Kink-kink correlations in nonlinear quenches across a quantum critical point

This work investigates the universality of kink-kink correlations in one-dimensional transverse Ising models under algebraic quenches across a quantum critical point and shows that while superlinear quenches are determined exclusively by the Kibble-Zurek length, sublinear quenches require an additional dephasing length and exhibit a continuously varying compressed exponential decay in their correlation functions.

The Gist

Imagine you have a huge, perfectly synchronized marching band (the quantum system). Each musician holds a flag, and everyone is supposed to look in the same direction (the "ground state").

Now imagine you want to change the music so that the band suddenly must look in the opposite direction. This is called a "quench." If you change the music slowly and gently, the band can perfectly adjust their steps, and at the end, everyone is looking in the right direction. This is an "adiabatic" process.

But what happens if you must change the music quickly? The musicians in the middle of the field (the "critical region") become confused. They cannot react fast enough to the changing beat. As a result, some musicians turn in the wrong direction, creating "defects" or "kinks" in the row.

This work examines exactly how these confused musicians behave when the music changes in a non-linear way. Instead of accelerating at a constant rate (a linear change), the beat might initially speed up slowly and then suddenly sprint, or vice versa.

Here is a breakdown of what the researchers found using simple analogies:

1. The "Kibble-Zurek" Rulebook

Scientists have a standard rulebook called the Kibble-Zurek (KZ) mechanism. It predicts how many errors (defects) a system makes based on how quickly you change the conditions.

• The old idea: If you know how fast you change the music, you can precisely predict how many confused musicians you will have.
• The new discovery: The authors found that this rulebook is incomplete. While it predicts the number of errors reasonably well, it fails to predict how these errors are arranged relative to each other.

2. The Two "Rulers" of Confusion

To understand how the confused musicians are distributed, the researchers found that two different rulers (length scales) are needed, not just one.

• Ruler A (The KZ scale): This is the standard ruler. It gives the average distance between errors based on how quickly the music changed.
• Ruler B (The dephasing scale): This is a new, longer ruler. It accounts for a "phase difference." Imagine the musicians trying to march in step, but since they reacted at slightly different times, their internal clocks are slightly out of sync. This "out-of-sync" feeling creates a second, longer distribution pattern that the old rulebook overlooked.

3. The Shape of Confusion (The "Compressed Exponential Function")

When the researchers examined how the correlation (the relationship) between two confused points changes as you move further apart, they discovered something surprising.

• Old expectation: They thought the relationship would decay like a standard exponential curve (like a ball rolling to a stop).
• Reality: The relationship decays much faster, in the form of a "compressed exponential function." Imagine a sponge being squeezed: it retains its shape for a while, then collapses very suddenly. The speed of this collapse depends entirely on how the music tempo was changed (the "quench exponent").

4. The "Superlinear" vs. "Sublinear" Twist

The researchers tested different ways of changing the beat:

• Sublinear (Slow start, fast end): The system becomes "dephased." The musicians' internal clocks are thrown so off-kilter that they eventually lose any connection to each other. The pattern of confusion becomes random.
• Superlinear (Fast start, slow end): The system remains "coherent." The musicians' internal clocks stay sufficiently synchronized so that the long-range pattern remains visible. In this case, you only need the standard KZ ruler; the second "dephasing" ruler is unnecessary because the confusion does not scramble the pattern.

5. The "Optimal" Speed

The work also asks: "Is there a perfect speed to change the music that produces the fewest errors?"

• They found that you get more errors if you change the music too slowly at the beginning or too quickly at the end.
• There is a "Goldilocks" zone (an optimal exponent) where the number of confused musicians is minimized. Interestingly, this same "Goldilocks" speed also helps to scramble the internal clocks (dephasing) the most, allowing the system to settle into a cleaner, more stable state.

6. The "Pause" Button

Finally, they tested what happens if you hit the "pause" button in the middle of the change (holding the field constant for a while while the system is in the ferromagnetic phase).

• Result: Pausing at the right spot helps to scramble the internal clocks even more. It is like letting the confused musicians stand still for a moment; this gives them time to lose their synchronization completely, which actually helps the system transition into a more random, stable state.

Summary

In short, this work shows that the "errors" a quantum system makes when pushed too quickly through a critical point are not just random noise. They follow complex patterns that depend on how you pushed it.

• If you push it in a certain way (superlinear), the errors remain organized.
• If you push it in another way (sublinear), the errors get scrambled and randomized.
• The old rules told us only how many errors there were; this work tells us how they are arranged, revealing that the arrangement depends on a second, hidden "scrambling" scale.

Technical Summary

Technical Summary: Kink-Kink Correlations in Nonlinear Quenches Across a Quantum Critical Point

Problem Statement
The Kibble-Zurek (KZ) mechanism provides a standard framework for understanding the nonequilibrium dynamics of quantum mechanical systems quenched across a second-order phase transition in finite time. While the KZ paradigm successfully predicts local observables, such as the mean defect density, recent studies on the one-dimensional transverse-field Ising model (TFIM) with linear quenches have shown that higher-order correlation functions exhibit behavior beyond the KZ description. Specifically, these correlations do not decay exponentially, as expected from the adiabatic-impulse-adiabatic (AIA) picture; instead, they decay as Gaussian functions and depend on a second, longer length scale (the dephasing length) resulting from finite phase differences between low-energy modes.

