Tunable Flux through a Synthetic Hall Tube of Neutral Fermions

Xi-Wang Luo,1 Jing Zhang,2, 3, ∗ and Chuanwei Zhang1, †

1Department of Physics, The University of Texas at Dallas, Richardson, Texas 75080-3021, USA2State Key Laboratory of Quantum Optics and Quantum Optics Devices,

Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, P.R.China3Synergetic Innovation Center of Quantum Information and Quantum Physics,

University of Science and Technology of China, Hefei, Anhui 230026, P. R. China

Hall tube with a tunable flux is an important geometry for studying quantum Hall physics, butits experimental realization in real space is still challenging. Here, we propose to realize a syntheticHall tube with tunable flux in a one-dimensional optical lattice with the synthetic ring dimensiondefined by atomic hyperfine states. We investigate the effects of the flux on the system topology andstudy its quench dynamics. Utilizing the tunable flux, we show how to realize topological chargepumping, where interesting charge flow and transport are observed in rotated spin basis. Finally,we show that the recently observed quench dynamics in a synthetic Hall tube can be explained bythe random flux existing in the experiment.

INTRODUCTION

Ultracold atoms are emerging as a promising platformfor the study of condensed matter physics in a clean andcontrollable environment [1, 2]. The capability of gen-erating artificial gauge fields and spin-orbit coupling us-ing light-matter interaction [3–18] offers new opportu-nity for exploring topologically nontrivial states of mat-ter [19–25]. One recent notable achievement was therealization of Harper-Hofstadter Hamiltonian, an essen-tial model for quantum Hall physics, using laser-assistedtunneling for generating artificial magnetic fields in two-dimensional (2D) optical lattices [26–28]. Moreover, syn-thetic lattice dimension defined by atomic internal states[29–38] provides a new powerful tool for engineering newhigh-dimensional quantum states of matter with versatileboundary manipulation [32, 33], using a low-dimensionalphysical system.

Nontrivial lattice geometries with periodic boundaries(such as a torus or tube) allow the study of many in-teresting physics such as the Hofstadter’s butterfly [39]and Thouless pump [40–45], where the flux through thetorus or tube is crucially important. In a recent exper-iment [37], a synthetic Hall tube has been realized in a1D optical lattice and interesting quench dynamics havebeen observed, where the flux effect was not considered.More importantly, the flux through the tube, determinedby the relative phase between Raman lasers, is spatiallynon-uniform and random for different iterations of the ex-periment, yielding major deviation from the theoreticalprediction. The physical significance and experimentalprogress raise two natural questions: can the flux throughthe synthetic Hall tube be controlled and tuned in real-istic experiments? If so, can such controllability lead tothe observation of interesting quench dynamics?

In this paper, we address these important questions byproposing a simple scheme to realize a controllable fluxΦ through a three-leg synthetic Hall tube and studyingits quench dynamics and topological pumping. Our main

results are:

i) We use three hyperfine ground spin states, each ofwhich is dressed by one far-detuned Raman laser, to re-alize the synthetic ring dimension of the tube. The fluxΦ can be controlled simply by varying the polarizationsof the Raman lasers [11]. The scheme can be applied toboth Alkali (e.g., potassium) [11–15] and Alkaline-earth(-like) atoms (e.g., strontium, ytterbium) [46].

ii) The three-leg Hall tube is characterized as a 2Dtopological insulator with Φ playing the role of the mo-mentum along the synthetic dimension. The system alsobelongs to a 1D topological insulator at Φ = 0 and π,where the winding number is quantized and protected bya generalized inversion symmetry.

iii) The tunable Φ allows the experimental observa-tion of topological charge pumping in the tube geome-try, which also probes the system topology. Interestingcharge flow and transport can be observed in rotated spinbasis.

iv) We study the quench dynamics with a tunable fluxand show that the experimental observed quench dynam-ics in [37] can be better understood using a random fluxexisting in the experiment.

