Scale-Invariant Continuous Entanglement Renormalization of a Chern Insulator

Su-Kuan Chu,1, 2 Guanyu Zhu,1 James R. Garrison,1, 2 Zachary Eldredge,1, 2

Ana Valdes Curiel,1 Przemyslaw Bienias,1 I. B. Spielman,1 and Alexey V. Gorshkov1, 2

1Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA2Joint Center for Quantum Information and Computer Science,

NIST/University of Maryland, College Park, Maryland 20742, USA

The multi-scale entanglement renormalization ansatz (MERA) postulates the existence of quan-tum circuits that renormalize entanglement in real space at different length scales. Chern insulators,however, cannot have scale-invariant discrete MERA circuits with finite bond dimension. In thisLetter, we show that the continuous MERA (cMERA), a modified version of MERA adapted for fieldtheories, possesses a fixed point wavefunction with nonzero Chern number. Additionally, it is wellknown that reversed MERA circuits can be used to prepare quantum states efficiently in time thatscales logarithmically with the size of the system. However, state preparation via MERA typicallyrequires the advent of a full-fledged universal quantum computer. In this Letter, we demonstratethat our cMERA circuit can potentially be realized in existing analog quantum computers, i.e., anultracold atomic Fermi gas in an optical lattice with light-induced spin-orbit coupling.

A quantum many-body system has a Hilbert spacewhose dimension grows exponentially with system size,making exact diagonalization of its Hamiltonian imprac-tical. Fortunately, tensor networks [1, 2] are capable ofefficiently representing the ground states of many sys-tems with local interactions [3–8]. Another powerful toolin many-body physics is the renormalization group (RG)[9, 10], which uses the fact that the description of a phys-ical system can vary at different length scales, forming ahierarchical structure. The RG provides a systematicprescription to transform an exact microscopic descrip-tion to an effective coarse-grained description. Appli-cations of RG range from critical phenomena in con-densed matter to the electroweak interaction in high-energy physics [11].

One approach which combines tensor networks andrenormalization group is called the multi-scale entangle-ment renormalization ansatz (MERA) [3, 7]. MERA pro-poses a quantum circuit acting on a state which is ini-tially entangled at many length scales. The two elemen-tary building-block tensors of the MERA, isometries anddisentanglers, are discrete unitary gates which physicallyimplement RG in real space by successively removing en-tanglement at progressively larger length scales. Interest-ingly, since the circuit depth only increases logarithmi-cally with the system size, a reversed MERA circuit canefficiently prepare a state with finer entanglement struc-ture from a weakly-entangled initial state. In practice,MERAs are most convenient when the disentanglers andisometries are independent of the length scale [12–18].The state that is left unchanged in the thermodynamiclimit by these scale-invariant unitaries is termed a fixed-point wavefunction.

Experimentally, a reversed MERA circuit might beused to prepare exotic states, such as chiral topologicalstates, which include integer quantum Hall states andcertain fractional quantum Hall states [19, 20]. Somefractional quantum Hall systems are believed to feature

anyons useful for topological quantum computation [21].Due to their great theoretical interest, it would be usefulto be able to study these systems under highly controlledsettings, such as in ultracold atomic gases. However, tocreate a chiral topological state in the lab, we must notonly engineer the parent Hamiltonian, but also cool thesystem down to the ground state. The latter is usuallyhard experimentally for topological states due to theirlong-range entanglement [22]. A reversed MERA circuitcan possibly resolve this issue by directly generating thetarget chiral topological state from another state that iseasier to obtain by cooling.

Here, as a first step towards finding a MERA for afractional quantum Hall state, we instead search for aMERA whose fixed-point wavefunction describes an (in-teger) Chern insulator. A Chern insulator is an inte-ger quantum Hall state on a lattice and is therefore asimpler system than the fractional quantum Hall state.However, there are no-go theorems stating that a MERAcannot have a Chern insulator ground state as its fixed-point wavefunction [23–26]. Since conventional MERAonly contains strictly local interactions, adding quasi-local interactions with quickly decaying tails could possi-bly circumvent the no-go theorems. A modified formal-ism of MERA adapted for field theories called continu-ous MERA (cMERA) [27] can include such quasi-localinteractions [28]. In contrast to the MERA paradigm,in which the renormalization circuit consists of discreteunitary gates, cMERA treats the circuit time, which cor-responds to the length scale, as a continuous variableand generates continuous entanglement renormalizationusing a Hermitian Hamiltonian.

In this Letter, we show that a type of Chern insula-tor wavefunction can be generated by a scale-invariantcMERA circuit. The Chern insulator model we consideris the Bernevig-Hughes-Zhang model in the continuumlimit [29]. In addition, we propose a possible experimen-tal realization of the cMERA circuit with neutral 171Yb

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atoms in an optical lattice by introducing spin-orbit cou-pling.

Our work complements and can be contrasted withRefs. [30, 31]. While Ref. [30] previously developed acMERA for the continuous Chern insulator model men-tioned above, our work uses a scale-invariant disentan-gler. Other prior work in Ref. [31] presented a scale-invariant entanglement renormalization for a two-bandChern insulator model. While Ref. [31] makes use ofthe lattice structure and quasi-adiabatic paths betweena series of gapped Hamiltonians, our cMERA approachallows smooth time evolution and emphasizes the contin-uum physics. Another difference is that the RG evolu-tion in Ref. [31] involves interactions decaying with dis-tance faster than any power-law function but slower thanan exponential, whereas our cMERA only needs an ex-ponentially decaying interaction. Other known methodsfor representing chiral topological states include artificialneural network quantum states [32–34], projected entan-gled pair states [25, 35–37], matrix product states [38],and polynomial-depth unitary circuits [39].

Review of cMERA.—Within the framework of con-ventional MERA [3, 7], disentanglers Vu and isome-tries Wu are strictly local discrete unitary operators em-ployed to renormalize entanglement at layer u ∈ Z+. IncMERA [27], we simply replace them by continuous uni-tary transformations, which are infinitesimally generatedby self-adjoint operators K(u) and L: Vu → e−iK(u)du,Wu → e−iLdu. The notation du denotes an infinitesimalRG step, and u ∈ (−∞, 0]. When the continuous variableu approaches zero, the system is said to be at the ultravi-olet (UV) length scale, possessing both short-range andlong-range entanglement. As u→ −∞, the system flowsto the infrared (IR) length scale, where short-range en-tanglement is removed and nearly all degrees of freedomare disentangled from each other. Note that the genera-tor of disentangler K(u) can in general depend on scale u.A cMERA is called scale-invariant if K(u) is independentof u.

To emulate the coarse-graining behavior of isometriesin conventional lattice MERA, L is chosen to be thescaling transformation in field theory. For example, fora single fermion field ψ(x) in d spatial dimensions, wepick L = − i

2

∫ (ψ†(x)x · ∇ψ(x)− x · ∇ψ†(x)ψ(x)

)ddx

[27, 30]; thereby, fermionic operators ψ(x) in real spaceand ψ(k) in momentum space satisfy the following

scaling transformations: e−iuLψ(x)eiuL = ed2uψ(eux),

e−iuLψ(k)eiuL = e−d2uψ(e−uk). One can check that the

anti-commutation relations ψ(x), ψ†(x′) = δ(x−x′) inreal space and ψ(k), ψ†(k′) = δ(k− k′) in momentumspace are preserved under the scaling transformation. Wewill sometimes abuse the terminology to call K(u) andL themselves the disentangler and the isometry ratherthan the verbose generator of disentangler and generatorof isometry.

