pith. sign in

arxiv: 2510.15111 · v2 · submitted 2025-10-16 · ❄️ cond-mat.str-el · hep-th

When Wannier centers jump: Critical points between atomic insulating phases

Yunchao Zhang , T. Senthil This is my paper

Pith reviewed 2026-05-18 05:56 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-th
keywords atomic insulatorsWannier centersquantum phase transitionsQED3monopole suppressionlattice symmetriesdeconfined criticalitybosonic insulators
0
0 comments X

The pith

Transitions between obstructed atomic insulators can realize a stable QED3 critical point when lattice symmetries suppress monopoles.

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

The paper examines phase transitions between distinct featureless bosonic atomic insulators in two spatial dimensions plus time. These phases are separated because their Wannier centers sit at different locations relative to the lattice sites, even though both lack topological order. The key finding is that for particular lattices the transition point between them can be a conformally invariant critical state governed by quantum electrodynamics in 2+1 dimensions. This emerges because the way the lattice symmetries fit into the continuum description prevents monopoles from proliferating and destroying the gauge field. The authors verify this by constructing explicit lattice models where the QED3 fixed point appears from a local Hamiltonian.

Core claim

The critical point separating two atomic insulating phases with shifted Wannier centers hosts an emergent QED3 theory when the microscopic lattice symmetries are embedded into the continuum theory in a way that suppresses monopole proliferation, and this QED3 state is realizable in local lattice models for the lattices considered.

What carries the argument

The embedding of microscopic lattice symmetries into the continuum QED3 theory, which through anomaly matching prevents monopole proliferation and stabilizes the deconfined critical point.

If this is right

  • Atomic insulators on certain lattices can exhibit an emergent gauge theory at their transition without additional tuning.
  • The absence of topology does not preclude rich critical phenomena such as conformal invariance and emergent electrodynamics.
  • Lattice symmetry constraints can be used to protect gauge theories in condensed matter systems.
  • Explicit lattice Hamiltonians exist that realize the QED3 fixed point at the insulator-insulator transition.

Where Pith is reading between the lines

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

  • This opens the possibility that other transitions between trivial phases might host emergent gauge fields if symmetry embeddings are chosen appropriately.
  • Similar mechanisms could apply to higher dimensions or different symmetry classes, suggesting a broader class of deconfined critical points.
  • Numerical studies of specific models could directly measure the conformal dimensions predicted by QED3.

Load-bearing premise

The embedding of the lattice symmetries into the continuum theory must suppress monopole proliferation enough to reach the QED3 fixed point rather than a confined phase.

What would settle it

A numerical simulation or exact diagonalization of one of the proposed lattice models that fails to show the expected scaling dimensions or correlation functions of QED3 at the critical point between the two atomic insulators.

Figures

Figures reproduced from arXiv: 2510.15111 by T. Senthil, Yunchao Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1: Two possible atomic insulating phases on the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: An illustration of the two phases of the SSH [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The phase diagram of the SSH chain the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a) The vanishing of the energy gap, as the [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: An illustration of the BBH model. There are [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: An illustration of the decorated triangular [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Phase diagram and band dispersions for the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Illustration of the fermion gap [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Schematic phase diagram and dispersions at [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Schematic phase diagram and dispersions as [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: (a) Green’s functions and photon-fermion [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: (a) Resummed effective gauge propagator and [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
read the original abstract

We study a class of quantum phase transitions between featureless bosonic atomic insulators in $(2+1)$ dimensions, where each phase exhibits neither topological order nor protected edge modes. Despite their lack of topology, these insulators may be ``obstructed'' in the sense that their Wannier centers are not pinned to the physical atomic sites. These insulators represent distinct phases, as no symmetry-preserving adiabatic path connects them. Surprisingly, we find that for certain lattices, the critical point between these insulators can host a conformally invariant state described by quantum electrodynamics in $(2+1)$ dimensions (QED$_3$). The emergent electrodynamics at the critical point can be stabilized if the embedding of the microscopic lattice symmetries suppresses the proliferation of monopoles, suggesting that even transitions between trivial phases can harbor rich and unexpected physics. We analyze the mechanism behind this phenomenon, discuss its stability against perturbations, and explore the embedding of lattice symmetries into the continuum through anomaly matching. In all the models we analyze, we confirm that the QED$_3$ is indeed emergeable, in the sense that it is realizable from a local lattice Hamiltonian.

