Topology in condensed matter physics

Magda Marganska- Lyzniak

(Dated: May 7, 2016)

The geometry of an atomic lattice and its chemical structure determine the electronicproperties of a solid. All the band structures of solids can be classified according to theirtopological properties. If these properties are nontrivial, we observe such extraordinaryphenomena as insulators with conducting surfaces, currents running without dissipationor particles which are their own antiparticles. In this course we aim to introduce thetopology as a tool for condensed matter physics, study its consequences in systemsof various symmetries and dimensionality, and explore the experimental signatures ofnontrivial topology.

CONTENTS

I. Preface 1

II. Equivalence relations 1

III. Topological spaces and invariants 6

IV. Geometric phase 12A. Two-level system 13B. Geometric phase in optics: Pancharatnam’s phase 15C. Berry phase in the Bloch bands 16D. Adiabatic current 17

I. PREFACE

This course is based on two books: Mikio Nakahara’s “Geometry, topology and physics”1 provides the mathemati-cal foundation, B. Andrei Bernevig’s “Topological insulators and topological superconductors” provides the physicalinterpretation. There are also several other books and scripts, where the ideas of the authors help to understand thetopic in question in more ways than one. The ones which I used most are

• Shun-Qing Shen “Topological insulators - Dirac equation in condensed matters”

• “Topological insulators”, edited by Marcel Franz and Laurens Molenkamp (2013)

• Raffaele Resta “Geometry and topology in electronic structure theory”, http://www-dft.ts.infn.it/~resta/gtse/draft.pdf

• Janos K. Asboth, Laszlo Oroszlany, Andras Palyi “A short course on topological insulators” http://arxiv.

org/abs/1509.02295

• Joel E. Moore “Notes for the MIT minicourse on topological phases”, http://socrates.berkeley.edu/

~jemoore/Moore_group,_UC_Berkeley/Welcome_to_the_Moore_group,_Department_of_Physics,_UC_Berkeley_

files/mitcourse.pdf

• Joel E. Moore “An introduction to topological phases of electrons” http://cmt.berkeley.edu/sites/default/

files/LH_merged.pdf

II. EQUIVALENCE RELATIONS

In this course we will use several powerful tools. The first one is the correspondence between the physical parametersor quantum numbers of the system and the Hilbert space of quantum states. This correspondence is a mapping, definedby the system’s Hamiltonian.

1 The German translation is available as an ebook from the university library.

2 Topology in physics

The second tool is the eqiuvalence, under certain conditions of the interesting objects whose properties we do notknow, and the familiar objects, whose properties we know well. This equivalence, in the context of topology, is calledhomeomorphism, which means that the two objects can be continuously deformed into each other.Let us start with maps.Definitions:

Let X, Y be sets. A map or mapping is the rule by which we assign to elements x ∈ X the elements y ∈ Y . It isdenoted by f : X → Y and f : x→ f(x). The set X is the domain of the map, the set Y the range of the map. Theset of all y’s such that y = f(x) for some x ∈ X is called the image of the map. For a given y belonging to the imageof the map, the inverse image of y is defined as f−1(y) = x ∈ X|f(x) = y.Note that a map is only fully defined when both its domain and range are specified. For instance, consider the followingmaps:

f : x→ exp(x) if X,Y = R, y = −1 has no inverse image

if X,Y = C, f−1(y) = (2n+ 1)π i|n ∈ Z

f : x→ sin(x) if X,Y = R, the range of the map is [−1, 1]

if X,Y = C, the range of the map is the whole complex plane.

Maps can have, among other, the following properties:

(i) A map f : X → Y is injective (or one to one) if x 6= x′ implies f(x) 6= f(x′).

(ii) A map f : X → Y is surjective (or onto) if eachy ∈ Y is the image of some x ∈ X.

(iii) If a map is both injective and surjective, it is bijective.

Examples:

f : R→ R defined by f : x→ ax (a ∈ R\0) is bijective,f : R→ R defined by f : x→ x2 is neither injective nor surjective,f : R→ R defined by f : x→ exp(x) is injective but not surjective,f : R→ R defined by f : x→ sin(x) is neither injective nor surjective, but if we restrict its domain and range so thatf : [−π/2, π/2]→ [−1, 1], it is bijective.

Definitions: Given two maps f : X → Y and g : Y → Z, the composite map of f and g is a map g f : X → Zdefined by g f(x) = g(f(x). A diagram of maps is called commutative if any composite maps between a pair ofsets do not depend on how they are composed. For instance, f, g : R → R, where f : x → x + n, n ∈ Z andg : x→ frac(x) := x− bxc form a commutative diagram of two maps, illustrated below.

f

R

RR

f

g

The identity map idX : X → X is defined by idX(x) = x. If f : X → Y defined by f : x → f(x) is bijective, thereexists an inverse map f−1 : Y → X such that f−1 : f(x)→ x, which is also bijective. The maps f and f−1 satisfyf f−1 = idY and f−1 f = idX . Conversely, if f : X → Y and g : Y → X satisfy f g = idY and g f = idX , thenf and g are bijections.

Examples:

f : R→ (0,∞) is a bijection defined by f(x) = exp(x). The inverse map is f−1 : x→ lnx.Let M be an element of the general linear group GL(n,R), whose matrix representation is given by n × n matriceswith non-vanishing determinant. Then M : Rn → Rn, x→Mx is bijective and the inverse map is given by the inverseof the matrix M , M−1. (If detM = 0, M would be neither injective nor surjective.)The n-dimensional Euclidean group En is made of an n-dimensional translation a : x → x + a (x, a ∈ Rn) and anO(n) rotation, R : x→ Rx, R ∈ O(n). A general element (R, a) of En acts on x by (R, a) : x→ Rx+ a. The productis defined by (R2, a2) (R1, a1) : x → R2(R1x + a1) + a2, i.e. (R2, a2) (R1, a1) = (R2R1, R2a1 + a2). The inversemaps are a−1 : x→ x− a, R−1 : x→ R−1x, (R, a)−1 : x→ R−1x−R−1a− a.

Topology in physics 3

Definitions: Suppose that the setsX,Y are endowed with some algebraic structures (product, addition...). If f : X → Ypreserves these structures, it is called a homomorphism. For instance, if the structure in question is a product, thenf : ab→ f(a)f(b). Note that ab is a product in X, while f(a)f(b) is a product in Y . If a homomorphism is bijective,it is called an isomorphism and X is said to be isomorphic with Y , X ∼= Y .The elements of a set can be in a relation, which is also a mapping: ∼: X × X → True, False. The examples ofrelations are >,<,=,∈. If the relation satisfies the three following conditions:

(i) a ∼ a (reflective)

(ii) if a ∼ b then b ∼ a (symmetric)

(iii) if a ∼ b and b ∼ c then a ∼ c (transitive),

the relation is called an equivalence relation. One example of such a relation is introduced by the division modulo2: x ∼ y if (x− y)|mod 2 = 0.

Given a set X and an equivalence relation ∼, we can partition X into mutually disjoint sets called equivalenceclasses. A class [a] is made of all the elements x ∈ X such that x ∼ a: [a] = x ∈ X|x ∼ a2The set of all equivalence classes is called the quotient space, denoted by X/ ∼. The element a (or any element in[a]) is called a representative of [a].

Examples:

The division modulo 2 separates Z into even and odd integers. We can choose the representatives to be 0 and 1. Thenthe quotient space Z/mod 2 is isomorphic with the cyclic group Z2, whose algebra is defined by 0+0=0, 0+1=1=1+0,1+1=0. If the division is modulo n, the quotient space Z/mod n is isomorphic with Zn.Another important quotient space is R, where we have impose the equivalence of all points which are separated by2πn, where n ∈ Z ( x ∼ y if there exists such n ∈ Z that y = x+ 2πn). The class [x] is the set ...x−2π, x, x+ 2π, ....All classes in R have their representatives in the interval [0, 2π), which is the quotient space R/ ∼. We can picture itas sketched below.

0 2π

∼

x x+ 2π

relation

2π

0 ∼ 2π

0

quotient space

R/ ∼ S1

∼=

R with the equivalence

Note that the points x = ε and x = 2π − ε lie far from each other on R, but close in the quotient space. The notionof the closeness of points is one of the central ones in topology.

Similar equivalence relation can be defined on R2, and they yield several quotient spaces which are either veryinteresting, or very useful, or both.

(i) Let (x1, y1) ∼ (x2, y2) if x2 = x1 + 2πn, where n ∈ Z. The resulting quotient space is a cylinder.

2 Prove that two classes [a] and [b] satisfy either [a] = [b] or [a] ∩ [b] = ∅.

4 Topology in physics

−π π

x

y

(ii) Let (x1, y1) ∼ (x2, y2) if x2 = x1 + 2πn ∧ y2 = y1 + 2πm, where n,m ∈ Z. The resulting quotient space is atorus, denoted T 2. We will encounter it very often, as it is the Brillouin zone of a 2D square lattice.

x

y

(iii) Let (x1, y1) ∼ (x2, y2) if x2 = x1 + 2πn ∧ y2 = (−1)ny1, where n ∈ Z. The resulting quotient space is called aMobius strip or Mobius band. It is non-orientable - it has only one surface, and is in fact a subspace of all othernon-orientable spaces3.

x

y

(iv) Let (x1, y1) ∼ (x2, y2) if x2 = x1+2πn ∧ y2 = (−1)ny1+2πm, where n,m ∈ Z. The quotient space is a so-calledKlein bottle, which is actually impossible to embed in our R3 space without it intersecting itself. In higherdimensions the Klein bottle has no intersections with itself - just as a twisted 1D loop has a self-intersection ifit is embedded in R2, but in R3 we can lift one of the loop’s sides above the plane.

3 You are invited to try to cut the Mobius band along the central line or at one third and two thirds of the strip’s height. Interestingfigures result.

Topology in physics 5

x

y

(v) Let (x1, y1) ∼ (x2, y2) if x2 = (−1)mx1 + 2πn ∧ y2 = (−1)ny1 + 2πm, where n,m ∈ Z. The resulting quotientspace is the real projective plane, denoted RP 2, also called a cross-cap.

x

y

The cross-cap is self-intersecting in R3. It can be visualized as a hemisphere where the antipodal points on theequator are identical. The cross-cap represents also the set of directions which a non-oriented rod (i.e. not anarrow, which has a head and a tail) can take. Because the rod is not oriented, the directions differing by πcorrespond to the same position of the rod.

=P’ P

The twisted quotient spaces, the Mobius band, the Klein bottle and the cross-cap are so unusual that they keepinspiring artists of any ilk. See below a doubly twisted architectonic Mobius band, a glass-blown Klein bottle and acrocheted cross-cap.

Design: Vincent Callebaut, source: VincentCallebaut Architectures

Design: Clifford Stoll,Photo: CC BY-SA 3.0,source: Wikipedia

Design: Matthew Wright, http://mrwright.name/crochet/

6 Topology in physics

III. TOPOLOGICAL SPACES AND INVARIANTS

There are several definitions of topological spaces. Here I give two: one which emphasizes the closeness of points (theneighbourhood definition), one which emphasizes their separability (the open sets definition).Definitions:

Open set definition. Let X be any set and T denote a certain collection of subsets of X. The pair (X, T ) is atopological space if T satisfies the following requirements:

(i) ∅, X ∈ T

(ii) the union (finite or infinite) of any collection of subsets from T also belongs to T

(iii) the intersection of a finite number of subsets from T also belongs to T .

T is then said to give a topology to X.

Neighbourhood (Hausdorff) definition. Let X be a set and N a function assigning to each element from X (henceforthcalled a point) a non-empty collection of subsets of X. The elements n ∈ N(x) are called neighbourhoods of x withrespect to N . N is called a neighbourhood topology if the axioms below are satisfied:

(i) if n ∈ N(x), x ∈ n (each point belongs to all its neighbourhoods)

(ii) if m ∈ X and contains a neighbourhood of x, m ∈ N(x) (a superset of a neighbourhood of x is again aneighbourhood of x)

(iii) the intersection of two neighbourhoods of x is a neighbourhood of x

(iv) any neighbourhood n of x contains a neighbourhood m of x such that n is a neighbourhood of each point in m.

Then (X,N) is called a topological space.

Examples:

Let X = A,B,C. We can define several T ’s:

A B C

A B C

A B C

A B C

∅

∅

∅

∅T1 = ∅, X

T2 = ∅, A,X

T3 = ∅, A, A,B, X

T4 = ∅, A,B, B,C, X

The sets T1, T2 and T3 define indeed topologies. The set T4 does not, because it violates the third axiom of the openset definition, B = A,B ∩ B,C /∈ T4.

There are several special ways in which we can introduce a topology. If T = ∅, X, we have a trivial topology -there is only one neighbourhood and all points belong to it. If T contains all subsets of X (including the intersectionsof infinite collection of open sets), the topology is discrete. The usual topology on R contains all open intervals(a, b) and their unions (a and b can be −∞,+∞). By extension, on Rn the usual topology contains all open intervals(a1, b1)× (a2, b2)...× (an, bn) and their unions.The set X can be endowed with a mapping d : X ×X → R, such that

i d(x, y) = d(y, x)

ii d(x, y) ≥ 0, and d(x, y) = 0 if and only if x = y

iii d(x, y) + d(y, z) ≤ d(x, z) (triangle inequality).

Topology in physics 7

This map is then called a metric and it gives X a topology defined as the collection of all open discs Uε(x) = y ∈X|d(x, y) < ε. This is the metric topology. The topology on X may be inherited from its superset; if (X, T ) is atopological space, we can define a relative topology on X ′ ⊂ X, given by T ′ = Ui ∈ T |Ui ∈ X ′. In this way thesphere Sn, defined as x ∈ Rn+1|d(x, 0) = 1, inherits the topology defined by the metric on Rn+1.

When are two topological spaces equivalent, and what does “equivalence” mean in this context? The central ideain topology is that of homeomorphism. A map f : X → Y is said to be continuous if the inverse images of opensets in Y are open sets in X. (Note that this relation is not symmetric: the image of an open set in X does not haveto be open, consider e.g. f : R → R such that f(x) = x2. It is continuous, and the image of the open set (−ε, ε) is[0, ε2) which is neither open nor closed.)If the continuous map f : X → Y has an inverse map f−1 : Y → X, which is also continuous, it is called a homeo-morphism. It defines an equivalence relation, dividing all topological spaces into equivalence classes.

This is the source of the common statement that “topology is the science of objects made of rubber” - the equiva-lence classes consist of spaces which can be continuously deformed into each other, without cutting or pasting. Twoexamples are shown below - the first one is the famous coffe cup/doughnut equivalence. The second is more subtle:two linked rings (a so-called Hopf link) are homeomorphic to two unlinked rings. We can continuously deform thesetwo objects into each other only when we embed them into R4, which requires some stretching of the imagination.We can also define the homeomorphism directly.

linked rings are homeomorphic

with unlinked rings

a cup is homeomorphic

with a doughnut

If we do not know the homeomorphism though, in general there is no way to determine whether two spaces are home-omorphic. We can say what needs to be conserved when deforming one space to the other, i.e. define topologicalinvariants. It does not help us to make a positive statement that two spaces are homeomorphic, because even ifsome invariants are the same, others may be different, and we can never be certain that we know all of them. Whatwe can say instead, is that if we find a topological invariant which is different in two spaces, they are certainly nothomeomorphic.

Definitions:

Connectedness. A topological space X is connected if it cannot be written as a disjoint sum of other spaces. It isarcwise connected if for any two points x, y ∈ X we can define a continuous mapping f : [0, 1] → X such thatf(0) = x and f(y) = 1. For all spaces with which we will be concerned in this course the two definitions will beequivalent.A loop is a map f : [0, 1] → X such that f(1) = f(0). If any loop in X can be continuously shrunk to a point, Xis simply connected. For instance: R\0 is disconnected. R2\0 is connected, but not simply connected, andR3\0 is simply connected.

Compactness. A set A ⊂ X is called closed if its complement X\A is open. The smallest closed set containing A iscalled a closure of A and denoted A. The largest open set contained in A is called the interior of A and denoted A,while A\A is the boundary. In Rn the set is compact if it is closed and bounded. This is an important property - infact, we need it for most of our band structure calculations, when we apply periodic boundary conditions on a infinitebulk lattice, turning it effectively from an Rn into Tn. Note, however, that the procedure of compactification changesa topological invariant - Rn and Tn are not homeomorphic. We are allowed to work on a Tn instead of Rn, becauseour Rn has an equivalence relation ∼, the set of translations by lattice vectors. Now, the quotient space Rn/ ∼ ishomeomorphic to T 2.

Basing on these two properties we can already identify some pairs of spaces which are not homeomorphic.

8 Topology in physics

1. The closed interval [0, 1] is not homeomorphic to the open interval (0, 1) even though both are simply connected,because one is compact and the other not.

2. Interestingly, an open interval is homeomorphic with the whole R. As an example consider X = (−π/2, π, 2), Y =R and the map f : X → Y, f(x) = tan(x). This map is continuous and bijective, and has an inverse f−1(y) =arctan(y), which is also continuous and bijective given our choice of X and Y . Thus boundedness is not atopological invariant.

3. The circle S1 and the closed interval [0, 1] are not homeomorphic, even though both are compact and connected,because the latter is simply connected while the former isn’t.

4. A sphere S2\N (without the North Pole) and the plane R2 are homeomorphic, where the homeomorphism isdefined by the stereographic projection (see below). Both are simply connected and non-compact.

N

S

P

P’

If we close the sphere by adding the North Pole and close R2 by adding ∞, we obtain again two homeomorphicspaces, S2 and R2 ∪ ∞, this time both doubly connected and compact. This procedure is called one-pointcompactification.

The properties defined above are topological invariants, but there are also others, which are more quantifiable.

One of them is based on the equivalence of maps. Let f, g : X → Y be continuous maps. If we can define acontinuous map F : X × [0, 1]→ Y such that F (x, 0) = f(x) and F (1) = g(x), the map is said to be homotopic to g(denoted f ∼ g) and F is the homotopy between f and g. If there exist continuous maps f : X → Y and g : Y → Xsuch that f g ∼ idY and g f ∼ idX , Y is said to be of the same homotopy type as X. The map f is then thehomotopy equivalence, and g the homotopy inverse.

Example:

An interval X = (0, 1) is of the same homotopy type as a point Y = 0. We can define f : X → Y, f(x) = 0 andg : Y → X, g(y) = 1/2. The composite map g f = idY , and f g ∼ idX , because we can find a continuous mapF : X × [0, 1]→ X, given by F (x, a) = ax+ (1− a)/2 and indeed F (x, 0) = 1/2 and F (x, 1) = x.

If two spaces X and Y are homeomorphic, they are also of the same homotopy type, but the reverse is not true.For instance, a disc is of the same homotopy type as a point (disc can be continuously shrunk to a point), but the twoare not homeomorphic.Nevertheless, “of the same homotopy type” defines an equivalence relation in the set of topological spaces, albeitweaker than the homeomorphism does.

The mapping f : [0, 1]→ X defines a path in X. If on top of that f(0) = f(1), the path is a loop. The concatena-tion of paths defines a group structure in the space of all paths on X, with the neutral element being the constantpath (or constant loop) which maps [0, 1] onto a single point. The relation of homotopy defines the equivalenceclasses of loops which can be continuously deformed into each other. The set of all the equivalence classes defines thefundamental (or first) homotopy group of X, π1(X).

Examples:

1. All loops in R2 can be continuously shrunk to a point, therefore π1(R2) = 0.

2. All loops on S1 fall into the homotopy classes defined by the number of times they encircle S1, called theirwinding number, as illustrated below. The black paths are some arbitrary loops, and the blue ones are therepresentants of their homotopy class.

Topology in physics 9

w=0 w=1 w=2

The paths which do not encircle S1 are homotopic with a constant loop and have winding number w = 0. Thosewhich encircle it once (w = 1) are all in the same homotopy class, and they cannot be deformed to those whichencircle it twice (w = 2). The first homotopy group of S1 is then π1(S1) = Z.(Juan-Diego has a nice analytical insight here: if we parametrize each path as exp(iωt), with t ∈ [0, 1], the loopsare closed only if ω = 2πn, with n ∈ Z.)

3. The closed paths on the real projective plane, shown below, fall into two homotopy classes: those which crossthe equator (∼ β) and those which don’t (∼ α). Those which cross the equator twice (∼ γ), or indeed any evennumber of times, can be again shrunk to a point and thus are homotopic to α. The first homotopy group of RP 2

is π1(RP 2) = Z2.

γ

β

α

If two spaces X,Y are homeomorphic, their first homotopy groups are equal. Again, the reverse statement is nottrue.Also higher homotopy groups can be defined, consisting of the equivalence classes of closed surfaces and hypersurfaces.We’ll return to the homotopy groups when we arrive at the classification of topological defects (if we have time).

Euler characteristic. Let us now restrict ourselves to R3 and the polyhedra embedded into it. A polyhedron isa closed object consisting of NF faces, NE edges and NV vertices - we shall call them all simplexes. Note that theboundary of a simplex is a simplex one level lower, e.g. the boundary of a face is a set of edges.Let X be a figure in R3 which is homeomorphic to a polyhedron K. The Euler characteristic of X is then definedas

χ(X) = NF (K)−NE(K) +NV (K). (1)

The Poincare-Alexander theorem states that the Euler characteristic χ(X) does not depend on what polyhedron Kwe choose to calculate it, as long as K is homeomorphic with X.

Examples:

1. The Euler characteristic of a point is χ(point) = 0 − 0 + 1 = 1. The Euler characteristic of a line is χ(line) =0− 1 + 2 = 1. This example shows that the equality of Euler characteristics does not imply the homeomorphismbetween sets.

2. The Euler characteristic of a cube is χ(cube) = 6 − 12 + 8 = 2. The Euler characteristic of a sphere can befound from our quotient-space picture: the sphere is homeomorphic to a square patch with the edges identifiedas shown below. (The edges of the same colour become one edge, the vertices of the same colour become onevertex, and there is only one face.)

10 Topology in physics

χ(S2) = 1−2 + 3 = 2, which is indeed the same as that of the cube. That was the content of the original Euler’stheorem: for every polyhedron homeomorphic to S2, NF −NE +NV = 2.

3. We can apply the same method to finding the Euler characteristic of a cylinder (diagram shown below).

χ(cylinder) = 1 − 3 + 2 = 0, which is actually the same as that of the Mobius strip. The Euler characteristicdoes not know about the orientability of the surface in question.

4. Let us take the torus T 2. We can continuously deform it into a polyhedron K shown below, with χ(T 2) =χ(K) = 12− 24 + 12 = 0.

We can also calculate it from the geometric representation of the torus as a quotient space of R2 in the previouslecture. The four vertices of the square patch are identified into one, out of four edges only two remain, and thepatch has one face. Then χ(T 2) = 1− 2 + 1 = 0.

5. In the same way we can calculate the χ of a Klein bottle (χ(Klein bottle) = 1) and of a real projective plane(χ(RP 2) = 1).

The connected sum X#Y of two spaces X and Y is constructed by removing a disc from each of the two spacesand connecting the created holes by a cylinder, as shown below for two spheres.

removed faces

X = S Y = S22

Topology in physics 11

The Euler characteristic of a connected sum is given by

χ(X#Y ) = χ(X) + χ(Y )− 2.

This is easy to prove, if we deform the cylinder to a trigonal prism, which we are allowed to do. In the process ofremoving the discs (now triangles) we have added to each space three vertices and three edges, while removing oneface. By connecting them with the prism we have added three edges and three faces. In the end

χ(X#Y ) = (χ(X)− 1− 3 + 3) + (χ(Y )− 1− 3 + 3) + 3− 3 = χ(X) + χ(Y )− 2.

The Euler characteristic can also be defined for smooth surfaces, and there it is given by the Gauss-Bonnet theoremas the integral of the local curvature over the whole surface. We leave now the domain of pure topology and enterthat of differential manifolds. Since the differential geometry is usually treated in the courses on electrodynamics andfield theory, I’ll assume most of the basic definitions to be known.

Let us consider a smooth (i.e. infinitely differentiable) closed surface M of arbitrary shape. At every point p on thesurface we can define a tangent plane Tp. In the local system of Cartesian coordinates on the plane the equation of thesurface is given by z = z(x, y), with z(0, 0) = 0 since (x, y)p = (0, 0). The equation of the surface in local coordinatescan be approximated (locally) as

z(x, y) ≈ 1

2(x y)

∂2z∂x2

∂2z∂x∂y

∂2z∂y∂x

∂2z∂y2

( x

y

). (2)

The Hessian matrix contains only the second derivatives because the tangency of the plane requires∂z

∂x|p =

∂z

∂y|p = 0.

The matrix is real and symmetric. It can be diagonalized and has two eigenvalues λ1, λ2 which correspond to twoeigendirections in the local coordinate system. The magnitude of λi is the inverse radius of curvature and the signtells whether the surface curves towards the positive z direction in our local system, or away from it. The Gaussiancurvature Ωp is the determinant of the Hessian at the point p. In general it can be positive, negative, or zero. TheGauss-Bonnet theorem states that

1

2π

∫M

dσΩp = 2(1− g) = χ(M). (3)

The number g is the so-called genus of the surface, equivalent to the number of “handles” in the surface4.

Example:

Let us consider a sphere S2 with radius R. The equation of the sphere in the local coordinates of the tangent planeat point p is

z = R−√R2 − x2 − y2 ' x2 + y2

2R,

and the Hessian matrix is

Hp =

(1/R 0

0 1/R

).

We obtain then Ωp = 1/R2, independent of p. Its integral over the sphere is∫ 2π

0

dϕ

∫ π

0

dθR2 sin2 θ1

R2= 4π = 2π(2− 0).

The genus of the sphere is confirmed to be 0.

4 The Euler characteristic can also be defined for surfaces with a boundary. We calculate it by first capping off each boundary componentwith a disc, and subtracting the number of discs (χ(disc) = 1) from the Euler characteristic of the closed surface.

12 Topology in physics

IV. GEOMETRIC PHASE

The Euler characteristic was our first topological invariant which could be expressed in terms of an integral overour topological space. We are going to see now another, directly related to physics, and it will be directly related tothe so-called geometric or Berry phase.

The original 1984 paper of Michael Berry (now Sir Michael Berry) described the evolution of a quantum systemunder an adiabatic variation of its parameters. An adiabatic evolution means that the system at every moment intime is in an eigenstate of the current Hamiltonian. The variation in time happens through a manipulation of theparameters which define the Hamiltonian - those parameters may be the electric field, the magnetic field, the pressure,the deformation of the atomic lattice and many others. In the following we are going to use a generic set of parametersR = R(t). We introduce a so-called snapshot basis, defined by

H(R(t)) |n(R(t))〉 = εn(R(t)) |n(R(t))〉 .The adiabatic evolution along a closed loop in the parameter space brings the state to the same as the initial one butwith a possible difference in phase.

|ψn(t)〉 = exp(iγn(t)) exp

− i~

∫ t

t0

dt′ εn(t′)

|n(R(t))〉 .

The second phase factor is the usual dynamical phase, the first phase factor is the geometric phase. Using theSchrodinger equation

i~∂

∂t|ψn(t)〉 = H(R(t)) |ψn(t)〉 ,

we obtain

LHS : i~ ∂∂t |ψn(t)〉 = −~ ˙γn(t) |ψn(t)〉+ εn(t) |ψn(t)〉+ i~ exp(iγn(t)) exp

− i~

∫ t

t0

dt′ εn(t′)

∣∣∣∣ ∂∂tn(R(t))

⟩,

RHS : H(R(t)) |ψn(t)〉= εn(t) |ψn(t)〉 ,

⇒ γn(t) = i 〈n(R(t))| ∂∂t|n(R(t))〉 = i 〈n(R)| ∇R |n(R)〉 ∂R

∂t,

where the whole dependence on time is now included in the parameters R. The integration over the cycle C in theparameter space yields the total Berry phase

γn(C) = i

∫ tC

t0

dt′ 〈n(R)| ∇R |n(R)〉 ∂R

∂t′= i

∮C

dR 〈n(R)| ∇R |n(R)〉 .

The integrand in this formula is the so-called Berry connection,

An(R) = i 〈n(R)| ∇R |n(R)〉 .The meaning of a “connection” is exactly what it says. It provides us with the information on how the objectschange when they are transported from one location on our manifold to another. One type of a connection is theparallel transport, which tells us how to transport vectors from one point on our surface to another so that theyremain parallel. The recipe, in a nutshell, is “connect the two points with a geodesic line and transport the vectorin such a way that the angle between the vector and the geodesic is constant”. On some surfaces, such as that of acylinder, it is possible to transport the vector on any closed path so that its final direction is parallel to the initial one.On the surface of a sphere this is possible only when the closed path is a geodesic one, i.e. a great circle, as shown below.

parallel transport

on a cylinder

with the "Berry phase" of

π/2

over a great circle

parallel transport on a sphere

Topology in physics 13

The connection An(R) is gauge-dependent. If we transform |n(R)〉 → eiζn(R) |n(R)〉,

An(R)→ i 〈n(R)| e−iζn(R)∇Reiζn(R) |n(R)〉 = −∇Rζn(R) 〈n(R) |n(R)〉+i 〈n(R)| ∇R |n(R)〉 = An(R)−∇Rζn(R).

The total Berry phase changes by ζ(R(T ))− ζ(R(0)). If we impose that the phase must be smooth and single-valued,this phase difference must be an integer multiple of 2π. This implies that the Berry phase is defined only modulo 2π,i.e. it has an integer ambiguity.From the conservation of the norm it follows that the Berry connection, as well as the Berry phase, is purely imaginary:

0!= ∂t 〈n(t) |n(t)〉 = 〈∂tn(t) |n(t)〉+ 〈n(t) |∂tn(t)〉 → 〈∂tn(t) |n(t)〉 = 〈n(t) |∂tn(t)〉∗ = −〈n(t) |∂tn(t)〉 ∈ iR.

Basing on our connection, we can define in analogy to electrodynamics a gauge field tensor, called Berry curvature

Ωnµν(R) =∂

∂RµAnν (R)− ∂

∂RνAnµ(R)

= i

[⟨∂n

∂Rµ

∣∣∣∣ ∂n∂Rν⟩−⟨∂n

∂Rν

∣∣∣∣ ∂n∂Rµ⟩]

.

In contrast to the Berry connection, Berry curvature is gauge-invariant, which is easily checked. In the particular caseof R3 we can define the Berry field vector, Ωn(R) = ∇R ×An(R), which is connected to the Berry curvature tensorby Ωnµν(R) = εµνλ(Ωn(R))λ. Using Stokes theorem we can also express Berry phase as a surface integral

γn(C) =

∮C

dR · An(R) =

∫∫SC

dRµ ∧ dRν 1

2Ωnµν(R) =

R3

∫∫SC

dS ·Ωn(R),

where SC is any surface whose boundary is the loop C.The Berry connection and Berry curvature can be recast in a form which does not require the differentiation of thesnapshot basis, which is especially a boon when our basis states have been obtained numerically. Since the phaseγn(C) is purely imaginary, we can write it as

γn(C) = −Im

∫∫SC

dS · ∇R × 〈n |∇Rn〉 = −Im

∫∫SC

dS · 〈∇Rn| × |∇Rn〉

= −Im

∫∫SC

dS ·∑m6=n

〈∇Rn |m〉 × 〈m |∇Rn〉 . ← 〈n |∇Rn〉 is imaginary

From the Schrodinger equation we have

〈m|H∇R |n〉 = 〈m| ∇R(H |n〉)− 〈m| (∇RH) |n〉 ,

Em 〈m |∇Rn〉 = En 〈m |∇Rn〉+∇R 〈m |n〉 − 〈m| (∇RH) |n〉 ,

⇒ 〈m |∇Rn〉 =〈m| (∇RH) |n〉En − Em

,

and in consequence

Ωn(R) = Im∑m 6=n

〈n| (∇RH) |m〉 × 〈m| (∇RH) |n〉(En − Em)2

.

We have just obtained a very useful result - we have shifted the differentiation with respect to R from the eigenstatesto the Hamiltonian itself. Since the dependence of the Hamiltonian on R is known, this will simplify considerably anyfurther calculations of the Berry effects.

A consequence of the above formula is that the sum of all Berry curvatures of all eigenstates vanishes for every R,∑n

Ωnµν(R) = 0.

A. Two-level system

The simplest system in which the Berry phase may appear is a two-level one. Its Hamiltonian is a 2× 2 Hermitianmatrix, and as such it can be written as a linear combination of Pauli matrices,

H = h(R) · σ =

(hz hx − ihy

hx + ihy −hz

)= h

(cos θ sin θe−iϕ

sin θeiϕ − cos θ

),

14 Topology in physics

where we have parametrized our Hamiltonian with h = h(sin θ cosϕ, sin θ sinϕ, cos θ). The eigenvalues are ±h(R),and the eigenvectors can be chosen as

u+ =

(cos θ2e

−iϕ

sin θ2

), u− =

(sin θ

2e−iϕ

− cos θ2

).

These vectors are well defined everywhere except at θ = π (the south pole) for u+ and θ = 0 (north pole) for u−.Since the upper component of these eigenvectors vanishes, the phase ϕ is ill-defined.Let us calculate Berry connection and curvature.

A+θ = i(cos θ2e

iϕ, sin θ2 ) ·

(− 1

2 sin θ2e−iϕ

12 cos θ2

)= 0, A+

ϕ = i(cosθ

2eiϕ, sin

θ

2) ·(−i cos θ2e

−iϕ

0

)= cos2

θ

2,

A−θ = i(sin θ2eiϕ,− cos θ2 ) ·

(12 cos θ2e

−iϕ

12 sin θ

2

)= 0, A−ϕ = i(sin

θ

2eiϕ,− cos

θ

2) ·(−i sin θ

2e−iϕ

0

)= sin2 θ

2,

and the non-vanishing components of the Berry curvature are

Ω+θϕ = ∂θA+

ϕ − ∂ϕA+θ = 2 cos

θ

2

(− sin

θ

2

)1

2= −1

2sin θ,

Ω−θϕ = ∂θA−ϕ − ∂ϕA−θ = 2 sinθ

2

(cos

θ

2

)1

2=

1

2sin θ.

When we calculate the Berry phase over a closed path C, we can express it as

γ±(C) =

∫∫SC

dS ·Ω±(R) = ∓1

2

∫∫SC

sin θdθdϕ.

The Berry phase is equal to half of the solid angle which the path C encompasses when viewed from the degeneracypoint h = 0.There are several important points here.

• When expressed in the coordinates x, y, z, the Berry curvature is Ω± = ∓1

2

h

h3, i.e. it is similar to the field

produced by a monopole (this is one of the problems to be calculated in class). Since the field is a curl of a“vector potential”, it corresponds to an abstract magnetic, not electric field. The monopole is placed at thedegeneracy point h = 0 - indeed, the degeneracy points act as sources and drains of the Berry curvature.

• The Berry curvature is written as a covariant tensor. If it is indeed covariant, it should transform as

Ωαβ =∂Rµ

∂Rα∂Rν

∂RβΩµν ,

where R are the components of R in a new parametrization (e.g. (x, y, z) versus (h, θ, ϕ)). I was not able toprove the covariance of Ωµν for this two-level system, but if such a relation indeed holds, it would be very useful.

• When we integrated over the parameters θ, ϕ, we did not attach any additional sin θ factor, apart from the onearising from the Berry curvature. This is because we were not integrating over the surface of a sphere, but overa parameter region - see the diagram below. Our parameter space has the topology of a sphere S2, but not itsmetric.

0

whole edge identified

with a point

2π

π

ϕ

θ

The solid edges are identified in a normal way, but each dashed edge corresponds to a single point in ourparameter space.

Topology in physics 15

When we express the Berry phase in terms of a surface integral, we have the freedom of choosing the surface. Theloop C is the boundary of both S1 and S2 shown below.

C

S1

S2

Using the result above, we know that the integral of the Berry phase over the whole sphere, which spans the solidangle of 4π, should be ∓2π. Then

γ+(S1)− γ+(S2) = −2π → γ+(S1) = γ+(S2)− 2π,

and we encounter again the modulo 2π ambiguity of the Berry phase. The integral of the Berry curvature over thewhole parameter space is

γ±(S2) = (∓1) 2π.

In later lectures we will identify the ∓1 as the Chern number of the ± bands.

Note on the interpretation of the Berry curvature. It is tempting to try and imagine the Berry curvatureas a sort of generalized Gaussian curvature, with a tangential plane at each point of our parameter surface, equippedwith a local basis represented as two orthogonal vectors. The association with the parallel transport enhances thistemptation even more.The method which we apply in both cases is indeed similar - we take a parameter space B (base space), for each pointin B we find another space F and we assign one to another, constructing local products B × F 5. For the Gaussiancurvature B was the sphere and F was the local tangential plane. For the Berry curvature the parameter surface wasagain a sphere, but F is instead a whole Hilbert space for the Hamiltonian defined by the values of its parameters ata particular point. We can certainly try to find a pictorial representation for the local Hamiltonian eigenstates on thesphere, but we must remember that this picture is only a metaphor for the real thing. And as usual with metaphors,we can push them too far. For instance, the parallel transport on the sphere over a closed path turns both vectorsspanning the local tangential plane in the same direction. The parallel transport of the eigenstates u+ and u− overthe same loop in the parameter space “rotates” them in the opposite directions (Berry phases have opposite signs).

B. Geometric phase in optics: Pancharatnam’s phase

The geometric phase which Berry described appears so often throughout many domains of physics, that it hadalready been noticed several times. What Berry did was to provide a mathematical context (that of topology) in whichall these remarkable occurrences can be seen as different manifestations of the same principle. At the time of writinghis famous 1984 paper he knew of two such occurrences, the Aharonov-Bohm effect and the so-called dynamicalJahn-Teller effect in molecular physics. After the publication he learned that thirty years earlier, in 1956, an Indianphysicist, S. Pancharatnam, had a very similar insight. His interest was sparked by the interference patterns which hesaw when looking through a biaxial absorbing crystal, iolite. The patterns were strongest when he inserted a linearpolarizer in front of and behind the crystal, which also makes the geometrical interpretation clearer. The experimentis sketched on the left below. The first polarizer sets the beam to a polarization A. The second polarizer sets itto the polarization D, but the beams recombining on our retinas were propagating through different environmentsin the anisotropic medium of iolite; say, the beam coming from the right went through a sequence of polarizationsA → B → D, but the beam coming from the left through a sequence A → C → D. The resulting pattern musthave looked similar to what you can see in the photo of the sky, taken by Berry through a sandwich of two linearpolarizers with an overhead transparency in between. The acetate from which the transparency was made, is indeedan anisotropic medium.

5 In a more technical language, the space F is called a fibre, and the base space, together with the set of all the fibres and the rules bywhich we assign them, is called a fibre bundle.

16 Topology in physics

A

D

A

A

B

D

D’C

Pancharatnam used a Poincare sphere to parametrize the path which each beam traced. For monochromatic light,travelling in the z direction, the electric field vector lies in the xy plane and the state of polarization is described by

d = (dx, dy), |d| = 1, dx, dy ∈ C.

We can combine both polarizations in a spinor |ψ〉 = (ψ+, ψ−), with ψ± = (dx ± idy)/√

2. The polarization matrix is

H(r) = r · σ =

(z x− iy

x+ iy −z

)=

(cos θ sin θe−iϕ

sin θeiϕ − cos θ

).

Each point rA = (θA, ϕA) defines a matrix H, whose eigenvector with + sign is the polarization |A〉. At the polesθ = 0 and θ = π we have ψ− = 0 and ψ+ = 0, respectively. These two points denote the two circular polarizations.On the equator, θ = π/2, |ψ+| = |ψ−| and these points correspond to different directions of the linear polarization.Everywhere else the polarization is elliptical.Pancharatnam found a way in which to determine the phase difference between two beams with different polarizations,or in other words he found the Pancharatnam connection. His reference point was the situation in which the intensityof the combined beam was maximum:

(〈A|+ 〈B|)(|A〉+ |B〉) = 2 + 2 |〈A |B〉| cos(ph(A,B)).

A

B

C

D

With this definition of phase difference he was able to determine the relative phase accumulated by the two beamstravelling on the paths ABD and ACD on the Poincare sphere (above on the right). This phase difference turned outto be half of the solid angle which the loop ABCDA encompassed when viewed from the centre of the sphere.

C. Berry phase in the Bloch bands

In solid state physics one of the most prominent dependencies of the electronic Hamiltonian is that on the crystalmomentum. It can also be manipulated by electromagnetic fields, or by deforming the lattice. The Hamiltonian can

Topology in physics 17

be written in several forms, and we’ll begin by the most basic one,

H =p2

2m+ V (r), where V (r + a) = V (r),

with a the Bravais lattice vectors. The Bloch theorem states that the eigenstates of such a Hamiltonian fulfill thecondition

ψnq(r + a) = eiq·aψnq(r).

We have now a q-independent Hamiltonian and q-dependent boundary conditions. Since differentiating the Hamilto-nian is usually easier than differentiating the eigenstates, we can apply a gauge transformation H(q) = e−iq·rHeiq·r

and obtain

H(q) =(p+ ~q)2

2m+ V (r),

with a transformed eigenstate unq(r) = e−iq·rψnq(r). The boundary condition which unq obeys is strict,

unq(r + a) = unq(r),

and the unq functions have the same periodicity as the atomic lattice. The Brillouin zone is the parameter space ofthe transformed Hamiltonian H(q), with |unq〉 as the basis functions. Its topology is that of a torus; since the states|ψn(q)〉 and |ψn(q +G)〉 satisfy the same boundary condition, they may differ by at most a phase. We may keep ourtorus unclosed by setting no condition on the phase (that can sometimes be convenient), or we can close the torus byfixing |ψn(q)〉 = |ψn(q +G)〉. Consequently, the relation holding for |un(q)〉 is

unq(r) = eiG·runq+G(r).

This choice of gauge is called the periodic gauge.If the crystal momentum q is forced to vary, the Bloch state will pick up a Berry phase

γn =

∮C

dq · 〈un(q)| i∇q |un(q)〉 .

The Berry phase has a meaning only over a closed path, but Berry connection and curvature are defined at everypoint in the parameter space. In particular, the Berry curvature of the energy bands is

Ωn(q) = ∇q × 〈un(q)| i∇q |un(q)〉 .

The integral of the Berry curvature over the Brillouin zone is a so-called Zak’s phase,

γn =

∮BZ

dq · 〈un(q)| i∇q |un(q)〉 .

