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Space of conformal boundary conditions from the view of higher Berry phase: Flow of Berry curvature in parametrized BCFTs

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Pith's one-line read Continuously parametrized conformal boundary conditions make a BCFT ground state carry 2-form Berry curvature whose exterior derivative is a closed 3-form higher Berry curvature with integral 2π over S³, realizing a Chern number pump in…

desk verdict Clean explicit BCFT example of higher Berry curvature flow; the broad gapped-state claim is a conjecture, not a consequence. read the letter →

arxiv 2507.12546 v1 pith:6J4QXH5B submitted 2025-07-16 hep-th cond-mat.str-elmath-phmath.MPquant-ph

Xueda Wen This is my paper
classification hep-thcond-mat.str-elmath-phmath.MPquant-ph
keywords BoundaryconformalfieldtheoryHigherBerryphasecurvatureflowChernnumberpumpSpectralEntanglementHamiltonianDiracfermionBCFTParametrizedgappedsystems
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 connects the space of conformal boundary conditions in (1+1)-dimensional boundary conformal field theories to the space of gapped systems organized by higher Berry phases. The author considers a Dirac fermion BCFT whose boundary condition varies continuously over a three-sphere of parameters, as obtained from a gapped system in a nontrivial higher Berry class. The ground state of this BCFT carries ordinary two-form Berry curvature $\omega^{(2)}=(\alpha/\pi-1)\frac{\sin\theta}{2}\,d\theta\wedge d\phi$, and its exterior derivative is a closed 3-form higher Berry curvature $\Omega^{(3)}$ with quantized integral $2\pi$ over $S^3$. This is a Chern number pump in Fock space: increasing $\alpha$ by $2\pi$ while wrapping $(\theta,\phi)$ around $S^2$ returns the spectrum and fermion number while shifting the total Chern number. The paper further argues that any family of gapped states in a nontrivial higher Berry class should exhibit the same multi-parameter spectral flow in its entanglement Hamiltonians.

What carries the argument

The load-bearing object is the family of BCFT ground states $|G\rangle_{\alpha,\theta,\phi}=\prod_{r\le0}\tilde\psi^\dagger_{1,r}\prod_{s\le0}\tilde\psi^\dagger_{2,s}|\mathrm{vac}\rangle$ with twisted mode indices $r\in\mathbb{Z}+\alpha/2\pi$ and $s\in\mathbb{Z}-\alpha/2\pi$. The identity that carries the argument is $\Omega^{(3)}:=d\omega^{(2)}$, the exterior derivative of the total two-form Berry curvature of the filled Fermi sea; it measures the flow of ordinary Berry curvature across a chosen reference energy as the parameters vary. The quantization $\int_{S^3}\Omega^{(3)}=2\pi$ encodes the Chern number pump, and the $SU(2)$ twisting matrix from the mass parameters, diagonalized by $U(\theta,\phi)$, is what converts the parametrized boundary condition into two opposite spectral flows.

What would settle it

Compute the full entanglement spectrum of a finite-size lattice model in a nontrivial higher Berry class, such as the two-flavor Dirac model with mass parameters on $S^3$, and check whether over the closed cycle with $\alpha\in[0,2\pi]$ and $(\theta,\phi)$ wrapping $S^2$ the spectrum returns with total Chern number shifted by two; if that multi-parameter spectral flow is absent, the claimed connection to BCFT would be refuted.

Watch

Extended reading notes

Core claim

The central discovery is that Berry curvature can flow in the Fock space of a BCFT rather than in real space. For a two-flavor Dirac fermion BCFT, the boundary scattering matrix inherited from a gapped system with mass parameters on $S^3$ becomes an $SU(2)$ twisting matrix $\Psi_R=M\cdot\Psi_L$. After a unitary rotation depending only on $(\theta,\phi)$, this boundary condition diagonalizes into two opposite single-parameter spectral flows, $\tilde\psi_{1,R}=e^{i\alpha}\tilde\psi_{1,L}$ and $\tilde\psi_{2,R}=e^{-i\alpha}\tilde\psi_{2,L}$. Each filled single-particle mode carries two-form Berry curvature $\Omega^{(2)}_{\pm}=\pm\frac{\sin\theta}{2}\,d\theta\wedge d\phi$, and a zeta-function-regularized count of the filled Fermi sea gives the total curvature $\omega^{(2)}=(\alpha/\pi-1)\frac{\sin\theta}{2}\,d\theta\wedge d\phi$. Its exterior derivative $\Omega^{(3)}=d\omega^{(2)}=\frac{1}{2\pi}\sin\theta\,d\alpha\wedge d\theta\wedge d\phi$ is closed but not exact, because $\omega^{(2)}$ is not globally defined at $\alpha=0,\pi$, and its integral over $S^3$ is $2\pi$. The author concludes that this higher Berry curvature describes a Chern number pump in the BCFT Fock space, and that the same phenomenon appears as multi-parameter spectral flow in the entanglement Hamiltonians of gapped ground states belonging to nontrivial higher Berry classes.

