Coherence dynamics in quantum many-body systems with conservation laws

This paper investigates how conservation laws influence the spreading of quantum coherence in many-body systems, demonstrating that they replace the logarithmic saturation seen in unconstrained circuits with hydrodynamic relaxation and distinct local peak structures that differentiate symmetry-constrained circuits from ergodic Hamiltonians.

The Gist

Imagine you are at a massive, crowded music festival. This paper is essentially a study of how "vibes" (which the scientists call quantum coherence) spread through a crowd, and how certain "rules" (which they call conservation laws) act like security checkpoints or narrow corridors that change how those vibes move.

Here is the breakdown of the paper using everyday analogies.

1. The "Vibe" (Quantum Coherence)

In the quantum world, particles don't just exist in one state; they can exist in a "superposition"—a blurry, magical state of being in multiple places or conditions at once. This "blurriness" is coherence.

Think of it like a group of people at a festival. If everyone is standing perfectly still in a grid, the "vibe" is zero. But if people start dancing and interacting, a "vibe" spreads. The paper asks: If we start with a very boring, organized crowd, how long does it take for the "vibe" to spread to everyone, and how does it change if the crowd has strict rules?

2. The "Rules" (Conservation Laws)

In a normal, chaotic crowd (what scientists call "unconstrained"), people can move anywhere. The vibe spreads incredibly fast, like a wildfire.

However, the paper looks at systems with Conservation Laws. These are like rules at the festival:

• The U(1) Rule (The "Ticket" Rule): Imagine every person has a specific colored ticket. You can move around, but the total number of red, blue, and green tickets in the whole festival must stay exactly the same. You can swap tickets with a neighbor, but you can't create new ones.
• The Dipole Rule (The "Buddy System" Rule): This is even stricter. Not only must the number of tickets stay the same, but the balance of where they are must be preserved. It’s like saying if a red ticket moves left, a blue ticket must move right to keep the "center of gravity" of the colors in the same spot. This makes movement very clunky and slow.

3. What the Researchers Found

The Global View: The Slow Burn

In a normal crowd, the "vibe" hits everyone almost instantly (logarithmic growth). But when you add the "Ticket Rules," the vibe spreads through diffusion—much like a drop of ink spreading in a glass of water. It’s a slow, steady soak rather than a sudden explosion. The more people you add to the festival, the longer it takes for the vibe to reach everyone.

The Local View: The "Rise and Fall"

The researchers looked at small groups (subsystems) within the crowd. They noticed a fascinating pattern in the "local vibe":

1. The Rise: As the dance starts, the local group gets very excited (coherence increases).
2. The Peak: The vibe reaches a maximum.
3. The Fall: Eventually, the group becomes so "entangled" with the rest of the crowd that they lose their unique local rhythm and just become part of the background noise (coherence decays).

The "Plateau" Surprise (Hamiltonian Dynamics)

When they studied "real-world" physics (Hamiltonian dynamics) instead of just random "circuit" models, they saw something weird. Instead of the vibe peaking and then falling, it hit a plateau. It’s like a group of dancers who reach a peak level of excitement and then just stay in that high-energy state for a long, long time instead of cooling down.

Summary: Why does this matter?

The researchers proved that coherence is a "thermometer" for symmetry.

By watching how the "vibe" spreads, we can tell exactly what kind of rules are governing a system. If the vibe spreads like a wildfire, there are no rules. If it spreads like ink in water, there are conservation laws. If it gets stuck in a plateau, it’s a complex, interacting physical system.

This helps scientists understand how to control quantum information—the "vibes" of the future—in quantum computers, where keeping that coherence alive is the ultimate goal.

Technical Summary

Technical Summary: Coherence Dynamics in Quantum Many-Body Systems with Conservation Laws

Authors: Sreemayee Aditya, Emanuele Tirrito, Piotr Sierant, and Xhek Turkeshi
Date: April 28, 2026 (arXiv:2604.23192)

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1. Problem Statement

The paper investigates how conservation laws (symmetries) fundamentally alter the spreading of quantum coherence in isolated many-body systems. While unconstrained random circuits scramble information and entanglement ballistically, leading to a rapid (logarithmic) saturation of complexity, the presence of conserved charges (like total magnetization or dipole moment) is expected to slow down this process via hydrodynamic modes.

