www.sciencemag.org/content/
358/
6367/
1155/suppl/DC1
Supplementary
Material
s for
Stripe order in the underdoped region of the two
-dimensional Hubbard model
Bo
-Xiao Zheng,*† Chia-Min Chung,* Philippe Corboz,* Georg Ehlers,* Ming-Pu Qin,*
Reinhard M. Noack, Hao Shi,* Steven R. White, Shiwei Zhang, Garnet Kin-
Lic Chan†
*These authors contributed equally to this work.
†Corresponding author.
Email:
boxiao.zheng@gmai
l.com
(B.
-X.Z.);
gkc1000@gmail.com
(G
.K.
-L.C.)
Published
1 December
2017,
Science
358
, 1155
(201
7)
DOI:
10.1126/science.aam7127
This PDF file includes:
Materials and Methods
Figs. S1 to S
41
Tables S1 to S
10
References
S1 Structure of the supplementary information
Section S2 contains notes on the figures and main discussion. Section S3 contains a summary
of the best energy data used to compare the relative stripe energetics. Section S4 discusses
the estimate of the long-range Coulomb effects. The remaining sections describe in detail the
calculations performed using AFQMC, DMRG, hybrid DMRG, DMET, and iPEPS.
Materials and Methods
2
S2 Additional information for figures and main discussion
Fig. 1: The plotted energies (units of
t
) correspond to the following specific calculations.
•
AFQMC:
0
.
766
±
0
.
001
from extrapolation to
1
(in both length and width directions)
clusters with pinning fields.
•
DMRG:
0
.
7627
±
0
.
0005
from extrapolation to
1⇥
6
clusters with pinning fields using
the hybrid momentum/real-space representation (h-DMRG).
•
DMET:
0
.
77063
±
0
.
00001
from
8
⇥
2
clusters with spin-flip boundary conditions.
•
iPEPS:
0
.
7673
±
0
.
002
from
16
⇥
2
supercells with extrapolation to zero truncation
error.
Note that the error bars only reflect errors that can be estimated from the calculations them-
selves, and not all systematic errors.
Discussion of Fig. 2: The metastable DMET state on the
p
5
⇥
2
lattice is slightly higher in
energy than the ground-state. It is a “stripe-like” state, but appears to be frustrated at this unit-
cell size. It is a little hard to estimate the relative energy of this state and the vertical striped
ground state as the tilted impurity cluster energies are systematically shifted with respect to
the energies of the non-tilted clusters. We estimate the relative energy of the
p
5
⇥
2
state as
E
=
E
p
5
⇥
2
E
p
2
⇥
2
+
E
2
⇥
2
, which includes the difference between the
2
⇥
2
cluster uniform
d
-wave state energy and tilted
p
2
⇥
2
cluster uniform
d
-wave state. This gives an estimate of
⇠
0
.
005
t
above the vertical striped state.
Fig. 3 and Fig. 6: The plotted energies are summarized in Section S3, Tables S1 to S4.
Fig. 4: The plotted orders correspond to the following specific calculations.
•
DMET: from
8
⇥
2
calculations with spin-flip boundary condition. We plotted two unit-
cells to illustrate the complete spin pattern.
3
•
AFQMC: from
48
⇥
4
calculations with cylinder boundary conditions (periodic in the
shorter direction). We duplicated the pattern in a
8
⇥
2
rectangle to illustrate the complete
spin pattern.
•
iPEPS: from the
8
⇥
2
calculations with 16 independent tensors. The numbers are shown
for the largest bond dimension used. We plot the pattern in a
16
⇥
2
region to illustrate
the complete spin pattern.
•
DMRG: from a
32
⇥
6
calculation with cylinder boundary conditions. The results are
extrapolated to zero truncation error. We duplicated the pattern in a
8
⇥
2
rectangle to
illustrate the complete spin pattern.
Notes on Fig. 5(D): The plot is reproduced from part of Fig. S22. See the figure caption for
detail.
Wavelengths of stripes: A key feature of the stripes that we see is that each stripe acts as an anti-
ferromagnetic domain wall. As a well-known consequence, at
1
/
8
doping for half-filled stripes,
the wavelength associated with the AF periodicity (
8
) is twice that of the charge periodicity (
4
).
As an oversimplified but useful characterization of this periodicity, we describe it by labeling the
spin pattern along a fixed row, assuming the stripe is width
1
:
...
·
"#"
·
#"#
·
"#"
·
...