Load-bearing premise

The broad application to all nontrivial higher Berry classes rests on identifying the entanglement Hamiltonian of a gapped system near criticality with the physical Hamiltonian of a BCFT on an interval, a step the paper explicitly notes remains without rigorous justification.

Editorial extensions

If this is right

  • For any (1+1)-dimensional BCFT obtained from a gapped family in a nontrivial higher Berry class, the multi-parameter spectral flow carries Berry curvature in Fock space, analogous to real-space Chern number pumping.
  • The higher Berry invariant is quantized: $\int_{S^3}\Omega^{(3)}=2\pi$, with the equivalent Stokes-form evaluation $\int_{S^2_{\alpha=\pi}}\omega^{(2)}-\int_{S^2_{\alpha=0+}}\omega^{(2)}=2\pi$, so the invariant counts the pumped Chern number.
  • Regularized conformal boundary states $e^{-\beta H/2}|B\rangle\rangle_\lambda$ have entanglement Hamiltonians that become the physical Hamiltonians of BCFTs on a cylinder, so nontrivial higher Berry classes manifest as spectral flow in entanglement spectra.
  • The construction is claimed to generalize to compact free boson BCFTs with an effective four-fermion interaction, and to higher-dimensional BCFTs where higher Thouless pumps may appear.

Reading between the lines

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

  • If the entanglement-Hamiltonian identification survives scrutiny, the higher Berry invariant becomes readable from entanglement spectra alone, giving a wavefunction-based probe of higher Berry classes that does not require direct access to the bulk Hamiltonian.
  • The moduli space of conformal boundary conditions may itself carry a natural higher Berry class, suggesting a topological characterization of boundary-condition spaces that could connect to D-brane moduli and T-duality discussions.
  • Because $\omega^{(2)}$ is singular at $\alpha=0,\pi$, a fully global formulation likely requires a patched connection, meaning the higher Berry invariant may be equivalently captured by transition functions on patches of the boundary-condition parameter space.
  • Since the $SU(2)$ twist is directly a boundary scattering matrix, the quantized flow may be observable through interference experiments on the reflection amplitudes of the coupled gapped system.
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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

4 major / 4 minor

Summary. The paper proposes a connection between continuously parametrized conformal boundary conditions in (1+1)-dimensional BCFTs and higher Berry classes of gapped systems. The central construction sandwiches a free Dirac-fermion CFT between a fixed boundary and a family of gapped systems whose infinite-mass limit yields an SU(2)-valued boundary condition M(α,θ,φ) parametrized by S^3. After a unitary rotation the CFT Hamiltonian splits into two twisted-mode towers (Eq. 15), each single-particle wavefunction carries 2-form Berry curvature Ω_±^(2)=±(sinθ/2)dθ∧dφ, and the filled Fermi sea produces the total curvature ω^(2) in Eq. (21). Taking the exterior derivative gives the 3-form Ω^(3) of Eq. (22), with integral 2π over S^3 (Eq. 23) and an equivalent Stokes' theorem formulation between α=0+ and α=π−0+ (Eq. 24). The paper then asserts that regularized conformal boundary states are ground states of gapped Hamiltonians and that entanglement Hamiltonians of gapped states near criticality are BCFT interval Hamiltonians, leading to the advertised application to parametrized gapped states.

Significance. If taken as an exact free-fermion statement, the paper is a clear and explicit demonstration of Berry-curvature flow in the Fock space of a BCFT, with honest treatment of the singular points at α=0 and α=π and a clean zeta-regularized computation of the pumped charge. The paper also correctly emphasizes the distinction from chiral-CFT twisting and from the real-space flow of Ref. [21], and it names the structural analogy precisely. The broader application to arbitrary families of gapped ground states is potentially significant, but it depends on an identification that the paper itself marks as unproved; the significance of that part is therefore conditional rather than established.