The renormalized wavefunction is governed by theSchrodinger equation,

i∂

∂u|ΨS(u)〉 = [K(u) + L]|ΨS(u)〉, (1)

where the superscript S denotes the Schrodinger picture.In this Letter, we will focus on the interaction picturewhich provides a more convenient way to look at continu-ous entanglement renormalization. We treat L as a “free”Hamiltonian and K(u) as an “interaction” Hamiltonian,i.e., |ΨI(u)〉 = eiuL|ΨS(u)〉, where the superscript I de-notes the interaction picture. Substituting this equationinto Eq. (1), we obtain

i∂

∂u|ΨI(u)〉 = K(u)|ΨI(u)〉, (2)

where K(u)def= eiuLK(u)e−iuL is the disentangler in

the interaction picture. The renormalized wavefunction|ΨI(u)〉 at scale u can be formally written in terms of theIR state |ΩIIR〉 ≡ |ΨI(u→ −∞)〉 as

|ΨI(u)〉 = P exp

(−i∫ u

−∞K(u′)du′

)|ΩIIR〉, (3)

where P is the path ordering operator. Unless other-wise stated, we will only consider the interaction picture;therefore, we will drop the superscript I in the rest ofthis Letter.

A continuous Chern insulator model.—We beginwith a two-band continuous Chern insulator model intwo spatial dimensions [29] with Hamiltonian H =∫

d2kψ†(k)[R(k) · σ]ψ(k), where k = (kx, ky) ∈ R2,

R(k) = (kx, ky, m − k2), m > 0, k ≡ |k| =√k2x + k2

y,

and σ = (σx, σy, σz) is a vector of Pauli matrices.The fermionic operator ψ(k) is a two-component spinor

ψ(k) ≡(ψ1(k) ψ2(k)

)Twhose components satisfy

ψ†i (k), ψj(k′) = δij δ(k− k′) for i, j ∈ 1, 2.

The ground state, which has the lower band filled, is[30]

|Ψ〉 =∏k

(ukψ

†2(k)− vkψ†1(k)

)|vac〉, (4)

uk =1√Nk

((m− k2

)+√

(m− k2)2 + k2),

vk =1√Nk

(ke−iθk

),

where Nk is a k-dependent normalization factor suchthat |uk|2 + |vk|2 = 1, and the state |vac〉 is the vac-uum state annihilated by ψ1,2(k). The angle θk isdefined via kx = k cos θk and ky = k sin θk, i.e., itis the polar angle in momentum space. The Chernnumber of the bottom band of this two-band system

is C = 14π

∫R2 d2k n(k) ·

(∂n(k)∂kx

× ∂n(k)∂ky

)= 1, where

3

n(k) ≡ R(k)|R(k)| and where the integrand divided by two

is called the Berry curvature.

Now, we show how to obtain a scale-invariant cMERAfor this model.

Entanglement renormalization of the Chern insu-lator.—Following the convention in Refs. [27, 30,40], we take the Gaussian ansatz for the dis-entangler in the Schrodinger picture, K(u) =

i∫

d2k[g(k, u)ψ†1(k)ψ2(k) + g∗(k, u)ψ1(k)ψ†2(k)

][41].

If we require our disentangler to be scale-invariant,then g(k, u) should not have explicit u dependence,g(k, u) = g(k). We also take the ansatz that g(k) =H(k)e−iθk , where H(k) is a real-valued function tobe determined. Through rewriting the disentangleras K(u) =

∫d2kψ†(k)[H(k) · σ]ψ(k) with H(k) =

(H(k) sin θk,−H(k) cos θk, 0), we can intuitively under-stand its action by imagining an effective magnetic fieldof strength H(k) in a clockwise direction about the originapplied to the pseudo-spin at each momentum point. Inthe interaction picture, the disentangler becomes

K(u) = i

∫d2k

[H(e−uk)e−iθkψ†1(k)ψ2(k)

+H(e−uk)eiθkψ1(k)ψ†2(k)

]. (5)

Now, we start to renormalize the wavefunction and de-termine the form of the disentangler. We assume that therenormalized wavefunction at scale u can be expressed as

|Ψ(u)〉 =∏k

(Pk(u)ψ†2(k)−Qk(u)ψ†1(k))|vac〉, (6)

with |Pk(u)|2 + |Qk(u)|2 = 1. From Eq. (2), we get

Pk(u) = Ake−iϕ(e−uk) +Bke

iϕ(e−uk), (7)

Qk(u) = −ie−iθk[Ake

−iϕ(e−uk) −Bkeiϕ(e−uk)

].

CoefficientsAk andBk are complex numbers with |Ak|2+|Bk|2 = 1

2 , and ϕ(e−uk) ≡∫∞ke−u

H (t) dtt . At UV scale

u = 0, the wavefunction should match the ground state inEq. (4); at IR scale u→ −∞, we would like the renormal-

ized wavefunction to be the product state∏

k ψ†1(k)|vac〉

or the product state∏

k ψ†2(k)|vac〉 [27, 30, 40]. By tak-

ing Ak = − 12i and Bk = 1

2i , the boundary conditions canbe satisfied by requiring

H (k) =k(m+ k2)

2 [k4 + k2(1− 2m) +m2]. (8)

Substituting Eq. (8) into Eqs. (6) and (7), we attain an

UV

IRu=-∞

u=0kxky

FIG. 1. Berry curvature of the renormalized wavefunction inthe interaction picture at different scales u, drawn schemati-cally in momentum space. The blue arrow corresponds to thedirection of the reversed cMERA circuit. The area contribut-ing to the Chern number expands as one approaches the UVscale.

explicit form of the renormalized wavefunction,

|Ψ(u)〉 =∏k

1√Nk,u

×[((m− k2e−2u) +

√(m− k2e−2u)2 + k2e−2u

)ψ†2(k)

−k e−ue−iθk ψ†1(k)

]|vac〉, (9)

where Nk,u is a normalization factor that depends onk and u. The Berry curvature of the renormalizedwavefunction at different u is shown schematically inFIG. 1. The IR state is |ΩIR〉 = limu→−∞ |Ψ(u)〉 =∏

k e−iθkψ†1(k)|vac〉, which is equal to

∏k ψ†1(k)|vac〉 =∏

x ψ†1(x)|vac〉 up to an overall phase. Note that the

nonzero Chern number does not survive in the IR statebecause the integration operation does not commute withthe limit u → −∞. However, at any finite u, the Chernnumber is always one. Therefore, there is no phase tran-sition during the entanglement renormalization process,consistent with the result in Ref. [30].

To analyze the spatial structure of the disentangler, werewrite the expression for H (k). We first define λ+ andλ− as the two roots of the equation x2+(1−2m)x+m2 =

0, λ± = −1+2m±√

1−4m2 . They are real and negative when

0 < m < 1/4. Although setting this restriction on m isnot necessary for our disentangler, we will assume it inthe following in order to assist our experimental realiza-tion. Now, the expression H (k) can be rewritten as

H (k) =

(−1 +√

1− 4m

4√

1− 4m

)k

k2 − λ+

+

(1 +√

1− 4m

4√

1− 4m

)k

k2 − λ−. (10)

By inserting this expression into Eq. (5) and perform-ing a Fourier transform, it can be shown that the disen-

4

|g1i|g2i

|e1i|e2i

spin-orbit interaction

FIG. 2. A scheme to engineer the cMERA circuit in the in-teraction picture. The two excited states are coupled by spin-orbit interaction to each other and by off-resonant lasers tothe two ground states.

tangler in real space decays exponentially with charac-teristic length e−umax

√−λ+,

√−λ−. Therefore, our

cMERA involves quasi-local interactions.

