PHYSICAL REVIEW LETTERS 122, 120502 (2019)

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

2

Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
Joint Center for Quantum Information and Computer Science, NIST/University of Maryland,College Park, Maryland 20742, USA(Received 14 September 2018; published 27 March 2019)

The multiscale entanglement renormalization ansatz (MERA) postulates the existence of quantum
circuits that renormalize entanglement in real space at different length scales. Chern insulators, however,cannot have scale-invariant discrete MERA circuits with a finite bond dimension. In this Letter, we showthat the continuous MERA (cMERA), a modified version of MERA adapted for field theories, possesses a
fixed point wave function with a nonzero Chern number. Additionally, it is well known that reversed
MERA circuits can be used to prepare quantum states efficiently in time that scales logarithmically with thesize of the system. However, state preparation via MERA typically requires the advent of a full-fledged
universal quantum computer. In this Letter, we demonstrate that our cMERA circuit can potentially berealized in existing analog quantum computers, i.e., an ultracold atomic Fermi gas in an optical lattice withlight-induced spin-orbit coupling.
DOI: 10.1103/PhysRevLett.122.120502

A quantum many-body system has a Hilbert space
whose dimension grows exponentially with system size,
making exact diagonalization of its Hamiltonian impractical. Fortunately, tensor networks [1,2] are capable ofefficiently representing the ground states of many systems 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 aphysical system can vary at different length scales,
forming a hierarchical structure. The RG provides asystematic prescription to transform an exact microscopic
description to an effective coarse-grained description.
Applications of RG range from critical phenomena incondensed matter to the electroweak interaction in highenergy physics [11].
One approach, which combines tensor networks and therenormalization group, is called the multiscale entanglement renormalization ansatz (MERA) [3,7]. MERA proposes a quantum circuit acting on a state that is initiallyentangled at many length scales. The two elementarybuilding-block tensors of the MERA, isometries anddisentanglers, are discrete unitary gates that physically
implement RG in real space by successively removing
entanglement at progressively larger length scales.
Interestingly, since the circuit depth only increases logarithmically with the system size, a reversed MERA circuit
can efficiently prepare a state with finer entanglement
structure 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 thermodynamic limit by
0031-9007=19=122(12)=120502(7)these scale-invariant unitaries is termed a fixed-point
wave function.
Experimentally, a reversed MERA circuit might be used
to prepare exotic states, such as chiral topological states,
which include integer quantum Hall states and certain
fractional quantum Hall states [19,20]. Some fractional
quantum Hall systems are believed to feature anyons usefulfor topological quantum computation [21]. Due to theirgreat theoretical interest, it would be useful to be able tostudy these systems under highly controlled settings, suchas in ultracold atomic gases. However, to create a chiral
topological state in the lab, we must not only engineer theparent Hamiltonian, but also cool the system down to theground state. The latter is usually hard experimentally for
topological states due to their long-range entanglement[22]. A reversed MERA circuit can possibly resolve thisissue by directly generating the target chiral topologicalstate from another state that is easier to obtain by cooling.
Here, as a first step towards finding a MERA for afractional quantum Hall state, we instead search for a
MERA whose fixed-point wave function describes an(integer) Chern insulator. A Chern insulator is an integer
quantum Hall state on a lattice and is therefore a simpler
system than the fractional quantum Hall state. However,there are no-go theorems stating that a MERA cannot havea Chern insulator ground state as its fixed-point wave
function [23–26]. Since conventional MERA only contains
strictly local interactions, adding quasilocal interactions
with quickly decaying tails could possibly circumvent the
no-go theorems. A modified formalism of MERA adapted
for field theories called continuous MERA (cMERA) [27]120502-1© 2019 American Physical Society

