pith. sign in

arxiv: 2604.14146 · v2 · submitted 2026-04-15 · ❄️ cond-mat.str-el · cond-mat.mes-hall

Topological anisotropic non-Fermi liquid from a Berry-dipole semimetal

Konstantinos Ladovrechis This is my paper

Pith reviewed 2026-05-10 11:52 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hall
keywords Berry-dipole semimetalnon-Fermi liquidtopological quantum critical pointCoulomb interactionsrenormalization groupanisotropic scalingHopf insulator
0
0 comments X

The pith

Coulomb interactions turn a Berry-dipole semimetal into a spatially anisotropic non-Fermi liquid with an enhanced topological moment.

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

The paper examines the fate of a three-dimensional Berry-dipole semimetal that sits exactly at the topological transition separating a Hopf insulator from a trivial insulator when long-range Coulomb interactions are added. A controlled renormalization-group calculation that combines the large-N_f limit with an epsilon expansion shows that the non-interacting critical point is unstable and flows to a new infrared fixed point. At this fixed point the dispersion becomes anisotropic while the Berry-dipole moment that measures the topology grows larger than its bare value. The resulting state is a non-Fermi liquid whose thermodynamic and transport quantities obey distinctive power-law scalings in energy and temperature that encode both the anisotropy and the strengthened topology.

Core claim

Starting from a Berry-dipole semimetal at the topological quantum critical point between a Hopf insulator and a trivial insulator, long-range Coulomb interactions drive the system, within a large-N_f and epsilon-expansion renormalization-group analysis, to a stable infrared fixed point that describes a spatially anisotropic non-Fermi liquid in which the Berry-dipole moment is enhanced relative to the non-interacting case.

What carries the argument

The renormalization-group flow of the interacting Berry-dipole semimetal, which stabilizes an anisotropic non-Fermi liquid fixed point with an increased Berry-dipole moment and yields explicit scaling relations for observables.

If this is right

  • Specific heat and density of states acquire anomalous power-law dependences on temperature and energy set by the anisotropic dispersion and the fixed-point value of the Berry dipole.
  • Electrical conductivity and other transport coefficients become direction-dependent, with distinct scaling along different crystal axes.
  • Response functions that probe topology, such as orbital magnetization, are expected to show stronger signals because the Berry-dipole moment is enlarged.
  • The scaling relations provide a direct experimental test for the enhanced topological character through energy- and temperature-dependent spectroscopy or transport.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same mechanism may apply to other topological semimetals whose low-energy structure is governed by higher-order multipole moments rather than simple Berry curvature.
  • Real-material searches could target compounds near Hopf-insulator transitions where long-range interactions dominate, checking for the predicted anisotropy in thermodynamic quantities.
  • Adding short-range interactions or weak disorder on top of the Coulomb term would test how robust the anisotropic fixed point remains.

Load-bearing premise

The large-N_f and epsilon-expansion approximations are assumed to give a reliable description of the infrared fixed point and its scaling even for the small numbers of fermion flavors that occur in real materials.

What would settle it

A measurement that finds isotropic scaling exponents in specific heat or conductivity, or that extracts an unenhanced Berry-dipole moment from response functions, would rule out the predicted anisotropic non-Fermi liquid fixed point.

Figures

Figures reproduced from arXiv: 2604.14146 by Konstantinos Ladovrechis.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic phase diagram of the non-interacting [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Phase portrait at [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. The expression [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The expression [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The expression [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Evolution of the ratios of fermionic anisotropy parameters [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Evolution of the RG parameters [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
read the original abstract

The interplay among topology and interactions has been a promising path towards identifying novel phases of condensed matter beyond these predicted by the established classification paradigms. In the present work, we propose such a novel phase of matter by studying the fate of a three-dimensional Berry-dipole semimetal, lying at the topological quantum critical point separating a Hopf insulator from a trivial insulator, in the presence of long-range Coulomb interactions. Utilizing large-$N_f$ analysis and an $\epsilon$-expansion within the renormalization-group scheme, we predict the emergence of a spatially anisotropic non-Fermi liquid with enhanced Berry-dipole moment. The corresponding scaling relations of certain physical observables are derived as functions of the probed energy and temperature scale, and we conclude by providing an observational test for probing the enhanced topological features of the anisotropic non-Fermi liquid.

