Numerical exploration of trial wavefunctions for the particle-hole-symmetric Pfaffian
Ryan V. Mishmash,
1, 2, 3
David F. Mross,
4
Jason Alicea,
1, 2
and Olexei I. Motrunich
1, 2
1
Department of Physics and Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA
2
Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
3
Department of Physics, Princeton University, Princeton, New Jersey 08540, USA
4
Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, 76100, Israel
(Dated: April 17, 2018)
We numerically assess model wavefunctions for the recently proposed particle-hole-symmetric Pfaffian (‘PH-
Pfaffian’) topological order, a phase consistent with the recently reported thermal Hall conductance [Banerjee
et al., arXiv:1710.00492] at the ever enigmatic
ν
= 5
/
2
quantum-Hall plateau. We find that the most natural
Moore-Read-inspired trial state for the PH-Pfaffian, when projected into the lowest Landau level, exhibits a
remarkable numerical similarity on accessible system sizes with the corresponding (compressible) composite
Fermi liquid. Consequently, this PH-Pfaffian trial state performs reasonably well energetically in the half-filled
lowest Landau level, but is likely not a good starting point for understanding the
ν
= 5
/
2
ground state. Our
results suggest that the PH-Pfaffian model wavefunction either encodes anomalously weak
p
-wave pairing of
composite fermions or fails to represent a gapped, incompressible phase altogether.
Introduction.
The half-filled Landau level has long stood
as a paradigmatic example of an inherently strongly inter-
acting quantum many-body system displaying various exotic
phenomena. For the half-filled lowest Landau level (LLL), it
is now experimentally [1–4] and numerically [5–7] well es-
tablished that the Coulomb-interacting ground state exhibits a
Fermi sea of composite fermions [8, 9]—emergent degrees of
freedom each consisting of an electron bound to two fictitious
flux quanta that, on average, cancel the applied magnetic field.
The theoretical description for this remarkable gapless ‘com-
posite Fermi liquid’ (CFL) phase was pioneered by Halperin,
Lee, and Read (HLR) [10]. Recently, the issue of particle-hole
symmetry resurfaced as an important aspect of the problem
following Son’s proposal that composite fermions are Dirac
particles [11]—a picture that has since found numerical sup-
port [7, 12–14].
An even subtler topic concerns the nature of the
ν
= 5
/
2
plateau seen at the half-filled second Landau level (SLL) [15].
Here, numerics generally support [6, 16, 17] either Moore-
Read (MR) Pfaffian topological order [18] that emerges upon
p
x
−
ip
y
pairing composite fermions [19], or its particle-
hole conjugate, the anti-Pfaffian [20, 21]. The Coulomb-
interacting problem projected into the SLL exhibits an ex-
act particle-hole symmetry; in this setting the MR-Pfaffian
and anti-Pfaffian are exactly degenerate, and the emergence
of one over the other requires spontaneous symmetry break-
ing [22, 23]. For more experimentally realistic models that
include, for example, Landau level mixing (which explic-
itly breaks particle-hole symmetry), it is now believed—after
some debate [24–27]—that the anti-Pfaffian state is favored
microscopically [28].
The experimental status of the
ν
= 5
/
2
quantum Hall state
is similarly complex; for a review, see Ref. [29]. In a re-
cent breakthrough, Banerjee et al. [30] measured the thermal
Hall conductance and found
κ
xy
≈
5
/
2
[in units of
π
2
k
2
B
3
h
T
].
Assuming all edge modes have equilibrated,
κ
xy
probes the
edge’s total chiral central charge [31]; a half-integer value
directly implies an odd number of Majorana edge modes
and concomitant non-Abelian Ising anyon bulk quasiparticles
[32]. Intriguingly,
κ
xy
= 5
/
2
is half-integer yet corresponds
to the edge structure of neither the MR-Pfaffian nor the anti-
Pfaffian, but rather is consistent with
particle-hole-symmetric
Pfaffian
(PH-Pfaffian) topological order recently elucidated
by Son [11]. (Importantly, PH-Pfaffian topological order is
compatible
with particle-hole symmetry, but does not require
it [33]; cf. Refs. [34, 35].) This result is confounding, par-
ticularly in light of the extensive numerical work summarized
above, which has not revealed any evidence for such a state.
