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Topological Insulator Part II: Berry Phase and Topological Index

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Topological Insulator Part II: Berry Phase and Topological Index Phys620.nb 11 3 Topological insulator part II: Berry Phase and Topological index 3.1. Last chapter Topological insulator: an insulator in the bulk and a metal near the boundary (surface or edge) Quantum Hall insulator: sxx = 0 2D metal (weak lattice or no lattice potential), e = k2 2 m. As long as m>0, metal (assuming there is no impurities and no interactions) 1 B field: Landau levels: en,k = n + Ñ w (very similar to energy bands). If m in the gap, it is an insulator. If m coincides with a Landau level, 2 the system is a metal. These insulating states are topologicalI M insulators (bulk insulator with chiral edge states). 2 Chiral edge states gives quantized Hall conductivity sxy = n e h Q: Why is Hall conductivity nonzero? A: chiral edge states. For a chiral edge state, the top and lower edges have opposite velocities (say electrons near the top edge goes to the right and the lower edge to the left). If we apply +V/2 for top edge and -V/2 for bottom edge. We will have a higher electron density at the top edge and a lower density at the lower edge (because electrons like positive voltage). So we have more electrons moving to the right (top edge has more electrons) and less electrons moving to the left (low edge has less electrons). So there is a net current in the x direction. Another way to see this: if we pass a current (from left to right), we need to put more charge near the top edge, and less charge near the bottom edge. Charge imbalance induces a E field in the y direction. Q: Why Hall conductivity is quantized? A: chiral edge states. One edge contributes to e2 h. If we have n edge states: n e2 h Q: Why are impurities unimportant? A: chiral edge states. Electrons cannot be reflected by an impurity. 3.2. Some insulators have chiral edge state but others don’t. Why? A piece of plastic is an insulator. a IQH state is also an insulator. Why does the IQH system have chiral edge states but a piece of plastic doesn’t? Q: Why insulators are different from each other? How many different types of insulators do we have in this world? What tool should we use? Where should we start to look for an answer? Hint 1: sxy depends on Ñ. So this phenomenon must be due to some quantum physics. Hint 2: Transport is the motion of charge. In quantum mechanics, charge is closely related with one quantity. What is that? Hint 2.5: It is the conjugate variable of charge (like p is the conjugate variable of r) Phase: invariant under f®f+df implies the conservation of charge. Hint 3: Phase can often lead to quantization (e.g. quantization of Lz = n Ñ) Wavefunction is single valued. If we go around a circle, the change of the phase can only be 2 p n where n is an integer. 12 Phys620.nb Hint 4: Phase of a wavefunction: quantum physics. (This is good, because we know that the quantum Hall effect is a quantum phenomena). So: we will focus on “phase”. Q: Phase of a wavefunction: important or not? A: The relative phase (the phase difference between two states) is important (which leads to interference), but the absolute value of a phase is not important If y is an eigenstate, y' = eä f y is the wavefunction for the same state. All the observables remains the same. y A y = y ã-ä f A ãä f y = y' A y' (3.1) \ \ \ Relative phase is important X \ Y ] X \ 1 1 Y = y1 + y2 (3.2) 2 2 \ _ _ y1 P y1 + y2 P y2 y1 P y2 + y2 P y1 Y P Y = + (3.3) 2 2 X \ X \ X \ X \ The Xphysical \meaning of the first term: system has half of the probably on state y1 and half on y2 . So the momentum of Y is the average of y1 and y2 , just like in classical systems. The physical meaning of the second term: interference between the two quantum state.