Correlated low-dimensional electrons

Phase diagram and minimal model of quasi-1D organic unconventional superconductors. (a) Generic phase diagram of the organic Bechgaard and Fabre salts, consisting of many parallel chains, each formed from a 1D stack of cationic molecules. Depending on the temperature, tuning the (weak) interchain electron tunneling drives phase transitions from a localized phase to a 2D metal (at intermediate temperatures), or even from a 3D magnetically ordered phase to a 3D unconventionally superconducting (USC) phase (low temperature). (b) The 2D U-V Hubbard model, the minimal model of the organics thought - but not yet proven - to contain the essential physics of these materials, including the magnetically ordered and the USC phases. Studying this issue is a focus of this groups work using pDMRG. It consists of 2D lattice-electrons at half-filling with strong anisotropy in the tunneling (much stronger along the 1D chains than in-between them), strong on-site repulsion U and weaker on-chain nearest-neighbour repulsion V.
General scheme for experiments on 2D exciton condensate. (a) Two realizations of 2D electron gases, such as e.g. graphene, at close enough proximity that electron-hole pairs may form via screened Coulomb attraction, are each backgated. (b) The dispersion relations of the sheets are shifted via the backgating, s.t. one sheet is hole doped and the other electron doped. (c) Due to the formation of excitons and their condensation, a condensate gap opens where the dispersion relations of the non-interacting system would otherwise intersect. Yet, outside the Quantum Hall regime, no exciton condensation has been observed in 2D.
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-Tc 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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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-Tc 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 dxy-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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Evidence for an USC ground state in the U-V model from pDMRG. From forthcoming publication with Michele Dolfi, Matthias Troyer and Thierry Giamarchi. (a) The spin gap of infinitely long strips as a function of the strips' width Nch (different line-colours represent two different ways of accounting for numerical errors). Here, U=4t, and V has two different values. As correlations between chains decrease rapidly with distance, it is highly unlikely that the spin gap would start to decrease towards zero if Nch were further increased, the only scenario that would be compatible with a magnetically ordered ground state in the 2D-limit of diverging Nch. As charge-ordering can be ruled out on other grounds, USC order is the only plausible remaining option for the ground state, as a finite spin gap is also incompatible with any Fermi-liquid scenario. (b) At short ranges rs, correlation functions will increasingly resemble those of the full 2D limit as Nch grows. We find that an only in the dxy-symmetry channel do the correlations become strongly enhanced at or below rs with increasing Nch (black: Nch=2, red: Nch=4, green: Nch=6, blue: Nch=8). In all other channels, as well as for charge ordering, they continue to decay strongly even at short distances.
Exciton condensate in 1D bilayer system. From Phys. Rev. Lett. 119, 37601 (2017). (a) The setup is analogous to that proposed for 2D - two 1D electron systems, each backgated and separated by a spacer material thin enough to permit small yet finite inter-layer tunneling of electrons. (b) One system has an electron-like band (blue dashed curve) near the Fermi-edge, the other a hole-like band (red dashed curve). Backgating allows to control their relative position. Weak-interlayer tunneling results in a (trivial) single-particle gap opening up at the shared Fermi-level, the bands hybridize (shaded solid lines). (c) Using DMRG numerics, we show quasi-exactly that taking realistic long-range interactions into account (both inside and between layers), a many-body condensate state with off-diagonal long-range order forms. The condensate gap is shown here as a function of interlayer tunneling for three different scenarios (details in the article), and can reach up to 300 K in temperature.
Exciton condensate in 1D

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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 - such condensates could be used for ultra low-power logical circuits, showing only contact-limited resistance.

Yet, these states have not yet been realized outside the Quantum-Hall regime (low temperatures, high magnetic field).

Together with D. S. Abergel, my work shows that 1D bilayer systems have unique advantages over previous 2D proposals to realize exciton condensates at high temperatures and without strong magnetic fields [Phys. Rev. Lett. 119, 37601 (2017)]. This is not least due to the fact that screening is much reduced in 1D, as compared to 2D, where it is reducing achievable critical temperatures substantially or even very substantially (depending on the employed approximation).

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 temperatures on the order of hundreds of Kelvin for an exciton condensate to appear (c.f. figure).

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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