wide range of U/t indicates that striped orders
with low-energy fluctuations of domain walls
remain a robust feature in the moderate to
strongly coupled underdoped region.
Connection to stripe order in HTSCs
In HTSCs, the accepted stripe wavelength at 1/8
doping (e.g., in LaSrCuO) is l ≈ 4.3 (close to half-filled) (40). However, we find that the l = 4 stripe
is not favored in the 2D Hubbard model for the
coupling range (U/t = 6 to 12) normally considered
most relevant to cuprate physics. This implies that
the detailed charge ordering of real materials arises
from even stronger coupling or, more likely, quantitative corrections beyond the simple Hubbard
model. With respect to the latter, one possibility
is long-range hopping (such as a next-nearest-neighbor hopping), which has been seen to change
the preferred stripe wavelength in the frustrated
t-J model (45). Another possibility is the long-range Coulomb repulsion. Long-range repulsion
can play a dual role, in both driving charge inhomogeneity as well as smoothing it out. In the
Hubbard model, where stripes naturally form,
the latter property can help drive the ground state
toward shorter stripe wavelengths. We have estimated the effect of the long-range interactions on
the stripe energetics by computing the Coulomb
energy of the charge distributions in Fig. 4. We
use a dielectric constant of 15.5 [in the range
proposed for the cuprate plane (50)]. This gives
a contribution favoring the shorter wavelength
stripes that is on the order of ~0.01t per site for
the l = 4 versus l = 8 stripe (30). Although this
is only an order-of-magnitude estimate, it is on
the same energy scale as the stripe energetics in
Fig. 2 and thus provides a plausible competing
mechanism for detailed stripe physics in real
In this work, we have used state-of-the-art
numerical methods to determine the ground
state of the 1/8 doping point of the 2D Hubbard
model at moderate to strong coupling. Through
careful convergence of all the methods, and
exhaustive cross-checks and validations, we are
able to eliminate several of the competing orders
that have been proposed for the underdoped
region in favor of a vertically striped order with
wavelength near l ≈ 8. The striped order displays a remarkably low-energy scale associated
with changing its wavelength, which implies
strong fluctuations either at low temperature
or in the ground state itself. This low-energy
scale can roughly be accounted for at the mean-field level with a strongly renormalized U. We
find coexisting pairing order with a strength
dependent on the stripe wavelength, indicating a coupling of stripe fluctuations to superconductivity. The stripe degeneracy is robust,
as the coupling strength is varied.
It has long been a goal of numerical sim-
ulations to provide definitive solutions of mi-
croscopic models. Our work demonstrates that
even in one of the most difficult condensed
matter models, such unambiguous simulations
are now possible. In so far as the 2D Hubbard
model is a realistic model of high-temperature
superconductivity, the stripe physics observed
here provides a firm basis for understanding the
diversity of inhomogeneous orders seen in the
materials, as well as a numerical foundation
for the theory of fluctuations and its connec-
tions to superconductivity. However, our work
also enables us to see the limitations of the
Hubbard model in understanding real HTSCs.
Unlike the stripes at this doping point in real
materials, we find filled stripes rather than
near-half-filled stripes. Given the very small
energy scales involved, terms beyond the
Hubbard model, such as long-range Coulomb
interactions, will likely play a role in the de-
tailed energetics of stripe fillings. The work
we have presented provides an optimistic per-
spective that achieving a comprehensive numer-
ical characterization of more-detailed models
of the HTSCs will also be within reach.
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Work performed by B.-X.Z., C.-M.C., M.-P.Q., H.S.,
S.R. W., S.Z., and G.K.-L.C. was supported by the Simons
Foundation through the Simons Collaboration on the
Many Electron Problem. S.R. W. acknowledges support
from the NSF (DMR-1505406), as do S.Z. and H.S.
(DMR-1409510). M.-P.Q. was also supported by
the U.S. Department of Energy (DOE) (DE-SC0008627).
G.K.-L.C. acknowledges support from a Simons
Investigators Award and the DOE (DE-SC0008624).
DMET calculations were carried out at the National
Energy Research Scientific Computing Center, a DOE
Office of Science User Facility supported by
DE-AC02-05CH11231. AFQMC calculations were
carried out at the Extreme Science and Engineering
Discovery Environment, supported by the NSF
(ACI-1053575); at the Oak Ridge Leadership
Computing Facility at Oak Ridge National Lab; and
at the computational facilities at the College of
William and Mary. P.C. was supported by the European
Research Council under the European Union’s
Horizon 2020 research and innovation program
(grant no. 677061). G.E. and R.M.N. acknowledge
support from the Deutsche Forschungsgemeinschaft
(DFG) through grant no. NO 314/5-1 in Research
Unit FOR 1807. Data used in this work are in the
supplementary materials and online at github.com/zhengbx/
stripe_data. The DMET calculations were performed
with the DMET library, available online at bitbucket.org/
zhengbx/libdmet. The real-space DMRG calculations
were performed with ITensor, available online at ITensor.org.
Access to the other computer codes can be arranged
with the authors upon reasonable request: For the AFQMC
code, please contact S.Z.; for the hybrid DMRG code,
please contact R.M.N.; and for the iPEPS code, please
Materials and Methods
Figs. S1 to S41
Tables S1 to S10
5 January 2017; accepted 17 October 2017
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