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REVIEW 2 major objections 4 minor 56 references

Hidden Quantum Geometry in Bilayer Exciton Condensates

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Pith's one-line read Exciton condensation alone creates hidden quantum geometry in trivial bands, and the telltale signal is an out-of-plane second-harmonic polarization that obeys an inverse-square scaling law.

desk verdict A clean derivation of a new Δ^{-2} second-harmonic response for ideal bilayer exciton condensates, with an overbroad generality claim that needs tempering. read the letter →

arxiv 2608.03049 v1 pith:2XRI7F54 submitted 2026-08-04 cond-mat.mes-hall

Xuzhe Ying , Benjamin T. Zhou This is my paper · ORCID
classification cond-mat.mes-hall
keywords bilayerexcitoncondensatehiddenquantumgeometryin-planeBerrycurvatureout-of-planepolarizationsecond-harmonicresponsenon-perturbativescalingkineticequationMoyalproduct
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that inter-layer excitonic order is itself a source of quantum geometry: even with trivial non-interacting electron and hole bands, the condensate order parameter rebuilds the quasiparticle wavefunctions and gives the correlated bands a hidden in-plane Berry curvature. Concretely, an in-plane AC electric field then drives the out-of-plane polarization to oscillate at the second harmonic, with strength $\chi^{(2)}_{z;\alpha\beta} \approx C e^2/\Delta^2\,\delta_{\alpha\beta}$ and $C\approx -0.01326$. Because the order parameter appears in the denominator, this response is non-perturbative in the condensate, making it a direct experimental signature of inter-layer coherence. The result matters because it turns a subtle geometric object into a measurable optical signal, and the authors argue it extends beyond bilayer exciton condensates to van der Waals stacked materials.

What carries the argument

The load-bearing object is the hidden in-plane Berry curvature, defined as the in-plane curl of the out-of-plane Berry connection: $\Omega_n^\alpha(k)=\epsilon_{\alpha\beta z}\partial_{k_\beta} A_z^n(k)$, with $A_z^n(k)=u_n^\dagger(k)(-\sigma_z/2)u_n(k)$. This quantity is 'hidden' because there is no Bloch momentum in the stacking direction; the out-of-plane direction enters through the layer-polarization operator rather than through a third momentum component. The second piece of machinery is the Moyal quantum kinetic equation, which dresses the Bloch wavefunctions order by order in the electric field and produces the generalized third-rank quantum geometric tensor integrands in Eq. (12). Together these convert a static geometric object, inter-layer coherence, into a dynamical observable, the second-harmonic out-of-plane polarization.

What would settle it

Measure the second-harmonic out-of-plane polarization of an exciton-condensed double layer as a function of temperature through $T_c$ and test whether $\chi^{(2)}_{z;xx} \propto \Delta^{-2}$ with the independently measured order parameter; the claim fails if the signal disappears when inter-layer coherence is suppressed in a system with otherwise trivial non-interacting bands, or if the exponent departs measurably from $-2$.

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Extended reading notes

Core claim

The paper's central claim is that spontaneous inter-layer coherence generates genuine quantum geometry in the correlated electron-hole bands even when the bare bands are trivial. In the mean-field two-band description, the condensate order parameter $\Delta$ hybridizes the layer degrees of freedom. The authors define an out-of-plane 'Berry connection' $A_z^n(k)=u_n^\dagger(k)\hat z u_n(k)$ with $\hat z=-\sigma_z/2$, whose in-plane curl gives a hidden in-plane Berry curvature $\Omega_n^\alpha(k)=\epsilon_{\alpha\beta z}\partial_{k_\beta} A_z^n(k)$. Using a Moyal-based quantum kinetic equation, they compute the out-of-plane polarization response to a uniform in-plane AC field up to second order, obtaining $\chi^{(2)}_{z;\alpha\beta} = -\frac{e^2}{8}\int \frac{d^2k}{(2\pi)^2} \frac{\Delta^2}{\epsilon^5(k)} \left[\partial^2_{k_\alpha k_\beta}\xi(k) - \frac{5}{2}\frac{\xi(k)-\mu}{\epsilon^2(k)}\partial_{k_\alpha}\xi(k)\partial_{k_\beta}\xi(k)\right]$, which in the long-wavelength, small-$\Delta$ limit reduces to $\chi^{(2)}_{z;\alpha\beta} \approx C e^2/\Delta^2\,\delta_{\alpha\beta}$ with $C\approx -0.01326$. The same mechanism produces both a DC and a second-harmonic component, and the paper interprets this as a hidden Berry-phase effect driven by electron-hole correlations.

