Reentrant phase transitions involving glassy and superfluid orders in the random hopping Bose-Hubbard model

Using the replica trick and Trotter-Suzuki expansion, the authors demonstrate that a Bose-Hubbard model with random hopping exhibits a family of reentrant phase transitions between glassy, superfluid, and disordered phases as a function of on-site interaction, driven by the interplay of thermal energy and hopping disorder.

The Gist

Imagine a crowded dance floor where everyone is trying to move in perfect sync (that's the superfluid state) or trying to freeze in place in a chaotic, frozen pattern (that's the glass state). Usually, you'd think that if you turn up the heat, everyone just starts dancing wildly and the order disappears forever. If you turn up the "friction" between dancers (the interaction), they might get stuck in a grid.

But this paper discovers something weird and counter-intuitive: sometimes, adding a little bit of chaos or friction actually creates order, only to lose it again if you add too much. It's like a "reentrant" phase transition—a fancy way of saying the system goes from Order → Chaos → Order → Chaos as you tweak a single knob.

Here is the story of their discovery, broken down simply:

The Setting: A Chaotic Dance Floor

The scientists are studying a group of particles called bosons (think of them as tiny, polite dancers who love to be in the same spot).

• The Rules: They are on a grid (like a dance floor with tiles).
• The Chaos: The "hopping" between tiles is random. Some tiles are slippery, some are sticky, and the connections between them are a mess. This is called off-diagonal disorder.
• The Knob: They can turn a dial called $U$ (on-site interaction). This represents how much the dancers dislike being on top of each other.
  • Low $U$: They don't mind crowding; they flow freely.
  • High $U$: They hate crowding; they try to stay apart.

The Big Surprise: The "Goldilocks" Zone

Usually, scientists expect that if you start with a messy, disordered system and turn up the interaction knob ($U$), the system will just get more disordered or freeze into a solid block.

But this team found that as they turned the knob, the system did a three-step dance:

1. Disordered: At first, it's just a mess.
2. Ordered: As they turn the knob up a bit, the mess suddenly organizes itself into a specific pattern (Glass or Superfluid).
3. Disordered Again: If they turn the knob too high, the order breaks down, and it goes back to being a mess.

This is the Reentrant Phase Transition. It's like putting on a coat to get warm, but if you put on too many coats, you overheat and have to take them all off again.

The Three Specific Dances

The paper found this happening in three different scenarios:

1. The "Glass" Dance:

   • What happens: The system goes from a liquid mess $\rightarrow$ a frozen, chaotic glass $\rightarrow$ back to a liquid mess.
   • The Analogy: Imagine a crowd of people running randomly. You add a rule that they must avoid each other. Suddenly, they get stuck in a frozen, jumbled pile (Glass). But if you make the rule too strict, they panic and start running around again, breaking the frozen pile.
   • Why: The "heat" (thermal energy) and the "randomness" of the floor are fighting. At just the right interaction strength, they accidentally lock into a pattern.

2. The "Superfluid" Dance:

   • What happens: The system goes from a liquid mess $\rightarrow$ a super-coordinated flow (Superfluid) $\rightarrow$ back to a liquid mess.
   • The Analogy: Imagine a traffic jam. You tell cars to slow down and keep distance. Suddenly, the traffic starts flowing smoothly in a perfect stream. But if you tell them to keep too much distance, the flow breaks, and they get stuck in a jam again.
   • Note: This was known in clean systems, but finding it in a messy, random system was a new discovery.

3. The "Superglass" Dance:

   • What happens: This is the wildest one. The system goes from a Superfluid (flowing) $\rightarrow$ a Superglass (flowing and frozen in a chaotic pattern at the same time) $\rightarrow$ back to a Superfluid.
   • The Analogy: Imagine a river that is flowing perfectly, but the water molecules are also frozen in a weird, jagged crystal shape. It's flowing and frozen simultaneously. Then, if you push the interaction too hard, it loses the crystal shape and just becomes a normal river again.

Why Does This Happen?

The scientists found that this "re-entry" into order happens at temperatures slightly higher than you'd expect.

• For the Glass: The heat energy matches the "spread" of the random floor. It's like the dancers are jittering just enough to find a spot where they can lock into a pattern before the heat melts it.
• For the Superfluid: The heat energy matches the average speed of the dancers. The interaction acts like a conductor, organizing the chaos just enough to create a flow, until the interaction gets too strong and crushes the flow.

The Takeaway

This paper is important because it shows that disorder and interaction can work together to create order, rather than just destroying it. It's a reminder that in complex systems (like materials, or even perhaps economies or ecosystems), adding a constraint or a bit of friction doesn't always make things worse; sometimes, it's the exact thing needed to make the system work, provided you don't overdo it.

In short: Sometimes, a little bit of friction is the glue that holds the party together, but too much friction makes the party fall apart.

Technical Summary

1. Problem Statement

The paper investigates the phase behavior of a system of strongly correlated bosons subject to off-diagonal disorder (randomness in the kinetic energy/hopping terms). While diagonal disorder (randomness in potential energy) is known to produce the Bose Glass (BG) phase, off-diagonal disorder is less understood but theoretically linked to spin-glass physics.