The observed
κ
xy
= 5
/
2
can nevertheless be plausibly ex-
plained by disorder-induced PH-Pfaffian behavior [36–38] or
incomplete thermal equilibration of an anti-Pfaffian edge [39].
In this paper we focus on the clean limit and numerically
explore minimal trial PH-Pfaffian wavefunctions projected
into a fixed Landau level. We specifically extract their de-
gree of particle-hole symmetry, pair correlation functions, en-
tanglement spectra, and overlap with exact ground states of
model Hamiltonians and other trial wavefunctions, in systems
with up to
N
= 12
electrons. These diagnostics all reveal
a striking similarity between our trial PH-Pfaffian wavefunc-
tions and, surprisingly, the compressible CFL state. We dis-
cuss several possible interpretations of these results and asso-
ciated conundrums that they raise.
Trial wavefunctions.
The hallmark of PH-Pfaffian topolog-
ical order is a reversed chirality of the Majorana edge mode
relative to that of the MR-Pfaffian. Therefore, in analogy with
the celebrated MR-Pfaffian wavefunction [40],
Ψ
MR
−
Pf
(
{
z
i
}
) = Pf
[
1
z
i
−
z
j
]
∏
i<j
(
z
i
−
z
j
)
2
,
(1)
we may view the PH-Pfaffian as a
p
x
+
ip
y
superconductor of
composite fermions [11] naturally described by [33, 41]
Ψ
PH
−
Pf
(
{
z
i
}
) =
P
LLL
Pf
[
1
z
∗
i
−
z
∗
j
]
∏
i<j
(
z
i
−
z
j
)
2
.
(2)
arXiv:1804.01107v2 [cond-mat.str-el] 15 Apr 2018
2
Importantly, the presence of antiholomorphic terms on the
right-hand side necessitates an explicit LLL projection, con-
trary to the MR-Pfaffian. Our primary goal is to numerically
characterize the trial PH-Pfaffian wavefunction in Eq. (2), as-
sessing its efficacy for describing gapped PH-Pfaffian topo-
logical order in the half-filled Landau level. To this end it will
prove very useful to also consider a model CFL wavefunction
based on the HLR construction (also putatively able to capture
some aspects of Dirac composite fermions [12–14, 42–44]):
Ψ
CFL
(
{
z
i
}
) =
P
LLL
det[
e
i
k
j
·
r
i
]
∏
i<j
(
z
i
−
z
j
)
2
,
(3)
with
det[
e
i
k
j
·
r
i
]
a Slater determinant of plane-wave orbitals.
For our numerics, we consider a spherical geometry [45] in
which
N
electrons moving on the surface experience a radial
magnetic field produced by a monopole of strength
Q >
0
at
the origin. The sphere has radius
√
Q`
0
, with
`
0
=
√
~
c/eB
the magnetic length, and
N
φ
= 2
Q
flux quanta penetrate its
surface. Quantum-Hall states defined in the plane, such as
Eqs. (1) through (3), can be translated to the sphere using a
standard procedure [45, 46]. Due to finite curvature of the
sphere, states at a given filling
ν
are characterized by their
‘shift’ quantum number
S
[47] via
N
φ
=
ν
−
1
N
−S
= 2
N
−S
(taking
ν
= 1
/
2
). The MR-Pfaffian and PH-Pfaffian states re-
spectively occur at shifts
S
MR
= 3
and
S
PH
= 1
. Note that
the latter shift corresponds to a LLL Hilbert space of dimen-
sion
N
orb
=
N
φ
+ 1 = 2
N
, a clear zeroth-order condition for
a wavefunction to exhibit particle-hole symmetry.
We can define CFL states on the sphere following Rezayi
and Read [5]. Specifically, upon attaching two flux quanta
to each electron that oppose the external field, each such
composite fermion feels a total average magnetic flux of
2
q
=
N
φ
−
2(
N
−
1)
. Thus, at the respective shifts
S
MR
and
S
PH
, we have
q
MR
=
−
1
/
2
and
q
PH
= +1
/
2
[48].
CFL wavefunctions are then obtained by replacing the plane
waves in Eq. (3) by appropriate monopole harmonics [49, 50]:
det[
e
i
k
j
·
r
i
]
→
det[
Y
q
`
j
,m
j
(
θ
i
,φ
i
)]
.
