Dissipative quantum North-East-Center model: steady-state phase diagram, universality and nonergodic dynamics

This paper investigates the dissipative quantum North-East-Center model using a cluster mean-field approach to reveal a bistable-normal phase diagram, characterize nonergodic dynamics through the constant-velocity reabsorption of minority spin islands, and propose a linear-order equation of motion for this reabsorption velocity.

The Gist

Imagine a vast, two-dimensional grid of tiny magnets (spins), each pointing either up or down. Now, imagine these magnets are part of a game with very specific, quirky rules about how they can change their minds. This is the "North-East-Center" (NEC) model, a system the authors studied to understand how quantum systems behave when they are constantly losing energy to their environment (dissipation).

Here is a breakdown of their findings using everyday analogies:

1. The Game Rules: The "Majority Vote" with a Twist

In this model, every magnet looks at its North neighbor and its East neighbor (and itself).

• The Rule: If the majority of these three are pointing "Up," the magnet at the corner is forced to point "Up." If the majority is "Down," it must point "Down."
• The Twist: This rule is chiral (handed). The magnet only listens to its North and East neighbors, ignoring its South and West neighbors. It's like a person who only listens to advice coming from the top and the right, completely ignoring everyone else.
• The Noise: Sometimes, the magnets make mistakes due to "thermal noise" (like a gust of wind) or "quantum noise" (like a sudden, random quantum jump).

2. The Two Main Outcomes: The "Bistable" vs. The "Normal"

The authors discovered that depending on how much noise is in the system, the magnets settle into one of two distinct behaviors:

• The Normal Phase (The "Melting Pot"): If the noise is too high, the system forgets its history. No matter how you start the game, the magnets eventually mix together and settle into a single, uniform state. It's like stirring a cup of coffee; eventually, the cream and coffee blend into one color.
• The Bistable Phase (The "Memory Bank"): If the noise is low enough, the system becomes bistable. This means it has two stable states it can settle into: one where almost everything is "Up" and one where almost everything is "Down."
  • The Analogy: Think of a ball in a landscape with two deep valleys separated by a hill. If you push the ball gently, it will roll into one valley or the other and stay there. Crucially, which valley it ends up in depends entirely on where you started. The system "remembers" its initial condition.

3. The Universal Discovery: It Doesn't Matter How You Shake It

The researchers tested this system with different types of "shaking" (quantum fluctuations):

• Free Shaking: Randomly flipping spins without any rules.
• Constrained Shaking: Flipping spins only if their neighbors are in a specific state (mimicking the same rules as the dissipation).
• The Result: Surprisingly, the Bistable Phase (the memory bank) appeared in all cases. Whether the quantum noise was chaotic or followed the same strict rules as the dissipation, the system still managed to maintain two distinct stable states.
• The Caveat: While the existence of the two states is universal, the size of the "safe zone" (where bistability works) changes. If the shaking is too strong or too "free" (unconstrained), the memory bank collapses, and the system melts into the Normal Phase.

4. The "Island" Experiment: Eating the Mistakes

To understand how this memory works, the authors simulated a scenario where a small "island" of magnets pointing the "wrong" way (e.g., a square of "Down" spins) was placed inside a sea of "Up" spins.

• In the Normal Phase: The island of "Down" spins quickly dissolves and spreads out until the whole grid becomes a uniform mix. The system forgets the island existed.
• In the Bistable Phase: The island does not spread. Instead, the surrounding "Up" spins act like a vacuum cleaner, reabsorbing the island.
  • The Key Finding: The island shrinks at a constant speed, regardless of how big the island is. A tiny speck and a large square both get eaten at the same rate.
  • Why this matters: This suggests the system has a built-in mechanism to correct errors. If a few magnets accidentally flip to the wrong state, the "majority vote" rule (with its North-East bias) will systematically push them back, restoring the original order.

5. The Speed of Correction

The authors derived a formula for how fast these islands get eaten. They found that:

• Thermal noise (heat) and Quantum noise both slow down the reabsorption process.
• However, they act independently. You can think of them as two different people slowing down a runner; one pushes from the left, one from the right, but they don't necessarily work together to stop the runner completely.
• Interestingly, thermal noise is a much stronger "brake" on this process than quantum noise.

Summary

The paper shows that a simple set of rules (listen only to North and East neighbors) creates a robust system that can "remember" its initial state even in a noisy quantum world. This system can automatically correct small errors (islands of wrong spins) by swallowing them at a steady pace. This behavior is surprisingly robust, surviving even when the underlying quantum rules change, suggesting that such "chiral" (handed) systems could be very stable for storing information in quantum devices.

Technical Summary

Technical Summary: Dissipative Quantum North-East-Center Model

Problem Statement
This work investigates the non-equilibrium physics of the dissipative quantum North-East-Center (NEC) model, a two-dimensional spin-1/2 lattice system. The model combines chiral, kinetically constrained dissipation with coherent quantum interactions. The primary objective is to determine the steady-state phase diagram of this system under various Hamiltonians and to characterize the nature of its bistable phase. Specifically, the authors address whether the robust bistability observed in the classical NEC model persists in the presence of quantum fluctuations, how different microscopic sources of quantum noise (constrained vs. unconstrained) affect the extent of the bistable region, and the dynamical properties of the system, particularly the non-ergodic behavior of minority spin islands.

