Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals

Alexander Avdoshkin; Joel E. Moore; Johannes Mitscherling

Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals

T0 review · 0 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that second-order DC photocurrents and the third polarization cumulant reduce to gauge-invariant two-state projector tensors, generalizing established formulas to degenerate bands and metals.

desk verdict A solid, checkable methods paper; the flagged degenerate-cumulant tension doesn't hold up, but Eq. (77) is asserted rather than derived for degenerate bands. read the letter →

arxiv 2412.03637 v2 pith:3PAXJMO2 submitted 2024-12-04 cond-mat.str-el cond-mat.mes-hallcond-mat.mtrl-sciquant-ph

Johannes Mitscherling , Alexander Avdoshkin , Joel E. Moore This is my paper · ORCID

classification cond-mat.str-elcond-mat.mes-hallcond-mat.mtrl-sciquant-ph

keywords quantumstategeometryprojectorcalculusgaugeinvarianceinjectioncurrentshiftgeometricconnectiontorsiontensorpolarizationcumulants

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 argues that the geometric information controlling several crystal observables can be organized by gauge-invariant projectors onto Bloch bands, without ever choosing wavefunction phases. The key claim is that second-order DC photocurrents, the injection current and the shift current, can be written exactly as sums over band pairs of simple spectral factors times two-state geometric tensors built from projectors. The paper also shows that the third cumulant of the polarization distribution equals the volume times the Brillouin-zone integral of the imaginary part of the quantum geometric connection. If these results are correct, formulas previously known for non-degenerate bands extend cleanly to degenerate bands and to metals with Fermi-function weights, and a single formalism unifies polarization theory with nonlinear optical response.

What carries the argument

The load-bearing objects are the band projectors $\hat P_n(k)$ and the two-state tensors $Q^{mn}_{\alpha\beta} = \operatorname{tr}[\hat P_n (\partial_\alpha \hat P_m)(\partial_\beta \hat P_n)]$ and $C^{mn}_{\alpha;\beta\gamma} = \operatorname{tr}[\hat P_n(\partial_\beta \hat P_m)((\partial_\alpha\partial_\gamma \hat P_n)+(\partial_\alpha \hat P_m)(\partial_\gamma \hat P_n))]$. Because projectors are built from bra-ket outer products, they are invariant under U(1) and U(M) gauge transformations, so derivatives are well defined without gauge fixing. The paper supplies algebraic identities, such as trace-reversal rules, vanishing projector combinations, and decompositions of Hamiltonian derivatives, that let one reduce the two-lifetime conductivity expression into sums of geometric tensors multiplied by spectral factors. It further shows how the two-state connection decomposes into quantum metric and Berry curvature dipoles plus a torsion tensor, and how the interband Wilson loop and shift vector emerge from the same objects.

What would settle it

Compute the full two-lifetime conductivity and the simplified projector expressions for a two-band model with a tunable direct gap; for $\gamma/|\epsilon_{mn}|$ around 0.1 or larger, any discrepancy in the $1/\gamma$-divergent injection current or in the shift current would show the limit's scope. Alternatively, evaluate the third-cumulant identity for a model with degenerate bands against the independent definition of the cumulant.

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Extended reading notes

Core claim

At a fixed momentum, the central claim is that the local multi-state geometry of Bloch states is captured by a small set of projector objects: the single-state and two-state quantum geometric tensors, the quantum geometric connection, and the torsion tensor, all defined as traces of projector derivatives. These objects are gauge invariant by construction and identical in form for degenerate and non-degenerate bands, provided projectors onto degenerate subspaces are used. Applied to the two-lifetime model of optical response, the paper derives that, in leading order in $\gamma/|\epsilon_{mn}| \ll 1$, the injection current is proportional to $\delta(\omega-\epsilon_{nm}) f_{nm} (\partial_a \epsilon_{nm}) Q^{mn}_{bc}$, and the shift current to $i\pi$ times the analogous sum of $(C^{mn}_{a;cb}-C^{nm}_{a;bc})$, where $Q$ and $C$ are the two-state quantum geometric tensor and connection. The same formalism yields the third cumulant of the polarization distribution as the Brillouin-zone integral of $\operatorname{Im} Q_{\alpha;\beta\gamma}$. These expressions generalize prior photocurrent formulas to degenerate bands and metals, and they are exact within the stated limit.

Load-bearing premise

The derivation assumes the scattering rate is much smaller than every direct energy gap between bands, so an operator can be replaced by a simple complement projector; near-degenerate or small-gap bands violate this.

Editorial extensions

If this is right

Injection and shift currents can be computed numerically from projectors alone, eliminating gauge-fixing and simplifying material calculations.
The same two-state tensor objects appear in both the shift current and the polarization cumulants, making the geometric link between the two observables explicit.
Degenerate bands are treated on equal footing with non-degenerate bands through projectors of rank $M$, so the formulas apply to materials with band degeneracies and to metals with Fermi-function weights.
In two-band systems the two-state quantities reduce to single-state ones and the torsion vanishes, so the formalism automatically recovers known simpler limits.