.Here
the
·
’s indicate the positions of the stripes, and the patterns
"
·
#
or
#
·
"
signify the do-
main wall nature of the stripe. Consider a charge wavelength which is an odd integer, e.g.
5
:
...
·
"#"#
·
"#"#
...
We see that the ratio of AF and charge wavelengths is one in this case,
not two! This odd-even alternation is potentially confusing, particularly if one has non-integer
charge periodicity.
However, experimentally, one looks at structure factors, noting peaks near
(
⇡
,
⇡
)
. The loca-
tions of the peaks nearest
(
⇡
,
⇡
)
do not show any odd/even alternation. To see this note that the
shift of the
k
-space origin to
(
⇡
,
⇡
)
, for one particular row, is equivalent to an alternating sign
4
chain
1
x
in the AF pattern, e.g. for charge wavelength
4
,
...
·
"#"
·
#"#
·
"#"
·
...
!
...
·
"""
·
###
·
"""
·
...
and for charge wavelength
5
...
·
"#"#
·
"#"#
...
!
...
·
""""
·
####
...
In both the even and odd cases, the distance of peaks from
(
⇡
,
⇡
)
corresponds to an AF “wave-
length” of twice the charge wavelength.
5
S3 Summary of stripe energy results
Table S1: Best estimates of energy (per site) of stripes and competing states for
U/t
=8
.
AFQMC numbers obtained as described in section S5, DMRG numbers obtained as described
in section S6, Hybrid (h-) DMRG numbers obtained as described in section S7, iPEPS numbers
obtained as described in section S8, DMET numbers obtained as described in section S9. For
the AFQMC calculations (PBC) denotes periodic boundary conditions used on both the short-
and long-axes of the cylinder. For the DMRG (real-space) calculations, periodic boundary
conditions were used along the short axis, open boundary conditions on the long axis. For the h-
DMRG calculations, periodic or anti-periodic boundary conditions were used on the short axis,
denoted PBC or APBC. SF denotes that the DMET correlation potential in the spin-channel
is flipped, doubling the spin wavelength. (Thus the
8
⇥
2
(SF) pattern in DMET has a charge
wavelength of 8 but a spin wavelength of 16.)
Method
Size
Wavelength Energy (
t
) Error (
t
)
AFQMC
12
⇥
8
(PBC)
6
0
.
7653
0.0002
AFQMC
14
⇥
8
(PBC)
7
0
.
7653
0.0002
AFQMC
16
⇥
8
(PBC)
8
0
.
7668
0.0002
AFQMC
18
⇥
8
(PBC)
9
0
.
7655
0.0002
AFQMC
20
⇥
8
(PBC)
10
0
.
7653
0.0002
AFQMC
1⇥
48
0
.
7680
0.0001
AFQMC
1⇥
68
0
.
7653
0.0003
AFQMC
1⇥
88
0
.
7656
0.0004
DMRG
1⇥
48
0
.
76598
0.00003
DMRG
1⇥
65
0
.
7615
0.0004
DMRG
1⇥
68
0
.
762
0.001
DMRG
1⇥
77
0
.
762
0.001
DMRG
1⇥
69
0
.
751
0.0016
h-DMRG
1⇥
6
(PBC)
5
0
.
76210
0.00005
h-DMRG
1⇥
4
(APBC)
8
0
.
76057
0.00007
h-DMRG
1⇥
4
(PBC)
8
0
.
7657
0.0003
h-DMRG
1⇥
4
(av.)
a
8
0
.
7631
0.0003
h-DMRG
1⇥
6
(PBC)
8
0
.
7627
0.0005
iPEPS
2
⇥
2
b
2
0
.
7560 0
.
0025
iPEPS
5
⇥
25
0
.
7632 0
.
0018
iPEPS
7
⇥
27
0
.
7629 0
.
0026
iPEPS
16
⇥
28
0
.
7673 0
.
002
a
Average of APBC and PBC results.
b
Using 2 independent tensors.
6
Method
Size
Wavelength Energy (
t
) Error (
t
)
iPEPS
16
⇥
16
c
diag.
4
p
2
0
.
7581 0
.
0014
DMET
2
⇥
2
d
-wave
0
.
7580 0
.
0005
DMET
3
⇥
23
0
.
7437 0
.
0009
DMET
4
⇥
2
(SF)
4
0
.
7614 0
.
00005
DMET
5
⇥
25
0
.
7691 0
.
001
DMET
6
⇥
2
(SF)
6
0
.
7706 0
.
00007
DMET
7
⇥
27
0
.
7704 0
.
0003
DMET
8
⇥
2
(SF)
8
0
.
