🔬 condensed matter

Spin subdiffusion in perturbed infinite-U Hubbard chain

This paper investigates spin dynamics in a perturbed infinite-U Hubbard chain, demonstrating that while integrability yields ballistic or anomalous diffusion, breaking integrability while preserving Hilbert space fragmentation leads to charge-mediated spin subdiffusion with a mechanism distinct from disorder or dipole-conserving systems.

Original authors: Jakub Rękas, Marcin Mierzejewski, Zala Lenarčič, Peter Prelovšek

Published 2026-03-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jakub Rękas, Marcin Mierzejewski, Zala Lenarčič, Peter Prelovšek

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 a long, narrow hallway filled with people. In this hallway, there are two types of people: Red Shirts and Blue Shirts. They are holding hands in a tight line, and there is a strict rule: no two people can ever stand on the same spot. If they try to jump, they bounce off each other.

This hallway represents a Quantum Chain (specifically, the infinite-𝑈 Hubbard model), and the people represent electrons. In the world of physics, understanding how these "people" move and interact is the key to understanding how electricity and magnetism work in materials like superconductors.

Here is the story of what happens when we try to get these people to move, based on the research in the paper.

1. The Frozen Line (The "Integrable" Case)

In the perfect, un-perturbed version of this hallway, the rules are very rigid.

The Charge (Movement): The people can slide down the hallway freely, like a train on a frictionless track. They move perfectly fast. This is called Ballistic Transport.
The Spin (Color): Here is the weird part. Even though the people are sliding, their order never changes. If you have a line like Red-Blue-Red-Red-Blue, that specific pattern of colors is "frozen" in place. The Red person can't swap places with the Blue person because they can't pass each other.

The Analogy: Imagine a line of dancers holding hands, sliding across an ice rink. They can glide forward together, but they can never let go to switch places with their neighbor. The "dance formation" (the spin sequence) is stuck.

The Result:

If the hallway has an equal number of Red and Blue shirts (Zero Magnetization), nothing happens. No net flow of color. The system is stuck in a way that prevents any steady flow of "spin."
However, if you look at the average behavior over many different possible lineups, you find something surprising: the colors spread out in a strange, "ghostly" way. It's not a normal flow; it's Anomalous Diffusion. It's like the colors are spreading, but without losing any energy, as if the hallway has no friction at all.

2. The Chaos (The "Perturbed" Case)

Now, imagine we introduce a little bit of chaos. Maybe the floor isn't perfectly smooth, or the people have slightly different shoe sizes (this is the perturbation).

The Charge: The perfect sliding stops. The people start bumping into each other and slowing down. The movement becomes Normal Diffusion (like a drunk person stumbling down a hallway).
The Spin: Because the order of colors is still frozen (they still can't swap places), the "color flow" is still stuck. But now, because the people are stumbling (diffusing), the frozen color patterns are dragged along with them.

The Big Discovery:
The researchers found that when you mix this "stumbling" charge movement with the "frozen" spin order, something very unusual happens to the spin.
Instead of spreading out normally, the spin spreads out extremely slowly. This is called Subdiffusion.

The Creative Metaphor: The Porous Sponge
Think of the spin spreading like water soaking into a sponge.

Normal Diffusion: Water soaks in at a steady, predictable rate.
Subdiffusion: Imagine the sponge is made of a material that gets harder the more water tries to get in. The more the water spreads, the more it resists. The water moves, but it slows down drastically the further it goes.

In this quantum hallway, the "Red" and "Blue" regions act like this sponge. The "Red" people are stuck in a cluster, and the "Blue" people are stuck in another. Because they can't swap, the only way for the colors to mix is for the whole group to shuffle forward. But as the groups get bigger, it gets harder for them to shuffle. The result is a Subdiffusive spread: the colors move, but they move slower and slower over time.

3. Why This Matters

Usually, when physicists see things moving very slowly (subdiffusion), they blame it on disorder (like a hallway full of random obstacles) or conservation laws (like a rule that says "you can't move your hands").

