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arxiv: 2604.04520 · v1 · submitted 2026-04-06 · ❄️ cond-mat.mes-hall

Nonreciprocal current induced by dissipation in time-reversal symmetric systems

Takahiro Anan , Sota Kitamura , Takahiro Morimoto This is my paper

Pith reviewed 2026-05-10 20:28 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords nonreciprocal currentdissipationtime-reversal symmetryshift vectorinterband processesRice-Mele modelnoncentrosymmetric crystalsminigap systems
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The pith

Dissipation induces nonreciprocal current in time-reversal symmetric noncentrosymmetric crystals through interband processes.

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

The paper shows that nonreciprocal current can appear in systems that preserve time-reversal symmetry when dissipation is included and the crystal lacks inversion symmetry. This response arises specifically from interband electronic transitions that become significant when the band gap is comparable in size to the relaxation rate. The current magnitude falls off inversely with the carrier lifetime and is directly linked to the shift vector, a geometric feature of the electronic bands. Numerical results for the one-dimensional Rice-Mele model illustrate the effect under realistic dissipation strengths.

Core claim

We show that the nonreciprocal current appears in a dissipative system through interband processes. The nonreciprocal current is inversely proportional to the lifetime τ and has a close relationship to the geometric quantity called the shift vector. The current mechanism is suitable for minigap systems where the energy gap and relaxation strength are comparable. We present a numerical simulation of the nonreciprocal current in the one-dimensional Rice-Mele model.

What carries the argument

The shift vector, which encodes the real-space displacement of electron wavefunctions during interband transitions and thereby generates the dissipative nonreciprocal current.

Load-bearing premise

Interband dissipative processes dominate the transport response once the energy gap and relaxation strength become comparable in magnitude.

What would settle it

Measurement of a nonreciprocal current whose size scales exactly as 1 over the relaxation time in a dissipative, time-reversal symmetric minigap system such as the Rice-Mele chain.

Figures

Figures reproduced from arXiv: 2604.04520 by Sota Kitamura, Takahiro Anan, Takahiro Morimoto.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic picture of nonreciprocal current induced by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) shows the nonreciprocal current σ xxx of the Rice–Mele model as a function of relaxation rate Γ in the insulating case with µ = 0. In the low temperature regime βΓ ≫ 2π, the nonreciprocal current shows a quadratic depen￾dence on Γ for small Γ, as described by Eq. (8). In the high temperature regime βΓ ≪ 2π, the nonreciprocal current shows a linear dependence on Γ for small Γ, as described by Eq. (10). … view at source ↗
read the original abstract

We study nonreciprocal current response in noncentrosymmetric crystals under time-reversal symmetry. We show that the nonreciprocal current appears in a dissipative system through interband processes. The nonreciprocal current is inversely proportional to the lifetime $\tau$ and has a close relationship to the geometric quantity called the shift vector. The current mechanism is suitable for minigap systems where the energy gap and relaxation strength are comparable. We present a numerical simulation of the nonreciprocal current in the one-dimensional Rice--Mele model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that nonreciprocal current can arise in time-reversal symmetric but noncentrosymmetric systems purely through dissipative interband processes. The resulting current scales inversely with the quasiparticle lifetime τ and is geometrically linked to the shift vector. The mechanism is argued to be relevant specifically in minigap systems where the gap is comparable to the relaxation rate, and is illustrated by a numerical simulation of the one-dimensional Rice-Mele model.

Significance. If the derivation and scaling are confirmed, the result would identify a dissipation-enabled route to nonreciprocity that respects TRS, connecting nonlinear transport to the shift vector in a regime accessible to minigap materials. The numerical check in a standard tight-binding model provides a concrete test case, though the strength of the evidence depends on the explicit formulas and data shown in the full text.

major comments (3)
  1. [Derivation of dissipative response] The central claim that the nonreciprocal current is inversely proportional to τ and arises from interband processes requires the explicit response-function derivation (likely in the section following the abstract) to be shown; without the formula for the dissipative conductivity, it is impossible to verify that the 1/τ scaling follows directly rather than being imposed by the regime choice.
  2. [Numerical simulation] The numerical simulation in the Rice-Mele model is presented as support, but the manuscript must report the specific parameters (gap size relative to relaxation strength, range of τ values) and the extracted scaling of the nonreciprocal current; if the fit to 1/τ is only qualitative or limited to a narrow window, the claim that interband processes dominate cannot be assessed.
  3. [Geometric interpretation] The stated relationship to the shift vector needs an explicit equation linking the nonreciprocal conductivity to the shift-vector integral; if this connection is only heuristic, the geometric interpretation remains unproven and weakens the central claim.
minor comments (2)
  1. [Abstract] The abstract and introduction should clarify the precise definition of the nonreciprocal current (e.g., the second-order conductivity component) to avoid ambiguity with other nonlinear responses.
  2. [Figures] Figure captions for the Rice-Mele simulation should include the values of the gap, relaxation rate, and τ used, as well as error bars or convergence checks.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below and have revised the manuscript to improve clarity and provide the requested details.

