Phys620.nb 95 6 Gauge theory of insulators 6.1. Gauge theory (E&M) for Chern insulators In an crystal, we have nucleons, electrons, and they couple together through gauge fields (E and B). If we don’t consider the spin degree of freedom (ignoring magnetic ordering and magnetization, etc), the only degrees of freedom are the motion of charge and the gauge field. In an insulator, because charged particles cannot move, the only useful degrees of freedom are the gauge fields. So let’s consider the gauge field in an insulator. 6.1.1. Hamiltonian for E and B fields E and B field carries energy. In vacuum, the energy (Hamiltonian) is 1 B2 Ø 2 H = „ r e0 E + (6.1) 2 m0 In an insulator with time-reversal symmetry, the energy of gauge fields must be a function of E and B. For weak fields (small E and B) we can do a power series expansion: 1 Ø 1 H = „ r eij Ei E j + Bi B j + higher order terms (6.2) 2 mij We cannot have any other quadratic terms, like E ÿB, because it violates the time-reversal symmetry (B is odd under time-reversal but E is even). Higher order terms are in principle allowed, and they are responsible for non-linear E&M responses. These nonlinear effects are negligible if E and B are weak enough. So we will ignore them here. The bottom line: the energy is still E2 + B2. The coefficients can be renoramlized, but the gauge theory in an insulator is qualitatively the same as in vacuum. Q: What will happen if we break the T-reversal symmetry? A: Additional terms are allowed by symmetry, e.g. E ÿ B if the lattice also breaks the inversion symmetry. If the Hamiltonian contains this term, it means that we will induce magnetization by applying E field (and induce electric dipoles by applying a B field). However, this is a “small” effect, which we will not consider. In addition to E ÿ B, the system can get another term which is the Chern-Simon’s gauge term. 6.1.2. the Chern-Simon’s gauge theory in 2+1D For simplicity, we use the theorists’ unit: e=m = everything = 1. The Cherm-Simon’s theory is easier to handle using Lagrangian, instead of the Hamiltonian. The Lagrangian is k Ø m n l LCS = „ r emnl A ∑ A (6.3) 4 p Here, k is the coupling constant; m, n and l run over t, x and y. Ax, Ay is the vector potential and At is the electric potential f. etxy =exyt =eytx= 1 and etyx =eyxt =exty =-1 and e=0 for any other subindcies. The action is 96 Phys620.nb k Ø m n l SCS = „ tL= „ r „ t emnl A ∑ A (6.4) 4 p The coupling constant k is an integer for topological reasons, which will be discussed below. Gauge invariance: Let’s consider a gauge transformation: Am Ø Am ' = Am +∑m c (6.5) k k Ø m m n l l Ø m m n l SCS Ø SCS ' = „ r „ t emnl A +∑ c ∑ A +∑ c = „ r „ t emnl A +∑ c ∑ A = 4 p 4 p (6.6) k k k Ø m n l Ø m n l Ø m n l „ r „ t emnl A ∑ A + „ r „ t emnl ∑ c∑ A = SCS + „ r „ t emnl ∑ c∑ A = SCS + surface term 4 p 4 p 4 p Here, the last term is an integral of a total deriviative, which is just a surface integral, according to the Stokes’ theorem. If we don’t worry about the surface term (assuming the system has no boundary), the action is gauge invariant. So this is a valid gauge theory (if there is no boundary). The surface term will be consider later. In fact, the surface term of the Chern-Simon’s gauge theory violates the gauge symmetry. But luckily, the edge metallic state (chiral state) that we have at the edge also violates the gauge symmetry through parity anomaly. And these two effects cancels each other, so the whole system is gauge invariant. For the whole insulator: 1 Ø 1 S = SCS + „ r eij Ei E j - Bi B j + higher order terms (6.7) 2 mij If the field A varies very slowly in space and time, ∑A is