetic. The expression -0". ~

is ox' course short hand for

0". (j. • ~z JZ

We assume the numbers J(ij) satisfy the inequality

l:j€G(i) J(ij) ::: K

where K is a constant and G(i) is the set of vertices j € £ connected

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in i by a line (ij) of G. The expression (2) for the Hamiltonian

is in general not well defined but with assumption (3) H can be used

to define a *-derivation 6 given by

5(A) i !:(ij) €G J( ij)[I (4)

From (3) it follows that the above sum converges for all A € ~O' with ~o

in the *-subalgebra of ~ spanned by all polynomials in the (rr.; i € £). 1

From the rapidly developing theory of unbound *-derivations discussed in

Professor Sakai's lecture it follows that the closure of 6 in the generator

of a strongly continuous reparameter group of *-automorphisms at' We will

call at a *-automorphism group associated with the Hamiltonian H.

We will be interested in two types of states associated with the

Hamiltonian H. One type is states of finite energy. These are simply

states W of ~ satisfying

The second type is (at'~) KMS states of ~. If G is finite and ~ is

(2n x 2

n)-matrix algebra a (at)~)-KMS state of A is the state given by

(5)

( ~H W(A) = tr A e H ) (6)

tree ~ )

where tr is the trace of ~. When £ is infinite (at)~)-KMS states

use a generalization of the above notion to infinite systems. We refer to

Professor Araki's lecture for a description of KMS states.

The Heisenberg model was invented to explain, or give a model for, the

existence of magnets. In an iron crystal the spin of neighboring iron atoms

interact in accordance with Hamiltonian (1). Naturally this is simplification

of an actual iron crystal but the model is believed to be realistic enough

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to explain why magnets exist in nature. The mathematical property corres-

pending to the existence of a magnet is long range order. What we want to

prove is that if w is on (at'~) KMS state then for certain physically

reasonable Hamiltonians w has long range order, i.e. w(ai

· rf.) > £ >-0 J -

for all i,j E J:. Long range order corresponds to an alignment of spins

and physically the alignment of the spins of 10~~ atoms results in a

magnet.

Quite recently some very important work has been done on the Heisenberg

model. Frohlich, Simon and Spencer have proved the existence of long range

order for the 3 dimensional classical Heisenberg model [1]. Their argu-

ments rely.heavily on the symmetry of 3 dimensional Heisenberg model.

It is my hope that by using ideas connected with reSistance their results

can be extended to more general systems.

Next we discuss the calculation of the resistance of an electrical

network consisting of resistors. Consider a finite graph G with vertices

t. We image that for each line (i,j) E G there is resistor of J(ij)-l

ohms. Given two vertices iO,jo E ~ we wish to calculate the electrical

resistance R(io,jo)

ampere of current at

between

iO and

io and jO' We imagine injecting one

extracting one ampere of current at jo' The

current flows from iO to jo along the lines (ij) of G. The current

flow is governed by two laws. One is Ohn's law which states that the

voltage difference between two vertices i and j connected by a line

(ij) EGis given by

V(i) - V(j) J(ij) -1 I (ij)

where I(ij) is the current flowing along the line (ij): (then I(ij) =

-I(ji». The second law is the Kirchhoff law which states that the total

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current flowing into a vertice k is zero unless k = iO or k = jO'

The Kirchhoff law can be stated

where

~jEG(ij) I(ij) = 5io

(i) - 5jo (i)

o. (k) = 0 if k I iO and O. (i ) = 1. ~O ~O 0

These two equations can be combined as follows. If f is a function

defined on £ we define ~ by

(~)(i) = ~j€G(i) J(ij)(f(j) - rei»~

We say a function f is harmonic if ~ = O. If G is finite one can

easily show that the only harmonic functions are constant. To calculate

the resistance between iO,jO E £ one solves the equation

(8)

(9)

(10)

The current flow I(ij) = J(ij)(V(i) - V(j» for (i,j) E G satisfies

Ohm's law and the Kirchhoff law. The resistance R(io,jo) is given by the

voltage difference

(11.)

Since two different solutions to equation (10) differ by a constant function

equation (10) determines the resistance R(io,jO) uniquely.

For infinite graphs we define the resistance as the limit

where Rn(iO,jO) is the resistance as calculated from a finite subgraph

Gn of G and the graphs Go increase up to G as n - 00. One can show

that the limit is independent of the increasing sequence G • n

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An equivalent definition of resistance is as follows. Given G is

finite or infinite graph with vertices ! and a function f on ! define

Q(f) = ~(ij)€G J(ij)(f(i) - f(j»)2 (12)

Then the resistance R(i,j) is given by

We also give a second definition of resistance

where ~o is the set of the functions on ! with finite support. For all

finite graphs and most graphs of physical interest Rl(i,j) = R(ij). In fact,

if Rl(io,jO) < R(io,jO) for some io,jo €! then there must exist a oon­

constant bounded harmonic function on !.

Let ! =!Il;n for n = 1,2, ... and let G be the graph obtained by

connecting nearest neighbors of ! with one ohm resistors. For these graphs

Rl(ij) = R(ij) for all i,j €!. For n = 1 the resistance R(ij) = Ii - jl, the distance between i and j. For n = 2 the reSistance R(ij) grows

logarithmically with the distance between i and j. For n = 3 the resis­

tance R(ij) is bounded, in fact

We now state some resistance inequalities for the Heisenberg model.

THEOREM 1. Suppose w is a state of the Heisenberg spin algebra ~

and H is a Hamiltonian given in (2). Then

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COROLLARY. Suppose w is a state of the Heisenberg model in three

dimensions with nearest neighbor interaction of unit strength. Then if

w is a state of finite energy then for every

set S so that

for all i,j I S.

w(O'1' CT,) :> 1 - E J -

€ there is a finite

THEOREM 2. Suppose w is an (at'~)-KMS state of the Heisenberg

spin algebra ~ associated with a Hamiltonian given by equation (2). Then

This result generalizes the well known theorem of Mermin and Wagner [2].

It follows frem this theorem or was shown by Mermin and Wagner that one and

2-dimensional Heisenberg models do not have long range order since in one

and 2-dimensions R(ij) grows without bound on Ii - jl ~ 00.

We conclude with a conjecture.

CONJECTURE. There is a constant KO (independent of G) so that if

w is a.n (<\ ,~ )-KMS state of )ll then

where

w(l - cr' .. cr'.) < KO ~-l Rl'(i,j) ~ J -

inf{L .. G J(ij)w(O' .. cr.)(f(i) - f(j»2 f € /)0' f(iO) - f(jO) = l} lJE 1 J

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The paper of Frohlich, Simon and Spencer when combined with the resistance

calculations described here shows this conjecture is true for the Heisenberg

model in n-dimensions with unit nearest neighbor interaction with the

constant KO = 3/2 .

The truth of this conjecture would establish the existence of long

range order for a large class of physical models. The truth of this con­

jecture plus Theorem 2 would show that the question of long range order for

the isotropic Reisemberg model could be determined from resistance

calculations.

REFERENCES

1. Frohlicb. Simon and Spencer, preprint.

2. Mermin and Wagner, Phys. Rev. Letters 17, 1133 (1966).

3. R. Powers, Jour. of Math. Phys. 17, 1910 (l976).

4. R. Powers, Comm. of Math. Phys. 51, 151 (1976).