PHYSICAL REVIEW LETTERS 122, 120502 (2019)
can include such quasilocal interactions [28]. The distinction between strictly local and quasilocal interactions is thatthe interaction range of the former is finite, while the latterare a broader class that includes interactions decaying fasterthan any power law with respect to distance, e.g., exponentially decaying interactions. In contrast to the MERA
paradigm, in which the renormalization circuit consistsof discrete unitary gates, cMERA treats the circuit time,
which corresponds to the length scale, as a continuous
variable and generates continuous entanglement renormalization using a Hermitian Hamiltonian.
In this Letter, we show that a type of Chern insulator
wave function can be generated by a scale-invariant
cMERA circuit. The Chern insulator model we consider
is the Bernevig-Hughes-Zhang model in the continuum
limit [29]. In addition, we propose a possible experimental
realization of the cMERA circuit with neutral 171Yb atoms
in an optical lattice by introducing spin-orbit coupling.
Our work complements and can be contrasted with
Refs. [30,31]. While Ref. [30] previously developed acMERA for the continuous Chern insulator model mentioned above, our work uses a scale-invariant disentangler.
Other prior work in Ref. [31] presented a scale-invariant
entanglement renormalization for a two-band Chern insulator model. While Ref. [31] makes use of the lattice
structure and quasiadiabatic paths between a series ofgapped Hamiltonians, our cMERA approach allows
smooth time evolution and emphasizes the continuum
physics. Another difference is that the RG evolution in
Ref. [31] involves interactions decaying with distancefaster than any power-law function but slower than an
exponential, whereas our cMERA only needs an exponentially decaying interaction. Other known methods forrepresenting chiral topological states include artificial
neural network quantum states [32–34], projectedentangled pair states [25,35–37], matrix product states[38], and polynomial-depth unitary circuits [39].
Review of cMERA.—Within the framework of conventional MERA [3,7], disentanglers V u and isometries W u
are strictly local discrete unitary operators employed torenormalize entanglement at layer u ∈ Zþ. In cMERA[27], we simply replace them by continuous unitary transformations, which are infinitesimally generated by selfadjoint operators KðuÞ and L: V u → e−iKðuÞdu , W u →e−iLdu . The notation du denotes an infinitesimal RG step,and u ∈ ð−∞; 0. When the continuous variable u
approaches zero, the system is said to be at the ultraviolet(UV) length scale, possessing both short-range and longrange entanglement. As u → −∞, the system flows to theinfrared (IR) length scale, where short-range entanglement
is removed and nearly all degrees of freedom are disentangled from each other. Note that the generator of
disentangler KðuÞ can in general depend on scale u. A
cMERA is called scale invariant if KðuÞ is independentof u.

To emulate the coarse-graining behavior of isometries inconventional lattice MERA, L is chosen to be the scalingtransformation in field theory. For example, for a single
fermionR field ψðxÞ in d spatial dimensions, we pick L ¼−ði=2Þ ½ψ † ðxÞx · ∇ψðxÞ − x · ∇ψ † ðxÞψðxÞdd x [27,30];
thereby, fermionic operators ψðxÞ in real space and ψðkÞ inmomentum space satisfy the following scaling transformations: e−iuL ψðxÞeiuL ¼ eðd=2Þu ψðeu xÞ, e−iuL ψðkÞeiuL ¼e−ðd=2Þu ψðe−u kÞ. One can check that the anticommutation
relations fψðxÞ; ψ † ðx0 Þg ¼ δðx − x0 Þ in real space andfψðkÞ; ψ † ðk0 Þg ¼ δðk − k0 Þ in momentum space are preserved under the scaling transformation. We will sometimes
abuse the terminology to call KðuÞ and L themselves thedisentangler and the isometry rather than the verbose
generator of disentangler and generator of isometry.
The renormalized wave function is governed by the
Schrödinger equation,i∂jΨS ðuÞi ¼ ½KðuÞ þ LjΨS ðuÞi;∂u

ð1Þ

where the superscript S denotes the Schrödinger picture. In
this Letter, we will focus on the interaction picture, which
provides a more convenient way to look at continuous
entanglement renormalization. We treat L as a “free”
Hamiltonian and KðuÞ as an “interaction” Hamiltonian,i.e., jΨI ðuÞi ¼ eiuL jΨS ðuÞi, where the superscript I
denotes the interaction picture. Substituting this equation
into Eq. (1), we obtaini∂jΨI ðuÞi ¼ K̂ðuÞjΨI ðuÞi;
∂u