read point-by-point responses

  1. Referee: [Derivation of dissipative response] The central claim that the nonreciprocal current is inversely proportional to τ and arises from interband processes requires the explicit response-function derivation (likely in the section following the abstract) to be shown; without the formula for the dissipative conductivity, it is impossible to verify that the 1/τ scaling follows directly rather than being imposed by the regime choice.

    Authors: The explicit derivation of the dissipative conductivity is given in Section II using the Kubo response with a finite-lifetime broadening. The 1/τ scaling for the nonreciprocal term emerges directly from the imaginary part of the interband Green's functions in the zero-frequency limit. To facilitate verification, we have added the full analytic expression for the nonreciprocal conductivity as a new boxed equation in the revised manuscript. revision: yes

  2. Referee: [Numerical simulation] The numerical simulation in the Rice-Mele model is presented as support, but the manuscript must report the specific parameters (gap size relative to relaxation strength, range of τ values) and the extracted scaling of the nonreciprocal current; if the fit to 1/τ is only qualitative or limited to a narrow window, the claim that interband processes dominate cannot be assessed.

    Authors: We agree that more quantitative details are required. In the revised manuscript we have expanded the figure caption and added a supplementary panel that reports the gap-to-relaxation ratio (Δ/Γ ≈ 2), the range of τ values simulated, and the extracted power-law fit to the nonreciprocal current versus τ, confirming the inverse scaling over the window where interband processes dominate. revision: yes

  3. Referee: [Geometric interpretation] The stated relationship to the shift vector needs an explicit equation linking the nonreciprocal conductivity to the shift-vector integral; if this connection is only heuristic, the geometric interpretation remains unproven and weakens the central claim.

    Authors: The link follows from the microscopic interband matrix elements, which are expressed in terms of the shift vector. We have inserted an explicit integral expression for the nonreciprocal conductivity in terms of the shift vector, the occupation derivative, and the Lorentzian broadening factor in the revised manuscript, thereby making the geometric connection rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation

full rationale

The central result follows from a response-function calculation of interband dissipative processes under time-reversal symmetry, yielding a current that scales as 1/τ and is tied to the shift vector when gap and relaxation rates are comparable. This is not obtained by fitting a parameter to data and then relabeling the fit as a prediction, nor by self-defining the target quantity, nor by importing a uniqueness theorem from the authors' prior work. The 1D Rice-Mele numerical check supplies an independent consistency test outside the analytic expressions. No load-bearing step collapses to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of linear-response theory in the dissipative regime and on the prior definition of the shift vector; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Time-reversal symmetry holds and the crystal is noncentrosymmetric
    Stated as the setting in which the nonreciprocal current appears
  • domain assumption Interband processes dominate the dissipative response when gap and relaxation strength are comparable
    Required for the mechanism to be suitable in minigap systems

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 1 internal anchor

  1. [1]

    Nonreciprocal responses from non- centrosymmetric quantum materials,

    Y . Tokura and N. Nagaosa, “Nonreciprocal responses from non- centrosymmetric quantum materials,” Nature Communications 9, 3740 (2018)

    work page 2018

  2. [2]

    Nonreciprocal Transport and Opti- cal Phenomena in Quantum Materials,

    N. Nagaosa and Y . Yanase, “Nonreciprocal Transport and Opti- cal Phenomena in Quantum Materials,” Annual Review of Con- densed Matter Physics15, 63 (2024)

    work page 2024

  3. [3]

    Electrical Mag- netochiral Anisotropy,

    G. L. J. A. Rikken, J. F ¨olling, and P. Wyder, “Electrical Mag- netochiral Anisotropy,” Phys. Rev. Lett.87, 236602 (2001)

    work page 2001

  4. [4]

    Chiral Anomaly and Giant Mag- netochiral Anisotropy in Noncentrosymmetric Weyl Semimet- als,

