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The Integer Quantum Hall E↵ect Rhine Samajdar Harvard University December 14, 2016 Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 1 / 41 Introduction Table of Contents 1 Introduction The Classical Hall E↵ect Integer Quantum Hall E↵ect 2 Landau Levels Review Conductivity: A quick calculation 3 5 6 Büttiker’s Theory Edge modes Disorder and Localization Rhine Samajdar (Harvard University) 4 7 Laughlin’s Gedankenexperiment Gauge Invariance Spectral Flow Topology Linear Response Quantized Hall Conductivity Particles on a Lattice The TKNN formula Peierls substitution Chern Numbers References The Integer Quantum Hall E↵ect December 14, 2016 2 / 41 Introduction The Classical Hall E↵ect The Classical Hall E↵ect A magnetic field causes charged particles to move in circles. Let B = (0, 0, B), E = (E, 0, 0), v = (ẋ, ẏ, 0 ). Figure: The setup for the Hall e↵ect. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 3 / 41 Introduction The Classical Hall E↵ect The Classical Hall E↵ect A magnetic field causes charged particles to move in circles. Let B = (0, 0, B), E = (E, 0, 0), v = (ẋ, ẏ, 0 ). Figure: The setup for the Hall e↵ect. Rhine Samajdar (Harvard University) Edwin Hall, 1879 In equilibrium, a current in the xdirection requires an electric field with a component in the y-direction! The Integer Quantum Hall E↵ect December 14, 2016 3 / 41 Introduction The Classical Hall E↵ect The Classical Hall E↵ect Drude model: m dv = dt eE ev ⇥ B Rhine Samajdar (Harvard University) mv ⌧ The Integer Quantum Hall E↵ect December 14, 2016 4 / 41 Introduction The Classical Hall E↵ect The Classical Hall E↵ect Drude model: m dv = dt J= = 1 ✓ ne2 ⌧ m 2 + !B xx xy xy ⌧2 mv ⌧ ev ⇥ B eE ✓ yy 1 !B ⌧ Rhine Samajdar (Harvard University) ◆ E !B ⌧ 1 ◆ The Integer Quantum Hall E↵ect December 14, 2016 4 / 41 Introduction The Classical Hall E↵ect The Classical Hall E↵ect Drude model: m dv = dt J= = 1 Hall coefficient: ✓ ne2 ⌧ m 2 + !B xx xy xy ⌧2 mv ⌧ ev ⇥ B eE ✓ yy 1 !B ⌧ Rhine Samajdar (Harvard University) ◆ E !B ⌧ 1 Ey ⇢xy = Jx B B !B 1 = = B DC ne RH = ◆ The Integer Quantum Hall E↵ect December 14, 2016 4 / 41 Introduction The Classical Hall E↵ect The Classical Hall E↵ect Drude model: m dv = dt J= = 1 Hall coefficient: ✓ ne2 ⌧ m 2 + !B xx xy xy ⌧2 mv ⌧ ev ⇥ B eE ✓ yy 1 !B ⌧ Rhine Samajdar (Harvard University) ◆ Ey ⇢xy = Jx B B !B 1 = = B DC ne RH = E !B ⌧ 1 ◆ ⇢xx = The Integer Quantum Hall E↵ect m ; n e2 ⌧ ⇢xy = B ne December 14, 2016 4 / 41 Introduction Integer Quantum Hall E↵ect The Integer Quantum Hall E↵ect Figure: Klitzing, Dorda, and Pepper, PRL 45, 494 (1980). Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 5 / 41 Introduction Integer Quantum Hall E↵ect The Integer Quantum Hall E↵ect The Hall resistivity ⇢xy sits on a plateau for a range of magnetic fields. On these plateaus, ⇢xy = eh2 ⌫1 ; ⌫ 2 Z. Figure: Klitzing, Dorda, and Pepper, PRL 45, 494 (1980). Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 5 / 41 Introduction Integer Quantum Hall E↵ect The Integer Quantum Hall E↵ect The Hall resistivity ⇢xy sits on a plateau for a range of magnetic fields. On these plateaus, ⇢xy = eh2 ⌫1 ; ⌫ 2 Z. When ⇢xy sits on a plateau, the longitudinal resistivity vanishes: ⇢xx = 0. Figure: Klitzing, Dorda, and Pepper, PRL 45, 494 (1980). Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 5 / 41 Introduction Integer Quantum Hall E↵ect The Integer Quantum Hall E↵ect The Hall resistivity ⇢xy sits on a plateau for a range of magnetic fields. On these plateaus, ⇢xy = eh2 ⌫1 ; ⌫ 2 Z. When ⇢xy sits on a plateau, the longitudinal resistivity vanishes: ⇢xx = 0. Figure: Klitzing, Dorda, and Pepper, PRL 45, 494 (1980). Rhine Samajdar (Harvard University) The center of each of plateau h occurs at B = n⌫ 0 ; 0 = e. The Integer Quantum Hall E↵ect December 14, 2016 5 / 41 Landau Levels Table of Contents 1 Introduction The Classical Hall E↵ect Integer Quantum Hall E↵ect 2 Landau Levels Review Conductivity: A quick calculation 3 5 6 Büttiker’s Theory Edge modes Disorder and Localization Rhine Samajdar (Harvard University) 4 7 Laughlin’s Gedankenexperiment Gauge Invariance Spectral Flow Topology Linear Response Quantized Hall Conductivity Particles on a Lattice The TKNN formula Peierls substitution Chern Numbers References The Integer Quantum Hall E↵ect December 14, 2016 6 / 41 Landau Levels Review Landau Levels Landau gauge: A = x B ŷ n,k (x, y) Symmetric gauge: A = 2 ⇠ eiky Hn (x + k lB ) 2 2 ⇥ e (x+k lB )/2lB The wavefunctions look like strips, extended along ŷ but