Load-bearing premise

The calculation assumes that the out-of-plane position operator is exactly the layer-density difference, $\hat z=-\sigma_z/2$, and that the AC field only dresses the wavefunctions while the quasiparticle distribution stays at equilibrium; if a real bilayer has interlayer position matrix elements beyond $\sigma_z$, or if the field redistributes quasiparticles among the reconstructed bands, the predicted $\Delta^{-2}$ scaling would not be the full response.

Editorial extensions

If this is right

  • - An in-plane AC field produces an out-of-plane dipole oscillation at both DC and twice the driving frequency, with the second-harmonic component equal to half of the static response; this is a directly measurable optical signature.
  • - Since the response scales as $\Delta^{-2}$, cooling through $T_c$ should produce a sharp rise in the second-harmonic signal, followed by a $\sim 1/(T_c-T)$ increase if the order parameter follows the mean-field square-root law.
  • - The effect occurs even when the non-interacting bands carry no quantum geometry, so it isolates the correlations' contribution from single-particle band-structure effects.
  • - The same formulas apply to van der Waals stacked materials generally, which means an in-plane field can manipulate out-of-plane polarization through the quadratic response.
  • - Because the order parameter sits in the denominator, the response cannot be obtained by a perturbative expansion in the condensate; it is a non-perturbative fingerprint of inter-layer coherence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • - A finite-frequency generalization of the kinetic-equation result should show resonances when $2\omega$ approaches the quasiparticle gap; such resonances would separate the geometric contribution from a trivial background response.
  • - In a realistic bilayer the position operator may contain interlayer tunneling matrix elements beyond $-\sigma_z/2$; computing the same response with such a generalized operator would test how robust the clean $\Delta^{-2}$ law is away from the idealized model.
  • - The non-perturbative $\Delta^{-2}$ scaling implies that any effective theory truncated at low powers of the order parameter will miss the effect, so the second-harmonic signal is a probe of wavefunction geometry rather than of free-energy curvature.
  • - In lower-symmetry or time-reversal-broken bilayers the linear response also becomes nonzero; separating the linear and quadratic channels may be necessary before the inverse-square scaling can be used as a clean experimental fingerprint.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. This Letter studies the nonlinear out-of-plane polarization response of a bilayer exciton condensate to an in-plane AC electric field. Within a mean-field two-band model with order parameter Δ, the authors define a 'hidden' in-plane Berry curvature from the z-position operator, and use a Moyal-product quantum kinetic formalism to derive linear and second-order response functions (Eqs. (11), (12)). For a square-lattice tight-binding model with no interlayer hopping, they obtain χ^(2)_{z;αβ} = C e²/Δ² δ_{αβ} with C≈−0.01326, an inverse-square dependence on the excitonic order parameter, and propose a second-harmonic vertical dipole oscillation as an experimental signature of interlayer coherence. The main technical derivation is contained in the Supplemental Material.

Significance. If the result holds as stated, the paper gives a genuinely interaction-generated quantum geometric response in a trivial-band system and a concrete, falsifiable experimental prediction. The analytic calculation is internally consistent: Eq. (12) is the standard quantum-kinetic second-order response, and the square-lattice integral leading to C≈−0.01326 is evaluated exactly in the SM. The proposal to use a 2ω out-of-plane response as a probe of interlayer coherence is attractive and could be tested in graphene/TMD double layers. The main limitation, discussed below, is that the specific Δ^{-2} law is derived only for the equilibrium-distribution (ideal fully occupied) case; the paper's unqualified claim that it holds generally for bilayer ECs is not supported. With that scope fixed, the result would be a worthwhile contribution.

