Disorder-induced localisation in the Mott-Hubbard model

Here is an explanation of the paper using simple language and creative analogies.

The Big Picture: A Crowded Dance Floor

Imagine a massive, perfectly organized dance floor (a crystal lattice) where every single spot is occupied by exactly one dancer. In physics, this is called a Mott Insulator. The dancers are electrons. Because they are so crowded and repel each other strongly, they can't move around freely; they are stuck in their spots, like a traffic jam where no one can budge.

Now, imagine we want to see what happens if we introduce a "dancer" who is missing a partner (a hole) or a "dancer" who has grabbed an extra partner (a doublon). These are the "quasi-particles" the scientists are studying. They want to know: If we shake up the dance floor, will these special dancers get stuck in one spot, or will they be able to roam the whole floor?

The researchers tested two ways of "shaking up" the floor: Charge Disorder and Spin Disorder.

1. The Two Types of "Shaking"

Type A: Charge Disorder (The "VIP Section" Analogy)

Imagine that some random spots on the dance floor suddenly get a VIP sign. Maybe a VIP spot has a free drink (low energy), while a regular spot is expensive (high energy).

The Setup: The scientists randomly assigned these "VIP" spots to about 26% of the dancers.
The Result: The dancers who wanted to move into the VIP spots got stuck there. They formed small, isolated clusters.
The Surprise: The dancers who stayed on the regular spots didn't care about the VIPs. They kept dancing freely across the whole floor.
The Takeaway: This created a split personality in the system. You have two distinct groups:
1. The Trapped: Dancers stuck in the VIP zones (localized).
2. The Free: Dancers roaming the regular zones (delocalized).
- Analogy: It's like a city where a few neighborhoods are gated communities. People inside the gates stay put, but people outside the gates can drive anywhere.

Type B: Spin Disorder (The "Left-Handed vs. Right-Handed" Analogy)

Now, imagine the dance floor rules change based on which way the dancers are facing.

The Setup: Some dancers are facing North, some South. The rules say: "If you are facing North, you can only dance with a North-facing partner. If you are facing South, you can only dance with a South-facing partner."
The Result: Because the background is a random mix of North and South faces, every dancer finds themselves in a slightly different environment.
The Takeaway: In this scenario, everyone gets stuck. No matter where you are on the floor or how much energy you have, the mismatched rules prevent you from moving far.
Analogy: Imagine a game of musical chairs where the chairs are randomly labeled "Left" or "Right." If you are a "Left" player, you can only sit in "Left" chairs. If the chairs are scattered randomly, you might get stuck in a corner with no other "Left" chairs nearby. Everyone ends up isolated.

2. The Tools Used to Watch the Dance

The scientists used two different methods to predict this behavior, like using two different cameras to film the dance:

The "Correlation Hierarchy" (The High-Definition Camera):
This method looks at how dancers influence their neighbors. It's a complex mathematical way of saying, "If I move, how does that ripple out to the person next to me?" This method gave them a very detailed picture of the split between the "VIP" dancers and the "Free" dancers.
The "Perturbation Theory" (The Sketch Artist):
This is a simpler, older method. It assumes the dancers barely move at all and tries to guess the result based on small nudges.
- The Comparison: The sketch artist (Perturbation Theory) got the general idea right (that VIPs get stuck), but the high-definition camera (Correlation Hierarchy) showed much more detail, revealing the specific shapes of the clusters and the subtle differences between the two types of disorder.

3. Why Does This Matter?

You might ask, "Why do we care about stuck electrons?"

Quantum Computers: To build a quantum computer, we need particles to stay in specific places (to hold information) without losing their energy to the environment. This research helps us understand how to "trap" information using disorder.
Insulators vs. Conductors: It helps us understand the boundary between materials that conduct electricity and those that don't.
The "Many-Body" Mystery: For a long time, physicists thought that if particles interacted strongly (like in a crowded dance floor), they would eventually mix and thermalize (reach a uniform temperature). This paper shows that disorder can stop this mixing, keeping the system "frozen" in a specific state even at higher temperatures. This is a step toward understanding Many-Body Localization (MBL).

Summary in One Sentence

The paper shows that if you randomly change the "cost" of spots on a crowded dance floor, you get a mix of stuck and free dancers, but if you randomly change the "rules" of who can dance with whom, everyone gets stuck, effectively freezing the system in place.

Here is a detailed technical summary of the paper "Disorder-induced localisation in the Mott-Hubbard model" by Knipšis et al.

1. Problem Statement

The paper investigates the behavior of quasi-particle excitations (specifically doublons and holons) in a Mott insulator phase of the Fermi-Hubbard model when subjected to disorder. While Anderson localization is well-understood for non-interacting systems, the interplay between strong electron-electron interactions (characteristic of the Mott phase) and disorder remains a complex area of study, particularly regarding the stability of localization at finite temperatures and the nature of Many-Body Localization (MBL).

The authors specifically aim to distinguish between the effects of two distinct types of disorder on a half-filled 2D lattice:

Charge Disorder: Random on-site potentials (energy shifts) that affect both spin states equally.
Spin Disorder: Random on-site magnetic fields (spin-splitting potentials) that energetically favor specific spin orientations, effectively freezing the spin background.

