Logarithmic Entanglement Growth

In the quantum realm, systems of interacting particles are typically expected to thermalize, scrambling information and erasing all memory of their initial state. However, a fascinating exception exists in the form of many-body localization (MBL), where strong disorder prevents a system from reaching thermal equilibrium, effectively trapping information and halting the transport of energy. This raises a profound question: If an MBL system is a perfect insulator where nothing appears to move, how does quantum information, in the form of entanglement, spread? This article delves into the slow, yet persistent, creep of entanglement that defines these exotic systems.

The first chapter, "Principles and Mechanisms", will demystify the origin of logarithmic entanglement growth, introducing the concept of quasi-local integrals of motion (l-bits) and explaining how their ghostly, long-range dephasing governs the system's dynamics. Following this, the "Applications and Interdisciplinary Connections" chapter will broaden our perspective, revealing how this same logarithmic behavior emerges as a unifying principle in diverse physical scenarios, from periodically driven systems and quantum critical points to the new frontier of measurement-induced phase transitions.

Imagine a vast room filled with people, each humming a note. In a normal, "thermal" room, if you start a new conversation in one corner, the sound waves spread out, and soon everyone is chattering about the new topic. Information scrambles, and the room reaches a new, uniform state of noise. This is the world of thermalization. But what if the room were filled with a strange, sound-absorbing fog? What if the people were so intensely focused that they could only faintly hear their immediate neighbors? This is the strange, quiet world of many-body localization (MBL). Here, a local secret remains a local secret, information gets stuck, and the system stubbornly refuses to thermalize, preserving a memory of its initial state for incredibly long times.

But if nothing moves, and no energy or particles are transported, is the system simply frozen? Not quite. A subtle, purely quantum "conversation" still takes place, leading to one of the most fascinating phenomena in modern physics: the slow, logarithmic growth of entanglement. Let's peel back the layers and see how this works.

The first key to unlocking the MBL puzzle is to realize that the fundamental particles we start with—say, the spins on a lattice—are not the most useful characters in our story. In the presence of strong disorder and interactions, the system finds a new set of "true" degrees of freedom. Physicists call them ​​quasi-local integrals of motion​​, or ​​l-bits​​ for short. You can think of an l-bit as a "dressed" version of the original spin. It’s still localized near a specific site, but its structure is smeared out a little by the interactions with its neighbors.

The incredible thing about these l-bits is that they form a hidden, simpler order within the complex system. Unlike physical spins, the orientation of each l-bit along a specific axis (let's call it the z-axis, $\tau_i^z$) is a ​​conserved quantity​​. This means once you set the value of $\tau_i^z$ for the l-bit at site $i$, it stays that way forever. This immediately explains why MBL systems are perfect insulators: since the l-bits can't flip, they can't carry charge, spin, or energy across the system.

The dynamics of the system can be almost entirely described by an ​​effective Hamiltonian​​ written in terms of these l-bits:

This equation looks a bit intimidating, but the idea is simple. The first term, $\sum_i \tilde{h}_i \tau_i^z$, just gives each l-bit its own energy. The crucial part is the interaction terms, like $\sum_{i<j} \tilde{J}_{ij} \tau_i^z \tau_j^z$. This term tells us that the total energy depends on the relative orientation of pairs of l-bits. And here is the most important ingredient of all: the interaction strength $\tilde{J}_{ij}$ between two l-bits separated by a distance $r = |i-j|$ is not constant; it falls off exponentially fast, something like $\tilde{J}_{ij} \propto \exp(-r/\xi)$, where $\xi$ is a special parameter called the ​​localization length​​. The l-bits are gossiping, but their voices are just whispers that fade incredibly quickly with distance.

So, if the l-bit orientations $\tau_i^z$ are all frozen, how can any entanglement possibly grow? This is where the magic of quantum mechanics comes in. While the population of each l-bit state is fixed, its phase is not.

Let's go back to our analogy of humming people, but now let's make them quantum spinning tops (our l-bits). The conservation of $\tau_i^z$ means they can't flip from pointing "up" to "down". But they can still spin around the z-axis, like a clock hand. The effective Hamiltonian tells us that the speed of this spinning—the precession frequency—for a top at site $i$ depends on the orientation of all the other tops! A top at site $j$ being "up" or "down" will slightly speed up or slow down the spinning of the top at site $i$.