Experimental realization of the cMERA circuit.—Wepropose a way to realize our reversed cMERA circuit toprepare a Chern insulator state in an optical lattice withneutral 171Yb, which are fermionic atoms with two outerelectrons. From now on, we will drop the word “reversed”for our cMERA circuit when the context is clear. Recallthat the cMERA circuit starts with an initial IR state.As discussed above, the IR state at u → −∞ does nothave the correct Chern number; therefore, we start froma near-IR state with large negative u. In addition, thecMERA circuit is only valid on a lattice when the contin-uum approximation holds. Therefore, throughout the cir-cuit, the physical length scale e−umax

√−λ+,

√−λ−

should be significantly larger than the lattice spacing,but significantly smaller than the total size of the lattice.Going forward, we begin with a near-IR state and useour cMERA circuit to obtain the UV state without everviolating the requirements of the continuum approxima-tion.

Here, we assume that we already have an initial near-IR state waiting to be inserted into the cMERA circuit.Since, in finite-size systems, the Berry curvature is con-centrated on a few discrete momentum points near k = 0,the preparation of this near-IR state should be fast if wecan individually create states at each point in momen-tum space. In the Supplemental Material, we provideone possible method for generating this initial state.

We now present the cMERA circuit engineeringscheme (see Supplemental Material for details). We use|g1〉 and |g2〉 as shorthand notations for the two sta-ble hyperfine ground states |F = 1/2, mF = −1/2〉 and|F = 1/2, mF = 1/2〉 in 1S0; these form the basis of our

spinor ψ(k) ≡(ψ1(k) ψ2(k)

)T. We find that if we have

two metastable excited states |e1〉 and |e2〉 (e.g. from the

3P manifold) with quadratic dispersions coupled by spin-orbit interaction and couple them off-resonantly to therespective ground states as shown in FIG. 2, then thedisentangler in the interaction picture can be engineered.Intuitively, the spin-orbit interaction allows us to gener-ate a momentum-dependent effective magnetic field forEq. (5), whereas the off-resonant couplings to quadraticdispersive bands induce quadratic terms in the denom-inators of Eq. (10). To accomplish this, we utilize thescheme detailed in Refs. [42–45] to create two dressedexcited states coupled by spin-orbit interaction. How-ever, as the two dressed states are linear combinations ofbare excited states, the dressed states do not have goodquantum numbers to have clear selection rules to forbidthe transitions |g1〉 ←→ |e2〉 and |g2〉 ←→ |e1〉. Never-theless, by carefully choosing the driving fields to coupleground states to the bare excited states, we can createinterferences to generate synthetic selection rules. Byvarying the laser parameters as the circuit progresses, wecan engineer the disentangler in the interaction picture.

When the UV state is generated by the cMERA circuit,one can then use the experimental techniques introducedin Refs. [46–48] to measure the Chern number and theBerry curvature.

Discussion.—In this work, we found a quasi-localcMERA whose fixed-point wavefunction is a Chern in-sulator. This is a novel and unexpected way to rep-resent systems with chiral topological order. We alsodemonstrate that our quasi-local quantum circuit can berealized experimentally in a cold atom system, despitethe common intuition that a quantum circuit should bestrictly local to allow easier implementation.

In our realization, we only explored one possibilityto engineer spin-orbit coupling, but it may be possibleto engineer the interaction in other ways, such as usingmagnetic fields on a chip [49] or microwaves [50]. Otheralkaline-earth atoms could also provide promising experi-mental platforms. Although our experimental realizationtook place in the interaction picture, one could in prin-ciple use the Schrodinger picture for cMERA, where thelattice constant must continuously contract [51, 52].

It is also interesting that the Chern insulator groundstate is a fixed point of our cMERA with finite correla-tion length. This observation seems to contradict theusual intuition that the fixed point correlation lengthmust be zero or infinity, as the correlation length mustdecrease under rescaling of each strictly local RG step inreal space. However, since our cMERA involves contin-uous time evolution and quasi-local interactions, it haspotential to restore the original correlation length aftera finite time evolution. The no-go theorems in Refs. [23–26] are similarly circumvented by a cMERA construction.Our work suggests that quasi-local RG transformationsare a more powerful framework than strictly local RGtransformations. It also might shed light on some of thekey properties of MERA-like formalisms for a wide range

5

of chiral topological states. In the future, we hope toextend the methods of this Letter to fractional quantumHall states.

We are grateful to Bela Bauer, Yu-Ting Chen, Ze-Pei Cian, Ignacio Cirac, Glen Evenbly, Zhexuan Gong,Norbert Schuch, Brian Swingle, Tsz-Chun Tsui, Bray-den Ware, and Xueda Wen for helpful discussions. Thisproject is supported by the AFOSR, NSF QIS, ARLCDQI, ARO MURI, ARO, NSF PFC at JQI, and NSFIdeas Lab. S.K.C. partially completed this work duringhis participation in the long-term workshop “Entangle-ment in Quantum Systems” held at the Galileo Galilei In-stitute for Theoretical Physics as well as “Boulder School2018: Quantum Information,” which is supported by theNational Science Foundation and the University of Col-orado. He is also funded by the ACRI fellowship underthe Young Investigator Training Program 2017. G.Z. isalso supported by ARO-MURI, YIP-ONR and NSF CA-REER (DMR431753240). J.R.G. acknowledges supportfrom the NIST NRC Research Postdoctoral Associate-ship Award. Z.E. is supported in part by the ARCSFoundation. I.B.S. and A.V.C. acknowledge the addi-tional support of the AFOSR’s Quantum Matter MURIand NIST.

[1] R. Orus, “A practical introduction to tensor networks:Matrix product states and projected entangled pairstates,” Ann. Phys. (N.Y.) 349, 117–158 (2014).

[2] J. C. Bridgeman and C. T. Chubb, “Hand-waving andinterpretive dance: an introductory course on tensor net-works,” J. Phys. A 50, 223001 (2017).

[3] G. Vidal, “Class of Quantum Many-Body States ThatCan Be Efficiently Simulated,” Phys. Rev. Lett. 101,110501 (2008).

[4] F. Verstraete and J. I. Cirac, “Matrix product states rep-resent ground states faithfully,” Phys. Rev. B 73, 094423(2006).

[5] F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I.Cirac, “Criticality, the area law, and the computationalpower of projected entangled pair states,” Phys. Rev.Lett. 96, 220601 (2006).

[6] M. B. Hastings, “An area law for one-dimensional quan-tum systems,” J. Stat. Mech.: Theor. Exp. 2007, P08024(2007).

[7] G. Vidal, “Entanglement renormalization,” Phys. Rev.Lett. 99, 220405 (2007).

[8] M. M. Wolf, F. Verstraete, M. B. Hastings, and J. I.Cirac, “Area Laws in Quantum Systems: Mutual Infor-mation and Correlations,” Phys. Rev. Lett. 100, 070502(2008).