major comments (2)
  1. [Nonlinear response; Eq. (12) and the sentence after Eq. (14)] Equation (12) is derived from Eq. (10) by setting f_n(k;t)=f^eq_n(k) (paragraph after Eq. (7)). This is exact when the lower band is completely filled and the upper band is empty, because then ∂_k f^eq=0 and the anomalous-velocity term in Eq. (5) vanishes. For the general bilayer ECs invoked in the sentence after Eq. (14) and in the abstract, f^eq is not momentum-independent: near the Fermi surface the Boltzmann equation (8) generates f^(1)≈eτ E·∂_k f^eq, and the products f^(1)(U^(0)† ẑ U^(1)+h.c.) in Eq. (10) are second order in E but are absent from Eq. (12). These terms can scale as e²τ/(Δ T) or as transport-type contributions, not necessarily as Δ^{-2}; the assertion that χ^(2)∼Δ^{-2} 'hold[s] generally for bilayer ECs' is therefore unproven. Please either restrict the universality claim to the ideal T=0 exciton insulator with full occupancy, or include the distribution-correction terms and show that they are subleading for the regimes of interest.
  2. [Conclusion] The prediction that χ^(2)_{z,xx} ∼ 1/(T_c−T) below T_c combines the mean-field Δ∼(T_c−T)^{1/2} with Eq. (14). However, close to T_c the order parameter is small (Δ≪T), and the equilibrium-distribution assumption used to obtain Eq. (14) fails because ∂_k f^eq has a thermal width ∼T, so the Boltzmann-drift corrections f^(1)∼eτ E·∂_k f^eq are not parametrically small. The distribution-generated second-order terms scale as e²τ/(Δ T); with T∼Δ near T_c this is the same order as the wavefunction-dressing term e²/Δ². A quantitative estimate, or an explicit restriction to the deep superfluid regime, is required.
minor comments (4)
  1. [Supplemental Material, Eq. (31)] The off-diagonal entry of A^α_{mn} contains a stray closing bracket: the numerator should read −iΔ ∂_{kα}ξ(k)/(2ϵ²(k)), not '−i∆∂kαξ(k)]/(2ϵ²(k))'.
  2. [Paragraph after Eq. (7)] The statement that the equilibrium distribution is 'almost constant' at low temperature is imprecise: in an exciton insulator f^eq_n is 0 or 1 over most of the Brillouin zone but has a step of width T at the gap. Please state explicitly that ∂_k f^eq=0 exactly at T=0 for full occupancy, and otherwise quote the small parameter controlling the distribution corrections.
  3. [Fig. 2 and Eq. (14)] The inset caption says the same data are plotted 'taking absolute value', but the main text reports a negative C≈−0.01326. Please state the sign convention for χ^(2)_{z,xx} and the units of Δ (presumably t) used in the figure, so that the reader can directly compare the numerical plot with Eq. (14).
  4. [References] Reference [44] is incomplete: it gives the journal, volume, and article number but not the author list or a complete publication year; please update it to a fully citable form.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Δ^{-2} scaling is obtained by explicit integration from the input mean-field model, not fitted or assumed.

full rationale

The central claim, Eq. (14), is a direct evaluation of the general second-order response kernel, Eq. (12), for the two-band mean-field exciton-condensate Hamiltonian, Eq. (1). The order parameter Δ is an input of the model, and the inverse-square scaling arises from the explicit momentum integral in Eq. (13), whose reduction to C e²/Δ² with C≈-0.01326 is carried out analytically in the Supplemental Material. No parameter is fitted to the target observable, and no equation is assumed in the form of the answer. The identification of a 'hidden' in-plane Berry curvature, Eq. (4), is a definition in terms of the layer-polarization operator ẑ=-σ_z/2; computing the response from the same matrix elements is a calculation, not a tautology, because the nontrivial content is the resulting Δ^{-2} law and its coefficient. The self-citation to Ref. [43] supports the physical interpretation of the out-of-plane Berry connection, is accompanied by Ref. [44] by independent authors, and is not needed to derive Eqs. (11)-(14). The stated restriction 'we focus on the exciton insulators and take f_n(k;t)=f^eq_n(k)' is a genuine approximation that may limit the claimed generality of the Δ^{-2} law for partially filled or finite-temperature ECs, but this is a correctness/scope concern, not circularity. The derivation is self-contained: the only external ingredients are the Moyal/quantum-kinetic framework, whose use is summarized in the paper and SM, and the standard mean-field description of bilayer exciton condensation.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a mean-field model with input parameters Δ, μ and t, on identifying the layer pseudospin as the out-of-plane position, and on the equilibrium-distribution assumption in the nonlinear response calculation. No parameters are fitted to data; the leading scaling law is independent of the non-universal parameters. The hidden in-plane Berry curvature is a derived quantity rather than a new physical entity.