ð2Þ

def

where K̂ðuÞ¼ eiuL KðuÞe−iuL is the disentangler in theinteraction picture. The renormalized wave functionjΨI ðuÞi at scale u can be formally written in terms of
the IR state jΩIIR i ≡ jΨI ðu → −∞Þi as

 Z
u
0
0
K̂ðu Þdu jΩIIR i;jΨ ðuÞi ¼ P exp −i
I

−∞

ð3Þ

where P is the path ordering operator. Unless otherwise
stated, we will only consider the interaction picture;
therefore, we will drop the superscript I in the rest of thisLetter.
A continuous Chern insulator model.—We begin with a
two-band continuous Chern insulator model inR two spatial
dimensions [29] with Hamiltonian H ¼ d2 kψ † ðkÞ½RðkÞ · σψðkÞ, where k ¼ ðkx ; ky Þ ∈ R2 , RðkÞ ¼ ðkx ; ky ;qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
m − k2 Þ; m > 0; k ≡ jkj ¼ k2x þ k2y , and σ ¼ ðσ x ; σ y ; σ z Þis a vector of Pauli matrices. The fermionic operator ψðkÞis a two-component spinor ψðkÞ ≡ ½ ψ 1 ðkÞ ψ 2 ðkÞ T whose

120502-2

PHYSICAL REVIEW LETTERS 122, 120502 (2019)−ucomponents satisfy fψ †i ðkÞ;ψ j ðk0 Þg ¼ δij δðk − k0 Þ fori; j ∈ f1; 2g.
The ground state, which has the lower band filled, is [30]jΨi ¼

Y½uk ψ †2 ðkÞ − vk ψ †1 ðkÞjvaci;k
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
12
uk ¼ pﬃﬃﬃﬃﬃﬃ ðm − k Þ þ ðm − k2 Þ2 þ k2 ;Nk1
vk ¼ pﬃﬃﬃﬃﬃﬃ ðke−iθk Þ;
Nkð4Þwhere N k is a k-dependent normalization factor such thatjuk j2 þ jvk j2 ¼ 1, and the state jvaci is the vacuum stateannihilated by ψ 1;2 ðkÞ. The angle θk is defined via kx ¼
k cos θk and ky ¼ k sin θk ; i.e., it is the polar angle inmomentum space. The Chern number of the Rbottom bandof this two-band system is C ¼ ð1=4πÞ R2 d2 knðkÞ·f½∂nðkÞ=∂kx  × ½∂nðkÞ=∂ky g ¼ 1,
wherenðkÞ ≡½RðkÞ=jRðkÞj and where the integrand divided by two
is called the Berry curvature.
Now, we show how to obtain a scale-invariant cMERA
for this model.
Entanglement renormalization of the Chern insulator.—
Following the convention in Refs. [27,30,40], we take theGaussian ansatz for the disentangler in the Schrödinger
R
picture,KðuÞ ¼ i d2 k½gðk; uÞψ †1 ðkÞψ 2 ðkÞ þ g ðk; uÞψ 1 ðkÞψ †2 ðkÞ [41]. If we require our disentangler to bescale invariant, then gðk; uÞ should not have explicit u
dependence, gðk; uÞ ¼ gðkÞ. We also take the ansatz
that gðkÞ ¼ HðkÞe−iθk , where HðkÞ is a real-valued function to be determined.
R 2 † Through rewriting the disentangler
as KðuÞ ¼ d kψ ðkÞ½HðkÞ · σψðkÞ with HðkÞ ¼½HðkÞ sin θk ; −HðkÞ cos θk ; 0, we can intuitively understand its action by imagining an effective magnetic field ofstrength HðkÞ in a clockwise direction about the origin
applied to the pseudo-spin at each momentum point. In theinteraction picture, the disentangler becomesZ
K̂ðuÞ ¼ i−u

Pk ðuÞ ¼ Ak e−iφðe kÞ þ Bk eiφðe kÞ ;−u−u

Qk ðuÞ ¼ −ie−iθk ½Ak e−iφðe kÞ − Bk eiφðe kÞ :

