Dynamical Fermionization and Emergent Bethe Rapidity Structure in the Spatial Density of Cold quenched Lieb-Liniger gas

Using ab initio quantum Monte Carlo simulations, this study demonstrates that the long-time spatial density profile of a Lieb-Liniger gas following a geometric quench directly encodes the system's underlying Bethe rapidity distribution, thereby establishing ballistic expansion as a practical method for mapping momentum-space structures to real-space observables.

The Gist

Imagine a crowded dance floor where everyone holds hands and moves in perfect, chaotic synchrony. Now imagine the walls of the room suddenly vanish, and the dancers are free to run out into a vast, empty hall.

This is the core scenario of the paper by Sumita Datta and her colleagues. They investigated what happens when a group of tiny particles (bosons), which normally prefer to stay together, are suddenly released from a small box into a larger one.

Here is a breakdown of their discovery using simple analogies:

1. The Setup: A Sudden Release

Imagine the particles as a tightly packed crowd in a small room with hard walls (a "box"). They press against each other because they repel one another.

• The Quench: At a specific moment, the walls of the small room disappear, and the particles are allowed to run into a much larger space. This is called a "geometric quench."
• The Goal: The researchers wanted to see how the crowd spreads out over time and whether the way it spreads reveals something about how it moved before the walls vanished.

2. The Big Discovery: The "Shadow" of the Past

Normally, when you watch a crowd run out of a room, you only see it spreading further and becoming less dense. One might think that the details of their original motion are lost.

However, the researchers found something surprising. If you look at the crowd not at where it is, but at how fast it is moving (which they calculate by dividing the distance traveled by the time elapsed), a hidden pattern emerges.

• The Analogy: Imagine taking a photo of a sprinter at the starting line and another as they cross the finish line. Looking at the photo at the finish line, you cannot tell how fast they started. But if you examine the pattern of their motion in relation to time, you can actually reconstruct their starting speed.
• The Result: The paper shows that the "shape" of the crowd in this "velocity view" remains constant once they have run for a while. This stable shape is a direct map of the hidden "momenta" or "velocity distributions" the particles had when they were trapped.

3. The "Fermi" Transformation

Here comes the most magical part. These particles are bosons (a type of particle that normally likes to clump together, like a choir singing the same note). However, if they are compressed strongly enough to repel each other intensely and then released, they begin to behave like fermions (particles that hate being in the same place, like people who refuse to stand next to each other).

• The Metaphor: It is like a group of shy people who, when forced to run in panic, suddenly act like a row of disciplined soldiers, refusing to touch one another.
• The Paper's Claim: The researchers call this "Dynamical Fermionization." They found that the crowd in the "velocity view" (momentum space) looks exactly like a group of non-interacting fermions, even though they are still bosons.

4. The Secret Code: Bethe Rapidities

In the world of quantum physics, there is a complex mathematical code called "Bethe rapidities" that describes the hidden velocities of these particles. For a long time, scientists could only calculate this code on paper or in very specific, simple limiting cases.

• The Breakthrough: This paper claims that by observing how the particles spread in real space (the large room), one can "read" this secret code. The shape of the spreading cloud is a direct translation of these hidden mathematical numbers.
• The Analogy: It is as if you could look at the waves on a pond after a stone has fallen in and immediately recognize the exact shape of the stone, without ever having seen the stone itself.

5. How They Did It

They did not just guess; they used a powerful computer method called "Quantum Monte Carlo."

• The Method: Imagine simulating millions of random "walks" for the particles to see which paths are most probable. By running these simulations, they tracked the density of the particles over time.
• The Finding: They tested two scenarios:
  1. Moderate Repulsion: The particles spread out, and the "velocity pattern" slowly settled into a stable shape.
  2. Strong Repulsion: The particles repelled each other very strongly. In this case, they almost immediately settled into the stable "velocity pattern," and the pattern looked very similar to the "soldier-like" fermion behavior.

Summary

The paper shows that when a quantum gas is suddenly released from a trap, it does not simply scatter randomly. It spreads in a highly organized, "ballistic" manner. If you view this spreading through the lens of "velocity" rather than "position," you can see a frozen, stable pattern that serves as a fingerprint of the particles' hidden quantum velocities.

This proves that the chaotic motion of the particles actually encodes a deep, mathematical order (the Bethe rapidities) that can be observed in the real world, effectively transforming a complex quantum puzzle into a visible, measurable form.

