Metastability in a long range one-dimensional Ising model

Volume 75A, number 5

PHYSICS LETTERS

4 February 1980

METASTABILITY IN A LONG RANGE ONE-DIMENSIONAL ISING MODEL~ R.J. McCRAW1

Physics Department, Indiana University, Bloomington, IN, USA and Physics Department, Technion, Haifa, Israel Received 21 June 1979 Revised manuscript received 5 December 1979

Decay of the metastable state of the one-dimensional Ising model with an x~ potential is investigated using instanton (critical droplet) techniques. Calculations indicating that the analytic structure of the free energy is modified by droplet— droplet interactions for 1
1. Introduction. First order phase transitions in one dimension are generally considered unrealistic because of the long range of the interactions. We find however that the x potential (1
z~f=[2/I’(a + 1)] (j3~y/2ir)hI2[2/(a where ~ =

1)IHI]0/2(c~1)exp{_(fl/7)[4/(2 a)] [2/(a l)IHI] (2—a)I(a_1)} (1) 1/kT. For a droplet with a sharp edge (no extended surface region and which we shall henceforth call —

—

—

,

—

a “zero temperature droplet”) the critical droplet picture gives F/V’~—exp{—(j3/y)[4/(2 a)] [2/(a 1)JHI](2—a)I(~~)}. —

(2)

—

Since eq. (1) is calculated for nonzero temperature the agreement between it *2 and eq. (2) implies that the surface region of the low-T droplet is small compared to the overall size of the droplet, for small IHI. Instanton techniques assume that the instantons, or droplets, are noninteracting more specifically that the action of multiinstanton solutions is the sum of single instanton contributions. To check the validity of this assumption in the presence of the long range x—a potential we consider a gas of zero temperature critical droplets —

* 1

* *2

Supported in part by the U.S.—Israel Binational Science Foundation. Lady Davis Fellow 1977—1979. Present address: Physics Department, University of Southern California, Los Angeles, CA, USA. The existence of a phase transition is proved by Dyson [1]. The dynamical factor would not affect the (dominant) exponential behavior [6].

379

Volume 75A, number 5

PHYSICS LETTERS

4 February 1980

(justified by the above arguments). A new singularity structure of the free energy, caused by the interaction of droplets, may be assumed to arise if this gas undergoes a phase transition. A van der Waals calculation suggests that this occurs for a < a0 1.2 (independently of H for small HI). For a0
J>0, n is the gamma function, and ‘y will be taken small. For large x, v(x) field the energy of the system is ~

±1which inter-

(3)

XaleXlXIdX

E=—7

=

Jx

—°.

In the presence of a magnetic

v(7Ii-fI)p~/-HE~~.

1~i
(4)

i=1

v(x) may be written as (for J = 1) 1--- lim ~ akexp(—akIxI), v(x)— a m—*~ok’1 where ak = (k/m)°~/mand 0k = k/rn. For small ‘y the free energy/spin is given by [12,131 ~

(5)

where ~ m 0 is the lowest eigenvalue of the equation m

—

E

m +

~_-~

~

—

ln cosh

(

m

E (akak)l/2 ~k + 1311) 1P

=

m~

(6)

This equation may be solved for ~m0 using instanton techniques, where (‘y/f3)~is the required large parameter. For j3 >13c = {[2/T(a)] ~k(ak/ak)}’ the negative of the potential in eq. (6)(the relevant function for instanton techniques) has two maxima, unequal in height for H ‘~ 0 and associated with the stable and metastable states. To solve for the instanton we make a low temperature approximation to the potential, giving m

m

m

(7) The zero of energy has been shifted to the metastable maximum located (forH< 0) at = (2/a~)[akuk/F(a)]1I2. By standard instanton arguments (including the assumption of noninteracting droplets [5,9—11] the imaginary part of the ground state energy is ilm ~

=

mO

+ —

1 lim

~ exp [—S(ç 2S(ç 2S(~ 1)13/7] f ~ exp [— ~ 213 L—’°” L exp [—S(~0)j3/7]f ~ exp [—~& 1)13/7] 0)13/’y]

(8)

(the right hand side appropriately analytically continued) where S(~)is the classical action of ~ (x) for the negative of the potential calculated over the interval [—L,L] and ti(x) is a small variation, ç0(x) and ~1(x) are, respectively, 380

Volume 75A, number 5

PHYSICS LETTERS

4 February 1980

the constant and single instanton solutions of 6S = 0. ç1(x) satisfies ~1(±L) ~ and at some intermediate x makes an excursion towards the stable maximum at ~ ( Explicitly, for the potential in eq. (7)

—c).

