Evidence for a charged vortex lattice in an exciton superfluid

Ryan L. Lee; Jungwoo Lee; Viliam Vaňo; Bo Zou; Cheng-Li Chiu; Kenji Watanabe; Takashi Taniguchi; Allan MacDonald; Ali Yazdani

AI summary · arXiv: 2610.37781v1 · cond-mat.str-el

Evidence for a charged vortex lattice in an exciton superfluid

Ryan L. Lee, Jungwoo Lee, Viliam Vaňo, Bo Zou, Cheng-Li Chiu, Kenji Watanabe, Takashi Taniguchi, Allan MacDonald,

and 1 more author

Ali Yazdani

Original paper Submitted to arXiv on 27 Oct. 2026 License: NONEXCLUSIVE-DISTRIB 1.0 Also in: fr

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The essentials

Scanning tunnelling microscopy reveals charge lattices in graphene bilayer exciton condensates whose density dependence supports a lattice of charged vortices and antivortices.

Question

Can a weakly charged exciton condensate form the theoretically predicted vortex–antivortex lattice, and can its signatures be distinguished from a conventional Wigner crystal? The study tests whether lattice periodicity follows total excess charge and remains unchanged when charge is redistributed between layers.

Method

The authors used scanning tunnelling microscopy to image a graphene double layer separated by approximately 3.2 nm of hexagonal boron nitride, at 9 T and 200 mK. By varying gate and interlayer voltages, they measured electron- and hole-doped states and density imbalance, then extracted lattice spacings from the structure factors of spatial tunnelling-current maps.

Findings

Result Electron doping produces a charge lattice whose spacing follows the predicted dependence on total excess filling at low doping and approaches the top-layer Wigner-crystal spacing near a filling of approximately 0.17.
See the passage in the paper
The extracted lattice constant (Fig. 2o, black symbols) closely agrees with the predicted vortex lattice constant 𝑎=#4π𝑙!"/√3ν+#$#%& (red curve), where 𝑙! is the magnetic length and ν+#$#%& is filling relative to the neutral exciton condensate.
Result At fixed total carrier density, changing the interlayer density imbalance leaves the measured lattice spacing essentially unchanged, despite changes in the individual layer fillings.
See the passage in the paper
Instead, the experimentally extracted lattice constant (see Methods) remains essentially invariant (Fig. 3n), in quantitative agreement with the vortex lattice prediction (solid line).
Authors’ interpretation The lattice contrast changes with density imbalance, consistent with fractional charge shifting between vortices and antivortices in opposite layers.
See the passage in the paper
Microscopically, tuning the density imbalance redistributes fractional charge between vortices and antivortices residing in opposite layers, modifying their layer-resolved charge contrast without altering their spatial positions (see SI section 13).
Result Hole doping also produces an ordered lattice, and its spacing follows total excess filling while remaining insensitive to interlayer imbalance.
See the passage in the paper
Quantitatively, the extracted lattice spacing follows the total excess filling factor across both layers and remains insensitive to interlayer density imbalance (Fig. 4n).
Authors’ interpretation The authors interpret the observed lattices and their charge-dependent behavior as evidence for charged vortices in an interlayer-coherent exciton condensate.
See the passage in the paper
Together, the invariance of the lattice constant under density imbalance and the systematic evolution of the layer-resolved contrast provide evidence for an interlayer-coherent exciton condensate whose charged excitations are vortices of a U(1) order parameter.

Limitations

The measurements do not permit a precise quantitative determination of the exciton coherence length from vortex-core size.
See the passage in the paper
Although the present data do not allow a precise quantitative extraction, the observed trend supports a finite exciton coherence length, which future higher-resolution experiments could measure more precisely.
The study focuses on a particular graphene bilayer condensate and does not establish vortex-lattice behavior in other exciton-superfluid materials.