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

1 major / 2 minor

Summary. The manuscript examines the renormalization-group fate of a three-dimensional Berry-dipole semimetal at the topological quantum critical point separating a Hopf insulator from a trivial insulator when long-range Coulomb interactions are included. Employing a simultaneous large-N_f expansion and an ε-expansion, the authors identify an infrared fixed point that generates spatial anisotropy, non-Fermi-liquid scaling, and an enhanced Berry-dipole moment; they derive the corresponding energy- and temperature-dependent scaling forms for observables and propose an observational signature of the enhanced topology.

Significance. If the reported fixed point is stable, the work would establish a concrete example of a topologically nontrivial non-Fermi liquid whose anisotropy and Berry-dipole enhancement are controlled by the interplay of Coulomb interactions and the underlying band topology. The explicit scaling relations and suggested experimental test constitute concrete, falsifiable predictions that could guide searches in candidate materials near Hopf-insulator transitions.

major comments (1)
  1. [Renormalization-group analysis (flow equations for the anisotropy and Berry-dipole couplings)] The central claim that an anisotropic NFL with enhanced Berry-dipole moment emerges rests on the leading-order large-N_f and ε-expansion results for the beta functions. For the physically relevant values N_f = 1 or 2 the expansions are uncontrolled; the manuscript does not provide either higher-order terms, a resummation analysis, or a numerical check of fixed-point stability at finite N_f. If the fixed-point value of the anisotropy parameter or the sign of the Berry-dipole renormalization changes at O(1/N_f^2) or at ε = 1, the predicted phase and its scaling relations cease to exist.
minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction refer to 'large-N_f analysis' without stating the range of N_f for which the expansion is expected to be reliable or the value adopted for the physical case.
  2. [Model and RG setup] Notation for the ε-expansion parameter and the precise upper critical dimension assumed for the Coulomb interaction should be stated explicitly when the flow equations are first introduced.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their careful reading of our manuscript and for the positive assessment of its potential significance. We address the major comment below and have revised the manuscript to incorporate additional discussion of the limitations of our perturbative approach.

read point-by-point responses

  1. Referee: The central claim that an anisotropic NFL with enhanced Berry-dipole moment emerges rests on the leading-order large-N_f and ε-expansion results for the beta functions. For the physically relevant values N_f = 1 or 2 the expansions are uncontrolled; the manuscript does not provide either higher-order terms, a resummation analysis, or a numerical check of fixed-point stability at finite N_f. If the fixed-point value of the anisotropy parameter or the sign of the Berry-dipole renormalization changes at O(1/N_f^2) or at ε = 1, the predicted phase and its scaling relations cease to exist.

    Authors: We agree that the leading-order large-N_f and ε expansions are uncontrolled for small N_f=1 or 2. Our analysis is performed in the controlled limit of large N_f and small ε, where the beta functions yield a stable infrared fixed point with spatial anisotropy and an enhanced Berry-dipole moment. We have revised the manuscript by adding a dedicated paragraph in the Discussion section that explicitly states the perturbative nature of the results, cautions against direct extrapolation to N_f=1,2 without further checks, and outlines possible future directions such as higher-order calculations or numerical simulations. While these leading-order findings provide a consistent and falsifiable framework with concrete scaling predictions, we acknowledge that quantitative details could be modified by higher-order corrections. revision: partial

standing simulated objections not resolved
  • Explicit higher-order terms in the 1/N_f expansion, resummation analysis, or numerical verification of fixed-point stability at N_f=1 or 2, as these require substantial additional work beyond the present study.