~1k(X)

=

—2~sinh(crkx0) e(~k~+

xl ~xo,

~,

2~exp(—akxo)cosh(akx) xI~x0, wherex0 is given by [2/n(a)]~~1 [(ak/uk)exp(—2ukxo)] +H 0. The term limL~{exp[—S(~1)j3/7]/ exp [—S(ç0)13/’y]} in eq. (8) gives the exponential term in eq. (1). 2S is diagonalized by means of the auxilicalculate the small oscillation contribution, the second variation ~ aryTo equation: ~1k(X)

— ~,

2~bzk(i)[~(x

—~(i;x)+ o~~~(i;x) —

—

x 0)

+

~(x +xo)]n?)(j;x) = 4X5(i)nr~(i;x),

(10)

with boundary conditions i~(i; ±L)= 0, where i = I for ~ (x), i = 0 for ~0(x)and blk(i) =

6i1 (a~a~a~ cr 1) h/2/~ [a1sinh (a1x0) exp (— a1x0)].

This equation has X0(1) <0 and X1(I) = 0 as L -÷ oo, corresponding to the usual growth and translation modes of the instanton. Their contributions to the small oscillation integral are

f

2I~(1)I13/y)dz= ±~i(~y/IX exp(z

2

(11)

0(1)I13)”

and 2LIl(d/dx)~

2 1(x)II= 2L~f2[S(~1)]’I respectively. The continuum contribution is [5,9—11] [13/(iry)]I~(1)I112 [x

(12)

,

1(1)]l/2Ifl~[X5(l)/X5(0)] Ih/2. Combining this with eqs. (11) and (12) eliminates X0(l). X1(1) is calculated by observing that eq. (10) is an uncoupled equation except at x = ±x0, for i = 1, and can be solved exactly in the uncoupled regions. This results in an explicit equation for the eigenvalues which, to lowest order in exp(—akL) gives 8

m

“1O)s(~)r(a) k~lk

(ukxo)exp(—2okL).

(13)

The product IH~[X5(1)/X5(0)]I equals [14]ID(l)/D(0)I where D(i) = det g/k(i; L) and g/~(i;x)is a solution of the Jacobi equation (eq. (10) with the right-hand side equal to zero) with initial conditionsg/~(i;—L) = 0 and k1k(~—L) = ~/k~ The evaluation of ID(l)/D(0)l involves a lengthy calculation which is simplified by the fact that only two directions contribute. We find to lowest order in exp(—okL) m =

m

2(~ akexP(_ukxO))[~—

m sinh2(akxo)exp(_2a,~i~)]/[E aksinh(akxo)exP(_akxO)]

2

.(14)

Combining these gives the small oscillation contribution to jIm dm0’ eq. (8). Taking into account the change of variable and energy shift (by which ~ mo) i~fis given through eq. (5) and for small IHI is the result quoted in eq. (1).

381

Volume 75A, number 5

PHYSICS LETTERS

4 February 1980

3. Droplet—droplet interactions. As mentioned previously instanton calculations assume that the instantons do not interact. The underlying problem in our case is one of statistical physics where instantons are identified with critical droplets. The usual justification for neglecting their interactions is that the size of a droplet and hence the energy of formation grow inversely as powers of HI. The density of droplets then decreases exponentially,p exp(—~E/kT),with decreasing IHI, and so for short range forces justifies a dilute gas approximation. For long range forces this is not necessarily true. It is possible however to test this assumption in the case of the x~ model. The interaction of two zero temperature droplets (whose use is supported by comparison of eqs. (1) and (2)) is given by ~(x) = yf~cdyf~~1cdzv(7[z y]), where lc(11) is the droplet length. For large x, this potential goes as 2x (1 / Hl)2/~a—1)x a, indicating a possible phase transition in a gns of critical droplets. This would be exl~ pected to change the singularity of the free energy. We find evidence that this does occur by assuming a van der Waals gas of critical droplets where the van der Waals parameters are determined by ~(x) and ‘c~The critical temperature is found to go as T~~~~(1/lHI)(2—a)/(a—l) so that, for small IHI, any T smaller than Tc of the original x° system is also less than T~.The specific volume of the droplet gas Udr = p~ goes as —~

—

exp{(13/’y)[4/(2

—

—

—

a)] [2/(a

—

l)lHlI~21~~}

(15)

,

for small HI and T. Whether or not this falls in the two phase region of the van der Waals curve is determined by its value relative to Vc, the specific volume at which the gas condenses. In the limit of low-T Vc

exp {(13/7)[2(22

—

l)/(a

—

l)(2

—

aX3

—

a)] [2/(a

—

l)lHl](2_~a~~1)} .

(16)

These specific volumes have the same IHI dependence. This indicates that for small 1111, the importance of droplet— droplet interactions depends on a. We find that Vdr
382