Lecture 22 : SSH Winding and Boundary Localization Where this fits
Geometric phase becomes a condensed - matter diagnostic only when symmetry , filling , boundary , and allowed deformation are specified . The SSH chain supplies the smallest setting in which those ingredients can be followed from a bulk family of states to an explicit boundary recursion . That construction becomes a controlled prototype for later bulk – boundary arguments , while also showing why termination and symmetry breaking can change an edge claim without changing every bulk energy . Topology does not attach to a band diagram in isolation ; it relates a gapped bulk object , an equivalence rule , and a boundary problem under declared protecting conditions .
Trans - polyacetylene supplies the historical mechanism : a Peierls distortion turns equal carbon – carbon bonds into alternating stronger and weaker bonds , so the electronic hopping alternates as well . Its two dimerization patterns have the same bulk spectrum in the idealized chain but differ at a domain wall or a specified end . Real polyacetylene also has lattice dynamics , spin , interactions , chemical end groups , and soliton physics . We therefore use an engineered electronic or cold - atom dimer chain as the literal realization of the minimal Hamiltonian , because there the hopping ratio and termination can be prepared and local spectral weight or site occupation can be resolved .
The two bulk dimerizations can have the same energy spectrum while a specified termination treats them differently . When does a bulk winding actually license a localized end state ? The qualified answer applies to electronic , photonic , mechanical , and electrical dimer chains , but only after symmetry , unit cell , and termination have been stated .
We will Fourier transform the declared dimerized chain , calculate its winding , and solve the open - boundary zero - mode recursion for both terminations and a symmetry - breaking variation .
With one frozen chain , let 𝑡 1 / 𝑡 2 = 0 . 5 t 1 / t 2 = 0.5 , take positive hoppings and an exposed A 1 A 1 site , and traverse the Brillouin zone with increasing 𝑘 k . Predict whether the hopping loop encloses the origin and whether a zero - energy amplitude launched at the left edge grows or decays . The lecture will make those two answers meet and then deliberately break one of their shared assumptions .
In five minutes , Fourier transform only ∑ 𝑗 ˆ 𝑐 † A , 𝑗 + 1 ˆ 𝑐 B , 𝑗 ∑ j c ^ A , j + 1 † c ^ B , j , decide when a circle of radius | 𝑡 2 | | t 2 | centered at 𝑡 1 t 1 contains the origin , and state when ∑ 𝑗 ≥ 0 𝑟 2 𝑗 ∑ j ≥ 0 r 2 j converges . These retrieve the lattice transform from Lecture 7 , two - band geometry from Lecture 4 , and loop reasoning from Lecture 21 . The local changes are the off - diagonal hopping function and the open - boundary recursion .
22.1 Fourier transform the declared chain The SSH winding and its end state live in two related but different descriptions . The winding belongs to a chiral Bloch Hamiltonian with an open gap , | 𝑡 1 | ≠ | 𝑡 2 | | t 1 | ≠ | t 2 | , and fixed orientation and unit - cell conventions ; the localized mode belongs to a terminated real - space operator . Both calculations are exact for the declared alternating - hopping model . Mapping an actual chain or device to that model is phenomenological unless omitted onsite , longer - range , and interaction terms are bounded . Changing the termination or breaking chiral symmetry is therefore a decisive physical test , not an afterthought to the topological label .
The Brillouin zone is a circle and the Bloch operator acts on a two - orbital space . Declare the chiral operator , unit - cell convention , complex hopping function , winding orientation , finite - chain Hilbert space , termination , and boundary - state normalizability domain . A continuously chosen phase is a local branch , whereas the winding is globally defined .
The fiber - operator / matrix distinction is the one fixed in the Bloch Hamiltonian definition . The later polarization comparison reuses the Zak - phase definition , including its origin , unit - cell , orientation , and endpoint qualifications .
The bulk object is a gapped symmetry - constrained Hamiltonian family , and its topological statement is an obstruction to a symmetry - preserving gapped deformation . A winding number is one representation of that obstruction after the chiral basis and unit - cell convention are declared . Moving a unit - cell boundary can relabel a winding representative ; physically changing a boundary termination can change the boundary spectrum . Edge localization is therefore a conditional bulk – boundary consequence , not the definition of the bulk phase .
Choose cells 𝑗 j of length 𝑎 a , with orbitals A 𝑗 , B 𝑗 A j , B j . Let real 𝑡 1 t 1 connect A 𝑗 A j to B 𝑗 B j , and real 𝑡 2 t 2 connect B 𝑗 B j to A 𝑗 + 1 A j + 1 . Here A A and B B are orbital names , so their equation subscripts are upright . The real - space operator is
ˆ 𝐻 = ∑ 𝑗 ( 𝑡 1 ˆ 𝑐 † A , 𝑗 ˆ 𝑐 B , 𝑗 + 𝑡 2 ˆ 𝑐 † A , 𝑗 + 1 ˆ 𝑐 B , 𝑗 + h . c . ) . H ^ = ∑ j ( t 1 c ^ A , j † c ^ B , j + t 2 c ^ A , j + 1 † c ^ B , j + h . c . ) .
For 𝑁 c N c periodic cells at positions 𝑅 𝑗 = 𝑗 𝑎 R j = j a , use the cell - coordinate Fourier convention
ˆ 𝑐 𝛼 , 𝑘 = 1 √ 𝑁 c ∑ 𝑗 e − i 𝑘 𝑅 𝑗 ˆ 𝑐 𝛼 , 𝑗 , 𝑘 ∈ [ − 𝜋 / 𝑎 , 𝜋 / 𝑎 ) . c ^ α , k = 1 N c ∑ j e − i k R j c ^ α , j , k ∈ [ − π / a , π / a ) .
Before transforming , predict which bond remembers a cell displacement . The intracell term has no phase ,
∑ 𝑗 ˆ 𝑐 † A , 𝑗 ˆ 𝑐 B , 𝑗 = ∑ 𝑘 ˆ 𝑐 † A , 𝑘 ˆ 𝑐 B , 𝑘 , ∑ j c ^ A , j † c ^ B , j = ∑ k c ^ A , k † c ^ B , k ,
whereas 𝑅 𝑗 + 1 = 𝑅 𝑗 + 𝑎 R j + 1 = R j + a gives
∑ 𝑗 ˆ 𝑐 † A , 𝑗 + 1 ˆ 𝑐 B , 𝑗 = ∑ 𝑘 e − i 𝑘 𝑎 ˆ 𝑐 † A , 𝑘 ˆ 𝑐 B , 𝑘 . ∑ j c ^ A , j + 1 † c ^ B , j = ∑ k e − i k a c ^ A , k † c ^ B , k .
Let B A B = ( | A ⟩ , | B ⟩ ) B AB = ( | A ⟩ , | B ⟩ ) be the ordered orthonormal basis of the two - orbital Bloch fiber . The resulting operator representation , its short numerical name , and its spectrum are
(22.1)
[ ˆ 𝐻 ( 𝑘 ) ] B A B = : H S S H ( 𝑘 ) = ( 0 𝑡 1 + 𝑡 2 e − i 𝑘 𝑎 𝑡 1 + 𝑡 2 e i 𝑘 𝑎 0 ) = [ 𝑡 1 + 𝑡 2 c o s ( 𝑘 𝑎 ) ] 𝜎 𝑥 + 𝑡 2 s i n ( 𝑘 𝑎 ) 𝜎 𝑦 , [ H ^ ( k ) ] B AB =: H SSH ( k ) = ( 0 t 1 + t 2 e − i k a t 1 + t 2 e i k a 0 ) = [ t 1 + t 2 cos ( k a ) ] σ x + t 2 sin ( k a ) σ y , 𝜀 ± ( 𝑘 ) = ± | 𝑡 1 + 𝑡 2 e i 𝑘 𝑎 | . ε ± ( k ) = ± | t 1 + t 2 e i k a | .
Definition 22.1 (Chiral symmetry)
Let ˆ 𝐻 H ^ be a self - adjoint operator on a Hilbert space H H with domain D ( ˆ 𝐻 ) D ( H ^ ) , and fix an energy origin 𝐸 0 E 0 . A chiral symmetry is a unitary , self - adjoint involution ˆ Γ : H → H Γ ^ : H → H such that ˆ Γ 2 = ˆ 𝐼 , ˆ Γ D ( ˆ 𝐻 ) = D ( ˆ 𝐻 ) , ˆ Γ ( ˆ 𝐻 − 𝐸 0 ˆ 𝐼 ) ˆ Γ − 1 = − ( ˆ 𝐻 − 𝐸 0 ˆ 𝐼 ) . Γ ^ 2 = I ^ , Γ ^ D ( H ^ ) = D ( H ^ ) , Γ ^ ( H ^ − E 0 I ^ ) Γ ^ − 1 = − ( H ^ − E 0 I ^ ) . Equivalently , { ˆ Γ , ˆ 𝐻 − 𝐸 0 ˆ 𝐼 } = 0 { Γ ^ , H ^ − E 0 I ^ } = 0 on the common domain . The definition depends on the declared energy zero and acts on states ; after a basis is chosen , its numerical matrix has no hat . It is an operator relation , not a property of one eigenvector representative .
The numerical Pauli matrices act on the declared A / B A / B sublattice coordinates and carry no hats . The corresponding Hilbert - space chiral operator ˆ Γ Γ ^ is + 1 + 1 on A A and − 1 − 1 on B B , so [ ˆ Γ ] B A B = 𝜎 𝑧 [ Γ ^ ] B AB = σ z . The off - diagonal form gives
𝜎 𝑧 H S S H ( 𝑘 ) 𝜎 𝑧 = − H S S H ( 𝑘 ) , { ˆ Γ , ˆ 𝐻 } = 0 . σ z H SSH ( k ) σ z = − H SSH ( k ) , { Γ ^ , H ^ } = 0.
Chiral symmetry pairs every bulk energy 𝐸 E with − 𝐸 − E . The gap is 2 m i n 𝑘 | 𝑡 1 + 𝑡 2 e i 𝑘 𝑎 | 2 min k | t 1 + t 2 e i k a | and closes when | 𝑡 1 | = | 𝑡 2 | | t 1 | = | t 2 | at the corresponding zone point . Including orbital positions in the Fourier phase would change matrix representatives but not the spectrum ; Berry and winding representatives would have to transform consistently .
22.2 Winding from a continuously unwrapped phase The lower - left matrix entry fixes the ordinary complex function
𝑞 ( 𝑘 ) = 𝑑 𝑥 ( 𝑘 ) + i 𝑑 𝑦 ( 𝑘 ) = 𝑡 1 + 𝑡 2 e i 𝑘 𝑎 . q ( k ) = d x ( k ) + i d y ( k ) = t 1 + t 2 e i k a .
The upper - right entry is 𝑞 ∗ ( 𝑘 ) q ∗ ( k ) ; changing either entry without changing the orientation would conjugate the loop and reverse the reported sign .
Definition 22.2 (SSH winding number)
Fix an oriented Brillouin circle B Z BZ , a unit cell , an ordered sublattice basis , and a Bloch trivialization in which 𝐻 ( 𝑘 ) = ( 0 𝑞 ∗ ( 𝑘 ) 𝑞 ( 𝑘 ) 0 ) . H ( k ) = ( 0 q ∗ ( k ) q ( k ) 0 ) . If the ordinary continuously differentiable function 𝑞 : B Z → ℂ ∖ { 0 } q : BZ → C ∖ { 0 } never vanishes , the SSH winding number is the degree of 𝑞 / | 𝑞 | : B Z → 𝑈 ( 1 ) q / | q | : BZ → U ( 1 ) , 𝜈 [ 𝑞 ] = 1 2 𝜋 i ∫ B Z 𝑞 ( 𝑘 ) − 1 𝜕 𝑘 𝑞 ( 𝑘 ) d 𝑘 ∈ ℤ . ν [ q ] = 1 2 π i ∫ BZ q ( k ) − 1 ∂ k q ( k ) d k ∈ Z . Equivalently it is the total continuously unwrapped change of a local branch of a r g 𝑞 arg q , divided by 2 𝜋 2 π . It is undefined when 𝑞 = 0 q = 0 , reverses sign with the orientation , and its representative depends on the unit - cell trivialization . The function 𝑞 q , its circle - valued phase , and a local real argument branch are three different mathematical objects ; none is a distribution .
For the declared branch convention , the same definition can be displayed as the total unwrapped angle :
(22.2)
𝜈 = 1 2 𝜋 ∫ B Z d 𝑘 𝜕 𝑘 a r g 𝑞 ( 𝑘 ) . ν = 1 2 π ∫ BZ d k ∂ k arg q ( k ) .
Differentiating a principal - value branch directly would create artificial jumps . A branch - independent evaluation is
𝜈 = 1 2 𝜋 i ∫ B Z d 𝑘 𝑞 ( 𝑘 ) − 1 𝜕 𝑘 𝑞 ( 𝑘 ) , 𝑞 ( 𝑘 ) ≠ 0 . ν = 1 2 π i ∫ BZ d k q ( k ) − 1 ∂ k q ( k ) , q ( k ) ≠ 0.
The SSH image is a circle of radius | 𝑡 2 | | t 2 | centered at 𝑡 1 t 1 . For positive hoppings and increasing 𝑘 k , | 𝑡 1 / 𝑡 2 | < 1 | t 1 / t 2 | < 1 encloses the origin once and gives 𝜈 = + 1 ν = + 1 ; | 𝑡 1 / 𝑡 2 | > 1 | t 1 / t 2 | > 1 gives 𝜈 = 0 ν = 0 . Reversing the Brillouin - zone orientation reverses the sign . If 𝑞 = 0 q = 0 , the inverse is singular , the bulk gap closes , and the gapped winding is undefined rather than noninteger .
The argument principle evaluates the claim without relying on the drawing . Set 𝑧 = e i 𝑘 𝑎 z = e i k a ; as 𝑘 k increases through the Brillouin zone , | 𝑧 | = 1 | z | = 1 is traversed counterclockwise . Then
𝜈 = 1 2 𝜋 i ∮ | 𝑧 | = 1 𝑡 2 d 𝑧 𝑡 1 + 𝑡 2 𝑧 . ν = 1 2 π i ∮ | z | = 1 t 2 d z t 1 + t 2 z .
The integrand has one pole at 𝑧 = − 𝑡 1 / 𝑡 2 z = − t 1 / t 2 . Its residue is one , so the integral is + 1 + 1 when | 𝑡 1 / 𝑡 2 | < 1 | t 1 / t 2 | < 1 and zero when the pole lies outside . At equality the pole lies on the contour and the invariant is not defined . This pole - in - contour test fixes both the magnitude and the orientation sign .
Close the text . For 𝑡 1 / 𝑡 2 = 0 . 5 t 1 / t 2 = 0.5 , reconstruct the intercell Fourier phase , the pole in the 𝑧 z plane , its winding , and the boundary ratio you expect before seeing the boundary equation . A correct magnitude with the wrong winding sign calls for an orientation check ; disagreement between pole location and expected decay calls for a unit - cell / termination check .
The anchor now has a bulk prediction : the declared 0 . 5 0.5 loop winds once and stays gapped . Keep that same cell and cut fixed while asking whether real - space normalizability makes the identical inequality reappear .
For the occupied SSH band in this chiral two - band convention , 𝛾 Z a k = 𝜋 𝜈 ( m o d 2 𝜋 ) γ Zak = π ν ( mod 2 π ) . The Berry connection remains gauge dependent even though the holonomy and winding relation is stable modulo its quantum . A unit - cell redefinition can shift the displayed winding together with the polarization and boundary bookkeeping ; changing only the bulk convention is inconsistent .
To see the factor of one half , write 𝑞 = | 𝑞 | e i 𝜑 q = | q | e i φ on a local phase branch . A normalized negative - energy eigenvector and its connection are
| 𝑢 − ( 𝑘 ) ⟩ = 1 √ 2 ( − e − i 𝜑 ( 𝑘 ) 1 ) , A − ( 𝑘 ) = i ⟨ 𝑢 − | 𝜕 𝑘 𝑢 − ⟩ = 1 2 𝜕 𝑘 𝜑 ( 𝑘 ) . | u − ( k ) ⟩ = 1 2 ( − e − i φ ( k ) 1 ) , A − ( k ) = i ⟨ u − | ∂ k u − ⟩ = 1 2 ∂ k φ ( k ) .
The unwrapped change in 𝜑 φ around the Brillouin circle is 2 𝜋 𝜈 2 π ν , giving the stated Zak phase modulo 2 𝜋 2 π . The spinor uses e − i 𝜑 e − i φ , which is single valued even when no single real phase branch is periodic . This calculation uses the fixed cell - periodic orbital convention above ; an embedding change must also be included in the physical polarization .
Worked example 22.1 — A winding from the declared SSH bonds
Use the real bonds 𝑡 1 , 𝑡 2 > 0 t 1 , t 2 > 0 of Equation ( 22 . 1 ) and the forward transform ˆ 𝑐 𝛼 , 𝑘 = 𝑁 − 1 / 2 c ∑ 𝑗 e − i 𝑘 𝑅 𝑗 ˆ 𝑐 𝛼 , 𝑗 c ^ α , k = N c − 1 / 2 ∑ j e − i k R j c ^ α , j . The inverse has e + i 𝑘 𝑅 𝑗 e + i k R j . Take increasing 𝑘 k on the Brillouin circle . Transform the two bonds , identify the chiral off - diagonal function 𝑞 ( 𝑘 ) = 𝑡 1 + 𝑡 2 e i 𝑘 𝑎 q ( k ) = t 1 + t 2 e i k a , and calculate its winding for 𝑡 1 ≠ 𝑡 2 t 1 ≠ t 2 . Check the two atomic limits before making a boundary claim .
The intracell bond has no cell displacement . The bond ˆ 𝑐 † A , 𝑗 + 1 ˆ 𝑐 B , 𝑗 c ^ A , j + 1 † c ^ B , j contributes e − i 𝑘 𝑎 e − i k a , so the upper off - diagonal element is 𝑞 ∗ q ∗ , not 𝑞 q . The resulting matrix anticommutes with 𝜎 𝑧 σ z . Set 𝑧 = e i 𝑘 𝑎 z = e i k a ; increasing 𝑘 k traverses | 𝑧 | = 1 | z | = 1 counterclockwise . Then 𝜈 = 1 2 𝜋 i ∮ | 𝑧 | = 1 𝑡 2 d 𝑧 𝑡 1 + 𝑡 2 𝑧 = { 1 , 𝑡 1 < 𝑡 2 , 0 , 𝑡 1 > 𝑡 2 . ν = 1 2 π i ∮ | z | = 1 t 2 d z t 1 + t 2 z = { 1 , t 1 < t 2 , 0 , t 1 > t 2 . The only zero is 𝑧 = − 𝑡 1 / 𝑡 2 z = − t 1 / t 2 , inside the contour precisely in the first regime . At 𝑡 1 = 0 t 1 = 0 , one intercell bond winds once ; at 𝑡 2 = 0 t 2 = 0 , the constant intracell bond winds zero times . Independently the spectral gap is 2 | 𝑡 2 − 𝑡 1 | 2 | t 2 − t 1 | , so the contour hits its zero exactly at the gap closing . Using 𝑞 ∗ q ∗ or reversing the coordinate orientation would reverse the signed winding , not change the spectrum . This result has a fixed chiral basis and cell convention ; the next recursion must use the corresponding physical termination rather than assume an edge from a circle drawing .
22.3 Boundary recursion and localization length Definition 22.3 (Boundary-state localization length)
Let 𝜓 = ( 𝜓 𝑗 ) 𝑗 ∈ ℕ ψ = ( ψ j ) j ∈ N be a nonzero normalizable boundary solution in ℓ 2 ( ℕ ) ⊗ ℂ 𝑠 ℓ 2 ( N ) ⊗ C s for cells separated by 𝑎 > 0 a > 0 , each with 𝑠 s orbitals ( 𝑠 = 2 s = 2 for SSH ) , so that 𝜓 𝑗 ∈ ℂ 𝑠 ψ j ∈ C s . If its amplitude has a positive asymptotic decay rate , its boundary - state localization length is 1 𝜉 l o c = − l i m 𝑗 → ∞ 1 𝑗 𝑎 l n ‖ 𝜓 𝑗 ‖ 𝐶 > 0 , 1 ξ loc = − lim j → ∞ 1 j a ln ‖ ψ j ‖ C > 0 , where any fixed 𝐶 > 0 C > 0 gives the same limit . For a constant amplitude ratio 0 < 𝑟 < 1 0 < r < 1 , 𝜉 l o c = − 𝑎 / l n 𝑟 ξ loc = − a / ln r ; the adjacent probability ratio is 𝑟 2 r 2 , not 𝑟 r . The localization length 𝜉 l o c ξ loc diverges as 𝑟 → 1 − r → 1 − ; for 𝑟 ≥ 1 r ≥ 1 the geometric sequence is not a normalizable boundary state , and for 𝑟 = 0 r = 0 it has compact support with the limiting convention 𝜉 l o c = 0 ξ loc = 0 . A finite - chain fit estimates this semi - infinite quantity only after opposite - edge hybridization and bulk isolation have been checked .
Take a semi - infinite chain beginning with A 1 A 1 , with no B 0 B 0 bond . The zero - energy equations permit a state with all B B - sublattice amplitudes set to zero . The remaining B 𝑗 B j equation is
(22.3)
𝑡 1 𝜓 A , 𝑗 + 𝑡 2 𝜓 A , 𝑗 + 1 = 0 , 𝜓 A , 𝑗 + 1 = − 𝑡 1 𝑡 2 𝜓 A , 𝑗 . t 1 ψ A , j + t 2 ψ A , j + 1 = 0 , ψ A , j + 1 = − t 1 t 2 ψ A , j .
Thus
𝜓 A , 𝑗 = 𝜓 A , 1 ( − 𝑡 1 𝑡 2 ) 𝑗 − 1 . ψ A , j = ψ A , 1 ( − t 1 t 2 ) j − 1 .
Predict the boundary criterion before normalizing : the amplitude must shrink into the bulk . With 𝑟 = | 𝑡 1 / 𝑡 2 | r = | t 1 / t 2 | , normalizability requires 𝑟 < 1 r < 1 , and
1 = ∞ ∑ 𝑗 = 1 | 𝜓 A , 𝑗 | 2 = | 𝜓 A , 1 | 2 1 − 𝑟 2 , | 𝜓 A , 1 | = √ 1 − 𝑟 2 . 1 = ∑ j = 1 ∞ | ψ A , j | 2 = | ψ A , 1 | 2 1 − r 2 , | ψ A , 1 | = 1 − r 2 .
The probability ratio of adjacent cells is 𝑟 2 r 2 . Writing the amplitude as
| 𝜓 A , 𝑗 | = | 𝜓 A , 1 | e x p [ − ( 𝑗 − 1 ) 𝑎 / 𝜉 l o c ] , 𝜉 l o c = − 𝑎 l n 𝑟 , | ψ A , j | = | ψ A , 1 | exp [ − ( j − 1 ) a / ξ loc ] , ξ loc = − a ln r ,
shows that the localization length diverges as 𝑟 → 1 − r → 1 − , exactly where the bulk gap closes . In a finite chain of length 𝐿 L , left and right boundary modes overlap by order e − 𝐿 / 𝜉 l o c e − L / ξ loc and can split symmetrically around zero without breaking chiral symmetry . A semi - infinite zero mode and a finite - chain pair therefore answer different questions .
22.4 A bulk – boundary audit for two hopping ratios 22.4.1 Bulk winding for both hopping ratios
Worked example 22.2 — Normalizing an SSH boundary mode
Cut the chain of Example 22 . 1 at a left A 1 A 1 site . For a zero - energy mode supported only on the A sublattice , derive and normalize the boundary recursion . Evaluate it for 𝑡 1 / 𝑡 2 = 0 . 5 t 1 / t 2 = 0.5 and 1 . 5 1.5 , and estimate how a finite chain changes the zero - energy conclusion .
The B - site equation is 𝑡 1 𝑎 𝑗 + 𝑡 2 𝑎 𝑗 + 1 = 0 t 1 a j + t 2 a j + 1 = 0 , so 𝑎 𝑗 = 𝐶 𝑟 𝑗 − 1 a j = C r j − 1 with the signed ratio 𝑟 = − 𝑡 1 / 𝑡 2 r = − t 1 / t 2 ; its magnitude | 𝑟 | | r | is the amplitude ratio of Definition 22 . 3 . On the half - infinite chain ∑ 𝑗 | 𝑎 𝑗 | 2 = | 𝐶 | 2 / ( 1 − | 𝑟 | 2 ) = 1 ∑ j | a j | 2 = | C | 2 / ( 1 − | r | 2 ) = 1 , requiring | 𝑟 | < 1 | r | < 1 and 𝐶 = √ 1 − | 𝑟 | 2 C = 1 − | r | 2 after a phase choice . Writing the modulus as | 𝑎 𝑗 | = 𝐶 e − ( 𝑗 − 1 ) 𝑎 / 𝜉 l o c | a j | = C e − ( j − 1 ) a / ξ loc , with cell spacing 𝑎 a , gives 𝜉 l o c = − 𝑎 / l n | 𝑟 | ξ loc = − a / ln | r | ; the probability decays twice as fast .
Keep 𝑡 2 > 0 t 2 > 0 , increasing 𝑘 k , the declared cell , and the left A 1 A 1 termination .
For 𝑡 1 / 𝑡 2 = 0 . 5 t 1 / t 2 = 0.5 , the 𝑞 ( 𝑘 ) q ( k ) circle encloses the origin and has 𝜈 = + 1 ν = + 1 . The signed boundary ratio is 𝑟 = − 0 . 5 r = − 0.5 , so the adjacent probability ratio is 0 . 2 5 0.25 , the first - cell probability is 1 − 0 . 5 2 = 0 . 7 5 1 − 0.5 2 = 0.75 , and
𝜉 l o c = − 𝑎 l n 0 . 5 ≈ 1 . 4 4 𝑎 . ξ loc = − a ln 0.5 ≈ 1.44 a .
For 𝑡 1 / 𝑡 2 = 1 . 5 t 1 / t 2 = 1.5 , the circle misses the origin and has 𝜈 = 0 ν = 0 . The same left - edge recursion grows into the bulk and is not normalizable . At 𝑡 1 = 𝑡 2 t 1 = t 2 , the loop passes through the origin at 𝑘 𝑎 = 𝜋 k a = π , the bulk gap closes , a r g 𝑞 arg q is undefined there , and 𝜉 l o c ξ loc diverges . These are three representations of one transition .
For 𝑁 N cells , the truncated vector has 𝐶 2 𝑁 = ( 1 − 𝑟 2 ) / ( 1 − 𝑟 2 𝑁 ) C N 2 = ( 1 − r 2 ) / ( 1 − r 2 N ) for real 𝑟 r . Its unsatisfied far - end equation leaves amplitude 𝑡 1 𝐶 𝑁 𝑟 𝑁 − 1 t 1 C N r N − 1 . Projecting onto the normalized right B - edge vector gives coupling 𝑡 1 𝐶 2 𝑁 𝑟 𝑁 − 1 t 1 C N 2 r N − 1 , of order 𝑡 2 | 𝑟 | 𝑁 t 2 | r | N , and an exponentially small splitting when the edges are well separated . It is not an exactly isolated zero mode at arbitrary finite size . The geometric - series norm and the 𝑡 1 → 0 t 1 → 0 single - site limit check normalization and termination independently . Normalizability was derived in real space , not inferred from the winding . Breaking chirality later tests energy pinning without necessarily destroying localization .
22.4.2 Measured decay in a momentum - space chain The recursion predicts that weakening dimerization spreads a boundary state . Does the measured profile show that trend , and what prevents an exact identification with the ideal eigenstate ? Meier , An and Gadway implemented an SSH chain using 21 momentum states of 8 7 R b 87 Rb atoms , rather than atomic positions in a polymer . After a 1 ms preparation ramp they inferred decay lengths from measured populations on even sites [MAG16, fig. 4(d) , p. 5, and Adiabatic preparation] .
Before reading the decay plot , trace Figure 22 . 1 from the physical atom cloud to the measured populations . Which objects occupy real space , and which labels denote states rather than distinct trapped atoms ?
A cold-atom momentum lattice and its population readout. Panel (a) shows representative atoms in a dilute condensate and the field directions; panel (b) separates the synthetic chain from the measurement and fit. Neither panel contains measured intensities.
An irregular three-dimensional cloud of representative rubidium-87 atom symbols depicts a dilute condensate of about 50,000 atoms in the experiment. Many atoms share one coherent matter-wave mode. The symbols are not measured positions. Counterpropagating optical fields couple a separate abstract chain of 21 momentum states, not 21 atoms. Release, 18 ms time of flight, absorption imaging and an even-site population fit connect the system to the decay length.
Here the starting material is a Bose - Einstein condensate of approximately 5 × 1 0 4 5 × 10 4 rubidium - 87 atoms , prepared by all - optical evaporative cooling and held in an optical trap . It is a dilute , ultracold gas , not a solid droplet . Condensation means that a macroscopic number of atoms occupy the same single - particle matter - wave mode ; it does not mean that they sit at one point . The small atom symbols make the gas visible schematically . Their count , positions , sizes and the cloud shape are not experimental measurements or a map of localized quantum particles .
Counterpropagating 1064 nm fields couple momentum states 𝑝 𝑛 = 2 𝑛 ℏ 𝑘 p n = 2 n ℏ k , 𝑛 = 0 , … , 2 0 n = 0 , … , 20 , where 𝑘 k here is the optical wave number , not the SSH Bloch label . Multiple frequency components address the 20 links ; the displayed left termination starts with a weak link . A two - photon Bragg process transfers momentum 2 ℏ 𝑘 2 ℏ k between adjacent states . The driving coherently redistributes the condensate among these momentum components of its matter wave . Thus a site labels a momentum state that many atoms can populate , not a particular atom or a separate trap in real space . Field axes are offset only for visibility ; no chamber , camera orientation or optical hardware layout is inferred .
After preparation the confinement is removed . During 18 ms time of flight , different momenta separate approximately as 𝑥 𝑛 − 𝑥 0 ≃ ( 𝑝 𝑛 − 𝑝 0 ) 𝑇 / 𝑚 x n − x 0 ≃ ( p n − p 0 ) T / m . Absorption imaging then yields resolved site populations ; the even - site exponential fit yields the population - decay length . This ballistic relation explains the readout principle , not the full imaging calibration . The 18 ms readout interval is distinct from the 1 ms preparation ramp . The other STM schematic later in this lecture describes a different platform and is not the provider for these data .
The length convention matters . Write the site spacing as 𝑑 d , so our cell spacing is 𝑎 = 2 𝑑 a = 2 d , and put 𝛿 = Δ f i n a l / 𝑡 δ = Δ final / t , 𝑡 1 = 𝑡 − Δ f i n a l t 1 = t − Δ final , 𝑡 2 = 𝑡 + Δ f i n a l t 2 = t + Δ final . Then 𝑟 = ( 1 − 𝛿 ) / ( 1 + 𝛿 ) r = ( 1 − δ ) / ( 1 + δ ) . On even sites 𝑛 = 2 ( 𝑗 − 1 ) n = 2 ( j − 1 ) , 𝑃 𝑛 ∝ 𝑟 𝑛 = e x p ( − 𝑛 𝑑 / 𝜉 𝑃 ) P n ∝ r n = exp ( − n d / ξ P ) , so
𝜉 𝑃 𝑑 = 1 l n [ ( 1 + 𝛿 ) / ( 1 − 𝛿 ) ] , 𝜉 l o c = 2 𝜉 𝑃 . ξ P d = 1 ln [ ( 1 + δ ) / ( 1 − δ ) ] , ξ loc = 2 ξ P .
Thus the fitted population length must not be compared directly with the amplitude length in Definition 22 . 3 . Inspect Figure 22 . 2 with this conversion in place . For its physical system and signal - to - population mapping , return to Figure 22 . 1 , panels ( a , b ) , respectively .
The recovered inputs are supplied here so the comparison does not depend on reading tiny markers . Uncertainties in this table are the reported standard errors recovered from the panel , rounded at its usable precision ; the last column has no fit to the measurements .
𝛿 δ , reported estimate and s . e . 𝜉 𝑃 / 𝑑 ξ P / d , reported estimate and s . e . Ideal 𝜉 𝑃 / 𝑑 ξ P / d
0 . 0 6 0 ± 0 . 0 0 5 0.060 ± 0.005 4 . 2 1 ± 1 . 9 8 4.21 ± 1.98 Not compared : ramp limitation
0 . 1 7 0 ± 0 . 0 0 7 0.170 ± 0.007 2 . 7 1 ± 0 . 5 6 2.71 ± 0.56 2 . 91
0 . 3 7 5 ± 0 . 0 1 0 0.375 ± 0.010 1 . 4 4 ± 0 . 1 3 1.44 ± 0.13 1 . 27
0 . 5 4 4 ± 0 . 0 1 2 0.544 ± 0.012 0 . 9 2 ± 0 . 1 2 0.92 ± 0.12 0 . 82
0 . 8 0 6 ± 0 . 0 0 8 0.806 ± 0.008 0 . 5 0 ± 0 . 0 5 0.50 ± 0.05 0 . 45
For the row near 𝛿 = 0 . 3 7 5 δ = 0.375 , the recursion gives 𝑟 = 0 . 6 2 5 / 1 . 3 7 5 ≃ 0 . 4 5 5 r = 0.625 / 1.375 ≃ 0.455 , hence 𝜉 𝑃 / 𝑑 ≃ 1 . 2 7 ξ P / d ≃ 1.27 , compared with the recovered 1 . 4 4 ± 0 . 1 3 1.44 ± 0.13 . The trend and scale agree , but the points do not sit exactly on the ideal line . A finite - time preparation and a fit to measured populations need not return the exact eigenstate profile ; the smallest - imbalance point is especially vulnerable as the gap shrinks . The quoted uncertainties do not justify a precision goodness - of - fit test without the original fit details and correlations . This cold - atom record tests the localization mechanism , not an STM signal or a measurement on polyacetylene . The next calculation returns to our declared real - space chain and its termination and finite - length corrections .
22.4.3 Boundary recursion and termination The finite - chain calculation should approach these semi - infinite predictions as its length grows . It can show two near - zero levels in the winding phase , but their exponentially small splitting is not a failure of the recursion . This equal - cell chain has the same number of A and B orbitals . It differs from an ideal 21 - site open bipartite chain with 11 A and 10 B orbitals : setting B amplitudes to zero leaves ten recursions for eleven A amplitudes , with no final B - site condition . That ideal odd - site chain therefore has an exact zero mode . A state prepared in finite time in the experiment need not be this exact eigenstate ; do not use its odd - site count to infer the paired - end splitting of the equal - cell chain .
Reuse the finite - chain normalization and residual end coupling in Example 22 . 2 : with 𝑟 = − 𝑡 1 / 𝑡 2 r = − t 1 / t 2 , the magnitude is | 𝑀 L R | = 𝑡 2 𝐶 2 𝑁 | 𝑟 | 𝑁 | M LR | = t 2 C N 2 | r | N , and the approximate levels are 𝐸 ± = ± | 𝑀 L R | E ± = ± | M LR | . For | 𝑟 | = 0 . 5 | r | = 0.5 and 𝑁 = 1 0 N = 10 , | 𝑀 L R | ≃ 7 . 3 2 × 1 0 − 4 𝑡 2 | M LR | ≃ 7.32 × 10 − 4 t 2 , so the level separation is about 1 . 4 6 × 1 0 − 3 𝑡 2 1.46 × 10 − 3 t 2 . Increasing 𝑁 N by one halves the leading splitting , a quantitative localization check rather than a visual judgment of “ near zero . ”
Return now to an engineered - chain spectrum rather than leaving the result in units of 𝑡 2 t 2 . Take 𝑡 2 = 1 0 0 m e V t 2 = 100 meV , 𝑡 1 = 5 0 m e V t 1 = 50 meV , 𝑎 = 0 . 5 n m a = 0.5 nm , and 𝑁 = 1 0 N = 10 . The bulk band gap is
𝐸 g a p = 2 | 𝑡 2 − 𝑡 1 | = 1 0 0 m e V , E gap = 2 | t 2 − t 1 | = 100 meV ,
the edge - state localization length is 1 . 4 4 𝑎 = 0 . 7 2 n m 1.44 a = 0.72 nm , and the finite - size level separation is approximately 0 . 1 4 6 m e V 0.146 meV . The local spectral density at orbital 𝑗 j ,
𝜌 𝑗 ( 𝐸 ) = ∑ 𝑛 | 𝜓 𝑛 ( 𝑗 ) | 2 𝛿 ( 𝐸 − 𝐸 𝑛 ) , ρ j ( E ) = ∑ n | ψ n ( j ) | 2 δ ( E − E n ) ,
therefore predicts an in - gap peak whose integrated weight is concentrated near the exposed end : 7 5 % 75 % of the semi - infinite zero - mode probability lies on the first A A orbital and each subsequent occupied - sublattice probability is smaller by a factor 0 . 2 5 0.25 . Local tunneling spectroscopy tests the energy and spatial decay together ; bulk spectroscopy tests the 1 0 0 m e V 100 meV gap . A peak near zero without the predicted end localization , length dependence , and bulk gap would not establish this bulk – boundary mechanism .
The measurement problem starts far outside the tunnel junction . For a simple vacuum - barrier model , 𝐼 ∝ e x p ( − 2 𝜅 𝑧 ) I ∝ exp ( − 2 κ z ) , so a small relative motion 𝛿 𝑧 δ z gives
𝛿 𝐼 𝐼 ≃ − 2 𝜅 𝛿 𝑧 . δ I I ≃ − 2 κ δ z .
With a representative 𝜅 ∼ 1 0 n m − 1 κ ∼ 10 nm − 1 , even 1 p m 1 pm of tip – sample motion produces a current change of order 2 % 2 % . The building foundation , pneumatic isolation , acoustic and RF shielding , remote pumps , flexible services , UHV chamber , scanner , and feedback loop are therefore parts of one measurement chain rather than optional laboratory decoration .
Before reading Figure 22 . 3 , trace that chain from the building to the junction . Then predict how a local spectrum at the exposed end differs from one deep in the bulk , and state what must be measured in addition to a near - zero - bias peak .
Before reading Figure 22 . 4 , predict which loop encloses the origin , which bulk spectrum closes at the transition , and which boundary probability sequence decays . Then check that all panels use the same hopping and termination conventions .
Use the boundary panel quantitatively : its adjacent probability ratio should be 0 . 2 5 0.25 , and its decay should give 𝜉 l o c ≈ 1 . 4 4 𝑎 ξ loc ≈ 1.44 a for the anchor . The figure is the only simultaneous audit of loop , bulk gap , and boundary profile ; removing it requires a numerical table carrying the same hoppings , orientation , cell , termination , and finite length .
Theorem 22.1 (SSH bulk--boundary correspondence)
Consider the nearest - neighbor SSH operator on ℓ 2 ( ℕ ) ⊗ ℂ 2 ℓ 2 ( N ) ⊗ C 2 with the declared A 1 A 1 termination , real nonzero 𝑡 1 , 𝑡 2 t 1 , t 2 , energy origin zero , ˆ Γ = + 1 Γ ^ = + 1 on A A and − 1 − 1 on B B , and the lower - left function 𝑞 ( 𝑘 ) = 𝑡 1 + 𝑡 2 e i 𝑘 𝑎 q ( k ) = t 1 + t 2 e i k a . Assume the bulk is gapped , | 𝑡 1 | ≠ | 𝑡 2 | | t 1 | ≠ | t 2 | . Then this boundary has exactly one normalizable zero mode of chirality + 1 + 1 when | 𝑡 1 | < | 𝑡 2 | | t 1 | < | t 2 | , and none when | 𝑡 1 | > | 𝑡 2 | | t 1 | > | t 2 | ; in the frozen orientation these cases have 𝜈 = 1 ν = 1 and 𝜈 = 0 ν = 0 , respectively .
More generally , for a local finite - range chiral operator asymptotic to gapped bulks , the stable statement is the Fredholm index : the signed number of zero modes of opposite chiralities equals the winding difference across the interface while locality , chiral symmetry , and the bulk gaps preserve the Fredholm property . It is a net chiral index , not an unconditional count of all in - gap states . Extra zero modes may occur in opposite - chirality pairs . A finite chain has two interfaces and its paired end modes may split exponentially ; if chiral symmetry or a bulk gap is lost , this theorem does not apply .
22.4.4 The symmetry - breaking perturbation test This is the exact point where “ topological ” can be overextended .
If the bulk winding is one in the displayed cell , must every physical cut contain a zero - energy end state ? Move the cut by one bond : the leftmost site is now B 1 B 1 , followed by the retained 𝑡 2 t 2 bond to A 2 A 2 . Write the first zero - energy boundary equation , then test normalizability at 𝑡 1 / 𝑡 2 = 0 . 5 t 1 / t 2 = 0.5 before reading the calculation below .
For this changed cut , set all A amplitudes to zero and denote the amplitude on B 𝑗 B j by 𝑏 𝑗 b j . The equation at A 𝑗 + 1 A j + 1 gives
𝑡 2 𝑏 𝑗 + 𝑡 1 𝑏 𝑗 + 1 = 0 , 𝑏 𝑗 + 1 = − 𝑡 2 𝑡 1 𝑏 𝑗 . t 2 b j + t 1 b j + 1 = 0 , b j + 1 = − t 2 t 1 b j .
Thus a normalizable B - edge state requires | 𝑡 2 / 𝑡 1 | < 1 | t 2 / t 1 | < 1 , the opposite inequality to the original A - edge state . At 𝑡 1 / 𝑡 2 = 0 . 5 t 1 / t 2 = 0.5 , successive amplitudes grow by magnitude two , so this cut has no normalizable zero mode . The bulk hoppings have not changed . A winding quoted in the original cell cannot specify a boundary without also specifying its physical cut and the matching interface convention .
A chiral - preserving perturbation can deform the complex loop without changing its winding unless the loop crosses the origin . By contrast , an onsite sublattice imbalance adds a numerical 𝜎 𝑧 σ z term . The spectral gap may remain open , but chiral anticommutation is lost , the planar winding no longer classifies the full Hamiltonian , and a localized boundary level may move away from zero . Persistence of localization and symmetry protection of exact energy are different claims . Cutting the opposite bond likewise requires a new boundary recursion and can exchange which sublattice is exposed .
The uniform staggered perturbation makes the energy shift explicit . Add 𝑚 ˆ Γ m Γ ^ , taking ˆ Γ = + 1 Γ ^ = + 1 on the exposed A A sublattice . The semi - infinite left mode remains an eigenstate because its 𝐵 B amplitudes vanish and the original hopping Hamiltonian annihilates it , but its energy is
𝐸 l e f t = + 𝑚 . E left = + m .
The right 𝐵 B - sublattice mode shifts to − 𝑚 − m , while the bulk spectrum is 𝐸 ± ( 𝑘 ) = ± √ | 𝑞 ( 𝑘 ) | 2 + 𝑚 2 E ± ( k ) = ± | q ( k ) | 2 + m 2 . Localization can therefore survive even though zero - energy pinning and the planar chiral winding no longer apply . A boundary - only onsite perturbation gives the corresponding first - order shift ⟨ 𝜓 | ˆ 𝑉 e d g e | 𝜓 ⟩ ⟨ ψ | V ^ edge | ψ ⟩ , weighted by the first - cell probability 1 − 𝑟 2 1 − r 2 .
For the same illustrative chain , a uniform 𝑚 = 2 0 m e V m = 20 meV imbalance shifts the left - edge peak to + 2 0 m e V + 20 meV , while the minimum positive bulk energy becomes √ ( 5 0 m e V ) 2 + ( 2 0 m e V ) 2 = 5 3 . 9 m e V ( 50 meV ) 2 + ( 20 meV ) 2 = 53.9 meV . The peak can remain localized and in the gap even though it is no longer pinned to zero and the planar chiral winding no longer classifies the full Hamiltonian . This is the experimental discrimination the symmetry discussion was designed to produce : a moved end peak is compatible with persistent localization but incompatible with exact chiral protection .
