Kibble–Zurek Scaling and Defect Production

When a control parameter crosses a continuous quantum critical point at finite speed, the diverging relaxation time prevents the state from following the instantaneous ground state arbitrarily closely. Kibble–Zurek reasoning turns that loss of adiabaticity into freeze-out scales and, with additional assumptions about defect formation, a testable excitation-density law.

Required background. The protocol language comes from quenches and relaxation, while quantum phase-transition scales supplies $\nu$ and $z$. Helpful background. Dynamic universality classes clarifies when classical dissipation changes the quantum-ramp prediction.

Freeze-out from critical slowing down

Write the dimensionless distance from criticality as $\epsilon=(\lambda-\lambda_c)/\lambda_c$. In the scaling regime,

$$\xi(\epsilon)=\xi_0\lvert\epsilon\rvert^{-\nu}, \qquad \tau(\epsilon)=\tau_0\lvert\epsilon\rvert^{-z\nu}.$$

For a linear crossing $\epsilon(t)=t/\tau_Q$, compare the intrinsic relaxation time with the time remaining to the critical point:

$$\tau(\hat\epsilon) =\left\lvert\frac{\epsilon}{\dot\epsilon}\right\rvert_{\hat t} =\lvert\hat t\rvert.$$

Solving gives

$$\hat t=(\tau_0\tau_Q^{z\nu})^{1/(1+z\nu)}, \qquad \hat\epsilon=\left(\frac{\tau_0}{\tau_Q}\right)^{1/(1+z\nu)},$$

and hence

$$\hat\xi =\xi_0\left(\frac{\tau_Q}{\tau_0}\right)^{\nu/(1+z\nu)}.$$

The “impulse” picture—adiabatic evolution outside $[-\hat t,\hat t]$ and a frozen state inside—is a mnemonic, not a literal discontinuity in the dynamics. The robust content is the scaling estimate set by the equality of timescales.

The causal-domain idea originates with Kibble 1976; its quantum-ramp implementation requires the critical relaxation law written above rather than a literal frozen interval.

If independent domains of size $\hat\xi$ choose order, a defect of dimension $D$ in $d$ spatial dimensions has density per transverse volume

$$n_{\mathrm{def}} \sim \hat\xi^{-(d-D)} \sim \tau_Q^{-(d-D)\nu/(1+z\nu)}.$$

For point defects $D=0$. The proportionality constant is nonuniversal, and the identification of an excitation with a topological defect is model dependent. In a quantum Ising chain, $d=1$, $z=\nu=1$, so the density of kinks scales as $\tau_Q^{-1/2}$. The exact Jordan–Wigner solution reduces the long-wavelength modes to Landau–Zener crossings and reproduces this exponent, while fixing a protocol-dependent prefactor.

The solvable Ising-chain crossing and its finite-rate scaling are developed by Zurek, Dorner, and Zoller 2005; Dziarmaga 2010 reviews the broader quantum-transition framework and its corrections.

What a measured defect density contains

The density recorded after a ramp need not equal the freeze-out density. Defects can annihilate, coarsen, escape the observation region, or be produced after the critical crossing. A controlled comparison therefore records the measurement delay $t_m$, uses one operational defect definition at every ramp rate, and tests whether

$$\ell_{\mathrm{obs}}(\tau_Q,t_m) \simeq \hat\xi(\tau_Q)$$

before post-ramp growth becomes dominant. Correlations or excitation energy can be cleaner observables when individual defects cannot be resolved.

The fitted interval must also lie inside the critical scaling window. Very fast ramps probe microscopic rather than universal dynamics. Very slow ramps encounter finite size, residual temperature, loss, or noise. The expected log–log curve is consequently a central power-law window bounded by two crossovers, not a power law extending to both limits.

For a system of linear size $L$, Kibble–Zurek growth saturates when $\hat\xi\sim L$. The corresponding ramp scale is

$$\tau_Q^{(L)} \sim \tau_0\left(\frac{L}{\xi_0}\right)^{(1+z\nu)/\nu}.$$

Beyond this scale a finite isolated system can become nearly adiabatic, and fitting the saturated points biases the exponent toward zero. A finite-temperature crossing adds a thermal length and relaxation law; it is not described simply by inserting the zero-temperature $z$ and $\nu$.

