Interaction-driven quantum phase transitions between topological and crystalline orders of electrons

By leveraging electrostatic control of Landau-level crossings in high-quality bilayer graphene devices, this study demonstrates that enhanced Landau-level mixing near orbital crossings stabilizes electronic crystals over topological fractional quantum Hall liquids at constant filling factors, revealing distinct transition characteristics and evidence for a paired composite fermion state.

The Gist

Imagine a crowded dance floor where the dancers are electrons. Usually, when you put a magnetic field on this floor, the dancers get organized into a very specific, fluid pattern called a Fractional Quantum Hall (FQH) state. Think of this like a perfectly choreographed ballet: everyone moves in sync, flowing smoothly without bumping into each other, creating a state of "topological order" that is incredibly stable and conductive.

However, if the dancers start pushing and shoving each other too hard (strong electrical repulsion), they might decide to stop dancing and instead stand still in a rigid grid, like soldiers in formation. This is a Wigner Crystal (WC). It's a "crystalline order" where the electrons freeze into a solid lattice, making the material resistive (like an insulator) because the electrons can't flow freely.

The Big Question:
For a long time, scientists thought you could only switch between these two states (fluid ballet vs. rigid crystal) by changing how many dancers were on the floor (the electron density). But this paper asks: Can we force them to switch states without adding or removing any dancers, just by changing the "rules of the dance floor"?

The Experiment: The "Tilted" Dance Floor
The researchers used a special material called Bilayer Graphene (two sheets of carbon atoms stacked like a sandwich). They placed this in a strong magnetic field and applied an electric "displacement field."

Think of the displacement field as a tilt applied to the dance floor.

• The Setup: The electrons live in "energy levels" (like floors in a building). Usually, the bottom floor (Level N=0) is full, and the next one up (Level N=1) is empty.
• The Tilt: As they increased the tilt (the electric field), they made the bottom floor rise and the next floor sink until they crossed each other.
• The Magic: At the exact moment these two floors cross, the electrons get confused. They can't decide which floor to stand on. They start mixing their "outfits" (quantum wave functions) from both floors. This mixing creates a chaotic environment where the electrons are forced to choose: Do I keep flowing like a fluid, or do I freeze into a crystal?

What They Found:
By carefully tuning this tilt, they watched the electrons switch back and forth between the fluid ballet and the rigid crystal, all while keeping the number of dancers exactly the same.

1. The Switch: Near the crossing point, the "fluid" state (FQH) becomes unstable. The electrons, feeling the strong push from their neighbors, suddenly snap into a rigid crystal (Wigner Crystal).
2. The Re-entrant State: At a specific filling factor (7/3), they saw something cool: The electrons formed a crystal on top of a perfect fluid background. It's like having a solid ice cube floating in a perfectly smooth stream. This is called a "Re-entrant Integer Quantum Hall" state.
3. The "Half-Filled" Mystery: At exactly half-filling (where the dance floor is half-empty), they saw a new, strange state emerge right before the crystal formed. They suspect this is a paired state, where electrons pair up like dance partners before freezing. This is a potential candidate for a very exotic, "non-Abelian" state that could be useful for future quantum computers.

Why This Matters:

• Control: They proved you can control the state of matter (fluid vs. solid) just by applying an electric field, without changing the material or the number of electrons.
• The "Glue": The key was the Landau Level Mixing. When the energy levels cross, the electrons borrow energy from the "next floor up," which acts like a super-strong glue that forces them to crystallize.
• Future Tech: Understanding how electrons switch between these topological (fluid) and crystalline (solid) states is a huge step toward building better quantum computers. These states are robust and could store information in a way that is immune to errors.

In a Nutshell:
The researchers built a "tunable dance floor" for electrons. By tilting the floor, they forced the electrons to argue over whether to dance in a fluid line or stand in a rigid grid. They found that at the tipping point, the electrons don't just slowly change; they undergo a dramatic phase transition, revealing new, exotic states of matter that could power the next generation of technology.

Technical Summary

Here is a detailed technical summary of the paper "Interaction-driven quantum phase transitions between topological and crystalline orders of electrons."

1. Problem Statement

The paper addresses the fundamental competition between two distinct phases of matter in two-dimensional electron systems under strong magnetic fields:

• Fractional Quantum Hall (FQH) Liquids: Topologically ordered states characterized by incompressible fluids and fractional charge excitations.
• Wigner Crystals (WC): Crystalline states arising from the spontaneous breaking of translational symmetry due to strong Coulomb repulsion.

While previous studies have observed transitions between these phases by varying the filling factor ($\nu$) or disorder levels, a critical gap remained in understanding interaction-driven transitions at a fixed filling factor. Specifically, the role of Landau-Level Mixing (LLM)—where electrons virtually excite into nearby Landau levels, modifying the interaction potential—was theoretically predicted to stabilize Wigner crystals, but had not been continuously tuned in a single sample to observe the phase transition dynamics. The authors aim to map the phase diagram between topological (FQH) and crystalline (WC) orders by continuously tuning the interaction strength via an external parameter while keeping the electron density constant.

