4D Topological Physics with Synthetic Dimensions Hannah Price - PowerPoint PPT Presentation

4D Topological Physics with Synthetic Dimensions Hannah Price University of Birmingham, UK

Synthetic Dimensions General Concept: 1. Identify a set of states and reinterpret as sites in a synthetic dimension 0 1 2 3 4 2. Couple these modes to simulate a tight-binding “hopping” J e − i φ J e i φ w 0 1 2 3 4 WHY? • Implement artificial gauge fields Boada et al., PRL, 108, 133001 (2012), • Reach higher-dimensional models Celi et al., PRL, 112, 043001 (2014)

Outline 1. Reminder of the 2D Quantum Hall Effect 2. 4D Topological Physics 3. 4D Quantum Hall in Synthetic Dimensions

2D Quantum Hall Effect j x = − q 2 E y X ν n h E y 1 j x Ω n n ∈ occ. xy band insulator Bloch states Geometrical Berry curvature ψ n, k ( r ) = e i k · r u n,k ( r )  � h ∂ u n | ∂ u n i � h ∂ u n | ∂ u n i Ω n xy = i ∂ k x ∂ k y ∂ k y ∂ k x Topological First Chern number 1 = 1 Z Topological transitions only ν n Ω n xy d k x d k y when band-gap closes 2 π BZ

How to get a 2D QH system? Minimal two-band model, e.g. spinless atoms on lattice with two-site unit cell: H ( k ) = ε ( k )ˆ I + d ( k ) · σ Topological transitions: e.g. at Dirac points H ( q ) ≈ v x q x σ x + v y q y σ y H ( q ) ≈ v x q x σ x + v y q y σ y + m σ z Dirac q y cone E m m q x -ve +ve m = 0 m

Berry curvature xy = 1 2 ✏ abc ˆ d a @ q x ˆ d b @ q y ˆ H ( q ) ≈ d ( q ) · σ Ω − d c d ( q ) ≈ ( v x q x , v y q y , m ) q y E m q x +ve -ve m = 0 m d 3 = − m → d 3 = m Berry curvature flips across transition as Ω − Type 1 : same signs —> increases d 1 , d 2 xy d 1 , d 2 Type 2 : opposite signs —> decreases Ω − xy

Chern Number Time-reversal symmetry for spinless particles d 1 , 3 ( k )= d 1 , 3 ( − k ) H ∗ ( k ) = H ( − k ) implies d 2 ( k )= − d 2 ( − k ) σ ∗ as y = − σ y So have TRS pairs of opposite type 1 = 1 Z K − K Ω − xy d k x d k y ν − 4 π BZ Type 1 Type 2 transitions are topologically trivial with TRS

