Interaction-induced asymmetry in infinite-temperature dynamical correlations of hard-core anyons

This paper demonstrates that while infinite-temperature density-density correlations in interacting hard-core anyons remain insensitive to fractional statistics and follow standard XXZ transport regimes, single-particle Green's functions exhibit a pronounced interaction-induced left-right asymmetry and statistical-angle-dependent decay, revealing dynamical correlations as direct probes of fractional statistics in high-entropy quantum systems.

The Gist

Imagine a crowded dance floor where everyone is moving randomly because the music is so loud and chaotic that no one remembers the steps or the rules of the dance. This is what physicists call an "infinite-temperature" state: a system so hot and energetic that every possible arrangement of particles is equally likely.

In this paper, the authors study a very specific type of dancer on a one-dimensional line (like a single-file queue). These dancers are called Hard-Core Anyons.

Here is the breakdown of the story using simple analogies:

1. The Characters: What is an "Anyon"?

In our world, particles are usually either Bosons (like polite people who can all stand in the same spot) or Fermions (like grumpy people who refuse to stand next to each other).

Anyons are the "in-between" dancers. They are a bit of both. When two anyons swap places, they don't just say "hello" (like bosons) or "get lost" (like fermions). Instead, they whisper a secret code to each other. This code is a fractional phase (represented by the angle $\theta$).

• The Catch: In a one-dimensional line, you can't swap two people without one of them "jumping over" the other. In quantum mechanics, this "jump" leaves a ghostly trail behind them, called a Jordan-Wigner string. It's like a long, invisible rope attached to every dancer that stretches back to the start of the line.

2. The Experiment: The "Maximally Mixed" Dance Floor

The researchers asked: If we throw these dancers onto a chaotic, high-energy dance floor (infinite temperature), does their secret code (the fractional phase) still matter?

Usually, when things get hot and chaotic, specific details get washed out. You'd expect the secret code to disappear. However, the authors found something surprising: The secret code is still there, but it changes how the dancers move.

3. The Main Discovery: The "Left-Right" Bias

The most exciting finding is about Chirality (handedness).

• The Non-Interacting Case (No Talking): If the dancers just move around without talking to their neighbors (no interaction), the dance floor looks perfectly symmetrical. If you look at the dance from the left or the right, it looks the same. The secret code is hidden.
• The Interacting Case (Talking): But, if the dancers start bumping into each other and interacting (adding a "repulsion" force), something magical happens. The dance floor suddenly becomes lopsided.
  • For dancers with a "secret code" between the two extremes, the movement becomes asymmetric. They prefer to drift left or right depending on their specific code.
  • The Analogy: Imagine a crowd of people trying to walk through a narrow hallway. If they are just walking, they spread out evenly. But if they start elbowing each other (interacting) and holding invisible ropes (the anyonic strings), the crowd suddenly starts flowing more strongly to the left than the right. The "secret code" of the dancers dictates which way the crowd leans.

4. The Three Zones of Behavior

The authors found that the behavior changes depending on how strongly the dancers interact:

• Weak Interaction (The Gentle Nudge): The asymmetry is strongest here. The dancers are just starting to bump into each other, and the invisible ropes twist the crowd into a distinct left-right bias.
• Strong Interaction (The Atomic Limit): If the dancers are forced to stay far apart (very strong repulsion), they get "stuck" in their spots. The secret code fades away again. The dance floor becomes symmetric once more, but the movement slows down and follows a universal, boring rhythm (decaying as $1/t$).
• The Middle Ground: This is where the magic happens. The competition between the desire to move (hopping) and the desire to stay apart (interaction) creates the most dramatic left-right imbalance.

5. The "Density" vs. The "Dancer"

The paper makes a crucial distinction between two things:

1. The Dancer's Path (Green's Function): This tracks a single particle. This is where the secret code (fractional statistics) shows up as a left-right bias.
2. The Crowd Density (Density-Density Correlation): This tracks how many people are in a specific spot, regardless of who they are. Surprisingly, this part of the dance does not care about the secret code at all. It behaves exactly like a standard crowd of people, following known rules of traffic flow (ballistic, super-diffusive, or diffusive).

