Topological States of Matter - PowerPoint Presentation

1. Topological States of MatterGirish S SetlurDepartment of PhysicsIIT Guwahati

2. COPYRIGHT DISCLAIMER: ALL ILLUSTRATIONS AND SOME PASSAGES IN THESE SLIDES HAVE BEEN DOWNLOADED FROM VARIOUS INTERNET SOURCES.LISTING EACH SOURCE SEPARATELY WILL TAKE UP ALL MY TIME SO I SHALL DESIST FROM DOING SO.THE ONLY ORIGINAL QUALITY IN THESE SLIDES IS THE CHOICE OF TOPICS AND THE RELATIVE EMPHASIS GIVEN TO VARIOUS CONCEPTS.

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4. What is a topological property?Any property of a space that is unchanged when that space is continuously deformed. The minimum requirement for this “space” is that it allows its description in terms of “open sets” . An open set has the property that intersection of two open sets is open and the union of two open sets is also open. A continuous deformation is one where any open set belonging to the deformed space is obtainable by deforming an open set of the original space.

5. Algebraic topologyIn the earlier slide I spoke of the “property” of a space without explaining what that means. The most useful properties of a space are algebraic notions that we may attribute to the space. This leads to the study of algebraic topology which is the study of (topological) spaces through the algebraic structures associated with them.What is an algebraic structure? It is a collection of objects that are algebraic – by this one means notions such as multiplication, addition, inverting e.t.c. of two such objects have been made sense of. A simple and important example of an algebraic structure is a group. For a finite group there are a finite number of distinct elements. This number then becomes, for example a numerical algebraic property associated with a topological space. If these properties are unchanged under continuous deformation of the space we say that these algebraic notions are topological properties.

6. Differential topologyIt is the study of topological spaces by associating structures with them that enable the use of notion of differential and integral calculus.The main example is a differential manifold: A topological space that is obtained by smoothly gluing together pieces each of which look like a piece of an ordinary two dimensional plane (for example) on which we know how to do calculus.

7. Gauss-Bonnet theorem for closed two dimensional surfaces: is the number of “holes” (for a torus ) 

8. What has all this got to do with physics?Physics is an effort to understand the world around us. It is critically important to appreciate that while the world around us is physical and real, the effort to understand it and describe its working is decidedly mental and not physical. Typically this mental activity which leads to a feeling of understanding takes the form of mathematical reasoning. Thus there is no such thing as a “physical explanation” of a phenomenon. All explanations are “mental” and not physical and therefore also mathematical.

9. Branches of mathematics used in physicsNow that we know that understanding nature can only be achieved through mathematical reasoning, it is natural to ask what branches of mathematics have been employed to achieve this understanding. A detailed understanding of nature entails a description of space and time and the matter that inhabits it. It is no surprise that the mathematics subject of calculus developed concurrently with our quantitative understanding of the physics subject of mechanics. It is only in the early twentieth century that extensive use of complex numbers (more generally complex analysis) was made use of in physics spurred largely by the advent of quantum mechanics.

10. Hence real analysis (classical mechanics), complex analysis (quantum mechanics) were extensively employed until the mid twentieth century. The physics subject of quantum many body theory (quantum mechanics of many mutually interacting particles) also saw the use of mathematical subjects of functional analysis and operator theory.If we look at all the other branches of mathematics we see severe under representation from fields like number theory, topology (algebraic, differential) and the relatively modern fields of algebraic geometry employed to understand physical phenomena.A notable semi-exception is found in general relativity which requires differential geometry for its understanding. There the focus is on geometrical rather than topological notions: geometrical notions such as curvature are not topological since they change under continuous deformation.

11. What is to be done about it?One could take the point of view that the nature of the physics problems will tell us which branch of mathematics to use. Indeed this has been the strategy (if there was one) till now. A physical phenomenon known as the Integer Quantum Hall Effect was discovered in 1980 which appeared to require the subject of (differential) topology for its understanding (D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, 1982).Since then people have actively looked for and found physical phenomena and experimentally accessible properties of physical systems that may be described in terms of topological properties. This is the reason for all the excitement in the last two decades or so which culminated in the 2016 Nobel Prize in Physics.In passing we note that the arcane branch of physics called String Theory does occasionally make use of newer branches of mathematics such as algebraic geometry.

12. Topological invariants used in physicsWe list and describe some simple topological invariants (representing measurable qualities that are unchanged under continuous deformation of the parameters of the physical system). The most important of these are (they are closely related to each other):Winding numbersChern numbersMaybe more that I am not aware of (e.g. analytic and topological index in the context of Atiyah Singer Index theorem, knot invariants in topological quantum field theories)

13. Winding number

14. Winding number = +1Winding number = +2Winding number = -2

15. Winding numbers in PhysicsA) Flux quantization through a superconducting ring:Consider the following scenario: a superconducting wire in the shape of a ring carrying a supercurrent.

16. The problem with this definition of current density is it is not gauge invariant. Gauge transformation involves making the change i.e. local phase transformations. Since density is clearly gauge invariant and density and current form a 4-vector, we expect all components to have similar symmetries. In order to make the current gauge invariant we have to add a vector potential which means, in unchanged under the simultaneous transformation, &  

17. If the density is constant we may write down a formula for the current density in the superconductor as Ampere’s Law states Due to Meissner effect, deep inside the superconductor, = 0 and . Currents and magnetic fields live only on the surfaces.Note that  

18. Now a curious thing happens when you integrate over the circumference of the ring ()From Stokes theorem, the left hand side is just the magnetic flux passing through the empty region whose circumference is the ring.The right hand side is, ( ) 

19. Now and are the same points. Note that it is the wave function that is single valued which meansHence . Here represents the number of times the phase of the order parameter shifts by as I move along the ring once. This is a topological invariant since even if I distort the ring the idea that has many values that differ from each other by multiples of 2 still hold. Hence magnetic flux penetrating the empty interior of the region bounded by the ring is quantized. is a measurable physical quantity and n is the winding number which is a topological invariant.  

20. B) Vortices with hollow cores in superfluids Here is the velocity of the superfluid at . At points where is not zero, the velocity may be unambiguously defined in terms of the complex order parameter as, 

21. The vortex strength is defined as the circulation of the velocity around a closed loop.But If is contained in the region enclosed by the loop then , where n is the number of times the loop loops around the point at which   

22. Work done in IIT GuwahatiIn the earlier slide we say how the presence of cores causes the vortex strength to be zero even in the absence of a magnetic field. Not only that this vortex strength is quantised not because of quantum mechanics but because of topology.The antithesis of this situation has been examined (G.S. Setlur, Dynamics of Classical and Quantum Fields, CRC Press) where quantum vortices in a charged Bose fluid has been studied coupled to a quantum electromagnetic field with no sources. In this situation, the average vortex strength (operator) is zero but we may define the correlation function of two vortices:Geometrical meaning given to a physical quantity. 

23. C) The Su-Schrieffer-Heeger (SSH) model    In the bulk 

24. When (staggered) the two bands do not touch, i.e. h(k) = 0 is not allowed. 

25. Define the vector The path of the endpoint of , as goes through the Brillouin Zone is a closed path since On the plane, the origin is excluded from any path the endpoint takes since for a staggered system, is not possible for any value of in To a closed path on the plane that does not contain the origin, we can associate a winding number. This is the number of times the path encircles the origin, the winding of the complex number as the wavenumber is swept across the Brillouin Zone. More on this later 

26. A Geometrical Digression: Pancharatnam-Berry PhaseImagine a Hamiltonian where has multiple components like or . The wavefunction of a stationary state obeys [for a fixed ]We consider only the situation when is nondegenerate. It is well-known that the phase of cannot be fixed by solving this equation and so it is not measurable. Now imagine the vector is slowly changing with time so that the Hamiltonian is now weakly time dependent. In this case, we have to solve the time-dependent Schrodinger equation, 

27. If changes slowly with time, it is believable that (and can be proved) and differ only by a phase i.e. they correspond to the same state. Here is the stationary state at the new value of which is now Hence we may write,Substituting into the time dependent equation we get,But Hence,  

28. This phase is as unphysical as the phase of - indeed changing to changes to + . It was Berry’s deep insight to realise that this is not the case if the path is closed. That means if at = T, , the ambiguity in the choice of the phase does not affect the value of , in other words it is gauge invariant. Note that in this case does not depend on T but is given by the gauge invariant expressionThis is called Pancharatnam Berry phase. This phase is physics analog of the geometrical notion of holonomy as pointed out by Barry Simon. 

29. Holonomy on a sphereTransporting a tangent vector parallel to itself along a closed loop on a curved surface will lead to the final vector to be not parallel to the starting vector, the angle between the two is related to the solid angle subtended by spherical triangle.

