SciPost Phys. 12, 180 (2022) ·
published 1 June 2022

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By using variational quantum Monte Carlo techniques, we investigate the
instauration of stripes (i.e., charge and spin inhomogeneities) in the Hubbard
model on the square lattice at hole doping $\delta=1/8$, with both nearest
($t$) and nextnearestneighbor hopping ($t^\prime$). Stripes with different
wavelengths $\lambda$ (denoting the periodicity of the charge inhomogeneity)
and character (bond or sitecentered) are stabilized for sufficiently large
values of the electronelectron interaction $U/t$. The general trend is that
$\lambda$ increases going from negative to positive values of $t^\prime/t$ and
decreases by increasing $U/t$. In particular, the $\lambda=8$ stripe obtained
for $t^\prime=0$ and $U/t=8$ [L.F. Tocchio, A. Montorsi, and F. Becca, SciPost
Phys. 7, 21 (2019)] shrinks to $\lambda=6$ for $U/t\gtrsim 10$. For
$t^\prime/t<0$, the stripe with $\lambda=5$ is found to be remarkably stable,
while for $t^\prime/t>0$, stripes with wavelength $\lambda=12$ and $\lambda=16$
are also obtained. In all these cases, pairpair correlations are highly
suppressed with respect to the uniform state (obtained for large values of
$t^\prime/t$), suggesting that striped states are not superconducting at
$\delta=1/8$.
SciPost Phys. 7, 021 (2019) ·
published 12 August 2019

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The dualism between superconductivity and charge/spin modulations (the
socalled stripes) dominates the phase diagram of many stronglycorrelated
systems. A prominent example is given by the Hubbard model, where these phases
compete and possibly coexist in a wide regime of electron dopings for both weak
and strong couplings. Here, we investigate this antagonism within a variational
approach that is based upon JastrowSlater wave functions, including backflow
correlations, which can be treated within a quantum Monte Carlo procedure. We
focus on clusters having a ladder geometry with $M$ legs (with $M$ ranging from
$2$ to $10$) and a relatively large number of rungs, thus allowing us a
detailed analysis in terms of the stripe length. We find that stripe order with
periodicity $\lambda=8$ in the charge and $2\lambda=16$ in the spin can be
stabilized at doping $\delta=1/8$. Here, there are no sizable superconducting
correlations and the ground state has an insulating character. A similar
situation, with $\lambda=6$, appears at $\delta=1/6$. Instead, for smaller
values of dopings, stripes can be still stabilized, but they are weakly
metallic at $\delta=1/12$ and metallic with strong superconducting correlations
at $\delta=1/10$, as well as for intermediate (incommensurate) dopings.
Remarkably, we observe that spin modulation plays a major role in stripe
formation, since it is crucial to obtain a stable striped state upon
optimization. The relevance of our calculations for previous densitymatrix
renormalization group results and for the twodimensional case is also
discussed.
SciPost Phys. 6, 018 (2019) ·
published 5 February 2019

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Shortrange antiferromagnetic correlations are known to open a spin gap in
the repulsive Hubbard model on ladders with $M$ legs, when $M$ is even. We show
that the spin gap originates from the formation of correlated pairs of
electrons with opposite spin, captured by the hidden ordering of a spinparity
operator. Since both spin gap and parity vanish in the twodimensional limit,
we introduce the fractional generalization of spin parity and prove that it
remains finite in the thermodynamic limit. Our results are based upon
variational wave functions and Monte Carlo calculations: performing a finite
sizescaling analysis with growing $M$, we show that the doping region where
the parity is finite coincides with the range in which superconductivity is
observed in two spatial dimensions. Our observations support the idea that
superconductivity emerges out of spin gapped phases on ladders, driven by a
spinpairing mechanism, in which the ordering is conveniently captured by the
finiteness of the fractional spinparity operator.
Dr Tocchio: "The table of the energy differ..."
in Comments  comment on Stripes in the extended $tt^\prime$ Hubbard model: A Variational Monte Carlo analysis