The Big Picture

This paper tells us that even in a completely chaotic, high-energy system where you'd expect order to be lost, quantum weirdness (fractional statistics) can still leave a fingerprint.

It's like saying that even in a riotous mosh pit, if everyone is wearing a specific type of invisible magnet, the crowd will still swirl in a specific direction when they bump into each other. The authors have found a way to see this swirl, proving that dynamical correlations (how things move over time) are a perfect tool to detect these exotic quantum particles, even when the system is "hot" and messy.

In short: Interactions turn a symmetric, chaotic dance into a biased, chiral flow, revealing the hidden "secret language" of the particles.

Technical Summary

1. Problem Statement and Motivation

The paper investigates the dynamical correlations of interacting hard-core anyons on a one-dimensional lattice in the infinite-temperature limit ($T = \infty$).

• Context: Anyons are quasiparticles with fractional exchange statistics ($\theta$), interpolating between bosons ($\theta=0$) and fermions ($\theta=\pi$). While 1D anyon models are well-studied, their behavior in high-entropy (infinite temperature) regimes remains less explored.
• The Paradox: In hard-core anyon models, the many-body energy spectrum is independent of the statistical phase $\theta$ because the Hamiltonian can be mapped to a standard XXZ spin chain or hard-core boson model via Jordan-Wigner transformations. However, the authors hypothesize that dynamical correlation functions might retain sensitivity to $\theta$ due to non-local Jordan-Wigner strings, even when thermal fluctuations are maximal.
• Key Questions:
  1. How visible are fractional statistics in real-time correlation functions of a maximally mixed ensemble?
  2. How do nearest-neighbor interactions ($V$) reshape these correlations, specifically regarding their spreading, decay, and spectral structure?

2. Methodology

The authors employ a combination of analytical derivations and advanced numerical simulations:

• Model: A 1D lattice of $L$ sites with hard-core anyons described by the Hamiltonian:
  $$H = \sum_{j=1}^{L-1} J(a^\dagger_j a_{j+1} + \text{H.c.}) + V \sum_{j=1}^{L-1} \left(n_j - \frac{1}{2}\right)\left(n_{j+1} - \frac{1}{2}\right)$$
  where $a_j$ are anyonic operators related to fermions ($c_j$) and bosons ($b_j$) via generalized Jordan-Wigner strings involving the phase $\theta$.
• Ensemble: The infinite-temperature limit where the density matrix is proportional to the identity ($\rho_\infty = 1/D$). This simplifies expectation values to traces over the full Hilbert space.
• Observables Computed:
  • Single-particle Green's functions: $G^>(x, t) = -i\langle a_x(t) a^\dagger_0(0) \rangle$ and retarded versions.
  • Spectral functions: $A(q, \omega)$ and local density of states $\rho(\omega)$.
  • Density-density correlators: $C_{jk}(t) = \langle n_j(t) n_k(0) \rangle$.
• Numerical Techniques:
  • Exact Diagonalization (ED): Used for small systems ($L \sim 10$) to benchmark results.
  • Free-limit analytical solutions: Utilizing Fredholm determinants and properties of quadratic fermionic Hamiltonians for $V=0$.
  • Time-Evolving Block Decimation (TEBD):
    • Standard TEBD for operator evolution in the Fock space.
    • Fock-Liouville TEBD: A vectorization technique mapping the density matrix evolution to a Schrödinger-like equation in a doubled Hilbert space, allowing simulation of larger systems ($L=129$) with bond dimension $\chi=128$.