Circularity Check

0 steps flagged

No circularity: RG derivation starts from microscopic Hamiltonian

full rationale

The paper performs a standard large-N_f plus ε-expansion RG analysis on the microscopic Hamiltonian of the 3D Berry-dipole semimetal plus long-range Coulomb interactions. The abstract and context indicate that the anisotropic NFL fixed point, enhanced Berry-dipole moment, and scaling relations are obtained as outputs of the flow equations rather than being inserted by definition or by a self-citation chain. No fitted parameters are relabeled as predictions, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via citation. The derivation is therefore self-contained against external benchmarks; any concerns about the reliability of the expansions for small N_f belong to correctness risk, not circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on perturbative RG approximations whose validity for the physical case is assumed rather than derived.

free parameters (2)
  • N_f
    Large number of fermion flavors introduced to control the perturbative expansion.
  • epsilon
    Small expansion parameter around the upper critical dimension in the RG scheme.
axioms (1)
  • domain assumption Large-N_f and ε-expansion reliably describe the infrared physics of the Coulomb-interacting Berry-dipole semimetal.
    Invoked in the abstract to obtain the fixed-point behavior and scaling relations.

pith-pipeline@v0.9.0 · 5434 in / 1206 out tokens · 38609 ms · 2026-05-10T11:52:18.039769+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages

  1. [1]

    Qualitatively, the results in Eqs

    +N f B(γ ∗ 21, γ∗ 31) +N f B(γ ∗ 31, γ∗ 21) i |Nf=1 ϵ=1 ≈0.5.(24) Therefore, the renormalized bosonic Hamiltonian is ex- pected to scale asL −1 indicating a strong renormaliza- tion of the bosonic spectrum. Qualitatively, the results in Eqs. (23)–(24) agree with the conclusions derived using the large-Nf method, thus reinforcing the argument for the break...

    work page

  2. [2]

    M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys.82, 3045 (2010)

    work page 2010

  3. [3]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and su- perconductors, Rev. Mod. Phys.83, 1057 (2011)

    work page 2011

  4. [5]

    Rachel, Interacting topological insulators: a review, Reports on Progress in Physics81, 116501 (2018)

    S. Rachel, Interacting topological insulators: a review, Reports on Progress in Physics81, 116501 (2018)

    work page 2018

  5. [6]

    A. M. Turner and A. Vishwanath, Beyond band insu- lators: Topology of semi-metals and interacting phases (2013), arXiv:1301.0330 [cond-mat.str-el]

    work page arXiv 2013

  6. [7]

    C.-K. Chiu, J. C. Teo, A. P. Schnyder, and S. Ryu, Clas- sification of topological quantum matter with symme- tries, Reviews of Modern Physics88, 10.1103/revmod- phys.88.035005 (2016)

    work page doi:10.1103/revmod- 2016

  7. [8]

    N. P. Armitage, E. J. Mele, and A. Vishwanath, Weyl and dirac semimetals in three-dimensional solids, Rev. Mod. Phys.90, 015001 (2018)

    work page 2018

  8. [9]

    Zhu and R.-X

    P. Zhu and R.-X. Zhang, Delicate topology of luttinger semimetal, Phys. Rev. B110, 165120 (2024)

    work page 2024

  9. [10]

    Savary, E.-G

    L. Savary, E.-G. Moon, and L. Balents, New type of quan- tum criticality in the pyrochlore iridates, Phys. Rev. X4, 041027 (2014)

    work page 2014

  10. [11]

    Boettcher and I

    I. Boettcher and I. F. Herbut, Anisotropy induces non- fermi-liquid behavior and nematic magnetic order in three-dimensional luttinger semimetals, Phys. Rev. B95, 075149 (2017)

    work page 2017

  11. [12]

    Lai, Correlation effects in double-weyl semimetals, Phys

    H.-H. Lai, Correlation effects in double-weyl semimetals, Phys. Rev. B91, 235131 (2015)

    work page 2015

  12. [13]

    Jian and H

    S.-K. Jian and H. Yao, Correlated double-weyl semimet- als with coulomb interactions: Possible applications to hgcr2se4 andsrsi 2, Phys. Rev. B92, 045121 (2015)

    work page 2015

  13. [14]

    G. Y. Cho and E.-G. Moon, Novel quantum criticality in two dimensional topological phase transitions, Scientific Reports6, 10.1038/srep19198 (2016)

    work page doi:10.1038/srep19198 2016

  14. [15]

