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

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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

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SLIDE 1

4D Topological Physics with Synthetic Dimensions

Hannah Price University of Birmingham, UK

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SLIDE 2

General Concept:

  • 1. Identify a set of states and reinterpret as sites in a synthetic dimension

1 2 3 4

Synthetic Dimensions

1 2 3 4

  • 2. Couple these modes to simulate a tight-binding “hopping”

w J eiφ J e−iφ

Boada et al., PRL, 108, 133001 (2012), Celi et al., PRL, 112, 043001 (2014)

WHY?

  • Implement artificial gauge fields
  • Reach higher-dimensional models
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SLIDE 3
  • 1. Reminder of the 2D Quantum Hall Effect
  • 2. 4D Topological Physics
  • 3. 4D Quantum Hall in Synthetic Dimensions

Outline

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SLIDE 4

2D Quantum Hall Effect

Ey jx = −q2 h Ey X

n∈occ.

νn

1 band insulator

Ωn

xy

Topological First Chern number Geometrical Berry curvature

Ωn

xy = i

 h∂un ∂kx |∂un ∂ky i h∂un ∂ky |∂un ∂kx i

  • Bloch states

ψn,k(r) = eik·run,k(r)

Topological transitions only when band-gap closes

νn

1 = 1

2π Z

BZ

Ωn

xydkxdky

jx

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SLIDE 5

H(k) = ε(k)ˆ I + d(k) · σ

Minimal two-band model, e.g. spinless atoms on lattice with two-site unit cell:

How to get a 2D QH system?

H(q) ≈ vxqxσx + vyqyσy Dirac cone

qx qy E

Topological transitions: e.g. at Dirac points

m

+ve

  • ve

m m = 0 m

H(q) ≈ vxqxσx + vyqyσy + mσz

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SLIDE 6

Berry curvature

H(q) ≈ d(q) · σ Ω−

xy = 1

2✏abc ˆ da@qx ˆ db@qy ˆ dc

m = 0 m m

+ve

  • ve

qx qy E

d(q) ≈ (vxqx, vyqy, m)

Berry curvature flips across transition as Type 1: same signs —> increases Type 2: opposite signs —> decreases

d1, d2 d1, d2 d3 = −m → d3 = m Ω−

xy

Ω−

xy

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SLIDE 7

Chern Number

d2(k)=−d2(−k)

Time-reversal symmetry for spinless particles implies

H∗(k) = H(−k)

σ∗

y = −σy

as

d1,3(k)=d1,3(−k) K −K

Type 1 Type 2

So have TRS pairs of opposite type transitions are topologically trivial with TRS

ν−

1 = 1

4π Z

BZ

Ω−

xydkxdky

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SLIDE 8

Breaking Time-Reversal Symmetry

e.g. Landau levels / Hofstadter model:

H = J X

m,n

(ˆ c†

m+1,nˆ

cm,n + ei2πΦmˆ c†

m,n+1ˆ

cm,n) + h.c.

Φ

(m, n) (m + 1, n) (m, n + 1)(m + 1, n + 1)

14

ENERGY LEVELS AND %AVE FUNCTIONS OF BLQCH. . .

2241 q; hence one might expect the above condition to be satisfied in roughly q distinct regions of the e axis (one region centered on each root). This

is indeed the case, and is the basis for a very

striking (and at first disturbing)

fact about this problem:

when n =p/q, the Bloch band always

breaks up into i.-recisely q distinct energy bands. Since small variations

in the magnitude
  • f o. can
produce enormous fluctuations in the value of the denominator q, one is apparently faced with an unacceptable physical prediction. However, nature

is ingenious

enough to find a way out of this ap- pax'ent, anomaly. Befox'e we go into the x'esolution however, let us mention certain facts about the

spectrum belonging to any value of z. Most can be proven trivially: (i) Spectrum(tr)

and spectrum

(ci+N) are identical. (ii) Spectrum(n)

and spec-

trum(-tr) are identical. (iii) & belongs to spec- trum(a }if and only if -e belongs to spectrum(a}.

