Topological Insulators

Henry Hunt

March 25, 2020

Abstract

This is a review of topological insulators assuming the reader has anunderstanding of basic quantum mechanics but no background in con-densed matter physics. First, I provide background on the electronicstates of crystal lattices including a review of the Kronig-Penny model,the Classical Hall Effect, and some principles of topology. This reviewdoes not contain any topological proofs, but it does cite some results. Idiscuss the use of Berry Phase and the Z/2Z and Chern invariants derivedfrom it to characterize several examples of topological phases includingpolarization in the Su Schrieffer Heeger model, Thouless Pump, and theInteger Quantum Hall state.

1 Background

1.1 Bloch’s Theorem

1.1.1 Proposition in 1 Dimension

Bloch’s theorem states that if for some a, V (x) = V (x + a), then there existsan energy eigenbasis ψn,q such that

ψn,q(x) = e−iqxun,q(x) (1)

where un,q(x) = un,q(x+ a).

1.1.2 Proof

Consider the translation operator Ta such that Taf(x) = f(x + a). I claim Tais a linear operator. Let h(x) = αf(x) + g(x),

Tah(x) = h(x+ a) = αf(x+ a) + g(x+ a) = αTaf(x) + Tag(x) (2)

Therefore, Ta is a linear operator. Now calculate the commutator [Ta, H]

[Ta, H]f(x) = [Ta,−~2

2m

d2

dx2+ V (x)]f(x) (3)

1

=−~2

2m[f ′′(x+ a)− d2

dx2f(x+ a)] + [V (x+ a)− V (x)]f(x) = 0 (4)

Because Ta and H commute, they must be mutually diagonalizable. Therefore,the eigenbasis can be written as ψn,q where Taψn,q(x) = eiqaψn,q(x). Here qmight be any complex number. Let un,q(x) = e−iqxψn,q(x).

Taun,q(x) = e−iqae−iqxψn,q(x+ a) = e−iqxψn,q(x) = un,q(x) (5)

Therefore, un,q(x+ a) = un,q(x) and ψn,q(x) = e−iqxun,q(x). QED

1.1.3 Proposition in 3 Dimensions

Bloch’s theorem states that if for some ~a1, ~a2, and ~a3, V (~r) = V (~r+n1~a1+n2~a2+n3~a3) for any integers n1, n2, and n3, then there exists an energy eigenbasis ψn,~ksuch that

ψn,~k(x) = e−i~k·~run,~k(x) (6)

where un,~k(~r) = un,~k(~r + n1~a1 + n2~a2 + n3~a3).

1.1.4 Abrivated Proof

The proof extends trivially from the one given in 1D. Simply consider T~a1 ,T~a2 , and T~a3 which all commute with the Hamiltonian, and define un,~k(x) =

ei~k·~rψn,~k(x) which can easily be seen to have the above property.

1.1.5 The Bloch Hamiltonian

There is a useful reformulation of Bloch’s theorem where the ~k dependence istransferred to the Hamiltonian in the following way:

H(k) = ei~k·~rHe−i

~k·~r (7)

Because the eigenstates must have the same periodicity of the lattice, the Hilbertspace on which the Hamiltonian operates is a periodic unit cell of the lattice.This unit cell can be viewed as a 3-torus due to this boundary condition. Bythe results derived above, it is clear that there must be an eigenbasis, called

the Bloch states, of∣∣∣un,~k⟩, alternatively

∣∣∣un(~k)⟩

, and associated eigenvalues

En(k). To convert from this unit cell view to that of real space, Bloch’s theorem

shows you need only to multiply by the plane wave associated with ~k, while theenergy spectrum remains the same. For real-space eigenfunctions ψn,~k and the

corresponding Bloch eigenfunctions un(~k), the following relationship holds:

ψn,~k(~r) = e−i~k·~run(~k)(~r) (8)

This equation may seem like a meaningless mathematical maneuver, but creat-ing a Hamiltonian parameterized by a vector turns out to be significant. More-over, this notation corresponds to the conventional way that band structure plots

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are made showing E0(~k) and E1(~k). The range of the En(~k) energy eigenvaluesdefine the band structure of the solid as each eigenstate of the full Hamiltonian

corresponds to some∣∣∣un,~k⟩ with equal energy.