THE MODEL

We consider an experimental setup with cold atomstrapped in 1D optical lattices along the x-direction,where transverse dynamics are suppressed by deep opti-cal lattices, as shown in Fig. 1a. The bias magnetic fieldis along the z direction to define the quantization axis.Three far-detuned Raman laser fields ~Es, propagating inthe x-y plane, are used to couple three atomic hyperfineground spin states, with each state dressed by one Ramanlaser, as shown in Figs. 1b and 1c for alkaline-earth(-like)(e.g., strontium, ytterbium) and alkali (e.g., potassium)atoms, respectively. The three spin states form three legsof the synthetic tube system as shown in Fig. 1d, and the

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b 20

21

2

𝜃 𝜃

ℰ1

ℰ2

ℰ3

𝐵

𝑦

𝑥 𝑧

ℰ1𝜎

ℰ1𝜋

ℰ3𝜋

ℰ3𝜎

ℰ2𝜎

𝑆01

𝑃13

𝐹 =7

2

𝐹 =9

2

ℰ2𝜋

ℰ3𝜋

ℰ3𝜎

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(a) (b) (c)

|1⟩

|3⟩

|2⟩

|3⟩

|2⟩

|1⟩

|1⟩

|2⟩

|3⟩

𝑗

𝑗 + 1

𝑗 − 1

𝛷

𝐽

𝑥

𝜙0

(d)

𝑃3/22

𝑃1/22

𝑃03 or

FIG. 1: (a) Schematic of the experimental setup for tunable

flux through the synthetic Hall tube. The Raman lasers ~E1, ~E2and ~E3 generate the couplings along the synthetic dimensionspanned by atomic hyperfine states. (b) and (c) The cor-responding Raman transitions for alkaline-earth(-like) atomsand alkali, respectively. (d) Synthetic Hall tube with a uni-form flux φ0 on each side plaquette and Φ through the tube.

tight-binding Hamiltonian is written as

H =∑j;s6=s′

Ωss′;jc†j,scj,s′ −

∑j;s

(Jc†j,scj+1,s +H.c.), (1)

where Ωss′;j = Ωss′eiφj;ss′ , c†j,s is the creation operator

with j, s the site and spin index. J and Ωss′ are the tun-neling rate and Raman coupling strength, respectively.For alkaline-earth(-like) atoms, we use three states in the1S0 ground manifold to define the synthetic dimension.The long lifetime 3P0 or 3P1 levels are used as the inter-mediate states for the Raman process (see Fig. 1b) suchthat δmF = ±2 Raman process does not suffer the heat-ing issues [36]. While for alkaline atoms (see Fig. 1c), wechoose three hyperfine spin states |F,mF 〉, |F,mF − 1〉and |F −1,mF 〉 from the ground-state manifold to avoidδmF = ±2 Raman process, so that far-detuned Ramanlasers tuned between the D1 and D2 lines can be usedto reduce the heating [11, 12].

The laser configuration in Fig. 1a generates a uniformmagnetic flux penetrating each side plaquette as well asa tunable flux through the tube. Each Raman laser maycontain both z-polarization (responsible for π transition)and in-plane-polarization (responsible for σ transition)

components, which can be written as ~Es = eπEπs + eσEσs .For alkaline-earth(-like) atoms, we choose Eπ2 = 0 so

that Ω21;j ∝ Eπ1 Eσ∗2 , Ω32;j ∝ Eσ2 Eπ∗3 and Ω13;j ∝ Eσ3 Eσ∗1 .Therefore, the corresponding Raman coupling phases areφj;21 = (k1−k2)·xj+ϕπ1−ϕσ2 , where ks is the wave vectorof the s-th Raman laser, ϕπ,σs are the (π, σ)-componentphases of the s-th Raman laser at site j = 0 andxj = jdxx is the position of site j with dx the lattice con-stant. As a result, we have φj;21 = jφ0 + ϕπ1 − ϕσ2 where