But this paper shows something unique:

The hallway is perfectly clean (no disorder).
The rules are standard (no weird conservation laws).
The only reason the movement is so slow is because of the frozen order (Hilbert Space Fragmentation). The people are trapped in their specific lineups, and the only way to move is to drag the whole line along.

Summary

The Setup: Electrons in a 1D chain that can't pass each other.
The Twist: The order of their "spins" (colors) is frozen, but they can still move as a group.
The Finding:
- In the perfect world, the spins spread out in a weird, frictionless way.
- In the messy (perturbed) world, the spins get stuck in a "slow-motion" mode called Subdiffusion.
The Lesson: Even in a perfectly ordered system, if the particles are "jammed" into specific patterns they can't break, the transport of information (spin) can become incredibly sluggish. It's a new kind of traffic jam caused not by bad roads, but by the strict rules of the dance formation itself.

1. Problem Statement and Motivation

The paper investigates the transport properties of spin excitations in one-dimensional (1D) strongly correlated electron systems, specifically focusing on the infinite- $U$ Hubbard model (the $t$ -model).

Context: The 1D Hubbard model at $U \to \infty$ maps to the integrable $t$ -model. In this limit, double occupancy is forbidden, leading to Hilbert-space fragmentation. The sequence of spins becomes "frozen" (dynamically decoupled subspaces), while charge carriers (holes) move freely.
The Gap: While charge transport in the integrable $t$ -model is ballistic, the behavior of spin transport is non-trivial due to the frozen spin sequence. Previous studies on similar models (like the folded XXZ model) suggested anomalous diffusion, but the specific mechanism in the $t$ -model, particularly under perturbations that break integrability but preserve fragmentation, was not fully understood.
Key Question: How does spin transport behave in the integrable $t$ -model versus perturbed versions (non-integrable) where integrability is broken but Hilbert-space fragmentation persists? Does the system exhibit ballistic, diffusive, or subdiffusive transport?

2. Methodology

The authors employ a combination of analytical arguments and advanced numerical techniques:

Models Studied:
1. Integrable $t$ -model: $H_t = -\sum t_\sigma \tilde{c}^\dagger_{l+1,\sigma} \tilde{c}_{l,\sigma} + \text{H.c.}$ with $t_\uparrow = t_\downarrow$ .
2. Perturbed $t$ -model: Introduces spin-dependent hopping ( $t_\uparrow \neq t_\downarrow$ , parameterized by $\Delta t$ ) or $t-J_z$ interactions ( $J_z \sum S^z_i S^z_{i+1}$ ), which break integrability but maintain the impenetrability of particles (fragmentation).
3. Broken Fragmentation: Models including spin exchange ( $J_\perp$ ) to contrast results.
Analytical Tools:
- Thouless Level Sensitivity (LS): Analyzing the sensitivity of energy levels to spin-dependent magnetic flux ( $\varphi$ ) to determine spin current expectation values.
- Mazur Inequality: Used to establish lower bounds for spin stiffness based on overlaps between spin and charge currents.
- Porous Medium Equation: Used to derive the hydrodynamic equation for spin density spreading.
Numerical Methods:
- Microcanonical Lanczos Method (MCLM): Used to calculate dynamical correlation functions and spin diffusion constants for system sizes up to $L=20$ .
- Open Quantum Systems: Solving the Lindblad master equation for boundary-driven systems to study steady-state currents.
- Liouville Spectrum Analysis: Examining the eigenvalues of the Liouvillian superoperator to understand non-decaying oscillations and steady states.

3. Key Contributions and Results

A. Intrinsic Coupling of Spin and Charge Currents

The authors establish that in the $t$ -model, spin transport is strictly mediated by charge transport due to Hilbert-space fragmentation.

Zero Magnetization ( $m=0$ ): In any fixed magnetization sector with $m=0$ , the diagonal matrix elements of the spin current vanish ( $\langle n|j_s|n \rangle = 0$ ). Analytical arguments show that the energy levels are independent of spin-dependent flux at $m=0$ .
Open Systems: Numerical simulations of boundary-driven open systems confirm that without a mechanism to drive a charge current (e.g., charge exchange at boundaries), no steady-state spin current can be generated, even if spin-flipping terms are present.
Conclusion: Spin current is intrinsically coupled to charge current. If charge transport is blocked (or averages to zero in a specific sector), spin transport is also blocked.