30. Chern numberNow we are equipped to discuss another topological invariant known as the Chern number. Define a vector field, . This is real since . Berry’s phase is now given by,Using Stokes theorem we may write ( ), where is called the Berry curvature and is called the Berry connection. Also is any surface whose boundary is . 

31. Now imagine that is a closed surface. You may imagine this in a limiting manner so that is a tiny circle which then shrinks to a point since a closed surface should not have a boundary.The Chern number is defined as (where is a closed surface)You may think this has to be zero always since this is also the circulation of a vector field over a circle so tiny that is finally shrinks to a point. But there is a catch. If the vector field is singular at some point, the circulation of the vector field around a curve enclosing this point - no matter how tiny - may not vanish.  

32. It is easy to show thatis an integer. Imagine two surfaces and each bounded by a closed curve C. - = = = m  =  

33. Example of a situation where the Chern number is not zero: Imagine a wavefunction has a phase which makes : ; This Berry connection is well defined at all except at = 0 where the answer depends on the direction in which I approach = 0. Suppose I want to calculate Berry’s phase and = 0 is not enclosed within C,when = 0 is enclosed inside C, we may draw a tiny circle around = 0 and the region between the tiny circle and C contributes nothing to the Berry phase but the circulation of the vector field around = 0 is not zero since we may write andIf the surface R is compact and closed (like a torus) then it has no boundary so C is a tiny circle that shrinks to zero, but if the vector field is appropriately singular the Chern number can be non-trivial (Q = /2 = n ). 

34. Chern numbers in Physics Integer Quantum Hall Effect The integral quantum Hall effect was discovered in 1980 by Klaus von Klitzing, Michael Pepper, and Gerhardt Dorda. Truly remarkably, at low temperatures (~ 4 K), the Hall resistance of a two dimensional electron system is found to have plateaus at the exact values of . In the above expression, is an integer, h is Planck’s constant and e is the electron charge. At the same time, in the applied magnetic field range where the Hall resistance shows the plateaus, the magnetoresistance (i.e., the resistance measured along the direction of the current flow) drops to negligible values.  

35. The electronic states are given by the wavefunction It can be proved that for a fully filled band the transverse conductivity is given by the formula,     Q 

36. Chern Insulators: Bulk-boundary correspondenceChern insulators are insulators in the bulk (where the bands are gapped) and the bulk electronic states possess a non-vanishing Chern number. When these materials have a boundary, then the states on the boundary are zero energy gapless conducting states (in more than one bulk dimension) that are topologically protected by the non-vanishing Chern number.Topological protection refers to symmetries that the bulk possesses that are also shared by surface states that cannot be changed by local impurities or defects. Typically, time reversal non-invariant Chern insulators (like Quantum Hall Systems) have many gapless surface states, some propagating clockwise (say the number is and some propagating counterclockwise (say the number is Their difference is a topological invariant which is the Chern number of the bulk. 

37. How do gapless states emerge on surfaces of a topologically non-trivial bulk insulator?One dimensional bulk insulator (SSH model) with single site edge states with zero energy.Is the topologically trivial dimer phase v = 1, w = 0 (winding number is zero)Is the topologically nontrivial dimer phase with v = 0 and w = 1 (winding number is non-zero)For other values of v and w the system has edge states or not depending upon whether the winding number is trivial (zero) or nontrivial (not zero).

38. Time reversal invariant topological insulatorsTime-reversal symmetry protected edge states were predicted in 1987 to occur in quantum wells (very thin layers) of mercury telluride sandwiched between cadmium telluride and were observed in 2007. In 2007, they were predicted to occur in three-dimensional bulk solids of binary compounds involving bismuth. The first experimentally realized 3D topological insulator state (symmetry protected surface states) was discovered in bismuth-antimony. Shortly thereafter symmetry protected surface states were also observed in pure antimony, bismuth selenide, bismuth telluride and antimony telluride using ARPES. Many semiconductors within the large family of Heusler materials are now believed to exhibit topological surface states. In some of these materials the Fermi level actually falls in either the conduction or valence bands due to naturally occurring defects, and must be pushed into the bulk gap by doping or gating. The surface states of a 3D topological insulator is a new type of 2DEG (two-dimensional electron gas) where the electron's spin is locked to its linear momentum.

39. OutlookLook for other types of topological invariants and try to construct physical models that have physical quantities that are related to these invariants.Look at other branches of mathematics that are under represented in physics and try to apply those tools to study physical systems.THANK YOU