φ0 = (k1−k2)·x = kRdx cos(θ) gives rise to the magneticflux penetrating the side plaquette of the tube, with kRthe recoil momentum of the Raman lasers. Similarly, wehave φj;32 = jφ0 +ϕσ2 −ϕπ3 , φj;13 = −2jφ0 +ϕσ3 −ϕσ1 . Toobtain a uniform magnetic flux for each side plaquette ofthe tube, we choose the incident angle θ such that themagnetic flux for each side plaquette is φ0 = 2π/3. Thephases ϕσ,πs determine the flux through the tube which isΦ ≡ −φj;21−φj;32−φj;13 = ϕσ3−ϕπ3 +ϕπ3−ϕσ3 . Therefore,we obtain Φ = ∆ϕ3 −∆ϕ1, where ∆ϕs = ϕπs −ϕσs is thephase difference between two polarization components ofthe s-th Raman laser. We notice that the phase of eachRaman coupling (i.e., φj;ss′) depends on random phasedifference between two Raman lasers (e.g., φj;21 dependson ϕπ1 −ϕσ2 ); however, their summation (i.e., the flux Φ)only depends on the phase difference ∆ϕs between twopolarizations of the same Raman laser, which can be con-trolled at will using wave plates. Moreover, the phase dif-ferences ∆ϕs do not depend on the transverse positionsy and z. Basically, the reason why we can control theflux Φ is that each spin state is dressed by one and onlyone Raman laser, and the random global phases of theRaman lasers can be gauged out by absorbing it into thedefinition of the spin states. With proper gauge choice,we can set the tunneling phases as φj;21 = φj;32 = jφ0

and φj;13 = jφ0 + Φ, as shown in Fig. 1d.

For alkali atoms, we choose Eσ2 = 0, yielding Ω21;j =

α21Eσ1 Eπ∗2 , Ω32;j = α32Eπ2 Eπ∗3 and Ω13;j = β13Eπ3 Eσ∗1 +α13Eσ3 Eπ∗1 , with αs,s′ , βs,s′ determined by the transi-tion dipole matrix. We further consider (Eπ3 , Eσ1 ) (Eσ3 , Eπ1 ) Eπ2 and Eσ3 Eπ1 ' Eπ3 Eπ2 ' Eσ1 Eπ2 . Therefore,

we have Ω13;j ' α13Eπ∗1 Eσ3 with amplitudes Ω21 ' Ω32 'Ω13. Similar as the alkaline-earth(-like) atoms, we ob-tain uniform magnetic flux φ0 = 2π/3 for tube side bychoosing kRdx cos(θ) = 2π/3. The flux through the tubebecomes Φ = ∆ϕ3 + ∆ϕ1, which can also be tuned atwill through the polarization control.

PHASE DIAGRAM

The Bloch Hamiltonian in the basis [ck,1, ck,2, ck,3]T

reads

Hk =

−2J cos(k − φ0) Ω21 Ω13e−iΦ

Ω21 −2J cos(k) Ω32

Ω13eiΦ Ω32 −2J cos(k + φ0)

,(2)

with k the momentum along the real-space lattice. ForΩ21 = Ω32 = Ω13, the above Hamiltonian is nothingbut the Harper-Hofstadter Hamiltonian with Φ the ef-fective momentum along the synthetic dimension, andφ0 = 2π/3 is the flux per plaquette. The topology is

3

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2

0

2

4

Ω13

δ2N T N

(a)

𝐸

𝐸 𝐸

𝛷

𝛷 𝑘

(b)

(d) (c)

Edge states

Edge states

Closed at 𝑘 = 𝜋

Red lines:𝛷 = 𝜋

Blue lines:𝛷 = 0

Closed at 𝑘 = 0

𝐶1 = 1

𝐶2 = −2

𝐶3 = 1

𝑊1𝜋=0

𝑊2,3𝜋=±1

FIG. 2: (a) Band structures in the topological phase. (b)Phase diagram in the Ω13-δ2 plane, with solid (dashed) linesthe boundary between topological (T) and normal (N) phasesfor the upper (lower) gap. (c) and (d) Band structures at theleft phase boundary with δ2 = 0 and Ω13 = 1. Commonparameters: Ω = 2 with energy unit J .

characterized by the Chern number [47]

Cn =i

2π

∫dkdΦ〈∂Φun|∂k|un〉 − 〈∂kun|∂Φ|un〉, (3)

where |un〉 is the Bloch states of the n-th band, satis-fying Hk(Φ)|un(Φ, k)〉 = En(Φ, k)|un(Φ, k)〉. In Fig. 2a,we plot the band structures as a function of Φ with anopen boundary condition along the real-lattice direction.There are three bands, and two gapless edge states (oneat each ends) in each gap. The two edge states crossonly at Φ = 0 and Φ = π, where the tube belongs to a1D topological insulator with quantized winding number(Zak phase) [48]

W 0,πn =

1

π

∮dk〈un|∂k|un〉

∣∣Φ=0,π

. (4)

The winding number is protected by a generalized in-version symmetry IHkI−1 = H−k, where the inversionsymmetry I swaps spin states |1〉 and |3〉 [49].

