Characterizing the Many Body Localization Crossover as a Metal-Insulator Transition: Localization length from Polarization and Quantum Metric

This paper characterizes the Many-Body Localization crossover as a metal-insulator transition by utilizing the many-body quantum metric and polarization-based localization parameters to extract a consistent localization length that describes the real-space spread of wave functions in disordered insulating phases.

The Gist

Imagine a crowded dance floor where everyone is trying to move around. In a normal party (a "metal" or conductor), people mingle, bump into each other, and eventually, the whole room reaches a state of equilibrium where everyone is equally mixed up. This is called thermalization.

But now, imagine a chaotic party where the floor is covered in sticky glue spots (disorder) and the dancers are holding hands tightly (interactions). In this scenario, the dancers get stuck in their own little circles. They can't move freely, they don't mix with the crowd, and the party never reaches a "mixed" state. This is Many-Body Localization (MBL). It's a strange state of matter where a system refuses to settle down, even after a long time.

For a long time, physicists have struggled to find a simple way to tell the difference between a "stuck" party (insulator) and a "moving" party (conductor), especially when looking at highly excited states (like a party at its peak energy) where the rules get fuzzy.

This paper introduces a new, geometric way to measure this "stickiness" using two main tools:

1. The Two Rulers: Polarization and the Quantum Metric

The authors use two different "rulers" to measure how stuck the particles are:

• Ruler A (The Polarization Parameter): Think of this as measuring how far the dancers have drifted from their starting spots. If they are stuck in a small circle, this number stays small. If they are running wild across the whole room, this number grows huge.
• Ruler B (The Quantum Metric): This is a bit more abstract. Imagine the dance floor has a "twist" or a hidden knob you can turn. The Quantum Metric measures how much the dancers' positions change when you tweak this knob. It's like asking, "If I slightly change the rules of the room, how much does the dance pattern shift?"

2. The "Agreement" Test

Here is the clever part of the discovery:

• In a Conducting (Moving) System: The two rulers tell completely different stories. One says "they are moving," and the other says something else entirely. They don't agree.
• In an Insulating (Stuck) System: Even though the math is complex, these two rulers start to agree. They both say, "Yes, the dancers are stuck in a small area."

The authors created a simple score (let's call it the "Agreement Score") to see how much these two rulers match up.

• If the score is high (near 1), the system is conducting (moving).
• If the score is low (near 0), the system is insulating (stuck/MBL).

3. Why This is Special

Usually, these geometric tools only work for systems that have a "gap" (a clear separation between energy levels), like a calm, quiet room. But the authors showed that this trick works even in high-energy, chaotic systems (like a loud, crowded party) where there is no gap.

They tested this on two scenarios:

1. The Single Dancer (Anderson Insulator): A single particle in a messy room. They showed that even here, the two rulers agree when the particle is stuck.
2. The Crowd (Many-Body Localization): A group of interacting particles. They found that as they increased the "glue" (disorder), the system switched from a moving state to a stuck state, and their "Agreement Score" dropped perfectly to zero, marking the transition.

4. The Result: A New Map

Using this method, the authors were able to draw a map of the "stickiness" of the system. They found a specific localization length—a measure of exactly how big the "stuck circle" is for the dancers.

• In the MBL regime (the stuck phase), this length is finite and well-defined.
• In the ergodic regime (the moving phase), the length is effectively infinite.

The Bottom Line

The paper claims that by comparing these two geometric measurements, we can clearly see the line between a system that thermalizes (mixes) and one that localizes (stays stuck). This provides a new, consistent way to define the "size" of the localized region in these complex quantum systems, acting as a reliable compass to navigate the transition between order and chaos in the quantum world.

What the paper does NOT claim:

• It does not claim to cure diseases or solve climate change.
• It does not claim to build a working quantum computer today (though it mentions quantum processors could help prepare states in the future).
• It does not definitively say what happens in an infinitely large universe (the "thermodynamic limit"), but rather focuses on what we can observe in finite, real-world-sized systems.