Reading between the lines

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

A direct test would be to evaluate the simplified injection and shift current formulas against the full two-lifetime conductivity in a small-gap model, a check the paper does not perform and which would quantify the validity range of $\gamma \ll |\epsilon_{mn}|$.
Because the formulas are exact geometric decompositions at leading order in $\gamma$, subleading corrections could produce measurable signatures in materials with near-degenerate bands, signatures not present in the non-degenerate limit.
The same projector strategy may extend to third-order optical responses or to interacting systems through the exterior-power representation of Slater determinants, although the paper only sketches those directions.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

This is a careful methods paper that delivers on its promise: the projector calculus for multi-state quantum geometry is laid out in far more detail than in the companion paper, and the two applications—polarization cumulants and injection/shift currents—are worked through with enough intermediate steps that the algebra is checkable. The final photocurrent formulas in Eqs. (108)–(109) reproduce the benchmarks from Ahn et al. and Sipe–Shkrebtii, which is the right kind of validation for a paper whose main product is a tool.

On the stress-test note: I don't think the flagged tension survives a close read. The sentence after Eq. (79) says deviations in the expansion of A^k_α(k+q) between non-degenerate and degenerate cases start at "the third order." That is third order in q, which feeds the fourth cumulant, not the third. So the caveat is consistent with Eq. (79), which only depends on the q^2 term. The legitimate residue of the concern is that Eq. (77) for degenerate bands is asserted rather than derived here; the paper points to the companion and to Ref. 12, but a few lines showing how the matrix-valued inverse in Eq. (76) reduces to Q_{α;βγ} at order q^2 would have closed the gap. That is a presentation gap, not a load-bearing flaw.

The γ ≪ |ε| approximation in Sec. III B is the more substantive caveat. The paper is upfront that Eq. (87) replaces the complement projector with 1-P_n, valid only when γ is small compared to the direct gap. No quantitative error bound or small-gap test is given, so the injection-current formula should be used with care near degeneracies. It's a limitation, not an error; the benchmarks match where the limit applies.

The novelty is modest—most concepts are in the companion paper, and Ahn et al. already had the two-state quantum geometric tensor and its shift-current link. What this paper adds is a complete algebra, clean projector identities, and worked examples. That is genuinely useful for anyone computing geometric objects without dealing with gauge-dependent wavefunctions.

The paper deserves a serious referee and likely publication. I'd ask the authors to either derive or more explicitly cite the derivation of Eq. (77) for degenerate bands. It's a solid, reproducible contribution.

Referee Report

0 major / 4 minor

Summary. The paper develops a gauge-invariant projector calculus for quantum state geometry, introducing single-state and two-state geometric invariants such as the quantum geometric tensor, quantum geometric connection, torsion tensor, and shift vector. It applies this formalism to two physical problems: the relation between the third cumulant of the polarization distribution and the imaginary part of the quantum geometric connection (Eq. 79), and a derivation of the injection and shift currents in terms of two-state quantum geometric objects (Eqs. 108-109). The paper claims these results generalize known non-degenerate formulas to degenerate bands and to metals with Fermi-function weights.

Significance. The projector formalism is clean, gauge-invariant, and avoids explicit gauge fixing, which is a substantial practical advantage for both analytic and numerical work. The derivation in Section III B is explicit and the technical steps are relegated to appendices; the final photocurrent formulas agree with the benchmark results of Ref. 22, and the shift vector reduces to the standard expression in Appendix C. The paper also provides useful numerical recipes for evaluating projectors and their derivatives (Appendix A) and a systematic set of projector identities (Section II G). If the claimed generalization to degenerate bands is correct, it constitutes an important extension of quantum-geometric response theory. The paper is carefully written and the central derivations are sound.

minor comments (4)

[III A 2] The sentence following Eq. (79) states that deviations in the expansion of A^k_α(k+q) are expected to start in the third order. Since the expansion in Eq. (77) is in powers of q, 'third order' means the q^3 term (i.e., the fourth cumulant), not the third cumulant. As written, the statement is ambiguous and could be misread as contradicting Eq. (79). Please rephrase to specify 'third order in q' or 'starting with the fourth cumulant'.
[III B, Eq. (87)] The approximation Q^(γ)_n ≈ 1 - P_n is valid when P_n is understood as the projector onto a (possibly degenerate) energy eigenspace, so that the gaps ε_mn are between distinct eigenspaces. If P_n were taken as an individual band projector within a degenerate subspace, the approximation would fail because ε_mn = 0 for degenerate partners. Please add a clarifying remark to this effect.
[III A 2 / Appendix D] The expansion of the matrix-valued inverse in Eq. (76) to quadratic order in q is not shown explicitly for degenerate projectors. While the second-order correction vanishes in the trace due to identity (50), an explicit demonstration would strengthen the derivation of Eq. (79).
[Appendix A] The finite-difference formulas for projector derivatives are useful, but the choice of step size λ is only briefly mentioned. A short discussion of how to estimate the numerical error using the projector identities (50)-(52) would improve the practical guidance.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the photocurrent and polarization formulas are obtained by algebraic reduction from independent inputs, with only non-load-bearing self-citations.