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

2 major / 1 minor

Summary. The paper examines quantum phase transitions between distinct atomic insulating phases in (2+1)D that lack topological order or edge modes but are distinguished by the positions of their Wannier centers. It finds that for certain lattices, the critical point can realize a conformally invariant QED3 state. The emergent QED3 is stabilized when lattice symmetries are embedded in a way that suppresses monopole proliferation, and the authors confirm via anomaly matching that QED3 emerges from local Hamiltonians in the models studied.

Significance. If the results hold, this work reveals that rich emergent physics, including conformal invariance and gauge theories, can arise at critical points between seemingly trivial insulating phases. This has implications for the classification of quantum phases and the realization of exotic critical points in condensed matter systems. The use of anomaly matching to establish emergeability is a strength.

major comments (2)
  1. [Abstract and stabilization discussion] The stabilization of the QED3 fixed point relies on the lattice symmetry embedding rendering monopoles irrelevant. While anomaly matching confirms that the symmetries are compatible with QED3, the manuscript does not include an explicit analysis (such as RG flow or bootstrap bounds) to show that the scaling dimension of the lowest allowed monopole operator exceeds 3. This is load-bearing for the claim that QED3 can be stabilized at these critical points.
  2. [Analysis of models] The abstract states that QED3 is confirmed to be emergeable in all analyzed models. However, the provided text lacks specific derivations, tables of data, or detailed calculations supporting this confirmation, making it challenging to assess the strength of the evidence for emergence from local Hamiltonians.
minor comments (1)
  1. [Introduction] The definition of obstructed insulators and Wannier centers could benefit from a schematic figure to illustrate the jumping of centers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which help clarify the presentation of our results on emergent QED3 at critical points between obstructed atomic insulators. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and stabilization discussion] The stabilization of the QED3 fixed point relies on the lattice symmetry embedding rendering monopoles irrelevant. While anomaly matching confirms that the symmetries are compatible with QED3, the manuscript does not include an explicit analysis (such as RG flow or bootstrap bounds) to show that the scaling dimension of the lowest allowed monopole operator exceeds 3. This is load-bearing for the claim that QED3 can be stabilized at these critical points.

    Authors: We thank the referee for this important observation. Our central claim is that certain lattice symmetry embeddings are compatible with QED3 via anomaly matching and can suppress monopole proliferation at the level of symmetry selection rules. We do not perform an explicit RG analysis or conformal bootstrap calculation of monopole scaling dimensions in the present work; such a computation would require additional numerical or non-perturbative techniques beyond the symmetry-based approach we employ. We will revise the discussion section to reference existing bootstrap results on monopole operators in QED3 with discrete symmetries (where dimensions exceeding 3 have been reported for relevant representations) and to clarify that stabilization is argued on symmetry grounds, with a full dimension bound constituting a natural direction for future study. This constitutes a partial revision. revision: partial

  2. Referee: [Analysis of models] The abstract states that QED3 is confirmed to be emergeable in all analyzed models. However, the provided text lacks specific derivations, tables of data, or detailed calculations supporting this confirmation, making it challenging to assess the strength of the evidence for emergence from local Hamiltonians.

    Authors: We apologize if the supporting calculations were not sufficiently highlighted. The full manuscript presents explicit lattice model constructions, symmetry embeddings, and anomaly matching computations for each case in the main text and appendices. To improve accessibility, we will insert a summary table listing the models, their lattice symmetries, the emergent QED3 flavor content, and the key anomaly-matching results that establish emergeability from local Hamiltonians. We will also add cross-references to the relevant derivations. This revision will be made. revision: yes

Circularity Check

0 steps flagged

No significant circularity in QED3 emergence claim

full rationale

The paper's derivation proceeds from symmetry analysis of obstructed atomic insulators to the possibility of a QED3 critical point, conditioned on lattice symmetry embedding suppressing monopoles, with anomaly matching used to check consistency and explicit confirmation that QED3 is realizable from local lattice Hamiltonians in the models studied. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the stabilization argument is presented as conditional rather than tautological, and the emergeability confirmation is an independent verification step. This matches the default expectation of a self-contained theoretical derivation without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the domain assumption that lattice symmetries can be embedded into the continuum so as to suppress monopoles, together with standard QFT anomaly matching; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Lattice symmetries can be embedded into the continuum theory in a manner that suppresses monopole proliferation.
    This is the mechanism invoked to stabilize the QED3 fixed point.