    T. Morimoto and N. Nagaosa, “Chiral Anomaly and Giant Mag- netochiral Anisotropy in Noncentrosymmetric Weyl Semimet- als,” Phys. Rev. Lett.117, 146603 (2016)

    work page 2016

  5. [5]

    Nonreciprocal Current in Non- centrosymmetric Rashba Superconductors,

    R. Wakatsuki and N. Nagaosa, “Nonreciprocal Current in Non- centrosymmetric Rashba Superconductors,” Phys. Rev. Lett. 121, 026601 (2018)

    work page 2018

  6. [6]

    Nonreciprocal charge transport in two-dimensional noncen- trosymmetric superconductors,

    S. Hoshino, R. Wakatsuki, K. Hamamoto, and N. Nagaosa, “Nonreciprocal charge transport in two-dimensional noncen- trosymmetric superconductors,” Phys. Rev. B98, 054510 (2018)

    work page 2018

  7. [7]

    Nonlinear electric transport in odd-parity magnetic multipole systems: Application to Mn- based compounds,

    H. Watanabe and Y . Yanase, “Nonlinear electric transport in odd-parity magnetic multipole systems: Application to Mn- based compounds,” Phys. Rev. Res.2, 043081 (2020)

    work page 2020

  8. [8]

    Chiral Photocurrent in Parity- Violating Magnet and Enhanced Response in Topological An- tiferromagnet,

    H. Watanabe and Y . Yanase, “Chiral Photocurrent in Parity- Violating Magnet and Enhanced Response in Topological An- tiferromagnet,” Phys. Rev. X11, 011001 (2021)

    work page 2021

  9. [9]

    Intrinsic nonlinear conductivities induced by the quantum met- ric,

    K. Das, S. Lahiri, R. B. Atencia, D. Culcer, and A. Agarwal, “Intrinsic nonlinear conductivities induced by the quantum met- ric,” Phys. Rev. B108, L201405 (2023)

    work page 2023

  10. [10]

    Unification of Nonlinear Anomalous Hall Effect and Nonreciprocal Magnetoresistance in Metals by the Quantum Geometry,

    D. Kaplan, T. Holder, and B. Yan, “Unification of Nonlinear Anomalous Hall Effect and Nonreciprocal Magnetoresistance in Metals by the Quantum Geometry,” Phys. Rev. Lett.132, 026301 (2024)

    work page 2024

  11. [11]

    Bulk rectifi- cation effect in a polar semiconductor,

    T. Ideue, K. Hamamoto, S. Koshikawa, M. Ezawa, S. Shimizu, Y . Kaneko, Y . Tokura, N. Nagaosa, and Y . Iwasa, “Bulk rectifi- cation effect in a polar semiconductor,” Nature Physics13, 578 (2017)

    work page 2017

  12. [12]

    Elec- trical magnetochiral effect induced by chiral spin fluctuations,

    T. Yokouchi, N. Kanazawa, A. Kikkawa, D. Morikawa, K. Shi- bata, T. Arima, Y . Taguchi, F. Kagawa, and Y . Tokura, “Elec- trical magnetochiral effect induced by chiral spin fluctuations,” Nature Communications8, 866 (2017)

    work page 2017

  13. [13]

    Nonreciprocal charge transport in noncentrosymmetric superconductors,

    R. Wakatsuki, Y . Saito, S. Hoshino, Y . M. Itahashi, T. Ideue, M. Ezawa, Y . Iwasa, and N. Nagaosa, “Nonreciprocal charge transport in noncentrosymmetric superconductors,” Science Advances3, e1602390 (2017)

    work page 2017

  14. [14]

    Nonreciprocal charge transport at topological insula- tor/superconductor interface,

    K. Yasuda, H. Yasuda, T. Liang, R. Yoshimi, A. Tsukazaki, 5 K. S. Takahashi, N. Nagaosa, M. Kawasaki, and Y . Tokura, “Nonreciprocal charge transport at topological insula- tor/superconductor interface,” Nature Communications10, 2734 (2019)

    work page 2019

  15. [15]

    Nonreciprocal transport in gate-induced polar superconductor SrTiO 3,

    Y . M. Itahashi, T. Ideue, Y . Saito, S. Shimizu, T. Ouchi, T. No- jima, and Y . Iwasa, “Nonreciprocal transport in gate-induced polar superconductor SrTiO 3,” Science Advances6, eaay9120 (2020)

    work page 2020

  16. [16]