full rationale

The central derivations are self-contained. The injection and shift current expressions, Eqs. (108) and (109), start from the conductivity formula of Holder et al. (Ref. 25), an external input written in terms of Bloch-Hamiltonian derivatives, and are reduced by the projector identities of Sec. II G; the final objects Q^mn and C^mn were defined independently in Sec. II F before the reduction. The results are benchmarked against external works (Refs. 22 and 46), so the geometric re-expression is not a prediction forced by a fit. Likewise, Eq. (79) for the third cumulant follows from the generating function Eq. (75), whose derivation from Eq. (74) is reproduced in Appendix D; the identification of the expansion coefficients with Q_ab and Q_a;bc is algebraic, not definitional in a circular sense. The same-author citations that could raise a flag are Ref. 12 (minimality of three-point functions) and the companion Ref. 7, but neither is load-bearing: the photocurrent derivation does not invoke those uniqueness statements, and the polarization result is rederived here. The gamma << |eps_mn| limit in Eq. (87) is an explicit asymptotic assumption, and the caveat in Sec. III A 2 that "we expect deviations in the expansion of A^k_a(k+q) between the non-degenerate and degenerate cases starting in the third order" concerns q^3-and-higher terms in A^k (i.e., fourth and higher cumulants), not the q^2 coefficient used for the third cumulant in Eq. (79); in any case it is a validity caveat, not a circularity. No step was found in which a defined quantity is identical by construction to the predicted quantity, or in which a fitted parameter is renamed as a prediction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central formulas rest on the non-interacting projector description, an imported conductivity expression, and the leading-order gamma approximation. No free parameters are fitted to data; gamma and Gamma are phenomenological inputs. No new physical entities are postulated.

free parameters (2)

intraband relaxation rate gamma
Phenomenological constant entering Eq. (82); the injection current in Eq. (108) is proportional to 1/gamma. It is not fitted here, but the final formula's domain of validity depends on gamma being much smaller than the band gaps.
interband relaxation rate Gamma
Phenomenological constant entering Eq. (82); the resonant formulas are obtained in the delta-function limit, so Gamma drops out of Eqs. (108) and (109).

assumptions (6)

domain assumption The electrons are described by a non-interacting Bloch Hamiltonian with projectors satisfying idempotence, orthogonality and completeness.
Stated in Sec. II; all derivations rely on this single-particle band structure.
domain assumption The expression for the second-order conductivity sigma_{a;bc} from Ref. 25, with two independent relaxation rates gamma and Gamma, is a valid starting point.
Eq. (82) is imported without re-derivation; it defines the observable being decomposed.
domain assumption The intraband relaxation rate is much smaller than all relevant band gaps, gamma much less than |epsilon_mn|.
Used in Eq. (87) to replace Q^(gamma)_n by 1 - P_n and in Appendix E to drop subleading terms.
standard math Standard trace identities and matrix calculus, including cyclic trace, completeness relations, and Jacobi's formula for det derivatives, are valid.
Used throughout Sec. II G and Appendix D.
domain assumption The polarization ground state is a Slater determinant, so its generating function depends only on the occupied subspace via the Plucker embedding.
Sec. III A assumes a non-interacting insulator; the determinant structure is essential for Eqs. (74) and (75).
domain assumption The Fermi function f_n is treated as momentum-independent when integrating by parts in Appendix E, so the boundary term for sigma^(0) vanishes.
Appendix E, second paragraph: 'assuming a momentum-constant f_n'.

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Pith. "Pith review of Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals." pith.science (2026). https://pith.science/paper/3PAXJMO2

@misc{pith2026241203637,
  author       = {Pith},
  title        = {Pith review of: Gauge-invariant projector calculus for quantum state geometry and applications to observables in crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3PAXJMO2}},
  note         = {Machine review of arXiv:2412.03637}
}

read the original abstract

The importance of simple geometrical invariants, such as the Berry curvature and quantum metric, constructed from the Bloch states of a crystal has become well-established over four decades of research. More complex aspects of geometry emerge in properties linking multiple bands, such as optical responses. In the companion work [arXiv:2409.16358], we identified novel multi-state geometrical invariants using an explicitly gauge-invariant formalism based on projection operators, which we used to clarify the relation between the shift current and the theory of electronic polarization among other advancements for second-order non-linear optics. Here, we provide considerably more detail on the projector formalism and the geometrical invariants arising in the vicinity of a specific value of crystal momentum. We combine the introduction to multi-state quantum geometry with broadly relevant algebraic relationships and detailed example calculations, enabling extensions toward future applications to topological and geometrical properties of insulators and metals.

