(1) Topological terms and metallic transport (2) Dynamics as a probe of Majorana fermions

Transcription

1 (1) Topological terms and metallic transport (2) Dynamics as a probe of Majorana fermions Harvard, September 16, 2014 Joel Moore University of California, Berkeley, and Lawrence Berkeley National Laboratory

2 Outline 1. Approaching metallic transport in magnetic materials by statistical topology: a topological version of the anomalous Hall effect (J. Dahlhaus, R. Ilan, and JEM, unpublished; JEM and J. Orenstein, PRL 2010) Partly motivated by challenge of finding a use for new C*- algebra methods to compute topological invariants efficiently. 2. Quench dynamics in condensed matter and an orthogonality catastrophe probe of Majorana/parafermion zero modes (Romain Vasseur, J. Dahlhaus, and JEM, Physical Review X 2014; RV and JEM, PRL 2014)

3 Topological invariants Bloch s theorem: One-electron wavefunctions in a crystal (i.e., periodic potential) can be written Good news: for the invariants in the IQHE and topological insulators, we need one fact about solids (r) =e ik r u k (r) where k is crystal momentum and u is periodic (the same in every unit cell). Crystal momentum k can be restricted to the Brillouin zone, a region of k-space with periodic boundaries. As k changes, we map out an energy band. Set of all bands = band structure. The Brillouin zone will play the role of the surface as in the previous example, and one property of quantum mechanics, the Berry phase which will give us the curvature.

4 Berry phase in solids In a solid, the natural parameter space is electron momentum. The change in the electron wavefunction within the unit cell leads to a Berry connection and Berry curvature: (r) =e ik r u k (r) A = u k i k u k F = A We keep finding more physical properties that are determined by these quantum geometric quantities. The first was that the integer quantum Hall effect in a 2D crystal follows from the integral of F (like Gauss-Bonnet). Explicitly, n = bands i 2π σ xy = n e2 h d 2 k ( u k 1 u k 2 TKNN, 1982 u k 2 u k 1 ) first Chern number F = S. S. Chern First QAHE observation 2013, by Beijing-Stanford group A

5 How can we picture A? = A dk, A = k i k k To get a physical interpretation of what A means, note that if we consider a plane wave exp(i k r), then the vector potential just gives the position r. Now in a periodic crystal, the position can t be uniquely defined, but we nevertheless expect that A might reflect something to do with the position of the wavefunction within the unit cell.

6 Recall ordinary electrical polarization Electrical polarization: another simple Berry phase in solids (Can be used to give another picture of topological insulators) Sum the integral of A over bands: in one spatial dimension, P = X Z dq e 2 hu v(q) i@ q u v (q)i v Intuitive idea: think about the momentum-position commutation relation, A = hu k ir k u k i hri There is an ambiguity of e per transverse unit cell, the polarization quantum. Note: just as da=f is a closed form and very useful to define Chern number, in 4 dimensions there is a second Chern form Fact from cohomology: Odd dimensions have Chern-Simons forms that have a quantum ambiguity; Even dimensions have Chern forms that are quantized.

7 But what does F do in metals? It is useful to get some intuition about what the Berry F means: Its simplest consequence is that it modifies the semiclassical equations of motion of a Bloch wavepacket: dx a dt = 1 n a + F ab n (k) dk b dt. a magnetic field in momentum space. The anomalous velocity results from changes in the electron distribution within the unit cell: the Berry phase is connected to the electron spatial location. Example I: the intrinsic anomalous Hall effect in itinerant magnets still no universal agreement on its existence; wait one moment Example II: helicity-dependent photocurrents in optically active materials (Berry phases in nonlinear transport) (JEM and J. Orenstein, PRL 2010)

8 But what does F do? Example I: the anomalous Hall effect in itinerant magnets An electrical field E induces a transverse current through the anomalous velocity if F is nonzero averaged over the ground state. A nonzero Hall current requires T breaking; microscopically this follows since time-reversal symmetry implies dx a dt = 1 n a + F ab n (k) dk b dt. F ab (k) = F ab ( k). Smit s objection: in steady state the electron distribution is stationary; why should the anomalous velocity contribute at all? (In a quantum treatment, the answer is as if dk/dt resulted only from the macroscopic applied field, which is mostly consistent with experiment)

9 But what does F do? To try to resolve the question of what the semiclassical equation means: Example II: helicity-dependent photocurrents in optically active materials (Berry phases in nonlinear transport) In a T-symmetric material, the Berry phase is still important at finite frequency. Consider circular polarization: The small deviation in the electron distribution generated by the electrical field gives an anomalous velocity contribution that need not average to zero over the wave. k y ee v 0 v 1 dk/dt k x