ð7Þ

Coefficients Ak and Bk are complex
R ∞ numbers with
jAk j2 þ jBk j2 ¼ 12, and φðe−u kÞ ≡ keAt−u HðtÞðdt=tÞ.
UV scale u ¼ 0, the wave function should match theground state in Eq. (4); at IR scale u → −∞, we would
like the renormalized wave function to be the productQ
Q
state k ψ †1 ðkÞjvaci or the product state k ψ †2 ðkÞjvaci[27,30,40]. By taking Ak ¼ −ð1=2iÞ and Bk ¼ ð1=2iÞ, theboundary conditions can be satisfied by requiringHðkÞ ¼

kðm þ k2 Þ:
2½k4 þ k2 ð1 − 2mÞ þ m2 ð8Þ

Substituting Eq. (8) into Eqs. (6) and (7), we attain an
explicit form of the renormalized wave function,
1
pﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðm − k2 e−2u ÞjΨðuÞi ¼
N k;ukqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
þ ðm − k2 e−2u Þ2 þ k2 e−2u ψ †2 ðkÞ−u −iθk †− ke e ψ 1 ðkÞ jvaci;Yð9Þ

where N k;u is a normalization factor that depends on k andu. The Berry curvature of the renormalized wave function
at different u is shown schematically in Fig. 1. The IR
Q
state is jΩIR i ¼ limu→−∞ jΨðuÞi ¼ k e−iθk ψ †1 ðkÞjvaci,
Q †Q †which is equal to k ψ 1 ðkÞjvaci ¼ x ψ 1 ðxÞjvaci up toan overall phase. Note that the nonzero Chern number does
not survive in the IR state because the integration operation
does not commute with the limit u → −∞. However, at
any finite u, the Chern number is always one. Therefore,
there is no phase transition during the entanglementd2 k½Hðe−u kÞe−iθk ψ †1 ðkÞψ 2 ðkÞþ Hðe−u kÞeiθk ψ 1 ðkÞψ †2 ðkÞ:ð5Þ

Now, we start to renormalize the wave function anddetermine the form of the disentangler. We assume that therenormalized wave function at scale u can be expressed asjΨðuÞi ¼

Y½Pk ðuÞψ †2 ðkÞ − Qk ðuÞψ †1 ðkÞjvaci; ð6Þkwith jPk ðuÞj2 þ jQk ðuÞj2 ¼ 1. From Eq. (2), we get

FIG. 1. Berry curvature of the renormalized wave function in
the interaction picture at different scales u, drawn schematicallyin momentum space. The blue arrow corresponds to the directionof the reversed cMERA circuit. The area contributing to theChern number expands as one approaches the UV scale.

120502-3

PHYSICAL REVIEW LETTERS 122, 120502 (2019)
renormalization process, which is consistent with theresults in Refs. [30,42,43].
To analyze the spatial structure of the disentangler, we
rewrite the expression for HðkÞ. We first define λþ and λ−as the two roots of theequation x2 þ ð1 − 2mÞx þ m2 ¼ 0,
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
λ ¼ ½ð−1 þ 2m  1 − 4mÞ=2. They are real and negative when 0 < m < 1=4. Although setting this restriction
on m is not necessary for our disentangler, we will assume
it in the following in order to assist our experimental
realization. Now, the expression HðkÞ can be rewritten aspﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
−1 þ 1 − 4mkpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃHðkÞ ¼
2k − λþ
4 1 − 4m
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 þ 1 − 4mkpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
þ:2k − λ−
4 1 − 4m