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Reference graph

Works this paper leans on

105 extracted references · 59 canonical work pages · cited by 7 Pith papers

[1]

Quantum geometric tensor The (single-state) quantum geometric tensor in projec- tor form reads Qαβ ≡ tr [ ˆP (∂α ˆP ) (∂β ˆP ) ] , (16) 5 involving a projector onto a (non-)degenerate band, occu- pied states, or other sets of quantum states. Besides the quantum geometric tensor expressed in the Bloch states for a non-degenerate band, see Appendix C, the p...
[2]

Quantum geometric connection We continue by considering the second local geometric invariant in Eq. ( 15). For a given projector ˆP we have the (single-state) quantum geometric connection Qα ;βγ ≡ tr [ ˆP ( ∂α ˆP )( ∂β∂γ ˆP ) ] , (27) which involves a ﬁrst- and second-order derivative of the projector. Note that Qα ;βγ =Qα ;γβ . The quantum geo- metric co...
[3]

These projec- tors explicitly respect the required U (M ) gauge invari- ance of the degenerate band

to ( 6) hold. These projec- tors explicitly respect the required U (M ) gauge invari- ance of the degenerate band. In addition, it might be convenient to combine multiple bands n1 tonM into one subspace of quantum states of interest, ˆP(n1...n M )(k) = M∑ i=1 ˆPni (k), (11) which still satisfy the projector properties (
[4]

The most common example is the projector onto occupied states, which takes the form ˆPocc(k) =∑ n∈occ ˆPn(k)

and (4). The most common example is the projector onto occupied states, which takes the form ˆPocc(k) =∑ n∈occ ˆPn(k). C. Global and local geometric invariants Being gauge invariant and hermitian, projectors are, in principle, measurable and, thus, oﬀer minimal build- ing blocks to construct observables. The projectors ˆP (k) constructed from a given Bloc...
[5]

While one can still deﬁne the quan- tum geometric tensor according to Eq

Non-Abelian quantum geometric tensor A single m-degenerate state is represented by a pro- jector ˆP of rank m. While one can still deﬁne the quan- tum geometric tensor according to Eq. ( 16), it does not exhaust all geometric information contained in the pro- jector. A convenient way of representing the additional information is by considering its non-Abe...
[6]

and ( 54). It might be convenient to keep derivatives of the Hamiltonian explicit in order to avoid, at that point of the derivation, lengthy expressions aris- ing from identities such as those given in Eqs. (

work page
[7]

and explained how they characterize aspects of the Bloch states that go beyond the quantum metric and Berry cur- vature

In particular, we system- atically developed the theory of these geometrical objects ∗ These authors contributed equally. and explained how they characterize aspects of the Bloch states that go beyond the quantum metric and Berry cur- vature. In the following, we provide a detailed introduc- tion to local single- and multi-state geometric invariants in co...
[8]

(24) which we expressed in terms of ˆemn α =i ˆPm ( ∂α ˆPn ) ˆPn

and a purely two-state contribution, Qmn αβ =δnmQn αβ − tr [ ˆenm α ˆemn β ] . (24) which we expressed in terms of ˆemn α =i ˆPm ( ∂α ˆPn ) ˆPn. (25) Note that ˆenn α = 0 individually, which follows from pro- jector identity ( 4). The purely two-state contribution tr[ˆenm α ˆemn β ] yields the product of non-Abelian Berry con- nections, see Appendix C, an...

Show all 105 references

[9]

Graf and F

A. Graf and F. Pi´ echon, Berry curvature and quantum metric in N -band systems: An eigenprojector approach, Phys. Rev. B 104, 085114 (2021)

2021
[10]

Mera and T

B. Mera and T. Ozawa, K¨ ahler geometry and Chern in- sulators: Relations between topology and the quantum metric, Phys. Rev. B 104, 045104 (2021)

2021
[11]

Mera and J

B. Mera and J. Mitscherling, Nontrivial quantum geometry of degenerate ﬂat bands, Phys. Rev. B 106, 165133 (2022)

2022
[12]

Expressions like Eq

involving projectors at arbitrar- ily separated points on the Brillouin zone are called global geometric invariants. Expressions like Eq. ( 13) involving derivatives of projectors, that is, only inﬁnitesimally sep- arated in k, are called local geometric invariants . Both shou...
[13]

Torsion tensor The two-state quantum geometric connection yields novel geometric information about the Bloch state man- ifold in systems with more than two bands, the torsion. Following our insights presented in the joint submission [ 7], we introduce the torsion tensor in pro...
[14]

and ( 6); see Sec. II G 2. A cyclic summation of C(mn) α ;[βγ ] is proportional to the real part of the cyclic summation of the torsion tensor T mn α ;βγ , which relates the torsion tensor to the circular shift current and oﬀers a novel path for quantized non- linear optical r...
[15]