Worked example 22.3 — A localized edge without zero-energy pinning
Keep the normalized left - A solution of Example 22 . 2 and 0 < 𝑡 1 < 𝑡 2 0 < t 1 < t 2 , but add + 𝑚 + m on A and − 𝑚 − m on B with real 𝑚 m . Determine whether the half - infinite A - edge state survives and whether its energy remains protected at zero . Compare its energy with the bulk band edge .
Set 𝑏 𝑗 = 0 b j = 0 . The A equation becomes 𝐸 𝑎 𝑗 = 𝑚 𝑎 𝑗 E a j = m a j , so 𝐸 e d g e = 𝑚 E edge = m . The B equation is still 0 = 𝑡 1 𝑎 𝑗 + 𝑡 2 𝑎 𝑗 + 1 0 = t 1 a j + t 2 a j + 1 , giving the same normalized ratio 𝑟 = − 𝑡 1 / 𝑡 2 r = − t 1 / t 2 and localization length as before . The bulk energies now satisfy 𝐸 ± ( 𝑘 ) = ± √ | 𝑡 1 + 𝑡 2 e i 𝑘 𝑎 | 2 + 𝑚 2 , | 𝐸 b u l k | ≥ √ ( 𝑡 2 − 𝑡 1 ) 2 + 𝑚 2 > | 𝑚 | . E ± ( k ) = ± | t 1 + t 2 e i k a | 2 + m 2 , | E bulk | ≥ ( t 2 − t 1 ) 2 + m 2 > | m | . Thus this particular edge remains in the gap , while { 𝜎 𝑧 , 𝐻 } = 2 𝑚 𝐼 { σ z , H } = 2 m I , nonzero when 𝑚 ≠ 0 m ≠ 0 . Its localization survives ; zero - energy pinning by chiral symmetry does not . At 𝑚 = 0 m = 0 the previous solution is recovered , while 𝑡 1 → 0 t 1 → 0 leaves an isolated A site of energy 𝑚 m , independently checking the sign . Changing the physical cut would change the first boundary equation ; it is not merely a Bloch rephasing . Nor does persistence under this special mass prove protection under every symmetry - breaking perturbation . The following gauge ledger separates those distinct changes .
22.5 Gauge - dependent objects and invariant content Reconstruct the complete chain by classifying each object :
| 𝑢 𝑛 ( 𝑘 ) ⟩ | u n ( k ) ⟩ is a local section representative after a gauge choice ; 𝐴 𝑛 ( 𝑘 ) A n ( k ) is an ordinary local coefficient of a connection one - form and is gauge dependent ; e i 𝛾 e i γ is a circle - valued loop holonomy ; a real 𝛾 γ is a chosen branch modulo 2 𝜋 2 π ; 𝑞 ( 𝑘 ) q ( k ) is an ordinary complex function fixed by the unit - cell and Fourier convention ; 𝑞 / | 𝑞 | q / | q | is circle valued , whereas a r g 𝑞 arg q has only local real branches ; 𝜈 ν is an oriented integer while 𝑞 ( 𝑘 ) ≠ 0 q ( k ) ≠ 0 ; and 𝜓 A , 𝑗 ψ A , j is an ordinary sequence for a declared boundary termination .
No Dirac distribution is hidden in an angle jump : a jump of a chosen principal branch is a representation artifact . This classification prevents a gauge change from being mistaken for a topological transition and prevents a boundary claim from being detached from its physical cut .
For the declared unit cell and left termination , determine 𝜈 ν , normalizability , sublattice support , and adjacent probability ratio for 𝑡 1 / 𝑡 2 = 0 . 8 t 1 / t 2 = 0.8 . Then state what a chiral - symmetry - breaking onsite imbalance changes .
Qualification to retain . An end state is not guaranteed for every cut . The bulk invariant , unit cell , interface , termination , gap , and protecting symmetry form one convention - dependent but physically consistent statement .
22.6 From bulk winding to a qualified boundary claim The real - space hopping convention fixes the Bloch matrix . Its off - diagonal function traces a circle whose oriented enclosure is well defined only while the gap remains open . Under the matching termination , the zero - energy Schrödinger equation gives a geometric boundary sequence whose normalizability and localization length agree with that bulk criterion . Chiral symmetry and the bulk gaps are hypotheses of the correspondence , not decorative labels .
The larger - picture advance is a closed prediction loop . A real - space bond pattern fixes a Bloch loop ; the loop and symmetry constrain a boundary recursion ; the recursion predicts an energy , sublattice support , decay length , and finite - size trend that a local probe can test . Topology does not replace materials modeling — it identifies which conclusions survive continuous deformations once the symmetry , bulk gap , interface , and termination have all been declared .
The one - dimensional construction relates a loop to end localization under a declared symmetry and termination . It does not count oriented state transfer across a two - dimensional sample . Lecture 23 supplies a contrastive route : magnetic flux fixes a state count and a boundary twist moves those states , creating the spectral - flow operation later reformulated for lattice bands .
From one parameter set , reconstruct the bulk gap , winding , boundary recursion , and localization length together with the symmetry and termination conditions required to compare them . An SSH diagnosis must carry the chiral symmetry , unit - cell and orientation conventions , physical interface , and boundary termination throughout . The winding represents the bulk obstruction in that convention ; the localized recursion tests a conditional boundary consequence . Relabeling the cell can change representatives , whereas changing hoppings , symmetry , or the physical cut changes the model . An isolated edge level therefore cannot define the bulk phase without the complete deformation and termination audit .
22.7 Problems 1 . Bulk winding and boundary recursion Fourier transform the declared SSH chain , compute the winding without choosing a global angle branch , solve the zero - energy boundary recursion , and compare the two criteria .
Use positive 𝑡 2 t 2 , increasing 𝑘 : − 𝜋 / 𝑎 → 𝜋 / 𝑎 k : − π / a → π / a , the ordered basis ( A , B ) ( A , B ) , the lower - left convention 𝑞 ( 𝑘 ) = 𝑡 1 + 𝑡 2 e i 𝑘 𝑎 q ( k ) = t 1 + t 2 e i k a , the A 1 A 1 termination , and 𝑡 1 / 𝑡 2 = 0 . 5 t 1 / t 2 = 0.5 . Show the intercell phase , contour pole , normalization constant , adjacent probability ratio , and localization length . For ten cells , use | 𝑀 L R | = 𝑡 2 1 − 𝑟 2 1 − 𝑟 2 0 𝑟 1 0 , 𝑟 = | 𝑡 1 / 𝑡 2 | , | M LR | = t 2 1 − r 2 1 − r 20 r 10 , r = | t 1 / t 2 | , to estimate the positive near - zero energy and the full pair separation . Identify which intermediate quantity first exposes a reversed Fourier sign .
2 . Change termination or break chiral symmetry Shift the unit cell or add a staggered onsite potential . Predict which winding representative , end - state energy , and boundary localization statements remain meaningful .
