Correlated low-dimensional electrons

Systems in 1D and 2D are central to many intensely-studied phases of correlated electrons.
Among them are **unconventionally superconducting (USC) materials** resulting from repulsion-mediated pairing of electrons,
as well as **exciton condensates** based on pairing of electrons and holes. These macroscopic
quantum states are of great fundamental as well as technological interest, as they offer lossless transport
of current and/or energy.

In USC materials, the superconducting phase is generally in direct competition with Mott-insulating and magnetically ordered phases.**It presents a great challenge to understand this competition and the physics taking place
inside the USC phase**, as for example
in the ongoing effort to comprehend the mechanism behind high-T_{c} superconductivity.

I am working on a paradigmatic case in this domain, the minimal U-V Hubbard model of the organic Bechgaard and Fabre salts. In these materials, repulsively interacting electrons prefer to move along 1D chains (formed by stacks of cationic organic molecules). Weak interchain tunneling of electrons then is found to provide the critical ingredient to tune the system from magnetically ordered insulator to unconventional superconductor. The strong spatial inhomogeneity of this quasi-1D setup provides a critical technical advantage over the weakly doped 2D Hubbard model. This is so because this property allows to use parallel DMRG (pDMRG) to investigate the physics of an USC phase and how electron pairing may arise from purely repulsive interactions with reliable numerics in the limit of infinite system size. The results of large-scale calculations (with single simulations spread over dozens or even hundreds of nodes) on the Piz Daint parallel supercomputer offer**strong indirect evidence from reliable numerics that a minimal model of an USC material can enter a superconducting phase at strong coupling**,
something had proven to be very difficult so far.

In my research on bilayer excitons, the situation is somewhat reversed. Theory is certain in principle as to how these stable bound states of electrons and holes form and then may enter a condensate phase. However, outside the Quantum Hall regime (low temperature, high magnetic field) so far no experiment on 2D bilayers could show a condensate forming. While there is no canonical explanation for this yet, important factors are known to be the impact of screening on the electron-hole attraction that is critical for exciton formation, and the problem of having to achieve nesting of two 2D Fermi surfaces experimentally. My focus here is how**1D bilayer systems remove the problem of screening,
and provide a chance to observe exciton condensates at temperatures of up to a few hundred Kelvin**.

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In USC materials, the superconducting phase is generally in direct competition with Mott-insulating and magnetically ordered phases.

I am working on a paradigmatic case in this domain, the minimal U-V Hubbard model of the organic Bechgaard and Fabre salts. In these materials, repulsively interacting electrons prefer to move along 1D chains (formed by stacks of cationic organic molecules). Weak interchain tunneling of electrons then is found to provide the critical ingredient to tune the system from magnetically ordered insulator to unconventional superconductor. The strong spatial inhomogeneity of this quasi-1D setup provides a critical technical advantage over the weakly doped 2D Hubbard model. This is so because this property allows to use parallel DMRG (pDMRG) to investigate the physics of an USC phase and how electron pairing may arise from purely repulsive interactions with reliable numerics in the limit of infinite system size. The results of large-scale calculations (with single simulations spread over dozens or even hundreds of nodes) on the Piz Daint parallel supercomputer offer

In my research on bilayer excitons, the situation is somewhat reversed. Theory is certain in principle as to how these stable bound states of electrons and holes form and then may enter a condensate phase. However, outside the Quantum Hall regime (low temperature, high magnetic field) so far no experiment on 2D bilayers could show a condensate forming. While there is no canonical explanation for this yet, important factors are known to be the impact of screening on the electron-hole attraction that is critical for exciton formation, and the problem of having to achieve nesting of two 2D Fermi surfaces experimentally. My focus here is how

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Understanding and solving the U-V model of unconventional superconductors

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The minimal models of all unconventional superconductors are those abstracted from the complex structures of the underlying materials. For the organic Bechgaard and Fabre salts, this is the 2D U-V Hubbard model at half-filling (pictured above), for the high-T_{c} cuprates it is the the weakly doped 2D Hubbard model.

In the regime of strong electron repulsion that is relevant to explain experiments,**it is however a major challenge for theory to show
unambiguously that any of these minimal models actually can have a superconducting ground state**. The first major use in my work for
parallel DMRG has been to approach this problem for the 2D U-V model of the organics (with Michele Dolfi,
Matthias Troyer and Thierry Giamarchi, publication forthcoming).

Compared to the weakly doped 2D Hubbard model, the U-V model has the unique advantage that using pDMRG the spin gap can be extrapolated to infinite length for strips of finite width. In this way,**we find a parameter regime where the main competitor phases, the magnetic orders, are highly unlikely to occur**
(c.f. figure).
This strong indirect evidence for a USC ground state is supplemented by an increase of correlations
exclusive to the *d*_{xy}-symmetry channel as the strips' width increases (c.f. figure).

As companion work to this research on the 2D lattice, Thierry Giamarchi and myself are studying the surprising competition between superconducting and charge-density order on the 2-leg U-V ladder in the regime relevant to the organics (publication forthcoming). Contrary to intuition,**we show that an increase in transverse kinetic energy by coupling two U-V chains together
drives the system towards being a charge-ordered insulator** for a wide range of parameters.

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The minimal models of all unconventional superconductors are those abstracted from the complex structures of the underlying materials. For the organic Bechgaard and Fabre salts, this is the 2D U-V Hubbard model at half-filling (pictured above), for the high-T

In the regime of strong electron repulsion that is relevant to explain experiments,

Compared to the weakly doped 2D Hubbard model, the U-V model has the unique advantage that using pDMRG the spin gap can be extrapolated to infinite length for strips of finite width. In this way,

As companion work to this research on the 2D lattice, Thierry Giamarchi and myself are studying the surprising competition between superconducting and charge-density order on the 2-leg U-V ladder in the regime relevant to the organics (publication forthcoming). Contrary to intuition,

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A bilayer exciton is a stable bound state of an electron in one band of a solid with a hole in another band. These bands may be inside the same solid or in two different ones.

There is great fundamental and technological interest in having a macroscopic number of such excitons condense -

Together with D. S. Abergel, my work shows that

Moreover, all interactions can be treated quasi-exactly in 1D by using DMRG numerics. In this way, we can reliably predict that realistic system parameters for a bilayer system can yield

The key to enabling the condensate in 1D lies in inter-layer tunnelling quenching the phase fluctuations that would normally preclude the condensate from forming in 1D.

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