The interband Wilson loop reads W mn αβ (k, q) = tr [ ˆenm β (k) ˆemn α (k + q) ] , (37) involving the transition dipole moments in Eq

Interband Wilson loop and shift vector We present the connection between the projector for- malism and the gauge-invariant Pancharatnam-Berry phase over an interband loop, which has recently been introduced in the shift vector description of the shift cur- rent response [ 43–4...
[16]

The intrinsic objects for degenerate states are described in Sec

and ( 15). The intrinsic objects for degenerate states are described in Sec. II F 5. The geometry of multi-state objects is not yet suﬃciently un- derstood for separation into intrinsic and extrinsic in- variants. Progress can be made by considering a Taylor expansion of globa...
[17]

(23) We note that the two-state quantum geometric tensor decomposes into the diagonal part given by the single- state quantum geometric tensor given in Eq

into its symmetric and antisymmetric contribution naturally generalizes the quantum metric and Berry curvature to their two-state quantities, i.e., Qmn αβ =gmn αβ − i 2 Ω mn αβ (21) with gmn αβ ≡ 1 2 tr [ ( ∂α ˆPm(k) )( ∂β ˆPn(k) ) ] , (22) Ω mn αβ ≡i tr [ ˆPn ( ∂α ˆPm )( ∂β ˆ...
[18]

is recovered by tracing its non- Abelian version, i.e., Qαβ = tr Qαβ . The antisymmetric part of Qαβ relates to the non- Abelian Berry curvature Fαβ =i (Qαβ − Qβα ) (43) =∂αAβ −∂βAα − [Aα,A β ], (44) where Aα is the non-Abelian Berry connection, also known as the Wilczek-Zee c...
[19]

We focus on those identities relevant for the evaluations throughout this pa- per and the joint submission [ 7]

to ( 6). We focus on those identities relevant for the evaluations throughout this pa- per and the joint submission [ 7]. Further generalizations are straightforwardly obtained. We omit the momentum dependence throughout this section, i.e., ˆP ≡ ˆP (k)
[20]

ﬁxing the relation Qnn αβ = Qn αβ via the commonly used deﬁnition in Eq. ( 16). We con- clude by noticing that gmn αβ = g(mn) (αβ ) and Ω mn αβ = Ω [mn] [αβ ] forn ⁄=m, where we denote the symmetrization and an- tisymmetrization of the indices as ( αβ) and [αβ], respec- tively...
[21]

ˆPN ] = tr [ ˆP2

Trace manipulations and complex conjugation The cyclic property of the trace allows the cyclic per- mutation of the involved projectors, tr [ ˆP1 ˆP2... ˆPN ] = tr [ ˆP2... ˆPN ˆP1 ] , (47) and further operators such as the Bloch Hamiltonian. The invariance of the trace under ...
[22]

The remaining C(mn) α ;[βγ ] and C[mn] α ;(βγ ) deter- mine the shift current under linear and circular polarized illumination, respectively; see Ref

and Berry curvature ( 23), respectively. The remaining C(mn) α ;[βγ ] and C[mn] α ;(βγ ) deter- mine the shift current under linear and circular polarized illumination, respectively; see Ref. [ 7] and Sec. III B for a detailed discussion. The real and imaginary parts of Cmn α ...
[23]

ˆPN ] = tr [ ˆPN

lead to tr [ ˆP1 ˆP2... ˆPN ] = tr [ ˆPN... ˆP2 ˆP1 ] , (48) where the overline denotes complex conjugation. We note that complex conjugation eﬀectively reverses the order of the projectors under the trace. The same identity holds when projector derivatives and other Hermitian...
[24]

(49) Note that the projector and its derivative do not com- mute in general

Vanishing projector combinations The idempotence and orthogonality of the projectors formalized in the identity ( 4) directly implies ∂α ˆP = ˆP (∂α ˆP ) + (∂α ˆP ) ˆP . (49) Note that the projector and its derivative do not com- mute in general. This identity implies the vani...
[25]

Note that the interband Wilson loop vanishes for n = m

at separated momenta. Note that the interband Wilson loop vanishes for n = m. It reduces to the estab- lished deﬁnition [ 44] for non-degenerate bands; see Ap- pendix C 3. Expanding in small q leads to the two-state quantum geometric tensor and quantum geometric con- nection, ...
[26]

that ˆPm ( ∂α ˆPl ) ˆPn = 0 (53) for l ⁄=n,m
[27]

Derivatives of the Bloch Hamiltonian In the derivation of response functions, derivatives of the Bloch Hamiltonian are usually present. We summa- rize the most important identities arising from projector property ( 5), or equivalently, ˆH = ∑ n En ˆPn (54) where we omit the mo...
[28]