We calculate via Monte Carlo (MC) sampling all coeffi-
cients of the trial wavefunctions in the many-body Fock space
built out of LLL orbitals on the sphere, i.e.,
〈{
m
i
}|
Ψ
trial
〉
where
m
i
is the
z
-component of angular momentum of par-
ticle
i
. There are two reasons for this brute-force approach
(see also Ref. [12]). First, it is a completely general way to
perform LLL projection; notably, Eq. (2) is not amenable to
the Jain-Kamilla projection scheme [51]. Second, obtaining
the full second-quantized wavefunction constitutes the only
practical way to calculate important quantities such as the de-
gree of particle-hole symmetry (Fig. 1) and the entanglement
spectrum (Fig. 3).
In what follows, we closely compare PH-Pfaffian and CFL
model wavefunctions at shift
S
PH
, i.e., with
N
φ
= 2
N
−
1
.
Hereafter we will denote the CFL at shift
S
by CFL(
S
). The
CFL(
S
PH
) state has filled angular momentum shells at elec-
tron numbers
N
= 2
,
6
,
12
,
20
,...
[52]. The largest such
system amenable to our MC approach has
N
= 12
and
0
1
2
3
4
5
6
angular momentum
L
0
.
00
0
.
05
0
.
10
0
.
15
Θ = 1
−|〈
Ψ
|PH|
Ψ
〉|
2
CFL(
S
PH
)
PH
−
Pf
10
8
10
9
10
10
MC steps
0
.
000
0
.
005
Θ
P
L
=0
applied
FIG. 1. Measured particle-hole asymmetry
Θ
versus measured total
angular momentum
L
for MC evaluation of the
N
= 12
CFL(
S
PH
)
and PH-Pfaffian trial states. Different (filled) points correspond to
different numbers of MC samples, while the unfilled points at
L
= 0
represent the best MC state projected to
L
= 0
; the solid curves
are fits to a quadratic polynomial. In the inset, we show
Θ
after
projecting to
L
= 0
versus total number of MC steps.
N
φ
= 23
—hence, we focus extensively on this system. In this
case, the many-body Hilbert space in the
L
z
=
∑
i
m
i
= 0
sector contains 61108 basis states. Fully diagonalizing the
total-angular-momentum operator
ˆ
L
2
in this basis reveals that
there are
dim
H
L
=0
= 127
eigenstates with
L
= 0
. We use
these states to form a projection operator
P
L
=0
that we apply
to our MC-acquired wavefunctions to obtain final, perfectly
rotationally invariant trial states. While the above trial wave-
functions adapted to the sphere are exact
ˆ
L
2
eigenstates with
L
= 0
[46], statistical error introduced by our MC scheme
spoils this property;
P
L
=0
merely removes this error.
Physical properties.
Although the CFL(
S
PH
) and PH-
Pfaffian trial wavefunctions exist at the ‘correct’ shift, their
degree of particle-hole symmetry on finite-size systems is
not manifest. Figure 1 illustrates the particle-hole asymme-
try
Θ = 1
− |〈
Ψ
|PH|
Ψ
〉|
2
exhibited by these wavefunc-
tions as obtained via MC for the
N
= 12
system. The main
panel plots measured
Θ
versus measured
L
[defined through
〈
Ψ
|
ˆ
L
2
|
Ψ
〉
=
L
(
L
+ 1)
] for different MC runs with varying
numbers of MC steps
before applying
P
L
=0
; solid curves de-
pict fits to a second-order polynomial. In the inset, we mon-
itor
Θ
versus MC steps
after applying
P
L
=0
. These results
indicate
Θ =
O
(10
−
3
)
for both wavefunctions. At smaller
sizes
N
= 4
,
6
,
8
,
10
we find for the PH-Pfaffian trial state
Θ = 0
,
3
.
3
×
10
−
6
,
2
.
9
×
10
−
4
,
2
.
1(2)
×
10
−
4
(exact evalu-
ation is possible for
N
≤
8
). While we expect that
Θ
→
1
in the thermodynamic limit, the degree of particle-hole sym-
metry shown by both wavefunctions up to
N
= 12
is impres-
sively high; see Refs. [14, 41] for recent related discussions.
Next,
we
examine
the
‘pair
correlation
func-
tion’ evaluated along the sphere’s equator:
g
(
r
)
=
1
ρ
2
〈
ˆ
ψ
†
(
r
)
ˆ
ψ
†
(0)
ˆ
ψ
(0)
ˆ
ψ
(
r
)
〉
, where
ρ
is the 2D density
and
ˆ
ψ
is the LLL-projected electron operator.
