Modular Hamiltonian and Typicality of Degenerate States

In the quiet corners of the universe, from the cooling of a star to the flow of electricity in a wire, matter tends to settle into a state of balance known as thermal equilibrium. For over a century, physicists have understood that when a system reaches this balance, its particles follow a predictable pattern of energy distribution, much like a crowd of people spreading out evenly in a room. This behavior is so reliable that it forms the bedrock of thermodynamics. However, a deeper question has long puzzled scientists: how does a single, isolated quantum system, which never truly forgets its past, manage to look like it has forgotten everything and settled into this calm state? The answer lies in a concept called eigenstate typicality. This idea suggests that if you look at a tiny piece of a large, complex quantum system that is in a specific high-energy state, that small piece will behave exactly as if it were part of a hot, thermal bath, even if the whole system is perfectly isolated and cold. It is as if the rest of the universe acts as a heat bath for the small piece, simply because of how the particles are entangled with one another. This phenomenon is crucial for understanding why the world looks the way it does, and it has recently become a bridge between the physics of black holes and the theory of quantum information.

A team of researchers at Hofstra University has now taken a closer look at this phenomenon, specifically asking what happens when the rules of the game are slightly altered. They focused on a system of free fermions, which are particles like electrons that do not push or pull on each other. In a standard scenario, if you take a single, typical high-energy state of these non-interacting particles and examine a small section of it, the mathematics describing that section is surprisingly simple. It behaves like a collection of independent particles, and the information about the rest of the system is encoded in a way that looks like a standard thermal distribution. This simplicity is a hallmark of the "eigenstate typicality" mentioned earlier. However, the researchers wondered what would happen if the system was not in just one single state, but was instead a mixture of several different states that happen to have the exact same energy. In the quantum world, it is possible to have multiple distinct arrangements of particles that share the same total energy, known as degenerate states. When you mix these states together, you create a new, more complex reality.

The team set out to investigate this mixture by performing precise numerical simulations on a one-dimensional lattice of fermions. They constructed a system where the particles were arranged in a line and then created a specific quantum state by blending two different energy configurations that shared the same energy level. They then calculated the mathematical description, known as the modular Hamiltonian, which acts like a map of how the small section of the system is connected to the rest. In the case of a single, pure state, this map is simple and describes non-interacting particles. But when they introduced the mixture of degenerate states, the results changed dramatically. The map they found was no longer simple. Instead of describing independent particles, the new map revealed that the particles within the small section appeared to be interacting with one another, even though the fundamental laws governing the entire system stated that the particles should not interact at all.

This discovery suggests that the act of mixing degenerate states introduces a kind of effective interaction that was not present before. The researchers found that the mathematical description of the system became "non-Gaussian," a technical way of saying that the particles' behavior deviated from the standard, smooth distribution expected of a thermal system. Instead of following the familiar rules of free particles, the small section of the system began to exhibit correlations that looked like the particles were pushing and pulling on each other. These interactions were not local, meaning they did not just happen between neighbors; they spanned across the entire small section, linking particles that were far apart. The researchers confirmed this by running their calculations on small systems with up to twelve sites, where they could track the exact behavior of the particles. They observed that even a tiny amount of mixing between the degenerate states was enough to trigger these complex interactions, causing the simple, non-interacting picture to break down immediately.

The implications of this finding reach beyond the specific numbers of their simulation. The researchers propose that this effect might explain how information about the initial conditions of a system is preserved as it evolves toward equilibrium. In a typical thermal system, we assume that all memory of how the system started is lost, replaced by a uniform temperature. However, if the system is a mixture of degenerate states, the "interaction terms" that appear in the mathematical map of the small section could act as a hidden code. This code would encode the specific details of how the system was prepared, stored in the subtle correlations between the particles. While these correlations are too complex to measure directly in a real-world experiment, the theory suggests they are there, waiting to be decoded. This challenges the traditional view that equilibrium always means a complete loss of history, suggesting instead that the history might be written in the very fabric of the particle interactions, even in a system that appears to be perfectly balanced.

The study also highlights a tension between this new finding and the standard picture of how fermions behave at low energies. In the real world, fermions often have weak repulsive forces between them, but these forces usually become negligible as the energy drops, leaving the particles to move freely. The researchers' work suggests that even in a system where particles are fundamentally free, the way we choose to describe the system—specifically, by mixing different states of the same energy—can create the illusion of strong, complex interactions. This does not mean the particles are actually colliding, but rather that the mathematical tool used to describe the small section of the system, the modular Hamiltonian, acquires terms that look like interactions. This distinction is vital for understanding the limits of our current theories, particularly in fields like quantum error correction and the study of black holes, where the relationship between a small part of a system and the whole is central to the theory.

Ultimately, the work serves as a reminder that the behavior of a quantum system is deeply sensitive to how it is prepared. The researchers did not find a new force of nature, but rather a new way in which the rules of quantum mechanics can conspire to create complexity from simplicity. By mixing states that share the same energy, they showed that the "typical" behavior we expect from thermal systems is not the only possibility. The system can retain a fingerprint of its initial state, encoded in the strange, long-range connections between its particles. This finding opens a new door for understanding how information survives in a quantum world, suggesting that the path to equilibrium is not a straight line where all details are erased, but a landscape where the history of the system might be hidden in plain sight, written in the language of interactions that only appear when we look closely enough.