Bulk-edge correspondence in topological transport and pumping. Ken Imura Hiroshima University

Transcription

1 Bulk-edge correspondence in topological transport and pumping Ken Imura Hiroshima University

2 BEC: bulk-edge correspondence bulk properties - energy bands, band gap gapped, insulating edge/surface properties mid-gap edge states gapless, metallic - band structure & wave function topologically nontrivial vs. trivial - number of - presence/absence of edge states - topological invariant: bulk-edge correspondence C = 1 Z d k µ n [@ kµ k n] 8 Z vs. Z types

3 Topological vs. non-topological band structures TI: topological insulator OI: ordinary insulator inverted gap 4 normal band E 0 E kêp edge/surface states kêp D vs. 3D examples

4 D example: how to characterize the bulk The winding number mapping: stereographic projection

5 BEC in different formats here, two specific examples: Case 1: topological insulator thin films - correspondence in physical properties Phys. Rev. B 9, (015) Phys. Rev. B 94, (016) bulk edge penetration of top/bottom surface wave function into the <<bulk>> of auxiliary 3D system 1D helical modes circulating around a thin-film Case : topological quantum pump arxiv: Laughlin s argument, a version of BEC - pump version: more intuitive interpretation

6 Case 1: topological insulator thin films Model: standard Wilson-Dirac type H WD bulk(k) =m 3D (k) z gap/mass and Wilson terms m 3D (k) =m 0 µ=x,y,z b µ cos k µ - Topological classification periodic table (ten-fold way) Ryu & Schnyder, PRB 010 Present model: 3D, class AII Diagnosis: Z type 16 different types of topological phases: 8, 7 WTI, 1 OI 0, ( 1,, 3 ) Z indices: µ=x,y,z bzêb t µ sin k µ x µ OI H1;111L H1;110L H1;001L WTI H0;001L H1;000L spin & orbital OI WTI H0;111L WTI H0;111L OI m 0 êb H1;110L WTI H0;001L 4 4 H1;111L H1;000L H1;001L OI

7 The periodic table of topological insulators (ten-fold way) Symmetry =d D s AZ A Z 0 Z 0 Z 0 Z 0 1 AIII Z 0 Z 0 Z 0 Z 0 AI Z Z 0 Z Z 1 BDI Z Z Z 0 Z D Z Z Z Z 0 3 DIII Z Z Z Z 4 AII Z 0 Z Z Z CII Z 0 Z Z Z C Z 0 Z Z Z 0 7 CI Z 0 Z Z Z Ryu & Schnyder, PRB 010; Teo & Kane, PRB 010

8 /WTI: In reciprocal space two types of topological insulators strong vs. weak In real space N D mod = 1 N D mod = 0 N D mod 4 = N M =6 N M =4 WTI

9 - tight-binding construction: D gap/mass and Wilson terms: m D (k D )=m 0 µ=x,y b µ cos k µ Reduction to a thin film H WD film (k D )=1 Nz m D (k D ) 0 + b z t z µ=x,y t µ sin k µ µ 0 i i i i 0 TI thin film = stacked D QSH layers - Topological property as a quasi D system still Z type 3 Nz =1 =0 m 0 êb WTI WTI WTI Phys. Rev. B 9, (015)

10 Two characteristic patterns? brick vs. stripe Log10 E ) stripe pattern: Nz: even Nz: odd ) brick pattern: even-odd feature w.r.t. Nz hybridization of gapless helical edge modes formation of the hybridization gap =0 a single gapless combination remains =1 in an auxiliary 3D semi-infinite system WTI situation oscillation of the surface wave function size gap edge point of view situation oscillation of the surface wane function Lx Phys. Rev. B 89, 1545 (014) surface/bulk point of view

11 Oscillatory vs. over-damped regimes Phys. Rev. B 89, N 1545 (014) 0 TI-overdamped 3 Log10 E0 4 6 OI 1 OI TI-oscillatory Lx E 0 (L x ) 4 N Energy gap = + 1 ( ϵ x ) m x 1 m 1+ 1 ψ semi (x = L x + 1). m 1 oscillation of the surface wave function -

12 Case of Weyl semimetal thin films - Model Hamiltonian: H CI bulk(k) =m 3D (k) z + gap/mass and Wilson terms are the same as the TI case: similar phase diagram µ=x,y t µ sin k µ µ m 3D (k) =m 0 µ=x,y,z b µ cos k µ bzêb OI H1;111L H1;110L H1;001L WTI H0;001L H1;000L OI WTI H0;111L WTI H0;111L OI m 0 êb H1;110L WTI H0;001L H1;111L H1;000L H1;001L OI bzêb OI WSM I WSM I WSM III CI WSM III WSM IV WSM II WSM II WSM IV m 0 êb WSM III CI WSM III WSM I WSM I OI

13 Thin film case: brick and stripe patterns Phys. Rev. B 94, (016) Nz WSM CI m 0 êb WSM CI WSM straight regular pattern cf. stripe pattern CI (Chern insulator) H D = m D (k D ) z + = stacked QAH layers µ=x,y t µ sin k µ µ contributions from each layer: m D (k D )=m 0 µ=x,y b µ cos k µ xy = e h they all add up in the CI phase: xy = N e h N = N z