Nonlinear and inhomogeneous ramps

For $\epsilon(t)=\operatorname{sgn}(t)\lvert t/\tau_Q\rvert^r$ with $r>0$, the same calculation yields

$$\hat\xi \sim \xi_0 \left(\frac{\tau_Q}{r\tau_0}\right)^{r\nu/(1+r z\nu)},$$

up to a convention-dependent constant. Reporting only a “quench time” without the ramp shape is therefore insufficient.

In a trap or gradient, the critical point is crossed at different positions at different times. Let $v_F$ be the velocity of the critical front. If already ordered regions can communicate their choice ahead of the front, defect formation is suppressed relative to the homogeneous estimate. The comparison is between $v_F$ and the relevant information or sound speed near freeze-out, not between $v_F$ and an arbitrary microscopic velocity. Spatially resolved correlation functions are needed to distinguish this causal suppression from simple density averaging.

Nonlinear, inhomogeneous, and defect-specific refinements are synthesized by del Campo and Zurek 2014.

Validity checks

A convincing Kibble–Zurek test fixes the ramp variable and exponent $r$, obtains $z$ and $\nu$ independently, identifies the scaling window without using the same points to tune it, and varies size, initial temperature, and measurement delay. It reports covariance between a fitted amplitude and exponent and compares the result with at least one non-Kibble–Zurek alternative such as Landau–Zener production far from criticality or post-ramp coarsening.

Agreement of a single exponent is not enough to identify a universality class. The stronger test is joint consistency of $\hat t$, $\hat\xi$, correlations, and defect observables, including their finite-size crossover.

Exercises

1. Nonlinear crossing. Derive the freeze-out time for $\epsilon(t)=\operatorname{sgn}(t)\lvert t/\tau_Q\rvert^r$ and confirm the exponent of $\hat\xi$ quoted above.

Solution

Here $\lvert\epsilon/\dot\epsilon\rvert=\lvert t\rvert/r$. Equating it to $\tau_0\lvert t/\tau_Q\rvert^{-rz\nu}$ gives $\lvert\hat t\rvert^{1+rz\nu}=r\tau_0\tau_Q^{rz\nu}$. Substitution into $\xi_0\lvert\hat t/\tau_Q\rvert^{-r\nu}$ gives $\hat\xi\propto\tau_Q^{r\nu/(1+rz\nu)}$; constants depend on how $\tau_Q$ and $\epsilon$ are normalized.

2. Finite-size crossover. For the one-dimensional quantum Ising universality class, find the power of $L$ in $\tau_Q^{(L)}$. What happens to the kink-density fit for still slower ramps?

Solution

With $z=\nu=1$, $\tau_Q^{(L)}\sim\tau_0(L/\xi_0)^2$. For slower ramps the predicted $\hat\xi$ exceeds the sample size, so the independent-domain argument fails and the excitation probability crosses toward finite-size adiabatic behavior. Including that region in a single power-law fit produces an apparent exponent smaller in magnitude than $-1/2$.

References

• del Campo, Adolfo, and Wojciech H. Zurek. “Universality of Phase Transition Dynamics: Topological Defects from Symmetry Breaking.” International Journal of Modern Physics A 29, 1430018 (2014). DOI.
• Dziarmaga, Jacek. “Dynamics of a Quantum Phase Transition and Relaxation to a Steady State.” Advances in Physics 59, 1063–1189 (2010). DOI.
• Kibble, T. W. B. “Topology of Cosmic Domains and Strings.” Journal of Physics A: Mathematical and General 9, 1387–1398 (1976). DOI.
• Zurek, Wojciech H., Uwe Dorner, and Peter Zoller. “Dynamics of a Quantum Phase Transition.” Physical Review Letters 95, 105701 (2005). DOI.