The main
3
0
2
4
6
8
10
r/`
0
0
.
00
0
.
25
0
.
50
0
.
75
1
.
00
1
.
25
g
(
r
)
S
PH
:
N
= 12
,N
φ
= 23
LLL g
.
s
.
SLL g
.
s
.
CFL
PH
−
Pf
0
10
r/`
0
0
1
g
(
r
)
S
MR
:
N
= 12
,N
φ
= 21
MR
−
Pf
FIG. 2. Pair correlation function
g
(
r
)
for the LLL and SLL Coulomb
ground states as well as the CFL(
S
PH
) and PH-Pfaffian trial states at
the shift
S
PH
(main panel). The inset shows data for the analogous
states at
S
MR
; the first three legend entries in the main panel also
apply to the inset.
panel of Fig. 2 corresponds to shift
S
PH
. Specifically, we
show data for PH-Pfaffian and CFL(
S
PH
)
trial states, and
for the ground state of the Coulomb potential (defined in
terms of the chord distance) projected into either the LLL
or the SLL (implemented via Haldane pseudopotentials
[45, 53, 54]).
We again focus on
N
= 12
and apply
P
L
=0
to all MC-obtained trial states [55].
Remarkably,
the PH-Pfaffian and CFL(
S
PH
)
data are qualitatively in-
distinguishable. In fact, these two trial states exhibit very
high overlap:
|〈
Ψ
CFL
|
Ψ
PH
−
Pf
〉|
2
= 0
.
9106(2)
. At long
distances, we expect
g
(
r
)
to approach unity as an oscillatory
exponential (power law) for a gapped (gapless) phase. While
the asymptotic behavior exhibited by the PH-Pfaffian trial
state is not obvious at these sizes (it does have slightly
reduced oscillation amplitudes at the largest distances), its
similarity with the CFL wavefunction calls into question
whether
Ψ
PH
−
Pf
represents a gapped phase [56]. Finally, the
data for the LLL ground state unsurprisingly closely tracks
the PH-Pfaffian and especially CFL trial states [57], while
that for the SLL ground state behaves very differently.
For comparison, the inset of Fig. 2 presents analogous
g
(
r
)
data taken at shift
S
MR
with
N
= 12
and
N
φ
= 21
. The
MR-Pfaffian and CFL(
S
MR
)
data differ substantially as ex-
pected, and the overlap of the two wavefunctions is signifi-
cantly reduced to
0
.
384(4)
even though the
L
= 0
Hilbert
space contains fewer states than at shift
S
PH
(
dim
H
L
=0
= 52
versus 127). Additionally,
g
(
r
)
for the MR-Pfaffian is trend-
ing in the direction of the SLL ground state while saturating
to
g
(
r
)
→
1
at long distances (clearly observing incompress-
ibility in the Coulomb-interacting SLL system itself requires
N
≥
20
electrons [58]).
First introduced by Ref. [59] in the MR-Pfaffian context,
the entanglement spectrum (ES) has since become an in-
valuable tool for characterizing topological phases. Figure 3
30
40
L
A
z
0
2
4
6
8
ξ
CFL(
S
PH
)
PH
−
Pf
30
40
L
A
z
LLL g
.
s
.
CFL(
S
PH
)
30
40
L
A
z
SLL g
.
s
.
PH
−
Pf
FIG. 3. Orbital-partition entanglement spectra for the
N
= 12
,
N
orb
=
N
φ
+ 1 = 24
system with
N
A
orb
= 12
and
N
A
= 6
. In the
left panel, we depict the close similarity between the CFL(
S
PH
) and
PH-Pfaffian trial states, while in the other two panels we compare the
respective trial states to the LLL (middle) and SLL (right) Coulomb
ground states.
