Spin-Charge Subordination in the Infinite-$U$ $SU(N)$ Hubbard Chain

In the quantum world, particles often behave less like solid marbles and more like waves that can pass through one another. Yet, under extreme conditions, these waves can become so crowded that they are forced to line up in a single file, unable to overtake their neighbors. This phenomenon, known as the "no-passing" constraint, turns a chaotic gas into an orderly queue where the position of every particle is locked relative to the others. Scientists study these systems to understand how information and energy move through materials that are completely impenetrable, a scenario that is not just a theoretical curiosity but a reality in experiments with ultracold atoms trapped in laser grids. The key question researchers have long asked is how two different types of movement—how the particles move through space and how their internal identities change—interact when they are forced to move together in such a tight line.

A team of physicists has now provided a complete and exact answer to this question for a specific, highly complex model of quantum matter. By studying a one-dimensional chain of atoms where particles cannot double up on the same spot, they discovered a strict rule that governs how these particles transport their internal "flavors" or spin states. In this system, the movement of the particles themselves is relatively straightforward and fast, behaving like a stream of free, invisible runners. However, the transport of their internal labels is a much slower, more complicated process that is entirely dependent on the first one. The researchers found that the internal state of the system does not evolve on its own; instead, it is strictly subordinate to the number of particles that happen to cross a specific point in the chain.

To reach this conclusion, the team analyzed a mathematical model known as the infinite-U SU(N) Hubbard chain. In this model, the "U" represents a repulsive force so strong that two particles are forbidden from occupying the same location. The "SU(N)" part refers to the fact that each particle can exist in one of many possible internal states, or flavors, rather than just the two states (up or down) found in simpler magnetic systems. The researchers focused on a scenario where the system is at a very high temperature, meaning the particles are arranged in a completely random order of flavors at the start. They then simulated how this system evolves over time, tracking two things simultaneously: how many particles crossed a central dividing line, and what the total sum of their internal flavors was after they crossed.

The results revealed a precise and surprising relationship between these two quantities. Because the particles cannot pass one another, the sequence of their internal flavors remains frozen in place, like a necklace of colored beads that cannot be rearranged. As the particles move, they carry these fixed beads with them. Consequently, the total amount of "flavor" that crosses the dividing line is simply the sum of the specific beads carried by however many particles happened to cross. If ten particles cross, the flavor transport is the sum of the ten specific beads attached to those ten particles. This creates a direct link where the chaotic randomness of the internal state is entirely determined by the statistics of the particle crossings.

This connection allowed the researchers to derive an exact formula that predicts the full distribution of flavor transport based solely on the distribution of particle transport. They found that while the number of particles crossing the line follows a familiar, bell-curve pattern typical of random walks, the transport of the internal flavors behaves very differently. The fluctuations in the flavor transport are significantly slower and follow a distinct, non-Gaussian shape known as the M-Wright distribution. In practical terms, this means that while the particles themselves spread out quickly, their internal identities spread out much more slowly, growing only as the fourth root of time rather than the square root. This "subordination" mechanism explains why internal states in such crowded quantum gases move anomalously slowly compared to the particles carrying them.

The team verified these theoretical findings using powerful computer simulations for systems with two, three, and four different internal flavors. In every case, the data collapsed perfectly onto the predicted universal curve, confirming that the rule holds true regardless of how many different types of particles are involved. The researchers also noted that this behavior is not limited to the specific high-temperature conditions they modeled; the underlying mechanism of frozen flavor order remains valid even at lower temperatures, though the exact speed of the movement would change. This universality suggests that the slow transport of internal states is a fundamental feature of any impenetrable quantum gas, independent of the specific details of the particles involved.

These findings offer a clear explanation for a phenomenon that was previously observed only in simpler systems. By proving that the transport of internal states is kinematically tied to the transport of particles, the study provides a new way to understand how quantum information moves in constrained environments. The results are directly relevant to current experiments with ultracold atoms, particularly those using elements like ytterbium or strontium, which can mimic these multi-flavor quantum chains. Experimentalists can now look for the specific signature of this slow, M-Wright distributed transport in their own data, using the precise mathematical predictions from this work as a benchmark. The study confirms that in the quantum world, when particles are forced to move in a single file, their internal lives are bound by the same strict rules of order that govern their physical positions.