[9] K. G. Wilson, “The renormalization group and the ε ex-pansion,” Phys. Rep. 12, 75–199 (1974).

[10] K. G. Wilson, “The renormalization group: Critical phe-nomena and the Kondo problem,” Rev. Mod. Phys. 47,773–840 (1975).

[11] J. Zinn-Justin, Phase transitions and renormalizationgroup (Oxford, 2007).

[12] M. Aguado and G. Vidal, “Entanglement renormaliza-tion and topological order,” Phys. Rev. Lett. 100, 070404(2008).

[13] R. N. C. Pfeifer, G. Evenbly, and G. Vidal, “Entan-glement renormalization, scale invariance, and quantumcriticality,” Phys. Rev. A 79, 040301 (2009).

[14] R. Konig, B. W. Reichardt, and G. Vidal, “Exact en-tanglement renormalization for string-net models,” Phys.Rev. B 79, 195123 (2009).

[15] R. Konig and E. Bilgin, “Anyonic entanglement renor-malization,” Phys. Rev. B 82, 125118 (2010).

[16] S. Singh and G. Vidal, “Symmetry-protected entangle-ment renormalization,” Phys. Rev. B 88, 121108 (2013).

[17] G. Evenbly and S. R. White, “Entanglement Renormal-ization and Wavelets,” Phys. Rev. Lett. 116, 140403(2016).

[18] J. Haegeman, B. Swingle, M. Walter, J. Cotler, G. Even-bly, and V. B. Scholz, “Rigorous Free-Fermion Entan-glement Renormalization from Wavelet Theory,” Phys.Rev. X 8, 011003 (2018).

[19] T. H. Hansson, M. Hermanns, S. H. Simon, and S. F.Viefers, “Quantum Hall physics: Hierarchies and confor-mal field theory techniques,” Rev. Mod. Phys. 89, 025005(2017).

[20] X.-G. Wen, “Colloquium: Zoo of quantum-topologicalphases of matter,” Rev. Mod. Phys. 89, 041004 (2017).

[21] C. Nayak, S. H. Simon, A. Stern, M. Freedman, andS. Das Sarma, “Non-Abelian anyons and topologicalquantum computation,” Rev. Mod. Phys. 80, 1083–1159(2008).

[22] S. Bravyi, M. B. Hastings, and F. Verstraete, “Lieb-Robinson Bounds and the Generation of Correlationsand Topological Quantum Order,” Phys. Rev. Lett. 97,050401 (2006).

[23] T. Barthel, M. Kliesch, and J. Eisert, “Real-space renor-malization yields finite correlations,” Phys. Rev. Lett.105, 010502 (2010).

[24] J. Dubail and N. Read, “Tensor network trial states forchiral topological phases in two dimensions and a no-gotheorem in any dimension,” Phys. Rev. B 92, 205307(2015).

[25] T. B. Wahl, H. H. Tu, N. Schuch, and J. I. Cirac, “Pro-jected entangled-pair states can describe chiral topologi-cal states,” Phys. Rev. Lett. 111, 236805 (2013).

[26] Z. Li and R. S. K. Mong, “Entanglement renormalizationfor chiral topological phases,” arXiv:1703.00464 (2017).

[27] J. Haegeman, T. J. Osborne, H. Verschelde, andF. Verstraete, “Entanglement renormalization for quan-tum fields in real space,” Phys. Rev. Lett. 110, 100402(2013).

[28] Q. Hu and G. Vidal, “Spacetime Symmetries and Con-formal Data in the Continuous Multiscale EntanglementRenormalization Ansatz,” Phys. Rev. Lett. 119, 010603(2017).

[29] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quan-tum Spin Hall Effect and Topological Phase Transitionin HgTe Quantum Wells,” Science 314, 1757 (2006).

[30] X. Wen, G. Y. Cho, P. L.S. Lopes, Y. Gu, X. L. Qi, andS. Ryu, “Holographic entanglement renormalization oftopological insulators,” Phys. Rev. B 94, 075124 (2016).

[31] B. Swingle and J. McGreevy, “Renormalization groupconstructions of topological quantum liquids and be-yond,” Phys. Rev. B 93, 045127 (2016).

[32] Y. Huang and J. E. Moore, “Neural network rep-

6

resentation of tensor network and chiral states,”arXiv:1701.06246 (2017).

[33] R. Kaubruegger, L. Pastori, and J. C. Budich, “Chiraltopological phases from artificial neural networks,” Phys.Rev. B 97, 195136 (2018).

[34] I. Glasser, N. Pancotti, M. August, I. D. Rodriguez, andJ. I. Cirac, “Neural-Network Quantum States, String-Bond States, and Chiral Topological States,” Phys. Rev.X 8, 011006 (2018).

[35] T. B. Wahl, S. T. Haßler, H. H. Tu, J. I. Cirac, andN. Schuch, “Symmetries and boundary theories for chiralprojected entangled pair states,” Phys. Rev. B 90, 115133(2014).

[36] D. Poilblanc, J. I. Cirac, and N. Schuch, “Chiral topo-logical spin liquids with projected entangled pair states,”Phys. Rev. B 91, 224431 (2015).

[37] D. Poilblanc, N. Schuch, and I. Affleck, “SU(2)1 chiraledge modes of a critical spin liquid,” Phys. Rev. B 93,174414 (2016).

[38] M. P. Zaletel and R. S. K. Mong, “Exact matrix productstates for quantum Hall wave functions,” Phys. Rev. B86, 245305 (2012).

[39] P. Schmoll and R. Orus, “Kitaev honeycomb tensor net-works: Exact unitary circuits and applications,” Phys.Rev. B 95, 045112 (2017).

[40] M. Nozaki, S. Ryu, and T. Takayanagi, “Holographicgeometry of entanglement renormalization in quantumfield theories,” J. High Energy Phys. 2012, 193 (2012).

[41] In the cMERA literature, a momentum cutoff Λ is typi-cally provided [27, 30]. With a finite cutoff, the UV stategenerated by a cMERA circuit approximates the groundstate of the Hamiltonian up to O( 1

Λ) corrections. Here,

we work in the continuum limit Λ → ∞ to avoid thistechnical subtlety. In principle, these finite-Λ correctionscan be worked out explicitly.

[42] D. L. Campbell, G. Juzeliunas, and I. B. Spielman, “Re-alistic Rashba and Dresselhaus spin-orbit coupling forneutral atoms,” Phys. Rev. A 84, 025602 (2011).

[43] D. L. Campbell and I. B. Spielman, “Rashba realization:Raman with RF,” New J. Phys. 18, 033035 (2016).

[44] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang,L. Chen, D. Li, Q. Zhou, and J. Zhang, “Experimen-tal realization of a two-dimensional synthetic spin-orbitcoupling in ultracold Fermi gases,” Nat. Phys. 12, 540(2016).

[45] V. Galitski and I. B. Spielman, “Spin-orbit coupling inquantum gases,” Nature (London) 494, 49–54 (2013).

[46] P. Hauke, M. Lewenstein, and A. Eckardt, “Tomographyof Band Insulators from Quench Dynamics,” Phys. Rev.Lett. 113, 045303 (2014).