7706 0
.
00001
DMET
9
⇥
29
0
.
7658 0
.
0008
DMET
2
p
2
⇥
p
2
d
-wave
0
.
7620 0
.
00001
DMET
5
p
2
⇥
p
2
frustrated
d
0
.
7689 0
.
0008
Table S2: Energy (per site) of stripes with UHF using effective
U/t
=2
.
7
. The effective
U
is
determined by self-consistent AFQMC procedure, described in Section S5.
Size
Wavelength Energy (
t
) Error (
t
)
8
⇥
2
(TABC)
4
1
.
0912
0.0004
10
⇥
2
(TABC)
5
1
.
0930
0.0003
12
⇥
2
(TABC)
6
1
.
0944
0.0002
14
⇥
2
(TABC)
7
1
.
0979
0.0003
16
⇥
2
(TABC)
8
1
.
1004
0.0002
18
⇥
2
(TABC)
9
1
.
0993
0.0001
20
⇥
2
(TABC)
10
1
.
0984
0.0002
22
⇥
2
(TABC)
11
1
.
0974
0.0002
c
Using 16 independent tensors.
d
No clear pattern, order appears to be frustrated.
7
Table S3: Best estimates of energy (per site) of stripes and competing states for
U/t
=6
.
AFQMC numbers obtained as described in section S5, DMET numbers obtained as described
in section S9. Other details as above.
Method
Size
Wavelength Energy (
t
) Error (
t
)
AFQMC
12
⇥
8
(PBC)
6
0
.
8684
0.0001
AFQMC
14
⇥
8
(PBC)
7
0
.
8692
0.0001
AFQMC
16
⇥
8
(PBC)
8
0
.
8718
0.0001
AFQMC
18
⇥
8
(PBC)
9
0
.
8701
0.0001
AFQMC
20
⇥
8
(PBC)
10
0
.
8702
0.0001
DMET
2
⇥
2
d
-wave
0
.
8679 0
.
0007
DMET
3
⇥
23
0
.
85867 0
.
00004
DMET
4
⇥
24
0
.
85890 0
.
00004
DMET
5
⇥
25
0
.
86836 0
.
00001
DMET
6
⇥
2
(SF)
6
0
.
87247 0
.
00001
DMET
7
⇥
27
0
.
87363 0
.
00002
DMET
8
⇥
2
(SF)
8
0
.
87667 0
.
0007
8
Table S4: Best estimates of energy (per site) of stripes and competing states for
U/t
=12
.
AFQMC numbers with twist averaged boundary conditions (TABC) obtained as described in
section S5, DMET numbers obtained as described in section S9, DMRG numbers obtained as
described in section S6. Other details as above.
Method
Size
Wavelength Energy (
t
) Error (
t
)
AFQMC
10
⇥
8
(TABC)
5
0
.
6446
0.0006
AFQMC
12
⇥
8
(TABC)
6
0
.
6452
0.0004
AFQMC
14
⇥
8
(TABC)
7
0
.
6461
0.0006
AFQMC
16
⇥
8
(TABC)
8
0
.
6458
0.0006
AFQMC
18
⇥
8
(TABC)
9
0
.
6462
0.0006
AFQMC
20
⇥
8
(TABC)
10
0
.
6450
0.0006
DMET
2
⇥
2
d
-wave
0
.
63940 0
.
00001
DMET
4
⇥
2
(SF)
4
0
.
6505 0
.
0001
DMET
5
⇥
25
0
.
6531 0
.
0001
DMET
6
⇥
2
(SF)
6
0
.
6526 0
.
0002
DMET
8
⇥
2
(SF)
8
0
.
6514 0
.
0001
DMRG
1⇥
4
4
-0.641379 0.000052
DMRG
1⇥
4
5
-0.64269 0.00019
DMRG
1⇥
4
6
-0.64285 0.00021
DMRG
1⇥
4
8
0.64168 0.00023
DMRG
1⇥
6
4
-0.6383
0.0026
DMRG
1⇥
6
5
-0.64148 0.00059
DMRG
1⇥
6
6
-0.6418
0.0013
DMRG
1⇥
6
8
-0.6438
0.0019
9
S4 Long-range Coulomb interaction
We estimate the long-range Coulomb interaction in the vertical stripes by computing the elec-
trostatic potential energy from charge patterns obtained from the DMET calculations
e
Coul
=
1
N
c
X
i
2
imp
,j,i
6
=
j
(
h
i
̄
h
)(
h
j
̄
h
)
/
4
⇡"
0
"
r
ij
(S1)
where
N
c
is the number of impurity sites,
h
i
is the hole density on site
i
, and
̄
h
is the average
Figure S1: Energy landscape before and after adding the estimated long-range Coulomb inter-
action for vertical stripes of different wavelength. The charge distributions are from DMET
calculations.
hole density (
1
/
8
). In atomic units, i.e. if we express the energy in Hartrees, and distance in
Bohr,
1
/
4
⇡"
0
=1
. The appropriate dielectric constant to use in a statically screened Coulomb
interaction in the CuO
2
plane has been estimated to lie between about
4
and
27
(
50,51
). We use
a dielectric constant of
"
=15
.