free parameters (3)
  • mean-field order parameter Δ
    Input parameter of the model Hamiltonian; the paper computes the response as a function of Δ and does not fit it to data. It is the physical quantity the predicted signal is meant to probe.
  • chemical potential μ (or δμ = -4t - μ)
    Model parameter; the leading-order result Eq. (14) is independent of it within the regime Δ << |δμ| << t.
  • nearest-neighbor hopping t
    Sets the energy unit; cancels in the long-wavelength approximation and does not enter the scaling law.
assumptions (5)
  • domain assumption The bilayer exciton condensate is described by the mean-field Hamiltonian of Eq. (1) with a momentum-independent order parameter Δ.
    The entire calculation is performed within this mean-field description; fluctuations and the self-consistent determination of Δ are not treated. This enters at Eq. (1).
  • domain assumption The out-of-plane position operator is exactly the layer Pauli matrix ẑ = -σ_z/2, and the out-of-plane polarization is the expectation value of this operator in the quasiparticle bands (Eq. 2).
    This neglects finite layer thickness and interlayer position matrix elements; if the physical polarization differs from the layer-density difference, the response would change. Invoked in the definition of P_z.
  • domain assumption The distribution function remains the equilibrium Fermi distribution when computing the second-order response; the field only dresses the wavefunctions.
    The paper states this in the 'Nonlinear response' section. It is valid for a completely filled/empty band insulator in the low-frequency limit, but is an assumption that could fail near resonance or for metallic fillings.
  • domain assumption The long-wavelength and small-detuning approximation Δ << |μ+4t| << t permits replacing the tight-binding dispersion by a parabolic one and extending the energy integral to ±∞.
    Used to derive Eq. (14) from Eq. (13) in the SM; the leading result is independent of this approximation, but the exact coefficient C may change for generic band structures.
  • standard math The Moyal product expansion and quantum kinetic equation of Ref. [37] correctly describe the nonlinear response to a uniform electric field.
    The paper relies on the quantum kinetic equation framework of Ref. [37]; this is an external theoretical tool not proven within the paper.

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Cite this review

Pith. "Pith review of Hidden Quantum Geometry in Bilayer Exciton Condensates." pith.science (2026). https://pith.science/paper/2XRI7F54

@misc{pith2026260803049,
  author       = {Pith},
  title        = {Pith review of: Hidden Quantum Geometry in Bilayer Exciton Condensates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XRI7F54}},
  note         = {Machine review of arXiv:2608.03049}
}
read the original abstract

When an electron-doped layer is stacked with a hole-doped layer with approximately equal carrier density, inter-layer Coulomb interaction turns the bilayer system into an exciton condensate (EC). In this Letter, we reveal a fundamental property of bilayer ECs: the excitonic order gives rise to nontrivial hidden quantum geometric effects in the correlated electron-hole bands, even when the non-interacting bands are trivial. Such peculiar EC-driven quantum geometry manifests itself in a characteristic out-of-plane polarization response upon applying an in-plane AC electric field to the bilayer EC system. In particular, the second-order response exhibits a characteristic inverse square scaling with the bilayer EC order parameter. Our finding reveals a fundamental hidden Berry phase effect driven by electron-hole correlations, and establishes bilayer EC as a promising platform for rich nonlinear physics.

Figures

Figures reproduced from arXiv: 2608.03049 by the authors.

Figure 1
Figure 1. Bilayer system and exciton condensate. (a) Schematic sys [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Exciton condensate’s nonlinear out-of-plane dielectric re [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Reference graph

Works this paper leans on

56 extracted references · 44 canonical work pages

  1. [1]

    J. P. Eisenstein, Exciton condensation in bilayer quantum hall systems, Annual Review Condensed Matter Physics5, 159 (2014)

    work page 2014

  2. [2]

    J. P. Eisenstein and A. H. MacDonald, Bose–einstein conden- sation of excitons in bilayer electron systems, Nature432, 691–694 (2004)

    work page 2004

  3. [3]

    J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, K. W. West, and S. He, New fractional quantum hall state in double-layer two- dimensional electron systems, Phys. Rev. Lett.68, 1383 (1992)

    work page 1992

  4. [4]

    Y . W. Suen, L. W. Engel, M. B. Santos, M. Shayegan, and D. C. Tsui, Observation of aν=1/2 fractional quantum hall state in a double-layer electron system, Phys. Rev. Lett.68, 1379 (1992)

    work page 1992

  5. [5]