10 Smit vs. Luttinger The resulting formula has 3 terms, of which one is Smit-type (i.e., nonzero even with the full E) and two are Luttinger-type. j dc x ne 3 2~ 2 1 1/ h i(e x E y E y Ex)ˆx i +1/ (E x Ey + E y Ex)ˆx + E x 2 ŷ. (JEM and J. Orenstein, PRL 2010). See also Deyo, Golub, Ivchenko, and Spivak (arxiv, 2009). We believe that the circularly switched term actually explains a decade of experiments on helicity-dependent photocurrents in GaAs quantum wells. Bulk GaAs has too much symmetry to allow the effect; these quantum wells show the effect because the well confinement breaks the symmetry ( confinement-induced Berry phase ).

11 Confinement-induced Berry phases Bulk GaAs has too much symmetry to allow the effect; these quantum wells show the effect because the well confinement breaks the symmetry ( confinement-induced Berry phase ). Our numerics and envelope approximation suggest a magnitude of 1 na for incident power 1W in a (110) well, which is consistent with experiments by S. D. Ganichev et al. (Regensburg). Only one parameter of GaAs is needed to describe F at the Brillouin zone origin: symmetries force (a) (b) Ω k (Å 3 ) (a/2,-a/2,0) (0,0,a) Width (Å) 0.25 ev square 0.5 ev square 0.25 ev Gaussian 0.5 ev triangular Envelope approx. F = k x (k 2 y k 2 z),k y (k 2 z k 2 x),k z (k 2 x k 2 y), 410Å 3

12 Anomalous Hall effect from topology Problem with semiclassical (and other) approaches: Focus on 2D systems to make the problem especially clear. Without disorder, there are Bloch oscillations and transport samples all states, not just initially occupied states. With disorder, there is no true metal at T=0 without interactions in 2D with broken time-reversal symmetry. From Nagaosa et al., RMP 2011

13 T=0 approach to disorderaveraged Hall conductivity Focus on 2D systems to make the problem especially clear. Without disorder, there are Bloch oscillations and transport samples all states, not just initially occupied states. With disorder, there is no true metal at T=0 without interactions in 2D with broken time-reversal symmetry. From Nagaosa et al., RMP 2011

14 Take a supercell of many unit cells Φ 1 Phase boundary conditions = momentum in band structure Φ 2 Larger real-space unit cell = smaller momentum-space Brillouin zone Bands touch at zone boundaries since they came from 1 band in the original (small) unit cell E k

15 Supercell Chern number When bands touch, only the total Chern number is well-defined. E E k Adding disorder splits degeneracies and leads to minibands, each with its own Chern number. These must sum to the original Chern number of the whole band. The precise disorder distribution determines how the Chern numbers are assigned to each miniband, consistent with this rule. k

16 Supercell Chern number Simplest case: E E k k Now we can compute the Hall conductivity in each realization, as each realization is (almost) an insulator: since each miniband is very narrow in energy, almost every band is completely filled or completely empty. In each realization, compute total Chern number of the bands occupied at each filling.

17 Chern # = pumping conductance Pumping conductance Scattering conductance j (t) Adiabatic (if non-degenerate) Closed system No Fermi surface States are evolving Leads j Dissipative (metallic) Open system Fermi surface contributions States are fixed Adiabatic transport requires going infinitely slowly as system size increases. Conjecture: pumping selects out intrinsic part.

18 The Chern number shell game Simplest case: total Chern is 1; which mini band gets it? (This picture is not strictly correct, since there can be Chern numbers of either sign.) There are clearly strong fluctuations in the Chern number of each mini band. But what can determine its average? Idea: (like equipartition) at weak disorder, the resolving of anti crossings does not shift Chern weight on average; the average is just determined by the Chern density.

19 Obtaining the metal as a statistical limit of Chern insulators h xy (E F )i e2 h lim N1 n F /N =n(e F ) n F X i=1 hc i i = e2 h Z E(k x,k y )<E F F (k x,k y ) dk x dk y Here N is the (large) number of bands in the supercell and angle brackets mean averaging over disorder. The metallic limit is that disorder is weak (it only opens the minigaps). More precisely: In standard IQHE limit (Chalker-Coddington physics): disorder fixed as system size increases. Eventually IQHE transition becomes sharp, occurs at a fixed energy. Here we should scale disorder to 0 as size increases.

20 Preliminaries: IQHE and QSHE One prediction of the localization theory of the IQHE plateau transition is: as the system size L increases at fixed disorder, the transition width in energy or density shrinks as a power law in L. This can be measured by asking over what region of density the topological index is fluctuating for a given size. In the QSHE, this is esp. interesting as whether or not a U(1) is preserved determines whether transition collapses or not. Essin and Moore, PRB 2009: approached using Avron, Seiler, Simon projection operator method of computing Chern number, which still involves an integral over boundary fluxes. Hastings and Loring, EPL & Ann. Phys. 2011: developed new operator algebra approach via almost commuting matrices.

21 IQHE-type width collapse from Loring and Hastings, EPL 2011 Point of Hastings-Loring approach: computing the topological invariant for one disorder realization requires (usually) just one diagonalization of the Hamiltonian, not one for each point in the boundary-condition torus. Not as robust as disorder goes to 0, but computationally very nice.

22 Point of Hastings-Loring approach: computing the topological invariant for one disorder realization requires (usually) just one diagonalization of the Hamiltonian, not one for each point in the boundary-condition torus. The band-projected position operators almost commute and are almost unitary for a system with a mobility gap (satisfied?). Can they be deformed to exactly unitary and exactly commuting? P exp(i )P U Tr log VUV U, P exp(i )P =2 im + r V m 2 Z. m6= 0 means topologically nontrivial Which branch of log? Clearly not robust as disorder goes to 0 since then individual-band Chern number m isn t even defined, but algorithm forced to return an integer. from Loring and Hastings, EPL 2011,

23 Metallic limit of topology Our claim is that there is also an interesting limit different from the scaling collapse: suppose we let the effect of disorder get weaker. fractional filling of Chern band localized C=0 total topological invariant of filled bands disorder = localized C=1

24 Metallic limit of topology Our claim is that there is also an interesting limit different from the scaling collapse: suppose we let the effect of disorder get weaker. fractional filling of Chern band total topological invariant of filled bands disorder =

25 Metallic limit of topology Our claim is that there is also an interesting limit different from the scaling collapse: suppose we let the effect of disorder get weaker. fractional filling of Chern band total topological invariant of filled bands disorder =

26 Metallic limit of topology So, even when the topological invariants of a system are strongly fluctuating, the average has meaningful information. We believe this is the correct route to a precise theory of the anomalous Hall effect in ordinary metals, and that this concept might be generalized to a few other symmetry classes. h xy (E F )i e2 h lim N1 n F /N =n(e F ) n F X i=1 hc i i = e2 h Z E(k x,k y )<E F F (k x,k y ) dk x dk y Conclusion: even when the topological invariants of a system are strongly fluctuating, the average has meaningful information. We believe this is the correct route to a precise theory of the anomalous Hall effect in ordinary metals, and that this concept might be generalized to a few other symmetry classes.

27 Part II: Quenching and Majoranas Quantum quenches, where the Hamiltonian is suddenly changed, have been a very active topic in recent years, partly motivated by atomic physics experiments. Quantum quenches in condensed matter (leaving aside emergent qubits) require quite a short time scale, but this does happen in optical experiments: absorption of an X-ray in a metal suddenly creates a core hole that acts as a quantum quench for the electrons. First goal: What is the time dependence of the Loschmidt echo, i.e., the overlap L(t) = h (0) (t)i 2 t This overlap of the state just before the quench with the state after time t in the new Hamiltonian is a standard quantity in quantum non equilibrium physics.

28 Where we started: how to see a Kondo impurity in a QSHE edge? It is known (Tanaka, Furusaki, Matveev, PRL 2011) that the Kondo impurity actually does not modify the quantized conductance. We wanted to observe the Kondo effect using a new technique for accomplishing a quantum quench in condensed matter:

29 Quantum quenching with optical absorption The idea of these experiments is that absorbing a photon makes a sudden transition between two states in a quantum dot, one coupled to the reservoir and one not. The details of the correlated initial state are reflected in the absorption spectrum. The ordinary Kondo effect with a metal or Luttinger liquid leads to a power law in the absorption spectrum, which is observed in the ETH/Munich experiment. For an artificial Anderson impurity at the QSHE edge, the power law is universally fixed by the Luttinger parameter as there are fewer operators to modify in the QSHE edge. R. Vasseur and JEM, PRL 2014

30 R. Vasseur, J. Dahlhaus, JEM, PRX (2014) What if we quench coupling to a Majorana fermion? Suppose that a 1D spinless fermion wire is suddenly disconnected from a topological superconducting wire (e.g., Kitaev 1D model). What is the time dependence of the Loeschmidt echo, i.e., the overlap L(t) = h (0) (t)i 2 t Preview: the exponent is connected to the scaling dimension of a boundary-condition-changing operator in the wire. With Majoranas, =1/4 Without Majoranas, the exponent is small and non-universal (depends on SC gap). Easiest picture for Majorana case: write metallic wire as two Majoranas, one of which picks up a modified boundary condition from coupling to bound Majorana and one does not. This calculation can be checked by DMRG and also carried through for parafermion zero modes.