The Chern number and winding number are still welldefined even when the Raman couplings have detuningsand/or the coupling strength Ωss′ are nonequal. Thechanges in these Raman coupling parameters drive thephase transition from topological to trivial insulators.The detuning can be introduced by including additionalterms

∑j;s δsc

†j,scj,s in the Hamiltonian Eq. 1. Here-

after we will fix Ω21 = Ω32 ≡ Ω and δ1 = δ3 = 0for simplicity. The phase diagram in the Ω13-δ2 planeis shown in Fig. 2b. The solid (dashed) lines are thephase boundaries corresponding to the gap closing be-tween two lower (upper) bands, with topological phasesbetween two boundaries. At the phase boundaries, thecorresponding band gaps close at Φ = 0 (Φ = π) for thetwo lower (upper) bands, as shown in Fig. 2c. In addi-

(a) (b)

(c)

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ۃ 𝑗ۄ

𝑗

𝑗,2 𝑗,1 𝑗,3

𝛷 𝛷 𝛷

𝛷 𝛷

𝜏p/𝐽ۃ1− Δ

𝑗ۄ

𝑓 = 1

𝑓 = 2

𝑓 = 3

𝑛𝑗

FIG. 3: (a) Total density distribution during one pump cycle.Inset shows the non-adiabatic effects on the center-of-massshift. (b) Center-of-mass (red line) and the rotated-spin con-tributions (blue lines) during one pump cycle. The blue linesare

∑j jnj,f (t)/N with f = 1, 2, 3 as labeled. (c) Rotated

spin-density distributions during one pump cycle.

tion, for the two lower (upper) bands, the gap closes atk = 0 and k = π (k = π and k = 0) on the right andleft boundaries, respectively, as shown in Fig. 2d. Thegaps reopen in the trivial phase with the disappearanceof edge states.

TOPOLOGICAL PUMPING

The three-leg Hall tube is a minimal Laughlin’s cylin-der. When the flux through the tube is adiabaticallychanged by 2π, the shift of Wannier-function center isproportional to the Chern number of the correspond-ing band [40–43]. Therefore all particles are pumped byC site (with C the total Chern number of the occupiedbands) as Φ changes by 2π, i.e., C particles are pumpedfrom one edge to another. Given the ability of controllingthe flux through the tube, we can measure the topologicalChern number based on topological pumping by tuningthe flux Φ adiabatically (compared to the band gaps).

Here we consider the Fermi energy in the first gap withonly the lowest C = 1 band occupied, and study thezero temperature pumping process (the pumped parti-cle is still well quantized for low temperature compar-ing to the band gap) [50]. The topological pumpingeffect can be identified as the quantized center-of-massshift of the atom cloud [50–53] in a weak harmonic trapVtrap = 1

2vT j2. The harmonic trap strength vT = 0.008J

and the atom number N = 36 are chosen such that theatom cloud has a large insulating region with one atomper unit-cell at the trap center (In a realistic experiment,the above parameters are typical). For simplicity, weset Ωss′ = J and δs = 0 for all s, s′, choose the gaugeas φj;21 = φj;32 = jφ0, φj;13 = jφ0 − Φ, and changeΦ slowly (compared to the band gap) as Φ(t) = 2πt

τp.

4

In Figs. 3a and 3b, we plot the total density distribu-tion nj(t) and the center-of-mass 〈j(t)〉 ≡

∑j jnj(t)/N

shift during one pumping circle with τp = 40J−1, andwe clearly see the quantization of the pumped atom〈∆j〉 ≡ 〈j(τp)〉 − 〈j(0)〉 = 1. The atom cloud shifts asa whole with nj(t) = 1 near the trap center. The in-set in Fig. 3a shows non-adiabatic effect (finite pumpingduration τp) on the pumped atom. Typically, J/2π isabout several hundred Hz; therefore, the pumping dura-tion τp should be of the order of tens of ms to satisfythe adiabatic condition. This timescale is of the samemagnitude as the lattice loading and time-of-flight imag-ing duration [34, 35]. Starting from a degenerate Fermigas prepared at the optical dipole trap, the whole exper-imental duration is less than 1s (which is typical for coldatom experiments).