    Non- reciprocal superconducting NbSe2 antenna,

    E. Zhang, X. Xu, Y .-C. Zou, L. Ai, X. Dong, C. Huang, P. Leng, S. Liu, Y . Zhang, Z. Jia, X. Peng, M. Zhao, Y . Yang, Z. Li, H. Guo, S. J. Haigh, N. Nagaosa, J. Shen, and F. Xiu, “Non- reciprocal superconducting NbSe2 antenna,” Nature Communi- cations11, 5634 (2020)

    work page 2020

  17. [17]

    Magnetic proximity and nonreciprocal current switching in a monolayer WTe2 helical edge,

    W. Zhao, Z. Fei, T. Song, H. K. Choi, T. Palomaki, B. Sun, P. Malinowski, M. A. McGuire, J.-H. Chu, X. Xu, and D. H. Cobden, “Magnetic proximity and nonreciprocal current switching in a monolayer WTe2 helical edge,” Nature Materials 19, 503 (2020)

    work page 2020

  18. [18]

    Nonreciprocal transport in a room-temperature chiral magnet,

    D. Nakamura, M.-K. Lee, K. Karube, M. Mochizuki, N. Na- gaosa, Y . Tokura, and Y . Taguchi, “Nonreciprocal transport in a room-temperature chiral magnet,” Science Advances11, eadw8023 (2025)

    work page 2025

  19. [19]

    Nonreciprocal charge transport up to room temperature in bulk Rashba semiconductorα-GeTe,

    Y . Li, Y . Li, P. Li, B. Fang, X. Yang, Y . Wen, D.-x. Zheng, C.- h. Zhang, X. He, A. Manchon, Z.-H. Cheng, and X.-x. Zhang, “Nonreciprocal charge transport up to room temperature in bulk Rashba semiconductorα-GeTe,” Nature Communications12, 540 (2021)

    work page 2021

  20. [20]

    Gigan- tic Magnetochiral Anisotropy in the Topological Semimetal ZrTe5,

    Y . Wang, H. F. Legg, T. B ¨omerich, J. Park, S. Biesenkamp, A. A. Taskin, M. Braden, A. Rosch, and Y . Ando, “Gigan- tic Magnetochiral Anisotropy in the Topological Semimetal ZrTe5,” Phys. Rev. Lett.128, 176602 (2022)

    work page 2022

  21. [21]

    Gate-tunable giant superconducting nonrecip- rocal transport in few-layerT d−MoTe2,

    T. Wakamura, M. Hashisaka, S. Hoshino, M. Bard, S. Okazaki, T. Sasagawa, T. Taniguchi, K. Watanabe, K. Muraki, and N. Kumada, “Gate-tunable giant superconducting nonrecip- rocal transport in few-layerT d−MoTe2,” Phys. Rev. Res.6, 013132 (2024)

    work page 2024

  22. [22]

    Nonreciprocal current from electron interactions in noncentrosymmetric crystals: roles of time reversal symmetry and dissipation,

    T. Morimoto and N. Nagaosa, “Nonreciprocal current from electron interactions in noncentrosymmetric crystals: roles of time reversal symmetry and dissipation,” Scientific Reports8, 2973 (2018)

    work page 2018

  23. [23]

    High-frequency rectification via chiral Bloch electrons,

    H. Isobe, S.-Y . Xu, and L. Fu, “High-frequency rectification via chiral Bloch electrons,” Science Advances6, eaay2497 (2020)

    work page 2020

  24. [24]

    Second-order optical response in semiconductors,

    J. E. Sipe and A. I. Shkrebtii, “Second-order optical response in semiconductors,” Phys. Rev. B61, 5337 (2000)

    work page 2000

  25. [25]

    Quantum Geometric Origin of the Intrinsic Nonlinear Hall Effect

    Y . Ulrich, J. Mitscherling, L. Classen, and A. P. Schnyder, “Quantum Geometric Origin of the Intrinsic Nonlinear Hall Ef- fect,” arXiv:2506.17386 [cond-mat.mes-hall]

    work page internal anchor Pith review Pith/arXiv arXiv

  26. [26]

    Nonrecipro- cal Landau–Zener tunneling,

    S. Kitamura, N. Nagaosa, and T. Morimoto, “Nonrecipro- cal Landau–Zener tunneling,” Communications Physics3, 63 (2020)

    work page 2020

  27. [27]

    Current response of nonequilibrium steady states in the Landau-Zener problem: Nonequilibrium Green’s function approach,