Why it matters

The authors argue that imaging the charged vortex lattice provides evidence of global interlayer phase coherence and spontaneously broken U(1) symmetry in an exciton condensate. The approach may enable spatial studies of vortex dynamics and phase-sensitive tests of exciton superfluidity.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: Exciton condensate charged vortex lattice interlayer coherence graphene bilayers scanning tunnelling microscopy

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Comprehensive Summary
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Original abstract

The paper investigates whether a bilayer exciton condensate can form a lattice of charged vortices, as predicted for a weakly charged exciton superfluid in a strong magnetic field. Such a lattice would provide evidence of interlayer phase coherence. Unlike vortices in a conventional superconductor, these vortices carry electric charge, which is expected to localize in their cores. A central prediction is that the lattice spacing depends on the total excess charge across both layers, not on how that charge is divided between them. This distinguishes the proposed vortex lattice from a Wigner crystal formed in a single layer. The researchers studied two graphene monolayers separated by about 3.2 nanometres of hexagonal boron nitride, in a perpendicular magnetic field of 9 tesla and at temperatures as low as 200 millikelvin. The device’s backgate controlled the total carrier density, while an interlayer voltage redistributed charge between the layers. A scanning tunnelling microscope measured the top layer’s tunnelling response and mapped its spatial variations. The authors first identified an exciton-insulating regime near total filling factor −2, where the layers have matched electron and hole densities and a spectroscopic gap. They then doped the condensate and used real-space current maps and their spatial structure factors to determine the charge-order pattern and lattice spacing. For both electron and hole doping, the maps showed periodic charge modulations with sixfold Bragg peaks. As doping increased, the features became denser, and the measured lattice spacing followed the predicted dependence on total excess filling across the two layers. At low excess filling, the results disagreed with the spacing expected for a Wigner crystal confined to the probed top layer. The vortex lattice evolved from sparse, partly disordered arrangements at low doping to more ordered patterns at intermediate doping, then to a disordered or melted regime at higher doping. Around an excess filling of 0.17, the measured spacing approached that expected for a top-layer Wigner crystal. This occurred near the onset of an incompressible state, consistent with the predicted transition from a vortex lattice to a conventional Wigner crystal. To test the role of charge distribution, the researchers varied the interlayer density imbalance while holding the total density fixed. Although electrostatic modelling indicated that the individual layer fillings changed by as much as 30 percent, the observed lattice spacing remained essentially unchanged. By contrast, a top-layer Wigner crystal would be expected to change its spacing by roughly 20 nanometres over the tested range. The lattice contrast did change, as expected if charge shifts between vortex and antivortex cores in opposite layers. Hole doping produced a similar lattice with reversed tunnelling contrast. Its spacing also tracked total excess filling rather than the top-layer density alone. Together, these observations support the interpretation that the patterns are charged vortices and antivortices in an interlayer-coherent exciton condensate, rather than a single-layer charge crystal. The authors conclude that the vortex lattice provides evidence of spontaneous U(1) symmetry breaking and global interlayer phase coherence. The images extend over roughly 400 nanometres, providing a lower bound of comparable scale for the coherence length. The modest broadening of vortex sites relative to Wigner-crystal sites is also suggestive of a finite core extent. Important limitations remain. STM measures charge-related tunnelling contrast rather than the condensate phase directly, so the vortex interpretation is inferred from how the lattice depends on density and interlayer imbalance. Disorder distorts or melts the pattern at some dopings, and the data do not allow a precise measurement of the exciton coherence length. The authors therefore present the results as evidence for phase coherence and as a platform for future phase-sensitive studies, rather than as a direct interferometric measurement.

Evidence for a Charged Vortex Lattice in an Exciton Superfluid

What happens when two layers of graphene are brought close enough for electrons in one layer to pair with holes in the other? Under the right conditions, the pairs may act together as an exciton superfluid—a state with a shared quantum phase across the layers. A new study, “Evidence for a charged vortex lattice in an exciton superfluid,” by Ryan L. Lee, Jungwoo Lee, Viliam Vaňo, Bo Zou, Cheng-Li Chiu, Kenji Watanabe et al., reports a striking pattern that may reveal this phase: a lattice of electrically charged vortices.

The result matters because these vortices are not quite like the magnetic vortices familiar from conventional superconductors. In the proposed exciton superfluid, a vortex can carry electric charge, expected to gather in its core. The way the vortex lattice responds to added charge—and to charge moved between the two layers—offers a way to distinguish it from a more ordinary crystal of electrons.