small. The CS term contains only one derivative, but E2 and B2 term contain two derivatives (E and B both take the form of ∑ A). Two derivatives means it is much smaller than the CS term, if A changes very slowly. There- fore, the CS term gives the dominate contribution (so we can ignore the E2 and B2 terms). Electric Current: dS d k Ø m n l ji r0, t0 = = „ r „ t emnl A r, t ∑ A r, t = dAi r0, t0 dAi r0, t0 4 p k d Am r, t k d Al r, t Ø n l Ø m n „ r „ t emnl ∑ A r, t + „ r „ temnl A r,t ∑ 4 p dA r , t 4 p dA r , t i 0 0 i 0 0 (6.8) k Ø k Ø = „ r „ t e d d r - r d t - t ∑n Al r, t + „ r „ t e Am r, t ∑n d d r - r d t - t mnl m,i 0 0 mnl l,i 0 0 4 p 4 p k k k k Ø n l n m m n Ø = einl ∑ A r0, t0 - emni ∑ A r, t = eimn ∑ A r0, t0 =- ei μ E j 4 p 4 p 2 p 2 p Ø k m n k Ø Q: Why e ∑ A r0, t0 =- ei μ E j? 2 p imn 2 p A: let’s do i = x as an example: k m n jx r0, t0 = exmn ∑ A r0, t0 = 2 p (6.9) k k k k k t y y t t y y y exty ∑ A r0, t0 + exyt ∑ A r0, t0 = -∑ A r0, t0 +∑ f r0, t0 = ∑t A r0, t0 +∑y f r0, t0 =- Ey 2 p 2 p 2 p 2 p 2 p k Ø Ø Since, ji r0, t0 =- ei μ E j, the Hall conductivity is: 2 p k sxy = (6.10) 2 p Above, we set e and Ñ to 1 in the calculations above. Here we add them back Phys620.nb 97 k e2 e2 sxy = = k (6.11) 2 p Ñ h Because k is an integer for the CS gauge theory, sxy is quantized. Q: What happens if we have a metal? A: For metals, the charge degrees of freedom must also be considered. So we have both gauge fields and charge motions. The situation is much more complicated and in general, we cannot get a clear answer of sxy. 6.1.3. Electric charge and magnetic field in a quantum Hall system The Cherm-Simon’s gauge theory k Ø m n l SCS = „ r „ t emnl A ∑ A (6.12) 4 p Charge density: dS d k Ø m n l r r0, t0 = = „ r „ t emnl A r, t ∑ A r, t = dA0 r0, t0 dA0 r0, t0 4 p k d Am r, t k d Al r, t Ø n l Ø m n „ r „ t emnl ∑ A r, t + „ r „ temnl A r, t ∑ 4 p dA r , t 4 p dA r , t 0 0 0 0 0 0 (6.13) k Ø k Ø = „ r „ t e d d r - r d t - t ∑n Al r, t + „ r „ t e Am r, t ∑n d d r - r d t - t mnl m,0 0 0 mnl l,0 0 0 4 p 4 p k k k k Ø k n l n m m n = e0 nl ∑ A r0, t0 - emn0 ∑ A r, t = e0 mn ∑ A r0, t0 = “μA = B 4 p 4 p 2 p 2 p 2 p Bottom line: charge density is propotional to magnetic field k r r0, t0 = B (6.14) 2 p Integral on both sides: k „ r r r0, t0 = „ rB (6.15) D 2 p D So we find that electric charge is proportional to the magnetic flux k q = FB (6.16) 2 p Flux attachment: † Electrons in an IQH system carries magnetic flux FB = 2 p k (the Chern-Simon's gauge theory attaches a magnetic flux FB = 2 p k to each of the electrons). † Magnetic flux in an IQH system carries charge q=k FB 2 p Why? Q: Faraday’s law+Hall effect Let’s add an magnetic flux in an IQH system, by increasing the magnetic field in some region D. From time ti to t f , we increase the B field from B r, ti to B r, t f . If B t f > B ti , we add some magnetic flux into the system DFB = „ rBr, t f - „ rBr, ti (6.17) D D Increasing magnetic flux will create some electric field (Faraday’s law) „FB E ÿ„l =- (6.18) „ t 98 Phys620.nb If we have E, it will creates Hall current perpendicular to E (Hall effect). Because here E is in the circlar direction, the current j is along the radials direction, which means that charges are flowing away from (or towards) the place where we insert the magnetic flux j =sxy E (6.19) Total current flowing away from the region D: Ø Ø Ø Ø „FB I = j μ„ r =sxy E ÿ„ l =-sxy (6.20) „ t Total charge accumulated due to this current flow: „FB Dq = q t f - q ti =- „ tI t =sxy „ t =sxy FB t f -FB ti =sxy DFB (6.21) „ t k 2 If we create an flux FB in a IQH system, some charge will accumulate at this flux: q =sxy FB.