major comments (4)
  1. [Application section and Appendix E] The abstract and conclusion assert that any family of gapped states in a nontrivial higher Berry class has entanglement Hamiltonians exhibiting the same multi-parameter spectral flow. This is not established by the BCFT calculation. The paper itself states that the identification of the regularized boundary states e^{-βH/2}|B>>_λ with ground states of gapped Hamiltonians 'remains open', and Appendix E defers the lattice analysis to future work. Equation (26) and the conformal-mapping argument assume that H_E(λ) is exactly the physical Hamiltonian of a BCFT on an interval; if this identification fails for interacting gapped states, the global claim does not follow. To make the application load-bearing, the manuscript should either provide a lattice check (for example, an entanglement-spectrum computation for a finite chain in a nontrivial higher Berry class showing the α-dependent level flow), or explicitly downgrade the abstract/conclusion assertion to a conjecture.
  2. [Eqs. (7)-(10) and Appendix A] The BCFT boundary condition M is obtained in the limit m→∞ after also taking L→∞ in the scattering derivation of Appendix A. The Kapustin-Spodyneiko higher Berry curvature, however, is defined for uniformly gapped families with a finite spectral gap. The paper does not show that the m→∞ limit commutes with the higher Berry invariant, nor that the Fock-space curvature computed in the BCFT equals the limit of the curvature of the finite-gap family. This matters because the central interpretation is that Eq. (22) is the BCFT avatar of the gapped system's higher Berry class. A stability argument, or at least a finite-m computation followed by the limit, would close this gap.
  3. [Eqs. (21)-(24) and Fig. 2] The text and Fig. 2 describe increasing α by 2π, while Eq. (8) declares α∈[0,π] and the Stokes' evaluation in Eq. (24) integrates between α=0+ and α=π−0+. The two conventions can be reconciled by viewing α as a lifted coordinate that wraps S^3 once over each interval of length π, but this should be stated explicitly. As written, the reader cannot immediately tell whether the claimed 'total Chern number shifts by two' refers to the change between α=0+ and α=2π− or to the wrapping multiplicity of the S^3 parameterization.
  4. [Appendix B, Eq. (B3)] Equation (B3) contains an apparent typo: the normal-ordered sum is written with the mode operator ~ψ_{1,r}, but the quantity Q_- is defined from ~ψ_{2,s} in the preceding lines and in the main text. The mode index and field label should be corrected so that the zeta-regularized expression for Q_- is internally consistent.
minor comments (4)
  1. [After Eq. (22)] The sentence explaining why Ω^(3) is 'a closed form rather than an exact one' is potentially misleading: on the punctured manifold M^3 used in Eq. (24), Ω^(3) is exact by construction, while the intended statement is that it is not the exterior derivative of a globally well-defined two-form on all of S^3.
  2. [Notation around Eq. (20)] The zeta-regularized expression in Eq. (20) would benefit from stating explicitly that the sum over r∈Z+α/2π with r<0 uses the Hurwitz zeta identity for 0<α/2π<1 before extension to larger α; this would make the floor function in Eq. (20) less surprising.
  3. [Appendix D] The remark that in Eq. (D2) 'the same phase factor e^{iπ/4} appears in both equations' is unclear, since both equations contain that factor by definition; if the intended point is a relative sign or a convention choice, it should be stated explicitly.
  4. [General] There are several typographical errors, including 'untiary' in Appendix B and 'familied' in the Conclusion; these should be corrected during revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the BCFT higher Berry curvature is computed from the model's boundary conditions rather than assumed; the broad gapped-state application depends on an explicitly unproven HE=BCFT identification, which is a missing proof, not circularity.

full rationale

No circular step can be exhibited. The central result is a direct free-fermion computation: the boundary condition (10)-(11) is obtained by an m→∞ scattering limit (Appendix A), the mode expansion (15) and ground state (16) follow, and the single-mode curvatures Ω± (18)-(19) are computed from the explicit wavefunction (17), not assumed. The prefactor (α/π−1) in ω^(2) (21) comes from zeta-regularized mode counting (20) and (B1)-(B3), so it is derived rather than fitted. Ω^(3) (22) is defined as dω^(2), and the finite integral (23)-(24) follows from the non-global nature of ω^(2) at α=0,π; this is a construction, not a circular prediction. The application to arbitrary gapped states does rest on an identification of entanglement Hamiltonians with BCFT interval Hamiltonians, which the paper itself flags as unproven (“although a rigorous justification of this identification remains open” in the Application section; Appendix E: “A detailed analysis using lattice models will be presented in a future work”). That is an unproven premise and a correctness risk, not a circular reduction. Self-citations such as [21], [35], [36], and [66] provide context and interpretation and are not load-bearing for Eqs. (18)-(23); [21] is an independently published result. Accordingly, the derivation chain is self-contained for the BCFT calculation, and no circularity score above 0 is warranted.