1.2 Periodic Potentials and Band Structure

In most cases, Bloch Hamiltonians, that is ones which are invariant under aset of transformations, will have an associated band structure of their energyeigenvalues. A band structure is a grouping of allowed energy eigenvalues. Inother words, there will be some energy ranges where there is a very high densityof states and then large gaps where no energy eigenvalues appear.

1.2.1 The 1D Kronig-Penney model

This model gives the simplest demonstration of a Bloch Hamiltonian producingband gaps in its energy spectrum. Consider the following potential function:

V (x) = α

∞∑j=−∞

δ(x− ja) (9)

This function describes a series of delta spikes with period a and strength α.Because for x 6= ja, the potential is zero, V (x) = 0, the eigenstates at x 6= 0must be

ψ(x) = Asin(kx) +Bcos(kx) (10)

for some constants A and B. Applying Bloch’s theorem and the Schrodingerequation at divergent points,

limx→+0

d

dxψ(x)− lim

x→−0

d

dxψ(x) =

2mα

~2ψ(0) (11)

one finds that,

cos(qa) = cos(ka) +mα

~2ksin(ka) (12)

Therefore, k is an allowed crystal momentum when |cos(ka) + mαk sin(ka)| ≤ 1.

This equation is not analytically solvable but upon solving numerically, onefinds that the allowed values of k are grouped in bands of increasing width as k

increases. The energy corresponding to this crystal momentum is ~2k2

2m . Becausethe energy is proportional to k2, the energy of this system is also grouped intoallowed bands. You can see in Figure 1 an example of this Kronig-Penny bandstructure with α = 1.5 ~

ma .

1.2.2 Band Structure for 3D atomic lattices

The Kronig-Penney model gave a simple example of band structure and a BlochHamiltonian, but it isn’t immediately applicable to any real materials. In realatomic latices, the potential can generally be described by Coulomb wells, and

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Figure 1: This a plot of the energy band structure based on the Kronig Penneymodel with α = 1.5 ~

ma . This plot is simplified and only shows the allow energieswithout any depiction of k dependence.

the Hamiltonian will be 3 dimensional. In general, a direct theoretical approachlike the one shown above is not used to find band structures of real materials.For example, in determining the band structure of Si and Ge, a combinationof experimental measurements and theory are used to find band structure plotslike the one in figure 2.[2]

1.3 Brillouin Zone

Considering a Bloch Hamiltonian with lattice symmetry, i.e., V (~r) = V (~r +

n1~a1 +n2~a2 +n3~a3), it follows trivially from Bloch’s theorem that ~k and ~k′ suchthat

(~k − ~k′) · (n1~a1 + n2~a2 + n3~a3) = 2πn (13)

where n is an integer have an equivalent set of un,~k and therefore an equivalentset of ψn,~k. It is also simple to derive from this result that the full set of

translations of ~k which maintain equivalent wavefunctions are integer linearcombinations of

~bi =~aj × ~ak

2π~ai · (~aj × ~ak)(14)

where i, j, and k are distinct indices. These vectors describe the reciprocallattice. Therefore one can consider the space of non-equivalent ~k, which is~k = α1

~b1 + α2~b2 + α3

~b3 with −1 < αi ≤ 1. In Figure 3, you can see anexample Brillouin zone with face-centered cubic translational symmetry, i.e.,~a1 = a

2 (~y + ~z), ~a2 = a2 (~x+ ~z), and ~a3 = a

2 (~x+ ~y). This space of non-equivalentcrystal momentum is called the Brillouin Zone.

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Figure 2: A plot of the band structure of Germanium taken from Kettel’stextbook.[2] The x-axis here represents a line running through the Broullinzone from 2π

a ( 12 ,

12 ,

12 ) to 2π

a (1, 0, 0). Each curve represents an energy eigenvalueof the Bloch Hamiltonian. The area covered grey represents the energies withinthe Fermi surface, i.e., the filled first band.

1.4 The Classical Hall Effect

The classical Hall effect is the tendency for conductors in a magnetic field per-pendicular to the flow of current to produce a voltage along the axis perpen-dicular to both the current and the magnetic field. Thinking classically, thismatches the model of electrons as a gas flowing throw the conductor, becausethe application of a magnetic field to classical point charge would apply a forceperpendicular to its velocity and the field. From the classical Drude Model, youcan derive that the ratio between the electric field and the current perpendicularto the electric field is

RH =1

nec(15)

1.5 Topology and Band Theory

Topology is the study of properties of sets which are unaffected by bi-continuousmaps, with bi-continuous meaning a bijection which is continuous in both direc-tions. Technically. the definition for continuous involves the choice of open sets

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Figure 3: Brillouin Zone of a face centered cubic lattice. [4]

in the space. However, a discussion of the formal mathematical way of defin-ing topology is beyond the scope of this presentation. Topological insulatorsare characterized by some topological invariant if adiabatic continuous transfor-mations are considered to be the bi-continuous maps of the Hamiltonian. Anadiabatic transformation is one that can be represented as an adiabatic pro-cess, that is one in which the change in the Hamiltonian is far slower than thetime scale defined by the energy gap.[6] These topological invariants are analo-gous to the topological invariants in a euclidean space such as winding number,genus, or orientability. The significance of insulating phases with a topologicalinvariant is that the surface where these invariants change have non-adiabatictransformation in their Hamiltonian which can lead to band crossing edge statesbetween materials that both have ban gaps.[8] This is often significant for thetransport properties of these materials.

2 Berry Phase

2.1 Berry Phase, Connector, and Curvature [6]

Consider a Hamiltonian parameterized by some vector, H(~R), and its corre-

sponding eigenstates,∣∣∣un(~R)

⟩, and energies, En(~(R)). Because the eigenbasis

for each ~R is entirely separate and the overall phase of the a vector in the Hilbertspace is meaningless, transformations of the type∣∣∣un(~R)

⟩→ eiφ(

~R)∣∣∣un(~R)

⟩(16)

where φ(~R) is a real function, preserve the physics of the system. The intuitionfor the Berry Phase comes from making an analogy to classical electromagnetism

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in which you can transform the vector potential by the gradient of a real scalerfield just like φ(~R) and preserve the physics of the system. This is called a gauge

transformation. Therefore, we can describe a vector field in which φ(~R) in theequation above would transform by adding its gradient, and and that vectorfield can be written as:

An = −i⟨un(~R)

∣∣∣∇~R

∣∣∣un(~R)⟩

(17)

By simply using the product rule, it is easy to see that the transformation inthe eigenbasis corresponds to the following transformation in A:

A→ A+∇φ(~R) (18)

The name given to the vector field A is the Berry connection. Up to this point,the only things involved have been extra mathematical degrees of freedom, butthe result of this manipulation suggests a real physical observable related to A.In E&M, though A is not directly observable, its curl is the magnetic field, theflux of which can be easily measured. So does the curl of this Berry connectioncorrespond to a real property of the system? Sometimes it can be, which is theBerry curvature, defined as

F = ∇×A (19)

Finally, the equivalent of the magnetic flux is perhaps the most relevant factorcalled the Berry phase:

γn =

∫D

Fn · d2 ~R = −i∫∂D

⟨un(~R)

∣∣∣∇~R

∣∣∣un(~R)⟩d~R (20)

In the following subsection, I will give examples of systems where the Berryphase and Curvature is physically relevant.

2.2 Berry Phase in Two Level Hamiltonians [6]

As was shown in this class, all two-by-two Hermitian matrices can be writtenin the Pauli basis plus the identity matrix with all real coefficients. Because alltwo level Hamiltonian’s must be Hermitian,

H = E0I + ~R · ~σ (21)

by gauge transformationH(~R) = ~R · ~σ (22)

This provides us with a parametrized Hamiltonian whose eigenvalues are

E±(~R) = ±|~R| (23)

and eigenvectors are∣∣∣u±(~R)⟩

= cos(±θR/2) |0〉+ e∓iφRsin(±θR/2) |1〉 (24)

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where θR and φR are ~R’s angular spherical coordinates. One can now consider

the Berry phase of∣∣∣u±(~R)

⟩. By using the gradient in spherical coordinates, it

simple to derive that

F =1

2sinθR (25)

Therefore, the Berry phases is

γ =1

2

∫sinθRdθRdφR =

1

2

∫dΩ (26)

In other words, the Berry phases is one half the solid angle swept out in ~R.

2.3 The Chern Invariant [1]

Now consider a Bloch Hamiltonian, H(k) with energy eigenstates un~k(~r). If younow consider the integral of the Berry curvature over the Brillouin zone, youcan simplify the integral in the following way:∫

BZ

Fn(~k)dkxdky = i

∮∂BZ

d~k ·∫d2ru∗

n~k(~r)∇~kun~k(~r) (27)

because the boundary of the Brillouin zone contains equivalent points on itsopposite faces, it may seem like this integral should be zero. However, it is

not true that un,~k(~r) = un,~k+~bi(~r), only that ψn~k(~r) = e−iθ(~k)ψn,~k+~bi(~r), where

θn(~k) is a periodic phase factor in the Broullin zone. Applying this conditionto the equation above, one obtains:

=

∮∂BZ

d~k · ∇~kθn(~k) (28)

because the path around the outside of the Broullin zone is closed, the changein θn must be a multiple of 2π. In fact,

Cn =1

2π

∫BZ

Fn(~k)dkxdky (29)

is a topological invariant called the Chern number, as long as the first is notdegenerate with any others, which can be any positive integer. The proof thatthis value is topologically invariant under adiabatic continuous maps is beyondthe scope of this paper.

2.4 Polarization in 1D and Wannier States [6]

Imagine a 1D insulator with a lattice length a. It is obvious from Bloch’stheorem that the Brillouin zone of this lattice is a circle of circumference 2π/a.In this system, the Barry phase considered over the Bloch Hamiltonian, which isthe Hamiltonian Parametrized in terms of crystal momentum, has an importantmeaning; it is 2π/e times the polarization mod e. Note that the Berry Phasecan not be written in terms of a Berry Curvature. because the parameter spaceis 1D, there is no curl of the Berry connection. This implies that the Barryphase in this system is only defined up to mod 2π due to the phase ambiguity.

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At each unit cell you can consider the polarization as the product of averagedistance between the center of the potential well and the electron and the chargeof the electron. The issue is that the Bloch eigenstates for the electrons in thelattice are distributed. Calculating the average position of a distributed statedoesn’t have any real meaning. Consider instead taking the Fourier transposeof the Blochs states

|ϕn(R)〉 =

∮BZ

dk

2πe−ik(R−r) |un(k)〉 (30)

These states are not energy eigenstates in general, however they are completebasis and they are localized, which you can show from the fact that the Blochstates are localized in momentum space. These are the Wannier States overwhich you can now consider the polarization

P = e 〈ϕn(R)| r −R |ϕn(R)〉 =ie

2π

∮BZ

dk 〈un(k)| ∇k |un(k)〉 (31)

Therefore, the Berry phase of the nth Bloch state consider of over the Brillouinzone is the polarization of the nth Wannier state. because the material here isan insulator you can consider the ground states of the system should a full E0(k)band. Therefore the polarization in the ground state should be the polarizationof the zeroth Wannier state. Therefore, the polerization in the ground stateshould be

P =ie

2π

∮BZ

dk 〈u0(k)| ∇k |u0(k)〉 (32)

or e2π times the Berry phase. As I noted at the beginning, this formula is only

accurate mod e due to the 1D Broullin zone. However, if you consider theHamiltonian for this system changing with time due to some parameter λ(t),you can take a curl of the Berry connector over the 2D space parametric by k,λ, where the curl here is a scalar. This implies that the boundary integral ofthis space over the Berry connector should be well defined. Therefore,

∆P =ie

2π

∮BZ

dk(〈u0(k, λ(t))| ∇k |u0(k, λ(t))〉 − 〈u0(k, λ(0))| ∇k |u0(k, λ(0))〉)

(33)without any caveat.

2.5 The Su Schrieffer Heeger Model[6]

This is essentially an application of the formulas derived in the two previous sec-tions to a simplified model of polyacetalene which will give us our first exampleof a topological invariant. The Hamiltonian for the system is

H =∑i

(t+ δt)c†AicBi + (t− δt)c†Ai+1

cBi + h.c. (34)

The indices in this Hamiltonian correspond to C-H groups in the lattice. Whileit is not immediately obvious that Bloch’s theorem applies to this Hamiltonian,

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if you consider the operator which shifts the lattice by two atoms it clear that itshould commute with this Hamiltonian. Therefore, you can consider the BlochHamiltonian on the system of two atoms and the corresponding En(k). TheBloch Hamiltonian for this system is

H(k) = [(t+ δt) + (t− δt)cos(ka)]σx + (t− δt)sin(ka)σy (35)

because this is a two level system we can use the formula we derive earlierthat Berry phase of a two level system is the half the solid angle swept out bythe pauli vector. because this Bloch Hamiltonian is contained entirely insidethe xy-plane, the solid angle is just πn where n is the winding number aboutthe origin. because the Bouillon zone is only length 2π/a the winding numbercan only be 0 or 1. It simple to see that the vector sweeps out a single circle

Figure 4: The polyacetalene lattice in the Su Schrieffer Heeger model and thewinding of the Bloch Hamiltonian’s 3 vector

of radius (t − δt) centered at (t + δt)x; therefore the winding number is 0 for0 < δt and 1 for 0 > δt. Now considering the 1D polarization result, it clearthat the δt > 0 state has polarization P mod e = 0 while the δt < 0 statehas polarization P mod e = e/2. This polarization mod e is in fact the firstexample of a topological invariant and is called the Z/2Z invariant. This valuestopological invariant because it is a direct map from the winding number whichitself is a topological invariant. Note here that winding number is preservedunder bi-continuous maps if winding number is consider on the surface whichis the image of the xy-plane around a center which is the image of the origin.If we consider this in the context of the Hamiltonian, this image of the originis some random energy. However, because adiabatic continuity stipulates thatband gaps must not close, the origin must map to itself. This because everyother point will have two energy eigenvalues and if such a point where map tozero the Hamiltonian set to that point for all k would have a band gap closer.In order to change from one state to another, you would need the origin to lieinside the map of the Brillouin zone. In other words the there has to be a pointin that transition where the Hamiltonian is zero from some k. This implies thatthere must be a point at which the two bands corresponding meet: a band gapcloser. If we consider a transition between the two phases as seen in figure 5,it is not immediately obvious why there must be this guaranties the existenceof a double degenerate zero edge mode. The argument for the protection ofthis by the bulk band structure state can be found in Kane’s textbook chapter

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having to due modeling the block Hamiltonian around this zero point as a DiracHamiltonian where the effective mass changes sign.[7]

Figure 5: The boundary between the two topologically states in polyactelceleproducing a protected zero mode.

3 Thouless Charge Pump [6]

Consider a one dimensional Bloch Hamiltonian which changes with time H(k, t)such that H(k, t) = H(k, t + T ). Now that this is a Hamiltonian with twodimension of parameters the Berry Phase of this system is well defined. Asshown earlier, this value is the difference in polarization from the start to theend of a cycle. It is easy to come up with a Hamiltonian such that δP from 0 toT is positive, for example consider V (x) = E0sin(k(x+ t/c)). This implies thatan insulator with such a Hamiltonian should have a monotonically increasingpolarization in other words this system pumps charge from end to the other,hence the name Thouless Charge Pump. If we now consider the possible valuethat the amount of charge pumped in one cycle, because the the Hamiltonianat given t can be consider at as once dimensional Bloch Hamiltonian we cancalculate the polarization up to mod e. because the 1D Hamiltonian at the endand beginning of the cycle is the same it, ∆P mod e = 0. Therefore, ∆P = ne,where n is an integer. As shown above, this integer is defined by the integral

n =1

2π

∫BZ⊕T

Fdkdt (36)

where T is just a circle of circumference T and F is the Berry curvature. becausethe Broullin zone is also a circle, this is identical to integration over a torus.This value is topological invariant, separate from the Z/2Z invariant, called theChurn number. In this system the Chern number represent the change in thePolarization in a single cycle, but in other systems it may represent a differentinvariant. So why is this invariant still significant? We had a clear argumentearlier for why the polarization mod e must be invariant in a 1D insulator as itcorresponded to the winding number of the map of the Broullin zone.

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4 Integer Quantum Hall Effect

The integer quantum hall effect is the name for the hall conductance of 2Delectronic systems.[9] In these systems the hall conductance, ρxy = I/VHall, islimited to

ρxy = Ne2

h(37)

for some integer N. This system also has zero resitivity along the axis of theelectric field. So, why does this happen.

As we shown in class, a 2D electron free gas in a magnetic field has the same

energy spectrum as a simple harmonic oscillator with ωc = eBcme

and l =√

~ceB .

H =1

2me

[p2x + (py −

eB

cx)]

=p2x

2me+

1

2meωe(x− kyl2) (38)

The landau levels of this system are separated by large energy gaps and arehighly degenerate, which make them some what analogous to energy bands inBloch Hamiltonian. In fact, another, view of the electron gas system is asa Bloch Hamiltonian. While x and y operators no longer commute with theHamiltonian, shifts by ax and ay such that a2B = hc/e do commute withthe Hamiltonian.[1] That is translations which in close a quanta of magneticflux.[7] because these translation commute with Hamiltonian, it can be writtenas a Bloch Hamiltonian on a square unit cell of side length l. In this case, thelandau levels truly Bloch are energy bands, thus we can apply result havingto do with the conductivity of latices. In particular, a results from transporttheory states that for a perturbing potential H ′ = e ~E · ~r where ~E varies slowly

⟨un~k∣∣~v ∣∣un~k⟩ =

1

~∂En(~k)

∂~k+ ~va(n,~k) (39)

Note ~va is the anomalous velocity that is the first order correction to the ex-pected velocity due to the electric field causing band mixing.[1] You can derivethat the

~va(n,~k) =e

h~E × ~Fn(~k) (40)

where Fn is the Berry curvature. Consider an state where all electrons lie in thefirst band of the quantum hall Hamiltonian with an electric field applied to it.The terms for the expected velocity from the dispersion of the non-perturbedHamiltonian must be zero, as there should not be any net current in groundstate. Therefore the net current density in the ground state should be:

~Jn = − eA

∑~k∈BZ

~va(n,~k) =e2

h

1

2π

∫BZ

Fn(~k)dkxdky =e2

hCn (41)

where Cn is the Chern invariant. Therefore, this transport property in theinteger quantum hall state are in fact the derived from the topological phase

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of the band structure. Similar to the Su Schreef Heeger model this impliestopological invariant implies the existence of protected edge states. I will notderive this result here, but the edges between the IQH state and the vacuumare chiral mono-directional states. This means that what ever local deformationthere is in the edge of the insulator there is no to back scatter electron inthese states without changing their spin.[6] These state are the cause of thelongitudinal super conductance.

References

[1] S.M. Girvin and K. Yang, Modern Condensed Matter Physics (CambridgeUniversity Press, Cambridge, United Kingdom, 2019). p.304-361

[2] C. Kittel, Introduction to Solid State Physics, 8th ed. (Wiley, Hoboken, NJ,2005).

[3] D.J. Griffiths and D.F. Schroeter, Introduction to Quantum Mechanics(Cambridge University Press, Cambridge, 2019).

[4] Wikipedia (2020). (Only used for a vector graphic.)

[5] C. Kane, Simon’s Foundation Lectures.

[6] C. Kane, Topological Band Theory I-III.

[7] C.L. Kane, Topological Band Theory and the Z/2Z Invariant.

[8] M.Z. Hasan and C.L. Kane, REVIEWS OF MODERN PHYSICS 82, 3045(2010).

[9] K.I. Wysokinski, European Journal of Physics 21, 535 (2000).

Note: this bibliography used the American Institute of Physics citation style,which is at times sparse. All of the online source above are hyperlinks whichcan be clicked on and will take you directly to the source.

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