Exotic critical states as fractional Fermi seas in the one-dimensional Bose gas

This paper demonstrates that cyclic interaction changes in an integrable one-dimensional Bose gas drive the system into a novel critical phase characterized by fractional Fermi seas, which exhibit correlation signatures incompatible with conventional Tomonaga-Luttinger liquid behavior.

The Gist

Imagine a crowded dance floor where everyone is trying to move in perfect sync. In the world of quantum physics, this "dance floor" is a one-dimensional tube filled with atoms. Usually, if these atoms are bosons (a type of particle that loves to clump together), they behave like a calm, flowing liquid. If they are fermions (particles that hate sharing space), they behave like a rigid, organized army.

For decades, physicists have known that under certain conditions, even bosons can start acting like fermions, forming what is called a Fermi Sea. Think of a Fermi Sea as a dance floor where every single dancer has their own specific spot, and no two dancers can ever swap places. This creates a very specific, predictable pattern of movement.

The Big Discovery: The "Fractional" Dance Floor

This paper, written by a team of physicists, introduces a brand-new, exotic state of matter they call a "Fractional Fermi Sea."

Here is the simple breakdown of how they did it and what it means:

1. The "Magic Cycle" (The Shaker)

Imagine you have a jar of marbles (the atoms).

• Step 1: You push the marbles apart (repulsion).
• Step 2: You suddenly pull them together so hard they almost crash into each other (attraction).
• Step 3: You let them go completely free.

The researchers did this over and over again in a specific loop. In the quantum world, this isn't just shaking a jar; it's a precise, mathematical dance called an "interaction cycle."

2. The "Fractional" Result

Normally, if you do this, the atoms just heat up or settle back into their original state. But because these atoms are in a "perfect" quantum system (called an integrable system), they don't just heat up. Instead, they get trapped in a weird, in-between state.

The authors found that after repeating this cycle a few times, the atoms settle into a state where only a fraction of the available "dance spots" are occupied.

• The Analogy: Imagine a concert hall with 100 seats.
  • Normal Fermi Sea: Every single seat is filled. No empty seats.
  • Fractional Fermi Sea: Only every third seat is filled. The rest are empty, but the people sitting there are still following strict rules about where they can move. It's like a dance floor where the dancers are spaced out in a specific, repeating pattern that doesn't exist in nature under normal conditions.

3. Why is this "Exotic"?

In standard physics, if you have a gapless system (one where particles can move freely), the patterns of how they interact usually follow a rule called the Tomonaga-Luttinger Liquid (TLL) theory. Think of TLL as the "standard operating manual" for 1D quantum liquids.

The researchers found that these "Fractional Fermi Seas" break the manual.

• They still show "critical" behavior (meaning they are highly sensitive and connected over long distances).
• However, the way they wiggle and interact (measured by something called correlation functions) is different. They show a unique pattern of ripples (Friedel oscillations) that don't match the standard rules.

It's like discovering a new type of music that uses the same instruments as a symphony but plays a rhythm that no one has ever heard before.

4. The "Time Travel" Aspect (Reversibility)

One of the coolest parts of the paper is about reversibility.

• If you run the cycle forward, the atoms settle into this Fractional state.
• If you try to run the cycle backward to return to the start, the system refuses to go back. It gets stuck in a new configuration with "bound states" (atoms sticking together in clumps).

This is like pouring water into a glass, then trying to pour it back into the pitcher. Usually, you can do it. But in this quantum experiment, the "water" turns into ice clumps when you try to pour it back, making it impossible to return to the original liquid state. This "staircase" of irreversibility is a key signature of this new state.

Why Should We Care?

This isn't just a math puzzle.

1. New Physics: It proves that we can create "fractional" states of matter out of equilibrium (when things are changing fast) without needing exotic particles like anyons.
2. Cold Atoms: The team predicts this can be done right now in labs using ultracold atoms (like cesium). In fact, they mention a companion paper where they likely did this experimentally.
3. Future Tech: Understanding these "exotic critical states" helps us understand how quantum information behaves, which is crucial for building future quantum computers.

In a Nutshell:
The researchers found a way to "program" a gas of atoms to occupy only a fraction of their available space, creating a new, stable, and exotic quantum state that behaves like a hybrid between a liquid and a solid, breaking the standard rules of quantum physics. It's a new chapter in the story of how matter can behave when pushed to its limits.

Technical Summary

1. Problem Statement

The paper addresses the fundamental question of whether fractional Fermi seas (FFS)—states where the occupancy of quantum states is reduced such that only a fraction $1/\alpha$ of states are occupied—can be realized in interacting bosonic systems beyond the ground state.

• Context: In one-dimensional (1D) quantum systems, the Tomonaga-Luttinger Liquid (TLL) theory describes low-energy critical behavior, characterized by power-law correlations and Friedel oscillations (FO) arising from an effective Fermi sea.
• Gap: While FFS are theoretically predicted as ground states of systems with Generalized Exclusion Statistics (GES) (Haldane's proposal), realizing them as highly excited, non-equilibrium states in a standard integrable Bose gas has been an open challenge.
• Goal: The authors aim to demonstrate that FFS can be engineered in the Lieb-Liniger (LL) model (1D bosons with contact interactions) through non-equilibrium interaction cycles, and to characterize their unique critical properties which differ from conventional TLL.

2. Methodology

The study combines Generalized Hydrodynamics (GHD), Quantum Adiabaticity (QA) concepts, and Monte Carlo (MC) simulations.