2. Methodology

• Material System: The study utilizes high-quality Bilayer Graphene (BLG) encapsulated in hexagonal Boron Nitride (hBN). BLG is chosen for its unique Landau level (LL) spectrum, which is highly tunable via an out-of-plane electric displacement field ($D$).
• Device Fabrication: Two devices (D1 and D2) were fabricated using dry-transfer stacking techniques with top and bottom graphite gates. This allows for independent control of the carrier density (filling factor $\nu$) and the displacement field ($D$).
• Experimental Conditions: Measurements were conducted in a dilution refrigerator at a base temperature of $T = 10$ mK and magnetic fields up to $B = 12$ T.
• Tuning Mechanism: The researchers applied a large displacement field ($D$) to lift the valley degeneracy of the spin- and orbital-polarized Landau levels. This induces crossings between the $N=0$ and $N=1$ Landau levels (which share the same spin and valley quantum numbers).
  • Near the crossing point, the wavefunctions of the $N=0$ and $N=1$ orbitals mix significantly, enhancing LLM.
  • Far from the crossing, the system resides in a pure $N=0$ or $N=1$ level with minimal mixing.
• Measurements: Longitudinal ($R_{xx}$) and transverse ($R_{xy}$) resistances were measured as a function of $\nu$ and $D$. Activation energies were extracted from temperature-dependent transport data to characterize the energy gaps of the phases.
• Theoretical Modeling: The authors performed Hartree-Fock calculations and variational Monte Carlo simulations using trial wavefunctions for Laughlin states (FQH) and Inter-Landau-Level Coherent Wigner Crystals (ILLC WC) to compare energies and predict phase stability.

3. Key Contributions

1. Continuous Tuning of LLM: The work demonstrates a method to continuously tune Landau-level mixing in a single device by sweeping the displacement field across an $N=0$ to $N=1$ level crossing, rather than comparing discrete samples with different densities.
2. Observation of Interaction-Driven Transitions: The authors successfully observed phase transitions between FQH liquids and Wigner crystals (or Reentrant Integer Quantum Hall states) at fixed filling factors ($\nu = 1/3, 2/5, 7/3$), proving that interactions alone can drive these transitions.
3. Discovery of a Universal WC Phase: They identified a single Wigner crystal phase that extends over the entire range of fractional fillings ($0 < \nu < 1$) near the level crossing, suggesting a robust electron crystal stabilized by strong mixing.
4. Characterization of Transition Order: The study distinguishes between the nature of transitions at different levels of the displacement field, observing continuous (symmetric) vanishing of gaps at lower $|D|$ and discontinuous transitions at higher $|D|$.
5. Emergence of Paired States: At half-fillings ($\nu = 1/2, 5/2$), the authors observed the formation of incompressible states (quantized plateaus) between the composite Fermi liquid and the Wigner crystal, suggesting a paired composite fermion state stabilized by LLM.

4. Key Results

• Phase Diagram at $\nu = 1/3$:
  • At low $|D|$ (pure $N=0$), a robust FQH state is observed ($R_{xx} \to 0$, quantized $R_{xy}$).
  • As $|D|$ increases toward the level crossing, the system transitions into a resistive state ($R_{xx}$ increases, $R_{xy}$ loses quantization), identified as a Wigner Crystal.
  • Upon further increasing $|D|$ (entering the $N=1$ regime), the system re-enters an FQH state.
  • Activation Energies: The activation gap of the FQH state ($\Delta$) and the activation energy of the WC ($E_A$) vanish smoothly and symmetrically at the transition on the lower $|D|$ side, indicating a continuous phase transition.
• Phase Diagram at $\nu = 7/3$:
  • Similar FQH–WC transitions are observed.
  • The WC phase here manifests as a Reentrant Integer Quantum Hall (RIQH) state, where the Hall conductance returns to an integer value ($G_{xy} = 2e^2/h$) with zero longitudinal resistance, corresponding to a WC in the partially filled level atop a fully filled integer background.
  • The transition at higher $|D|$ (entering the $N=1$ regime) appears discontinuous, with a sharp drop in the WC energy scale.
• Half-Filled States ($\nu = 1/2, 5/2$):
  • Near the level crossings, the authors observed the emergence of quantized plateaus in $R_{xy}$ (e.g., at $\nu=1/2$) accompanied by minima in $R_{xx}$.
  • These states appear as intermediate phases between the composite Fermi liquid and the Wigner crystal.
  • Theoretical analysis suggests these are paired composite fermion states (potentially non-Abelian 221 parton states), stabilized by the mixing of $N=0$ and $N=1$ orbitals which favors pairing.
• Theoretical Validation:
  • Hartree-Fock calculations confirm that the Inter-Landau-Level Coherent Wigner Crystal (ILLC WC), formed by coherent superpositions of $N=0$ and $N=1$ orbitals, has a lower energy than the Laughlin state near the crossing.
  • The calculations show that the wavefunction spread is minimized at a specific mixing angle, making the crystal energetically favorable.
  • While the theory qualitatively matches the experiment, it overestimates the range of the WC phase, likely due to the limitations of the trial wavefunctions in capturing the true ground state near the crossing.

5. Significance

• Fundamental Physics: This work provides the first experimental evidence of a continuous, interaction-driven phase transition between topological and crystalline orders at fixed density. It validates the theoretical prediction that Landau-level mixing is a powerful knob to stabilize electron crystals.
• Control of Quantum Phases: It establishes BLG under a displacement field as a premier platform for exploring the interplay between symmetry-breaking (crystalline) and topological orders, offering a level of tunability not available in traditional semiconductor heterostructures.
• New States of Matter: The observation of paired composite fermion states at half-fillings near level crossings opens new avenues for studying non-Abelian statistics and topological quantum computing candidates.
• Universal Behavior: The finding that a single WC phase spans all fractional fillings suggests a universal mechanism for crystallization in the presence of strong LLM, distinct from the disorder-driven transitions seen in other systems.

In summary, the paper demonstrates that by electrostatically tuning the orbital character of electrons in bilayer graphene, one can drive a system from a topological fluid to a crystalline solid and back again, revealing rich physics including continuous phase transitions and emergent paired states.