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

No numbers are fitted to data or chosen ad hoc. The mass scale m is sent to infinity to define conformal boundary conditions; beta and epsilon_0 are UV or regularization cutoffs whose values do not affect the quantized invariants; the parameter-space coordinates (alpha,theta,phi) are the variables of the family, not fit parameters. No new particles, forces, dimensions, or conserved quantities are introduced. The 'higher Berry curvature in BCFT' is a new interpretive quantity built from standard single-particle Berry curvature, not a new entity.

assumptions (5)
  • standard math Free Dirac fermion mode expansion on an interval with twisted boundary conditions, with zeta-function regularization of ground-state observables.
    Used in Eqs (4)-(6), (15), (20), Appendix B. Standard CFT technology; the zeta regularization is the specific method that turns divergent sums into the finite charges in Eq (21).
  • domain assumption The infinite mass limit of the gapped Hamiltonian (1)/(7) produces exact conformal boundary conditions (3)/(10) via the reflection matrix.
    Appendix A derives the reflection coefficient r and takes m to infinity. This assumes no residual energy dependence or subgap degrees of freedom survive the limit, which is standard but not proven at finite L.
  • standard math The unitary transformation U(theta,phi) in (13) diagonalizes the boundary condition and the resulting single-particle modes carry the standard spin-1/2 monopole Berry curvature.
    Eqs (12)-(14), (17)-(19). This is a direct computation; the monopole curvature sin(theta)/2 dtheta dphi is the standard result for an SU(2) rotation parametrized by (theta,phi).
  • domain assumption The parameter space for the spectral flow is taken as S^3, with alpha extended periodically beyond [0,pi] so that increasing alpha by 2 pi is a valid cycle.
    Eq (8), Fig 2, discussion near Eq (23). On the standard S^3 coordinates alpha lies in [0,pi]; the paper's alpha cycle from 0 to 2 pi 'wraps S^3 twice', an interpretation that requires the boundary condition M to be 2 pi-periodic in alpha.
  • domain assumption Regularized conformal boundary states (25) can be interpreted as ground states of gapped Hamiltonians, and entanglement Hamiltonians of gapped systems map to BCFT Hamiltonians (Appendix E).
    Application section, Eq (26), Appendix E, citing Refs [57-62] and [58,70]. The paper explicitly states rigorous justification remains open.

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Pith. "Pith review of Space of conformal boundary conditions from the view of higher Berry phase: Flow of Berry curvature in parametrized BCFTs." pith.science (2026). https://pith.science/paper/6J4QXH5B

@misc{pith2026250712546,
  author       = {Pith},
  title        = {Pith review of: Space of conformal boundary conditions from the view of higher Berry phase: Flow of Berry curvature in parametrized BCFTs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6J4QXH5B}},
  note         = {Machine review of arXiv:2507.12546}
}
read the original abstract

In this work, we study the connection between two subjects: the space of conformal boundary conditions in boundary conformal field theories (BCFTs) and the space of gapped systems characterized by higher Berry phases. We explore this connection by analyzing multi-parameter spectral flow in Dirac fermion BCFTs with continuously parametrized conformal boundary conditions, which are introduced by coupling a CFT to a family of gapped systems. When the gapped systems belong to a nontrivial higher Berry class, the associated conformal boundary conditions induce a flow of the ordinary Berry curvature, resulting in a Chern number pump in the Fock space of the BCFT. This phenomenon is the BCFT analog of Berry curvature flow in one-dimensional parametrized gapped systems, where the flow occurs in real space. Building on this correspondence, we introduce the notions of higher Berry curvature and higher Berry invariants within the BCFT framework. Our results provide a new perspective for studying the topological properties of families of conformal boundary states and gapped ground states: if a family of gapped states belongs to a nontrivial higher Berry class, then the corresponding entanglement Hamiltonians exhibit a multi-parameter spectral flow that carries Berry curvature in the Fock space.

Figures

Figures reproduced from arXiv: 2507.12546 by the authors.

Figure 1
Figure 1. FIG. 1. (a) A CFT defined in the interval [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Berry curvature flow and Chern number pump in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Left: Reduced density matrix [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. A parametrized family of gapped systems in the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Mapping from a BCFT with parametrized confor [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. A cylinder in [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Characterizing gapped phases by smeared boundary conformal field theories: Duality in unusual ordering with spontaneously broken generalized symmetries

    hep-th 2026-05 unverdicted novelty 7.0 of 10

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Reference graph

Works this paper leans on

70 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [56]

    Chern number pump

    In Ref.[21], the term “Chern number pump” is used for a closely related system overX =S2×S1, but the essential physics of both systems is the same. Here we use the term “pumping” forX =S3 somewhat more loosely

    work page

  2. [21]