Flaschner et al, Nat. Phys. 14, 265 (2018) … . 
 Jotzu et al, Nature 515, 237 (2014) Cold atom experiments: e.g. y x Breaking Time-Reversal Symmetry Haldane model: J 0 e � i φ Haldane, PRL 61, 2015 (1988) J 0 e i φ A B J ENERGY LEVELS AND %AVE FUNCTIONS OF BLQCH. . . 14 2241 q; hence one might expect the above condition to IV. RECURSIVE STRUCTURE OF THE GRAPH be satisfied in roughly q distinct regions of the e axis (one region centered on each root). This This graph has some vexy unusual properties. ( m, n + 1) ( m + 1 , n + 1) is indeed the case, and is the basis for a very The large gaps form a very striking pattern some- H = J Aidelsburger et al., Nat. Phys, 11,162. (2015) … . Miyake et al, PRL, 111, 185302 (2013), Aidelsburger et al., PRL, 111, 185301 (2013), Cold atom experiments: equally strik- (and at first disturbing) fact about this what resembling a butterfly; striking perhaps ing are the delicacy and beauty of the fine-grained problem: when n = p/q, the Bloch band always structure. These are due to a very intricate breaks up into i. -recisely q distinct energy bands. Since small variations of o. can scheme, by which bands cluster into groups, which in the magnitude ( m, n ) ( m + 1 , n ) in the value of the themselves may cluster into laxger groups, produce enormous fluctuations and so on. The exact rules of formation of these hier- denominator q, one is apparently faced with an (II's) are unacceptable physical prediction. However, nature archically organized clustering patterns Landau levels / Hofstadter model: of 0's m,n X is ingenious enough to find a way out of this ap- what we now wish to cover. Our description will be based on three statements, each of which pax'ent, anomaly. Befox'e we go into the x'esolution however, let us mention certain facts about the describes some aspect of the structure of the spectrum belonging to any value of z. Most can All of these statements are based on ex- graph. (ˆ Φ tremely close examination of the numex ical data, be proven trivially: (i) Spectrum(tr) and spectrum c † (ci+N) are identical. and spec- and are to be taken as "empirically proven" theo- (ii) Spectrum(n) m +1 ,n ˆ trum(-tr) are identical. (iii) & belongs to spec- rems of mathematics. It would be preferable to trum(a } if and only if -e belongs to spectrum(a}. have a rigorous proof but that has so far eluded (iv) If e belongs to spectrum (a) for any a, then capture. Before we present the three statements, -4 ~ &~+4. The last property is a little subtler let us first adopt some nomenclature. A "unit cell" is any portion of the graph located between than the previous three; it can be proven in dif- One proof has been published. " successive integers N and N +1 — ferent ways. in fact we will c m,n + e i 2 π Φ m ˆ From properties (i) and (iv), it follows that a call that unit cell the N th unit cell. Every unit cell has a "local variable" P which runs from 0 to 1. graph of the spectrum need only include values of & between + 4 and -4, and values of e in any unit P is defined to be the fractional part in particular, We shall look at the interval [0, 1]. Fur of rt, usually denoted as (a). At P=O and P= I, interval. thermore, as a consequence of pxoperties, there is one band which stretches across the full the graph inside the above-defined rectangular region width of the cell, separating it from its upper and namely the hor- must have two axes of reflection, lower neighbors; this band is therefore called a "cell wall. " It turns out that eex'tain rational val- izontal line z= &, and the vertical line &=0. A Hofstadter, PRB, 14 , 2239, 1976 − 4 J plot of spectrum(o. ), with n along the vertical axis, ues of I3 play a very important role in the descrip- in Fig. 1. (Only rational values of a with appears tion of the structure of a unit cell; these are the E 4 J "pure cases" less than 50 are shown. ) denominator 0 c † m,n +1 ˆ − ~ FIG. 1. Spectrum inside a unit cell. & is the hori- π Φ zontal variable, ranging and -4, and Φ 0 between+4 c m,n ) + h.c. p=(n) is the vertical vari- able, ranging from 0 to 1. 1

Outline 1. Reminder of the 2D Quantum Hall Effect 2. 4D Topological Physics 3. 4D Quantum Hall in Synthetic Dimensions

Second Chern Number Ω = 1 2 Ω µ ν ( k )d k µ ∧ d k ν  � h ∂ u n | ∂ u n i � h ∂ u n | ∂ u n i Ω µ ν n = i ∂ k µ ∂ k ν ∂ k ν ∂ k µ k z k w First Chern ν 1 = 1 Z Ω = 1 Z number Ω xy d k x d k y 2 π 2 π 2DBZ 2DBZ 1 Z Second Chern ν 2 = Ω ∧ Ω ∈ Z number 8 π 2 4DBZ 1 Z [ Ω xy Ω zw + Ω wx Ω zy + Ω zx Ω yw ] d 4 k = 4 π 2 4DBZ (and then the third Chern number in 6D… ) for 6DQH see Petrides, HMP , Zilberberg arXiv:1804.01871 and references there-in