(iv) If e belongs to spectrum (a) for any a, then
  • 4 ~ &~+4. The last property is a little subtler
than the previous three; it can be proven in dif-

ferent ways.

One proof has been published. "

From properties (i) and (iv), it follows that a

graph of the spectrum need only include values of & between + 4 and -4, and values of e in any unit

interval. We shall look at the interval [0,1]. Fur thermore, as a consequence

  • f pxoperties,
the graph inside the above-defined rectangular region must have two axes of reflection, namely the hor- izontal line z= &, and the vertical line &=0. A plot of spectrum(o. ), with n along the vertical axis, appears in Fig. 1. (Only rational values of a with denominator

less than 50 are shown. )

  • IV. RECURSIVE STRUCTURE OF THE GRAPH

This graph has some vexy unusual properties. The large gaps form a very striking pattern some-

what resembling

a butterfly;

perhaps equally strik- ing are the delicacy and beauty of the fine-grained

structure. These are due to a very intricate scheme,

by which bands cluster into groups, which themselves may cluster into laxger groups, and

so on. The exact rules of formation of these hier- archically organized clustering patterns (II's) are

what we now wish to cover. Our description
  • f 0's
will be based on three statements, each of which

describes some aspect of the structure

  • f the
graph. All of these statements are based on ex- tremely close examination
  • f the numex ical data,
and are to be taken as "empirically proven" theo-

rems of mathematics.

It would be preferable to

have a rigorous proof but that has so far eluded

capture. Before we present the three statements, let us first adopt some nomenclature.

A "unit

cell" is any portion of the graph located between

successive integers N and N +1—

in fact we will

call that unit cell the N th unit cell. Every unit cell

has a "local variable" P which runs from 0 to 1. in particular, P is defined to be the fractional part
  • f rt, usually denoted as (a). At P=O and P= I,

there is one band which stretches across the full

width of the cell, separating

it from its upper and

lower neighbors; this band is therefore called a

"cell wall. " It turns out that eex'tain rational val-

ues of I3 play a very important role in the descrip- tion of the structure

  • f a unit cell; these are the

"pure cases"

  • FIG. 1. Spectrum inside
a unit cell. & is the hori- zontal variable, ranging between+4 and -4, and

p=(n) is the vertical vari-

able, ranging from 0 to 1.

E

1

4J −4J

−~ π Φ Φ0

Hofstadter, PRB, 14, 2239, 1976 Cold atom experiments: Aidelsburger et al., PRL, 111, 185301 (2013), Miyake et al, PRL, 111, 185302 (2013), Aidelsburger et al., Nat. Phys, 11,162. (2015)….

Haldane model:

x y A B

J0eiφ J0eiφ

Haldane, PRL 61, 2015 (1988) J

Cold atom experiments: Jotzu et al, Nature 515, 237 (2014) Flaschner et al, Nat. Phys. 14, 265 (2018) …. 


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SLIDE 9

Outline

  • 1. Reminder of the 2D Quantum Hall Effect
  • 2. 4D Topological Physics
  • 3. 4D Quantum Hall in Synthetic Dimensions
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SLIDE 10

Second Chern Number

Ω= 1 2Ωµν(k)dkµ∧dkν Ωµν

n = i

 h∂un ∂kµ |∂un ∂kν i h∂un ∂kν |∂un ∂kµ i

  • kz

kw

Second Chern number First Chern number (and then the third Chern number in 6D… )

