We combine matrix-product-state (MPS) and mean-field (MF) methods to model the real-time evolution of a three-dimensional (3D) extended Hubbard system formed from one-dimensional (1D) chains arrayed in parallel with weak coupling in-between them. This approach allows us to treat much larger 3D systems of correlated fermions out-of-equilibrium over a much more extended real-time domain than previous numerical approaches. We deploy this technique to study the evolution of the system as its parameters are tuned from a charge-density wave phase into the superconducting regime, which allows us to investigate the formation of transient non-equilibrium superconductivity. In our ansatz, we use MPS solutions for chains as input for a self-consistent time-dependent MF scheme. In this way, the 3D problem is mapped onto an effective 1D Hamiltonian that allows us to use the MPS efficiently to perform the time evolution, and to measure the BCS order parameter as a function of time. Our results confirm previous findings for purely 1D systems that for such a scenario a transient superconducting state can occur.
We combine matrix-product-state (MPS) and mean-field (MF) methods to model the real-time evolution of a three-dimensional (3D) extended Hubbard system formed from one-dimensional (1D) chains arrayed in parallel with weak coupling in-between them. This approach allows us to treat much larger 3D systems of correlated fermions out-of-equilibrium over a much more extended real-time domain than previous numerical approaches. We deploy this technique to study the evolution of the system as its parameters are tuned from a charge-density wave phase into the superconducting regime, which allows us to investigate the formation of transient non-equilibrium superconductivity. In our ansatz, we use MPS solutions for chains as input for a self-consistent time-dependent MF scheme. In this way, the 3D problem is mapped onto an effective 1D Hamiltonian that allows us to use the MPS efficiently to perform the time evolution, and to measure the BCS order parameter as a function of time. Our results confirm previous findings for purely 1D systems that for such a scenario a transient superconducting state can occur.
Materials with strong electronic correlations often exhibit a superconducting phase in close competition with other insulating phases, which is outstandingly difficult to resolve, e.g., for a group of quasi-two-dimensional (Q2D) materials such as the cuprates, even for the simplified minimal model of these materials, the doped 2D Hubbard model. The present work shows how quasi-one-dimensional (Q1D) systems, 2D and three-dimensional (3D) arrays of weakly coupled 1D correlated electrons, are much more amenable to resolving such competition, treating both instabilities on equal footing. Using the recently established matrix product state plus mean field ( MPS + MF ) approach for fermions [Bollmark , ], we demonstrate that large systems can be reached readily in these systems, which opens up the thermodynamic regime via extrapolation. Focusing on basic model systems, 3D arrays of negative- U Hubbard chains with additional nearest-neighbor interaction V , we show that despite the MF component of the MPS + MF technique, we can reproduce the expected coexistence of the superconductivity and charge density wave at V = 0 for density n = 1 . We then show how we can tune away from coexistence by both tuning V and doping the system. This work thus paves the way to deploy two-channel MPS + MF theory on some highly demanding high- T c superconducting systems, such as 3D arrays of repulsive- U doped Hubbard ladders; we recently characterized the properties of such arrays in single-channel MPS + MF calculations [Bollmark , ].
Correlated electron states are at the root of many important phenomena including unconventional superconductivity (USC), where electron pairing arises from repulsive interactions.Computing the properties of correlated electrons, such as the critical temperature T c for the onset of USC, efficiently and reliably from the microscopic physics with quantitative methods remains a major challenge for almost all models and materials.In this theoretical work, we combine matrix product states (MPS) with static mean field (MF) to provide a solution to this challenge for quasi-one-dimensional (Q1D) systems: two-and threedimensional materials comprised of weakly coupled correlated 1D fermions.This MPS þ MF framework for the ground state and thermal equilibrium properties of Q1D fermions is developed and validated for attractive Hubbard systems first, and further enhanced via analytical field theory.We then deploy it to compute T c for superconductivity in 3D arrays of weakly coupled, doped, and repulsive Hubbard ladders.The MPS þ MF framework thus enables the quantitative study of USC and high-T c superconductivity-and potentially many more correlated phases-in fermionic Q1D systems based directly on their microscopic parameters, in ways inaccessible to previous methods.This approach further allows one to treat competing macroscopic orders, such as superconducting and insulating ones, on an equal footing.Benchmarks of the framework using auxiliary-field quantum Monte Carlo techniques show that the overestimation of, e.g., T c due to its mean-field component, is near constant in microscopic parameters.These features of the MPS þ MF approach to correlated fermions open up the possibility of designing deliberately optimized Q1D superconductors, from experiments in ultracold gases to synthesizing new materials.
We study quasi-one-dimensional strongly correlated materials using a multi-step approach based on density functional theory, downfolding techniques, and tensor-network simulations. The downfolding procedure yields effective multiband Hubbard models that capture the competition between electron hopping and local Coulomb interactions relevant to the system's low-energy properties. The resulting multiband Hubbard models are solved using matrix product states. