From Microscopic Lattice Hamiltonians to Effective Theories

The three arrows

We insist on the following factorization:

Here $L$ is a low-energy or spectral-reduction map. The map $I$ extracts a stable invariant. The dashed map $R$ is a realization problem. It is not a formal inverse of $I\circ L$.

The definitions prevent a common reversal error. The existence of an invariant map does not provide a microscopic model for every target class. Agreement of invariants does not imply a microscopic gapped path unless a completeness theorem has been proved in the chosen domain.

Contributions

The paper makes six claims of different strength.

1. We prove a controlled functional-calculus theorem for compact families of bounded free-fermion Hamiltonians with a common Fermi gap.

2. We define a relative operator K-class with respect to a chosen reference Hamiltonian and prove homotopy invariance and stacking additivity.

3. We give a precise retention and loss table for spectral flattening. Positive energy rescaling supplies a simple nonfaithfulness proof.

4. We extend the controlled construction to covariant disordered one-particle systems represented in $C(\Omega)\rtimes\mathbb Z^d$, under a genuine C*-spectral gap.

5. We compute the SSH transition charge on a linking circle and prove that a staggered mass provides a gapped symmetry-breaking detour.

6. We use the BDI interaction reduction to show why the free result has a sharp domain boundary.

The paper does not claim a general microscopic/EFT equivalence. It also does not claim that every bordism or homotopy class has a local gapped lattice representative.

Status vocabulary

Every substantive result is assigned one of the following labels.

The functional-calculus statements below are established. The condensed packaging is proposed. A general lattice/EFT equivalence is open. An unchanged extension of free K-theory to all interacting BDI phases is obstructed.

Microscopic quadratic systems

Let $\Gamma$ be a countable metric lattice and let $\mathcal H$ be the one-particle Hilbert space. In the simplest translation-invariant case, $$\mathcal H=\ell^2(\mathbb Z^d)\otimes\mathbb C^N.$$ A finite-range one-particle Hamiltonian is a bounded self-adjoint operator whose matrix elements vanish beyond a fixed lattice distance. Exponential or summable decay gives broader controlled classes.

After Fourier transform, a translation-invariant finite-range Hamiltonian is a matrix-valued trigonometric polynomial $$h:\mathbb T^d\longrightarrow M_N(\mathbb C), \qquad h(k)^*=h(k).$$ More generally, one works with a self-adjoint element $$h\in M_N(\mathcal A)$$ of a unital C*-algebra $\mathcal A$ encoding position, covariance, disorder, or symmetry.

We set $\mu=0$ after replacing $h$ by $h-\mu$. The algebra and the representation class are part of the problem. Changing them can change the relevant K-group.

Interacting lattice systems

An interacting microscopic system is not generally described by a bounded element $h\in\mathcal A_\Gamma$. It is described by an interaction $$\Phi:X\Subset\Gamma\longmapsto\Phi(X)\in\mathcal A_X^{\mathrm{sa}},$$ finite-volume Hamiltonians, and thermodynamic dynamics. The formal infinite sum of all $\Phi(X)$ need not converge in the quasi-local algebra.

This distinction matters for spectral flattening. Continuous functional calculus applies directly to a bounded self-adjoint element of a C*-algebra. It does not turn an extensive many-body Hamiltonian into a bounded quasi-local observable. Many-body phase equivalence instead uses a uniform finite-volume or bulk gap, quasi-adiabatic continuation, and stabilization by atomic systems.

What counts as an effective theory

The term effective field theory will be used in a restricted way. An effective description records fields, symmetries, couplings, and a regime of momenta or energies in which its correlation functions approximate those of a microscopic model to a stated accuracy. A topological field theory is a still coarser target, usually intended to retain only long-distance invertible response or defect data.

Spectral flattening is not such an approximation. It is a homotopy inside the space of gapped bounded operators. It preserves a stable topological class while deliberately destroying the energy scale.

An equivalence audit

Any claimed comparison must answer more than whether a map can be written. We use the following audit throughout.

Calling a comparison an equivalence requires affirmative answers in the chosen category. The controlled free-fermion map below proves construction, gapped-homotopy invariance, and stacking compatibility. It is intentionally not faithful to microscopic spectra.

A common spectral gap

Let $S$ be compact Hausdorff and $\mathcal A$ a unital complex C*-algebra. Write $$\mathcal A_S=C(S,\mathcal A).$$ An element $h\in M_N(\mathcal A_S)$ is equivalently a norm-continuous family $s\mapsto h_s\in M_N(\mathcal A)$.

Because $S$ is compact and $h$ is norm-continuous, pointwise invertibility is enough to produce some common positive gap. We state the quantitative constant because later parameter objects need not be compact, and because estimates depend on it.

Spectral flattening

Define the sign function on $\mathbb R\setminus\{0\}$ by $$\operatorname{sgn}(x)= \begin{cases} 1,&x>0,\\ -1,&x<0. \end{cases}$$ It is continuous on the spectrum of any invertible self-adjoint element.

Relative operator K-theory

The projection $p$ determines a class $$[p]\in K_0(C(S,\mathcal A)).$$ Following the difference-group viewpoint emphasized by Thiang [Thiang2016], we compare it with a reference system.

The word relative here refers to a difference from a reference phase. It should not be confused with the relative group $K^*(B,U)$ of a topological pair, which appears in the transition calculation later.

Chiral symmetry and the odd class

Suppose a grading $$\Gamma= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}$$ satisfies $\Gamma h\Gamma=-h$. Functional calculus preserves this oddness: $\Gamma q\Gamma=-q$.

Real symmetries and particle-hole structures require the appropriate real or graded K-group. The complex formula must not be reused after discarding those structures. Kitaev’s periodic table [Kitaev2009] and the operator-algebraic treatment of Thiang [Thiang2016] explain these symmetry-sensitive targets.

Naturality in the parameter space

Let $f:T\to S$ be continuous between compact Hausdorff spaces. Pullback gives a star-homomorphism $$f^*:C(S,\mathcal A)\longrightarrow C(T,\mathcal A), \qquad (f^*a)(t)=a(f(t)).$$

This naturality is the reliable part of the condensed interpretation. For a representable condensed parameter object $\underline S$, one evaluates the family on a compact or profinite test space and pulls back along maps of tests. Extending the construction to arbitrary condensed anima requires a sheaf or stack model of the Hamiltonian domain and is proposed, not proved by the elementary theorem above.

What the class retains

Within the controlled domain, spectral flattening retains the following stable data.

1. It retains the occupied versus empty grading determined by the chosen Fermi level.

2. It retains the connected component of the gapped self-adjoint operator in the stabilized topology.

3. It retains symmetry constraints that were built into the real, graded, or twisted target algebra.

4. It retains the direct-sum law represented by addition in K-theory.

5. Through pairings with cyclic cocycles or geometric cycles, it may retain quantized response coefficients under the additional hypotheses of those index theorems [Bellissard1994, ProdanSchulzBaldes2016].

The last item is not automatic from the abstract K-class alone. One must supply the pairing, regularity algebra, trace, and covariance data.

What the class forgets

The same map forgets several physically meaningful quantities.

This one-line counterexample is enough to rule out any claim that the flattened K-class reconstructs the microscopic Hamiltonian.

A contract, not a reconstruction theorem

We summarize the controlled map as $$(\mathcal A,h,h_{\mathrm{ref}},\mathcal S,\gamma) \longmapsto \kappa(h,h_{\mathrm{ref}}).$$ Its contract is:

This contract matches the formal Lean representation

That library records an Aoki comparison provider and C*-conditions without pretending to prove the analytic theorem inside Lean’s standard library.

Linearization near a crossing

Suppose a Bloch Hamiltonian depends smoothly on momentum $k$ and a mass parameter $m$. Near an isolated crossing $(k_0,0)$, one may have $$h(k_0+p,m) = \sum_{j=1}^d v_jp_j\Gamma_j +m\Gamma_0 +O(|p|^2+|m||p|+m^2),$$ where the $\Gamma_j$ satisfy suitable Clifford relations. The first-order term is a Dirac Hamiltonian. Any scalar term proportional to the identity has already been removed by the Fermi-level convention $\mu=0$ before the crossing subspace is expanded.

The proposition controls a local matrix approximation. It does not prove convergence of an interacting lattice theory to a relativistic quantum field theory. Even in the quadratic setting, the crossing subspace, momentum window, symmetry action, and neglected bands must be specified.

Scaling limits require more data

A genuine microscopic-to-EFT theorem would need at least:

1. a sequence of lattice spacings or energy cutoffs;

2. a renormalization prescription for fields and couplings;

3. convergence of a stated class of correlation functions or operator algebras;

4. control of irrelevant operators in the chosen topology;

5. a proof that the microscopic symmetry and anomaly data descend;

6. treatment of boundaries and defects if the target theory includes them.

Condensed mathematics can organize the parameter object on which such data vary. It does not supply any of these estimates.

Many-to-one behavior of effective descriptions

Two lattice Hamiltonians can share the same linear Dirac term while differing in quadratic dispersion, remote bands, and short-distance interactions. For example, adding $$\delta h(p)=a|p|^2\Gamma_0$$ does not change the first-order Dirac operator but changes the ultraviolet spectrum. Likewise, adding a high-energy atomic band changes the microscopic Hilbert space without changing the low-energy crossing.

Thus $L$ is generally many-to-one even before applying $I$. This is expected behavior for an effective theory. It blocks reconstruction unless extra ultraviolet data are retained.

From EFT data to stable classes

Kitaev’s free-fermion table [Kitaev2009] and related tenfold-way constructions [Schnyder2008, Ryu2010] extract stable classes after fixing dimension and symmetry. Teo and Kane extend the controlled free classification to defect families $H(k,r)$ [TeoKane2010]. These are strong results in their stated free-fermion domains.

Freed and Hopkins classify deformation classes of reflection-positive invertible extended field theories using stable homotopy and bordism data [FreedHopkins2021]. Their application to lattice phases assumes the existence and validity of low-energy effective-field-theory approximations. The bordism calculation is not, by itself, a theorem producing a local lattice Hamiltonian.

A profinite disorder hull

Let $D$ be a finite set of local labels and set $$\Omega=D^{\mathbb Z^d}.$$ With the product topology, $\Omega$ is compact, Hausdorff, and profinite. The translation action $$\alpha:\mathbb Z^d\curvearrowright\Omega$$ shifts a configuration.

The covariant bulk algebra is $$\mathcal A_{\Omega} = C(\Omega)\rtimes_{\alpha}\mathbb Z^d.$$ Because $\mathbb Z^d$ is amenable, full and reduced crossed products agree. A magnetic field may require a twisted crossed product. This noncommutative Brillouin-zone viewpoint underlies rigorous treatments of disordered topological phases and the quantum Hall effect [Bellissard1994, ProdanSchulzBaldes2016].

Finite-range covariant Hamiltonians

A finite-range covariant one-particle Hamiltonian can be written schematically as $$h = \sum_{|x|\leq R} h_xu_x, \qquad h_x\in M_N(C(\Omega)),$$ where the $u_x$ implement translations. The coefficients satisfy the adjoint relations that make $h=h^*$. The finite sum is an element of $M_N(\mathcal A_\Omega)$.

This result justifies the disorder algebra where a genuine spectral gap is present. It does not claim that every configuration has a gap merely because $\Omega$ is compact. The gap is a spectral property of the crossed-product element.

Mobility gaps are not spectral gaps

Disordered topological phases can remain quantized when the Fermi energy lies in a localized regime even though it belongs to the spectrum. That situation requires mobility-gap hypotheses, Sobolev or localization algebras, trace estimates, and index pairings. The discontinuous characteristic function is then handled in a more refined regularity framework.

What profinite approximation does and does not prove

Locally constant functions that factor through finite clopen quotients are sup-norm dense in $C(\Omega)$. This gives finite-resolution approximations to coefficient functions. It does not imply that every K-class, spectral gap, or response coefficient is determined by one finite quotient.