The authors specifically ask:

• How do conservation laws change the global spreading of the wavefunction in the computational basis?
• How do they affect the local coherence retained within a subsystem?
• Is there a qualitative difference between the coherence dynamics in random circuits (Floquet systems) versus ergodic Hamiltonians?

2. Methodology

The study employs a multi-pronged numerical and analytical approach to probe three paradigmatic dynamical settings:

1. $U(1)$-symmetric random circuits: Conserving total magnetization (spin-1/2 and spin-1).
2. Charge- and dipole-conserving circuits: A "fractonic" setting where both total charge and dipole moment are conserved, leading to Hilbert-space fragmentation.
3. Mixed-field Ising Hamiltonian (MFIM): A representative of local ergodic Hamiltonian dynamics.

Quantities of Interest:

• Global Probe: Rényi-2 Participation Entropy ($S_d$), which measures how broadly the many-body wavefunction is distributed across the computational basis.
• Local Probe: Relative Entropy of Coherence ($C_d$), defined as the difference between the subsystem diagonal entropy ($S_d(\rho_A)$) and the subsystem entanglement entropy ($S_R(\rho_A)$). This quantifies the "non-classicality" of a subsystem.

Numerical Techniques:

• Exact State-Vector Evolution: For small system sizes.
• Replica Tensor Networks (RTN): To compute second-moment quantities (purities) for large-scale systems.
• Matrix Product States (MPS) & TDVP: Specifically for the Hamiltonian dynamics to reach large subsystem sizes.
• Analytical Methods: Haar-averaging formulas for purities and a "rare-region" analysis using the Symmetric Simple Exclusion Process (SSEP) effective model.

3. Key Contributions and Results

A. Global Dynamics: From Logarithmic to Power-Law Relaxation
In unconstrained circuits, the deviation from saturation ($\Delta S_d$) decays exponentially and saturates in $O(\log L)$ time. In all symmetric systems studied, this is replaced by:

• Two-stage relaxation: An intermediate algebraic (power-law) decay ($\Delta S_d \sim t^{-\beta}$) followed by a late-time exponential decay once finite-size effects dominate.
• Hydrodynamic Scaling: The saturation time $t_{\epsilon}$ scales as a power law ($L^{\alpha}$) rather than $\log L$. This confirms that the slowest modes in the system are the hydrodynamic modes associated with the conserved charges.

B. Local Dynamics: The Rise–Peak–Fall Structure
In $U(1)$ and dipole-conserving circuits, the local coherence $C_d(t)$ exhibits a distinct rise–peak–fall profile:

• Rise: Driven by the initial generation of coherence.
• Peak: Occurs at a time $\tau_c^m$ that scales algebraically with subsystem size ($\tau_c^m \propto L_A^{\alpha_m}$). This peak is the result of the competition between the algebraic relaxation of $S_d$ and the ballistic growth of entanglement $S_R$.
• Fall: Driven by the buildup of entanglement between the subsystem and its environment, eventually leading to an incoherent (diagonal) state.

C. Hamiltonian vs. Circuit Dynamics
A major finding is the qualitative distinction in Hamiltonian systems:

• While symmetric circuits show a sharp coherence peak, ergodic Hamiltonians exhibit a peak-to-plateau crossover. As the subsystem size $L_A$ increases, the sharp peak broadens into an extended plateau. This signals a fundamentally different local relaxation mechanism compared to the fast scrambling seen in random circuits.

D. Hilbert-Space Fragmentation (Fractonic Dynamics)
In the charge-and-dipole-conserving circuits, the authors demonstrate that the dynamics are confined to specific Krylov fragments. While the qualitative "rise-peak-fall" remains, the exponents are modified by the constrained motion of the dipolar degrees of freedom, providing a signature of fractonic behavior.

4. Significance

This work establishes quantum coherence as a sensitive probe of symmetry-constrained thermalization. By linking the dynamics of a quantum resource (coherence) to the underlying hydrodynamics of conserved charges, the paper provides a new framework for understanding how information and resources spread in complex many-body systems. It bridges the gap between Quantum Information Theory (resource theories) and Condensed Matter Physics (hydrodynamics and transport).