⚛️ quantum physics

Kardar-Parisi-Zhang dynamics in an open integrable system: beyond the spontaneous-symmetry-breaking ansatz

This paper demonstrates that the Kardar-Parisi-Zhang (KPZ) dynamics, rather than the diffusion predicted by spontaneous symmetry breaking, emerges in the open integrable B3 model due to its equivalence to interacting asymmetric XXZ spin chains, thereby challenging the universality of the spontaneous symmetry breaking ansatz for charge transport in open quantum systems.

Original authors: Guo-Qiang Wang, Chang-Ling Zou, Guang-Can Guo, and Xu-Bo Zou

Published 2026-07-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Guo-Qiang Wang, Chang-Ling Zou, Guang-Can Guo, and Xu-Bo Zou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a crowd of people move through a busy train station. In physics, scientists try to predict how "charges" (like electric charge or spin) move through similar crowds of particles. For a long time, there was a popular theory called the Spontaneous Symmetry Breaking Ansatz (SSBA). Think of this theory as a very confident weather forecaster who says, "No matter what the specific crowd looks like, if they are just moving around randomly, the traffic will always slow down and spread out in a predictable, smooth way called 'diffusion'." It's like predicting that a drop of ink in water will always spread out evenly and slowly.

However, the authors of this paper found a specific "train station" (a mathematical model called the B3 model) where this weather forecaster was completely wrong.

Here is the breakdown of their discovery using simple analogies:

1. The Double-Decker Bus Analogy

The B3 model is an "open system," meaning it interacts with its environment. To understand it, the authors used a clever trick called the Choi isomorphism. Imagine the system isn't just one line of people, but a double-decker bus.

The Bottom Deck: Represents one side of the math.
The Top Deck: Represents the other side.
The Stairs: Represent the interaction between the decks (the "jump" operators).

The old theory (SSBA) looked at the whole bus and assumed the stairs were the most important part. It predicted that the "traffic" on the bus would spread out slowly (diffusion), just like the ink in water.

2. The "Ghost" in the Machine

The authors discovered that for a specific starting condition (a specific arrangement of passengers), the stairs become useless. The people on the top deck and the bottom deck stop interacting with each other. They become two separate, independent lines of people.

When the stairs are ignored, the old theory (SSBA) fails because it was only looking at the stairs. It missed what was happening on the decks themselves.

3. The One-Way Street (The Real Discovery)

Once the decks are separated, the authors realized that each deck behaves like a one-way street with a very specific rule:

People can hop forward easily.
People can hop backward, but the rules are different (asymmetric).

In physics, this is called an asymmetric XXZ spin chain. The authors showed that when you have this kind of one-way street, the traffic doesn't spread out slowly like ink. Instead, it moves in a wild, turbulent way known as KPZ dynamics.

The KPZ Analogy:
Think of the difference between pouring honey (diffusion) and watching a pile of sand grow as you pour more sand on top.

Diffusion (SSBA prediction): The sand spreads out flat and smooth.
KPZ (Actual result): The sand piles up, forms peaks and valleys, and the surface gets rough and bumpy. The "roughness" grows in a specific, faster way than the smooth spreading.

4. The "Negative Speed" Surprise

One of the most surprising parts of the paper is what happens when they tweak a dial in the model (changing a parameter called $\gamma$ ).

Usually, people hop forward with a positive speed.
In this model, under certain conditions, the math says the hopping speed becomes negative.

Imagine a traffic rule that says, "Cars must drive backward to move forward." It sounds impossible, but in this quantum world, it works. Even with this "negative speed," the traffic still forms those bumpy, KPZ-style piles. The authors found that the "roughness" of the traffic still follows the KPZ pattern, even when the direction of movement seems to flip.

5. Why the Old Theory Failed

The SSBA theory failed because it was like looking at a map that only showed the stairs of the double-decker bus. It assumed the stairs controlled the traffic. But in this specific case, the stairs were locked, and the real traffic was happening on the decks, which followed a different set of rules (the one-way street rules).

The Bottom Line

The paper proves that the popular "SSBA" theory is not a universal law for all open quantum systems. There are hidden structures (like the double-decker bus becoming two separate one-way streets) that can cause traffic to behave in a much more complex, "bumpy" way (KPZ scaling) rather than the smooth, slow spreading (diffusion) that everyone expected.

They also suggest that this specific model (B3) is a bridge between the messy world of quantum physics and the well-understood world of classical statistics (like how sand piles or traffic jams behave), showing that even "negative" rules in quantum mechanics can create real, observable patterns.

Technical Summary: Kardar-Parisi-Zhang dynamics in an open integrable system

Problem Statement
The universality of dynamical scaling laws is a cornerstone for understanding quantum many-body systems, particularly in non-equilibrium settings. Recent progress in open quantum systems has relied heavily on the Spontaneous Symmetry Breaking Ansatz (SSBA) [20], a phenomenological framework that utilizes Feynman-Bijl variational wave functions to approximate low-energy excitations near steady states. The SSBA predicts universal diffusive transport (dynamical exponent $z=2$ ) for open systems with local interactions and no constraints, successfully capturing behaviors in long-range interacting and constrained models. However, it remains unclear under what conditions this ansatz fails to capture emergent hydrodynamics, and specifically, whether nontrivial dynamics emerge when the approximation breaks down. The SSBA neglects terms in the Lindblad master equation that drive variational wave functions out of the variational subspace, leaving open the question of whether anomalous dynamics can arise in integrable open systems.