The safe condensed statement is functorial: the profinite space $\Omega$ is a legitimate test object, covariant families pull back along maps of disorder spaces, and their K-classes are natural when the analytic hypotheses survive pullback. The sheaf condition does not manufacture the crossed-product spectrum or its gap.

Bloch Hamiltonian and spectrum

The spinless SSH Bloch Hamiltonian is $$h(k;t_1,t_2) = (t_1+t_2\cos k)\sigma_x +t_2\sin k\,\sigma_y.$$ It anticommutes with the chiral operator $\Gamma=\sigma_z$. Writing $$q(k)=t_1+t_2e^{ik},$$ the energies are $$E_{\pm}(k)=\pm|q(k)|.$$ The model descends from the dimerized chain introduced by Su, Schrieffer, and Heeger [SSH1979].

Flattening and winding

When the model is gapped, define $$u(k)=\frac{q(k)}{|q(k)|}\in U(1).$$ The flattened Hamiltonian is $$Q(k) = \begin{pmatrix} 0&\overline{u(k)}\\ u(k)&0 \end{pmatrix}.$$

Choose a trivial reference with $|t_1|>|t_2|$. Then the relative AIII class of a topological sample is one. This is a concrete instance of the relative $K_1$ construction in theorem 3.8.

The gapless discriminant

The global coupling discriminant is $$\Sigma = \{(t_1,t_2):|t_1|=|t_2|\}.$$ Near the branch $t_1=t_2>0$ and momentum $k=\pi$, write $$m=t_1-t_2, \qquad k=\pi+p.$$ Then $$q(\pi+p) = m-it_2p+O(p^2).$$ The combined momentum-mass plane is essential. Looking only at the coupling line misses the linking circle around the isolated degeneracy in $(p,m)$.

Topologically, the normalized map defines a class in $$H^1(S^1;\mathbb Z)\cong\mathbb Z.$$ The connecting morphism for the pair $(D^2,S^1)$ sends it to $$\partial\nu \in H^2(D^2,S^1;\mathbb Z).$$ This is the elementary model for a relative transition charge. Teo and Kane’s free-fermion defect theory gives the broader controlled precedent [TeoKane2010].

A symmetry-breaking detour

Add a staggered sublattice potential $$\Delta\sigma_z.$$ The Hamiltonian becomes $$h_{\Delta}(k) = (t_1+t_2\cos k)\sigma_x +t_2\sin k\,\sigma_y +\Delta\sigma_z.$$ For $\Delta\neq0$, it no longer anticommutes with $\sigma_z$.

The theorem shows that the SSH charge is relative to the AIII symmetry problem. If the symmetry is removed from the category, the two endpoints are gapped-homotopic. This is not a failure of the AIII invariant. It is a reminder that symmetry is part of its domain.

Executable finite checks

The accompanying Haskell program implements the exact two-band formulas. Its checks include:

The code compiles with strict warnings and exits nonzero if a check fails. It is not evidence for a general thermodynamic limit.

The free integer invariant

Class BDI describes one-dimensional free fermions with particle-hole symmetry, time-reversal symmetry squaring to $+1$, and their product chiral symmetry. In the stable free classification, the phase index is an integer. Stacking adds the integers.

This statement is different from the complex AIII SSH invariant, even though both use winding in simple matrix representatives. The real symmetry structure changes the classification problem.

Fidkowski-Kitaev reduction

Fidkowski and Kitaev constructed a symmetry-preserving interacting path that connects free BDI phases whose integer labels differ by eight [FidkowskiKitaev2009]. They later proved that the eight residue classes are distinct and exhaustive for the specified one-dimensional interacting fermionic problem [FidkowskiKitaev2011].

What the reduction proves

The reduction proves that the free classification is not a complete invariant of the larger interacting category. It does not invalidate the free functional-calculus theorem. That theorem remains correct on its quadratic domain.

The categorical picture is a quotient-like comparison $$\mathbb Z \longrightarrow \mathbb Z/8$$ after the class of allowed paths is enlarged to include interactions. The kernel is not visible within free gapped homotopy.