    Yang, E.-G

    B.-J. Yang, E.-G. Moon, H. Isobe, and N. Nagaosa, Quantum criticality of topological phase transitions in three-dimensional interacting electronic systems, Nature Physics10, 774–778 (2014)

    work page 2014

  15. [16]

    S. Han, C. Lee, E.-G. Moon, and H. Min, Emergent anisotropic non-fermi liquid at a topological phase tran- sition in three dimensions, Phys. Rev. Lett.122, 187601 (2019)

    work page 2019

  16. [17]

    Wang, G.-Z

    J.-R. Wang, G.-Z. Liu, and C.-J. Zhang, Topological quantum critical point in a triple-weyl semimetal: Non- fermi-liquid behavior and instabilities, Phys. Rev. B99, 195119 (2019)

    work page 2019

  17. [18]

    J. E. Moore, Y. Ran, and X.-G. Wen, Topological surface states in three-dimensional magnetic insulators, Phys. Rev. Lett.101, 186805 (2008)

    work page 2008

  18. [19]

    Deng, S.-T

    D.-L. Deng, S.-T. Wang, C. Shen, and L.-M. Duan, Hopf insulators and their topologically protected surface states, Phys. Rev. B88, 201105 (2013)

    work page 2013

  19. [20]

    Nelson, T

    A. Nelson, T. Neupert, A. Alexandradinata, and T. c. v. Bzdušek, Delicate topology protected by rotation symme- try: Crystalline hopf insulators and beyond, Phys. Rev. B106, 075124 (2022)

    work page 2022

  20. [21]

    Zhuang, C

    Z.-Y. Zhuang, C. Zhang, X.-J. Wang, and Z. Yan, Berry- dipole semimetals, Phys. Rev. B110, L121122 (2024). 6

    work page 2024

  21. [22]

    Q. Mo, R. Zheng, C. Lu, X. Huang, Z. Liu, and S. Zhang, Observation of oriented landau levels and helical zero modes in berry dipole acoustic crystals, Phys. Rev. Lett. 134, 116604 (2025)

    work page 2025

  22. [23]

    X.-X. Yuan, L. He, S.-T. Wang, D.-L. Deng, F. Wang, W.-Q. Lian, X. Wang, C.-H. Zhang, H.-L. Zhang, X.-Y. Chang, and L.-M. Duan, Observation of topological links associated with hopf insulators in a solid-state quantum simulator, Chinese Physics Letters34, 060302 (2017)

    work page 2017

  23. [24]

    Wang, X.-T

    Z. Wang, X.-T. Zeng, Y. Biao, Z. Yan, and R. Yu, Real- ization of a hopf insulator in circuit systems, Phys. Rev. Lett.130, 057201 (2023)

    work page 2023

  24. [25]

    Herbut,A Modern Approach to Critical Phenomena (Cambridge University Press, 2007)

    I. Herbut,A Modern Approach to Critical Phenomena (Cambridge University Press, 2007)

    work page 2007

  25. [26]

    F.HerbutandL.Janssen,Topologicalmottinsulator in three-dimensional systems with quadratic band touching, Phys

    I. F.HerbutandL.Janssen,Topologicalmottinsulator in three-dimensional systems with quadratic band touching, Phys. Rev. Lett.113, 106401 (2014)

    work page 2014

  26. [28]

    A. C. Tyner and S. Sur, Dipolar weyl semimetals, Phys. Rev. B109, L081101 (2024)

    work page 2024

  27. [29]

    Vojta, Quantum phase transitions, Reports on Progress in Physics66, 2069–2110 (2003)

    M. Vojta, Quantum phase transitions, Reports on Progress in Physics66, 2069–2110 (2003)

    work page 2069

  28. [30]

    Ladovrechis and S

    K. Ladovrechis and S. Sur, Correlated hopf insulators, Phys. Rev. B112, 235148 (2025)

    work page 2025

  29. [31]

    Kopietz,Bosonization of Interacting Fermions in Ar- bitrary Dimensions(Springer Berlin Heidelberg, 1997)

    P. Kopietz,Bosonization of Interacting Fermions in Ar- bitrary Dimensions(Springer Berlin Heidelberg, 1997)

    work page 1997

  30. [32]