5
, and a lattice constant
a
=3
.
78
̊
A
=7
.
14
Bohr corresponding
to the lattice constant of La
2
CuO
4
. We transform the computed Coulomb energy (per site) to
units of
t
, using
t
⇠
3000
K
⇠
0
.
01
Hartree.
10
In 2D, the Coulomb summation converges reasonably fast. We choose a cutoff radius as
300 lattice spacings and converge the Coulomb energy to the fourth digit in units of
t
. The
results for the DMET vertical stripes are shown in Fig. S1. The long-range Coulomb interaction
favors shorter wavelength stripes and the homogeneous
d
-wave state, shifting the ground state to
wavelength
5
and making the uniform
d
-wave state also more competitive. Of course the above
treatment of the Coulomb term is quite crude, neglects the effect of relaxation in the presence of
the Coulomb interaction, and there is significant uncertainty in the dielectric. Nevertheless, the
calculation provides an energy scale,
O
(0
.
01
t
)
, over which the long-range Coulomb interaction
is important.
11
S5 AFQMC calculations
S5.1 Details of the AFQMC calculations
We studied cylinders of dimension
l
y
⇥
l
x
(
l
x
>l
y
) with several different boundary condi-
tions. In the first set of calculations, which allow for direct comparison with the finite system
DMRG calculations, we used open boundary conditions (OBC) along the
x
direction and pe-
riodic boundary conditions (PBC) along the
y
direction. We also applied pinning fields to pin
the underlying spin structures. Several types of pinning fields were used depending on the
targeted structure, as shown in Fig. S2. Along each edge, the pinning fields were always anti-
ferromagnetic. With FM (AFM) pinning fields, we targeted an odd (even) number of nodes
(
⇡
phase shifts) in the system (
l
x
is always even in our calculations). In some cases, we also
applied pinning fields only on one edge to accommodate states with different wavelengths. The
strength of the pinning fields is
|
h
|
=0
.
5
(units of
t
) for all calculations. All these calculations
used constrained path AFQMC method with self-consistent optimized trial wavefunctions (
33
).
In the second set of calculations, we used PBC along both directions, or twist averaged
boundary conditions (TABC) along both directions. The twist averaging allows us to reduce
the finite size errors in the total energy. These calculations used an unrestricted Hartree-Fock
(UHF) trial wavefunction generated by an effective
U
. In the following, DMRG results shown
for comparison are from section S6. Results are for
U
=8
unless otherwise stated.
S5.2 Wavelength
8
striped ground-state
The spin and hole structure of the
4
⇥
16
,
6
⇥
16
,
4
⇥
24
, and
6
⇥
32
systems are plotted in Figs.
S3, S4, S5, and S6 respectively. All the results are obtained with the self-consistent AFQMC
method starting from free electron trial wavefunctions.
Exhaustive comparisons were made between AFQMC and DMRG in these systems. The
ground state energies for these systems are listed in Table S5. The systematic error compared
12
Figure S2: Different types of pinning fields. The relative phase between the pinning fields on
the two edges is positive in (1) and negative in (2). We denote (1) and (2) by FM and AFM
pinning fields respectively. In some cases, to accommodate states with different wavelengths,
we also apply pinning fields on only one edge. We also studied systems with PBC and TABC
along both edges, to reduce the finite size effects. Notice that the pinning field along each edge
is always anti-ferromagnetic.
with DMRG from the constraint in AFQMC is about
0
.
4%
.
For the
6
⇥
16
system, two different pinning fields, i.e., AFM and FM, were tried. The energy
with AFM pinning field (wavelength 8) is lower. For the
6
⇥
32
system, self-consistent AFQMC
finds a ground state structure with wavelength
8
(
4
nodes) from a free electron trial wavefunc-
tion. We can also construct trial wavefunctions from the density obtained by DMRG. We calcu-
lated the energy using the two different trial wavefunctions constructed from the DMRG density
for the two states (
4
and
6
nodes). The energy comparison of the two stripe states is shown in
Table S5. We also did an AFQMC calculation by setting the trial wavefunction as an equal
linear combination of the two trial wavefunctions. The state with wavelength
8
survives after
convergence. The energy difference between the two states is
⇠
0
.
