Quantum Transport in Interacting Spin Chains: Exact Derivation of the GUE Tracy-Widom Distribution

Technical Summary: Quantum Transport in Interacting Spin Chains: Exact Derivation of the GUE Tracy-Widom Distribution

Problem Statement
The paper addresses the long-standing question of whether the Gaussian Unitary Ensemble (GUE) Tracy-Widom distribution, a universal distribution characterizing the largest eigenvalue of random matrices and the integrated current in the Kardar-Parisi-Zhang (KPZ) universality class, emerges in the dynamics of interacting quantum many-body systems. While this behavior was previously established for the non-interacting XX model (equivalent to free fermions via the Jordan-Wigner transformation), its presence in interacting models like the XXZ model has remained elusive. Previous numerical studies on the XXZ model yielded conflicting results, with some suggesting the absence of GUE Tracy-Widom behavior and others indicating its presence. Furthermore, exact analytical derivations for interacting models that cannot be mapped to non-interacting fermions have been lacking.

Methodology
The authors employ a two-pronged approach combining exact analytical solutions and numerical simulations:

1. Exact Analytical Derivation (Folded XXZ Model):

   • Model: The study focuses on the "folded" XXZ model, an effective Hamiltonian derived for the standard XXZ model in the limit of large anisotropic interaction ($\Delta \gg 1$). Crucially, this model cannot be mapped to a non-interacting fermion Hamiltonian via the Jordan-Wigner transformation.
   • Initial State: An alternating domain-wall state is considered, where up-spins occupy every other site in half of the system ($x \ge 2$).
   • Technique: The authors utilize the Bethe ansatz technique, specifically adapting methods developed by Schütz, Tracy, and Widom for the asymmetric simple exclusion process and the XXZ model.
   • Derivation Steps:
     • They derive an integral formula for the many-body wavefunction $\Phi(x_1, \dots, x_N, t)$.
     • Using the Bethe ansatz and properties of the six-vertex model (Izergin-Korepin determinant), they derive a determinantal formula for the complementary cumulative distribution function $F(x,t)$ of finding the left-most up-spin at site $x$.
     • By taking the limit $\Delta \to \infty$, they simplify the scattering amplitudes to obtain a specific determinant form compatible with random matrix theory.
     • Asymptotic analysis is performed using a scaling variable $s$ defined by $x = 2 + \lfloor -t - s(t/2)^{1/3} \rfloor$. This involves analyzing the kernel of the determinant in the long-time limit ($t \to \infty$) using techniques related to Toeplitz operators and Bessel functions, ultimately converging to the Airy kernel.

2. Numerical Investigation (Standard XXZ Model):

   • Model: The standard XXZ model with open boundary conditions and finite anisotropy $\Delta$.
   • Technique: Time-Evolving Block Decimation (TEBD) is used to simulate the real-time dynamics of the Schrödinger equation.
   • Analysis: The probability $P(x,t)$ of finding the left-most up-spin is computed for various $\Delta$ values (5, 10, 20). The data is rescaled according to the KPZ scaling exponents to compare against the GUE Tracy-Widom distribution.

Key Contributions and Results

1. Exact Derivation for Interacting Models: The primary contribution is the first exact derivation demonstrating that the probability distribution of the left-most up-spin in an interacting quantum spin model (the folded XXZ model) converges to the GUE Tracy-Widom distribution ($F_2(s)$) in the long-time limit. This confirms that the GUE Tracy-Widom behavior survives in systems that are not mappable to non-interacting fermions.
2. Mathematical Connection: The authors establish a mathematical link between the folded XXZ model and the XX model. They show that the many-body wavefunction of the folded XXZ model possesses a determinantal structure similar to that of the XX model, providing a mathematical origin for the emergence of the same universal distribution despite the presence of interactions.
3. Numerical Signatures in XXZ: For the standard XXZ model with finite $\Delta$, numerical results show that while the full distribution does not perfectly match the GUE Tracy-Widom curve for small $\Delta$ at accessible times, a specific "universal signature" emerges. Specifically, in the right tail of the distribution ($s > 0$), the probability $P(x,t)$ exhibits a curve independent of $\Delta$. This curve is characterized by the diagonal Airy kernel $K_{Ai}(s,s)$.
4. Fast Convergence Scaling: The paper introduces a refined scaling variable with a parameter $a=1/2$ (as opposed to the standard $a=0$) for the coordinate rescaling. Numerical tests demonstrate that this choice leads to significantly faster convergence of the finite-time distribution to the asymptotic Tracy-Widom limit.

Significance and Claims
The paper claims to provide the first exact proof that the GUE Tracy-Widom distribution arises in the dynamics of an interacting quantum model that cannot be reduced to free fermions. This resolves the ambiguity regarding the universality of KPZ physics in interacting quantum systems.

The authors modestly suggest that their findings offer a theoretical basis for observing these universal behaviors in current experimental setups. They note that recent experiments using ultracold atoms and superconducting qubits have realized XXZ-like models and accessed time scales where spins flip multiple times (approximately 5 times), which aligns with the time scales where the GUE Tracy-Widom signature appears in their numerical simulations for large $\Delta$.

The work opens future directions for exploring Tracy-Widom distributions in non-integrable models and other integrable systems, such as the phase model, and suggests that generalized hydrodynamics and ballistic macroscopic fluctuation theory could be used to analytically understand the universal behavior of the XXZ model.