    R. A. Ferrell and D. J. Scalapino, Order-parameter cor- relations within the screening approximation, Phys. Rev. Lett.29, 413 (1972)

    work page 1972

  31. [33]

    A. A. Abrikosov, Calculation of critical indices for zero- gap semiconductors, J. Exp. Theor. Phys.66, 1443 (1974)

    work page 1974

  32. [34]

    Y. Wang, A. Abanov, B. L. Altshuler, E. A. Yuzbashyan, and A. V. Chubukov, Superconductivity near a quantum- critical point: The special role of the first matsubara fre- quency, Phys. Rev. Lett.117, 157001 (2016)

    work page 2016

  33. [35]

    H. Wang, Y. Wang, and G. Torroba, Superconductivity versus quantum criticality: Effects of thermal fluctua- tions, Phys. Rev. B97, 054502 (2018)

    work page 2018

  34. [36]

    F.Schrodi, A.Aperis,andP.M.Oppeneer,Increasedper- formance of matsubara space calculations: A case study within eliashberg theory, Phys. Rev. B99, 184508 (2019)

    work page 2019

  35. [37]

    Snygg,Clifford Algebra: A Computational Tool for Physicists(OxfordUniversityPressNewYork, NY,1997)

    J. Snygg,Clifford Algebra: A Computational Tool for Physicists(OxfordUniversityPressNewYork, NY,1997)

    work page 1997

  36. [38]

    P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. Mod. Phys.49, 435 (1977)

    work page 1977

  37. [39]

    J. Li, L. Wang, Y. Wang, Z. Tao, W. Zhong, Z. Su, S. Xue, G. Miao, W. Wang, H. Peng, J. Guo, and X. Zhu, Observation of the nonanalytic behavior of op- tical phonons in monolayer hexagonal boron nitride, Nature Communications15, 10.1038/s41467-024-46229- 4 (2024)

    work page doi:10.1038/s41467-024-46229- 2024

  38. [40]

    H. Zhou, J. Choi, S. Choi, R. Landig, A. M. Douglas, J. Isoya, F. Jelezko, S. Onoda, H. Sumiya, P. Cappel- laro, H. S. Knowles, H. Park, and M. D. Lukin, Quantum metrology with strongly interacting spin systems, Physi- cal Review X10, 10.1103/physrevx.10.031003 (2020)

    work page doi:10.1103/physrevx.10.031003 2020

  39. [41]

    Sahin, M

    H. Sahin, M. B. A. Jalil, and C. H. Lee, Topolectrical cir- cuits—recent experimental advances and developments, APL Electronic Devices1, 10.1063/5.0265293 (2025)

    work page doi:10.1063/5.0265293 2025

  40. [42]

    D. E. Sheehy and J. Schmalian, Quantum critical scaling in graphene, Phys. Rev. Lett.99, 226803 (2007)

    work page 2007

  41. [43]

    Isobe, B.-J

    H. Isobe, B.-J. Yang, A. Chubukov, J. Schmalian, and N. Nagaosa, Emergent non-fermi-liquid at the quantum critical point of a topological phase transition in two di- mensions, Phys. Rev. Lett.116, 076803 (2016)

    work page 2016

  42. [44]

    Zhuang and Z

    Z.-Y. Zhuang and Z. Yan, Extrinsic and intrinsic nonlin- ear hall effects across berry-dipole transitions, Phys. Rev. B107, L161102 (2023). Supplemental Material: Topological anisotropic non-Fermi liquid from a Berry-dipole semimetal Konstantinos Ladovrechis Theoretische Physik III, Ruhr-Universität Bochum, D-44801 Bochum, Germany Details of the large-N f...

    work page 2023

  43. [45]

    + Ω2 t4 1 i2 × × coth t1p |Ω| s b2p(β21β31) q2 p β2 31 +b 2z(β21β31)β2 31q2z ! q a2p(β21β31)q2p +a 2z(β21β31)β4 31q2z ,(28) where we have substituted the expressionβ=β 21β31 to make the dependence on the ratios of fermionic anisotropy parameters explicit. Through the variable transformationsX=q p/ p Ω/t2 1 andY=q z/ p Ω/t2 1, the above expression becomes ...