A. Theoretical Framework: Generalized Hydrodynamics (GHD)

• Model: The system is described by the Lieb-Liniger Hamiltonian with tunable coupling $g_{1D}$.
• Protocol: The system undergoes a cyclic protocol where $g_{1D}$ is slowly varied:
  1. From weakly repulsive to strongly repulsive.
  2. Quenched through the Tonks-Girardeau (TG) to super Tonks-Girardeau (sTG) transition (crossing infinite attraction).
  3. Returned to the non-interacting point ($g_{1D}=0$) and back to the start.
  4. This cycle is repeated $W$ times.
• Dynamics: The evolution is governed by GHD equations for the occupancy function $\vartheta(\lambda)$ (fraction of occupied rapidity states).
  • The continuity of the root density $\rho(\lambda)$ across the non-interacting point ($g_{1D}=0$) imposes a specific mapping condition.
  • The cycle acts as a projector, mapping the initial Generalized Gibbs Ensemble (GGE) to a new GGE with reduced maximum occupancy:
    $$\vartheta_{max} \leq \frac{1}{2W + 1}$$
  • This realizes an effective fractional GES with parameter $\alpha = 2W + 1$.

B. Numerical Simulation: Monte Carlo Sampling

• To compute physical observables (specifically the one-particle correlation function $g^{(1)}(x)$), the authors employed a Monte Carlo algorithm based on the Bethe Ansatz.
• Procedure:
  1. Sample microstates $\{ \lambda_i \}$ consistent with the GGE occupancy derived from GHD.
  2. Compute form factors (matrix elements of the field operator) using determinant formulas.
  3. Calculate the momentum distribution $P(p)$ and obtain $g^{(1)}(x)$ via Fourier transform.
• Validation: The method was validated against known limits (non-interacting and TG regimes) and checked for convergence with system size ($N=30, 50$).

C. Comparison with Quantum Adiabaticity (QA)

• The authors compared GHD predictions with QA. While both predict the same energy densities and reduced occupancy, GHD captures irreversibility absent in QA.
• Irreversibility: In the reverse cycle, crossing $g_{1D}=0$ from attractive to repulsive leads to the formation of Bethe strings (bound states), causing a jump in entropy and energy, breaking the "staircase reversibility" seen in QA.

3. Key Contributions

1. Realization of Fractional Fermi Seas (FFS): Demonstrated that FFS can be generated in an integrable Bose gas as highly excited non-equilibrium states (GGEs) rather than ground states, via interaction cycles.
2. Novel Critical Phase: Identified a new critical phase where the system exhibits power-law correlations and Friedel oscillations but with signatures incompatible with standard TLL theory.
3. Bimodal Power-Law Decay: Discovered that the one-particle correlation function $g^{(1)}(x)$ exhibits a crossover between two distinct power-law regimes (short-distance vs. long-distance), a feature not present in conventional TLL.
4. Experimental Relevance: Provided quantitative predictions (correlation functions, energy densities) directly applicable to cold atom experiments (specifically ultracold cesium atoms), which are currently being realized in a joint experimental submission.

4. Key Results

• Occupancy Reduction: After $W$ cycles, the maximum occupancy of rapidity states is strictly reduced to $\vartheta \leq 1/(2W+1)$, effectively creating a "fractional" Fermi sea.
• Correlation Functions ($g^{(1)}(x)$):
  • Friedel Oscillations: The system exhibits prominent FO with a frequency $k_{FO}$ that deviates from the standard $2\pi n$ (where $n$ is density) for $W \geq 1$, depending on interaction strength $\gamma$.
  • Bimodal Decay: $g^{(1)}(x)$ decays as a power law $x^{-\eta}$ but with a crossover distance $\bar{x}$.
    • Short Distance (SD): Slower power-law decay.
    • Long Distance (LD): Faster power-law decay.
    • As interaction strength $\gamma$ increases, $\bar{x}$ decreases, and the decay eventually transitions to an oscillating exponential in the TG regime.
• Exponents: The power-law exponents increase with interaction strength but decrease with the number of cycles $W$ (as kinetic energy dominates).
• Irreversibility: The cycle is irreversible due to bound state formation upon crossing $g_{1D}=0$ in the reverse direction, leading to a "staircase" increase in entropy $S \propto n[(1+2W)\log(1+2W) - 2W\log(2W)]$.

5. Significance

• Beyond TLL: The work challenges the universality of TLL by showing that interacting bosons can exhibit critical behavior that cannot be described by a single Luttinger parameter. The "fractional" nature of the Fermi sea introduces new degrees of freedom in the bulk, not just at the Fermi edges.
• Non-Equilibrium Engineering: It establishes a protocol for engineering exotic quantum states (FFS) using integrability and cyclic driving, offering a new paradigm for creating states with tunable statistical properties.
• Bridge to Experiment: The predictions are directly testable with current cold-atom technology. The observation of the specific bimodal power-law decay and modified Friedel oscillations would provide the first experimental evidence of a non-TLL critical phase in a 1D Bose gas.
• Theoretical Insight: It clarifies the relationship between Generalized Exclusion Statistics (GES) and non-equilibrium GGEs, suggesting that while they share occupancy features, their correlation functions differ due to the presence of virtual states outside the projected Hilbert subspace.

In summary, the paper provides a rigorous theoretical and numerical demonstration that fractional Fermi seas can be dynamically engineered in 1D bosonic gases, leading to a novel critical phase characterized by unique correlation signatures that distinguish it from standard Luttinger liquids.