B. Transport in the Integrable $t$ -Model

Finite Magnetization ( $m \neq 0$ ): The system exhibits ballistic transport. The spin stiffness $D_s$ is finite and scales as $m^2$ .
Grand Canonical (GC) Average ( $m=0$ ): When averaging over all magnetization sectors, the system exhibits dissipationless (anomalous) diffusion.
- Although $D_s \propto m^2$ vanishes for a specific $m=0$ sector, the GC average $\langle m^2 \rangle_{GC} \propto 1/L$ combined with the boundary scattering time $\tau_L \propto L$ yields a finite, size-independent diffusion constant $D^0_s \approx 0.6t$ .
- This is analogous to the easy-axis XXZ model.

C. Transport in the Perturbed $t$ -Model (Main Discovery)

When integrability is broken (via $\Delta t \neq 0$ or $J_z \neq 0$ ) but fragmentation is preserved:

Finite Magnetization ( $m \neq 0$ ): Transport transitions from ballistic to normal diffusion. The diffusion constant scales as $D_s(m) \propto m^2 / \Delta t^2$ .
Grand Canonical Average ( $m=0$ ): The system exhibits spin subdiffusion.
- The diffusion constant for a specific sector scales as $D_s(m) \propto m^2$ .
- In the hydrodynamic limit, the local magnetization density $u(x,t)$ evolves according to a nonlinear equation:
  $\frac{\partial u}{\partial t} \propto \frac{\partial}{\partial x} \left( u^2 \frac{\partial u}{\partial x} \right)$
  This is the Porous Medium Equation.
- Scaling: The solution to this equation predicts a spreading profile $u(x,t) \sim t^{-1/4}$ , leading to a local spin correlation function scaling as $C_l(\omega) \sim \omega^{-3/4}$ .
- Numerical Verification: MCLM calculations for the perturbed $t$ -model and $t-J_z$ model confirm the exponent $\gamma \approx 3/4$ in the frequency dependence of local correlations, providing strong evidence for subdiffusion.

D. Role of Fragmentation Breaking

The authors demonstrate that if the perturbation allows for spin exchange ( $J_\perp \neq 0$ ), the Hilbert-space fragmentation is destroyed. In this case, the system reverts to normal diffusion even at $m=0$ , confirming that the subdiffusive behavior is a direct consequence of the preserved fragmentation.

4. Significance

Novel Mechanism: This work identifies a unique mechanism for subdiffusion in translationally invariant, clean quantum systems. Unlike subdiffusion in disordered systems (Many-Body Localization) or dipole-conserving systems (fractons), this subdiffusion arises from Hilbert-space fragmentation combined with integrability breaking.
Universality: The findings link the $t$ -model to the folded XXZ model, suggesting a broader universality class for fragmented systems where spin transport is mediated by diffusing holes through frozen spin domains.
Physical Insight: The paper clarifies the crucial role of the coupling between charge and spin currents. It shows that in fragmented systems, spin transport cannot be understood in isolation; it is entirely dependent on the dynamics of the charge carriers (holes).
Experimental Relevance: The results are relevant for cold-atom experiments simulating the Hubbard model and for understanding transport in high-temperature superconductors (cuprates) where strong correlations and $U \to \infty$ limits are often invoked.

Summary Conclusion

The paper demonstrates that while the integrable infinite- $U$ Hubbard chain exhibits dissipationless spin diffusion in the grand canonical ensemble, introducing perturbations that break integrability (but preserve Hilbert-space fragmentation) drives the system into a subdiffusive regime. This subdiffusion is governed by the porous medium equation, where the diffusion coefficient depends on the local magnetization squared ( $D \propto u^2$ ), a unique phenomenon distinct from disorder-induced or dipole-conserving subdiffusion.