for 6DQH see Petrides, HMP , Zilberberg arXiv:1804.01871 and references there-in

ν2 = 1 8π2 Z

4DBZ

Ω ∧ Ω ∈ Z = 1 4π2 Z

4DBZ

[ΩxyΩzw + ΩwxΩzy + ΩzxΩyw] d4k ν1 = 1 2π Z

2DBZ

Ω = 1 2π Z

2DBZ

Ωxydkxdky

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SLIDE 11

Fw

Ωyw

Bxw

jx

4D Quantum Hall Effect

Fw Ez

Ωzx

Zhang et al, Science 294, 823 (2001)…. HMP, Zilberberg, Ozawa, Carusotto & Goldman, PRL 115, 195303 (2015) HMP, Zilberberg, Ozawa, Carusotto & Goldman, PRB 93, 245113 (2016)

jx

Very simplest example: 4D Harper-Hofstadter Model

ν2 Bxw = ∂xAw − ∂wAx

Response to two perturbations:

Ez

jy

jy = − q3 h2 EzBxwνn

2

Z = 1 4π2 Z

4DBZ

[ΩxyΩzw + ΩwxΩzy + ΩzxΩyw] d4k

ν2 = νzx

1 νyw 1

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SLIDE 12

What do we need for a 4D QH system?

  • Quantized non-linear response
  • Bands labelled by integer second Chern numbers
  • Different classes of 4D QH systems

1. Preserved TRS for fermions: particles in spin-dependent gauge fields 2. Broken TRS: 4D Harper-Hofstadter model 3. Preserved TRS for spinless particles: just lattice connectivity!

Zhang et al, Science 294, 823 (2001), Qi et al, Phys. Rev. B 78, 195424 (2008).…. Kraus et al, Phys. Rev. Lett. 111, 226401 (2013), HMP et al. 115, 195303 (2015)… HMP, arXiv:1806.05263

δ Class T C S 1 2 3 4 5 6 7 A Z Z Z Z AIII 1 Z Z Z Z AI þ Z 2Z Z2 Z2 BDI þ þ 1 Z2 Z 2Z Z2 D þ Z2 Z2 Z 2Z DIII − þ 1 Z2 Z2 Z 2Z AII − 2Z Z2 Z2 Z CII − − 1 2Z Z2 Z2 Z C − 2Z Z2 Z2 Z CI þ − 1 2Z Z2 Z2 Z

Dimensions Symmetries

Kitaev, arXiv:0901.2686 Ryu et al., NJP, 12, 2010, Chiu et al RMP, 88, 035005 (2016)…

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SLIDE 13

What do we need for a 4D QH system?

  • Quantized non-linear response
  • Bands labelled by integer second Chern numbers
  • Different classes of 4D QH systems

1. Preserved TRS for fermions: particles in spin-dependent gauge fields 2. Broken TRS: 4D Harper-Hofstadter model 3. Preserved TRS for spinless particles: just lattice connectivity!

Zhang et al, Science 294, 823 (2001), Qi et al, Phys. Rev. B 78, 195424 (2008).…. Kraus et al, Phys. Rev. Lett. 111, 226401 (2013), HMP et al. 115, 195303 (2015)… HMP, arXiv:1806.05263

δ Class T C S 1 2 3 4 5 6 7 A Z Z Z Z AIII 1 Z Z Z Z AI þ Z 2Z Z2 Z2 BDI þ þ 1 Z2 Z 2Z Z2 D þ Z2 Z2 Z 2Z DIII − þ 1 Z2 Z2 Z 2Z AII − 2Z Z2 Z2 Z CII − − 1 2Z Z2 Z2 Z C − 2Z Z2 Z2 Z CI þ − 1 2Z Z2 Z2 Z

Dimensions Symmetries

Kitaev, arXiv:0901.2686 Ryu et al., NJP, 12, 2010, Chiu et al RMP, 88, 035005 (2016)…

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SLIDE 14

4D Dirac points

H(k) = ε(k)Γ0 + d(k) · Γ

Minimal four-band model:

d(q) ≈ (vxqx, vyqy, vzqz, vwqw, m)