ð10Þ

By inserting this expression into Eq. (5) and performing a
Fourier transform, it can be shown that the disentangler
in real space decays
with characteristicpﬃﬃﬃﬃﬃﬃﬃﬃﬃexponentiallypﬃﬃﬃﬃﬃﬃﬃﬃ−ulength e =maxf −λþ ; −λ− g. Therefore, our cMERA
involves quasilocal interactions.
Experimental realization of the cMERA circuit.—We
propose 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. Recall thatthe cMERA circuit starts with an initial IR state. As
discussed above, the IR state at u → −∞ does not havethe correct Chern number; therefore, we start from a nearIR state with large negative u. In addition, the cMERA
circuit is only valid on a lattice when the continuum
approximation holds. Therefore, throughout
the circuit,pﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ−uthe physical length scale e = maxf −λþ ; −λ− g should
be significantly larger than the lattice spacing. At the sametime, this length scale should be significantly smaller than
the total size of the lattice so that boundary effects do not
dominate the bulk physics. Going forward, we begin with anear-IR state and use our cMERA circuit to obtain the UV
state without ever violating these requirements.
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 concentrated
on a few discrete momentum points near k ¼ 0, thepreparation of this near-IR state should be fast if we can
individually create states at each point in momentum space.
In the Supplemental Material [44], we provide one possible
method for generating this initial state [44].
We now present the cMERA circuit engineering scheme(see Supplemental Material [44] for details). We use jg1 i
and jg2 i as shorthand notations for the two stable hyperfine
ground states jF ¼ 1=2; mF ¼ −1=2i and jF ¼ 1=2;
mF ¼ 1=2i in 1S0 ; these form the basis of our spinorψðkÞ ≡ ½ ψ 1 ðkÞ ψ 2 ðkÞ T . We find that if we have two

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

metastable excited states je1 i and je2 i (e.g., from the 3P
manifold) with quadratic dispersions coupled by spin-orbit
interaction and couple them off resonantly to the respective
ground states as shown in Fig. 2, then the disentangler in
the interaction picture can be engineered. Intuitively, thespin-orbit interaction allows us to generate a momentumdependent effective magnetic field for Eq. (5), whereas theoff-resonant couplings to quadratic dispersive bands induce
quadratic terms in the denominators of Eq. (10). To
accomplish this, we utilize the scheme detailed inRefs. [53–56] to create two dressed excited states coupledby spin-orbit interaction. However, as the two dressedstates are linear combinations of bare excited states, thedressed states do not have good quantum numbers to haveclear selection rules to forbid the transitions jg1 i ↔ je2 iand jg2 i ↔ je1 i. Nevertheless, by carefully choosing thedriving fields to couple ground states to the bare excitedstates, we can create interferences to generate synthetic
selection rules. By varying the laser parameters as thecircuit progresses, we can engineer the disentangler in theinteraction picture.
When the UV state is generated by the cMERA circuit,
one can then use the experimental techniques introduced inRefs. [57–59] to measure the Chern number and the Berrycurvature.
Discussion.—In this Letter, we found a quasilocal
cMERA whose fixed-point wave function is a Chern
insulator. This is a novel and unexpected way to represent
systems with chiral topological order. We also demonstratethat our quasilocal quantum circuit can be realized experimentally in a cold atom system, despite the common
intuition that a quantum circuit should be strictly local toallow easier implementation.
In our realization, we only explored one possibility toengineer spin-orbit coupling, but it may be possible toengineer the interaction in other ways, such as using
magnetic fields on a chip [60] or microwaves [61].
Other alkaline-earth atoms could also provide promising120502-4