This contribution investigates whether these findings are universal or specific to linear quenches. The authors examine the TFIM where the transverse field varies algebraically (nonlinearly) near the critical point, characterized by an exponent $\omega$. The main objective is to determine how the kink-kink correlation function behaves under such nonlinear protocols and to identify the relevant length scales governing the dynamics.

Methodology
The study focuses on the 1D TFIM on a ring, quenched from a paramagnetic phase ($g_i > 1$) into a ferromagnetic phase ($g_f = 0$). The quench protocol is defined such that the transverse field $g(t)$ varies as a power law near the critical point $g_c=1$: $|g(t) - g_c| \sim |t - t_c|^\omega$.

Since the problem of the nonlinear quench is not exactly solvable, the authors employ a multi-stage approach:

1. Adiabatic Perturbation Theory: Used to derive analytical expressions for the excitation probability and the resulting correlation functions in the limit of slow quenches ($\tau \to \infty$).
2. Analytical Arguments: Scaling arguments and stationary phase approximations are applied to evaluate the integrals determining the correlation functions.
3. Exact Numerical Integration: The relevant time-dependent differential equations (derived from the Bogoliubov-de Gennes equations for the fermionic modes) are solved numerically to verify the analytical predictions.

The authors analyze the equal-time longitudinal kink-kink correlation functions, $K(r, t)$, which are decomposed into a "dephased" part ($K_{on}$), related to the excitation probability, and an "off-diagonal" part ($K_{off}$), related to quantum interference and phase coherence.

Main Contributions and Results

1. Dephased Correlator ($K_{on}$):

   • The dephased correlator, which depends on the excitation probability $p_k = |u'_k|^2$, decays superexponentially with distance for all quench exponents $\omega > 0$.
   • Specifically, it decays as a "compressed exponential function," $f(r/\hat{\xi}) \sim \exp(-(r/\hat{\xi})^{\omega+1})$, where $\hat{\xi} \sim \tau^{\omega/(1+\omega)}$ is the KZ length scale.
   • This result confirms that the AIA approximation does not capture the correlation decay even for nonlinear quenches.

2. Off-diagonal Correlator ($K_{off}$) and Length Scales:

   • The behavior of the off-diagonal correlator depends critically on the quench exponent $\omega$.
   • Sublinear and Linear Quenches ($\omega \leq 1$): The system requires two length scales to describe the correlations: the KZ length $\hat{\xi}$ and a dephasing length $\ell$. For $\omega < 1$, $\ell \sim \tau^{1/(1+\omega)}$, and for $\omega = 1$, $\ell \sim \sqrt{\tau \ln \tau}$. In these regimes, the phase difference between modes diverges with time, allowing the system to eventually dephase. The correlator scales with the dephasing length $\ell$.
   • Superlinear Quenches ($\omega > 1$): The system does not dephase in the same way. The phase difference remains finite on the KZ scale, and the dephasing length $\ell$ becomes comparable to the KZ length ($\ell \sim \hat{\xi}$). Consequently, a single length scale (the KZ length) suffices to describe the kink-kink correlator.

3. Excitation Probability:

   • It is found that the excitation probability $p_k$ decays exponentially with a scaled variable $X \propto k \tau^{\omega/(1+\omega)}$ for most $\omega$, with a transition to power-law decay at very large $X$ for specific cases.
   • The amplitude of the oscillations in $p_k$ depends on $\omega$, exhibiting different behavior for odd integers, inverse even integers, and other values.

4. Optimal Quench Protocols:

   • The authors investigate whether an optimal quench exponent exists that minimizes the mean defect density or maximizes dephasing (randomization of phases).
   • They find that for a fixed quench time, the peak height of the off-diagonal correlator (a proxy for phase information) varies non-monotonically with $\omega$, suggesting the existence of an optimal exponent (at $\omega \sim 4-5$ for the parameters studied) where phase randomization is most effective.
   • They also discuss "hold" protocols (waiting in the ferromagnetic phase) and note that waiting times significantly enhance dephasing in sublinear quenches but have a negligible effect in superlinear quenches.

Significance and Claims
The contribution claims to establish the universality of the "two-length-scale" phenomenon observed in linear quenches and extend it to nonlinear protocols. The key findings are:

• The decay of correlation functions is not universally exponential; for nonlinear quenches, it follows a compressed exponential form with an exponent that varies continuously with the quench rate parameter $\omega$.
• The necessity of a dephasing length scale is conditional: it is required for sublinear and linear quenches ($\omega \leq 1$) but becomes redundant for superlinear quenches ($\omega > 1$), where the KZ length alone suffices.
• The results challenge the quantitative accuracy of the AIA picture for correlation functions and underscore that adiabatic perturbation theory and exact numerical methods are necessary for a complete description of nonequilibrium dynamics.

The authors conclude that while the mean defect density is well described by KZ scaling, the full correlation structure reveals richer physics dependent on the specific functional form of the quench protocol. They suggest that future work could investigate the long-time behavior of these systems (generalized Gibbs ensemble) and the behavior of entanglement entropy under nonlinear quenches.