As Type 1: even no/ minus signs —> increases integrand Type 2: odd no/ minus signs —> decreases integrand

d1, d2, d3, d4 d1, d2, d3, d4 d5 = −m → d5 = m

Qi et al, Phys. Rev. B 78, 195424 (2008)

m = 0 Z = 3 8⇡2 Z

BZ

d4k✏abcde ˆ da@kx ˆ db@ky ˆ dc@kz ˆ dd@kw ˆ de ⌫−

2

Γ1 = B B @ 1 0 1 1 0 1 0 1 C C A , Γ2 = B B @ 0 0 i 0 0 i i 0 0 i 1 C C A , Γ3 = B B @ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 C C A , Γ4 = B B @ 0 i i 0 i 0 i 1 C C A , Γ5 = B B @ 1 0 0 1 0 0 1 0 0 1 1 C C A ,

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SLIDE 15

implies

4D Dirac points

Again TRS for spinless particles

Γ1 = B B @ 1 0 1 1 0 1 0 1 C C A , Γ2 = B B @ 0 0 i 0 0 i i 0 0 i 1 C C A , Γ3 = B B @ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 C C A , Γ4 = B B @ 0 i i 0 i 0 i 1 C C A , Γ5 = B B @ 1 0 0 1 0 0 1 0 0 1 1 C C A ,

HMP, arXiv:1806.05263

H∗(k) = H(−k) d1,3,5(k)=d1,3,5(−k) d2,4(k)=−d2,4(−k)

Γ∗

2,4 = −Γ2,4

as

K −K

Type 1 Type 1

So have TRS pairs of the same type

) ∈ 2Z

can have topological transition with TRS!

⌫−

2

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SLIDE 16

4D Brickwall Lattice

(a)

H(k) = J [(2 cos kx + cos ky)Γ1 + sin kyΓ2 + (2 cos kz + cos kw)Γ3 + sin kwΓ4 + mΓ5]

Hopping terms Onsite energies

HMP, arXiv:1806.05263

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SLIDE 17

4D Brickwall Lattice

(a)

b) tr(Ω− ∧ Ω−)

m

Trivial Trivial

ν−

2 = 0

ν−

2 = 0

m = 0

m = −J/2 m = J/2

tr(Ω− ∧ Ω−)

HMP, arXiv:1806.05263

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SLIDE 18

Add real long-range hoppings, e.g.

(a)

4D Class AI Model

b) c) d)

m

Topological Trivial Trivial

ν−

2 = −2

ν−

2 = 0

ν−

2 = 0

m = J0 m = −2J0

tr(Ω− ∧ Ω−) tr(Ω− ∧ Ω−) tr(Ω− ∧ Ω−)

HMP, arXiv:1806.05263

H0(k) = 2J0 cos(2kx+ 2kz)Γ5

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SLIDE 19

Key points about 4D QH Systems

  • Bands labelled by integer second Chern numbers
  • Quantized non-linear response
  • Different classes of 4D QH systems

jy = − q3 h2 EzBxwνn

2 δ Class T C S 1 2 3 4 5 6 7 A Z Z Z Z AIII 1 Z Z Z Z AI þ Z 2Z Z2 Z2 BDI þ þ 1 Z2 Z 2Z Z2 D þ Z2 Z2 Z 2Z DIII − þ 1 Z2 Z2 Z 2Z AII − 2Z Z2 Z2 Z CII − − 1 2Z Z2 Z2 Z C − 2Z Z2 Z2 Z CI þ − 1 2Z Z2 Z2 Z

Dimensions Symmetries

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SLIDE 20

Outline

  • 1. Reminder of the 2D Quantum Hall Effect
  • 2. 4D Topological Physics
  • 3. 4D Quantum Hall in Synthetic Dimensions
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SLIDE 21

General Concept:

  • 1. Identify a set of states and reinterpret as sites in a synthetic dimension

1 2 3 4

Synthetic Dimensions

1 2 3 4

  • 2. Couple these modes to simulate a tight-binding “hopping”

w J eiφ J e−iφ

Boada et al., PRL, 108, 133001 (2012), Celi et al., PRL, 112, 043001 (2014)

HOW?