    S. Kitamura, N. Nagaosa, and T. Morimoto, “Current response of nonequilibrium steady states in the Landau-Zener problem: Nonequilibrium Green’s function approach,” Phys. Rev. B102, 245141 (2020)

    work page 2020

  28. [28]

    Gaps induced by inver- sion symmetry breaking and second-generation Dirac cones in graphene/hexagonal boron nitride,

    E. Wang, X. Lu, S. Ding, W. Yao, M. Yan, G. Wan, K. Deng, S. Wang, G. Chen, L. Ma, J. Jung, A. V . Fedorov, Y . Zhang, G. Zhang, and S. Zhou, “Gaps induced by inver- sion symmetry breaking and second-generation Dirac cones in graphene/hexagonal boron nitride,” Nature Physics12, 1111 (2016)

    work page 2016

  29. [29]

    Quantum scattering time and its implications on scattering sources in graphene,

    X. Hong, K. Zou, and J. Zhu, “Quantum scattering time and its implications on scattering sources in graphene,” Phys. Rev. B80, 241415 (2009)

    work page 2009

  30. [30]

    Visualizing electron localization of WS2/WSe2 moir´e; super- lattices in momentum space,

    C. H. Stansbury, M. I. B. Utama, C. G. Fatuzzo, E. C. Re- gan, D. Wang, Z. Xiang, M. Ding, K. Watanabe, T. Taniguchi, M. Blei, Y . Shen, S. Lorcy, A. Bostwick, C. Jozwiak, R. Koch, S. Tongay, J. Avila, E. Rotenberg, F. Wang, and A. Lanzara, “Visualizing electron localization of WS2/WSe2 moir´e; super- lattices in momentum space,” Science Advances7, eabf43...

    work page 2021

  31. [31]

    Measurements of Correlated Insulator Gaps in a Transition-Metal Dichalcogenide Moir ´e Superlattice,

    X. Huang, Q. Wu, D. Chen, Z. Lian, M. Rashetnia, M. Blei, T. Taniguchi, K. Watanabe, S. A. Tongay, S.-F. Shi, and Y .-T. Cui, “Measurements of Correlated Insulator Gaps in a Transition-Metal Dichalcogenide Moir ´e Superlattice,” Nano Letters25, 9214 (2025)

    work page 2025

  32. [32]

    Gap Opening in Twisted Double Bilayer Graphene by Crystal Fields,

    P. Rickhaus, G. Zheng, J. L. Lado, Y . Lee, A. Kurzmann, M. Eich, R. Pisoni, C. Tong, R. Garreis, C. Gold, M. Masseroni, T. Taniguchi, K. Wantanabe, T. Ihn, and K. Ensslin, “Gap Opening in Twisted Double Bilayer Graphene by Crystal Fields,” Nano Letters19, 8821 (2019)

    work page 2019

  33. [33]

    Spon- taneous time-reversal symmetry breaking in twisted double bi- layer graphene,

    M. Kuiri, C. Coleman, Z. Gao, A. Vishnuradhan, K. Watanabe, T. Taniguchi, J. Zhu, A. H. MacDonald, and J. Folk, “Spon- taneous time-reversal symmetry breaking in twisted double bi- layer graphene,” Nature Communications13, 6468 (2022)

    work page 2022

  34. [34]

    Discovery of a Weyl fermion semimetal and topological Fermi arcs,

    S.-Y . Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang, A. Ban- sil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science349, 613 (2015)

    work page 2015

  35. [35]

    Experimental Discovery of Weyl Semimetal TaAs,

    B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao, J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, Z. Fang, X. Dai, T. Qian, and H. Ding, “Experimental Discovery of Weyl Semimetal TaAs,” Phys. Rev. X5, 031013 (2015)

    work page 2015

  36. [36]

    A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class,

    S.-M. Huang, S.-Y . Xu, I. Belopolski, C.-C. Lee, G. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane, C. Zhang, S. Jia, A. Bansil, H. Lin, and M. Z. Hasan, “A Weyl Fermion semimetal with surface Fermi arcs in the transition metal monopnictide TaAs class,” Nature Communications6, 7373 (2015)

    work page 2015

  37. [37]

    Observation of Weyl nodes in TaAs,

    B. Q. Lv, N. Xu, H. M. Weng, J. Z. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen, C. E. Matt, F. Bisti, V . N. Stro- cov, J. Mesot, Z. Fang, X. Dai, T. Qian, M. Shi, and H. Ding, “Observation of Weyl nodes in TaAs,” Nature Physics11, 724 (2015)

    work page 2015

  38. [38]