    Choi and K

    Y. Choi and K. Ohmori, Higher berry phase of fermions and index theorem, Journal of High Energy Physics 2022, 10.1007/jhep09(2022)022 (2022)

    work page doi:10.1007/jhep09(2022)022 2022

  3. [1]

    J. L. Cardy, Boundary conditions, fusion rules and the verlinde formula, Nuclear Physics B324, 581 (1989)

    1989

  4. [2]

    boundary

    (20) Here we have considered the zeta function regulariza- tion to sum up the ground state charges by using∑ r∈Z+a,r<0 =ζ(0, 1−(a−[a])) =a−[a]−1/2, where[a] is the greatest integer less than or equal toa, and: :rep- resents the normal ordering. The total fermionsQ− with each fermion carryingΩ(2) − can be written down similarly [52]. Then the total 2-form ...

    work page

  5. [3]

    Friedan, The space of conformal boundary conditions for the c=1 gaussian model (1993), https://www.physics.rutgers.edu/~friedan/papers/ boundary_c=1_1999.pdf

    D. Friedan, The space of conformal boundary conditions for the c=1 gaussian model (1993), https://www.physics.rutgers.edu/~friedan/papers/ boundary_c=1_1999.pdf

    work page 1993

  6. [4]

    Friedan, The space of conformal boundary con- ditions for the c=1 gaussian model (more) (2003), https://www.physics.rutgers.edu/~friedan/papers/ boundary_c=1_2003.pdf

    D. Friedan, The space of conformal boundary con- ditions for the c=1 gaussian model (more) (2003), https://www.physics.rutgers.edu/~friedan/papers/ boundary_c=1_2003.pdf

    work page 2003

  7. [5]

    C. G. Callan, I. R. Klebanov, A. W. W. Ludwig, and J. M. Maldacena, Exact solution of a boundary con- formal field theory, Nuclear Physics B422, 417 (1994), arXiv:hep-th/9402113 [hep-th]

    arXiv 1994

  8. [6]

    Recknagel and V

    A. Recknagel and V. Schomerus, Boundary deformation theory and moduli spaces of D-branes, Nuclear Physics B 545, 233 (1999), arXiv:hep-th/9811237 [hep-th]

    arXiv 1999

Show all 70 references
  1. [7]

    M. R. Gaberdiel and A. Recknagel, Conformal bound- ary states for free bosons and fermions, Journal of High Energy Physics2001, 016 (2001), arXiv:hep-th/0108238 [hep-th]

    2001 arXiv

  2. [8]

    Gaberdiel, A

    M. Gaberdiel, A. Recknagel, and G. Watts, The confor- mal boundary states for su(2) at level 1, Nuclear Physics B 626, 344 (2002)

    2002

  3. [9]

    Recknagel and V

    A. Recknagel and V. Schomerus, Boundary conformal field theory and the worldsheet approach to D-branes (Cambridge University Press, 2013)

    2013

  4. [10]

    A. Kitaev, Toward a topological classification of many- body quantum states with short-range entanglement (2011), talk at Simons Center for Geometry and Physics http://scgp.stonybrook.edu/archives/1087

    2011

  5. [11]

    Kitaev, On the classification of short-range entan- gled states (2013), talk at Simons Center for Geometry and Physics http://scgp.stonybrook.edu/archives/ 16180

    A. Kitaev, On the classification of short-range entan- gled states (2013), talk at Simons Center for Geometry and Physics http://scgp.stonybrook.edu/archives/ 16180

    2013

  6. [12]

    A. Kitaev, Homotopy-theoretic approach to SPT phases in action: Z16-classification of three-dimensional su- perconductors (2015), talk at workshop Symmetry and Topology in Quantum Matter, Institute for Pure & Ap- plied Mathematics, University of California Los Angeles

    2015

  7. [13]

    Kitaev, Differential forms on the space of statistical mechanics models (2019), talk at the conference in cele- bration of Dan Freed’s 60th birthday:https://web.ma

    A. Kitaev, Differential forms on the space of statistical mechanics models (2019), talk at the conference in cele- bration of Dan Freed’s 60th birthday:https://web.ma. utexas.edu/topqft/talkslides/kitaev.pdf

    2019

  8. [14]

    Kapustin and L

    A. Kapustin and L. Spodyneiko, Higher-dimensional gen- eralizations of berry curvature, Physical Review B101, 10.1103/physrevb.101.235130 (2020)

    2020 doi

  9. [15]

    Kapustin and L

    A. Kapustin and L. Spodyneiko, Higher-dimensional generalizations of the thouless charge pump (2020), arXiv:2003.09519 [cond-mat.str-el]

    2020 arXiv

  10. [16]

