Instantons in Quantum Mechanics and the Ising Model

Let us describe the symmetry properties in the quantum mechanical system described by the action (for imaginary time):

2 a /■) (4.1)

with A The only interest of this model for us is that it provides the simplest demonstration of a phenomenon present in more complicated systems. The point we intend to examine is that this model in any finite order of perturbation theory has apparently broken symmetry, whereas in reality the symmetry is restored. To see this we expand:

(p= ± ^ - ^ X (4.2)

Expanding the action near, say, the left-hand minimum we have:

dtj^ f + (4.3)

We see that we have (for small A) almost harmonic oscillations near the bottom of the left-hand well. The left-right symmetry cp-^ —cp is broken and this will remain so in any finite order in A. At the same time we know from quantum mechanics that the ground state of this theory is described by an even (/^-function and therefore is nondegenerate, with

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DOI: 10.1201/9780203755082-4

restored symmetry. It is clear that restoration of symmetry occurs because a particle placed in the left well will (with finite probability) tunnel to the right one and back. Therefore if we wait long enough there will be equal probability of finding the particle in either of the wells.

There exists an interesting way to describe tunnelling which we briefly discuss now. It is easy to see (by rescaling q> q>) that for small X, S is very large, being of the order A" ^ That means that in the functional integral:

Z = ^ (pe-S[<pl (4.4)

we can use a saddle point approximation. It is crucial that together with the trivial minimum <p = ± the classical equations of motion for imaginary time:

Sq)(t)= (p pL^cp — Xcp^ = 0

have a solution:

(4.5)

(4.6) (io is an arbitrary constant). This solution (being a local minimum for 5[(p]) connects the left well at — oo with the right one at +oo.

Substituting (4.6) into (4.1) we find a classical action:

^cl ~ 3A (4.7)

At first glance the contribution of the trajectory (4.6) to the functional integral has the factor

g - S c i _ ^q-(2^2)h^I3X (48)

and is negligible. However this is not so. The reason is that we have not one classical solution but a continuum set of them distinguished by the value of ìq- Therefore we have to expect that the contribution to Z has the form:

-(2v/2)mV3Ajdio (4.9)

This means that for relatively short periods of time, t < e (2v/2)ii3/3A^

contribution of our trajectory is indeed irrelevant, but in the large time

limit it becomes very large. This is in complete agreement with the tunnelling interpretation of our solution, namely the characteristic time we have introduced, is just the tunnelling time and hence the time for restoration of symmetry.

Let us perform now a more quantitative analysis.

Let us expand our field near the classical solution — Iq):

<p(0 = <Pcl(t - io) + Z - t o ) (4.10) In (4.10) we have introduced functions ij/„ which are normal modes for the oscillations near They are to be found from

dt.d < p { t ) d ( p { t i ) W ii) =

<P = <Pclit)

(4.11) or

In the complete set of functions there exists i/^o for which col = 0- fis existence, being a consequence of translation invariance, is easily checked directly by differentiating equation (4.5) with respect to t.

In the expansion (4.10) we did not include i^o th® sum, introducing instead the parameter The reason for this is that while fluctuation of the C„ are small, bounded by the action S, this action does not depend on io and this degree of freedom has to be treated separately. In order to do this let us pass from the integration over cp(t) to the integration over C„ and io- The easiest way to find the corresponding Jacobian is to examine the metric in the functional space:

II S( pf = dt % (Sto)^ dt cpli

+ Z (SCf ⁺ OHStof)

^(4.12)

Therefore:

^ (pit) = A H dC„ dio A = dt

-1/2 (4.13)

and with this knowledge we can proceed to calculate the one kink contribution to the correlation function:

^^0 ^clih ~ ^o)^c\i^2 ~ ^o)

1 -h ß e

A

|d t„

^^0 (^cl (tl ~ to)(/>ci(t2 ~ to) —

(4.14)

ß = A J ---ndi„e-^<o^^

((o„ o are eigenfrequencies for the trivial minima Substituting (4.6) into (4.14) we obtain:

<<p(t.)</>(t2)> = y ( l - C e - * - | f i - f2l) C= - B

2B

C / ux u(x —1)

dx j tanh ^ \t^ — t2\ tanh---- — |ii — ^2! “ 1

V2' Ui - ^2! > ^ ^

V2

(4.15)

As was expected, for large times the kink solution (4.6) gives a large contribution. Moreover, it is clear that we have to take into account multi-kink configurations, which in the “tunnelling” language corre­

spond to trajectories travelling from left to right and back several times, for which (p(t) has the following structure:

It, t. h t r

--- --- 1^1

There is no exact classical solution of such a kind, because there is a tendency for kink and anti-kink to annihilate. The attractive force between them is easily estimated. Since the tails of kinks are exponen­

tially small:

<Pci(t)^ ± ^ ( 1 \ t \ >l (4.16)

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a mere superposition of kink and anti-kink will have the action:

Sik,k) ^ 5(fc) ^ 5(fc)

= -f (4.17)

where i i2 is the distance between k and k. This result implies that the interaction between kinks is exponentially small. Since for small A the average time distance between our objects is of the order of c~^

g2y2/i3/3A ^ ^ - 1 above-mentioned interaction can be neglected.

Another simplification, possible for the same reason, is that the width of the kink can be neglected. We can approximate the general configura­

tion by:

( p i T i O = - 7 T _yl^j=l

n

^{s g n (t - T j )}

= N

3A

(4.18) (4.19) Ti < T2 < ... < Tn

The total effect on the correlation function is given by:

= T I (-C)^e-*l>’ idT,...dT^ = ^e-^l--'^l (4.20)

A N=0 J ^

min(ii, ¹²) < < Tj < '" < Tff < max(ti, ¹²), A = C e

We have found that tunnelling trajectories (which are also called instantons) remove the degeneracy of the ground state present on a perturbative level. The symmetry —cp gets restored and the system acquires a finite, though large, correlation length As we discussed m Chapter 1, in the limit of large correlation length the quantum theory with action (4.1) must be equivalent to the ^ = 1 Ising model, t This equivalence is quite obvious from the present consider­

ation: the moment we replaced the (p-field by the step function (4.16) we actually started counting configurations of the Ising model described by the picture:

t__i__ L

T ~ T ~ 1 v ~ r _ L J

and the counting of kinks in (4.20) is exactly the counting of spin reversals in (1.18)

t This fact was realized long ago by Vaks and Larkin.

Im Dokument Gauge Fields and Strings (Seite 60-65)