(57) where we denote the two band projectors as ˆP±

Two-band systems We have a closer look at the case of a two-band system with non-degenerate bands, which are highly constraint via the completeness relation ( 6), i.e., ˆP+ + ˆP− = 12. (57) where we denote the two band projectors as ˆP±. This relation implies ∂α ˆP+ = −∂α ˆP− ...
[29]

However, this does not generalize to arbitrary subsets

and ( 33) for a two-band system. However, this does not generalize to arbitrary subsets. As we discuss in the joint submission [ 7], the summed-over two-state quantum geometric connection ∑ n∈occ m∈unocc Cmn αβ does not, in general, reduce to a ground state property, i.e., can...
[30]

    

Slater determinants and the Pl¨ ucker map The Slater determinant is a natural way of construct- ing a many-body fermionic state out of a collection of m single-body states |ψi⟩ via Ψ(x1,x 2,x 3, · · · ) = 1 m! det      ψ1(x1) ψ2(x1) ψ3(x1) · · · ψ1(x2) ψ2(x2) ψ3(x2) · · ·...
[31]

The corresponding ground state wavefunction is |Ψ ⟩ =∏ n, kun(k) ˆc† nk|0⟩ with vacuum |0⟩ and fermionic cre- ation operators ˆ c† nk

Polarization Consider an insulator with Nocc ﬁlled bands. The corresponding ground state wavefunction is |Ψ ⟩ =∏ n, kun(k) ˆc† nk|0⟩ with vacuum |0⟩ and fermionic cre- ation operators ˆ c† nk. We notice that eiq· ˆX|Ψ ⟩ = 10 ∏ n, kun(k) ˆc† n, k−q|0⟩. Thus, the generating func...
[32]

and (27) with respect to the projector onto the Nocc ﬁlled bands for the linear and quadratic term of the expan- sion, respectively. This leads to the explicit expressions for the second and third cumulants of the polarization distribution, ⟨XαXβ ⟩c =V ∫ BZ ReQαβ (k), (78) ⟨Xα...
[33]

Instead, we expect deviations in the expansion of Ak α (k+q) between the non-degenerate and degenerate cases starting in the third order

by replacing the non-degenerate with the degenerate projectors. Instead, we expect deviations in the expansion of Ak α (k+q) between the non-degenerate and degenerate cases starting in the third order. B. The injection and shift currents We demonstrate the projector calculus f...
[34]

We provided a detailed discussion on anticipated further applications of the formalism within the com- panion article [ 7]

General strategy Before performing the derivation in detail, we sketch the main steps, which we anticipate to apply to other observables as well, as long as the starting point is sim- ilar. We provided a detailed discussion on anticipated further applications of the formalism ...
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(82) We notice that it is suﬃcient to evaluate Babc(ω) ≡ tr [ ˆPocc [[ ∂a ˆH, ∂b ˆH ǫ −iγ ] , ∂c ˆH ω +ǫ +iΓ ] ] , (84) sinceσa;bc (2) (ω) = Bcab(ω) +Bbac(−ω)

Step-by-step evaluation of Eq. (82) We notice that it is suﬃcient to evaluate Babc(ω) ≡ tr [ ˆPocc [[ ∂a ˆH, ∂b ˆH ǫ −iγ ] , ∂c ˆH ω +ǫ +iΓ ] ] , (84) sinceσa;bc (2) (ω) = Bcab(ω) +Bbac(−ω). Following the ﬁrst step of the general strategy, we focus on the two fractions involvi...
[38]

Using the identity (

Combining all contributions to identify the relevant geometric invariants We start with the term that cannot be further re- duced. Using the identity (
[39]

II G 2, we identity the two-state quantum geometric tensor, i.e., tr [ ( ∂c ˆH ) ˆPn ( ∂b ˆPm ) ˆPm ] =ǫnmQnm cb

to evaluate the deriva- tive of the Bloch Hamiltonian and further simplify the resulting combinations of projectors via our results in Sec. II G 2, we identity the two-state quantum geometric tensor, i.e., tr [ ( ∂c ˆH ) ˆPn ( ∂b ˆPm ) ˆPm ] =ǫnmQnm cb . (101) Note that the ba...
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When diagonalized for a ﬁxed momentum, we obtain the eigenvalues En(k) and corresponding orthonormal eigenvectors |un(k)⟩

Numerical projector construction Consider a Bloch Hamiltonian Norb × Norb matrix ˆH(k) for Norb orbitals as a function of momentum k. When diagonalized for a ﬁxed momentum, we obtain the eigenvalues En(k) and corresponding orthonormal eigenvectors |un(k)⟩. For each index n, we...
[43]

If necessary, projectors onto degenerate or multiple bands are constructed by summing the respective ˆPn

to ( 5). If necessary, projectors onto degenerate or multiple bands are constructed by summing the respective ˆPn. No speciﬁc gauge choice is required for |un⟩ within this construction as long as ⟨un| is directly obtained from the corresponding |un⟩ by complex transposition
[44]