This effect is known as ​​dephasing​​. An initial state, like a product state where every l-bit is in a superposition of up and down, will evolve in a complex way. The parts of the wavefunction corresponding to different l-bit configurations (up-up-down..., up-down-down..., etc.) accumulate phase at different rates. Over time, the state of l-bit $i$ becomes inextricably linked to the state of l-bit $j$ through these phase relationships. This growing web of phase correlations is quantum entanglement. It’s a ghostly conversation where no "sound" (energy or particles) is exchanged, yet the tops become aware of each other's state.

This dephasing mechanism gives us a way to calculate the speed at which entanglement spreads. The logic is simple and beautiful.

Two l-bits, $i$ and $j$, separated by a distance $r$, will only become significantly entangled when the phase difference they accumulate from their interaction, which is roughly $\tilde{J}_{ij} \cdot t$, becomes noticeable (say, of order 1). This defines a characteristic ​​dephasing time​​:

Since the interaction $\tilde{J}(r)$ decays exponentially, $\tilde{J}(r) \sim \exp(-r/\xi)$, the time required to entangle over a distance $r$ grows exponentially!

To entangle with a neighbor just 10 sites away might take a thousand times longer than entangling with a neighbor just one site away. To travel a linear distance $L$, information must wait for an exponentially long time, $t_{\text{travel}} \propto \exp(L/\xi')$ for some constant $\xi'$.

Now, let's turn the question around: at a given time $t$, how far has the entanglement "front" propagated? We just need to invert the relationship. The characteristic distance $r(t)$ over which entanglement has developed is:

This is the heart of the matter. The entanglement doesn't spread out ballistically like a shockwave; it crawls outward at a painfully slow ​​logarithmic​​ pace. The total bipartite entanglement entropy, $S(t)$, across a cut is proportional to the size of this entangled region. Therefore, it also grows logarithmically:

The prefactor $\chi$, sometimes called the "entanglement velocity", is directly proportional to the localization length $\xi$. A larger $\xi$ means the l-bits have a longer "reach," and entanglement can spread a bit faster, but the growth remains fundamentally logarithmic. If multiple interaction mechanisms are present, the one with the longest reach (the largest $\xi$) will inevitably dominate the entanglement growth at long times.

This same ghostly conversation has another, equally important consequence. Imagine we focus not on the whole system, but on just a single l-bit at the center. We prepare it in a perfect quantum superposition state. What happens to it?

It ​​decoheres​​. As its phase becomes entangled with the ever-growing number of l-bits in its environment, its local quantum nature fades away. The purity of its initial state is lost to the non-local correlations being built across the system. And here's the beautiful unity: the rate of this decoherence is governed by the exact same physics as the growth of entanglement. The decay of the local superposition can be shown to follow a power law in time, $| \rho_{\uparrow\downarrow}(t) | \propto t^{-\gamma}$. The decay exponent $\gamma$ is directly proportional to the very same entanglement velocity $\chi$ that governs the logarithmic growth. The loss of coherence in one place is just the mirror image of the spread of entanglement everywhere else.

This slow, logarithmic dance of entanglement is a delicate quantum effect. It lives on a knife's edge, and it's easy to break the spell. What does it take?

One way is to make the interactions between l-bits too long-ranged. If instead of decaying exponentially, the interaction decays as a slower power-law, $V(r) \sim \frac{1}{r^{\alpha}}$, the dephasing can be much faster. In fact, if the interaction is too strong (if $\alpha$ is too small), it can trigger a "thermal avalanche." A single resonant pair of l-bits can destabilize their neighbors, which in turn destabilize their neighbors, and the whole system melts from a localized state into a thermal one. For a system in $d$ dimensions, theory predicts that the MBL phase and its logarithmic growth are only stable if the interactions are sufficiently short-ranged, specifically, if $\alpha > 2d$.

Another way to break the spell is to simply listen in from the outside. If the system is coupled to an external environment, this environment introduces random noise, causing each l-bit to decohere at a certain rate $\gamma_d$. This sets a maximum time, $t_{\text{coh}} \sim \frac{1}{\gamma_d}$, for any coherent quantum process to occur. The logarithmic growth of entanglement is cut short at this time. If the external noise is too strong, the coherence time becomes so short that entanglement doesn't even have time to spread to the nearest-neighbor l-bit. The MBL-driven dynamics are completely washed out, and the system behaves like a conventional, uninteresting insulator.