D. Adiabatic current

We want now to find out about the response of our system to an external perturbation. We’ll assume that theperturbation is adiabatic and cyclic in time, though the requirement of time periodicity can later be dropped.Let us consider a general state |ψ(t)〉, obeying the usual Schrodinger equation i~∂t |ψ(t)〉 = H(t) |ψ(t)〉. Using thesnapshot basis, we can express it in terms of Hamiltonian eigenstates as

|ψ(t)〉 =∑n

exp

− i~

∫ t

0

dt′ εn(t′)

an(t) |n(t)〉 .

Inserting it into the Schrodinger equation we obtain∑n

an(t) exp

− i~

∫ t

0

dt′ εn(t′)

|n(t)〉 = −

∑n

an(t) exp

− i~

∫ t

0

dt′ εn(t′)

|∂t n(t)〉

Acting on both sides with 〈n′(t)| we arrive at

an′(t) = −∑n

an(t) exp

− i~

∫ t

0

dt′ (εn(t′)− εn′(t′))

〈n′(t) |∂t n(t)〉 .

18 Topology in physics

We shall now choose a certain gauge, called the parallel transport gauge, which will make our calculations easier. Thegauge corresponds to taking

|n(t)〉 → |n(t)〉 = exp

i

∫ t

0

〈n(t) |∂t n(t)〉|n(t)〉 , or equivalently 〈n(t) |∂t n(t)〉 = 〈n(R) |∇R n(R)〉 ∂R

∂t

!= 0.

The last factor ∂R/∂t may be considered as an adiabaticity parameter, and as such ought to be as small as possible.We shall use it for the expansion of our wave functions.By adopting the parallel transport gauge we have sacrificed the idea that our wave function ought to be single-valued.The phase of |n(t)〉 depends now on the way by which t was reached, so the value of the wave function is path-dependent. After the cyclic evolution we shall recover again a Berry phase γn(C), but it will now arise from themulti-valuedness of the wave function, not from the Berry connection6. If our final expressions are gauge-invariantthough, we can use them in any convenient basis.If the system starts in the state |n(0)〉, we have an = 1 and an′ 6=n = 0. Because of the parallel transport gauge whichwe adopted, an(0) = 0, but an′(0) 6= 0. At t = dt, up to the first order in the adiabaticity parameter ∂R/∂t we’ll havestill an(dt) = 1, but an′(dt) is

an′(dt) = −〈n′(dt) |∂t n(dt)〉 exp

− i~

∫ dt

0

dt′ (εn(t′)− εn′(t′))

.

One ansatz for the an′ which would obey the differential equation for an′(t), is

an′(t) = −i~ 〈n′ |∂tn〉

εn − εn′exp

− i~

∫ t

0

dt′ (εn(t′)− εn′(t′))

.

Differentiating it over time, we obtain

an′(t) =

−i~∂t 〈n

′ |∂tn〉εn − εn′

+ i~〈n′ |∂tn〉

(εn − εn′)2∂t(εn − εn′)− 〈n(t) |∂t n(t)〉

exp

− i~

∫ t

0

dt′ (εn(t′)− εn′(t′))

.

The first two terms are of the second order in adiabaticity on account of ∂t 〈n′ |∂tn〉 and ∂t(εn − εn′). The last termis indeed the right side of the equation for an′(t). The state which started out as |n(0)〉 becomes then |ψ(t)〉, given by

|ψ(t)〉 = exp

(− i~

∫ t

0

dt′ εn(t′)

)|n(t)〉 − i~∑n 6=n′

|n′(t)〉 〈n′(t) |∂tn(t)〉

εn(t)− εn′(t)

.

Let us now return to our Bloch Hamiltonian. The system we will investigate is a 1D (for ease of calculation) bandinsulator, so that while integrating over our parameter surface we won’t encounter any degeneracies. The perturbationis time-dependent, slow (for adiabaticity) and periodic (H(t+T ) = H(t)) so that the parameter surface can be closed.If the time-dependent Hamiltonian still has the translational symmetry of the crystal, its instantaneous eigenstateshave the Bloch form

|ψn(q, t)〉 = eiqx |un(q, t)〉 .

The state which started out as |un(q, 0)〉 will become after a time t

|un(q, t)〉 = |un(q, t)〉 − i~∑n′ 6=n

|un′(q, t)〉 〈un′(q, t) |∂tun(q, t)〉εn − εn′

.

The velocity operator associated with |un(q)〉 is

vq =i

~[H(q, t), r] =

i

2m~[(p+ ~q)2, r] =

i

2m~((p+ ~q)[p+ ~q, r] + [p+ ~q, r](p+ ~q)) =

1

m(p+ ~q) =

∂H(q, t)

~∂q.

The average velocity in a state |un(q)〉 is, to the first order in adiabaticity,

vn(q, t) = 〈un(q, t)| ∂H(q, t)

~∂q|un(q)〉 = 〈un|

∂H

~∂q|un〉

6 As in the Problem 2.3, where in the tangential gauge the boundary condition incorporated the Aharonov-Bohm flux directly, but in theradial gauge only through the singularity at x = 0 = 2π.

Topology in physics 19

−i∑n′ 6=n

1

εn − εn′〈un| ∂qH |un′〉 〈un′ |∂tun〉 − 〈un′ | ∂qH |un〉 〈un |∂tun′〉 .

Using the identity∑n′ |un′〉 〈un′ | = 1, and

〈un| ∂qH |un′〉 = 〈un| ∂q(H |un′〉)− 〈un|H∂q |un′〉 = 〈un| ∂q(εn′ |un′〉)− εn 〈un |∂qun′〉

=∂εn′

∂q〈un |un′〉+ (εn − εn′) 〈un |∂qun′〉 ,

we obtain

vn(q) =∂εn~ ∂q

− i∑n′ 6=n

〈un |∂qun′〉 〈un′ |∂tun〉 − 〈un′ |∂qun〉 〈un |∂tun′〉

=∂εn~ ∂q

− i 〈∂qun |∂tun〉 − 〈∂tun |∂qun〉 − 〈un |∂qun〉 〈un |∂qun〉+ 〈un |∂qun〉 〈un |∂qun〉

=1

~∂εn∂q− Ωnqt.

The first term on the right hand side is the usual one for the velocity; the second part is an additional, anomalousvelocity, responsible for the off-diagonal response in the various types of quantum Hall effect, which we shall see later.