14 brick regions WSM (Weyl semimetal) cross sections at kz = fixed in the reciprocal space QAH, i.e., C(k z )=±1 if k 0 <k z <k 0 OI, i.e., C(k z )=0 otherwise WSM = partially broken CI case of WSM thin films N <N z D topological character of the constituent QAH layers are only partially maintained Similarly, stripe pattern brick pattern in TI thin films WTI topological nature of constituent layers is fully respected in the stacked system partially broken as partially broken WTI

15 Hosur & Qi, CRP 13 kz

16 What is the precise relation between the two systems? TI vs. WSM cases - both 3D bulk & D thin-film phase diagrams look very similar Remark: In the limit of t z 0 All the phase boundaries forming the brick and stripe patterns coincide in TI and WSM thin-film cases TI: Nz m 0 êb WSM: Nz m 0 êb

17 The reason is simple 1) TI thin films: H WD film (k D )=1 Nz m D (k D ) 0 + ) WSM case: b z H CI film(k D )=1 Nz m D (k D ) z + µ=x,y µ=x,y t µ sin k µ µ 0 + t z t µ sin k µ µ 0 i i i i 0 3 b z z m D (k D )=m 0 µ=x,y b µ cos k µ common in the two cases So, the two systems are indeed very similar

18 Then, how is the nature of brick patterns in the WSM case? Reminder: nature of brick patterns oscillation of the surface wave function in an auxiliary 3D semi-infinite system However, in the WSM model, there is no surface state on top and at bottom Answer: As approaching the limit t z 0 The surface Dirac cones in the WTI/ model sink into the bulk, transforming into a pair of Dirac/Weyl k z = ± arccos m 0 b z b b z ±k 0 N D = N W N D = N W = # of surface Dirac cones in the WTI/ model # of Weyl pairs in the CI/WSM model

19 A short summary - Relation between the /WTI vs. WSM type models A thin-film point of view WTI CI WSM stripe region brick region - How about the role of <<bulk-edge>> correspondence? bulk penetration of the surface wave function into the bulk (auxiliary 3D system) edge number of chiral edge modes in thin-film systems One-to-one correspondence in <<physical properties>> at the edge and in the bulk

20 A short detour on the role of disorder

21 J. Phy Downloaded from jou Phase diagram of D J. Phys. Soc.vs. Jpn. 80TI (011) L BHZ + Rashba thin film FIG. 7. (Color online) Conductance maps with (a) truncated and (b) periodic boundaries on the y sides. Parameters are the same as in TIin the presence of potential disorder (W = 3). The average of the conductance over 1000 samples is plotted. Figs.disordered (a) and (b) but, here, on calculation of the localization length. In the structure of phase. The phase boundary with a quantized conductance of -G =No direct transition between the phase diagram shown in Fig. 8, the existence of a diffusive is the D version of the Dirac semimetal line studied than the in Ref. [7]. At topological an m0 /m slightly larger different phases inlocation D metal phase is characteristic of systems of class AII symmetry. of this Dirac semimetal line, a system in the OI phase is Here, the initial D model, the Nz = 1 case of Eq. (5) with converted to a QSH insulator upon the addition of disorder on-site disorder, belongs to class A, while stacking more than potential disorder marginal in D (topological Anderson insulator behavior). At W! 7 the two two layers converts the system to class AII. In contrast to insulating phases are both overwhelmed by a diffusive metal this, only in the addition of Rashba-type spin-orbit coupling metal in between (symplectic phase, a region of large and nonquantized conductance. For in the D setup, the system is converted to a model of class the strongly disordered regime W! 15, the system re-enters AII symmetry in Ref. [45]. The setup of the present nanofilm symmetry class) the OI (Anderson insulator) phase. These features are most construction is much simpler, and it is an alternative way reminiscent of a similar phase diagram obtained earlier for a to realize a class AII QSH system without assuming an sz BHZ + Rashba (localization length) purely D system of the same class AII symmetry [45], based nonconserving term. JPSJ 80, (011) Fig.. (Color online) Phase diagram of disordered BHZ model in th presence (# ¼ 0:5, blue triangles) and absence (# ¼ 0, red circles) of sz non conserving SOC. Lines connecting the data points are guide to eyes. systems of different circumference L. Overall behavior o! L =L looks similar to that of the upper panel. Namely, as! increases, the system evolves as: localized! metallic! localized. Note that the first insulating phase:! <!1! "0:8 is of trivial nature (" ¼ 0), whereas the second on with! >!! "0:5) is Z nontrivial (" ¼ 1). Thes different kinds disorder are separated by a finite metallic region thin-film calculation (conductance) Nz = FIG.38. (Color online) Conductance map for a disordered slab (Nz = 3) with varyingphys. strengthbw9,. Other settings are(015) the same as Rev !1 <! <!. The second insulating phase with Z -numbe

22 cf. Phase diagram of 3D disordered TI Direct transitions! between different topological phases in 3D potential disorder irrelevant in 3D Phys. Rev. Lett. 110, (013)