This gives a direct counterexample to the identification of microscopic phases with free K-theory.

Stable free representatives

In controlled free-fermion problems, Bott periodicity and Clifford-module constructions often provide representatives for stable K-classes [Kitaev2009, Thiang2016]. Lattice Dirac models can realize many table entries after fixing a symmetry class and allowing stabilization.

Even there, one must distinguish:

1. a continuous family of finite matrices on a Brillouin torus;

2. a finite-range or rapidly decaying real-space Hamiltonian;

3. a chosen boundary termination;

4. a material realization with fixed orbital constraints.

A homotopy class of maps may have a continuous representative whose Fourier coefficients decay but are not finite range. Approximation can often recover finite range while preserving a positive gap, but the approximation theorem and symmetry constraints must be stated.

Interacting and bordism classes

For an abstract invertible field-theory or bordism class, a microscopic realization needs more:

1. local degrees of freedom and an on-site or spatial symmetry action;

2. a uniformly local or almost-local interaction;

3. a ground-state and boundary convention;

4. a gap uniform in system size;

5. a computation identifying the low-energy or response class;

6. an inverse under stacking if invertibility is claimed.

The bordism group does not package these witnesses. It classifies field-theoretic deformation data under its own axioms. Freed and Hopkins explicitly frame the lattice application through an EFT assumption [FreedHopkins2021].

Absence of a known model means open, not obstructed.

Why surjectivity is hard

Essential surjectivity of a realization map would require every abstract class to admit a microscopic witness. Three independent obstructions can intervene.

First, the abstract target may assume relativistic locality or full extension to defects that a lattice construction has not supplied. Second, anomaly constraints may force a boundary interpretation rather than an autonomous lattice bulk in the stated dimension. Third, a formal phase label may have no known system-size-independent gap proof for its proposed Hamiltonian.

These are mathematical requirements, not matters of terminology.

Why injectivity is hard

Injectivity would say that two microscopic phases with the same abstract class are connected by an allowed stabilized gapped path. The target class may omit crystalline data, noninvertible excitations, boundary conditions, or unstable finite-band information. If any omitted datum is retained by the microscopic equivalence, injectivity fails.

The BDI reduction shows the opposite possibility as well. Enlarging the microscopic path category can identify free classes that were distinct before interactions were allowed. The answer depends on the domain and its morphisms.

Representable parameter objects

For a compactly generated space $S$, write $$\underline S(T)=\operatorname{Cont}(T,S)$$ on profinite test spaces $T$. The precise full-faithfulness statement uses the compact-generation and cardinal conventions of condensed mathematics [Scholze2026].

A compact parameter family of controlled Hamiltonians is an element $$h\in M_N(C(S,\mathcal A))_{\mathrm{sa}}.$$ The uniformly gapped locus consists of those $h$ for which one lower bound works on all of $S$. Theorem 3.3 gives a natural map $$\operatorname{Flat}_S: \mathfrak{Ham}^{\mathrm{free,gap}}(S) \longrightarrow \operatorname{Proj}(M_\infty(C(S,\mathcal A))).$$

This is an exact statement about representable profinite tests. It does not prove that the spectral gap itself descends from arbitrary local gap constants unless the constants have a common positive lower bound.

The proposed comparison diagram

The controlled map and the broader comparison program are displayed as two separate chains:

The first arrow is the theorem proved in the controlled free sector. In the second chain, $I$ is solid only on the specified reflection-positive invertible-EFT domain classified by Freed and Hopkins. The low-energy map $L$ and the realization map $R$ require separate comparison and realization theorems. No map $K_{\mathrm{op}}\to\mathfrak{EFT}^{\mathrm{invertible}}$ and no commutative comparison square are claimed here.

Solidification is a K-theory bridge

Aoki proves that solidification of algebraic K-theory recovers connective topological K-theory for real and complex Banach algebras, with the stated connective and Bott-periodic qualifications [Aoki2024]. This supplies a genuine condensed-to-operator K-theory bridge.