    P.-S. Hsin, A. Kapustin, and R. Thorngren, Berry phase in quantum field theory: Diabolical points and bound- ary phenomena, Phys. Rev. B 102, 245113 (2020), arXiv:2004.10758 [cond-mat.str-el]

    2020 arXiv

  11. [17]

    Cordova, D

    C. Cordova, D. Freed, H. T. Lam, and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications i, SciPost Physics 8, 10.21468/scipostphys.8.1.001 (2020)

    2020 doi

  12. [18]

    Cordova, D

    C. Cordova, D. Freed, H. T. Lam, and N. Seiberg, Anomalies in the space of coupling constants and their dynamical applications II, SciPost Physics 8, 10.21468/scipostphys.8.1.002 (2020)

    2020 doi

  13. [19]

    D. V. Else, Topological goldstone phases of matter, Phys- ical Review B104, 10.1103/physrevb.104.115129 (2021)

    2021 doi

  14. [20]

    Bachmann, W

    S. Bachmann, W. De Roeck, M. Fraas, and T. Jappens, A classification of G-charge Thouless pumps in 1D in- vertible states, arXiv e-prints , arXiv:2204.03763 (2022), arXiv:2204.03763 [math-ph]

    2022 arXiv

  15. [22]

    X. Wen, M. Qi, A. Beaudry, J. Moreno, M. J. Pflaum, D. Spiegel, A. Vishwanath, and M. Hermele, Flow of higher Berry curvature and bulk-boundary correspon- dence in parametrized quantum systems, Phys. Rev. B 108, 125147 (2023), arXiv:2112.07748 [cond-mat.str-el]

    2023 arXiv

  16. [23]

    Aasen, Z

    D. Aasen, Z. Wang, and M. B. Hastings, Adiabatic paths of hamiltonians, symmetries of topological or- der, and automorphism codes, Physical Review B106, 10.1103/physrevb.106.085122 (2022)

    2022 doi

  17. [24]

    Hsin and Z

    P.-S. Hsin and Z. Wang, On topology of the moduli space of gapped hamiltonians for topological phases, Journal of Mathematical Physics64, 041901 (2023)

    2023

  18. [25]

    Kapustin and N

    A. Kapustin and N. Sopenko, Local Noether theorem for quantum lattice systems and topological invariants of gapped states, Journal of Mathematical Physics63, 091903 (2022), arXiv:2201.01327 [math-ph]

    2022 arXiv

  19. [26]

    Shiozaki, Adiabatic cycles of quantum spin systems, Physical Review B 106, 10.1103/physrevb.106.125108 (2022)

    K. Shiozaki, Adiabatic cycles of quantum spin systems, Physical Review B 106, 10.1103/physrevb.106.125108 (2022)

    2022 doi

  20. [27]

    Ohyama, K

    S. Ohyama, K. Shiozaki, and M. Sato, Generalized thou- less pumps in (1 + 1)-dimensional interacting fermionic systems, Phys. Rev. B106, 165115 (2022)

    2022

  21. [28]

    Ohyama, Y

    S. Ohyama, Y. Terashima, and K. Shiozaki, Discrete higher berry phases and matrix product states (2023), arXiv:2303.04252 [cond-mat.str-el]

    2023 arXiv

  22. [29]

    Artymowicz, A

    A. Artymowicz, A. Kapustin, and N. Sopenko, Quan- tization of the higher Berry curvature and the higher Thoulesspump,arXive-prints,arXiv:2305.06399(2023), arXiv:2305.06399 [math-ph]

    2023 arXiv

  23. [30]

    Beaudry, M

    A. Beaudry, M. Hermele, J. Moreno, M. Pflaum, M. Qi, and D. Spiegel, Homotopical foundations of parametrized quantum spin systems (2023), arXiv:2303.07431 [math- ph]

    2023 arXiv

  24. [31]

    Ohyama and S

    S. Ohyama and S. Ryu, Higher structures in matrix product states, arXiv e-prints , arXiv:2304.05356 (2023), arXiv:2304.05356 [cond-mat.str-el]

    2023 arXiv

  25. [32]

    M. Qi, D. T. Stephen, X. Wen, D. Spiegel, M. J. Pflaum, A. Beaudry, and M. Hermele, Charting the space of ground states with tensor networks, SciPost Phys.18, 168 (2025)

    2025

  26. [33]

    Shiozaki, N

    K. Shiozaki, N. Heinsdorf, and S. Ohyama, Higher Berry 7 curvature from matrix product states, arXiv e-prints , arXiv:2305.08109 (2023), arXiv:2305.08109 [quant-ph]

    2023 arXiv

  27. [34]

    Spodyneiko, Hall conductivity pump, arXiv e-prints , arXiv:2309.14332 (2023), arXiv:2309.14332 [cond- mat.mes-hall]

    L. Spodyneiko, Hall conductivity pump, arXiv e-prints , arXiv:2309.14332 (2023), arXiv:2309.14332 [cond- mat.mes-hall]

    2023 arXiv

  28. [35]

    Debray, S

    A. Debray, S. K. Devalapurkar, C. Krulewski, Y. L. Liu, N. Pacheco-Tallaj, and R. Thorngren, A Long Ex- act Sequence in Symmetry Breaking: order parame- ter constraints, defect anomaly-matching, and higher Berry phases, arXiv e-prints , arXiv:2309.16749 (2023), arXiv:2309.1674...