4D Quantum Hall Effect Z Very simplest example: 4D Harper-Hofstadter Model 1 Z [ Ω xy Ω zw + Ω wx Ω zy + Ω zx Ω yw ] d 4 k = ν 2 E z 4 π 2 4DBZ j x Ω zx 1 ν yw ν 2 = ν zx 1 F w j x B xw Response to two perturbations: B xw = ∂ x A w − ∂ w A x E z F w j y Ω yw j y = − q 3 h 2 E z B xw ν n Zhang et al, Science 294, 823 (2001) … . 2 HMP, Zilberberg, Ozawa, Carusotto & Goldman, PRL 115, 195303 (2015) HMP, Zilberberg, Ozawa, Carusotto & Goldman, PRB 93, 245113 (2016)

What do we need for a 4D QH system? Kitaev, arXiv:0901.2686 Ryu et al., NJP, 12, 2010, Symmetries Dimensions δ Chiu et al RMP, 88, 035005 (2016)… Class T C S 0 1 2 3 4 5 6 7 A 0 0 0 Z 0 Z 0 Z 0 Z 0 • Quantized non-linear response AIII 0 0 1 0 Z 0 Z 0 Z 0 Z þ AI 0 0 Z 0 0 0 2 Z 0 Z 2 Z 2 þ þ Z 2 Z 2 Z Z 2 BDI 1 0 0 0 0 þ D 0 0 Z 2 Z 2 Z 0 0 0 2 Z 0 •Bands labelled by integer second Chern numbers þ DIII 1 0 Z 2 Z 2 Z 0 0 0 2 Z − AII 0 0 2 Z 0 Z 2 Z 2 Z 0 0 0 − CII 1 0 2 Z 0 Z 2 Z 2 Z 0 0 − − C 0 0 0 0 2 Z 0 Z 2 Z 2 Z 0 • Different classes of 4D QH systems − þ 2 Z Z 2 Z 2 Z CI 1 0 0 0 0 − 1. Preserved TRS for fermions: particles in spin-dependent gauge fields Zhang et al, Science 294, 823 (2001), Qi et al, Phys. Rev. B 78, 195424 (2008). … . 2. Broken TRS: 4D Harper-Hofstadter model Kraus et al, Phys. Rev. Lett. 111, 226401 (2013), HMP et al. 115, 195303 (2015) … 3. Preserved TRS for spinless particles: just lattice connectivity! HMP, arXiv:1806.05263

What do we need for a 4D QH system? Kitaev, arXiv:0901.2686 Ryu et al., NJP, 12, 2010, Symmetries Dimensions δ Chiu et al RMP, 88, 035005 (2016)… Class T C S 0 1 2 3 4 5 6 7 A 0 0 0 Z 0 Z 0 Z 0 Z 0 • Quantized non-linear response AIII 0 0 1 0 Z 0 Z 0 Z 0 Z þ AI 0 0 Z 0 0 0 2 Z 0 Z 2 Z 2 þ þ Z 2 Z 2 Z Z 2 BDI 1 0 0 0 0 þ D 0 0 Z 2 Z 2 Z 0 0 0 2 Z 0 •Bands labelled by integer second Chern numbers þ DIII 1 0 Z 2 Z 2 Z 0 0 0 2 Z − AII 0 0 2 Z 0 Z 2 Z 2 Z 0 0 0 − CII 1 0 2 Z 0 Z 2 Z 2 Z 0 0 − − C 0 0 0 0 2 Z 0 Z 2 Z 2 Z 0 • Different classes of 4D QH systems − þ 2 Z Z 2 Z 2 Z CI 1 0 0 0 0 − 1. Preserved TRS for fermions: particles in spin-dependent gauge fields Zhang et al, Science 294, 823 (2001), Qi et al, Phys. Rev. B 78, 195424 (2008). … . 2. Broken TRS: 4D Harper-Hofstadter model Kraus et al, Phys. Rev. Lett. 111, 226401 (2013), HMP et al. 115, 195303 (2015) … 3. Preserved TRS for spinless particles: just lattice connectivity! HMP, arXiv:1806.05263