The atoms are equally distributed on the three spinstates [i.e., nj,s(t) ≡ 〈c†j,scj,s〉 = 1

3nj(t)]. To see thepumping process more clearly, we can examine the spindensities in the rotated basis nj,f (t) ≡ 〈c†j,f cj,f 〉, with

cj,f = 1√3

∑s cj,se

is 2fπ−Φ3 . The Hamiltonian in these ba-

sis reads H =∑3f=1Hf , where

Hf = 2J cos(Kj,f,Φ)c†j,f cj,f + (Jc†j,f cj,f+1 +H.c.) (5)

is the typical Aubry-Andre-Harper (AAH) Hamilto-

nian [54, 55] with Kj,f,Φ = 2π(f+j)−Φ3 . Each spin com-

ponent contributes exactly one third of the quantizedcenter-of-mass shift (see Fig. 3b). The bulk-atom flowduring the pumping can be clearly seen from the spindensities nj,f (t), as shown in Fig. 3c. The quantizedpumping can also be understood by noticing that theAAH Hamiltonians Hs are permutated as H1 → H3 →H2 → H1 after one pump circle. Each Hf returns toitself after three pump circles with particles pumped bythree sites (since the lattice period of Hf is 3). There-fore, for the total Hamiltonian H, particles are pumpedby one site after one pump circle. The physics for dif-ferent values of Ωss′ and δs are similar, except that therotated basis cj,f may take different forms.

In the presence of atom-atom interactions, the ro-bust topological properties should not be affected if theinteraction strength is much weaker compared to theband gaps, and the charge pumping should remain quan-tized [56]. The interaction can be written as Hint =U2

∑j,s 6=s′ nj,snj,s′ with atom number operator nj,s =

c†j,scj,s and interaction strength U . Atom in site (j, s)would interact with atoms in all other sites (j, s′ 6= s)along the synthetic dimension. Therefore, the interac-tion is long ranged along the synthetic dimension. Atthe strong interaction region, the system may producefractional quantum Hall physics and support fractionaltopological pumping [44, 45]. Our scheme generates atunable flux through the tube and thus offers an idealplatform for studying quantized fractional charge trans-

(c)

(b)

(d)

(a)

𝜏3

𝜏2 𝜏3𝜏2

𝑛𝑠

𝑛𝑠𝜋

𝑛𝑠𝜋

𝑡/𝐽−1

𝑡/𝐽−1

𝑡/𝐽−1

𝑡/𝐽−1

(𝜋)

FIG. 4: Quench dynamics for Φ = 0 (solid lines) and aver-aged over random Φ (dashed lines). Time evolution of spinpopulations (a) and averaged momenta (b) with Ω13 = Ω.Time evolution of spin populations at k = π with Ω13 = 5 in

(c) and Ω13 = 7.5 in (d). nπs = ns(π)∑s′ ns′ (π)

, τ2 and τ3 cross at

Ωc13 = 5.8. Both gaps are topological in (a) and (b), and onlythe upper (lower) gap is topological in (c) [(d)]. Commonparameters: Ω = 6.15, δ2 = −0.4Ω with energy unit J .

port and probing the fractional many-body Chern num-bers.

QUENCH DYNAMICS

Besides topological pumping, the quench dynamics ofthe system can also be used to demonstrate the presenceof gauge field φ0 and detect the phase transitions [37].Here we study how Φ affects the quench dynamics byconsidering that all atoms are initially prepared in state|1〉, then the inter leg couplings are suddenly activated byturning on the Raman laser beams. In Figs. 4a and 4b, weshow the time evolution of the fractional spin populationsns = 1

N

∫dkns(k), as well as the momenta 〈k〉 =

∑s〈ks〉

and 〈∆k〉 = 〈k2〉 − 〈k3〉 (both can be measured by time-of-flight imaging) for Φ = 0, where ns(k) = 〈cs(k)†cs(k)〉and 〈ks〉 = 1

N

∫kns(k)dk with N the total atom number.

