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Supplementary Information for Entanglement in the quantum
phases of an unfrustrated Rydberg atom array
Matthew J. O’Rourke
1
and Garnet Kin-Lic Chan
1
1
Division of Chemistry and Chemical Engineering, California Institute of Technology,
Pasadena, CA 91125, USA
Supplementary Methods
This section gives additional details regarding the con-
vergence of the numerical methods used in this work.
Γ
-point DMRG: finite size errors
There are a two types of finite size errors in the energy
in the
Γ
-point formulation of the bulk Rydberg system.
These can be associated with the Rydberg interaction
energy and the (emergent) kinetic energy.
As discussed in the Methods section (main text), the use
of the
Γ
-point basis induces a periodicity in the density-
density correlation function and thus in the numerator of
the Rydberg interaction term. This relation is exact for
classical crystals and it is also exact for quantum states
with such correlations (those that can be expressed ex-
actly in the supercell Bloch basis, which obviously need
not be classical crystals). However, one can imagine
that such periodic correlations are inaccurate for certain
quantum phases, such as the disordered phase.
As a metric for the error
per site
induced by the con-
strained form of the correlations, we compute the quan-
tity
e
=
2
·
R
6
b
ρ
ex
·
min(
L
x
,L
y
)
6
(
ˆ
n
i
〉−〈
ˆ
n
i
2
)
.
(1)
This is a measure of error for quantum crystals whose
correlations do not match those induced by the Bloch
basis. Note that
e
is always positive, and it can be
systematically reduced by increasing the supercell size.
The other source of systematic error comes from the
effective itinerancy of the Rydberg atoms arising from
the
ˆ
σ
x
operator [1]. The error in the kinetic energy
of fermionic systems when using a Bloch basis is well
studied and understood to converge superalgebraically
with the supercell size
L
x
×
L
y
(see e.g. Ref. [2] and
references within). We expect a similarly rapid conver-
gence here, although the precise quantitative effect can
only be directly assessed through simulations. We have
carried out such checks extensively to ensure conver-
gence of our calculations, as discussed in the following
subsection and Supplementary Fig. 1.
Γ
-point DMRG: convergence and physical
strategy
Despite the finite size effects discussed above, we find
that we can converge our calculations to sufficiently
high accuracy with reasonable bond dimensions and
manageable supercell sizes. Even in the very compli-
cated region of the phase diagram near
δ
= 5
.
0
6
.
0
,R
b
= 2
.
3
, we can distinguish the ground-state or-
ders using a bond dimension of
D
= 1200
, as shown
in Supplementary Fig. 1. However, although this is
enough to identify the ground state order, higher bond
dimensions would be needed to capture the phase transi-
tions with high precision; given the large region of phase
space explored here, we leave such detailed calculations
to future work.
The strategy used to generate the bulk phase diagram in
main text Fig. 2a, as well as the truncated interaction
phase diagram Fig. 2b, is as follows.
• For a given point in phase space
(
δ,R
b
)
, run a
D
max
= 1000
simulation for all reasonable super-
cell sizes between
4
×
4
and
10
×
10
, as well as
12
×
9
.
• Identify all supercells for which the ground state
has an energy per site within
10
2
of the lowest
energy.
• If there are competing orders, ensure these solu-
1