[47] E. Alba, X. Fernandez-Gonzalvo, J. Mur-Petit, J. K. Pa-chos, and J. J. Garcia-Ripoll, “Seeing Topological Orderin Time-of-Flight Measurements,” Phys. Rev. Lett. 107,235301 (2011).

[48] N. Flaschner, B. S. Rem, M. Tarnowski, D. Vogel, D.-S.Luhmann, K. Sengstock, and C. Weitenberg, “Experi-mental reconstruction of the Berry curvature in a FloquetBloch band,” Science 352, 1091–1094 (2016).

[49] B. M. Anderson, I. B. Spielman, and G. Juzeliunas,“Magnetically generated spin-orbit coupling for ultracoldatoms,” Phys. Rev. Lett. 111, 125301 (2013).

[50] F. Grusdt, T. Li, I. Bloch, and E. Demler, “Tunable spin-orbit coupling for ultracold atoms in two-dimensional op-tical lattices,” Phys. Rev. A 95, 063617 (2017).

[51] J. H. Huckans, I. B. Spielman, B. L. Tolra, W. D. Phillips,and J. V. Porto, “Quantum and classical dynamics of aBose-Einstein condensate in a large-period optical lat-tice,” Phys. Rev. A 80, 043609 (2009).

[52] S. Al-Assam, R. A. Williams, and C. J. Foot, “Ultracoldatoms in an optical lattice with dynamically variable pe-riodicity,” Phys. Rev. A 82, 021604 (2010).

[53] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau,P. Schauß, S. Hild, D. Bellem, U. Schollwock, T. Gia-marchi, C. Gross, I. Bloch, and S. Kuhr, “Quantumdynamics of a mobile spin impurity,” Nat. Phys. 9, 235EP – (2013).

[54] F. Nogrette, H. Labuhn, S. Ravets, D. Barredo,L. Beguin, A. Vernier, T. Lahaye, and A. Browaeys,“Single-atom trapping in holographic 2d arrays of micro-traps with arbitrary geometries,” Phys. Rev. X 4, 021034(2014).

[55] X. Zhang, M. Zhou, N. Chen, Q. Gao, C. Han, Y. Yao,P. Xu, S. Li, Y. Xu, Y. Jiang, Z. Bi, L. Ma, and X. Xu,“Study on the clock-transition spectrum of cold 171 Ybytterbium atoms,” Laser Phys. Lett. 12, 25501 (2015).

[56] T. Kohno, M. Yasuda, K. Hosaka, H. Inaba, Y. Naka-jima, and F.-L. Hong, “One-Dimensional Optical LatticeClock with a Fermionic 171 Yb Isotope,” Appl. Phys. Ex-press 2, 072501 (2009).

[57] N. D. Lemke, A. D. Ludlow, Z. W. Barber, T. M. Fortier,S. A. Diddams, Y. Jiang, S. R. Jefferts, T. P. Heavner,T. E. Parker, and C. W. Oates, “Spin-1/2 optical latticeclock,” Phys. Rev. Lett. 103, 063001 (2009).

[58] C. Y. Park, D.-H. Yu, W.-K. Lee, S. E. Park, E. B. Kim,S. K. Lee, J. W. Cho, T. H. Yoon, J. Mun, S. J. Park,T. Y. Kwon, and S.-B. Lee, “Absolute frequency mea-surement of transition of 1S0(F=1/2)-3P0(F=1/2) 171Ybatoms in a one-dimensional optical lattice at KRISS,”Metrologia 50, 119–128 (2013).

[59] A. Yamaguchi, Metastable State of Ultracold and Quan-tum Degenerate Ytterbium Atoms: High-Resolution Spec-troscopy and Cold Collisions, Ph.D. thesis, Kyoto Univer-sity (2008).

[60] U. Busch and K. A. Penson, “Tight-binding electronson open chains: Density distribution and correlations,”Phys. Rev. B 36, 9271–9274 (1987).

7

Supplemental Material

In this Supplemental Material, we provide details on the experimental realization. In Section I, we show how toengineer a synthetic selection rule between dressed states in the absence of any good quantum number. With thattechnique in mind, we show a scheme to realize the cMERA circuit in Section II. After that, in Section III, we provideone way to prepare the initial state for the cMERA circuit by using spatial light modulators [53, 54].

I. SYNTHETIC SELECTION RULES

In this section, we introduce a trick that will be useful for engineering the disentangler in a real atomic system.Suppose that we have a three-level system composed of states |s1〉, |s2〉, and |g〉. In the presence of an on-resonancedriving with Rabi frequency Ω between bare states |s1〉 and |s2〉, two dressed states |d1〉 and |d2〉 are formed. We aregoing to show that by fine-tuning the Rabi frequencies χ1 and χ2, we can generate a synthetic selection rule fromstate |g〉 to the two dressed states |d1〉 and |d2〉, e.g., |g〉 → |d2〉 is allowed while |g〉 → |d1〉 is forbidden. (Once weprove this, the converse case where |g〉 → |d1〉 is allowed and |g〉 → |d2〉 is forbidden is a trivial generalization.) Weconsider a driving Hamiltonian, which under rotating wave approximation is

h =

0 χ∗1ei(ω1−Ω+δ)t χ∗2e

i(ω2−Ω+δ)t

χ1e−i(ω1−Ω+δ)t ω1 Ωe−i(ω1−ω2)t

χ2e−i(ω2−Ω+δ)t Ωei(ω1−ω2)t ω2

. (S1)

The order of the columns (rows) is |g〉, |s1〉, |s2〉. We have assumed that |ω1 − ω2| Ω, allowing us to neglect sometransitions that are far off-resonant. The level diagram is illustrated in FIG. S1.

Going to the rotating frame defined by the unitary matrix

U =

1 0 00 e−i(ω1−ω2)t 00 0 1

, (S2)

we obtain the effective Hamiltonian

U†hU − i∂tU†U =

0 χ∗1ei(ω2−Ω+δ)t χ∗2e

i(ω2−Ω+δ)t

χ1e−i(ω2−Ω+δ)t ω2 Ω

χ2e−i(ω2−Ω+δ)t Ω ω2

. (S3)

After diagonalizing the 2× 2 block on the bottom right, we obtain the following Hamiltonian: 0 1√2(χ∗1 + χ∗2)ei(ω2−Ω+δ)t 1√

2(χ∗1 − χ∗2)ei(ω2−Ω+δ)t

1√2(χ1 + χ2)e−i(ω2−Ω+δ)t ω2 + Ω 0

1√2(χ1 − χ2)e−i(ω2−Ω+δ)t 0 ω2 − Ω

. (S4)

We denote the dressed state with energy ω2 + Ω as |d1〉, and the dressed state with energy ω2 −Ω as |d2〉. We cansee that if we fine-tune χ1 = −χ2, we synthesize a selection rule where only the transition between |d2〉 and |g〉 isallowed. The synthetic Rabi frequency is then

√2χ1.

This synthetic selection rule can be understood by considering two separate rotating frames with respect to states|s1〉 and |s2〉, as shown in FIG. S1. In each rotating frame, we have dressed states |d1〉 and |d2〉. We can couple |g〉to dressed states either by driving |g〉 to dressed states in the |s1〉 rotating frame or in the |s2〉 rotating frame. Bycreating interference between the two channels, we obtain a synthetic selection rule.