Methodology
The authors investigate the B3 model [28–30], a Yang-Baxter integrable open quantum system governed by the Lindblad master equation. The study employs a combination of analytical transformations and numerical simulations:

Choi Isomorphism: The Lindblad equation is mapped to a doubled Hilbert space, decomposing the Liouvillian superoperator $\mathcal{L}$ into three components: independent non-Hermitian evolutions on "left" and "right" chains ( $A \otimes \mathbb{1}$ and $\mathbb{1} \otimes B$ ) and an inter-chain coupling term ( $l \otimes l^*$ ) representing quantum jumps.
Variational Analysis: The authors analyze the SSBA within this doubled space, demonstrating that the single-chain terms ( $A$ and $B$ ) drive the system out of the variational subspace used by the ansatz, while the ansatz only captures the effects of the inter-chain coupling.
Initial State Selection: The system is prepared in a specific initial state $\rho_0 = 2n_{j'} |\Psi\rangle\langle\Psi| n_{j'}$ , where $|\Psi\rangle$ is a product state satisfying $l^* |\Psi^*\rangle = 0$ . This condition effectively decouples the two chains, rendering the inter-chain coupling dynamically irrelevant for the charge transport dynamics.
Numerical Simulation: Time-Evolving Block Decimation (TEBD) is used to simulate charge transport for a chain of length $N=256$ with bond dimension $\chi=64$ and parameter $\gamma=1.2$ .
Analytical Mapping: The single-chain dynamics are analytically mapped to an asymmetric XXZ spin chain model. The spectral gap of this effective Hamiltonian is calculated using the Bethe ansatz to determine the dynamical exponent.

Key Contributions and Results

Failure of SSBA: The paper demonstrates that the SSBA fails to describe the charge transport in the B3 model. While the SSBA predicts diffusive behavior ( $z=2$ ) based on the spectral gap of the full Liouvillian, the actual charge dynamics are governed by the single-chain terms, which the ansatz excludes.
Emergence of KPZ Scaling: For appropriate initial states, the B3 model exhibits Kardar-Parisi-Zhang (KPZ) scaling behavior ( $z=3/2$ ) rather than diffusion. Numerical results confirm that the charge density correlation function $G(x,t)$ decays as $t^{-2/3}$ and collapses onto the universal KPZ scaling function $f_{KPZ}(x/t^{2/3})$ .
Decoupling Mechanism: The study reveals that under specific initial conditions, the inter-chain coupling (which generates the diffusive spectrum predicted by SSBA) becomes irrelevant. The dynamics are dominated by the single-chain evolution, which is equivalent to an asymmetric XXZ model.
Asymmetric XXZ Origin: The single-chain operator $A_{j,j+1}$ is shown to be equivalent to the Hamiltonian of an asymmetric XXZ chain. Even when the hopping rate becomes negative (for $\gamma < 1$ ), the system maps to a statistical process described by a Burgers equation (the mean-field limit of KPZ), yielding the $z=3/2$ exponent.
Spectral Gap Discrepancy: The authors calculate the spectral gap of the effective single-chain Hamiltonian ( $H_{XXZ}$ ) via the Bethe ansatz, finding a scaling of $\Delta \propto N^{-3/2}$ . This contrasts with the $N^{-2}$ scaling of the full Liouvillian gap, explaining why the SSBA (which probes the full Liouvillian) misses the KPZ universality class.
Negative Hopping Rates: The work highlights that the effective hopping rate in the mapped XXZ model can be negative, yet the diffusion constant remains positive, allowing the Burgers equation to serve as an effective theory.

Significance
The paper claims that a simplistic understanding of open system hydrodynamics based solely on spontaneous symmetry breaking (SSB) is insufficient. It establishes that integrable open systems can exhibit KPZ dynamics, a universality class previously associated with isolated integrable models (like the Heisenberg magnet) or classical stochastic processes (ASEP), but not typically with open systems governed by the SSBA.

The work provides a pathway for connecting classical statistical mechanics with open quantum systems: if the inter-chain coupling vanishes and the single-chain operator corresponds to a statistical model generator, the quantum correlation functions coincide with the Green's functions of that classical process. This suggests that the B3 model reduces to the Asymmetric Simple Exclusion Process (ASEP) in the appropriate limit, offering a concrete example of how classical statistical tools can be applied to open quantum dynamics. The authors note that the emergence of KPZ scaling even with negative hopping rates points to new physical origins for such processes, though they do not propose specific experimental implementations beyond noting that advances in quantum simulation platforms (e.g., neutral atom arrays) could probe these phenomena.