2. Methodology

The study employs two complementary theoretical approaches to analyze the system:

A. Hierarchy of Correlations (Primary Method)

Framework: The authors use an expansion in orders of $1/Z $, where$ Z$ is the coordination number of the lattice.
Truncation: The hierarchy is truncated at the first order ($1/Z$). This involves considering one-point density matrices and two-point correlators, neglecting higher-order correlations.
Factorization Ansatz: The two-point correlators are factorized into products of single-site amplitudes ( $p^I_{\mu,s}$ ), reducing the many-body problem to a set of coupled linear differential equations.
Quasi-particles: The analysis focuses on dressed fermionic operators representing holons ( $I=0$ , removal of an electron) and doublons ( $I=1$ , addition of an electron).
Lattice Models: Simulations are performed on 2D lattices with varying coordination numbers: Hexagonal ( $Z=6$ ), Square ( $Z=4$ ), and Honeycomb ( $Z=3$ ).
Parameters: The system is in the strong-coupling limit ( $T \ll U$ ), with specific values $U = 0.35$ eV and $T/Z = 0.05$ eV. Disorder strengths ( $V$ for charge, $v$ for spin) are varied.

B. Strong-Coupling Perturbation Theory (Validation)

To validate the hierarchy of correlations results, the authors perform a first-order perturbation theory calculation in $T/U$ .
Constraints: This method restricts the Hilbert space to the single-hole subspace (ignoring double holes or complex multi-site excitations) to ensure tractability. It serves as a benchmark to check if the correlation hierarchy captures physics beyond simple perturbation theory.

C. Metrics for Localization

Inverse Participation Ratio (IPR): Defined as $\text{IPR} = \sum |p|^4$ . The inverse of the IPR (Participation Ratio) estimates the number of lattice sites a state occupies.
Bimodal Analysis: The distribution of participation ratios is analyzed to distinguish between localized (low participation ratio) and delocalized (high participation ratio) states.

3. Key Results

A. Charge Disorder (On-site Potential Randomness)

Band Splitting: The introduction of charge disorder ( $V_\mu$ ) splits the original holon and doublon bands into distinct sub-bands. The energy separation between these sub-bands is controlled by the disorder strength $V$ .
Spatial and Energetic Separation: A clear bimodal distribution emerges in the participation ratios.
- Localized States: Form distinct sub-bands at the edges of the spectrum (associated with doped sites where $V_\mu = V$ ). These states are spatially confined to clusters of disordered sites.
- Delocalized States: Occupy the remaining parts of the original bands (associated with sites where $V_\mu = 0$ ). These behave like plane waves.
Lattice Dependence: Reducing the coordination number $Z$ (from Hexagonal to Square to Honeycomb) increases the fraction of localized states due to percolation thresholds, making the separation between localized and delocalized regimes more pronounced.

B. Spin Disorder (Fixed Random Spin Background)

Uniform Localization: Unlike charge disorder, spin disorder results in localized states throughout the entire quasi-particle band, not just at the edges.
Mechanism: The random spin background restricts the available hopping paths for quasi-particles. Since the background spins are fixed (frozen on the timescale of the study), the quasi-particles cannot find a coherent path to delocalize, leading to localization at all energy levels within the band.
No Sub-band Splitting: The energy spectrum does not split into distinct sub-bands separated by energy gaps; instead, the entire band becomes "dirty" with localized states.

C. Combined Disorder

When both charge and spin disorders are present, the localization is enhanced, with the participation ratio distributions showing a stronger dominance of localized states compared to either disorder type alone.

D. Comparison with Perturbation Theory

Qualitative Agreement: Perturbation theory successfully reproduces the band splitting observed in charge disorder.
Quantitative Differences: Perturbation theory predicts a higher degree of localization for spin disorder compared to the hierarchy of correlations. The authors attribute this to the perturbative approach's restriction to the single-hole subspace, which excludes interactions with higher-order states that the $1/Z$ expansion naturally includes. This suggests the hierarchy of correlations captures more complex many-body effects even at first order.

4. Significance and Conclusions

Distinct Localization Mechanisms: The paper establishes a fundamental difference between charge and spin disorder in Mott insulators. Charge disorder leads to a spatially separated phase coexistence (localized clusters vs. delocalized bulk), whereas spin disorder induces global localization across the energy spectrum.
Timescale Validity: The results are valid for timescales shorter than the spin dynamics timescale ( $\sim U/T^2$ ), where the spin background can be treated as static.
Methodological Insight: The study demonstrates that the hierarchy of correlations (truncated at $1/Z$) is a powerful tool for studying disordered interacting systems, capturing effects (like the fine structure of localized sub-bands) that require higher-order corrections in standard perturbation theory.
Implications for MBL: The findings contribute to the understanding of how disorder interacts with strong correlations, suggesting that the nature of the disorder (charge vs. spin) critically determines whether the system exhibits a mixed phase of localized/delocalized states or a fully localized phase, which is relevant for the stability of Many-Body Localization.

In summary, the work provides a rigorous theoretical framework for understanding how different types of disorder induce localization in Mott-Hubbard systems, highlighting that the type of disorder dictates the spatial and energetic structure of the resulting localized states.