Experiments in 4 He

The experiments in 4_{He, conducted at Lancaster, follow Zurek’s original sug-}

gestion. The idea is to expand a sample of normal fluid helium, in a container with bellows, so that it becomes superfluid at essentially constant temperature. That is, we change 1_{− T/T}c from negative to positive by reducing the pressure,

thereby increasing Tc. As the system goes into the superfluid phase a tangle of

vortices is formed, because of the random distribution of field phases. The vor- tices are detected by measuring the attenuation of second sound within the bel- lows. Second sound scatters off vortices, and its attenuation gives a good mea- sure of vortex density. A mechanical quench is slow, with τQ some tens of mil-

In the first fair agreement was found with the prediction (3.25), although it was not possible to vary τQ. However, there were potential problems with hydrody-

namic effects at the bellows, and at the capillary with which the bellows were filled. A second experiment, designed to minimize these and other problems has failed to see any vortices whatsoever.

Some care is needed. Not only is the Landau-Ginzburg effective theory more suspect for 4_{He, but its Ginzburg regime is so wide, at O(1K) that the}

transition takes place entirely within it. Thus, the crucial question is whether enough long string would persist at the time of measurement to yield a positive signal. It has been proposed[32] that in order to measure a positive signal in these circumstances the measurements would have to be made sooner, after a much faster quench, or with a higher sensitivity. The validity of this suggestion will be confirmed later on.

Short summary and further motivation

The Zurek scenario is built on a mainly causal framework where the order parameter orders on the scale of some correlation length ξr, at a rate determined

by a relaxation time τ . Close to the transition point, ξr can be considered as a

constant simply because of the large value of τ which makes the field change very slowly. The explicit formulation of the parameters involved can be derived from the appropriate theory for our system. The TDGL equation for the free energy dynamics, can verify Zurek’s considerations in a mean field approximation for the potential and a gaussian thermal noise as we will see in the next chapter.

Within a certain fixed potential, the field configuration can be associated with various energy distributions which then should tend to evolve to the lowest possible stable energy state. In this picture, ϕ follows its own intrinsic laws. In Zurek’s platform, it is clear that one talks about the dynamics of the phase transition which impinge on the order parameter behavior. One can postulate that actually, ϕ will have to balance the external with the internal driving forces

of its consequent evolution.

The experiments performed in superfluid He had mixed results which could not provide watertight evidence in favor or against the Zurek scenario. One suggestion was to consider experiments in annular configurations where one would simply be able to measure the phase difference along the annulus A, and postulate about the number of defects that could have been created within the area bounded by A. The main assumption was that the field inside the annulus itself, orders to some lowest energy configuration. As we will see next, there are various length scales relevant to the defect density created. These are mainly the result of the geometry of the annulus.

Chapter 4

Kibble-Zurek: A Field Theoretic

Approach

In thermal equilibrium[31] the behavior of simple systems experiencing a contin- uous phase transition is generic, as manifest in the utility of Landau-Ginzburg theory. The early universe it is thought of as having proceeded through a se- quence of phase transitions whose consequences are directly observable, but whose detailed dynamics is unknown. Although it is difficult to measure an order parameter as it changes, many transitions generate topological charge or topological defects which can be detected. Motivated in part by Kibble’s mecha- nism for the formation of cosmic strings in the early universe, Zurek suggested[8] that we measure the density of vortices produced during a pressure quench of liquid 4_{He into its superfluid state, as well as the variance of superflow velocity.}

The scenario, as proposed by Zurek, is very simple. It is exemplified by assuming that the dynamics of the transition can be derived from an explicitly time-dependent Landau-Ginzberg free energy of the form

F (t) = Z d3x  −~2 2m|∇ϕ| 2_{+ a(t)} |ϕ|2+1 4β|ϕ| 4  . (4.1) In (4.1) ϕ = (ϕ1+ iϕ2)/ √

2 is the complex order-parameter field, whose magni- tude determines the superfluid density. We identify a(t) as an externally driven

time-dependent chemical potential. In equilibrium at temperature T , in a mean field approximation a(T ) takes the form a(T ) = a0(Tc), where  = (T /Tc− 1)

measures the critical temperature Tc relative to T . In a pressure quench at

approximately constant T , Tc will vary with time t, and we assume that  can

be written as

(t) = 0−

t τQ

θ(t) (4.2)

for −∞ < t < τQ(1 + 0), after which (t) = −1. 0 = (T /Tcin− 1) measures

the original critical temperature Tin

c against the temperature T at which the

quench takes place, and τQ defines the quench rate. The quench begins at time

t = 0 and the transition from the normal to the superfluid phase begins at time t = 0τQ.

With ξ2

0 = ~2/2ma0 and τ0 = ~/a0 setting the fundamental distance and

time scales, the equilibrium correlation length ξ(∆t) and the relaxation time τ (∆t) diverge at the relative time ∆t = t− 0τQ = 0 as

ξ(∆t) = ξ0  ∆t τQ −1/2 , τ (∆t) = τ0  ∆t τQ −1 . (4.3)

As we approach the transition, eventually the relaxation time will be so long that the system will not be able to keep up with the temperature change. We estimate the time tZ (and the relative time ¯t = ∆tZ = tZ − 0τQ) at which

the change from equilibrium to non-equilibrium behavior occurs by identifying τ (∆tZ) with −∆tZ i.e. −∆tZ =√t0τQ. After this time it is assumed that the

relaxation time is so long that the field correlation length ¯ξ = ξ(¯t) = ξ(∆tZ) =

ξ0(τQ/τ0)1/4 is more or less frozen in until the system is again changing slowly,

at time ∆t_{≈ +∆t}Z.

The correlation length of the field can only be measured indirectly. One of Zurek’s proposals, as yet unfulfilled, is to measure the variance in the superflow in an annulus after a quench. Since superflow velocity is proportional to the

gradient of the field phase θ, a random walk in phase would suggest that the measurable (∆θ)2 along a perimeter of length L has the form

(∆θ)2 = O  L ξvar  , (4.4)

where ξvar measures the effective phase-winding length. If, as Zurek does, we

assume that ξvar ≈ ¯ξ, then (∆θ)2 is large enough to be observed.

A more accessible experiment is to measure the density of vortices at their formation. If the initial density of vortices, the defects of 4_{He, is n}

def, with

separation ξdef, then

ndef = O  1 ξ2 def  . (4.5)

Zurek makes the assumption that ξdef ≈ ¯ξ whereby

ndef = O  1 ¯ ξ2  = O  1 ξ2 0 r τQ τ0  . (4.6)

Since ξ0 also measures cold vortex thickness, τQ  τ0 corresponds to a measur-

ably large number of widely separated vortices.

4.1

Reproducing the Zurek behavior for ξ

(t).

The primary ingredient for all those predictions is the qualitative picture of the freezing in of the correlation length as seen in figure (3.1). Since all equations of motion have causality built into them we should be able to confirm the first predictions of Kibble and Zurek explicitly, as we shall now see.