    2023 arXiv

  29. [36]

    O.E.Sommer, X.Wen,andA.Vishwanath,HigherBerry CurvaturefromtheWaveFunction.I.SchmidtDecompo- sition and Matrix Product States, Phys. Rev. Lett.134, 146601 (2025), arXiv:2405.05316 [cond-mat.str-el]

    2025 arXiv

  30. [37]

    O. E. Sommer, A. Vishwanath, and X. Wen, Higher Berry curvature from the wave function. II. Locally parametrized states beyond one dimension, Phys. Rev. B 111, 155110 (2025), arXiv:2405.05323 [cond-mat.str- el]

    2025 arXiv

  31. [38]

    Ohyama and S

    S. Ohyama and S. Ryu, Higher Berry connection for ma- trix product states, Phys. Rev. B111, 035121 (2025), arXiv:2405.05327 [cond-mat.str-el]

    2025 arXiv

  32. [39]

    Ohyama and S

    S. Ohyama and S. Ryu, Higher Berry phase from pro- jected entangled pair states in (2+1) dimensions, Phys. Rev. B 111, 045112 (2025), arXiv:2405.05325 [cond- mat.str-el]

    2025 arXiv

  33. [40]

    B. Liu, J. Zhang, S. Ohyama, Y. Kusuki, and S. Ryu, Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions, arXiv e-prints , arXiv:2410.08284 (2024), arXiv:2410.08284 [cond-mat.str-el]

    2024

  34. [41]

    Geiko, Parametrized topological phases in 1d and T-duality, arXiv e-prints , arXiv:2412.20905 (2024), arXiv:2412.20905 [math-ph]

    R. Geiko, Parametrized topological phases in 1d and T-duality, arXiv e-prints , arXiv:2412.20905 (2024), arXiv:2412.20905 [math-ph]

    2024 arXiv

  35. [42]

    Beaudry, M

    A. Beaudry, M. Hermele, M. J. Pflaum, M. Qi, D. D. Spiegel, and D. T. Stephen, A classifying space for phases of matrix product states (2025), arXiv:2501.14241 [math- ph]

    2025

  36. [43]

    Kapustin, Topological phases of matter and homotopy theory, inEncyclopedia of Mathematical Physics (Second Edition), edited by R

    A. Kapustin, Topological phases of matter and homotopy theory, inEncyclopedia of Mathematical Physics (Second Edition), edited by R. Szabo and M. Bojowald (Academic Press, Oxford, 2025) second edition ed., pp. 106–110

    2025

  37. [44]

    Manjunath and D

    N. Manjunath and D. V. Else, Anomalous continuous symmetries and quantum topology of goldstone modes, Physical Review B 111, 10.1103/physrevb.111.125151 (2025)

    2025 doi

  38. [45]

    A. Bose, A. Hardy, N. Manjunath, and A. Paramekanti, Symmetry constrained field theories for chiral spin liq- uid to spin crystal transitions (2025), arXiv:2505.01491 [cond-mat.str-el]

    2025

  39. [46]

    N. G. Jones, R. Thorngren, R. Verresen, and A. Prakash, Charge pumps, pivot hamiltonians and symmetry- protected topological phases (2025), arXiv:2507.00995 [cond-mat.str-el]

    2025

  40. [47]

    Kubota, Stable homotopy theory of invertible gapped quantum spin systems i: Kitaev’s ω-spectrum (2025), arXiv:2503.12618 [math-ph]

    Y. Kubota, Stable homotopy theory of invertible gapped quantum spin systems i: Kitaev’s ω-spectrum (2025), arXiv:2503.12618 [math-ph]

    2025

  41. [48]

    N. Doll, H. Schulz-Baldes, and N. Waterstraat,Spectral flow: A functional analytic and index-theoretic approach (De Gruyter, 2023)

    2023

  42. [49]

    Ji and X.-G

    W. Ji and X.-G. Wen, Categorical symmetry and nonin- vertible anomaly in symmetry-breaking and topological phase transitions, Phys. Rev. Res.2, 033417 (2020)