    Kellogg, I

    M. Kellogg, I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Observation of quantized hall drag in a strongly correlated bilayer electron system, Phys. Rev. Lett.88, 126804 (2002)

    work page 2002

  6. [6]

    Kellogg, J

    M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Bilayer quantum hall systems atν T = 1: Coulomb drag and the transition from weak to strong interlayer coupling, Phys. Rev. Lett.90, 246801 (2003)

    work page 2003

  7. [7]

    Nandi, J

    D. Nandi, J. P. Finck, A. D. K.and Eisenstein, L. N. Pfeiffer, and K. W. West, Exciton condensation and perfect coulomb drag, Nature488, 481–484 (2012)

    work page 2012

  8. [8]

    A. MacDonald, Superfluid properties of double-layer quan- tum hall ferromagnets, Physica B: Condensed Matter298, 129 (2001), international Conference on High Magnetic Fields in Semiconductors

    work page 2001

Show all 56 references
  1. [9]

    Kellogg, J

    M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Vanishing hall resistance at high magnetic field in a double- layer two-dimensional electron system, Phys. Rev. Lett.93, 036801 (2004)

    2004

  2. [10]

    Tutuc, M

    E. Tutuc, M. Shayegan, and D. A. Huse, Counterflow measure- ments in strongly correlated gaas hole bilayers: Evidence for electron-hole pairing, Phys. Rev. Lett.93, 036802 (2004)

    2004

  3. [11]

    Z. Wang, D. A. Rhodes, K. Watanabe, T. Taniguchi, J. C. Hone, J. Shan, and K. F. Mak, Evidence of high-temperature exciton condensation in two-dimensional atomic double layers, Nature 574, 76–80 (2019)

    2019

  4. [12]

    L. Ma, P. X. Nguyen, Y . Z. Zefang Wang, K. Watanabe, T. Taniguchi, A. H. MacDonald, K. F. Mak, and J. Shan, Strongly correlated excitonic insulator in atomic double layers, Nature598, 585–589 (2021)

    2021

  5. [13]

    J. Gu, L. Ma, S. Liu, K. Watanabe, T. Taniguchi, J. C. Hone, J. Shan, and K. F. Mak, Dipolar excitonic insulator in a moir ´e lattice, Nature Physics18, 395–400 (2022)

    2022

  6. [14]

    J. Wang, Q. Shi, E.-M. Shih, L. Zhou, W. Wu, Y . Bai, D. Rhodes, K. Barmak, J. Hone, C. R. Dean, and X.-Y . Zhu, Diffusivity reveals three distinct phases of interlayer excitons inmose 2/wse2 heterobilayers, Phys. Rev. Lett.126, 106804 (2021)

    2021

  7. [15]

    Shi, E.-M

    Q. Shi, E.-M. Shih, D. Rhodes, B. Kim, K. Barmak, K. Watan- abe, T. Taniguchi, Z. Papi ´c, D. A. Abanin, J. Hone, and C. R. Dean, Bilayer wse2 as a natural platform for interlayer exciton condensates in the strong coupling limit, Nature Nanotechnol- ogy17, 577–582 (2022)

    2022

  8. [16]

    Y . Zeng, D. Sun, N. J. Zhang, R. Q. Nguyen, Q. Shi, A. Ok- ounkova, K. Watanabe, T. Taniguchi, J. Hone, C. R. Dean, and J. I. A. Li, Observation of a superfluid-to-insulator transition of bilayer excitons, Nature650, 86–92 (2026)

    2026

  9. [17]

    K. Wang, K. D. Greve, L. A. Jauregui, A. Sushko, A. High, Y . Zhou, G. Scuri, T. Taniguchi, K. Watanabe, M. D. Lukin, H. Park, and P. Kim, Electrical control of charged carriers and excitons in atomically thin materials, Nature Nanotechnology 13, 128–132 (2018)

    2018

  10. [18]

    L. A. Jauregui, A. Y . Joe, K. Pistunova, D. S. Wild, A. A. High, Y . Zhou, G. Scuri, K. D. Greve, A. Sushko, C.-H. Yu, T. Taniguchi, K. Watanabe, D. J. Needleman, M. D. Lukin, H. Park, and P. Kim, Electrical control of interlayer exciton dynamics in atomically thin heterostru...