\ (relies on the relative\ phase between state 1 and\ 2) \ \ 1 ãä f Y' = y1 + y2 (3.4) 2 2 \ _ _ ä f -ä f y1 P y1 + y2 P y2 ã y1 P y2 + ã y2 P y1 Y' P Y' = + = 2 2 X \ X \ X \ X \ (3.5) X y1 P\ y1 + y2 P y2 y1 P y2 + y2 P y1 + cos f 2 2 X \ X \ X \ X \ Example: flux quantization an A+ answer: The absolute value of a phase is not important, but the fact that phase is unimportant is VERY IMPORTANT. 3.3. Example: Central charge and quantum group References: Weinberg, The quantum theory of fields, Vol. 1, Chapter 2.2 and 2.7. Polchinski, String theory, Vol. 1 Di Francesco, Mathieu, Senechal, Conformal Field theory Remarks on central charge: Not directly related to topological insulators Same physics principle: phase of a quantum state is unimportant, which is very important Useful when we discuss 1+1D quantum physics in later chapters It is always good to know the connections between different pieces of knowledge. 3.3.1. Commutation relations: Commutation relations are crucial for quantum systems. There are two types of commutation relations: Type 1: Obvious e.g. Lx, Ly = ä Lz Type 2: Not so obvious: central charge A E Phys620.nb 13 3.3.2. Example of the first type: Lx, Ly = ä Lz Q: Why angular momentum operators don’t commute with each other. A: Rotating a system along the x axisA first thenE along the y axis is different from rotating a system along the y axis first then along the x axis Angular momentum is relation with space rotation. (generators of rotations) Rx q = exp -ä q Lx (3.6) R q = exp -ä q L yH L H yL (3.7) q = -ä q Rz H L expI Lz M (3.8) L = ä ¶ R q xH L q xH q®0 L (3.9) L = ä ¶ R q y q yH L q®0 (3.10) L = ä ¶ R q z q z H L q®0 (3.11) All rotations form a “Group” (SO(3)), a set with product. H L Product: the product of two elements X and Y is another element of this set Z = X Y. (Closure) X Y and Y X don' t need to be the same. For rotations, for any two rotations R1 and R2, their product R2 R1 is defined as we first do the rotation R2 and then R1, which is another rotation R3 = R2 R1. For example Rx q1 Rx q2 = Rx q1 + q2 Associativity: A B C = A B C Identity element I. For any element X, I X = X I = X. H L H L H L For rotations, HI is rotationL H byL zero degree. I = R 0 . Inverse element: For any element X, there is another element X-1 which satisfies X X-1 = X-1 X = I H L For rotations, the inverse of Rx q is Rx -q , rotating along the same axis but we flip the sign of the angle. For rotations, the order of the product matters. H L H L Rx q1 Ry q2 ¹ Ry q2 Rx q1 (3.12) So H L H L H L H L Rx q1 , Ry q2 ¹ 0 (3.13) Rotations don’t commute with each other. This tells us that their generators cannot commute with each other. A H L H LE Lx, Ly ¹ 0 (3.14) If L , L = 0, we will have exp -ä q L , exp -ä q L = 0, which is obviously incorrect. Ax y E 1 x 2 y Using the commutation relation between rotations, we can find that: A E A H L I ME Lx, Ly = ä Lz (3.15) A geometry problem. No quantum physics invovled. A E 3.3.3. Example of the second type: Why is mass NOT an operator? Galilean transformation and space translation: commute or not? Space translations: T a changes x ® x' = x + a. The generator is the momentum operator p: T a = exp -ä p a (3.16) H L p = ä ¶ T a H L a H a®0 L (3.17) Galilean transformation: G v changes x ® x' = x + v t. The generator is called K. H L G v = exp -ä K v (3.18) H L K = ä ¶ G v H L v H v®0 L (3.19) H L 14 Phys620.nb Q: [K,p]=? A: Classical physics and quantum physics have different answer. Classical physics: T, G = 0 so [K,p]=0 Gv : x ® x' = x + v t (3.20) @ D Ta : x ® x' = x + a (3.21) Gv y x, t = y x + v t, t (3.22) T G y x, t = T y x + v t, t = y x + v t + a, t a vH L H a L (3.23) T y x, t = y x + a, t a H L H L H L (3.24) G T y x, t = G y x + a, t = y x + v t + a, t v Ha L H v L (3.25) T G y x, t = G T y x, t a v H L v aH L H L (3.26) T G = G T a v H v L a H L (3.27) Ta, Gv = 0 (3.28) So @ D (3.29) ¶a ¶v Ta, Gv = 0 (3.30) ¶ T , ¶ G = 0 a a@ v vD (3.31) ä ¶ T , ä ¶ G = 0 @ a a v D v (3.32) K, p = lim lim ä ¶ T , ä ¶ G = 0 @ a®0D v®0 a a v v (3.33) Quantum physics: @ D @ D Galilean transformation change velocity from u to u+v.
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