pith-pipeline@v0.9.0 · 5727 in / 1201 out tokens · 38294 ms · 2026-05-18T05:56:12.900117+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

88 extracted references · 88 canonical work pages · 1 internal anchor

  1. [1]

    (24) has been studied extensively, and we will briefly review the relevant details here

    A Review of the Critical Theory The quantum field theory in Eq. (24) has been studied extensively, and we will briefly review the relevant details here. We first remark that withoutδL E, it is believed that Eq. (24) flows to an interacting CFT fixed point [25– 30] forN f ≳4. Therefore, to analyze the QED3 critical point, one must classify the symmetry-all...

    work page

  2. [2]

    Monopole Quantum Numbers WefirstobservethattheQED 3 criticalpointhasanen- hanced symmetry group compared to that of the atomic insulators on either side of the transition. The QED3 critical point has infrared (IR) time reversal, reflection, charge conjugation, and Lorentz spacetime symmetries, in addition to an IR symmetry, SO(6)×U(1) top Z2 ,(32) describ...

    work page

  3. [3]

    emergeable

    The bosonic critical point The results of Table. I, along withU(1)b boson con- servation symmetry, forbid the proliferation of any triv- ial monopole operator, as the only singlet monpoles un- der the lattice and discrete symmetries are either non- Hermitian (such asϕ † 3 −ϕ 3) or breakU(1) b (such as Im[ϕ4]). The next most relevant operators are the ferm...

    work page

  4. [4]

    There aredimerizedhoppingamplitudescorrespondingtointra- and inter-site coupling

    The F ermionic BBH Model The BBH model [6, 7] consists of four orbitals de- scribing spinless fermions on a square lattice. There aredimerizedhoppingamplitudescorrespondingtointra- and inter-site coupling. Furthermore,πflux is threaded through each plaquette as shown in Fig. 5. The hopping Hamiltonian is given by, HBBH = X R h λ(c† 1,Rc3,R +c † 2,Rc4,R) +...

    work page

  5. [5]

    The Bosonic BBH model To analyze the bosonic BBH model, we will take a single species of hardcore bosonbhopping on the square lattice, and fractionalize it in terms of fermionic partons d1,2, b=d † 1d2.(A12) This rewriting reproduces the physical Hilbert space given we impose the constraintd† 1d1 +d † 2d2 = 1at each site/orbital and set the fillingνb =ν d...

    work page

  6. [6]

    Monopole quantum numbers For convenience, we will calculate the symmetry prop- erties of the monopole operators in the original BBH model with no particle-hole symmetry breaking pertur- bations as such perturbations do not affect the monopole quantum numbers. We adopt the same basis of monopole operators that was outlined in the main text, ϕ† 1,2,3 =f † i...

    work page

  7. [7]

    F ermionic We begin with a hopping model on the breathing hon- eycomb lattice with real couplings as shown in Fig. 6(a). The spectrum is gapped at half-filling, with two Dirac cones atk=Γ. Forλ > t, the phase hosts Wannier or- bitals at the hexagonal plaquette centers, while fort > λ, the Wannier centers are located at the hexagonal plaque- tte edges. At ...

    work page

  8. [8]

    10, and half-filling each fermionic parton

    Bosonic In the bosonic case, we repeat the steps done before, fractionalizing the hardcore boson on the lattice into fermionic partonsb=d † 1d2, having the partons each realize the Hamiltonian in Fig. 10, and half-filling each fermionic parton. The resulting theory is again that ofNf = 4QED 3, LE = X Nf=4 Ψ(−iγµ(∂µ +ia µ))Ψ + 1 4e2 f2 µν +δL E,(B2) whereδ...

    work page

  9. [9]

    Fromthe bipartite nature of the lattice, there is an adiabatic de- formation to an effective QCD3 theory that necessitates the existence of the trivial monopoleRe[ϕ1]

    Multicriticality As found in the case of the square lattice, there are two trivialmonopoles:Re[ϕ 1]andIm[ √ 3ϕ2/3+ϕ3]. Fromthe bipartite nature of the lattice, there is an adiabatic de- formation to an effective QCD3 theory that necessitates the existence of the trivial monopoleRe[ϕ1]. Then, the combination ofRe[ϕ 1]and the tuning mass leads to the existe...

    work page

  10. [10]