3. Key Contributions and Results

A. Non-Interacting Limit ($V=0$)

• Inversion Symmetry: Despite the presence of non-local Jordan-Wigner strings, the Green's functions are spatially inversion-symmetric ($|G(x,t)| = |G(-x,t)|$) for all $\theta$ at infinite temperature. This contrasts with finite-temperature results where symmetry is broken.
• Interpolation of Decay:
  • Bosons ($\theta=0$): Correlations decay as a Gaussian $e^{-J^2 t^2}$, showing strong localization and no ballistic light cone.
  • Fermions ($\theta=\pi$): Correlations exhibit ballistic spreading with a light cone velocity $v=2J$ and decay algebraically as $t^{-1/2}$ (Bessel function behavior).
  • Intermediate Anyons ($0 < \theta < \pi$): The decay is approximately exponential $e^{-\alpha(\theta)t}$ with oscillations at frequency $\approx 2J$. The decay rate $\alpha$ depends on $\theta$, bridging the Gaussian and power-law limits.

B. Interacting Regime ($V \neq 0$): Interaction-Induced Asymmetry

This is the central finding of the paper.

• Chiral Asymmetry: For anyons with $0 < \theta < \pi$, finite nearest-neighbor interactions break spatial inversion symmetry. The Green's functions develop a pronounced left-right asymmetry (chirality).
  • $G(x, t) \neq G(-x, t)$ for $x \neq 0$.
  • This effect arises because the interaction term $V$ does not commute with the non-local Jordan-Wigner strings.
• Coupling Dependence:
  • Intermediate Coupling ($V \sim J$): The chirality is strongest here, where hopping and interactions compete most effectively.
  • Strong Coupling ($V \gg J$): The system enters an "atomic limit." The dependence on $\theta$ is suppressed, and the dynamics become approximately inversion-symmetric again.
• Decay Laws:
  • Weak/Intermediate $V$: Exponential decay dominates for generic $\theta$.
  • Strong $V$: The Green's function decays universally as $t^{-1}$ for all $\theta$, independent of the statistical phase.

C. Spectral Functions

• Density of States (DOS):
  • At $V=0$, the DOS evolves from a single Gaussian peak (bosons) to van Hove singularities (fermions).
  • At strong coupling ($V \gg J$), the DOS develops a universal three-band structure centered at $\omega = 0$ and $\omega = \pm V$. The weights of these bands follow a $1:2:1$ ratio, corresponding to the local atomic configurations (0, 1, or 2 occupied neighbors).
• Momentum-Resolved Spectra:
  • In the strong-coupling limit, the spectral function shows three dispersive bands. The central band crosses zero energy at specific momenta, while side bands exhibit shifted dispersions dependent on $\theta$.

D. Density-Density Correlations

• Insensitivity to Statistics: Unlike single-particle Green's functions, density-density correlators do not depend on $\theta$. This is because the density operator $n_j$ does not contain Jordan-Wigner strings.
• XXZ Transport Regimes: The dynamics recover the known infinite-temperature transport phenomenology of the XXZ spin chain:
  • Ballistic ($V < 2J$): Correlation front spreads as $x \sim t$.
  • Superdiffusive ($V = 2J$): KPZ universality class with $x \sim t^{2/3}$ (dynamical exponent $z=3/2$).
  • Diffusive ($V > 2J$): $x \sim t^{1/2}$ (dynamical exponent $z=2$).

4. Significance and Conclusions

• Separation of Coherence and Transport: The work demonstrates a fundamental separation in high-entropy quantum systems:
  • One-body coherence (Green's functions) is highly sensitive to fractional statistics and interactions, exhibiting interaction-induced chirality.
  • Conserved density transport (density correlations) is insensitive to statistics and follows standard hydrodynamic universality classes.
• Probe for Fractional Statistics: The study identifies dynamical correlation functions as direct probes of fractional statistics in high-entropy (randomly populated) states, which are experimentally relevant for quantum simulators (e.g., cold atoms in optical lattices) that prepare high-energy states.
• Theoretical Insight: It resolves the apparent contradiction that while the energy spectrum is $\theta$-independent, the dynamics are not. The non-commutativity of interactions with non-local strings generates spatial asymmetries that are robust even at infinite temperature.

In summary, the paper establishes that interactions induce a chiral asymmetry in the dynamical correlations of hard-core anyons, a phenomenon that vanishes in the non-interacting limit and is washed out in the strong-coupling atomic limit, providing a unique signature of fractional statistics in thermalized quantum matter.