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
$\delta=1/8$.
Luciano Loris Viteritti, Francesco Ferrari, Federico Becca
SciPost Phys. 12, 166 (2022) ·
published 19 May 2022

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Neural networks have been recently proposed as variational wave functions for
quantum manybody systems [G. Carleo and M. Troyer, Science 355, 602 (2017)].
In this work, we focus on a specific architecture, known as Restricted
Boltzmann Machine (RBM), and analyse its accuracy for the spin1/2 $J_1J_2$
antiferromagnetic Heisenberg model in one spatial dimension. The ground state
of this model has a nontrivial sign structure, especially for $J_2/J_1>0.5$,
forcing us to work with complexvalued RBMs. Two variational Ans\"atze are
discussed: one defined through a fully complex RBM, and one in which two
different realvalued networks are used to approximate modulus and phase of the
wave function. In both cases, translational invariance is imposed by
considering linear combinations of RBMs, giving access also to the
lowestenergy excitations at fixed momentum $k$. We perform a systematic study
on small clusters to evaluate the accuracy of these wave functions in
comparison to exact results, providing evidence for the supremacy of the fully
complex RBM. Our calculations show that this kind of Ans\"atze is very flexible
and describes both gapless and gapped ground states, also capturing the
incommensurate spinspin correlations and lowenergy spectrum for
$J_2/J_1>0.5$. The RBM results are also compared to the ones obtained with
Gutzwillerprojected fermionic states, often employed to describe quantum spin
models [F. Ferrari, A. Parola, S. Sorella and F. Becca, Phys. Rev. B 97, 235103
(2018)]. Contrary to the latter class of variational states, the
fullyconnected structure of RBMs hampers the transferability of the wave
function from small to large clusters, implying an increase of the
computational cost with the system size.
SciPost Phys. 10, 012 (2021) ·
published 20 January 2021

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The recent progress in the optimization of twodimensional tensor networks
[H.J. Liao, J.G. Liu, L. Wang, and T. Xiang, Phys. Rev. X 9, 031041 (2019)]
based on automatic differentiation opened the way towards precise and fast
optimization of such states and, in particular, infinite projected entangledpair
states (iPEPS) that constitute a genericpurpose Ansatz for lattice problems
governed by local Hamiltonians. In this work, we perform an extensive study
of a paradigmatic model of frustrated magnetism, the J 1 − J 2 Heisenberg an
tiferromagnet on the square lattice. By using advances in both optimization
and subsequent data analysis, through finite correlationlength scaling, we re
port accurate estimations of the magnetization curve in the Néel phase for
J 2 /J 1 ≤ 0.45. The unrestricted iPEPS simulations reveal an U (1) symmetric
structure, which we identify and impose on tensors, resulting in a clean and
consistent picture of antiferromagnetic order vanishing at the phase transition
with a quantum paramagnet at J 2 /J 1 ≈ 0.46(1). The present methodology can
be extended beyond this model to study generic ordertodisorder transitions
in magnetic systems.
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.