    Renormalization Collecting the above results and restoring the valley index(ν)inΓσ νρ, we obtain the vertex function Γ = 1− 8 + 24ξ 3π2Nf ln k Λ /k+ X ν µνX ρ̸=σ Γσ νρkσγρ + Γσ νρ 56−120ξ 15π2Nf + Γνσ ρ 32 5π2Nf kσγρ ln k Λ X ν µνX ρ Γνρ ρ kργρ +  Γρ νρ 88−120ξ 15π2Nf − X λ̸=ρ Γλ νλ 64 15π2Nf   kργρ ln k Λ (C30) The renormalization two point vertex is...

    work page

  11. [11]

    Noteinthekagomeatomicinsulator, theUVsymmetryis no longerSO(3)×Z T 2 ×G lattice butSO(2)×Z T 2 ×G lattice due to the spin polarization of the vacuum in our parton description

    The form ofηandΩforG U V /IR To begin, we note that in all of our UV models, there is no lattice LSMOH theorem that applies, as we have triv- ialbosons, whichareintegerSO(3)spinandKramerssin- glets, occupying each unit cell and high symmetry point. Noteinthekagomeatomicinsulator, theUVsymmetryis no longerSO(3)×Z T 2 ×G lattice butSO(2)×Z T 2 ×G lattice du...

    work page

  12. [12]

    Calculating the Pullbacks We begin by performing anomaly matching for the square lattice critical point. We note for the square lat- tice (and all the cases we consider), it is clear from Ta- ble II that the symmetryO(6) T =O(3) v ×O(3) f fac- torizes intoO(3) v,f blocks for the valley and pseudospin symmetries acting on the monopole operators. Repeat- ed...

    work page

  13. [13]

    Bradlyn, L

    B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Nature547, 298 (2017)

    work page 2017

  14. [14]

    H. C. Po, H. Watanabe, and A. Vishwanath, Phys. Rev. 26 Lett.121, 126402 (2018)

    work page 2018

  15. [15]

    Khalaf, W

    E. Khalaf, W. A. Benalcazar, T. L. Hughes, and R. Queiroz, Phys. Rev. Res.3, 013239 (2021)

    work page 2021

  16. [16]

    Cano and B

    J. Cano and B. Bradlyn, Annual Review of Condensed Matter Physics12, 225 (2021)

    work page 2021

  17. [17]

    Y. Xu, L. Elcoro, Z.-D. Song, M. G. Vergniory, C. Felser, S. S. P. Parkin, N. Regnault, J. L. Mañes, and B. A. Bernevig, Phys. Rev. B109, 165139 (2024)

    work page 2024

  18. [18]

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Phys. Rev. B96, 245115 (2017)

    work page 2017

  19. [19]

    W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science357, 61 (2017)

    work page 2017

  20. [20]

    Schindler, A

    F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. Parkin, B. A. Bernevig, and T. Neupert, Science advances4, eaat0346 (2018)

    work page 2018

  21. [21]

    H. C. Po, A. Vishwanath, and H. Watanabe, Nature communications8, 50 (2017)

    work page 2017

  22. [22]

    W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett.42, 1698 (1979)

    work page 1979

  23. [23]

    This relative nontriviality is sometimes presented as an absolute one, but any such distinction requires fixing a convention for the unit cell or equivalently, fixing the locations of a background of positive ions. Mathemati- cally, these relative topological distinctions are captured by the mathematical structure of a torsor, not a group or cohomology ri...

    work page

  24. [24]

    C. Wang, A. Nahum, M. A. Metlitski, C. Xu, and T. Senthil, Physical Review X7, 031051 (2017)

    work page 2017

  25. [25]

    Senthil, 50 Years of the Renormalization Group: Ded- icated to the Memory of Michael E Fisher , 169 (2024)

    T. Senthil, 50 Years of the Renormalization Group: Ded- icated to the Memory of Michael E Fisher , 169 (2024)

    work page 2024

  26. [26]

    Senthil, A

    T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. Fisher, Science303, 1490 (2004)

    work page 2004

  27. [27]

    Senthil, L

    T. Senthil, L. Balents, S. Sachdev, A. Vishwanath, and M. P. A. Fisher, Phys. Rev. B70, 144407 (2004)

    work page 2004

  28. [28]

    Rantner and X.-G

    W. Rantner and X.-G. Wen, Physical Review Letters86, 3871 (2001)

    work page 2001

  29. [29]