presents orbital-partition entanglement spectra for the same
wavefunctions in the main panel of Fig. 2. We work at shift
S
PH
with
N
= 12
electrons and
N
orb
=
N
φ
+ 1 = 24
total orbitals; subsystem
A
is chosen as the 12 states with
positive angular momentum in the
z
direction (which have
predominant weight in the upper hemisphere). In Fig. 3 we
show the sector corresponding to
N
A
=
N/
2 = 6
electrons
in subsystem
A
and plot the ‘entanglement energies’
ξ
ver-
sus
L
A
z
, the total subsystem
z
-component angular momentum
[60]. A particle-hole transformation on a given wavefunction
takes
L
A
z
→
L
max
z
−
L
A
z
, where
L
max
z
=
1
2
(
1
2
+
Q
)
2
is the
maximum total
z
-component angular momentum of the en-
tire system. For the data in Fig. 3 with monopole strength
Q
=
N
φ
/
2 = 23
/
2
, we have
L
max
z
= 72
; hence, perfect
particle-hole symmetry implies a reflection symmetry in the
ES data about
L
A
z
=
L
max
z
/
2 = 36
.
Interpreting the ES of PH-Pfaffian topological order poses
an interesting open question. Here we simply observe that
for PH-Pfaffian model wavefunctions with up to
N
= 12
electrons, the ES exhibits a high degree of symmetry about
L
max
z
/
2
and closely tracks the CFL(
S
PH
) ES at the low-
est
ξ
.
These properties are subtler manifestations of the
near particle-hole symmetry of
Ψ
PH
−
Pf
and its high over-
lap with
Ψ
CFL
captured earlier. Figure 3 also shows com-
parisons between the CFL(
S
PH
) trial state and LLL Coulomb
ground state [which have near unit overlap: 0.9926(1)] and
between the PH-Pfaffian trial state and SLL Coulomb ground
state [which have near zero overlap: 0.01811(6)]. The pure
Coulomb interaction projected into the SLL likely sits at a
first-order phase transition between the MR-Pfaffian and anti-
Pfaffian [23], which would naturally explain the lack of clear
structure in the ES for that case.
The similarity between the PH-Pfaffian and CFL
(
S
PH
)
trial states uncovered above suggests that the LLL-projected
Coulomb interaction may be a reasonable starting point for re-
alizing the former model wavefunction as the best approxima-
tion of the ground state compared to other natural trial states.
As a preliminary exploration, Fig. 4 shows variational ener-
4
0
.
30
0
.
35
0
.
40
0
.
45
0
.
50
0
.
0
0
.
5
1
.
0
|〈
Ψ
trial
|
Ψ
0
〉|
2
CFL(
S
PH
)
PH
−
Pf
0
.
40
0
.
45
0
.
50
0
.
55
V
1
0
.
0
0
.
5
1
.
0
|〈
Ψ
trial
|
Ψ
0
〉|
2
0
2
4
(
E
trial
−
E
0
)
/
(
E
1
−
E
0
)
LLL
0
5
10
(
E
trial
−
E
0
)
/
(
E
1
−
E
0
)
SLL
FIG. 4. Overlaps (left axes, solid curves) and energies (right axes,
dashed curves) of the CFL(
S
PH
) and PH-Pfaffian trial states com-
pared with the exact ground state
|
Ψ
0
〉
at the shift
S
PH
for Coulomb
pseudopotentials in the LLL (top panel) and SLL (bottom panel) but
with variable
V
1
. As in Figs. 1 through 3, here
N
= 12
; the vertical
dashed lines indicate the respective pure Coulomb points.
gies (relative to the ground-state energy,
E
0
, and normalized
by the gap,
E
1
−
E
0
) and overlaps with the exact ground state
for both CFL
(
S
PH
)
and PH-Pfaffian trial states. Here we vary
the pseudopotential
V
1
and keep all other pseudopotentials
fixed at their Coulomb values in the LLL (top panel) and SLL
(bottom panel). In the LLL, adding short-distance attraction
to the potential by decreasing
V
1
relative to its Coulomb value
slightly improves the PH-Pfaffian trial energy and overlap, but
fails to overcome CFL
(
S
PH
)
in this parameter regime. We
reach similar conclusions upon varying
V
3
in addition to
V
1
.
Finally, both wavefunctions perform extremely poorly at the
Coulomb point for the SLL; increasing
V
1
eventually mimics
the LLL pseudopotentials, thereby stabilizing the CFL [6, 16].
We leave a more thorough investigation of energetics for the
PH-Pfaffian trial state for future work.
Discussion.