    2019

  11. [19]

    X. Liu, J. I. A. Li, K. Watanabe, T. Taniguchi, J. Hone, B. I. Halperin, P. Kim, and C. R. Dean, Crossover between strongly coupled and weakly coupled exciton superfluids, Science375, 205 (2022)

    2022

  12. [20]

    Y . Hu, J. W. F. Venderbos, and C. L. Kane, Fractional excitonic insulator, Phys. Rev. Lett.121, 126601 (2018)

    2018

  13. [21]

    Y . H. Kwan, Y . Hu, S. H. Simon, and S. A. Parameswaran, Ex- citonic fractional quantum hall hierarchy in moir ´e heterostruc- tures, Phys. Rev. B105, 235121 (2022)

    2022

  14. [22]

    Y . H. Kwan, Y . Hu, S. H. Simon, and S. A. Parameswaran, Ex- citon band topology in spontaneous quantum anomalous hall insulators: Applications to twisted bilayer graphene, Phys. Rev. Lett.126, 137601 (2021)

    2021

  15. [23]

    Z. Han, T. Wang, Z. Dong, M. P. Zaletel, and A. Vishwanath, Anyon superfluidity of excitons in quantum hall bilayers, Phys. Rev. X16, 021044 (2026)

    2026

  16. [24]

    Zhang, Z

    Y .-H. Zhang, Z. Zhu, and A. Vishwanath, Xy* transition and ex- traordinary boundary criticality from fractional exciton conden- sation in quantum hall bilayer, Phys. Rev. X13, 031023 (2023)

    2023

  17. [25]

    Z. Song, T. Cookmeyer, and L. Balents, A solvable model for strongly interacting nonequilibrium excitons, Proc. Natl. Acad. Sci. U.S.A.122, e2424663122 (2025)

    2025

  18. [26]

    Julku, G

    A. Julku, G. M. Bruun, and P. T ¨orm¨a, Quantum geometry and flat band bose-einstein condensation, Phys. Rev. Lett.127, 170404 (2021). 6

    2021

  19. [27]

    Verma, D

    N. Verma, D. Guerci, and R. Queiroz, Geometric stiffness in interlayer exciton condensates, Phys. Rev. Lett.132, 236001 (2024)

    2024

  20. [28]

    X. Hu, T. Hyart, D. I. Pikulin, and E. Rossi, Quantum-metric- enabled exciton condensate in double twisted bilayer graphene, Phys. Rev. B105, L140506 (2022)

    2022

  21. [29]

    Iskin, Pair size and quantum geometry in a multiband hub- bard model, Phys

    M. Iskin, Pair size and quantum geometry in a multiband hub- bard model, Phys. Rev. B111, 014502 (2025)

    2025

  22. [30]

    Ying and K

    X. Ying and K. T. Law, Flat band excitons and quantum metric (2024), arXiv:2407.00325 [cond-mat.mes-hall]

    2024 arXiv

  23. [31]

    Ying and K

    X. Ying and K. Li, Quantum metric driven transition between superfluid and incoherent fluid, Phys. Rev. B112, 014518 (2025)

    2025

  24. [32]

    I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Resonantly enhanced tunneling in a double layer quantum hall ferromagnet, Phys. Rev. Lett.84, 5808 (2000)

    2000

  25. [33]

    Xiao, M.-C

    D. Xiao, M.-C. Chang, and Q. Niu, Berry phase effects on elec- tronic properties, Rev. Mod. Phys.82, 1959 (2010)

    2010

  26. [34]

    Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, 2018)

    D. Vanderbilt,Berry Phases in Electronic Structure Theory: Electric Polarization, Orbital Magnetization and Topological Insulators(Cambridge University Press, 2018)

    2018

  27. [35]

    J. P. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Commun. Math. Phys.76, 289 (1980)

    1980

  28. [36]

    Cheng, Quantum geometric tensor (fubini-study metric) in simple quantum system: A pedagogical introduction, arXiv:1012.1337 (2010)

    R. Cheng, Quantum geometric tensor (fubini-study metric) in simple quantum system: A pedagogical introduction, arXiv:1012.1337 (2010)

    2010 arXiv

  29. [37]

    T. Park, X. Huang, L. Savary, and L. Balents, Quantum geom- etry from the moyal product: Quantum kinetic equation and nonlinear response, Phys. Rev. B113, 045146 (2026)

    2026

  30. [38]