    2020

  43. [50]

    D. S. Freed, G. W. Moore, and C. Teleman, Topo- logical symmetry in quantum field theory (2024), arXiv:2209.07471 [hep-th]

    2024 arXiv

  44. [51]

    Huang and M

    S.-J. Huang and M. Cheng, Topological holography, quantum criticality, and boundary states, arXiv e-prints , arXiv:2310.16878 (2023), arXiv:2310.16878 [cond- mat.str-el]

    2023 arXiv

  45. [52]

    D. J. Thouless, Quantization of particle transport, Phys. Rev. B27, 6083 (1983)

    1983

  46. [53]

    Hori,Mirror symmetry, Vol

    K. Hori,Mirror symmetry, Vol. 1 (American Mathemat- ical Soc., 2003)

    2003

  47. [54]

    A. G. Abanov and P. B. Wiegmann, Theta-terms in non- linear sigma-models, Nuclear Physics B570, 685 (2000), arXiv:hep-th/9911025 [hep-th]

    2000 arXiv

  48. [55]

    Our convention ofH is slightly different from those in Ref.[15, 54]

  49. [57]

    Miyaji, S

    M. Miyaji, S. Ryu, T. Takayanagi, and X. Wen, Bound- ary states as holographic duals of trivial spacetimes (2014), arXiv:1412.6226 [hep-th]

    2014 arXiv

  50. [58]

    G. Y. Cho, A. W. W. Ludwig, and S. Ryu, Universal en- tanglement spectra of gapped one-dimensional field theo- ries, Physical Review B95, 10.1103/physrevb.95.115122 (2017)

    2017 doi

  51. [59]

    G. Y. Cho, K. Shiozaki, S. Ryu, and A. W. W. Lud- wig, Relationship between symmetry protected topologi- cal phases and boundary conformal field theories via the entanglement spectrum, Journal of Physics A: Mathe- matical and Theoretical50, 304002 (2017)

    2017

  52. [60]

    Cardy, Bulk renormalization group flows and bound- ary states in conformal field theories, SciPost Physics3, 10.21468/scipostphys.3.2.011 (2017)

    J. Cardy, Bulk renormalization group flows and bound- ary states in conformal field theories, SciPost Physics3, 10.21468/scipostphys.3.2.011 (2017)

    2017 doi

  53. [61]

    B. Han, A. Tiwari, C.-T. Hsieh, and S. Ryu, Boundary conformal field theory and symmetry-protected topolog- ical phases in2 + 1dimensions, Phys. Rev. B96, 125105 (2017)

    2017

  54. [62]

    Li, C.-T

    L. Li, C.-T. Hsieh, Y. Yao, and M. Oshikawa, Boundary conditions and anomalies of conformal field theories in 1 + 1dimensions, Phys. Rev. B110, 045118 (2024)

    2024

  55. [63]

    Cardy and E

    J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, Journal of Statis- tical Mechanics: Theory and Experiment2016, 123103 (2016)

    2016

  56. [64]

    X.-L. Qi, H. Katsura, and A. W. W. Ludwig, General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states, Phys. Rev. Lett.108, 196402 (2012)

    2012

  57. [65]

    Das and S

    D. Das and S. Datta, Universal features of left-right en- tanglement entropy, Phys. Rev. Lett.115, 131602 (2015)

    2015

  58. [66]

    X. Wen, S. Matsuura, and S. Ryu, Edge theory approach to topological entanglement entropy, mutual informa- tion, and entanglement negativity in chern-simons theo- ries, Physical Review B93, 10.1103/physrevb.93.245140 (2016)

    2016 doi

  59. [67]

    C.-Y.LoandX.Wen,DetectinghigherBerryphaseswith boundary scattering (in preparation)

  60. [68]

    Y. Choi, H. Ha, D. Kim, Y. Kusuki, S. Ohyama, and S. Ryu, Higher Structures on Boundary Conformal Man- ifolds: Higher Berry Phase and Boundary Conformal Field Theory (To appear). 8

  61. [69]

    Datta, Electronic transport in mesoscopic systems (Cambridge university press, 1997)

    S. Datta, Electronic transport in mesoscopic systems (Cambridge university press, 1997)

    1997

  62. [70]

    unfolding

    R.-Z. Huang, L. Zhang, A. M. Lauchli, J. Haegeman, F. Verstraete, and L. Vanderstraeten, Emergent con- formal boundaries from finite-entanglement scaling in matrix product states, Physical Review Letters 132, 10.1103/physrevlett.132.086503 (2024). Appendix A: From boundary sca...

    2024 doi

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