II. THE CONTINUOUS MERA CIRCUIT ENGINEERING

In this section, we show that by using the scheme shown in FIG. S2(b), we can engineer the disentangler in theinteraction picture. Here, we choose the two hyperfine ground states of 171Yb shown in FIG. S3 as our spinor basisof the Chern insulator and effectively couple them to some dressed excited states by two pairs of driving fields. Themeaning of “dressed” excited states will become clear shortly. Additionally, the dressed excited states are coupled byspin-orbit interaction, while transitions |g1,k〉 ←→ |e2,k〉 and |g2,k〉 ←→ |e1,k〉 are forbidden. In order to implement

8

|s1i

|s2i

|gi

|d2i

|d1i

|d2i

|d1i

1

2

FIG. S1. A toy model of synthetic selection rules. Bare states |s1〉 and |s2〉 are driven by a field with Rabi frequency Ω, wherebytwo dressed states |d1〉 and |d2〉 are created. In view of the rotating frame, the dressed states are linear combinations of barestates. As a result, they do not have good quantum numbers to constitute a selection rule when coupling to another state, say|g〉. A synthetic selection rule can be generated through applying two driving fields from |g〉 to |s1〉 and |s2〉 with fine-tunedRabi frequencies χ1 and χ2, respectively. For example, we can forbid the transition from |g〉 to |d1〉 by choosing χ1 = −χ2.

(a)

-1/2 1/2-3/2 3/2mF =

F=1/21S0

F=5/2

F=3/23P2

-5/2 5/2

3P0 F=1/2

(b)

Δ1Δ2

|g1i

|g2i

|e1i|e2i

Ω1

Δ’1Δ’2

Ω2

Ω’1

Ω’2

spin-orbit interaction

FIG. S2. Disentangler engineering. (a) A magnetic field is applied to induce hyperfine splittings. The excited states are coupledby Raman beams (colored in blue) to generate an effective spin-orbit interaction. They are chosen from the hyperfine manifolds3P2 F = 5/2 and 3P0 F = 1/2, which are long-lived to circumvent dissipation issues. Their ultra-narrow linewidths are onthe order of tens of millihertz [55–59]. Additionally, we also have two sets of multiple lasers, colored in light and dark pink,coupling the ground states to the excited states to engineer the disentangler of our cMERA by creating synthetic selectionrules. (b) The effective couplings between ground states and the dressed excited states are generated from the scheme shownin (a). We ignore a third dressed state since it is far off-resonant. Now we effectively create two dressed excited states coupledby spin-orbit interaction, which are coupled to the ground states by two pairs of drivings colored in light and dark pink. Thesynthetic selection rules forbid |g1,k〉 ←→ |e2,k〉 and |g2,k〉 ←→ |e1,k〉. The effective Rabi frequencies and detunings for twopairs of effective drivings are labeled by unprimed and primed notation. The band structures are ignored in this picture, soby detunings we mean the detunings at k = 0. The light and dark purple arrows on the bottom right in (a) and (b) bothrepresent lasers used to cancel unwanted AC Stark shifts by coupling the ground states to some negative curvature bands ofsome excited state, e.g., an unused excited state in the 3P2 F = 5/2 hyperfine manifold.

9

F=3/2

F=1/2

F=1/2

F=1/2

-1/2 1/2-3/2 3/2

6s2 1S0

6s6p 3P0

6s6p 3P1

mF =

F=5/2

F=3/26s6p 3P2

-5/2 5/2

FIG. S3. Energy level diagram of neutral atom 171Yb. The hyperfine structure is shown. We employ the bottom two groundstates as our spinor basis of the Chern insulator.

this idea in neutral 171Yb atoms, we need to use techniques introduced in Refs. [42, 43] and Section I. To create statescoupled by spin-orbit coupling, we will utilize the method discussed in Refs. [42, 43]. However, the dressed statescreated by that scheme do not have good quantum numbers to enforce selection rules. Therefore, we use the techniqueoutlined in Section I to create a synthetic selection rule. In this part of the Supplemental Material, we show how tocombine those techniques consistently in neutral 171Yb.

First, we show how FIG. S2(b) arises from FIG. S2(a), inducing the disentangler interaction. We first consider thecase with the set of lasers colored in dark pink in FIG. S2(a) with additional Raman lasers coupling the bare excitedstates. This will give rise to the effective unprimed pair of drivings in FIG. S2(b). We will find that this schemegenerates one term in our disentangler with H (k) described by Eq. (10) in the main text. Therefore, to produceanother term, we will use another set of lasers with different parameters, which will effectively induce the primed pairof drivings in FIG. S2(b).

We assume that states |g1〉 and |g2〉 have flat bands, whereas the chosen bare excited states are weakly trapped. Inthe continuum, low-energy limit, atoms in the bare excited states can be described by non-relativistic particles withmass M . After appropriate Raman transitions for the bare excited states, we obtain the effective Hamiltonian in therotating frame of the basis |g1〉, |g2〉, |ebare,1〉, |ebare,2〉, and |ebare,3〉 under the rotating wave approximation:

h =

0 0 χ∗1,1e

i∆t χ∗1,2ei∆t χ∗1,3e

i∆t

0 0 χ∗2,1ei∆t χ∗2,2e

i∆t χ∗2,3ei∆t

χ1,1e−i∆t χ2,1e

−i∆t (k+k1)2

2M Ωeiφ1,2 Ωe−iφ3,1

χ1,2e−i∆t χ2,2e

−i∆t Ωe−iφ1,2 (k+k2)2

2M Ωeiφ2,3

χ1,3e−i∆t χ2,3e

−i∆t Ωeiφ3,1 Ωe−iφ2,3 (k+k3)2

2M

. (S5)

The order of the columns is |g1,k〉, |g2,k〉 |ebare,1,k + k1〉, |ebare,2,k + k2〉, and |ebare,3,k + k3〉. The notation ∆ isthe common detuning of all the lasers coupling the two ground states to the excited states, whereas χi,j representsthe Rabi frequencies of those lasers. We define the detuning at the zero momentum energy of the bare excitedstate. Here, k1, k2, and k3 are subject to the condition |k1| = |k2| = |k3| = kSOC, k1 + k2 + k3 = 0, andkj = kSOC[cos(2πj/3)ex + sin(2πj/3)ey].

We apply the following unitaries to conjugate the single body Hamiltonian

U =

1 0 0 0 00 1 0 0 0

0 0 e−i2π/3/√

3 e−i4π/3/√

3 1/√

3

0 0 e−i4π/3/√

3 e−i8π/3/√

3 1/√

3

0 0 s1/√

3 1/√

3 1/√

3

, (S6)

10

U ′ =

1 0 0 0 00 1 0 0 00 0 ei(φ1,2+φ2,3+φ3,1)/3 0 00 0 0 ei(−φ1,2+2φ2,3+2φ3,1)/3 00 0 0 0 eiφ3,1

, (S7)

and assume the following to obtain a synthetic selection rule:

χ1,2 = e2πi/3e−i (2φ1,2−φ2,3−φ3,1)/3χ1,1

χ1,3 = e−2πi/3e−i (φ1,2+φ2,3−2φ3,1)/3χ1,1

χ2,1 = e2πi/3ei (2φ1,2−φ2,3−φ3,1)/3χ2,2

χ2,3 = e−2πi/3ei (φ1,2−2φ2,3+φ3,1)/3χ2,2. (S8)