    Wen, Physical Review B70, 214437 (2004)

    M.Hermele, T.Senthil, M.P.A.Fisher, P.A.Lee, N.Na- gaosa, and X.-G. Wen, Physical Review B70, 214437 (2004)

    work page 2004

  30. [30]

    M. B. Hastings, Physical Review B63, 014413 (2000)

    work page 2000

  31. [31]

    Hermele, T

    M. Hermele, T. Senthil, and M. P. A. Fisher, Physical Review B72, 104404 (2005)

    work page 2005

  32. [32]

    Hermele, T

    M. Hermele, T. Senthil, and M. P. A. Fisher, Phys. Rev. B76, 149906 (2007)

    work page 2007

  33. [33]

    Y. Ran, M. Hermele, P. A. Lee, and X.-G. Wen, Physical Review Letters98, 117205 (2007)

    work page 2007

  34. [34]

    Hermele, Y

    M. Hermele, Y. Ran, P. A. Lee, and X.-G. Wen, Physical Review B77, 224413 (2008)

    work page 2008

  35. [35]

    Zhang, X.-Y

    Y. Zhang, X.-Y. Song, and T. Senthil, SciPost Phys. Core8, 024 (2025)

    work page 2025

  36. [36]

    Zhang, X.-Y

    Y. Zhang, X.-Y. Song, and T. Senthil, (2025), arXiv:2508.16725 [cond-mat.str-el]

    work page arXiv 2025

  37. [37]

    Appelquist, D

    T. Appelquist, D. Nash, and L. C. R. Wijewardhana, Phys. Rev. Lett.60, 2575 (1988)

    work page 1988

  38. [38]

    S. M. Chester and S. S. Pufu, Journal of High Energy Physics08, 019 (2016)

    work page 2016

  39. [39]

    Li, Journal of High Energy Physics11, 005 (2022)

    Z. Li, Journal of High Energy Physics11, 005 (2022)

    work page 2022

  40. [40]

    Albayrak, R

    S. Albayrak, R. S. Erramilli, Z. Li, D. Poland, and Y. Xin, Phys. Rev. D105, 085008 (2022)

    work page 2022

  41. [41]

    Y.-C. He, J. Rong, and N. Su, SciPost Phys.13, 014 (2022)

    work page 2022

  42. [42]

    Li, Physics Letters B831, 137192 (2022)

    Z. Li, Physics Letters B831, 137192 (2022)

    work page 2022

  43. [43]

    Song, Y.-C

    X.-Y. Song, Y.-C. He, A. Vishwanath, and C. Wang, Phys. Rev. X10, 011033 (2020)

    work page 2020

  44. [44]

    X.-Y. Song, C. Wang, A. Vishwanath, and Y.-C. He, Nature Communications10(2019)

    work page 2019

  45. [45]

    E. Lieb, T. Schultz, and D. Mattis, Annals of Physics 16, 407 (1961)

    work page 1961

  46. [46]

    Oshikawa, Phys

    M. Oshikawa, Phys. Rev. Lett.84, 1535 (2000)

    work page 2000

  47. [47]

    M. B. Hastings, Phys. Rev. B69, 104431 (2004)

    work page 2004

  48. [48]

    Zou, Y.-C

    L. Zou, Y.-C. He, and C. Wang, Phys. Rev. X11, 031043 (2021)

    work page 2021

  49. [49]

    W. Ye, M. Guo, Y.-C. He, C. Wang, and L. Zou, SciPost Phys.13, 066 (2022)

    work page 2022

  50. [50]

    M. V. Berry, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences392, 45 (1984)

    work page 1984

  51. [51]

    Zak, Phys

    J. Zak, Phys. Rev. Lett.62, 2747 (1989)

    work page 1989

  52. [52]

    F. D. M. Haldane, Journal of Physics C: Solid State Physics14, 2585 (1981)

    work page 1981

  53. [53]

    Voit, Reports on Progress in Physics58, 977 (1995)

    J. Voit, Reports on Progress in Physics58, 977 (1995)

    work page 1995

  54. [54]

    An introduction to bosonization,

    D. Sénéchal, “An introduction to bosonization,” in Theoretical Methods for Strongly Correlated Electrons, edited by D. Sénéchal, A.-M. Tremblay, and C. Bour- bonnais (Springer New York, New York, NY, 2004) pp. 139–186

    work page 2004

  55. [55]