We find that the time evolutions show similar oscillatingbehaviors for different Φ, but with different frequenciesand amplitudes. The difference between the momentaof atoms transferred to state |2〉 and |3〉 increase notice-ably at early time as a result of the magnetic flux φ0

penetrating the surface of the tube [37], which does notdepend on the flux Φ through the tube. In Fig. 4, wehave fixed Ω = 6.15J , δ2 = −0.4Ω and U = 0. Theinitial temperature is set to be T/TF = 0.3, with initialFermi temperature TF given by the difference betweenthe Fermi energy EF and the initial band minimum −2J(i.e., TF = EF + 2J). We use TF = 2J to get the sim-ilar initial filling and Fermi distribution as those in theexperiment [37] (the results are insensitive to TF ).

The quench dynamics can also be used to measure thegap closing at phase boundaries. Similar as Ref. [37], weintroduce two times τ2 and τ3, at which the spin-|2〉 andspin-|3〉 populations at k = 0 (or k = π) reach their firstmaxima, to identify the phase boundary. As we change δ2

5

𝑛𝑠𝜋

𝑛𝑠𝜋

𝑛𝑠𝜋

𝑛𝑠𝜋

(a) (b)

(c) (d)

𝑡/𝐽−1

𝑡/𝐽−1

𝑡/𝐽−1

𝑡/𝐽−1

𝛷 = 0

𝛷 = 0.1𝜋 𝛷 = −0.1𝜋

𝛷 = 𝜋

FIG. 5: (a)-(d) Time evolution of the spin populations atk = π for different values of Φ. All other parameters are thesame as Fig. 4c.

or Ω13 across the phase boundary (one gap closes and thedynamics is characterized by a single frequency), τ2 andτ3 cross each other. Notice that above discussions onlyapply to Φ = 0, π where the gap closing occurs. We findthat, even when no gap closing occurs for Φ away from0 and π, τ2 and τ3 would cross each other as we increaseΩ13, and the crossing point is generally away from thephase boundaries. Therefore, the measurement of gapclosing based on quench dynamics is possible only if Φcan be controlled. As an example, we consider Φ = 0 andplot the time evolution of the spin populations at k = πwith Ω13 around the left phase boundary Ωc13, as shownin Figs. 4c and 4d. The crossing points are different fordifferent Φ. In Fig. 5, we plot the time evolution of thespin populations at k = π for different values of Φ.

The ability to control the flux Φ is crucial for the studyof both topological properties and quench dynamics. Wenotice that for the experiment in [37], Φ cannot be con-trolled and may vary from one experimental realization toanother, Φ is also different for different tubes within oneexperimental realization. It is straightforward to showthat the flux Φ in [37] is determined by the difference be-tween the random phases of the Raman lasers at latticesites (see Appendix), which cannot be controlled sincethe Raman lasers propagate along different paths, not tomention that their wavelengths are generally not com-mensurate with the lattice. Moreover, in realistic exper-iments, arrays of independent fermionic synthetic tubesare realized simultaneously due to the transverse atomicdistributions in the y, z directions [36, 37], and the syn-thetic tubes at different y would have different Φ due tothe y-dependent ϕσ1 . For the parameters in Ref. [37], thedifference of Φ between neighbor tubes in y direction isabout 2π × 0.58. The random flux or phase problem isa common issue for schemes in which one spin state isdressed by two or more different lasers (different in wavevector) where the random global phases of the Ramanlasers can not be gauged out.

Due to the randomness of Φ, the dynamics in experi-ment [37] (averaged over enough samplings) should cor-respond to results averaged over Φ. In Fig. 4, we plot

𝑛𝑠

𝑡/𝐽−1 𝑡/𝐽−1

𝑛𝑠

(a) (b)

FIG. 6: Quench dynamics in the presence of a harmonic trapVtrap = 1

2vTj

2 for non-interacting fermions (U = 0) in (a) andinteracting fermions (U = 1.7J) in (b). The trap strength isvT ' 0.0158J . The results in (a) are averaged over Φ. Thesolid (dashed) lines in (b) are the results for Φ = 0 (averagedover Φ). All other parameters are the same as in Fig. 4a.

the corresponding dynamics averaged over Φ, which showdamped oscillating behaviors (with significant long-timedamping), as observed in the experiment. τ2 and τ3,which cross each other at different values of Ω13 for dif-ferent Φ with averaged value Ω13 = Ω 6= Ωc13, may notbe suitable to identify the phase boundaries (i.e., gapclosings) for a random flux Φ.