The Hamiltonian becomes

(U ′U)†hU ′U =

0 0 Ω∗1e

i∆t 0 00 0 0 Ω∗2e

i∆t 0

Ω1e−i∆t 0

k2+k2SOC

2M + 2Ω cos( 2π3 − φ) kSOC

M (kx − iky) kSOC

M (kx + iky)

0 Ω2e−i∆t kSOC

M (kx + iky)k2+k2SOC

2M + 2Ω cos( 4π3 − φ) kSOC

M (kx − iky)

0 0 kSOC

M (kx − iky) kSOC

M (kx + iky)k2+k2SOC

2M + 2Ω cos(φ)

,

(S9)where Ω1 ≡ −

√3e−iπ/3e−i(φ1,2+φ2,3+φ3,1)/3χ1,1, Ω2 ≡ −

√3e−iπ/3ei(φ1,2−2φ2,3−2φ3,1)/3χ2,2, and φ ≡ (φ1,2 + φ2,3 +

φ3,1)/3. The order of the columns is |g1,k〉, |g2,k〉, |e1,k〉, |e2,k〉, and |e3,k〉. States |e1,k〉, |e2,k〉, |e3,k〉 aredressed excited states which are linear combinations of the bare excited states |ebare,1,k + k1〉, |ebare,2,k + k2〉, and|ebare,3,k + k3〉. By adiabatically eliminating the dressed excited state representing the third column (row) to thezeroth order and expanding φ to the first order, we obtain the effective Hamiltonian

0 0 Ω∗1ei∆t 0

0 0 0 Ω∗2ei∆t

Ω1e−i∆t 0 k2

2M + ESOC +√

3Ωφ kSOC

M (kx − iky)

0 Ω2e−i∆t kSOC

M (kx + iky) k2

2M + ESOC −√

3Ωφ

, (S10)

where ESOC ≡ k2SOC/2M − Ω. The order of the columns is |g1,k〉, |g2,k〉, |e1,k〉, and |e2,k〉. By inspecting the

matrix elements, one can see that a spin-orbit interaction and a synthetic selection rule shown in FIG. S2(b) havebeen consistently generated as we claimed.

Now, we are going to show that with this Hamiltonian, we can almost generate the disentangler. First, we go to aframe in which |e1,k〉 and |e2,k〉 rotate with frequency ∆. The Hamiltonian becomes

0 0 Ω∗1 00 0 0 Ω∗2

Ω1 0 k2

2M + ESOC −∆ +√

3Ωφ kSOC

M (kx − iky)

0 Ω2kSOC

M (kx + iky) k2

2M + ESOC −∆−√

3Ωφ

. (S11)

For the sake of later convenience, we denote ∆1 ≡ ESOC −∆ +√

3Ωφ and ∆2 ≡ ESOC −∆−√

3Ωφ:0 0 Ω∗1 00 0 0 Ω∗2

Ω1 0 ∆1 + k2/2M kSOC

M (kx − iky)

0 Ω2kSOC

M (kx + iky) ∆2 + k2/2M

. (S12)

We can see that ∆1 and ∆2 correspond to the effective detunings at k = 0. Define α = kSOC/M and k, θk such thatk cos θk = kx and k sin θk = ky to simplify our calculations. Notice that we have chosen a different sign conventionof the detunings ∆1 and ∆2 from the normal convention. We will assume that ∆1,∆2 > 0 in our system so that theeffective drivings are red-detuned. Now we conjugate the Hamiltonian with the following unitary matrix:

1 0 0 00 1 0 0

0 0 1− α2k2

2(∆1−∆2)2 −αke−iθk∆1−∆2

0 0 αkeiθk∆1−∆2

1− α2k2

2(∆1−∆2)2

+O

((αk

∆1 −∆2

)3), (S13)

11

and the effective Hamiltonian to order(

αk∆1−∆2

)3

becomes0 0 Ω∗1

(1− α2k2

2(∆1−∆2)2

)−Ω∗1αke

−iθk

∆1−∆2

0 0Ω∗2αke

iθk

∆1−∆2Ω∗2

(1− α2k2

2(∆1−∆2)2

)Ω1

(1− α2k2

2(∆1−∆2)2

)Ω2αke

−iθk

∆1−∆2∆1 + α2k2

∆1−∆2+ k2/2M 0

−Ω1αkeiθk

∆1−∆2Ω2

(1− α2k2

2(∆1−∆2)2

)0 ∆2 − α2k2

∆1−∆2+ k2/2M

. (S14)

If we assume that M k2SOC

∆1−∆2, we can ignore the terms α2k2

∆1−∆2in the (3, 3) and (4, 4) entries. Now, we also drop

O

((αk

∆1−∆2

)2)

terms in the (1, 3), (2, 4), (3, 1), and (4, 2) entries. The remaining Hamiltonian is

0 0 Ω∗1 −Ω∗1αke

−iθk

∆1−∆2

0 0Ω∗2αke

iθk

∆1−∆2Ω∗2

Ω1Ω2αke

−iθk

∆1−∆2∆1 + k2/2M 0

−Ω1αkeiθk

∆1−∆2Ω2 0 ∆2 + k2/2M

. (S15)

We adiabatically eliminate the state in the first and second columns (rows). The remaining Hamiltonian of thesubspace spanned by dressed states |g1,k〉, and |g2,k〉 is − |Ω1|2

∆1+k2/2M −|Ω1|2

∆2+k2/2M

(αk

∆1−∆2

)2αke−iθkΩ∗1Ω2

(∆1−∆2)(∆1+k2/2M) −αke−iθkΩ∗1Ω2

(∆1−∆2)(∆2+k2/2M)

αkeiθkΩ1Ω∗2(∆1−∆2)(∆1+k2/2M) −

αkeiθkΩ1Ω∗2(∆1−∆2)(∆2+k2/2M) − |Ω2|2

∆1+k2/2M

(αk

∆1−∆2

)2

− |Ω2|2∆2+k2/2M

. (S16)

We have assumed ∆1, ∆2, Ω1, Ω2. A necessary condition of this assumption is that Ω Ω1, Ω2. Now, supposingthat we can tune ∆1 ∆2, and that the region of the Brillouin zone we consider satisfies ∆1 k2/2M , by droppingterms to quadratic order in αk

∆1−∆2, we obtain the Hamiltonian 0 − αke−iθkΩ∗1Ω2

∆1(∆2+k2/2M)

− αkeiθkΩ1Ω∗2∆1(∆2+k2/2M) − |Ω2|2

∆2+k2/2M

. (S17)

To make this approximation, we have assumed that the off-diagonal elements of Eq. (S17) are much greater than theterms in Eq. (S16) being dropped in Eq. (S17). There is a mismatch between the diagonal elements. To make states|g1,k〉 and |g2,k〉 rotate at the same speed, we might either couple the state |g1,k〉 to a band with positive curvatureto induce an AC Stark shift to cancel the first diagonal entry or couple the state |g2,k〉 to some band with negativecurvature to induce an AC Stark shift to cancel the second diagonal entry. The curvatures of those auxiliary bandshave to be tuned properly during the whole process.