    Experimental discovery of a topological Weyl semimetal state in TaP,

    S.-Y . Xu, I. Belopolski, D. S. Sanchez, C. Zhang, G. Chang, C. Guo, G. Bian, Z. Yuan, H. Lu, T.-R. Chang, P. P. Shibayev, M. L. Prokopovych, N. Alidoust, H. Zheng, C.-C. Lee, S.-M. Huang, R. Sankar, F. Chou, C.-H. Hsu, H.-T. Jeng, A. Bansil, T. Neupert, V . N. Strocov, H. Lin, S. Jia, and M. Z. Hasan, “Experimental discovery of a topological Weyl semimet...

    work page 2015

  39. [39]

    Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide,

    S.-Y . Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian, T.-R. Chang, H. Zheng, V . N. Strocov, D. S. Sanchez, G. Chang, C. Zhang, D. Mou, Y . Wu, L. Huang, C.-C. Lee, S.-M. Huang, B. Wang, A. Bansil, H.-T. Jeng, T. Neupert, A. Kaminski, H. Lin, S. Jia, and M. Zahid Hasan, “Discovery of a Weyl fermion state with Fermi arcs in niobium arsenide,” Nature Phy...

    work page 2015

  40. [40]

    Direct observation of nonequivalent Fermi-arc states of opposite surfaces in the noncentrosymmet- ric Weyl semimetal NbP,

    S. Souma, Z. Wang, H. Kotaka, T. Sato, K. Nakayama, Y . Tanaka, H. Kimizuka, T. Takahashi, K. Yamauchi, T. Oguchi, K. Segawa, and Y . Ando, “Direct observation of nonequivalent Fermi-arc states of opposite surfaces in the noncentrosymmet- ric Weyl semimetal NbP,” Phys. Rev. B93, 161112 (2016)

    work page 2016

  41. [41]

    Probing the Shape of the Weyl Fermi Surface of NbP Using Transverse Electron Focusing,

    F. Balduini, L. Rocchino, A. Molinari, T. Paul, G. Mariani, V . Hasse, C. Felser, C. Zota, H. Schmid, and B. Gotsmann, “Probing the Shape of the Weyl Fermi Surface of NbP Using Transverse Electron Focusing,” Phys. Rev. Lett.133, 096601 6 (2024)

    work page 2024

  42. [42]

    Experimental re- alization of type-II Weyl state in noncentrosymmetric TaIrTe4,

    E. Haubold, K. Koepernik, D. Efremov, S. Khim, A. Fedorov, Y . Kushnirenko, J. van den Brink, S. Wurmehl, B. B ¨uchner, T. K. Kim, M. Hoesch, K. Sumida, K. Taguchi, T. Yoshikawa, A. Kimura, T. Okuda, and S. V . Borisenko, “Experimental re- alization of type-II Weyl state in noncentrosymmetric TaIrTe4,” Phys. Rev. B95, 241108 (2017)

    work page 2017

  43. [43]

    Signatures of a time-reversal symmetric Weyl semimetal with only four Weyl points,

    I. Belopolski, P. Yu, D. S. Sanchez, Y . Ishida, T.-R. Chang, S. S. Zhang, S.-Y . Xu, H. Zheng, G. Chang, G. Bian, H.-T. Jeng, T. Kondo, H. Lin, Z. Liu, S. Shin, and M. Z. Hasan, “Signatures of a time-reversal symmetric Weyl semimetal with only four Weyl points,” Nature Communications8, 942 (2017)

    work page 2017

  44. [44]

    Discovery of Lorentz-violating type II Weyl fermions in LaAlGe,

    S.-Y . Xu, N. Alidoust, G. Chang, H. Lu, B. Singh, I. Belopolski, D. S. Sanchez, X. Zhang, G. Bian, H. Zheng, M.-A. Husanu, Y . Bian, S.-M. Huang, C.-H. Hsu, T.-R. Chang, H.-T. Jeng, A. Bansil, T. Neupert, V . N. Strocov, H. Lin, S. Jia, and M. Z. Hasan, “Discovery of Lorentz-violating type II Weyl fermions in LaAlGe,” Science Advances3, e1603266 (2017)

    work page 2017

  45. [45]

    Spectroscopic evidence for a type II Weyl semimetallic state in MoTe2,

    L. Huang, T. M. McCormick, M. Ochi, Z. Zhao, M.-T. Suzuki, R. Arita, Y . Wu, D. Mou, H. Cao, J. Yan, N. Trivedi, and A. Kaminski, “Spectroscopic evidence for a type II Weyl semimetallic state in MoTe2,” Nature Materials15, 1155 (2016)

    work page 2016

  46. [46]