Technical Summary

Technical Summary: Characterizing the Many-Body Localization Crossover via Quantum Metric and Polarization

Problem Statement
Many-body localization (MBL) represents a nonergodic phase where disorder and interactions prevent thermalization, offering a unique testing ground for understanding the breakdown of statistical mechanics. While MBL is well-studied, defining a robust, natural localization length that is consistent across various insulating phases remains challenging. Existing approaches, such as inverse participation ratios, spatial correlation decay, or phenomenological models based on local integrals of motion (LIOM), often lack clear experimental signatures or physical clarity. Furthermore, the existence of MBL as a true thermodynamic phase is debated, with numerical studies struggling to pinpoint critical disorder due to finite-size effects and system-size dependent drifts in indicators like entanglement entropy. A key theoretical gap exists in applying the modern theory of polarization and insulators—which defines localization length via the quantum metric—to highly excited, gapless states typical of MBL, as these formalisms traditionally require gapped ground states.

Methodology
The authors apply the formalism of the modern theory of insulators to the disordered hardcore Bose-Hubbard model in one dimension. They focus on two primary quantities defined in the parameter space of twist boundary conditions:

1. The Localization Parameter ($D_N$): Derived from the expectation value of the twist operator $e^{2i\pi X_{CM}/L}$, where $X_{CM}$ is the center-of-mass coordinate. In the thermodynamic limit, $D_N$ distinguishes insulators (finite) from conductors (divergent).
2. The Many-Body Quantum Metric (MBQM, $g$): A geometric property describing the response of the wavefunction to changes in the twist boundary phase (or vector potential). It is calculated using the sum-over-states formula involving energy differences and matrix elements of the derivative of the Hamiltonian.

The study investigates the relationship between $D_N$ and the scaled MBQM ($g_N = 4\pi^2 N g$) in finite-size systems ($L=8$ to $18$). The authors utilize the shift-and-invert method to calculate the MBQM for excited states in the middle of the spectrum (corresponding to "infinite temperature") without full diagonalization. They introduce a dimensionless agreement parameter, $\Delta = |g_N - D_N| / (g_N + D_N)$, to quantify the consistency between the two quantities. Theoretical arguments suggest that for insulators, $D_N \simeq g_N$ (leading to $\Delta \to 0$), whereas for conductors, the quantities are unrelated and $D_N$ diverges (leading to $\Delta \to 1$).

Key Results

• Single-Particle Anderson Insulator: The authors first validate the framework on a non-interacting Anderson insulator. They demonstrate that for sufficiently strong disorder, the MBQM, the localization parameter, and the variance of the center-of-mass coordinate converge to the same value, independent of system size. Discrepancies at low disorder are attributed to finite-size effects where the localization length exceeds the system size.
• Many-Body Localization Transition: Applying the method to the interacting Bose-Hubbard model, the authors observe a clear crossover. In the high-disorder regime, $g_N$ and $D_N$ agree, and $\Delta$ approaches zero, indicating an insulating MBL phase. In the low-disorder (ergodic) regime, the quantities diverge, and $\Delta$ approaches one.
• Phase Diagrams: By mapping $\Delta$ across the disorder ($W$) and interaction ($U$) plane, the authors construct phase diagrams that exhibit a dome-shaped ergodic region surrounded by an insulating MBL region. The phase boundary defined by $\Delta = 0.5$ aligns well with previous studies using dynamical exponents.
• Localization Length: The authors extract a characteristic localization length $\ell_N = \sqrt{g_N}/(2\pi n)$ (where $n$ is density) specifically within the MBL regime. They note that in the intermediate "quantum critical" region between the ergodic and fully localized phases, the lengths derived from $g_N$ and $D_N$ show discrepancies, a feature consistent with finite-size scaling limitations.

Significance and Claims
The paper claims to provide a geometric characterization of the MBL crossover as a metal-insulator transition. Its primary contributions are:

1. Extension of Insulator Theory: It demonstrates that the relationship between the localization parameter and the quantum metric, traditionally valid for gapped ground states, holds for highly excited, gapless states in finite disordered systems. This is possible because exact degeneracies are statistically improbable in finite disorder realizations.
2. New Diagnostic Tool: The agreement parameter $\Delta$ offers a method to distinguish between ergodic and MBL regimes in finite systems without requiring a thermodynamic limit extrapolation, which is often ambiguous for excited states.
3. Natural Localization Length: The work defines a localization length in the MBL regime consistent with Resta's definition for insulators. This length is theoretically grounded and, crucially, is linked to the MBQM, which the authors note is an experimentally measurable quantity (e.g., via AC response or static structure factor in cold atom systems).

The authors remain modest regarding the ultimate fate of MBL in the thermodynamic limit, acknowledging that their analysis is limited to finite sizes and that the critical regime requires further scaling analysis. However, they assert that their approach provides a complementary perspective focusing on the insulating properties of MBL and a direct experimental pathway to measure the localization length.