exponentially 2 in the x̂ localized around x = klB direction. Rhine Samajdar (Harvard University) LLL,m ⇠ ✓ z lB ◆m e 1 2r ⇥B 2 |z|2 /4lB The wavefunction with angular momentum mpis peaked on a ring of radius r = 2mlB . The Integer Quantum Hall E↵ect December 14, 2016 7 / 41 Landau Levels Review Landau Levels Landau gauge: A = x B ŷ n,k (x, y) Symmetric gauge: A = 2 ⇠ eiky Hn (x + k lB ) 2 LLL,m 2 ⇠ ✓ z lB ◆m e 1 2r ⇥B 2 |z|2 /4lB ⇥ e (x+k lB )/2lB The wavefunctions look like strips, The wavefunction with angular moextended along ŷ but exponentially mentum mpis peaked on a ring of 2 in the x̂ radius r = localized around x = klB 2mlB . direction. ✓ ◆ 1 AB En = ~ ! B n + ; Degeneracy: N = 2 0 Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 7 / 41 Landau Levels Review Landau Levels Figure: Landau level wavefunctions in the symmetric gauge (Figure courtesy Emil J. Bergholtz). Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 8 / 41 Landau Levels Conductivity: A quick calculation Conductivity in Filled Landau Levels ⇢xy = h 1 e2 ⌫ Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 9 / 41 Landau Levels Conductivity: A quick calculation Conductivity in Filled Landau Levels ⇢xy = h 1 e2 ⌫ Rhine Samajdar (Harvard University) ⌫ Landau levels are filled. The Integer Quantum Hall E↵ect December 14, 2016 9 / 41 Landau Levels Conductivity: A quick calculation Ansatz: Drude model On a plateau, the Hall resistivity takes the value ⇢xy = Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect h 1 . e2 ⌫ December 14, 2016 10 / 41 Landau Levels Conductivity: A quick calculation Ansatz: Drude model h 1 . e2 ⌫ B ⇢xy = n e . On a plateau, the Hall resistivity takes the value ⇢xy = From our classical calculation in the Drude model Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 10 / 41 Landau Levels Conductivity: A quick calculation Ansatz: Drude model h 1 . e2 ⌫ B ⇢xy = n e . B On a plateau, the Hall resistivity takes the value ⇢xy = From our classical calculation in the Drude model Thus, to get the resistivity of the ⌫ th plateau, n = 0 ⌫. This is exactly the density of electrons required to fill ⌫ Landau levels! Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 10 / 41 Landau Levels Conductivity: A quick calculation Ansatz: Drude model h 1 . e2 ⌫ B ⇢xy = n e . B On a plateau, the Hall resistivity takes the value ⇢xy = From our classical calculation in the Drude model Thus, to get the resistivity of the ⌫ th plateau, n = 0 ⌫. This is exactly the density of electrons required to fill ⌫ Landau levels! Further, when ⌫ Landau levels are filled, there is a gap in the energy spectrum. When we turn on a small electric field, the electrons are stuck in place like in an insulator. This means that the scattering time ⌧ ! 1 and ⇢xx = 0. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 10 / 41 Büttiker’s Theory Table of Contents 1 Introduction The Classical Hall E↵ect Integer Quantum Hall E↵ect 2 Landau Levels Review Conductivity: A quick calculation 3 5 6 Büttiker’s Theory Edge modes Disorder and Localization Rhine Samajdar (Harvard University) 4 7 Laughlin’s Gedankenexperiment Gauge Invariance Spectral Flow Topology Linear Response Quantized Hall Conductivity Particles on a Lattice The TKNN formula Peierls substitution Chern Numbers References The Integer Quantum Hall E↵ect December 14, 2016 11 / 41 Büttiker’s Theory Conductivity for a single free particle Particle velocity: m ẋ = p + e A Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 12 / 41 Büttiker’s Theory Conductivity for a single free particle Particle velocity: m ẋ = p + e A e P Total current: I = m filledstates h | i ~r + eA| i Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 12 / 41 Büttiker’s Theory Conductivity for a single free particle Particle velocity: m ẋ = p + e A e P Total current: I = m filledstates h | i ~r + eA| i Thus, with ⌫ Landau levels filled: ⌫ Ix = Iy = e XX m e m ⌧ n=1 k ⌫ X⌧ X n=1 k Rhine Samajdar (Harvard University) n,k i~ @ @x n,k i~ @ + exB @y =0 n,k The Integer Quantum Hall E↵ect n,k = e⌫ XE k B December 14, 2016 12 / 41 Büttiker’s Theory Conductivity for a single free particle Hence, ✓ ◆ ✓ E 0 E= )J= 0 e ⌫ E/ Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect 0 ◆ December 14, 2016 13 / 41 Büttiker’s Theory Conductivity for a single free particle Hence, ✓ ◆ ✓ E 0 E= )J= 0 e ⌫ E/ 0 ◆ Comparing to the definition of the conductivity tensor, we have xx Rhine Samajdar (Harvard University) =0 xy = e⌫ 0 The Integer Quantum Hall E↵ect December 14, 2016 13 / 41 Büttiker’s Theory Conductivity for a single free particle Hence, ✓ ◆ ✓ E 0 E= )J= 0 e ⌫ E/ 0 ◆ Comparing to the definition of the conductivity tensor, we have xx =0 xy = e⌫ 0 This is exactly the conductivity seen on the quantum Hall plateaus! Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 13 / 41 Büttiker’s Theory Edge modes Edge Modes Figure: Classical picture (skipping orbits). Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 14 / 41 Büttiker’s Theory Edge modes Edge Modes Particles move in one direction on one side of the sample, and in the other direction on the other side i.e. the particles have opposite chirality on the two sides. Figure: Classical picture (skipping orbits). Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 14 / 41 Büttiker’s Theory Edge modes Edge Modes Figure: E↵ective potential. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 15 / 41 Büttiker’s Theory Edge modes Edge Modes H= 1 p2 + (py + e B x)2 +V (x) 2m x If the potential is smooth over distance scales lB , V (x) ⇡ V (X) + @V (x @x X) + . . . Figure: E↵ective potential. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 15 / 41 Büttiker’s Theory Edge modes Edge Modes H= 1 p2 + (py + e B x)2 +V (x) 2m x If the potential is smooth over distance scales lB , V (x) ⇡ V (X) + Figure: E↵ective potential. Rhine Samajdar (Harvard University) @V (x @x X) + . . . Drift velocity along ŷ : vy = The Integer Quantum Hall E↵ect December 14, 2016 1 @V eB @x 15 / 41 Büttiker’s Theory Edge modes Filling the edge states... Figure: The bulk is an insulator. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 16 / 41 Büttiker’s Theory Edge modes Filling the edge states... Figure: Introduce a (chemical) potential di↵erence µ. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 16 / 41 Büttiker’s Theory Edge modes Filling the edge states... Z dk vy (k) 2⇡ Z e 1 @V = dx 2 eB @x 2⇡ lB e = µ h Iy = Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect e December 14, 2016 16 / 41 Büttiker’s Theory Edge modes Filling the edge states... Z dk vy (k) 2⇡ Z e 1 @V = dx 2 eB @x 2⇡ lB e = µ h Iy = e The Hall voltage is e VH = µ, giving us the Hall conductivity xy = Iy e2 = . VH h [Büttiker, PRB 38, 9375 (1988).] Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 16 / 41 Büttiker’s Theory Edge modes Filling the edge states... The e↵ective potential could be tilted by an electric field. . . Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 17 / 41 Büttiker’s Theory Edge modes Filling the edge states... . . . or simply be random—the Hall conductivity remains quantized. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 17 / 41 Büttiker’s Theory Edge modes Filling the edge states... With n filled Landau levels, there are . . . or simply be random—the Hall n chiral modes on each edge (as long conductivity remains quantized. as EF lies between levels). Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 17 / 41 Büttiker’s Theory Edge modes Robustness of the Hall State The calculations above show that if an integer number of Landau levels are filled, then the longitudinal and Hall resistivities are those observed on the plateaus. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 18 / 41 Büttiker’s Theory Edge modes Robustness of the Hall State The calculations above show that if an integer number of Landau levels are filled, then the longitudinal and Hall resistivities are those observed on the plateaus. Questions: Why do these plateaus exist in the first place? Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 18 / 41 Büttiker’s Theory Edge modes Robustness of the Hall State The calculations above show that if an integer number of Landau levels are filled, then the longitudinal and Hall resistivities are those observed on the plateaus. Questions: Why do these plateaus exist in the first place? Why are there sharp jumps between di↵erent plateaus? Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 18 / 41 Büttiker’s Theory Disorder and Localization The role of disorder Assume V ⌧ ~!B and |rV | ⌧ ~ !B lB (weak disorder). Figure: Density of states without [left panel] and with disorder [right panel]. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 19 / 41 Büttiker’s Theory Disorder and Localization Localization: Semiclassical argument Disorder turns many of the quantum states from extended to localized. Center of the cyclotron orbit: X=x ⇡y ⇡x ; Y =y+ m !b m !b Time evolution: @V @Y 2 @V ilB @X 2 i~Ẋ = [X, H + V ] = ilB i~Ẏ = [Y, H + V ] = Thus, the center of mass drifts in a direction (Ẋ, Ẏ ) which is perpendicular to rV i.e., the motion is along equipotentials. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 20 / 41 Büttiker’s Theory Disorder and Localization Localization due to disorder Figure: The localization of states due to disorder [left panel] and the resultant density of states [right panel]. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 21 / 41 Büttiker’s Theory Disorder and Localization Summary: Presence of plateaus Equipotentials which stretch from one side of a sample to another are relatively rare but exist on the edge of the sample. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 22 / 41 Büttiker’s Theory Disorder and Localization Summary: Presence of plateaus Equipotentials which stretch from one side of a sample to another are relatively rare but exist on the edge of the sample. The states at the far edge of a band (either of high or low energy) are localized. Only states close to the center of the band are extended. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 22 / 41 Büttiker’s Theory Disorder and Localization Summary: Presence of plateaus Equipotentials which stretch from one side of a sample to another are relatively rare but exist on the edge of the sample. The states at the far edge of a band (either of high or low energy) are localized. Only states close to the center of the band are extended. Since the localized states cannot contribute to the current, populating these states does not change the conductivity—this explains the plateaus that are observed. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 22 / 41 Laughlin’s Gedankenexperiment Table of Contents 1 Introduction The Classical Hall E↵ect Integer Quantum Hall E↵ect 2 Landau Levels Review Conductivity: A quick calculation 3 5 6 Büttiker’s Theory Edge modes Disorder and Localization Rhine Samajdar (Harvard University) 4 7 Laughlin’s Gedankenexperiment Gauge Invariance Spectral Flow Topology Linear Response Quantized Hall Conductivity Particles on a Lattice The TKNN formula Peierls substitution Chern Numbers References The Integer Quantum Hall E↵ect December 14, 2016 23 / 41 Laughlin’s Gedankenexperiment Gedankenexperiment geometry Figure: Gedankenexperiment considered by Laughlin [left panel] and the Corbino ring geometry [right] topologically equivalent to it [Laughlin, PRB 23, 5632 (1981)]. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 24 / 41 Laughlin’s Gedankenexperiment Gauge Invariance The role of Gauge Invariance Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of the ring. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 25 / 41 Laughlin’s Gedankenexperiment Gauge Invariance The role of Gauge Invariance Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of the ring. Suppose we increase EMF E = 0 /T. Rhine Samajdar (Harvard University) slowly from 0 to The Integer Quantum Hall E↵ect 0 = h/e. This induces an December 14, 2016 25 / 41 Laughlin’s Gedankenexperiment Gauge Invariance The role of Gauge Invariance Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of the ring. Suppose we increase EMF E = 0 /T. slowly from 0 to 0 = h/e. This induces an Assume that n electrons are transferred from the inner circle to the outer circle in this time. This would result in a radial current Ir = n e/T . Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 25 / 41 Laughlin’s Gedankenexperiment Gauge Invariance The role of Gauge Invariance Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of the ring. Suppose we increase EMF E = 0 /T. slowly from 0 to 0 = h/e. This induces an Assume that n electrons are transferred from the inner circle to the outer circle in this time. This would result in a radial current Ir = n e/T . Then, ⇢xy = Rhine Samajdar (Harvard University) E h 1 = 2 . Ir e n The Integer Quantum Hall E↵ect December 14, 2016 25 / 41 Laughlin’s Gedankenexperiment Gauge Invariance The role of Gauge Invariance Corbino ring: In addition to the background magnetic field B which penetrates the sample, we can thread an additional flux through the center of the ring. Suppose we increase EMF E = 0 /T. slowly from 0 to 0 = h/e. This induces an Assume that n electrons are transferred from the inner circle to the outer circle in this time. This would result in a radial current Ir = n e/T . Then, ⇢xy = E h 1 = 2 . Ir e n Our task, therefore, is to argue that n electrons are indeed transferred across the ring. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 25 / 41 Laughlin’s Gedankenexperiment Spectral Flow Spectral Flow in Landau Levels Using symmetric gauge, the wavefunctions in the lowest Landau level are 2 m |z|2 /4lB . m ⇠z e q 2. The mth wavefunction is strongly peaked at a radius r ⇡ 2mlB Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 26 / 41 Laughlin’s Gedankenexperiment Spectral Flow Spectral Flow in Landau Levels Using symmetric gauge, the wavefunctions in the lowest Landau level are 2 m |z|2 /4lB . m ⇠z e q 2. The mth wavefunction is strongly peaked at a radius r ⇡ 2mlB If we increase the flux from m( = 0) ! = 0 to = the wavefunctions shift as = m+1 ( = 0). q 2. Each state moves outwards, to radius r = 2(m + 1)lB Rhine Samajdar (Harvard University) m( 0, 0) The Integer Quantum Hall E↵ect December 14, 2016 26 / 41 Laughlin’s Gedankenexperiment Spectral Flow Spectral Flow in Landau Levels Using symmetric gauge, the wavefunctions in the lowest Landau level are 2 m |z|2 /4lB . m ⇠z e q 2. The mth wavefunction is strongly peaked at a radius r ⇡ 2mlB If we increase the flux from m( = 0) ! = 0 to m( = 0, the wavefunctions shift as 0) = m+1 ( = 0). q 2. Each state moves outwards, to radius r = 2(m + 1)lB Thus if all states in the Landau level are filled, a single electron is transferred from the inner ring to the outer ring as the flux is increased. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 26 / 41 Laughlin’s Gedankenexperiment Spectral Flow Spectral Flow in the Presence of Disorder 1 H = 2m ✓ ◆ ✓ @ @ ~ r + r @r @r 21 Rhine Samajdar (Harvard University) ~ @ eBr e i + + r@ 2 2⇡r The Integer Quantum Hall E↵ect ◆2 + V (r, ) December 14, 2016 27 / 41 Laughlin’s Gedankenexperiment Spectral Flow Spectral Flow in the Presence of Disorder 1 H = 2m ✓ ◆ ✓ @ @ ~ r + r @r @r 21 ~ @ eBr e i + + r@ 2 2⇡r ◆2 + V (r, ) We attempt to undo the flux by a gauge transformation ✓ ◆ e (r, ) ! exp i (r, ) h Always possible for localized states; possible for extended states only when is an integer multiple of 0 . Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 27 / 41 Laughlin’s Gedankenexperiment Spectral Flow Spectral Flow in the Presence of Disorder 1 H = 2m ✓ ◆ ✓ @ @ ~ r + r @r @r 21 ~ @ eBr e i + + r@ 2 2⇡r ◆2 + V (r, ) We attempt to undo the flux by a gauge transformation ✓ ◆ e (r, ) ! exp i (r, ) h Always possible for localized states; possible for extended states only when is an integer multiple of 0 . Conclusion Only the extended states undergo spectral flow; these alone must map onto themselves. The localized states don’t change as increases. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 27 / 41 Topology Table of Contents 1 Introduction The Classical Hall E↵ect Integer Quantum Hall E↵ect 2 Landau Levels Review Conductivity: A quick calculation 3 5 6 Büttiker’s Theory Edge modes Disorder and Localization Rhine Samajdar (Harvard University) 4 7 Laughlin’s Gedankenexperiment Gauge Invariance Spectral Flow Topology Linear Response Quantized Hall Conductivity Particles on a Lattice The TKNN formula Peierls substitution Chern Numbers References The Integer Quantum Hall E↵ect December 14, 2016 28 / 41 Topology Linear Response Linear Response xy Z 1 1 dt ei!t h0 |[Jy (0), Jx (t)]| 0i ~! 0 i X h0 |Jy | ni hn |Jx | 0i h0 |Jx | ni hn |Jy | 0i = ! ~! + En E0 ~! + E0 En = n6=0 Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 29 / 41 Topology Linear Response Linear Response xy Z 1 1 dt ei!t h0 |[Jy (0), Jx (t)]| 0i ~! 0 i X h0 |Jy | ni hn |Jx | 0i h0 |Jx | ni hn |Jy | 0i = ! ~! + En E0 ~! + E0 En = n6=0 Taking the DC (! ! 0) limit, we get Kubo formula xy = i~ X h0 |Jy | ni hn |Jx | 0i (En n6=0 Rhine Samajdar (Harvard University) h0 |Jx | ni hn |Jy | 0i E0 ) 2 The Integer Quantum Hall E↵ect December 14, 2016 29 / 41 Topology Linear Response The role of Topology Consider the Hall system on a spatial torus T2 . We thread a uniform magnetic field B through the torus. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 30 / 41 Topology Linear Response The role of Topology Consider the Hall system on a spatial torus T2 . We thread a uniform magnetic field B through the torus. Magnetic translation operators: ✓ ◆ ✓ d·p T (d) = exp i ! exp ~ Rhine Samajdar (Harvard University) ir + eA id · ~ The Integer Quantum Hall E↵ect ◆ December 14, 2016 30 / 41 Topology Linear Response The role of Topology Consider the Hall system on a spatial torus T2 . We thread a uniform magnetic field B through the torus. Magnetic translation operators: ✓ ◆ ✓ d·p T (d) = exp i ! exp ~ In Landau gauge, Ty Tx = e i e B Lx Ly /~ T ir + eA id · ~ x Ty , ◆ so we must have Dirac quantization condition B Lx Ly = Rhine Samajdar (Harvard University) h n; n 2 Z e The Integer Quantum Hall E↵ect December 14, 2016 30 / 41 Topology Linear Response Adding Flux Ax = x Lx ; Ay = H= y Ly +Bx X Ji i Li i=x,y Figure: Two fluxes are threaded through the torus. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 31 / 41 Topology Linear Response Adding Flux Ax = x Lx ; Ay = y Ly +Bx X Ji i Li H= i=x,y To first order in perturbation theory, | 0i 0 =| 0i + X hn| H| 0 i |ni En E0 n6= 0 Figure: Two fluxes are threaded through the torus. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 31 / 41 Topology Linear Response Adding Flux Ax = x Lx ; Ay = y Ly +Bx X Ji i Li H= i=x,y To first order in perturbation theory, | Figure: Two fluxes are threaded through the torus. Rhine Samajdar (Harvard University) 0i @ @ 0 =| 0 i The Integer Quantum Hall E↵ect 0i = + X hn| H| 0 i |ni En E0 n6= 0 1 X hn|Ji | 0 i |ni Li En E0 n6= 0 December 14, 2016 31 / 41 Topology Quantized Hall Conductivity Hall Conductivity and the Chern Number Introduce dimensionless angular variables ✓i = 2⇡ Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect i 0 , to parametrize T2 . December 14, 2016 32 / 41 Topology Quantized Hall Conductivity Hall Conductivity and the Chern Number Introduce dimensionless angular variables ✓i = 2⇡ ⌧ @ Berry connection: Ai ( ) = i 0 0 @✓i Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect i 0 , to parametrize T2 . December 14, 2016 32 / 41 Topology Quantized Hall Conductivity Hall Conductivity and the Chern Number Introduce dimensionless angular variables ✓i = 2⇡ ⌧ @ Berry connection: Ai ( ) = i 0 0 @✓i Berry curvature: ⌧ @Ax @Ay @ @ 0 Fxy = = i 0 @✓y @✓x @✓y @✓x Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect i 0 , to parametrize T2 . @ @✓x ⌧ 0 @ 0 @✓y December 14, 2016 32 / 41 Topology Quantized Hall Conductivity Hall Conductivity and the Chern Number Introduce dimensionless angular variables ✓i = 2⇡ ⌧ @ Berry connection: Ai ( ) = i 0 0 @✓i Berry curvature: ⌧ @Ax @Ay @ @ 0 Fxy = = i 0 @✓y @✓x @✓y @✓x xy = i~ @ @ y Rhine Samajdar (Harvard University) ⌧ 0 @ @ @ 0 x @ x ⌧ 0 The Integer Quantum Hall E↵ect @ @ i 0 , to parametrize T2 . @ @✓x 0 y ⌧ = 0 @ 0 @✓y e2 Fxy ~ December 14, 2016 32 / 41 Topology Quantized Hall Conductivity Hall Conductivity and the Chern Number Introduce dimensionless angular variables ✓i = 2⇡ ⌧ @ Berry connection: Ai ( ) = i 0 0 @✓i Berry curvature: ⌧ @Ax @Ay @ @ 0 Fxy = = i 0 @✓y @✓x @✓y @✓x xy = i~ @ @ y ⌧ 0 @ @ x Averaging over all fluxes, Z e2 d2 ✓ Fxy = xy = ~ T2 (2⇡)2 Rhine Samajdar (Harvard University) @ 0 @ x ⌧ 0 @ @ i 0 @ @✓x 0 ⌧ = y e2 1 C; C = h 2⇡ The Integer Quantum Hall E↵ect , to parametrize T2 . Z T2 0 @ 0 @✓y e2 Fxy ~ d2 ✓ Fxy December 14, 2016 32 / 41 Topology Quantized Hall Conductivity Hall Conductivity and the Chern Number Introduce dimensionless angular variables ✓i = 2⇡ ⌧ @ Berry connection: Ai ( ) = i 0 0 @✓i Berry curvature: ⌧ @Ax @Ay @ @ 0 Fxy = = i 0 @✓y @✓x @✓y @✓x xy = i~ @ @ y ⌧ 0 @ @ x Averaging over all fluxes, Z e2 d2 ✓ Fxy = xy = ~ T2 (2⇡)2 Rhine Samajdar (Harvard University) @ 0 @ x ⌧ 0 @ @ i 0 @ @✓x 0 ⌧ = y e2 1 C; C = h 2⇡ The Integer Quantum Hall E↵ect , to parametrize T2 . Z T2 0 @ 0 @✓y e2 Fxy ~ d2 ✓ Fxy 2 Z December 14, 2016 32 / 41 Particles on a Lattice Table of Contents 1 Introduction The Classical Hall E↵ect Integer Quantum Hall E↵ect 2 Landau Levels Review Conductivity: A quick calculation 3 5 6 Büttiker’s Theory Edge modes Disorder and Localization Rhine Samajdar (Harvard University) 4 7 Laughlin’s Gedankenexperiment Gauge Invariance Spectral Flow Topology Linear Response Quantized Hall Conductivity Particles on a Lattice The TKNN formula Peierls substitution Chern Numbers References The Integer Quantum Hall E↵ect December 14, 2016 33 / 41 Particles on a Lattice The TKNN formula TKNN invariants Lattice momenta take values in the Brillouin zone: ⇡ ⇡ < kx ; a a ⇡ ⇡ < ky b b The wavefunctions in a given band can be written in Bloch form as k (x) = ei k·x uk (x) with uk (x) periodic on a unit cell. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 34 / 41 Particles on a Lattice The TKNN formula TKNN invariants Lattice momenta take values in the Brillouin zone: ⇡ ⇡ < kx ; a a ⇡ ⇡ < ky b b The wavefunctions in a given band can be written in Bloch form as k (x) = ei k·x uk (x) with uk (x) periodic on a unit cell. Assumptions: The single particle spectrum decomposes into bands. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 34 / 41 Particles on a Lattice The TKNN formula TKNN invariants Lattice momenta take values in the Brillouin zone: ⇡ ⇡ < kx ; a a ⇡ ⇡ < ky b b The wavefunctions in a given band can be written in Bloch form as k (x) = ei k·x uk (x) with uk (x) periodic on a unit cell. Assumptions: The single particle spectrum decomposes into bands. The electrons are non-interacting. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 34 / 41 Particles on a Lattice The TKNN formula TKNN invariants Lattice momenta take values in the Brillouin zone: ⇡ ⇡ < kx ; a a ⇡ ⇡ < ky b b The wavefunctions in a given band can be written in Bloch form as k (x) = ei k·x uk (x) with uk (x) periodic on a unit cell. Assumptions: The single particle spectrum decomposes into bands. The electrons are non-interacting. There is a gap between bands and the Fermi energy EF lies in one of these gaps. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 34 / 41 Particles on a Lattice The TKNN formula TKNN invariants Berry connection: Ai (k) = Rhine Samajdar (Harvard University) ⌧ i uk @ uk @k i The Integer Quantum Hall E↵ect December 14, 2016 35 / 41 Particles on a Lattice The TKNN formula TKNN invariants Berry connection: Ai (k) = Berry curvature: @Ax @Ay Fxy = = @k y @k x Rhine Samajdar (Harvard University) i ⌧ ⌧ i uk @ uk @k i @u @u @k y @k x +i The Integer Quantum Hall E↵ect ⌧ @u @u @k x @k y December 14, 2016 35 / 41 Particles on a Lattice The TKNN formula TKNN invariants ⌧ i uk Berry connection: Ai (k) = Berry curvature: @Ax @Ay Fxy = = @k y @k x Chern number: C = Rhine Samajdar (Harvard University) i 1 2⇡ ⌧ Z @ uk @k i @u @u @k y @k x T2 +i ⌧ @u @u @k x @k y d2 k Fxy 2 Z The Integer Quantum Hall E↵ect December 14, 2016 35 / 41 Particles on a Lattice The TKNN formula TKNN invariants ⌧ i uk Berry connection: Ai (k) = Berry curvature: @Ax @Ay Fxy = = @k y @k x Chern number: C = i 1 2⇡ ⌧ Z @ uk @k i @u @u @k y @k x T2 +i ⌧ @u @u @k x @k y d2 k Fxy 2 Z More generally, assigning a Chern number to each band ↵, we get: TKNN formula xy = e2 X C↵ (Topological invariant) h ↵ [Thouless, Kohomoto, Nightingale, and den Nijs, PRL 49, 405 (1982).] Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 35 / 41 Particles on a Lattice Peierls substitution Adding a magnetic field Tight-binding model: H = Rhine Samajdar (Harvard University) t P P x j=1,2 |xihx The Integer Quantum Hall E↵ect + ej |+ h.c. December 14, 2016 36 / 41 Particles on a Lattice Peierls substitution Adding a magnetic field Tight-binding model: H = Rhine Samajdar (Harvard University) t P P x j=1,2 |xie The Integer Quantum Hall E↵ect ie a Aj (x)/~ hx + ej |+ h.c. December 14, 2016 36 / 41 Particles on a Lattice Peierls substitution Adding a magnetic field Tight-binding model: H = t P P x j=1,2 |xie ie a Aj (x)/~ hx + ej |+ h.c. Figure: Consider a particle which moves anti-clockwise around this plaquette. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 36 / 41 Particles on a Lattice Peierls substitution Adding a magnetic field Tight-binding model: H = t P P x j=1,2 |xie ie a Aj (x)/~ hx + ej |+ h.c. To leading order in t, it will pick up a phase e i , where eA A1 (x) + A2 (x + e1 ) ~ A1 (x + e2 ) A2 (x) ✓ ◆ e A2 @A2 @A1 eA2 B ⇡ = ~ @x1 @x2 ~ = Figure: Consider a particle which moves anti-clockwise around this plaquette. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 36 / 41 Particles on a Lattice Peierls substitution Adding a magnetic field Tight-binding model: H = t P P x j=1,2 |xie ie a Aj (x)/~ hx + ej |+ h.c. To leading order in t, it will pick up a phase e i , where eA A1 (x) + A2 (x + e1 ) ~ A1 (x + e2 ) A2 (x) ✓ ◆ e A2 @A2 @A1 eA2 B ⇡ = ~ @x1 @x2 ~ = Figure: Consider a particle which moves anti-clockwise around this plaquette. Rhine Samajdar (Harvard University) Aharonov-Bohm phase! The Integer Quantum Hall E↵ect December 14, 2016 36 / 41 Particles on a Lattice Peierls substitution Magnetic Brillouin zone Construct operators: T̃j = P x |xie ie a Ãj (x)/~ hx + ej |. New gauge field: @k Ãj = @j Ak . In Landau gauge, Ã1 = B x2 , Ã2 = 0. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 37 / 41 Particles on a Lattice Peierls substitution Magnetic Brillouin zone Construct operators: T̃j = P x |xie ie a Ãj (x)/~ hx + ej |. New gauge field: @k Ãj = @j Ak . In Landau gauge, Ã1 = B x2 , Ã2 = 0. We can build commuting operators by [T̃1n1 , T̃2n2 ], when p q Rhine Samajdar (Harvard University) December 14, 2016 The Integer Quantum Hall E↵ect n1 n2 2 Z. 37 / 41 Particles on a Lattice Peierls substitution Magnetic Brillouin zone Construct operators: T̃j = P x |xie ie a Ãj (x)/~ hx + ej |. New gauge field: @k Ãj = @j Ak . In Landau gauge, Ã1 = B x2 , Ã2 = 0. We can build commuting operators by [T̃1n1 , T̃2n2 ], when p q n1 n2 2 Z. Bloch-like eigenstates: H|ki = E(k) |ki with T1q |ki = eiq k1 a |ki. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 37 / 41 Particles on a Lattice Peierls substitution Magnetic Brillouin zone Construct operators: T̃j = P x |xie ie a Ãj (x)/~ hx + ej |. New gauge field: @k Ãj = @j Ak . In Landau gauge, Ã1 = B x2 , Ã2 = 0. We can build commuting operators by [T̃1n1 , T̃2n2 ], when p q n1 n2 2 Z. Bloch-like eigenstates: H|ki = E(k) |ki with T1q |ki = eiq k1 a |ki. Magnetic Brillouin zone ⇡ ⇡ < k1 ; qa qa Rhine Samajdar (Harvard University) ⇡ ⇡ < k2 a a The Integer Quantum Hall E↵ect December 14, 2016 37 / 41 Particles on a Lattice Peierls substitution Energy Spectrum The spectrum decomposes into q bands, each with a di↵erent range of energies. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 38 / 41 Particles on a Lattice Peierls substitution Energy Spectrum The spectrum decomposes into q bands, each with a di↵erent range of energies. Any energy eigenvalue in a given band is q-fold degenerate as |ki has the same energy as T̃1 |ki ⇠ |k1 , k2 + 2⇡p/qai. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 38 / 41 Particles on a Lattice Peierls substitution Energy Spectrum The spectrum decomposes into q bands, each with a di↵erent range of energies. Any energy eigenvalue in a given band is q-fold degenerate as |ki has the same energy as T̃1 |ki ⇠ |k1 , k2 + 2⇡p/qai. Rhine Samajdar (Harvard University) Figure: Hofstadter butterfly.[Hofstadter, PRB 14, 2239 (1976).] The Integer Quantum Hall E↵ect December 14, 2016 38 / 41 Particles on a Lattice Chern Numbers TKNN Invariants on a Lattice We can compute the Hall conductivity only for rational fluxes for which there exists a magnetic Brillouin zone. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect =p December 14, 2016 0 /q 39 / 41 Particles on a Lattice Chern Numbers TKNN Invariants on a Lattice We can compute the Hall conductivity only for rational fluxes = p 0 /q for which there exists a magnetic Brillouin zone. First consider the rth of the q bands. Then, to compute the Chern number, we have to solve the linear Diophantine equation q r = q sr + p tr ; |tr | . 2 Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 39 / 41 Particles on a Lattice Chern Numbers TKNN Invariants on a Lattice We can compute the Hall conductivity only for rational fluxes = p 0 /q for which there exists a magnetic Brillouin zone. First consider the rth of the q bands. Then, to compute the Chern number, we have to solve the linear Diophantine equation q r = q sr + p tr ; |tr | . 2 The Chern number of the rth band is C = tr Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect tr 1, with t0 ⌘ 0. December 14, 2016 39 / 41 Particles on a Lattice Chern Numbers TKNN Invariants on a Lattice We can compute the Hall conductivity only for rational fluxes = p 0 /q for which there exists a magnetic Brillouin zone. First consider the rth of the q bands. Then, to compute the Chern number, we have to solve the linear Diophantine equation q r = q sr + p tr ; |tr | . 2 The Chern number of the rth band is C = tr tr 1 , with t0 ⌘ 0. e2 If Er < EF < Er+1 , then xy = tr [TKNN, PRL 49, 405 (1982)]. h Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 39 / 41 References Table of Contents 1 Introduction The Classical Hall E↵ect Integer Quantum Hall E↵ect 2 Landau Levels Review Conductivity: A quick calculation 3 5 6 Büttiker’s Theory Edge modes Disorder and Localization Rhine Samajdar (Harvard University) 4 7 Laughlin’s Gedankenexperiment Gauge Invariance Spectral Flow Topology Linear Response Quantized Hall Conductivity Particles on a Lattice The TKNN formula Peierls substitution Chern Numbers References The Integer Quantum Hall E↵ect December 14, 2016 40 / 41 References References and Acknowledgments D. Tong, The Quantum Hall E↵ect, TIFR Infosys Lectures (2016). D. Yoshioka, The Quantum Hall E↵ect, Springer-Verlag (2002). Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 41 / 41 References References and Acknowledgments D. Tong, The Quantum Hall E↵ect, TIFR Infosys Lectures (2016). D. Yoshioka, The Quantum Hall E↵ect, Springer-Verlag (2002). I acknowledge useful discussions with Aavishkar A. Patel. Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 41 / 41 References References and Acknowledgments D. Tong, The Quantum Hall E↵ect, TIFR Infosys Lectures (2016). D. Yoshioka, The Quantum Hall E↵ect, Springer-Verlag (2002). I acknowledge useful discussions with Aavishkar A. Patel. Thank you! Rhine Samajdar (Harvard University) The Integer Quantum Hall E↵ect December 14, 2016 41 / 41