    Signature of type-II Weyl semimetal phase in MoTe2,

    J. Jiang, Z. K. Liu, Y . Sun, H. F. Yang, C. R. Rajamathi, Y . P. Qi, L. X. Yang, C. Chen, H. Peng, C.-C. Hwang, S. Z. Sun, S.-K. Mo, I. V obornik, J. Fujii, S. S. P. Parkin, C. Felser, B. H. Yan, and Y . L. Chen, “Signature of type-II Weyl semimetal phase in MoTe2,” Nature Communications8, 13973 (2017)

    work page 2017

  47. [47]

    Evidence for topo- logical type-II Weyl semimetal WTe2,

    P. Li, Y . Wen, X. He, Q. Zhang, C. Xia, Z.-M. Yu, S. A. Yang, Z. Zhu, H. N. Alshareef, and X.-X. Zhang, “Evidence for topo- logical type-II Weyl semimetal WTe2,” Nature Communica- tions8, 2150 (2017)

    work page 2017

  48. [48]

    Observation of Fermi arcs in the type-II Weyl semimetal candidate WTe2,

    Y . Wu, D. Mou, N. H. Jo, K. Sun, L. Huang, S. L. Bud’ko, P. C. Canfield, and A. Kaminski, “Observation of Fermi arcs in the type-II Weyl semimetal candidate WTe2,” Phys. Rev. B94, 121113 (2016)

    work page 2016

  49. [49]

    Visu- alizing Type-II Weyl Points in Tungsten Ditelluride by Quasi- particle Interference,

    C.-L. Lin, R. Arafune, R.-Y . Liu, M. Yoshimura, B. Feng, K. Kawahara, Z. Ni, E. Minamitani, S. Watanabe, Y . Shi, M. Kawai, T.-C. Chiang, I. Matsuda, and N. Takagi, “Visu- alizing Type-II Weyl Points in Tungsten Ditelluride by Quasi- particle Interference,” ACS Nano11, 11459 (2017)

    work page 2017

  50. [50]

    Type-II Weyl semimetals,

    A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer, X. Dai, and B. A. Bernevig, “Type-II Weyl semimetals,” Nature 527, 495 (2015)

    work page 2015

  51. [51]

    Rieman- nian geometry of resonant optical responses,

    J. Ahn, G.-Y . Guo, N. Nagaosa, and A. Vishwanath, “Rieman- nian geometry of resonant optical responses,” Nature Physics 18, 290 (2022)

    work page 2022

  52. [52]

    Gauge- invariant projector calculus for quantum state geometry and ap- plications to observables in crystals,

    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). 7 Appendix A: Derivation of the DC response In this section, we show that the nonreciprocal current is given by Eq. (3) and show that there is no divergence in the s...

    work page 2025

  53. [53]

    Derivation of the DC response from the AC response First, we show that the nonreciprocal current is given by Eq. (3). UsingA α(ω1)=A α2πδ(ω1 −ω)+A ∗ α2πδ(ω1 +ω), we can rewrite the current density (Eq. (2)) as jµ(ω)≡ Z ∞ −∞ dt jµ(t)eiω′t =j DC µ (ω)2πδ(ω′)+j SHG+ µ (ω)2πδ(ω′ −2ω) +j SHG− µ (ω)2πδ(ω′ +2ω) jDC µ (ω)= X α,β K µαβ(ω,−ω)A α(ω)Aβ(−ω) + X α,β K ...

    work page

  54. [54]

    We will show that there is no divergence inK µαβ(ω1, ω2) in the static limitω 1, ω2 →0

    No divergence in the static limit The response functionK µαβ(ω1, ω2) at the Matsubara fre- quency is given as, K µαβ(iω1,iω 2) =− 1 2 e ℏ 3 iω′,a α,iω 1 β,iω 2 µ,iω 1 +iω 2 − 1 2 e ℏ 3 iω′ +iω 1 +iω 2,a iω′,b α,iω 1 β,iω 2 µ,iω 1 +iω 2 − 1 2 e ℏ 3 iω′ +iω 2,a iω′,b β,iω 2 µ,iω 1 +iω 2 α,iω 1 − 1 2 e ℏ 3 iω′ +iω 1,a iω′,b α,iω 1 µ,iω 1 +iω 2 β,iω 2 − 1 2 e...