Keep 𝑡 1 / 𝑡 2 = 0 . 5 t 1 / t 2 = 0.5 . First move the physical cut by one bond so the leftmost site is B 1 B 1 and the first retained bond is the 𝑡 2 t 2 bond from B 1 B 1 to A 2 A 2 . Derive the zero - energy B B - sublattice recursion and decide normalizability . Separately , return to the original A 1 A 1 cut and add a uniform 𝑚 ˆ Γ m Γ ^ with 𝑚 = 0 . 2 0 𝑡 2 m = 0.20 t 2 . Calculate the left / right sublattice - resolved energy shifts and 𝐸 ± ( 𝑘 ) = ± √ | 𝑞 ( 𝑘 ) | 2 + 𝑚 2 E ± ( k ) = ± | q ( k ) | 2 + m 2 , including the minimum positive bulk energy . Separate four claims : winding representative , normalizability , localization , and exact zero pinning .
3 . Audit a bulk – boundary claim Given a finite spectrum and boundary weights for several lengths , decide whether the data establish an isolated bulk gap and a stable localized end state rather than a finite - size accident .
For the same 𝑡 1 / 𝑡 2 = 0 . 5 t 1 / t 2 = 0.5 , A 1 A 1 - terminated chain , the positive member 𝐸 + ( 𝐿 ) E + ( L ) of the near - zero pair and the smallest bulk band separation are
cells 𝐿 L 𝐸 + ( 𝐿 ) / 𝑡 2 E + ( L ) / t 2 bulk gap / 𝑡 2 / t 2
6 1 . 2 0 × 1 0 − 2 1.20 × 10 − 2 1 . 020
8 3 . 0 0 × 1 0 − 3 3.00 × 10 − 3 1 . 010
10 7 . 5 0 × 1 0 − 4 7.50 × 10 − 4 1 . 005
12 1 . 8 8 × 1 0 − 4 1.88 × 10 − 4 1 . 002
The measured adjacent boundary probability ratio is 0 . 2 5 0 ± 0 . 0 0 5 0.250 ± 0.005 . Fit 𝐸 + ( 𝐿 ) = 𝐶 e x p ( − 𝐿 𝑎 / 𝜉 𝐸 ) E + ( L ) = C exp ( − L a / ξ E ) , compare 𝜉 𝐸 ξ E with the localization length inferred independently from the probability ratio , and decide whether the bulk gap is size stable . State which chiral - symmetry and termination evidence must accompany the numerical fit .
Appendix 22.A (advanced): Soliton Charge at a Dimerization Domain Wall A slowly varying domain wall can be calculated in the continuum rather than inferred from a slogan . Let ˆ 𝐻 D = − i ℏ 𝑣 𝜎 𝑦 𝜕 𝑥 + Δ ( 𝑥 ) 𝜎 𝑥 o n H = 𝐿 2 ( ℝ , ℂ 2 ) , H ^ D = − i ℏ v σ y ∂ x + Δ ( x ) σ x on H = L 2 ( R , C 2 ) , with the usual Sobolev domain , 𝑣 > 0 v > 0 , and Δ ( 𝑥 ) = Δ 0 t a n h ( 𝑥 / ℓ ) , Δ 0 > 0 , ℓ > 0 . Δ ( x ) = Δ 0 tanh ( x / ℓ ) , Δ 0 > 0 , ℓ > 0. The ordinary function Δ : ℝ → ℝ Δ : R → R is the continuum dimerization mass ; it is not a distribution . The numerical matrix 𝜎 𝑧 σ z represents the chiral operator because { 𝜎 𝑧 , ˆ 𝐻 D } = 0 { σ z , H ^ D } = 0 in this spinor representation . The two limits Δ ( ± ∞ ) = ± Δ 0 Δ ( ± ∞ ) = ± Δ 0 represent the two asymptotic dimerizations .
Write the zero - energy spinor as 𝜓 0 = ( 𝜓 0 , A , 𝜓 0 , B ) 𝖳 ψ 0 = ( ψ 0 , A , ψ 0 , B ) T . The Dirac equation separates :
( ℏ 𝑣 𝜕 𝑥 + Δ ) 𝜓 0 , A = 0 , ( − ℏ 𝑣 𝜕 𝑥 + Δ ) 𝜓 0 , B = 0 . ( ℏ v ∂ x + Δ ) ψ 0 , A = 0 , ( − ℏ v ∂ x + Δ ) ψ 0 , B = 0.
Only the first solution decays at both spatial infinities . With 𝑝 = Δ 0 ℓ / ( ℏ 𝑣 ) > 0 p = Δ 0 ℓ / ( ℏ v ) > 0 ,
𝜓 0 ( 𝑥 ) = 𝐶 𝑝 s e c h 𝑝 ( 𝑥 / ℓ ) ( 1 0 ) , | 𝐶 𝑝 | 2 = Γ ( 𝑝 + 1 2 ) ℓ √ 𝜋 Γ ( 𝑝 ) . ψ 0 ( x ) = C p sech p ( x / ℓ ) ( 1 0 ) , | C p | 2 = Γ ( p + 1 2 ) ℓ π Γ ( p ) .
The normalization follows from ∫ ∞ − ∞ s e c h 2 𝑝 ( 𝑥 / ℓ ) d 𝑥 = ℓ √ 𝜋 Γ ( 𝑝 ) / Γ ( 𝑝 + 1 / 2 ) ∫ − ∞ ∞ sech 2 p ( x / ℓ ) d x = ℓ π Γ ( p ) / Γ ( p + 1 / 2 ) . Reversing the domain wall moves the normalizable solution to the other component and reverses its chiral eigenvalue . The asymptotic decay length is ℏ 𝑣 / Δ 0 ℏ v / Δ 0 , whereas ℓ ℓ controls the shape of the wall ; these are not the same length in general .
The localized mode and its electric charge are separate statements . Chiral symmetry pairs every nonzero single - particle level at + 𝐸 + E with one at − 𝐸 − E . For one spinless species , define the half - filled reference by filling all negative - energy states and assigning half of the unpaired zero - level spectral weight to the neutral background . If the zero mode has occupation 𝑛 0 ∈ { 0 , 1 } n 0 ∈ { 0 , 1 } , its excess electron number and conventional electric charge are
Δ 𝑁 = 𝑛 0 − 1 2 , 𝑄 = − 𝑒 Δ 𝑁 = − 𝑒 ( 𝑛 0 − 1 2 ) . Δ N = n 0 − 1 2 , Q = − e Δ N = − e ( n 0 − 1 2 ) .
Thus an empty zero mode carries 𝑄 = + 𝑒 / 2 Q = + e / 2 and a filled zero mode carries 𝑄 = − 𝑒 / 2 Q = − e / 2 , each understood relative to the declared half - filled background and modulo the addition of an integer electronic charge . Spin degeneracy , boundaries , chiral - breaking terms , or a different occupation convention change the charge ledger even when a localized in - gap state remains . The calculation therefore does not claim that every domain wall in real polyacetylene has an unqualified charge 𝑒 / 2 e / 2 .
The nonorthogonal - localized - basis Berry - geometry candidate is owned by this lecture with disposition defer; it has no source target and is not silently used in this continuum calculation . The appendix extends the core by adding a spatially varying mass and charge accounting , but it supplies no provider needed by Lectures 23 – 24 .
A reader who enters the appendix should reconstruct the two first - order equations , select the normalizable component , verify 𝐶 𝑝 C p , and compute 𝑄 Q for 𝑛 0 = 0 , 1 n 0 = 0 , 1 . Deleting the appendix removes no winding , boundary - state , or Lectures 23 – 24 dependency : the optional fractional - charge interpretation is not used as a core premise .
References
[MAG16] Meier, E. J., An, F. A., and Gadway, B. (2016).
“Observation of the topological soliton state in the Su–Schrieffer–Heeger model.” Nature Communications 7 , 13986.
https://doi.org/10.1038/ncomms13986 .