The logarithmic growth of entanglement is therefore more than just a mathematical curiosity. It is the definitive dynamical signature of a remarkable phase of matter, one that balances perfectly between the frozen stillness of a simple insulator and the chaotic scrambling of a thermal metal. It is a testament to the subtle, non-local, and profoundly quantum conversations that can happen even in a world where nothing seems to move at all.

So, we have discovered this peculiar law of nature: in certain disordered, interacting quantum systems—what we call many-body localized (MBL) systems—entanglement does not explode through the system as one might expect. Instead, it creeps. It spreads with the slow, deliberate pace of a logarithm, $S(t) \sim \ln(t)$. This is already a fascinating piece of physics, a departure from the "fast scrambling" of thermalizing systems.

But the real joy in physics is not just in discovering a new rule, but in seeing how it connects to everything else. Where does this strange behavior show up? What can we do with it? And does it hint at even deeper principles? This is our journey now: to trace the fingerprints of logarithmic entanglement growth across a breathtaking landscape of modern physics, from the heart of condensed matter to the frontiers of quantum information and measurement.

Let's first get a better feel for the central mechanism. Imagine an MBL system as a long chain of tiny, localized quantum "tops," which we call l-bits. Each l-bit is "stuck" near its own site, a prisoner of the system's disorder. But they are not entirely isolated. They can "talk" to each other through quantum interactions. The crucial feature, a direct consequence of localization, is that this talk is like a faint whisper that dies off exponentially with distance, $J(r) \sim \exp(-r/\xi)$, where $\xi$ is the all-important localization length.

Now, consider a partition down the middle of our chain. Entanglement grows when a top on the left side becomes quantum-mechanically linked with a top on the right. This linking, or dephasing, takes time. For nearest neighbors across the partition, the whisper is relatively loud, and they entangle quickly. But for a top deep inside the left region to entangle with a top on the right, it has to communicate across a large distance. The signal is exponentially weak, and so the time it takes is exponentially long!

If the time to entangle grows exponentially with distance, $t \sim \exp(d/\xi)$, then we can flip the question around: at a time $t$, what is the maximum distance $d$ over which entanglement has spread? A little bit of algebra tells you that the distance grows only as the logarithm of time, $d \sim \xi \ln(t)$. Since the total entanglement is roughly the number of entangled tops, and this number is proportional to the entangled distance, we arrive at our famous law: the entanglement entropy grows as $S(t) \sim \xi \ln(t)$.

The coefficient of this growth is proportional to the localization length $\xi$. This is a beautiful result. It tells us that the rate of information spreading is dictated by the degree of localization. The more localized the system (smaller $\xi$), the slower the entanglement grows. This simple, intuitive picture is incredibly robust. It doesn't matter if we model the entanglement "turning on" like a sudden switch or as a more realistic, smooth, and probabilistic process—the logarithmic law holds firm. It appears in a vast zoo of physical systems, from disordered spin chains to interacting bosons in quasiperiodic potentials, such as in the Aubry-Hubbard model. We can even abandon the discrete picture of l-bits and imagine a continuous quantum "fluid" of localized degrees of freedom; the mathematics, though more sophisticated, tells the same story.

Is this slow dance of entanglement a fragile thing, something that happens only in perfectly quiet, static systems? Not at all. Let's shake things up a bit. Imagine we don't just let the system be, but we "kick" it periodically with a laser, for example. We are now in the realm of Floquet systems, the quantum mechanics of periodically driven things. One might think this constant kicking would destroy the delicate localization and cause the system to heat up and thermalize. But remarkably, a state of "Floquet MBL" can exist. Even under this rhythmic driving, the system remains localized, it fails to absorb energy, and—you guessed it—the entanglement still grows logarithmically.

This picture also allows us to appreciate the role of randomness more deeply. In a real disordered material, the interaction strengths $J_{ij}$ are not all given by a single, clean exponential formula. They are themselves random variables, fluctuating from place to place. The interaction between two l-bits at distance $r$ is not a fixed number, but is drawn from a probability distribution. To find the entanglement growth, we must average over all possibilities for these couplings. And when we do, a wonderful piece of simplicity emerges from the complexity: the logarithmic law survives, with a growth rate now determined by the average decay rate of the interactions. The fundamental logarithmic heartbeat persists even in a world of statistical chance.