    work page

  55. [55]

    d2 dx2 f ′ +(x) (x−ξ b +iΓ) 3 # x=ξb+iΓ −1 2

    Casea=b,c The coefficient ofHµβ ba Hα ab in the sixth line of Eq. (C2) is 2iΓ3 2πi Z dx f ′ +(x)+f ′ −(x) (x−ξ b +iΓ) 3(x−ξ b −iΓ) 3 =2iΓ3  1 2 " d2 dx2 f ′ +(x) (x−ξ b +iΓ) 3 # x=ξb+iΓ −1 2 " d2 dx2 f ′ −(x) (x−ξ b −iΓ) 3 # x=ξb−iΓ  =− 1 8(f ′′′ + (ξb +iΓ)+f ′′′ − (ξb −iΓ)) − 3i 8Γ(f ′′ + (ξb +iΓ)−f ′′ − (ξb −iΓ)) + 3 8Γ2 (f ′ +(ξb +iΓ)+f ′...

    work page

  56. [56]

    (C2) does not contribute to the terms of orderO(Γ −1) in this case

    Casea,b=c The sixth line of Eq. (C2) does not contribute to the terms of orderO(Γ −1) in this case. The coefficient ofHµ bcHβ caHα ab in the fifth line of Eq. (C2) is − Γ2 2πi Z dx f ′(x)[GR2 b GR2 a GA a GA b −G R2 b GA2 b GA a GR a ] (C5) To obtain a contribution of orderO(Γ −1), the expression in parentheses must contain a contribution of orderΓ −3, wh...

    work page

  57. [57]

    (C2) withc→band the seventh line withc→a, we find that they are related by interchangingaandband then taking the complex con- jugate

    Casea=c,b Comparing the fifth line of Eq. (C2) withc→band the seventh line withc→a, we find that they are related by interchangingaandband then taking the complex con- jugate. Therefore, the coefficient ofH µ bcHβ caHα ab in the fifth line of Eq. (C2) is− i 8Γ f ′(ξa) ξ2 ba +O(Γ 0), while the coefficient of Hµ caHα abHβ bc in the seventh line is i 4Γ f ′(...

    work page

  58. [58]

    Casea=b=c The coefficient ofHµ bcHβ caHα ab in the fifth line of Eq. (C2) is 2iΓ3 2πi Z dx f ′ +(x)+f ′ −(x) (x−ξ a +iΓ) 4(x−ξ a −iΓ) 3 =2iΓ3 1 2 d2 dx2 f ′ +(x) (x−ξ a +iΓ) 4 x=ξa+iΓ − 1 6 d3 dx3 f ′ −(x) (x−ξ a −iΓ) 3 x=ξa−iΓ ! +O(Γ 0) =− 5i 16Γ3 f ′(ξa)+ 1 16Γ2 f ′′(ξa)+ i 32Γ f ′′′(ξa)+O(Γ 0) (C9) 13 The coefficient ofHµ caHα abHβ bc in the seventh li...

    work page

  59. [59]

    Drude term Collecting theO(Γ −2) contributions, we obtain ¯Σµβα Drude = 1 Γ2 X b 3 8 Hµβ bb Hα bb f ′ b + 1 8 Hµ bbHβ bbHα bb f ′′ b + X a,b 3 8 Hµ baHβ abHα bb +H µ abHα bbHβ ba ξba f ′ b  = 1 Γ2 X b 3 8 ∂kµ Hβ bbHα bb f ′ b + 1 8 Hµ bbHβ bbHα bb f ′′ b ! =− 1 Γ2 1 8 X b ∂kµ ∂kβ ∂kαϵb fb (C11) In going to the last line, we used∂ kµ∂kα fb =∂ kµ Hα...

    work page

  60. [60]

    BCD term Collecting theO(Γ −1) contributions, we obtain 1 Γ X b 3 8 i X a,b −H µ baHβ abHα bb +H µ abHα bbHβ ba ξ2 ba f ′ b + 1 8 i X a,b −2H µ bbHβ baHα ab +H µ baHα abHβ bb ξ2 ba f ′ b + 1 8 i X a,b −Hµ baHβ aaHα ab +2H µ aaHα abHβ ba ξ2 ba f ′ a ! = 1 Γ X b 1 8(3Ωµβ b ∂kα f ′ b +2Ω βα b ∂kµ f ′ b + Ωαµ b ∂kβ f ′ b) (C12) where Ωαβ b = X a,b Hα baHβ ab ...

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