PHYSICAL REVIEW LETTERS 122, 120502 (2019)
experimental platforms. Although our experimental realization took place in the interaction picture, one could in
principle instead use the Schrödinger picture for cMERA,
where the lattice constant must continuously contract inan experiment [62,63]. By using our cMERA circuit, theChern insulator bulk wave function can be prepared anddetected. We leave it for future work to study the edge
physics, for which one also needs to apply some unitaries
on the edge during the initial state preparation process andto carefully design the corresponding disentanglers; otherwise, the edge physics might not be preserved under thebulk unitary process [42,43,64].
It is also interesting that the Chern insulator ground stateis a fixed point of our cMERA with finite correlation length.
This observation seems to contradict the usual intuition thatthe fixed point correlation length must be zero or infinity, as
the correlation length must decrease under rescaling of eachstrictly local RG step in real space. However, since our
cMERA involves continuous time evolution and quasilocal
interactions, it has the potential to restore the originalcorrelation length after a finite time evolution. The no-go
theorems in Refs. [23–26] are similarly circumvented by acMERA construction. Our work suggests that quasilocal
RG transformations are a more powerful framework thanstrictly local RG transformations. It also might shed lighton some of the key properties of MERA-like formalisms
for a wide range of chiral topological states. In the future,
we hope to extend the methods of this Letter to fractional
quantum Hall 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, Brayden Ware, andXueda Wen for helpful discussions. This project is supported by the AFOSR, NSF QIS, ARL CDQI, ARO MURI,
ARO, NSF PFC at JQI, and NSF Ideas Lab. S. K. C.
partially completed this work during his participation in thelong-term workshop “Entanglement in Quantum Systems”
held at the Galileo Galilei Institute for Theoretical Physics
as well as “Boulder School: Quantum Information,” which
is supported by the National Science Foundation and the
University of Colorado. He is also funded by the ACRI
fellowship under the Young Investigator Training Program.
G. Z. is also supported by ARO-MURI, YIP-ONR, andNSF CAREER (DMR431753240). J. R. G. acknowledges
support from the NIST NRC Research Postdoctoral
Associateship Award. Z. E. is supported in part by the
ARCS Foundation. I. B. S. and A. V. C. acknowledge theadditional support of the AFOSR’s Quantum Matter MURI
and NIST.

[1] R. Orús, A practical introduction to tensor networks: Matrix
product states and projected entangled pair states, Ann.
Phys. (Amsterdam) 349, 117 (2014).

[2] J. C. Bridgeman and C. T. Chubb, Hand-waving and interpretive dance: An introductory course on tensor networks,
J. Phys. A 50, 223001 (2017).
[3] G. Vidal, Class of Quantum Many-Body States That Can
Be Efficiently Simulated, Phys. Rev. Lett. 101, 110501(2008).
[4] F. Verstraete and J. I. Cirac, Matrix product states represent 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 Computational Power ofProjected Entangled Pair States, Phys. Rev. Lett. 96, 220601(2006).
[6] M. B. Hastings, An area law for one-dimensional quantum
systems, J. Stat. Mech. 2007, P08024.
[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 Information andCorrelations, Phys. Rev. Lett. 100, 070502 (2008).
[9] K. G. Wilson, The renormalization group and the ϵ expansion, Phys. Rep. 12, 75 (1974).
[10] K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47, 773(1975).
[11] J. Zinn-Justin, Phase Transitions and Renormalization
Group (Oxford University Press, Oxford, 2007).
[12] M. Aguado and G. Vidal, Entanglement Renormalization
and Topological Order, Phys. Rev. Lett. 100, 070404(2008).
[13] R. N. C. Pfeifer, G. Evenbly, and G. Vidal, Entanglement
renormalization, scale invariance, and quantum criticality,
Phys. Rev. A 79, 040301 (2009).
[14] R. König, B. W. Reichardt, and G. Vidal, Exact entanglement renormalization for string-net models, Phys. Rev. B
79, 195123 (2009).
[15] R. König and E. Bilgin, Anyonic entanglement renormalization, Phys. Rev. B 82, 125118 (2010).
[16] S. Singh and G. Vidal, Symmetry-protected entanglement
renormalization, Phys. Rev. B 88, 121108 (2013).
[17] G. Evenbly and S. R. White, Entanglement Renormalizationand Wavelets, Phys. Rev. Lett. 116, 140403 (2016).
[18] J. Haegeman, B. Swingle, M. Walter, J. Cotler, G. Evenbly,
and V. B. Scholz, Rigorous Free-Fermion Entanglement
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 conformal
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, and S. Das
Sarma, Non-Abelian anyons and topological quantum
computation, Rev. Mod. Phys. 80, 1083 (2008).
[22] S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb-Robinson
Bounds and the Generation of Correlations and Topological
Quantum Order, Phys. Rev. Lett. 97, 050401 (2006).
[23] T. Barthel, M. Kliesch, and J. Eisert, Real-Space Renormalization Yields Finite Correlations, Phys. Rev. Lett. 105,
010502 (2010).