23 Case of disordered WSM thin films G m 0 /b W = 0.0 W = 0. W = 0.5 W = 0.7 W = 1.0 N - two-terminal conductance Phys. Rev. B 94, (016) co-propagating regime Co- vs. Counterpropagating regimes Nz N z = m 0 êb counter-propagating regime

24 Two-terminal vs. Hall conductances Chern number = Hall conductance G H =(N + while the two-terminal conductance N ) e h = N e h G =(N + + N ) e h measures the number of transmitting channels ( ) # of left- and rightgoing chiral modes They differ - in the presence of counter-propagating modes & - in the clean limit N ± Relaxation of counter-propagating modes at the edge recovers G =(N + N ) e h

25 Case : topological quantum pump

26 Why pumping? a b 1) Experiments in cold atoms 1.0 ϕ (π) 0.5 Nakajima et al., Nature Phys., 015; Lohse et al., ibid. x (d l ) x (d l ) ) Nobel prize in physics 016 TKNN vs. Thouless pump QHE D 1+1D topological pumping ϕ (π) Figure Centre-of-mass (COM) position of the atom cloud as a functi V l =0(1)E r,l. a, Detailed evolution of the measured COM position durin correspondence: lower-lying sites. The solid black line depicts the calculated COM motion error bar shows the error of the mean. COM positions are determined di same length, but with constant phase ' =0. One data set is obtained by COM positions. Inset, The motion of a localized Wannier function in the multiple pump cycles in the positive and negative pumping direction. Th standard deviation. The dashed black line shows the ideal motion of a lo assuming a finite pumping eciency of 97.9()% per each half of a pum between even and odd sites as a function of ' with n e (n o ) the fraction o and the error bars indicate the corresponding standard deviation. The gr function using the same model as for the COM displacement which yield - mathematically equivalent (classification, etc.) - physics different: different physical quantities, different physical pictures cf. Laughlin s argument: original: for QHE Thouless, PRB 1983 pump version? per cycle as expected for 1 =+1 and the steps appear arou ' =l,l Hatsugai Z, where & the Fukui, atomsprb tunnel016 from one side of the dou wells to the other. When performing multiple cycles the cloud ke moving to the right, whereas it propagates in the opposite direct for the reversed pumping direction ' < 0 (Fig. b). The sm deviation from the expected displacement for the motion of id

27 Topological pumping in the snapshot picture (adiabatic limit) Harper/AA model (pump version) Hatsugai & Fukui, PRB 016 AA = Aubry-Andre Laughlin s geometry - periodic in t - finite (w/ edges) in the x-direction What is related to the Chern number? V(x,t): periodic in time Ans.: pumped charge = change of polarization over the pumping cycle = center-of-mass position But, because of p.b.c. So, no pumping???

28 Polarization/center of mass half-integral jumps! filled states jumps: edge effects continuous part: bulk contribution bulk contribution is relevant skip jumps & reconstruct the continuous part: Recall: Hatsugai & Fukui, PRB 016

29 Consideration on Jumps vs. continuous part or edge vs. bulk contributions in the polarization curve or consideration on the adiabatic conditions: The adiabatic condition: T / t edge = / edge bulk: edge: bulk = g edge 0 Realistic situation: t bulk T exp t edge i.e., bulk: adiabatic edge: sudden Jumps due to the edge modes are not seen in experiments Rather, half-integral jumps emergent in the adiabatic limit are <<origin>> of the quantization of pumped (topological) charge

30 A short summary: Origin of quantization = half-integral jumps edge quantity bulk topological invariant - pump version of BEC filled bands BEC BEC: bulk-edge correspondence - pump version of Laughlin s argument A remaining issue: QHE vs. pump - check & quantify the robustness against disorder H(t) = L/ X x= L/ h i t x x +1ihx +(h.c.)+[v (x, t)+w (x)] xihx W (x) [ W/,W/] W: strength of impurity

31 In the presence of disorder arxiv: two types of jumps appear 1) quantized jumps - edge state origin ) non-quantized jumps - impurity origin Snapshot spectrum: E At weak disorder they appear separately in time Polarization: t/t - As far as they are separable, the pumped charge is still quantized x(t) t/t

32 Quantized vs. non-quantized jumps - Non-quantized jumps: jumps due to impurities - Quantized jumps: x = x imp [ sgn(slope)] occupy/empty - appear in pairs: appear/disappear - irrelevant to the pumped charge jumps due to edge states x = 1 sgn(x edge)[ sgn(slope)] = ± 1 R or L - also appear in pairs, but x edge = ±L/ quantized pumped charge x net = {j n } Edge quantity x jump (t jn )=0, ±1, ±, Bulk topological invariant

33 - two examples, in which Conclusions BEC manifests as a one-to-one relation between <<visible>> physical quantities in the bulk and at the edge - highlighted a rather specific role of bulk in case 1: topological insulator thin films cf. penetration of the surface wave function into the <<bulk>> in the auxiliary 3D system edge in case : topological quantum pumping cf. half-integral jumps in polarization as the <<origin>> of topological quantization