    work page

  44. [46]

    The summationTP iΩ Ω−1 exhibits a singularity at zero frequency

    + 1]2 × × coth s b2p(β21β31) X2 β2 31 +b 2z(β21β31)β2 31Y 2 ! q a2p(β21β31)X2 +a 2z(β21β31)β4 31Y 2 ,(29) where we used the fact that the integrand is invariant under the transformationY→ −Y. The summationTP iΩ Ω−1 exhibits a singularity at zero frequency. Since we are interested in the zero-temperature behavior of the fermionic self-energy, we transform ...

    work page

  45. [47]

    + 1]2 × × coth s b2p(β21β31) X2 β2 31 +b 2z(β21β31)β2 31Y 2 ! q a2p(β21β31)X2 +a 2z(β21β31)β4 31Y 2 .(33) t2 1-correction The correction to the fermionic anisotropy parametert2 1 is given by the expression δt2 1 = Tr σ1,2 ∂ 2t2 1∂kx,ykz Σ(k, iω) Tr(σ1,2) |k=0, ω=0 = 8πT X iΩ Z ∞ 0 qpdqp Z ∞ −∞ dqz 1 Nf t2 1 × × q8 pβ8 21 + 3q8 z β8 31 + 2q6 z q2 pβ4 31(−2...

    work page

  46. [48]

    + Ω2 t4 1 i3 × × coth t1p |Ω| s b2p(β21β31) q2 p β2 31 +b 2z(β21β31)β2 31q2z ! q a2p(β21β31)q2p +a 2z(β21β31)β4 31q2z .(34) Upon following the computation steps described in theω-correction case, we find δt2 1 =− lnΛ 0/µ Nf Σ1(β21, β31),(35) with Σ1(β21, β31) =− 4 Nf Z ∞ 0 XdX Z ∞ 0 dY coth s b2p(β21β31) X2 β2 31 +b 2z(β21β31)β2 31Y 2 ! q a2p(β21β31)X2 +a...

    work page

  47. [49]

    + 1]3 .(36) t2 2-correction Upon following the computation steps described in theω-correction case, the correction to the fermionic anisotropy parametert 2 2 reads δt2 2 = Tr σ3 ∂ t2 2∂kx,ykx,y Σ(k, iω) Tr(σ3) |k=0, ω=0 =− lnΛ 0/µ Nf Σ2(β21, β31),(37) with Σ2(β21, β31) = 8 Z ∞ 0 XdX Z ∞ 0 dY coth s b2p(β21β31) X2 β2 31 +b 2z(β21β31)β2 31Y 2 ! q a2p(β21β31...

    work page

  48. [50]

    + 2Y 6X2β2 31(−8 + 6β2 21β2 31 +β 4 21β4 31) +2(2Y 2 + 3X2β4 21)(−Xβ21 +Y β 31)(Xβ21 +Y β 31) +β 2 21 1 [X4β2 21 +Y 4β4 31 + 2X2Y 2(2−β 2 21β2

    work page

  49. [51]

    + 1]3 .(38) 8 t2 3-correction Upon following the computation steps described in theω-correction case, the correction to the fermionic anisotropy parametert 2 3 reads δt2 3 = Tr σ3 ∂ −t2 3∂kzkz Σ(k, iω) Tr(σ3) |k=0, ω=0 =− lnΛ 0/µ Nf Σ3(β21, β31),(39) with Σ3(β21, β31) = 8 Z ∞ 0 XdX Z ∞ 0 dY coth s b2p(β21β31) X2 β2 31 +b 2z(β21β31)β2 31Y 2 ! q a2p(β21β31)...

    work page

  50. [52]

    + 6Y 4X4β2 21β4 31(−6 +β 2 21 β2 31)−4Y 6X2β6 31(1 + 2β2 21β2

    work page

  51. [53]

    + 12Y 2X6β2 21(−4 + 3β2 21β2 31) +4(X4β2 21 −3Y 4β6 31 + 3Y 2X2β2 31(−1 +β 2 21β2 31)) +β 2 31 1 [X4β2 21 +Y 4β4 31 + 2X2Y 2(2−β 2 21β2

    work page

  52. [54]