001
in DMRG and
⇠
0
.
003
in AFQMC. Again the AFQMC energies are slightly lower than the DMRG energies due to the
13
Table S5: Ground state energies for different systems with pinning fields. DMRG results from
section S6.
Size
pinning field
state
DMRG
AFQMC
4
⇥
16
AFM
2
nodes / wavelength
8
-0.77127(2) -0.7744(1)
6
⇥
16
(meta-stable)
FM
3
nodes
-0.7682(3) -0.7692(1)
6
⇥
16
AFM
2
nodes / wavelength
8
-0.7691(5) -0.7725(2)
4
⇥
24
FM
3
nodes / wavelength
8
-0.76939(3) -0.7727(2)
6
⇥
32
(meta-stable)
AFM
6
nodes
-0.7648(3) -0.7663(1)
6
⇥
32
AFM
4
nodes / wavelength
8
-0.7658(7) -0.7691(2)
constrained path approximation. However, the results are consistent (the state with wavelength
8
has lower energy).
After convergence, for all the systems studied, the effective
U
in the self-consistent AFQMC
calculation is about
U
=2
.
7
.
S5.3 Comparison of different wavelengths
S5.3.1
4
⇥
40
We studied the
4
⇥
40
cylinder which accommodates the states with wavelengths of 5 and 8.
We used different pinning fields to favor states with different wavelengths.
We applied AFM pinning fields to favor states with an even number of nodes, that is states
with wavelengths of
5
,
10
or
20
. The result from the self-consistent AFQMC with a free trial
wavefunction is a state with wavelength
10
. If we used a trial wavefunction with wavelength
Table S6: Ground state energies of
4
⇥
40
system. Pinning fields are applied on only one edge.
The energies for the two states with wavelength
8
and
10
are very close, however, when we carry
out the AFQMC calculation using the equal linear combination of the two trial wavefunctions
as the initial wavefunction, the state with wavelength
8
survives.
pinning field
state
AFQMC
One edge
8
nodes / wavelength
5
(doesn’t survive in self-consistent AFQMC)
0
.
7657(2)
One edge
5
nodes / wavelength
8
0
.
7663(1)
One edge
4
nodes / wavelength
10
0
.
7665(3)
14
Figure S3: Spin and hole densities along
l
x
(with each panel for one value of
l
y
) in
4
⇥
16
system
with AFM pinning fields. Blue circles are for AFQMC and red triangles are for DMRG.
5
(
52
) to start the self-consistent AFQMC, the final pattern again converged to wavelength 10.
This suggests the stripe with wavelength
5
is a higher energy state. The spin and charge patterns
are plotted in Fig. S7.
We also tried FM pinning fields with which we obtain the ground state with wavelength
8
.
The spin and charge pattern are plotted in Fig. S8.
To compare the patterns with wavelength
8
and wavelength
10
, we studied the
4
⇥
40
system
with pinning fields on only left edge of the cylinder which accommodates both patterns. We find
that the energies for these two trial wavefunctions are very close:
0
.
7665(3)
for wavelength 10
15
Figure S4:
6
⇥
16
, with AFM pinning fields. Blue circles are for AFQMC and red triangles are
for DMRG.
and
0
.
7663(1)
for wavelength 8. However, if we use an equal linear combination of the two
as the initial wavefunction, the pattern with wavelength 8 survives in the AFQMC calculation.
This suggests that the pattern with wavelength 8 is the true ground state. We also calculated the
energy of this system using the unstable state with wavelength
5
as the trial wavefunction. The
energy is higher:
0
.
7657(2)
.
16
Figure S5:
4
⇥
24
, with FM pinning fields. Blue circles are for AFQMC and red triangles are
for DMRG.
S5.3.2
4
⇥
48
We studied the
4
⇥
48
cylinder which accommodates states with wavelengths of
6
and
8
. Unlike
the
4
⇥
40
case, the AFM pinning fields are compatible with both patterns.
From a trial wavefunction with wavelength
8
, the self-consistent result is plotted in Fig. S9.
The converged state has wavelength of 12 with energy
0
.
7701(1)
.
The self-consistent result from a free trial wavefunction is plotted in Fig. S10. The con-
verged energy is
0
.
7699(1)
.
17