So far, we have been passive observers, watching this phenomenon unfold. But can we interact with it? Can we control this slow spread of quantum information? The answer is a resounding yes, and it leads us to one of the most famous and counterintuitive ideas in quantum mechanics: the act of observation changes the observed.

Imagine we zoom in on our chain of l-bits and decide to continuously monitor just one of them, say at site $k$. We set up a detector that constantly measures its state (a so-called quantum nondemolition, or QND, measurement). According to the Quantum Zeno effect, a system that is continuously observed is "frozen" in its state. Our constant watching prevents the l-bit at site $k$ from evolving coherently.

This measured site now acts as a roadblock. For entanglement to spread from one side of the chain to the other, the quantum coherence has to "tunnel" past this watched site. The success of this tunneling is a competition between two timescales: the system's own interaction time, $t_{\text{prop}} \sim \frac{\hbar}{J}$, which drives the spreading, and the measurement time, set by the measurement rate $\Gamma$. If the measurement is very strong ($\Gamma$ is large), it repeatedly projects the l-bit's state before it has a chance to interact and pass the information along. The flow of quantum information is choked off. The logarithmic growth does not cease, but its rate is dramatically suppressed by a factor related to this tunneling probability, $P_{\text{tunnel}} = \exp(-\Gamma t_{\text{prop}})$. This is not just a theoretical curiosity; it's a profound demonstration that entanglement is a physical process that can be manipulated, forming a bridge between the study of MBL and the field of quantum control.

By now, you might be tempted to think that "logarithmic growth" is just another name for "many-body localization." But nature, in its boundless ingenuity, has more in store. Let's travel to a different corner of the quantum world: the realm of quantum phase transitions.

Consider a chain of tiny quantum magnets that can either point "up" or "down". They want to align with their neighbors, but a transverse magnetic field tries to flip them. This system—the random transverse-field Ising model (RTFIM)—has two phases: a ferromagnetic phase where the magnets align and a paramagnetic phase where they are scrambled. Right at the knife-edge between these two phases lies a quantum critical point. If we prepare the system in a simple state and suddenly "quench" it to this special critical point, we find that entanglement once again grows logarithmically with time.

But the reason is entirely different! It has nothing to do with l-bits dephasing. It stems from the very nature of criticality. The magic here is a deep connection, echoing principles from Conformal Field Theory, that links the non-equilibrium growth coefficient $\Gamma$ to an equilibrium property of the critical point itself: the effective central charge, $\tilde{c}$. This $\tilde{c}$, which you can think of as a measure of the number of gapless degrees of freedom, dictates the entanglement growth via the relation $\Gamma = \frac{\tilde{c}}{3}$. In these disordered systems, this effective central charge can be calculated using a powerful theoretical tool called the strong-disorder renormalization group (SDRG), which finds that $\tilde{c} = \frac{\ln 2}{2}$ for the RTFIM. This is a stunning unification: the way entanglement spreads in time is a direct reflection of the system's fundamental degrees of freedom in space at its most interesting point.

Let us take one final leap, into a world made not of particles and fields, but of information itself. Imagine a quantum computer, where time evolves in discrete steps. At each step, random quantum gates act on pairs of qubits, scrambling and entangling them. But interspersed with this unitary evolution, we, the observers, make projective measurements on the qubits with some probability.

This setup, known as a monitored random circuit, creates a fascinating battle. The unitary gates want to spread entanglement everywhere, leading to a "volume-law" phase characteristic of quantum chaos. The measurements, by collapsing quantum states, want to destroy entanglement, leading to an "area-law" phase of quantum order. What happens at the transition point between this chaos and order? Right at the critical measurement rate, the system compromises. It forgoes both the volume and the area law, and the entanglement entropy grows logarithmically with time.

The appearance of logarithmic growth in this abstract setting is remarkable enough. But the true astonishment lies in its origin. The universal coefficient, $\alpha$, that governs this growth is a fundamental constant of this measurement-induced phase transition. And its value can be calculated through an astonishing theoretical mapping: the problem of entanglement growth in this 1D quantum circuit is equivalent to a problem in 2D classical statistical mechanics—the critical Potts model, a textbook model used to describe magnetism.

From the quiet dephasing of localized spins to the critical point of a disordered magnet, and finally to the edge of chaos in a measured quantum circuit, the slow, logarithmic creep of entanglement emerges again and again. It is a profound and unifying motif, a thread that reveals the deep and often surprising connections woven throughout the very fabric of modern physics.