A test of interlayer coherence

The researchers studied two graphene monolayers separated by about 3.2 nanometres of hexagonal boron nitride. They applied a perpendicular magnetic field of 9 tesla and cooled the device to temperatures as low as 200 millikelvin. A scanning tunnelling microscope measured the tunnelling response of the top graphene layer and mapped how that response varied across the surface.

Two electrical controls let the team tune the system in different ways. A backgate changed the total carrier density across both layers, while an interlayer voltage shifted charge from one layer to the other. This distinction was essential: a vortex lattice tied to the exciton condensate should respond to the total excess charge, whereas a Wigner crystal—a regular arrangement of electrons confined to one layer—should depend on the charge in that layer alone.

First, the researchers identified an exciton-insulating regime near total filling factor −2. In this regime, the layers had matched electron and hole densities and showed a spectroscopic gap. They then added electrons or holes and looked for spatial patterns in the tunnelling maps.

Patterns that follow the total charge

With both electron and hole doping, the maps revealed periodic modulations with sixfold Bragg peaks, the signature of a hexagonal pattern. As the excess filling increased, the features became more closely spaced. Crucially, the measured lattice spacing followed the prediction based on the total excess filling of the two layers.

At low doping, this behavior did not match the spacing expected if the pattern were a Wigner crystal in the top layer alone. The observed arrangement instead supported the idea that the lattice involved charged vortices—or antivortices—of a coherent state spanning both layers.

The pattern changed as the researchers added more charge. At low excess filling, vortices appeared sparse and partly disordered. At intermediate filling, the lattice became more ordered; at higher filling, it became disordered or melted. Around an excess filling of 0.17, the spacing approached the value expected for a top-layer Wigner crystal. This happened near the onset of an incompressible state, consistent with a proposed transition from a vortex lattice to a conventional electron crystal.

Redistributing charge without moving the lattice

The most telling test came from changing the balance of charge between the layers while keeping the total density fixed. Electrostatic modelling indicated that the filling of each individual layer could shift by as much as 30 percent. Yet the measured lattice spacing remained essentially unchanged. A crystal determined by the top-layer density would instead be expected to change its spacing by roughly 20 nanometres across the tested range.

The lattice contrast did change as charge moved between layers. That is consistent with a picture in which charge shifts between vortex and antivortex cores in opposite layers, while the overall spacing remains set by the total excess charge. Hole doping produced a similar lattice with reversed tunnelling contrast, and its spacing also tracked total filling rather than the top-layer density.

The lattice spacing followed the total excess charge across both layers.
Changing the layer imbalance altered contrast but left the spacing nearly fixed.
At low doping, the pattern differed from the expected spacing of a top-layer Wigner crystal.

Evidence, with limits

Together, these observations support the interpretation that the patterns are charged vortices and antivortices in an interlayer-coherent exciton condensate. Such coherence would mean the system has spontaneously broken U(1) symmetry: the two layers share a collective phase. The images extend over roughly 400 nanometres, providing a lower bound of comparable scale for the coherence length. The vortex sites also appear modestly broader than sites in a Wigner crystal, a possible sign that the vortex cores have a finite extent.

Still, the measurement has limits. The microscope detects charge-related tunnelling contrast, not the condensate phase directly. The vortex interpretation is therefore inferred from how the pattern responds to total density and layer imbalance, rather than established through a direct phase-sensitive measurement. Disorder also distorts or melts the lattice at some dopings, and the data do not pin down the exciton coherence length precisely.

The study therefore offers evidence for interlayer phase coherence, not a direct interferometric proof. Its central finding is that the lattice behaves as a collective feature of both graphene layers, not simply as a crystal of charge in the layer being imaged. That makes the charged-vortex pattern a promising platform for future experiments designed to probe the phase of an exciton superfluid more directly.

Created on 02 Nov. 2026

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Evidence for a charged vortex lattice in an exciton superfluid

The essentials

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Method

Findings

Limitations

Why it matters

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Definitions

Evidence for a Charged Vortex Lattice in an Exciton Superfluid

A test of interlayer coherence

Patterns that follow the total charge

Redistributing charge without moving the lattice

Evidence, with limits

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