    Mangeolle, L

    L. Mangeolle, L. Savary, and L. Balents, Quantum kinetic equa- tion and thermal conductivity tensor for bosons, Phys. Rev. B 109, 235137 (2024)

    2024

  31. [39]

    Kamenev,Field theory of non-equilibrium systems(Cam- bridge University Press, Cambridge, 2023)

    A. Kamenev,Field theory of non-equilibrium systems(Cam- bridge University Press, Cambridge, 2023)

    2023

  32. [40]

    Resta and D

    R. Resta and D. Vanderbilt, Theory of polarization: a modern approach, Physics of Ferroelectrics , 31 (2007)

    2007

  33. [41]

    R. D. King-Smith and D. Vanderbilt, Theory of polarization of crystalline solids, Phys. Rev. B47, 1651 (1993)

    1993

  34. [42]

    Resta, Macroscopic polarization in crystalline dielectrics: the geometric phase approach, Rev

    R. Resta, Macroscopic polarization in crystalline dielectrics: the geometric phase approach, Rev. Mod. Phys.66, 899 (1994)

    1994

  35. [43]

    B. T. Zhou, V . Pathak, and M. Franz, Quantum-geometric origin of out-of-plane stacking ferroelectricity, Phys. Rev. Lett.132, 196801 (2024)

    2024

  36. [44]

    J. Chen, P. Cui, and Z. Zhang, Ferroelectricity-tuned band topology and superconductivity in 2d materials and related het- erostructures, Advanced Functional Materials34, 2408625

  37. [45]

    See Supplemental Material for detailed calculations for (i) Moyal diagonalization, (ii) linear response function, (iii) sec- ond order response function, (iv) application to simple model on square lattice

  38. [46]

    Barik and J

    T. Barik and J. D. Sau, Nonequilibrium nature of nonlinear opti- cal response: Application to the bulk photovoltaic effect, Phys. Rev. B101, 045201 (2020)

    2020

  39. [47]

    Altland and B

    A. Altland and B. D. Simons,Condensed matter field theory (Cambridge university press, 2010)

    2010

  40. [48]

    Avdoshkin, J

    A. Avdoshkin, J. Mitscherling, and J. E. Moore, Multistate ge- ometry of shift current and polarization, Phys. Rev. Lett.135, 066901 (2025)

    2025

  41. [49]

    Mitscherling, A

    J. Mitscherling, A. Avdoshkin, and J. E. Moore, Gauge- invariant projector calculus for quantum state geometry and ap- plications to observables in crystals, Phys. Rev. B112, 085104 (2025)

    2025

  42. [50]

    A. K. Geim and I. V . Grigorieva, Van der waals heterostructures, Nature499, 419–425 (2013)

    2013

  43. [51]

    K. S. Novoselov, A. Mishchenko, A. Carvalho, and A. H. C. Neto, 2d materials and van der waals heterostructures, Science 353, aac9439 (2016)

    2016

  44. [52]

    Li and M

    L. Li and M. Wu, Binary compound bilayer and multilayer with vertical polarizations: Two-dimensional ferroelectrics, multi- ferroics, and nanogenerators, ACS Nano11, 6382 (2017)

    2017

  45. [53]

    Yasuda, X

    K. Yasuda, X. Wang, K. Watanabe, T. Taniguchi, and P. Jarillo- Herrero, Stacking-engineered ferroelectricity in bilayer boron nitride, Science372, 1458 (2021)

    2021

  46. [54]

    M. V . Stern, Y . Waschitz, W. Cao, I. Nevo, K. Watanabe, T. Taniguchi, E. Sela, M. Urbakh, O. Hod, and M. B. Shalom, Interfacial ferroelectricity by van der waals sliding, Science 372, 1462 (2021)

    2021

  47. [55]

    Wu and J

    M. Wu and J. Li, Sliding ferroelectricity in 2d van der waals materials: Related physics and future opportunities, Proc. Natl. Acad. Sci. U.S.A.118, e2115703118 (2021)

    2021

  48. [56]

    (2,0) + (0,2)

    J. Ji, G. Yu, C. Xu, and H. J. Xiang, General theory for bilayer stacking ferroelectricity, Phys. Rev. Lett.130, 146801 (2023). Supplemental Material: Hidden Berry Curvature Effects in Bilayer Exciton Condensate Xuzhe Ying 1, 2 and Benjamin T. Zhou 3,∗ 1Department of Physics, ...

    2023

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.