Now, we have engineered one term in our disentangler with H (k) described by Eq. (10). We can choose a differentΩ′1, Ω′2, ∆′1, ∆′2 to generate the second term. We have to assume that the beat note between the two schemes satisfies

|∆2 −∆′2| ∣∣∣ αke−iθkΩ∗1Ω2

∆1(∆2+k2/2M)

∣∣∣ , ∣∣∣ αke−iθkΩ′∗1Ω′2∆′1(∆′2+k2/2M)

∣∣∣ to avoid crosstalk. Applying both of them at the same time, we have

the Hamiltonian in the |g1,k〉, |g2,k〉 basis: 0 − αke−iθkΩ∗1Ω2

∆1(∆2+k2/2M) −αke−iθkΩ′∗1Ω′2

∆′1(∆′2+k2/2M)

− αkeiθkΩ1Ω∗2∆1(∆2+k2/2M) −

αkeiθkΩ′1Ω′∗2∆′1(∆′2+k2/2M) 0

. (S18)

Now we list all the assumptions that have been made:

1. The energy splittings of the dressed excited states, which are of order Ω, have to be much smaller than thehyperfine splittings of all the states that we used. Otherwise, in FIG. S2(a), we cannot use frequency selectionto control each transition to engineer synthetic selection rules.

2. All the momentum kicks should allow atoms to be in the same Brillouin zone so that the continuum limit applies.That is, kSOC a 1, where a is the optical lattice constant.

12

3. αk∆1−∆2

= kSOCkM(∆1−∆2) 1 and

k2SOC

M(∆1−∆2) 1 as well as the primed version.

4. ∆1 ∆2, k2/2M as well as the primed version.

5. ∆1, ∆2 Ω1, Ω2 and ∆′1, ∆′2 Ω′1, Ω′2. These two conditions imply that Ω Ω1, Ω2, Ω′1, Ω′2.

6. The off-diagonal elements of Eq. (S17) are much greater than the terms in Eq. (S16) being dropped in Eq. (S17).

7. |∆2 −∆′2| ∣∣∣ αke−iθkΩ∗1Ω2

∆1(∆2+k2/2M)

∣∣∣ , ∣∣∣ αke−iθkΩ′∗1Ω′2∆′1(∆′2+k2/2M)

∣∣∣ to avoid crosstalk between the scheme determined by Ω1, Ω2,

∆1, ∆2 and the scheme determined by Ω′1, Ω′2, ∆′1, ∆′2.

We remind the readers that we engineer the cMERA circuit entirely in the interaction picture; therefore, the actionof the isometry is absorbed into that of the disentangler. The price that we have to pay is that the disentangler isnot scale-invariant at all in the interaction picture. In principle, one can also engineer the cMERA circuit in theSchrodinger picture. We leave this as a question for future research.

III. PREPARATION OF THE INITIAL NEAR-IR STATE

The near-IR state with a large but finite negative u is described by Eq. (9). We imagine the state to be infraredenough that the Berry curvature is concentrated on a few momentum points near k = 0. Here, we describe how itcan be created to use as input to the MERA circuit. A strong magnetic field should be applied to induce hyperfinesplitting in the ground-state manifold. We start with all states in the |g1〉 state, which is easy to prepare by dissipationtechniques. We then use a long-lived clock state 3P0 |F = 1/2, mF = 1/2〉 [55–58] as a “bus” state |e〉 to transferamplitude from |g1〉 to |g2〉. Seeing that S states and P states are well separated, we can use a two-dimensionaloptical lattice to tightly trap atoms in the S states and let the atoms in the P states propagate nearly freely. Weassume that the z direction is tightly confined for all states, so the corresponding degrees of freedom can be ignored.The energy bands of |g1〉 and |g2〉 are flat. Here, we assume that the |e〉 state is highly stable with a natural linewidthmuch smaller than the energy splitting between the spatial ground state and the first spatial excited state, allowingindividual momentum states to be resolved and manipulated.

In the following, we are going to use the spatial ground state of |e〉 as a bus state. Due to open boundary conditionsof optical lattices, the Bloch waves are no longer energy eigenstates for the excited state |e〉 and we must use standingwaves instead. Note that since the eigenstates in position space of the hyperfine ground states |g1〉 and |g2〉 are tightlytrapped and highly degenerate, we can still make superpositions of standing waves to create Bloch waves as energyeigenstates. Intuitively, since particles in the hyperfine ground states |g1〉 and |g2〉 are tightly trapped, particles farfrom the boundary cannot distinguish between different boundary conditions. Our procedure to prepare the IR stateis to transfer partial amplitude from state |g1〉 to |g2〉 in the Brillouin zone for each k. We denote the lowest energypoint of |e〉 as |e, 0〉, which is a standing wave with small amplitude on the boundary. We couple that state resonantlyto |g1,k〉 and |g2,k〉 successively by different light fields, i.e., |g1,k〉 ←→ |e, 0〉 and then |e, 0〉 ←→ |g2,k〉. Otherstanding waves of |e〉 are decoupled from the process due to driving frequency mismatch. Here, we also need to ensurethat other states |g1,k

′〉 and |g2,k′〉 with different momenta do not interfere with the process. As a consequence, the

light fields must create a momentum selection rule for the transitions |g1,k〉 ←→ |e, 0〉 and |e, 0〉 ←→ |g2,k〉.We imagine a square well with wavefunction amplitude vanishing on the periphery. This can be done by tuning

the potential with spatial light modulators [53, 54]. In the following, we work in the basis of the Wannier functionsof the ground states and the excited state, modeling the system by a N + 2 by N + 2 square lattice. We can label thelattice points by the vector x = (x1, x2), where 0 ≤ x1, x2 ≤ N + 1, while the wavefunction vanishes at points withx1 = 0, N + 1 or x2 = 0, N + 1. Therefore, the active degrees of freedom for the hyperfine ground states |g1〉 and|g2〉 will be at 1 ≤ x1, x2 ≤ N . In this case, the unnormalized single-particle wavefunction of the ground state |g1,k〉is [60]

ψg1(x) = 〈x|g1,k〉 = exp (ik · x) , (S19)

where k = 2π (n1, n2) /N with n1, n2 ∈ n | n ∈ Z,−N/2 < n ≤ N/2, and 1 ≤ x1, x2 ≤ N . The counterpart for theexcited state |e〉 is

ψe(x) = 〈x|e, 0〉 = sin

(π

N + 1x1

)sin

(π

N + 1x2

). (S20)

13

Using spatial light modulators [53, 54], we create the following light field:

Eg1(x) =exp (−iq · x)

sin(

πN+1x1

)sin(

πN+1x2

) , (S21)

where q = 2π (m1, m2) /N, m1,m2 ∈ Z. A momentum selection rule for |g1,k〉 ←→ |e, 0〉 can now be engineered:∑x

ψe(x)Eg1(x)ψg1(x) =∑x

sin

(π

N + 1x1

)sin

(π

N + 1x2

)exp (−iq · x)

sin(

πN+1x

)sin(

πN+1x2

) exp (ik · x)

=∑x

exp (i (k− q) · x) ∝ δk,q. (S22)

Notice that since the points where the denominator of E(x) becomes zero are excluded from our consideration, thelight field is well defined. A similar selection rule can be derived for |e, 0〉 ←→ |g2,k〉.

With this technique in mind, we can adjust the relative amplitude between |g1〉 and |g2〉 in the Brillouin zone tocreate the near-IR state described in Eq. (9) by fine-tuning phases and durations of the light field pulses. Given thatthe Berry curvature is concentrated on a few momentum points near k = 0, we can limit this procedure to only a fewsmall momentum points without too much error.