    Gepner, Nuclear Physics B252, 481 (1985)

    D. Gepner, Nuclear Physics B252, 481 (1985)

    work page 1985

  56. [56]

    Affleck, Nuclear Physics B265, 448 (1986)

    I. Affleck, Nuclear Physics B265, 448 (1986)

    work page 1986

  57. [57]

    Dempsey, I

    R. Dempsey, I. R. Klebanov, S. S. Pufu, B. T. Søgaard, and B. Zan, Phys. Rev. Lett.132, 031603 (2024)

    work page 2024

  58. [58]

    Ezawa, Phys

    M. Ezawa, Phys. Rev. Lett.120, 026801 (2018)

    work page 2018

  59. [59]

    W. A. Benalcazar, T. Li, and T. L. Hughes, Phys. Rev. B99, 245151 (2019)

    work page 2019

  60. [60]

    Borokhov, A

    V. Borokhov, A. Kapustin, and X. Wu, Journal of High Energy Physics2002, 049 (2002)

    work page 2002

  61. [61]

    Borokhov, A

    V. Borokhov, A. Kapustin, and X. Wu, Journal of High Energy Physics2002, 044 (2002)

    work page 2002

  62. [62]

    S. S. Pufu, Physical Review D89(2014)

    work page 2014

  63. [63]

    E. Dyer, M. Mezei, and S. S. Pufu, arXiv preprint arXiv:1309.1160 (2013)

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  64. [64]

    Franz, Z

    M. Franz, Z. Tešanović, and O. Vafek, Phys. Rev. B66, 054535 (2002)

    work page 2002

  65. [65]

    Vafek, Z

    O. Vafek, Z. Tešanović, and M. Franz, Phys. Rev. Lett. 89, 157003 (2002)

    work page 2002

  66. [66]

    TheZ 2 factor acts as−1on all local operators and can be chosen to be part ofSO(6)orU(1) top, de- pending on convention

    Note that there some ambiguity arising from the com- monZ 2 quotiented fromSO(6)×U(1) top global sym- metry. TheZ 2 factor acts as−1on all local operators and can be chosen to be part ofSO(6)orU(1) top, de- pending on convention. This is the same convention that determines whether a symmetry acts asψ→U ψfor U∈SU(4)orψ→iU, which differs by a factor of−1on ...

    work page

  67. [67]

    C. L. Kane and E. J. Mele, Phys. Rev. Lett.95, 146802 (2005)

    work page 2005

  68. [68]

    Cheng, M

    M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath, and P. Bonderson, Phys. Rev. X6, 041068 (2016)

    work page 2016

  69. [69]

    H. C. Po, H. Watanabe, C.-M. Jian, and M. P. Zaletel, Phys. Rev. Lett.119, 127202 (2017)

    work page 2017

  70. [70]

    Huang, H

    S.-J. Huang, H. Song, Y.-P. Huang, and M. Hermele, Phys. Rev. B96, 205106 (2017)

    work page 2017

  71. [71]

    G. Y. Cho, C.-T. Hsieh, and S. Ryu, Phys. Rev. B96, 195105 (2017)

    work page 2017

  72. [72]

    C.-M. Jian, Z. Bi, and C. Xu, Phys. Rev. B97, 054412 27 (2018)

    work page 2018

  73. [73]

    Thorngren and D

    R. Thorngren and D. V. Else, Phys. Rev. X8, 011040 (2018)

    work page 2018

  74. [74]

    Lin and T

    M. Lin and T. L. Hughes, Phys. Rev. B98, 241103 (2018)

    work page 2018

  75. [75]

    Luo and F

    X.-J. Luo and F. Wu, Phys. Rev. B108, 075143 (2023)

    work page 2023

  76. [76]

    W. E. Caswell, Phys. Rev. Lett.33, 244 (1974)

    work page 1974

  77. [77]

    Banks and A

    T. Banks and A. Zaks, Nuclear Physics B196, 189 (1982)

    work page 1982

  78. [78]

    Bi and T

    Z. Bi and T. Senthil, Phys. Rev. X9, 021034 (2019)

    work page 2019

  79. [79]

    S. K. Radha and W. R. L. Lambrecht, Phys. Rev. B103, 075435 (2021)

    work page 2021

  80. [80]

    Wen, Physical Review B65(2002)

    X.-G. Wen, Physical Review B65(2002)

    work page 2002

Showing first 80 references.