The specific focus is on identifying reentrant phase transitions. A reentrant transition occurs when a specific phase disappears and then reappears as a single system parameter (in this case, the on-site interaction strength, $U$) is varied. The authors aim to map the phase diagram of the random hopping Bose-Hubbard model to determine if and where such counter-intuitive transitions occur between glassy, superfluid, superglass, and disordered phases.

2. Methodology

The authors employ a theoretical framework combining statistical mechanics and quantum many-body physics techniques:

• Model: The system is described by the Bose-Hubbard Hamiltonian with random hopping integrals $J_{ij}$. The hopping terms are independent Gaussian-distributed random variables with mean $J_0/N$ and variance $J^2/N$, analogous to the Sherrington-Kirkpatrick model for spin glasses. This implies effectively long-range interactions.
• Analytical Approach:
  • Replica Trick: Used to average the free energy over the disorder distribution. The logarithm of the partition function is replaced using the limit $\lim_{n \to 0} \frac{1}{n}(Z^n - 1)$.
  • Trotter-Suzuki Expansion: Maps the quantum model onto a classical one by introducing a discrete imaginary time dimension ($M$), allowing the treatment of non-commuting operators.
  • Hubbard-Stratonovich Transformation: Applied twice to decouple site-mixing terms and replica-mixing terms, introducing auxiliary fields ($R, U, \Delta, q, u$).
  • Saddle Point Approximation: Used in the thermodynamic limit to derive self-consistent equations for the order parameters.
• Order Parameters:
  • $\Delta$: Superfluid order parameter.
  • $q$: Edwards-Anderson glass order parameter (measuring frozen disorder).
  • $u$: A secondary parameter distinguishing the superglass phase (where $0 < u < q$) from the pure superfluid phase (where $u = q$).
• Numerical Implementation: The self-consistent equations are solved numerically for finite Trotter slices ($M=4$ to $9$) and extrapolated to the limit $M \to \infty$.

3. Key Contributions

• Discovery of Triple Reentrance: The paper identifies a family of reentrant phase transitions occurring on three distinct boundaries as a function of on-site interaction $U$:
  1. Between the Glass (GL) and Disordered (DI) phases.
  2. Between the Superglass (SG) and Superfluid (SF) phases.
  3. Between the Superfluid (SF) and Disordered (DI) phases.
• Distinction from Spin Glasses: The authors demonstrate that on-site repulsion $U$ acts qualitatively differently in bosonic systems compared to transverse fields in spin glasses. While transverse fields typically destroy order monotonically, $U$ can stabilize ordered phases within a specific intermediate range before destroying them at higher values.
• Validation against Clean Systems: The study confirms that reentrant behavior on the SF-DI boundary, previously known in clean (non-disordered) systems, persists even in the presence of off-diagonal disorder.

4. Results

The phase diagrams were constructed in the $T/J$ vs. $U/J$ plane for fixed chemical potential $\mu$ and disorder strength ratios.

• Case 1: Weak Mean Hopping ($J_0 < J$)

  • A reentrant transition occurs between the Disordered (DI) and Glass (GL) phases.
  • As $U$ increases, the system transitions from DI $\to$ GL $\to$ DI.
  • Mechanism: This occurs when thermal energy is comparable to the spread of the hopping distribution (disorder). The interaction $U$ is necessary to stabilize the glassy order against thermal fluctuations, but excessive $U$ destroys it.
  • The critical temperature for this reentrance is slightly higher than the critical temperature of the non-interacting ($U=0$) system.

• Case 2: Strong Mean Hopping ($J_0 > J$)

  • Two types of reentrance are observed:
    1. SF-DI Boundary: The Superfluid phase appears between two Disordered regions (DI $\to$ SF $\to$ DI). This mirrors behavior seen in clean systems.
    2. SG-SF Boundary: The Superfluid phase appears both above and below the Superglass phase (SF $\to$ SG $\to$ SF).
  • Mechanism: For superfluidity, reentrance occurs when thermal fluctuations are comparable to the mean hopping ($J_0$). Intermediate interactions stabilize the superfluid/superglass order, while very low or very high interactions lead to disorder.
  • Temperature Scale: The reentrant transitions occur at temperatures slightly above the critical temperatures of the corresponding non-interacting systems ($T \approx J$ for GL-DI and $T \approx J_0$ for SF-related transitions).

• Order Parameter Analysis:

  • Numerical cross-sections confirm the transitions:
    • GL-DI: $q$ becomes non-zero and then returns to zero.
    • SF-DI: Both $q$ and $\Delta$ become non-zero and then vanish.
    • SG-SF: $q$ and $u$ become distinct (signaling SG) and then become equal again (signaling SF).

5. Significance

• Fundamental Physics: The work deepens the understanding of how kinetic disorder (off-diagonal) competes with interaction and thermal fluctuations in bosonic systems. It bridges the gap between Bose-Hubbard physics and spin-glass theory.
• Experimental Relevance: The proposed model is feasible for realization using current optical-lattice quantum simulators. The prediction of reentrant phases suggests that experimentalists could observe these transitions by tuning interaction strengths (via Feshbach resonances) while maintaining off-diagonal disorder.
• Theoretical Insight: The study highlights that interaction strength is not merely a destructive force for order; in disordered quantum systems, it can act as a stabilizing mechanism for complex phases (glassy and superglass) within a specific "window" of parameters, a phenomenon not captured by simple mean-field theories of clean systems.