In realistic experiments, a harmonic trap is usually ap-plied to confine the atoms. The above results for thequench dynamics still hold for atoms in a weak harmonictrap, as confirmed by our numerical simulations. Weconsider a harmonic trap frequency ωx = 2π × 57 Hzand J = 2π × 264 Hz as in the experiment [37], there-fore, the harmonic trap Vtrap = 1

2vT j2 has trap strengthvT ' 0.0158J . In Fig. 6a, we plot the time evolutionof spin populations, we see that the quench dynamicsare hardly affected by the harmonic trap. Moreover,atom-atom interactions may induce scattering betweendifferent momentum states that oscillate at different fre-quencies, leading to additional damping, which shouldbe minor due to the short evolution time . 1ms here. InFig. 6b, we plot the time evolution of spin populationsin the presence of weak interaction U = 1.7J [37]. Wehave adopted the mean-field approach and written the in-teraction as U

2

∑j,s 6=s′ nj,snj,s′ = U

2

∑j,s 6=s′ [〈nj,s′〉nj,s +

〈nj,s〉nj,s′ − 〈nj,s〉〈nj,s′〉], where the s-spin fermions in-teract with the average density of the s′-spin fermions,which should be valid for weak interaction and short evo-lution time (similar to the Hartree approximation). Theweak interaction can hardly affect the quench dynamicswhere the oscillation frequency (∼ Ω) is much larger thanthe interaction strength. Other experimental imperfec-tions such as spin-selective imaging error may also affectthe measured spin dynamics, and the final observed crosspoint in [37] is smaller than Ω and Ωc13.

CONCLUSION

In summary, we propose a simple and feasible schemeto realize a controllable flux Φ through the synthetic Halltube that can be tuned at will, and study the effects of theflux Φ on the system topology and dynamics. Previous

6

experimental quench dynamics may be better explainedby the results averaged over the random flux existing inthe experiment. Our scheme allows the study of interest-ing charge transport (e.g., the interesting charge flow andtransport for rotated spin states in our system) throughtopological pumping, which also probes the system topol-ogy. Moreover, it may open the possibility to study frac-tional charge pumping and probe the many-body Chernnumbers of fractional quantum Hall state at strong inter-actions. Our results provide a new platform for studyingtopological physics in a tube geometry with tunable fluxand may be generalized with other synthetic degrees offreedom, such as momentum states [57–59] and latticeorbitals [60, 61].

Acknowledgements: XWL and CZ are supportedby AFOSR (FA9550-16-1-0387, FA9550-20-1-0220), NSF(PHY-1806227), and ARO (W911NF-17-1-0128). JZ issupported by the National Key Research and Develop-ment Program of China (2016YFA0301602).

APPENDIX: RANDOM FLUX Φ IN PREVIOUSEXPERIMENTS

We would like to emphasize that our proposed setupto generate a tunable flux through the synthetic tube isdifferent from the setup in recent experimental work [37](as shown in Fig. 7 for comparison). The Raman laserconfigurations (directions, polarizations and frequencies)as well as the involved Raman processes are different. Itis straightforward to show that the flux Φ for the setupin Fig. 7 is Φ = 3ϕσ1 − 2ϕπ2 − ϕσ3 , which cannot be con-trolled since the Raman lasers propagate along differentpaths, not to mention that their wavelengths are gen-erally not commensurate with the lattice. The randomflux or phase problem is a common issue for schemesin which one spin state is dressed by two or more dif-ferent lasers (different in wave vector) where the randomglobal phases of the Raman lasers can not be gauged out.Moreover, in realistic experiments, arrays of independentfermionic synthetic tubes are realized simultaneously dueto the transverse atomic distributions in the y, z direc-tions, and the synthetic tubes at different y would havedifferent Φ due to the y-dependent ϕσ1 for the setup inFig. 7.

∗ Corresponding author.Email: jzhang74@sxu.edu.cn

† Corresponding author.Email: chuanwei.zhang@utdallas.edu

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(a)

ℰ2𝜋 + ℰ3

𝜎

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