A far more pressing matter concerns the nature
of the PH-Pfaffian model wavefunction itself given similar-
ities to the CFL. The stark difference between MR-Pfaffian
and CFL trial states at the same system sizes suggests that
these similarities are not merely finite-size artifacts. We have
also considered generalizations of
Ψ
PH
−
Pf
by including ‘sta-
bilization’ factors
∏
i<j
|
z
i
−
z
j
|
α
in Eq. (2) before projection.
Topological orders described by a
K
-matrix
K
=
(
n m
m n
)
with
n < m
actually necessitate such factors for thermody-
namic stability [61]. For
ν
= 5
/
2
, the 113 state is a plausible
candidate that requires stabilization [62]; a possible LLL sta-
bilized wavefunction is
Ψ
113
(
{
v
i
,w
i
}
) =
P
LLL
∏
i<j
|
v
i
−
v
j
|
4
|
w
i
−
w
j
|
4
×
∏
i<j
(
v
i
−
v
j
)(
w
i
−
w
j
)
∏
i,j
(
v
i
−
w
j
)
3
(4)
with
v
i
,w
i
complex coordinates for two species of distin-
guishable particles. Reference [63] argued that the 113 and
PH-Pfaffian topological orders share an intimate relation anal-
ogous to that between the Halperin 331 state [64] and the MR-
Pfaffian [18, 19]. This relationship is encoded in the wave-
functions studied here: Fully antisymmetrizing the 331 wave-
function over all coordinates yields
Ψ
MR
−
Pf
[65]; similarly,
associating
{
z
i
}
=
{
v
i
,w
i
}
, antisymmetrizing, and then LLL
projecting the
stabilized
wavefunction
Ψ
113
yields
Ψ
PH
−
Pf
modified by
∏
i<j
|
z
i
−
z
j
|
α
with
α
= 2
[66]. Surprisingly,
such factors very weakly affect the resulting LLL-projected
PH-Pfaffian trial state; e.g., for
N
= 12
electrons, wavefunc-
tions with
α
= 2
and
α
= 0
[Eq. (2)] have an overlap of
0.9931(2) [67]. This broader family of states thus also appears
closely related to the CFL.
One logical possibility is that our PH-Pfaffian trial states
describe gapped PH-Pfaffian topological order in the thermo-
dynamic limit, but with pairing that is significantly suppressed
by LLL projection. This interpretation raises an interest-
ing puzzle: What determines the anomalous pairing strength
given the absence of any obvious small parameter in the PH-
Pfaffian model wavefunctions? Another possibility is that
LLL projection obliterates the pairing entirely, and that in
the thermodynamic limit the PH-Pfaffian trial wavefunctions
describe a gapless state in the same university class as the
CFL. Here, too, a conundrum arises. Upon removing
P
LLL
,
Eq. (2) certainly describes gapped PH-Pfaffian topological or-
der (without particle-hole symmetry). In this scenario LLL
projection would qualitatively alter the universal properties of
the trial state—contrary to the typical situation—for possibly
fundamental reasons that are presently unclear. Landau-level
mixing might then, counterintuitively, be
required
to stabi-
lize PH-Pfaffian topological order. At present we cannot rule
out the possibility that alternative LLL-projected PH-Pfaffian
trial states do not suffer from the subtleties encountered here.
In this regard, it would be interesting to construct trial states
for the PH-Pfaffian with additional variational freedom—one
natural route is to consider more general pair wavefunctions
for a
p
x
+
ip
y
superconductor of composite fermions in the
spirit of Ref. [68].
Note added
: After completion of this work, Ref. [69] ap-
peared which contains some overlap with our results; see their
Appendix A.
R.V.M. gratefully acknowledges Mike Zaletel for valuable
discussions; we thank Ajit Balram for pointing out the con-
nection between the
α
= 2
generalized PH-Pfaffian state and
the states proposed in Ref. [70]. This work was supported by
grant No. 2016258 from the United States-Israel Binational
Science Foundation (BSF); the Minerva foundation with fund-
ing from the Federal German Ministry for Education and
Research (D.F.M.); the Army Research Office under Grant
Award W911NF-17-1-0323 (J.A.); the NSF through grants
DMR-1723367 (J.A.) and DMR-1619696 (O.I.M.); the Cal-
tech Institute for Quantum Information and Matter, an NSF
Physics Frontiers Center with support of the Gordon and Betty
Moore Foundation through Grant GBMF1250; and the Walter
5
Burke Institute for Theoretical Physics at Caltech.
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