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SLIDE 22

Ingredients:

  • 1. Reinterpret states as sites in synthetic dimension -> Internal atomic states
  • 2. Couple states to simulate a “hopping” term -> Coupling lasers

I = 5/2

173Yb

Synthetic dimension with internal atomic states

Florence: Mancini et al, Science, 349, 1510 (2015) Maryland: Stuhl et al. Science, 349, 1514 (2015)

For atomic hyperfine states:

Also now with clock transitions: Florence Livi et al, Phys. Rev. Lett. 117, 220401 (2016) Boulder: Kolkowitz et al, Nature, 542, 66 (2017)

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SLIDE 23

Ingredients:

  • 1. Reinterpret states as sites in synthetic dimension -> Harmonic oscillator states
  • 2. Couple states to simulate a “hopping” term -> Shaking of harmonic trap

Theory: HMP, T. Ozawa and N. Goldman, Phys. Rev. A 95, 023607 (2017) 1 2 3 4 1 2 3 4

Synthetic dimension with harmonic trap states

See poster by Grazia Salerno!

Also: synthetic dimensions for photons: Optomechanics: Schmidt et al, Optica 2, 7, 635 (2015) Optical cavities: Luo et al, Nature Comm. 6, 7704, (2015) Integrated photonics: Ozawa, HMP, Goldman, Zilberberg, & Carusotto, Phys. Rev. A 93, 043827 (2016),

  • L. Yuan, Y. Shi & S. Fan, Optics Letters 41, 4, 741 (2016)

Ozawa & Carusotto, PRL, 118, 013601 (2017) Waveguides: Lustig et al, arXiv:1807.01983

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SLIDE 24

hopping along a synthetic (internal-state) dimension

see Celi et al PRL ‘14

where (simply tune the Raman lasers!)

4D QH with Synthetic Dimensions

HMP, Zilberberg, Ozawa, Carusotto & Goldman, Phys. Rev. Lett. 115, 195303 (2015)

  • T. Ozawa, HMP, N. Goldman, O. Zilberberg, and I. Carusotto, Phys. Rev. A 93, 043827 (2016)

e.g. as in expt: Aidelsburger et al., Nat. Phys, 11,162. (2015)

{ { { {

With atoms With photons

e.g. as in expt: Hafezi et al, Nat. Photon. 7, 1001, (2013)

Observable: COM displacement…. Observable: Photonic steady-state….

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SLIDE 25

4D QH with Topological Pumping

With atoms With photons

See talk by Oded Zilberberg!

Lohse, Schweizer, HMP, Zilberberg, Bloch, Nature 553, 55–58 (2018) Zilberberg, Huang, Guglielmon, Wang, Chen, Kraus, Rechtsman., Nature 553, 59 (2018)

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SLIDE 26

(a) (b)

Summary

Synthetic dimensions for cold atoms or photons

1 2 3 4

Review: “Topological Photonics”

Tomoki Ozawa, Hannah M. Price, Alberto Amo, Nathan Goldman, Mohammad Hafezi, Ling Lu, Mikael Rechtsman, David Schuster, Jonathan Simon, Oded Zilberberg, Iacopo Carusotto

arXiv:1802.04173

(a)

Topological physics in four dimensions

PhD Position Available!

jy = − q3 h2 EzBxwνn

2

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SLIDE 27

In collaboration with:

Oded Zilberberg
 (Zurich) Nathan Goldman (Brussels) Iacopo Carusotto (Trento) Tomoki Ozawa (Riken, Japan) Immanuel Bloch Michael Lohse Christian Schweizer (Munich) (Munich) (Munich) Grazia Salerno (Brussels) Ioannis Petrides
 (Zurich)