Methodology
The authors employ a Cluster Mean-Field (CMF) approach to solve the Lindblad master equation governing the system's open quantum dynamics.

• Cluster Mean-Field (CMF): To systematically include short-range correlations, the lattice is partitioned into square clusters of size $\ell \times \ell$. The dynamics of a given cluster are governed by a CMF Hamiltonian and dissipator that include on-cluster terms and boundary terms treated at a mean-field level, derived from neighboring clusters. The authors verify convergence by comparing results for cluster sizes $\ell=2$ and $\ell=3$.
• Inhomogeneous CMF (iCMF): To study real-time dynamics of large systems with non-translationally-invariant initial conditions, the authors utilize an inhomogeneous generalization of the CMF ansatz. This allows for the simulation of "islands" of spins surrounded by a background of opposite orientation.
• Stability Analysis: A linear stability analysis is performed by introducing small spatial fluctuations (plane waves) around the homogeneous steady-state. The eigenvalues of the resulting super-operator determine the stability of the phase and the nature of instabilities near critical points.
• Models Investigated: The study compares three distinct classes of Hamiltonians representing different sources of quantum fluctuations:
  1. Unconstrained: A homogeneous transverse magnetic field ($H_X = \Omega \sum \sigma^x_j$).
  2. Constrained (2D PXP): A PXP model with Rydberg blockade constraints on all four neighbors.
  3. NEC-Symmetric (Chiral PXP): A PXP model where constraints respect the chiral NEC geometry (acting only on North and East neighbors).
     Additionally, the stability against unconstrained dissipative processes (incoherent spin flips) is tested.

Key Results

1. Steady-State Phase Diagram:

   • The system exhibits two distinct phases: a normal phase with a unique steady-state and a bistable phase characterized by two steady-states with opposite macroscopic magnetization.
   • The phase diagram is defined by the amplitude of classical fluctuations ($T$), their bias ($h$), and the amplitude of quantum fluctuations ($\Omega$).
   • Universality: The existence of the bistable phase is a universal feature of the NEC model, persisting across all Hamiltonians considered. However, the extent of the bistable region depends on the microscopic details of the quantum fluctuations.
   • Impact of Constraints: Hamiltonians with kinetic constraints (PXP models) preserve the bistable phase more effectively than unconstrained Hamiltonians ($H_X$). The unconstrained transverse field destroys bistability at lower fluctuation amplitudes compared to constrained models.
   • Phase Transitions: The transition from the bistable to the normal phase is continuous at zero bias ($h=0$) but first-order at non-zero bias ($h \neq 0$). The critical lines follow a quadratic dependence on bias, $T_c(h) \propto (1-|h|)^2$.

2. Stability and Criticality:

   • Linear stability analysis reveals that instabilities emerge at finite wavevectors near the critical point.
   • The structure of these instabilities differs depending on the driving mechanism: quantum fluctuations induce instability peaks at finite wavevectors ($|k_x| = \pi/4$), whereas thermal fluctuations show a peak at $k_x=0$.

3. Non-Ergodic Dynamics and Island Reabsorption:

   • In the normal phase, minority islands of spins mix with the background and disappear, leading to a unique steady-state.
   • In the bistable phase, minority islands are always reabsorbed by the surrounding majority phase, regardless of the island's size. This confirms the non-ergodic nature of the bistable region, where the system retains memory of initial conditions (magnetization direction).
   • Reabsorption Velocity: The reabsorption process occurs at a constant velocity independent of the island size (linear scaling of relaxation time $\tau$ with island size $\ell_\downarrow$).
   • Equation of Motion: The authors propose a phenomenological equation for the reabsorption velocity $v$:
     $$\frac{dr}{dt} = -C + Bh + DT + E\Omega$$
     where $C$ is the classical absorption rate, $B$ relates to the bias, and $D, E$ quantify the linear reduction in velocity due to thermal ($T$) and quantum ($\Omega$) fluctuations, respectively. The analysis suggests quantum fluctuations have a weaker effect ($E \approx 0.25$) compared to thermal fluctuations ($D \approx 2.4$).

Significance and Claims
The paper claims that the dissipative quantum NEC model provides a robust mechanism for genuine bistability in quantum many-body systems, challenging the notion that quantum fluctuations inevitably destroy such phases. The findings highlight that:

• Robustness: Bistability is stable against various microscopic coherent processes and even additional unconstrained dissipative channels, suggesting potential relevance for experimental implementations (e.g., Rydberg atoms) where perfect isolation is impossible.
• Non-Ergodicity: The model exhibits non-ergodic dynamics where the system's fate is determined by the absorption of error islands, a mechanism that could theoretically serve as a strategy for quantum error correction.
• Methodological Contribution: The work demonstrates the efficacy of the inhomogeneous cluster mean-field (iCMF) method for studying the dynamics of large, two-dimensional dissipative lattice systems, particularly for analyzing non-translationally-invariant phenomena like domain wall motion.

The authors conclude that the interplay between chiral kinetic constraints and dissipation creates a unique playground for non-equilibrium phases of matter, distinct from thermal equilibrium behaviors, and suggest future directions involving higher-dimensional generalizations and connections to Hilbert space fragmentation.