It does not reconstruct the C*-norm, positive cone, state space, or microscopic interaction. It also does not prove that every operator K-class is a physical phase. The bridge is at the level of K-theory after the Banach algebra is already given.

Claim ledger

Lean representation

The reviewed Lean foundation intentionally represents theorem interfaces, not deep analytic proofs. The declarations relevant here include:

• ClaimStatus and Provenance for the status ledger;

• FiniteVolumeGapWitness for an explicit common lower bound;

• ObjectwiseInvolution and ObjectwiseCStarNormConditions for operator-algebraic input;

• Stacking and PhaseEquivalence for the stable phase interface;

• AokiConnectiveComparisonInterface for the imported solidification theorem.

No Lean declaration in that foundation asserts a Lieb-Robinson estimate, thermodynamic limit, or Aoki comparison without a provider. The library builds without user axioms.

Haskell representation

The Haskell directory src/microscopic-effective-theories/ contains four modules.

All top-level bindings have explicit type signatures and each module has an explicit export list. The strict compiler flags are $$\texttt{-Wall -Wextra -Werror}.$$ The program uses no claim stronger than its finite computations.

Interacting spectral flattening

A general interacting infinite-volume system has no bounded one-particle Hamiltonian to which this construction can be applied. One can flatten finite-volume matrices, but the resulting operator may be highly nonlocal and need not converge to a quasi-local thermodynamic object. Spectral flow is the more appropriate interacting tool, provided a uniform gap and locality estimates have already been proved.

An open comparison problem is to relate interacting spectral-flow classes to operator-algebraic or generalized cohomology invariants without discarding the relevant higher defects.

Noninvertible topological order

A spectrum of invertible phases cannot classify general noninvertible topological order. Anyons, fusion, braiding, and higher defect categories require richer targets. Two systems can share an invertible response class while having different noninvertible excitation data.

The paper therefore restricts its bordism discussion to invertible sectors.

Boundaries

Bulk K-theory can pair with boundary maps and produce bulk-boundary correspondences under controlled hypotheses [ProdanSchulzBaldes2016, TeoKane2010]. A bulk class alone does not specify a boundary Hamiltonian, termination, or symmetry-breaking boundary condition. Any claimed microscopic/EFT equivalence with boundaries must include those objects and their morphisms.

Crystalline and spatial symmetries

Spatial symmetries are not ordinary on-site internal symmetries. Their action on momentum, position, disorder, and defects changes the target K-theory or equivariant homotopy theory. A comparison that forgets the spatial action can merge distinct crystalline phases.

The present paper treats symmetry abstractly through a fixed C*-algebraic structure and does not claim a complete crystalline classification.

Mobility-gap extension

A mobility-gap extension matters for disordered systems. It requires localization estimates that ensure the Fermi projection belongs to an algebra on which index pairings remain defined. Developing that extension in condensed families would require uniform localization control across the parameter object. Compactness or profiniteness alone does not provide it.

Realization tests

A useful next step is a machine-readable realization registry. Each row would record:

1. the abstract class and symmetry type;

2. an explicit microscopic interaction;

3. a locality certificate;

4. a uniform gap theorem;

5. a class computation;

6. the status realized, candidate, open, or obstructed.

Such a registry would expose which part of the inverse problem is missing for each example.

Continuity of the sign map on a gapped set

Fix $\gamma>0$ and $M>\gamma$. On the compact set $$X_{\gamma,M} = [-M,-\gamma]\cup[\gamma,M],$$ the sign function is continuous. By the Stone-Weierstrass theorem, polynomials can approximate it uniformly on $X_{\gamma,M}$. If $h$ is self-adjoint with $\operatorname{spec}(h)\subset X_{\gamma,M}$, then $\operatorname{sgn}(h)$ is the norm limit of the same polynomials in $h$.

For a norm-continuous family with a common gap and a common norm bound, the polynomial approximants are uniformly continuous in the family parameter. Their uniform limit is norm-continuous. This gives an elementary route to the continuity assertion in theorem 3.6.