31 Setup a) t<0 NL TSC t =0 b) NL L (0) = R (0) TSC (x) (x) L (0) = R (0)

32 Metallic wire plus 1D TSC Unfold metallic wire to ± and couple to Majorana. H = iv F Z 1 1 x + iapple + (0), (1) which describes a Majorana zero mode ( 2 = 1 and = ), coupled with strength apple / J 0 to the fermion (x) describing the normal lead, with regularization (0) (0 )+ (0+ ) 2. Will argue that effect of this perturbation is like that of a boundary magnetic field in an Ising model. Can show other expected boundary terms (Kondo, potential scattering, etc.) are irrelevant if Majorana is present

33 Write 1 Dirac as 2 Majoranas In the metallic wire, introduce real and imaginary parts of the original complex fermion. Let us introduce = +i 2,with and real Majorana fermions: { (x), (x0 )} = 2 (x x 0 ) and { (x), (x 0 )} =2 (x x 0 ). In terms of these Majorana fermions, H = i 4 i 4 Z 1 0 Z 1 0 dx ( x R x L ) dx ( x R x L )+iapple (0). (1) The boundary term on the second Majorana will flow onto the magnetic field boundary condition of the Ising model. What does this mean for a quench?

34 Scaling analysis of boundary This boundary term is relevant with scaling dimension 1/2. Boundary entropy drop of log(sqrt(2)). The boundary term on the second Majorana will flow onto the magnetic field boundary condition of the Ising model. What does this mean for a quench? Based on pure scaling, it is natural to expect L(t) to be a universal function of tapple 2 only, since t acts e ectively as the inverse of a typical energy scale. For large times, the fact that the (boundary of the) system is flowing to a completely new (boundary) RG fixed point leads to an algebraic decay L(t) t that can be interpreted as a time dependent version of the celebrated Anderson orthogonality catastrophe Compute using imaginary-time representation as a ratio of partition functions

35 Scaling analysis of boundary The key idea is to interpret the Loschmidt echo in imaginary time L(t = i ) = h 0 e H 0 i 2, (1) as the square of a partition function of a 2D statistical problem at its critical point two copies of the Ising model in our case in the half-plane x 2 [0, 1), y 2 ( 1, 1) The quench changes the boundary condition on the half plane (i.e., at x=0) over a region 0<y<tau. The scaling dimension of this boundary condition changing operator in the c=1 Dirac theory is known: =4h BCC, h BCC =1/16 and the same answer is obtained in the Ising representation.

36 Check from DMRG 1D dynamics is now quite approachable numerically thanks to advances in time-dependent DMRG: 10 0 g=1 µ =1.5 non topological L(t) g=3/2 µ =0.5 topological tt? J 0 =0.1 J 0 =0.2 J 0 =0.3 J 0 =0.4 Robust to Luttinger-liquid effects in the lead (though definition of T* is changed). Compare Lutchyn and Skrabacz (2013), Giuliano and Affleck (2013).

37 Non-topological case We expect a correction to the exponent going as the inverse square of the gap, which indeed is observed: 1.00 L(t) t ( ) µ =3 µ =5 µ =9 µ = 17 µ = t

38 Generalization: parafermions Consider a quench involving a Z3 parafermionic zero mode coupled to a gapless Z3 parafermionic theory. We expect the Z3 parafermionic edge zero mode to act as a boundary magnetic field perturbation, with dimension = 25, for the gapless Z3 parafermionic theory (describing the critical point of the 2D Q = 3-state Potts model, just like the gapless Majorana theory describes the critical point of the Ising model). The conformally invariant boundary conditions of the Potts model are well-known and classified, and the boundary condition changing operator from free to fixed in the Q = 3 Potts model has dimension hbcc = 18. Optical absorption experiment for Majoranas: a) NL QD L (0) = SC R (0) b a (x) (x) L (0) = R (0) L (0) = R (0) b) NL QD b a (x) (x) L (0) = R (0) TSC Worth trying, especially if one can avoid heating by the laser.

39 Conclusions 1. Metals If there is a limit where the intrinsic AHE formula is exact, it may be the disorder-averaged pumping conductance. This is an adiabatic process so the diagonal conductivity is effectively 0. The intrinsic AHE describes the mean Chern numbers in a weak-disorder regime where the Chern numbers fluctuate strongly between realizations. 2. Majoranas The Majorana fermion at the edge of a 1D superconducting wire (for example) produces a change of boundary condition on a tunnel-contacted metallic wire. This leads to a universal power-law in the Loschmidt echo, which could be measured in the absorption spectrum of a quantum dot.