    + 1]3 .(40) e-correction Finally, the correction to the electronic chargeereads [3] δe =e 2T X iΩ Z d3q (2π)3 Tr[G3(q, iΩ)G3(q, iΩ)] 2 D3,r(q, iΩ) = 2πT X iΩ Z ∞ 0 qpdqp Z ∞ −∞ dqz 1 Nf t2 1 q4 pβ4 21 +q 4 z β4 31 −2q 2 z q2 p(−2 +β 2 21β2 31)− Ω2 t4 1 h q4pβ4 21 +q 4z β4 31 + 2q2pq2z(2−β 2 21β2

    work page

  53. [55]

    + Ω2 t4 1 i2 × × coth t1p |Ω| s b2p(β21β31) q2 p β2 31 +b 2z(β21β31)β2 31q2z ! q a2p(β21β31)q2p +a 2z(β21β31)β4 31q2z =δ ω,(41) which is identical to the correction for the fermionic Matsubara frequency. The latter result is a proof of the Ward- Takahashi identity [4, 5] stating that the charge renormalization is equal to the frequency renormalization in ...

    work page

  54. [56]

    J. E. Moore, Y. Ran, and X.-G. Wen, Topological surface states in three-dimensional magnetic insulators, Phys. Rev. Lett. 101, 186805 (2008)

    work page 2008

  55. [57]

    Zhuang, C

    Z.-Y. Zhuang, C. Zhang, X.-J. Wang, and Z. Yan, Berry-dipole semimetals, Phys. Rev. B110, L121122 (2024)

    work page 2024

  56. [58]

    A. L. Fetter and J. D. Walecka, Quantum theory of many particle systems, Physics Today25, 54–55 (1972)

    work page 1972

  57. [59]

    J. C. Ward, An identity in quantum electrodynamics, Phys. Rev.78, 182 (1950)

    work page 1950

  58. [60]

    Takahashi, On the generalized ward identity, Il Nuovo Cimento6, 371–375 (1957)

    Y. Takahashi, On the generalized ward identity, Il Nuovo Cimento6, 371–375 (1957)

    work page 1957

  59. [61]

    G. D. Mahan,Many-Particle Physics(Springer US, 1990)

    work page 1990

  60. [62]

    Sachdev,Quantum Phase Transitions(Cambridge University Press, 2011)

    S. Sachdev,Quantum Phase Transitions(Cambridge University Press, 2011)

    work page 2011

  61. [63]

    H. v. Löhneysen, A. Rosch, M. Vojta, and P. Wölfle, Fermi-liquid instabilities at magnetic quantum phase transitions, Rev. Mod. Phys.79, 1015 (2007)

    work page 2007

  62. [64]

    Snygg,Clifford Algebra: A Computational Tool for Physicists(Oxford University PressNew York, NY, 1997)

    J. Snygg,Clifford Algebra: A Computational Tool for Physicists(Oxford University PressNew York, NY, 1997)

    work page 1997

  63. [65]

    A. A. Abrikosov, Calculation of critical indices for zero-gap semiconductors, J. Exp. Theor. Phys.66, 1443 (1974)

    work page 1974

  64. [66]

    Wang, G.-Z

    J.-R. Wang, G.-Z. Liu, and C.-J. Zhang, Topological quantum critical point in a triple-weyl semimetal: Non-fermi-liquid behavior and instabilities, Phys. Rev. B99, 195119 (2019). 20

    work page 2019

  65. [67]

    Fukuyama, A formula for the orbital magnetic susceptibility of bloch electrons in weak fields, Physics Letters A32, 111–112 (1970)

    H. Fukuyama, A formula for the orbital magnetic susceptibility of bloch electrons in weak fields, Physics Letters A32, 111–112 (1970)

    work page 1970

  66. [68]

    Goldenfeld,Lectures on Phase Transitions and the Renormalization Group(CRC Press, 2018)

    N. Goldenfeld,Lectures on Phase Transitions and the Renormalization Group(CRC Press, 2018)

    work page 2018

  67. [69]

    N. W. Ashcroft and N. D. Mermin,Solid State Physics(Holt-Saunders, 1976)

    work page 1976