Technical Summary

Technical Conclusion: Dynamic Fermionization and Emergent Bethe-Rapidities Structure in the Spatial Density of a Cold, Quenched Lieb-Liniger Gas

Problem Statement
The contribution addresses the challenge of accessing the underlying momentum (rapidity) distribution of an integrable one-dimensional quantum system via observables in real space. Specifically, it investigates the non-equilibrium dynamics of a Lieb-Liniger (LL) Bose gas following a geometric quench. The system is prepared in the interacting ground state of a box of length $L_0$ and suddenly expanded into a larger box of length $L > L_0$ at fixed interaction strength. The central question is whether the time-evolving spatial density in this ballistic expansion regime encodes the Bethe-rapidity structure and exhibits "Dynamic Fermionization" (DF), a phenomenon where strongly interacting bosons display fermionic expansion characteristics.

Methodology
The authors apply an ab-initio Quantum Monte Carlo (QMC) approach based on the generalized Feynman-Kac path integral representation.

• Model: A one-dimensional system of $N$ bosons interacting via a repulsive contact potential ($g > 0$) and confined by hard walls. The initial Hamiltonian describes the gas in a box of length $L_0$, which is instantaneously quenched into a box of length $L$.
• Numerical Procedure: The ground state and subsequent time evolution are computed via stochastic sampling of trajectories. The method utilizes a generalized Feynman-Kac formulation, where the wavefunction is represented as an expectation value over diffusion processes with a drift term derived from a trial function ($\phi_T$).
• Parameters: Simulations were performed for $N=100$ particles. Two interaction regimes were analyzed: intermediate coupling ($g=50$) and strong coupling ($g=162$), corresponding to the Tonks-Girardeau (TG) limit. Units are set such that $\hbar = m = 1$. The initial box length is $L_0 = 0.25$ and the final length is $L = 1.0$.
• Analysis: The authors calculate the many-body spatial density $\rho(x, t)$ and analyze its evolution both in real space and in the velocity scaling variable $v = x/t$.

Main Contributions and Results

1. Ballistic Expansion and Scaling: The study shows that the expansion following the quench is ballistic. When the spatial density is plotted against the velocity variable $x/t$, the profiles for different time points collapse onto a single, stationary curve. This confirms the scaling form $\rho(x, t) \sim \frac{1}{t} F(x/t)$ and distinguishes the dynamics from diffusive behavior (which would scale as $x/\sqrt{t}$).
2. Emergence of a Stationary Velocity Distribution: In the long-time limit, the density profile in velocity space assumes a time-independent form. This stationary distribution reflects the underlying rapidity structure of the system.
3. Dependence on Interaction:
   • At intermediate interaction ($g=50$), convergence to the stationary velocity profile occurs gradually.
   • At strong interaction ($g=162$), the system shows rapid convergence into the asymptotic regime. The density in velocity space systematically broadens with increasing interaction strength.
4. Dynamic Fermionization: In the strongly interacting regime ($g=162$), the rapid stabilization and the broadened velocity distribution are consistent with the phenomenon of Dynamic Fermionization. The results indicate that the expansion dynamics of the bosonic system, regarding spatial observables, approach those of non-interacting fermions.
5. Connection to Bethe Rapidities: Although the authors do not explicitly extract the Bethe rapidities from the data, they observe that the systematic broadening of the velocity distribution with increasing $g$ qualitatively agrees with the trends of exact Bethe Ansatz rapidity distributions. This provides numerical evidence that the spatial density in the long-time limit encodes the Bethe-rapidity structure.
6. Conservation Laws: The simulations confirm the conservation of particle number, as the area under the density curves remains invariant throughout the entire evolution.

Significance and Claims
The contribution claims to provide the first ab-initio numerical evidence via Quantum Monte Carlo for Dynamic Fermionization and the occurrence of Bethe-rapidity structures in the spatial density of a quenched Lieb-Liniger gas.

• Direct Mapping: The work establishes a practical route to access information from momentum space (Bethe rapidities) via spatial observables (spatial density) in the ballistic expansion limit.
• Numerical Validation: It validates theoretical predictions that the asymptotic spatial density profile is directly linked to the initial quasi-momentum (rapidity) distribution.
• Scope: The authors present these results as a numerical realization of ballistic integrable dynamics in a finite system. They explicitly clarify that they do not perform a direct extraction of rapidities, but rather demonstrate that the observed density evolution is consistent with theoretical expectations for DF and Bethe Ansatz structures.

The contribution concludes that the non-equilibrium dynamics of the quenched LL gas are ballistic and interaction-dependent, with the spatial density in the long-time limit serving as a proxy for the underlying integrable structure of the system. Future directions proposed include the explicit calculation of momentum distributions and extension to larger system sizes.