The ﬁrst derivative of the projector is obtained by the symmetric ﬁnite diﬀerence, ∂α ˆPn(k) = 1 2λ [ ˆPn ( k +λ eα ) − ˆPn ( k −λ eα ) ] + O(λ2), (A1) with an error of order λ2

Numerical derivative construction We denote the momentum unit vector in direction α as eα . The ﬁrst derivative of the projector is obtained by the symmetric ﬁnite diﬀerence, ∂α ˆPn(k) = 1 2λ [ ˆPn ( k +λ eα ) − ˆPn ( k −λ eα ) ] + O(λ2), (A1) with an error of order λ2. A comp...
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Quantum geometric tensor Expressed in terms of Bloch states, the quantum geo- metric tensor in Eq. (
[47]

and (48) are essential in the evaluation of diagrammatic ex- pansions; see, e.g., [ 47] and Sec. III B
[48]

takes the familiar form Qn αβ = ⟨∂αun|∂βun⟩ + ⟨un|∂αun⟩⟨un|∂βun⟩, (C1) from which the quantum metric and Berry curvature ex- pressed in Bloch states is straightforwardly derived. In order to derive the two-state quantum geometric tensor in terms of Bloch states we ﬁrst obtain ...
[49]

( 27), we obtain Qn α ;βγ = 1 2 ⟨∂αun|∂β∂γun⟩ + 1 2 ⟨un|∂αun⟩⟨un|∂β∂γun⟩ − ⟨∂αun|∂βun⟩⟨un|∂γun⟩ − ⟨un|∂αun⟩⟨un|∂βun⟩⟨un|∂γun⟩ + (β ↔γ)

Quantum geometric connection Inserting the explicit form of band projector onto a non-degenerate band into Eq. ( 27), we obtain Qn α ;βγ = 1 2 ⟨∂αun|∂β∂γun⟩ + 1 2 ⟨un|∂αun⟩⟨un|∂β∂γun⟩ − ⟨∂αun|∂βun⟩⟨un|∂γun⟩ − ⟨un|∂αun⟩⟨un|∂βun⟩⟨un|∂γun⟩ + (β ↔γ). (C6) The expression shows expl...
[50]

Appendix B: Strategy to determine closed analytic forms for few-band systems We describe how to relate the results of Ref

to ( 52). Appendix B: Strategy to determine closed analytic forms for few-band systems We describe how to relate the results of Ref. 8 to the presented formalism. Let us consider the generators ˆMα of SU(N) and expand the Bloch Hamiltonian, ˆH =h01N + ∑ α hα ˆMα (B1) where we ...
[51]

for n ⁄=m, we use Eq. ( C2) and obtain Cmn α ;βγ = −tr [ ˆenm β ∂α ˆemn γ ] (C7) = −(1 −δnm)2rβ nmtr [ |un⟩⟨um|∂α ( rγ mn|um⟩⟨un| )] (C8) = −(1 −δnm)rβ nm ( ∂α rγ mn −i ( ξα m −ξα n ) rγ mn ) (C9) = −(1 −δnm)rβ nmrγ mn;α (C10) with band Berry connection ξα n =rα nn =i⟨un|∂αun⟩...
[52]

Interband Wilson loop and shift vector We derive the explicit expression for the interband Wilson loop in Eq. (
[53]

and ˆPm(∂α ˆPm) ˆPn = − ˆPm(∂α ˆPn) ˆPn. Similarly, we obtain the decomposition of the second-order Hamil- tonian derivative ˆPm ( ∂α∂β ˆH ) ˆPn = 1 2ǫnm ˆPm ( ∂α∂β ˆPn ) ˆPn + ( ∂αǫnm ) ˆPm ( ∂β ˆPn ) ˆPn −Em ˆPm ( ∂α ˆPm )( ∂β ˆPn ) ˆPn − ∑ l⁄=n,m El ˆPm ( ∂α ˆPl ) ˆPl ( ∂β ...
[54]

for non-degenerate bands using 16 Eq. ( C2). We obtain W mn αβ (k, q) = tr [ ˆenm β (k) ˆemn α (k + q) ] (C11) = (1 −δnm)rβ nm(k)rα mn(k + q) × tr [ |un(k)⟩⟨un(k)|um(k + q)⟩⟨um(k + q)| ] (C12) = (1 −δnm)rα mn(k + q)rβ nm(k) × ⟨un(k)|um(k + q)⟩⟨un(k + q)|un(k)⟩, (C13) where we ...
[55]

(Step 2) We use projector identities such as those presented in Sec

and (56). (Step 2) We use projector identities such as those presented in Sec. II G to simplify the individual (prelim- inary) geometric contributions, for instance, when per- forming the commutators. This generically leads to con- straints for the band summations, further sim...
[56]