A resolvent formula

The negative spectral projection can also be written as a Riesz projection. Choose a contour $\mathcal C$ enclosing the negative spectrum and excluding the positive spectrum. Then $$p_h = \frac{1}{2\pi i} \int_{\mathcal C}(z-h)^{-1}\,dz.$$ If $h'$ is close enough to $h$, the same contour lies in the resolvent set of $h'$. The resolvent identity $$(z-h')^{-1}-(z-h)^{-1} = (z-h')^{-1}(h'-h)(z-h)^{-1}$$ gives norm continuity of the projection.

This formula also displays the role of the gap. As the contour approaches the spectrum, the resolvent norm grows and uniform control is lost.

Symmetry functoriality

Suppose an automorphism or anti-automorphism $\alpha$ encodes a symmetry and $h$ obeys a relation such as $$\alpha(h)=h, \qquad \alpha(h)=-h, \qquad\text{or}\qquad \alpha(h(k))=h(-k).$$ Functional calculus transports the corresponding relation to $q=\operatorname{sgn}(h)$ because the sign function is real and odd. The target K-group must still remember whether $\alpha$ is linear, antilinear, grading-preserving, or grading-reversing.

Why a reference is useful

The Grothendieck group of projections contains formal differences. Physically, choosing $h_{\mathrm{ref}}$ identifies which atomic or vacuum configuration counts as zero. The class $$[p_h]-[p_{h_{\mathrm{ref}}}]$$ then measures an obstruction between two gapped data rather than assigning an absolute label to one Hamiltonian.

Thiang’s formulation makes this relative viewpoint explicit [Thiang2016]. It avoids suggesting that the abstract group element contains a preferred microscopic representative.

Cocycle law

For three controlled systems $h_0,h_1,h_2$ in the same target, $$\kappa(h_0,h_2) = \kappa(h_0,h_1)+\kappa(h_1,h_2),$$ where $$\kappa(h_i,h_j)=[p_{h_i}]-[p_{h_j}].$$ The formula is immediate by cancellation in the abelian group. It is checked on finite SSH representatives by the Haskell program.

Reference changes

If the reference is changed from $r$ to $r'$, then $$\kappa(h,r') = \kappa(h,r)+\kappa(r,r').$$ Thus absolute coordinates change by a constant translation. Differences between two physical systems do not depend on that coordinate choice.

Orientation

With $$q(k)=t_1+t_2e^{ik},$$ the topological SSH winding is $+1$ for $|t_1|<|t_2|$. Near $k=\pi$, the local coordinate is $$q\simeq m-it_2p.$$ The positively oriented circle $(m,t_2p)=(r\cos\theta,r\sin\theta)$ maps to $e^{-i\theta}$ and therefore has degree $-1$. There is no contradiction. The local charge and the phase jump use different boundary orientations. Only the orientation-adjusted equality, or equality in magnitude, is canonical.

Discrete winding computation

For nonzero samples $q_j=q(k_j)$ with $k_j=2\pi j/N$, the code computes $$\nu_N = \operatorname{round} \left( \frac{1}{2\pi} \sum_{j=0}^{N-1} \operatorname{Arg} \left(\frac{q_{j+1}}{q_j}\right) \right).$$ The analytic gap check runs first, so division by zero is excluded. Set $g_\chi=||t_1|-|t_2||$ and $h=2\pi/N$. Since $$\left|\frac{d}{dk}\operatorname{Arg}q(k)\right| \leq \frac{|t_2|}{|q(k)|} \leq \frac{|t_2|}{g_\chi},$$ the code requires $h|t_2|/g_\chi<\pi$ before summing principal phase increments. This sufficient sampling condition prevents a true increment from crossing the branch cut; an explicit runtime check rejects an increment of magnitude $\pi$ as well. The result then agrees with theorem 7.2.

The computation is a regression test for the implementation. It is not used as the proof of the analytic proposition.

Finite gap sampling

The executable proof check compares the closed formula $$2\sqrt{(|t_1|-|t_2|)^2+\Delta^2}$$ with the minimum sampled band separation on a mesh that contains $k=\pi$. The analytic derivation remains the proof. The mesh catches sign, factor-of-two, and convention mistakes in code.