120502-5

PHYSICAL REVIEW LETTERS 122, 120502 (2019)[24] J. Dubail and N. Read, Tensor network trial states for chiral
topological phases in two dimensions and a no-go theorem
in any dimension, Phys. Rev. B 92, 205307 (2015).
[25] T. B. Wahl, H. H. Tu, N. Schuch, and J. I. Cirac, Projected
Entangled-Pair States can Describe Chiral TopologicalStates, Phys. Rev. Lett. 111, 236805 (2013).
[26] Z. Li and R. S. K. Mong, Entanglement renormalization forchiral topological phases, arXiv:1703.00464.
[27] J. Haegeman, T. J. Osborne, H. Verschelde, and F.
Verstraete, Entanglement Renormalization for Quantum
Fields in Real Space, Phys. Rev. Lett. 110, 100402 (2013).
[28] Q. Hu and G. Vidal, Spacetime Symmetries and Conformal Data in the Continuous Multiscale Entanglement
Renormalization Ansatz, Phys. Rev. Lett. 119, 010603(2017).
[29] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum
spin hall effect and topological phase transition in HgTe
quantum wells, Science 314, 1757 (2006).
[30] X. Wen, G. Y. Cho, P. L. S. Lopes, Y. Gu, X. L. Qi, and
S. Ryu, Holographic entanglement renormalization of topological insulators, Phys. Rev. B 94, 075124 (2016).
[31] B. Swingle and J. McGreevy, Renormalization groupconstructions of topological quantum liquids and beyond,Phys. Rev. B 93, 045127 (2016).
[32] Y. Huang and J. E. Moore, Neural network representation of
tensor network and chiral states, arXiv:1701.06246.
[33] R. Kaubruegger, L. Pastori, and J. C. Budich, Chiral
topological phases from artificial neural networks, Phys.
Rev. B 97, 195136 (2018).
[34] I. Glasser, N. Pancotti, M. August, I. D. Rodriguez, and J. 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, and N.
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 topologicalspin liquids with projected entangled pair states, Phys. Rev.
B 91, 224431 (2015).
[37] D. Poilblanc, N. Schuch, and I. Affleck, SUð2Þ1 chiral edge
modes of a critical spin liquid, Phys. Rev. B 93, 174414(2016).
[38] M. P. Zaletel and R. S. K. Mong, Exact matrix product statesfor quantum Hall wave functions, Phys. Rev. B 86, 245305(2012).
[39] P. Schmoll and R. Orús, Kitaev honeycomb tensor networks: Exact unitary circuits and applications, Phys. Rev. B
95, 045112 (2017).
[40] M. Nozaki, S. Ryu, and T. Takayanagi, Holographic
geometry of entanglement renormalization in quantum field
theories, J. High Energy Phys. 10 (2012) 193.
[41] In the cMERA literature, a momentum cutoff Λ is typicallyprovided [27,30]. With a finite cutoff, the UV state
generated by a cMERA circuit approximates the groundstate of the Hamiltonian up to Oð1=ΛÞ corrections. Here, we
work in the continuum limit Λ → ∞ to avoid this technicalsubtlety. In principle, these finite-Λ corrections can beworked out explicitly.

[42] M. D. Caio, N. R. Cooper, and M. J. Bhaseen, Quantum
Quenches in Chern Insulators, Phys. Rev. Lett. 115, 236403(2015).
[43] L. D’Alessio and M. Rigol, Dynamical preparation of
floquet chern insulators, Nat. Commun. 6, 8336 (2015).
[44] See Supplemental Material at http://link.aps.org/
supplemental/10.1103/PhysRevLett.122.120502 for details
on scale invariance as well as the experimental realization,
which includes Refs. [45–52].
[45] T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P.
Schauß, S. Hild, D. Bellem, U. Schollwöck, T. Giamarchi,
C. Gross, I. Bloch, and S. Kuhr, Quantum dynamics of amobile spin impurity, Nat. Phys. 9, 235 (2013).
[46] F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Béguin,
A. Vernier, T. Lahaye, and A. Browaeys, Single-Atom
Trapping in Holographic 2d Arrays of Microtraps with
Arbitrary Geometries, Phys. Rev. X 4, 021034 (2014).
[47] 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, Studyon the clock-transition spectrum of cold 171 Yb ytterbium
atoms, Laser Phys. Lett. 12, 025501 (2015).
[48] T. Kohno, M. Yasuda, K. Hosaka, H. Inaba, Y. Nakajima,
and F.-L. Hong, One-dimensional optical lattice clock with afermionic 171 Yb isotope, Appl. Phys. Express 2, 072501(2009).
[49] 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 Lattice Clock,Phys. Rev. Lett. 103, 063001 (2009).
[50] 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 measurement of transition of 1S0 ðF ¼ 1=2Þ − 3P0 ðF ¼ 1=2Þ 171Yb atoms in a
one-dimensional optical lattice at KRISS, Metrologia 50,
119 (2013).
[51] A. Yamaguchi, Metastable state of ultracold and quantum
degenerate Ytterbium atoms: High-resolution spectroscopy
and cold collisions, Ph.D. thesis, Kyoto University, 2008.
[52] U. Busch and K. A. Penson, Tight-binding electrons on
open chains: Density distribution and correlations, Phys.
Rev. B 36, 9271 (1987).
[53] D. L. Campbell, G. Juzeliunas, and I. B. Spielman, Realistic
Rashba and Dresselhaus spin-orbit coupling for neutral
atoms, Phys. Rev. A 84, 025602 (2011).
[54] D. L. Campbell and I. B. Spielman, Rashba realization:
Raman with RF, New J. Phys. 18, 033035 (2016).
[55] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang, L. Chen,
D. Li, Q. Zhou, and J. Zhang, Experimental realization of a
two-dimensional synthetic spin-orbit coupling in ultracold
Fermi gases, Nat. Phys. 12, 540 (2016).
[56] V. Galitski and I. B. Spielman, Spin-orbit coupling inquantum gases, Nature (London) 494, 49 (2013).
[57] P. Hauke, M. Lewenstein, and A. Eckardt, Tomography ofBand Insulators from Quench Dynamics, Phys. Rev. Lett.
113, 045303 (2014).
[58] E. Alba, X. Fernandez-Gonzalvo, J. Mur-Petit, J. K. Pachos,and J. J. Garcia-Ripoll, Seeing Topological Order in Timeof-Flight Measurements, Phys. Rev. Lett. 107, 235301(2011).

120502-6

PHYSICAL REVIEW LETTERS 122, 120502 (2019)[59] N. Fläschner, B. S. Rem, M. Tarnowski, D. Vogel, D.-S.
Lühmann, K. Sengstock, and C. Weitenberg, Experimental
reconstruction of the Berry curvature in a Floquet Blochband, Science 352, 1091 (2016).
[60] B. M. Anderson, I. B. Spielman, and G. Juzeliunas, Magnetically Generated Spin-Orbit Coupling for Ultracold
Atoms, Phys. Rev. Lett. 111, 125301 (2013).
[61] F. Grusdt, T. Li, I. Bloch, and E. Demler, Tunable spin-orbit
coupling for ultracold atoms in two-dimensional opticallattices, Phys. Rev. A 95, 063617 (2017).

[62] J. H. Huckans, I. B. Spielman, B. L. Tolra, W. D. Phillips,
and J. V. Porto, Quantum and classical dynamics of a BoseEinstein condensate in a large-period optical lattice, Phys.
Rev. A 80, 043609 (2009).
[63] S. Al-Assam, R. A. Williams, and C. J. Foot, Ultracold
atoms in an optical lattice with dynamically variable
periodicity, Phys. Rev. A 82, 021604 (2010).
[64] S. Mardanya, U. Bhattacharya, A. Agarwal, and A. Dutta,
Dynamics of edge currents in a linearly quenched haldane
model, Phys. Rev. B 97, 115443 (2018).

120502-7