Using the sep- aration of the fraction into two contribution as given in Eq

Vanishing of the contribution σ a;bc (0) We show that σa;bc (0) deﬁned as σa;bc (0) = tr [ ˆPocc ( ∂a∂b∂c ˆH + [ ∂b∂c ˆH, ∂a ˆH −ǫ +iγ ]) ] , (E1) vanishes in leading orders of γ ≪ |ǫnm|. Using the sep- aration of the fraction into two contribution as given in Eq. ( 86), we se...
[57]

( 81), which takes the form σa;bc (1) (ω) = Ab;ac(ω) + Ac;ab(−ω) with Aa;bc(ω) ≡ tr [ ˆPocc [ ∂a ˆH ω +ǫ +iΓ,∂ b∂c ˆH ] ]

Leading-order contributions to σ a;bc (1) (ω ) We evaluate the remaining contribution to Eq. ( 81), which takes the form σa;bc (1) (ω) = Ab;ac(ω) + Ac;ab(−ω) with Aa;bc(ω) ≡ tr [ ˆPocc [ ∂a ˆH ω +ǫ +iΓ,∂ b∂c ˆH ] ] . (E6) We decompose the fraction via Eq. ( 85). The ﬁrst term ...
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as previously reported in Ref. 26. Setting Γ = 0 in Eqs. ( 104) to ( 106) allows to determine oﬀ- resonant contributions to the currents [ 52]. IV. CONCLUSION AND OUTLOOK We have presented a general approach to handle the observable information encoded in the Bloch states of a...

2022

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Quantum geometric tensor The (single-state) quantum geometric tensor in projec- tor form reads Qαβ ≡ tr [ ˆP (∂α ˆP ) (∂β ˆP ) ] , (16) 5 involving a projector onto a (non-)degenerate band, occu- pied states, or other sets of quantum states. Besides the quantum geometric tensor expressed in the Bloch states for a non-degenerate band, see Appendix C, the p...

[2] [2]

Quantum geometric connection We continue by considering the second local geometric invariant in Eq. ( 15). For a given projector ˆP we have the (single-state) quantum geometric connection Qα ;βγ ≡ tr [ ˆP ( ∂α ˆP )( ∂β∂γ ˆP ) ] , (27) which involves a ﬁrst- and second-order derivative of the projector. Note that Qα ;βγ =Qα ;γβ . The quantum geo- metric co...

[3] [3]

These projec- tors explicitly respect the required U (M ) gauge invari- ance of the degenerate band

to ( 6) hold. These projec- tors explicitly respect the required U (M ) gauge invari- ance of the degenerate band. In addition, it might be convenient to combine multiple bands n1 tonM into one subspace of quantum states of interest, ˆP(n1...n M )(k) = M∑ i=1 ˆPni (k), (11) which still satisfy the projector properties (

[4] [4]

The most common example is the projector onto occupied states, which takes the form ˆPocc(k) =∑ n∈occ ˆPn(k)

and (4). The most common example is the projector onto occupied states, which takes the form ˆPocc(k) =∑ n∈occ ˆPn(k). C. Global and local geometric invariants Being gauge invariant and hermitian, projectors are, in principle, measurable and, thus, oﬀer minimal build- ing blocks to construct observables. The projectors ˆP (k) constructed from a given Bloc...

[5] [5]

While one can still deﬁne the quan- tum geometric tensor according to Eq

Non-Abelian quantum geometric tensor A single m-degenerate state is represented by a pro- jector ˆP of rank m. While one can still deﬁne the quan- tum geometric tensor according to Eq. ( 16), it does not exhaust all geometric information contained in the pro- jector. A convenient way of representing the additional information is by considering its non-Abe...

[6] [6]

and ( 54). It might be convenient to keep derivatives of the Hamiltonian explicit in order to avoid, at that point of the derivation, lengthy expressions aris- ing from identities such as those given in Eqs. (

work page

[7] [7]

and explained how they characterize aspects of the Bloch states that go beyond the quantum metric and Berry cur- vature

In particular, we system- atically developed the theory of these geometrical objects ∗ These authors contributed equally. and explained how they characterize aspects of the Bloch states that go beyond the quantum metric and Berry cur- vature. In the following, we provide a detailed introduc- tion to local single- and multi-state geometric invariants in co...

[8] [8]

(24) which we expressed in terms of ˆemn α =i ˆPm ( ∂α ˆPn ) ˆPn

and a purely two-state contribution, Qmn αβ =δnmQn αβ − tr [ ˆenm α ˆemn β ] . (24) which we expressed in terms of ˆemn α =i ˆPm ( ∂α ˆPn ) ˆPn. (25) Note that ˆenn α = 0 individually, which follows from pro- jector identity ( 4). The purely two-state contribution tr[ˆenm α ˆemn β ] yields the product of non-Abelian Berry con- nections, see Appendix C, an...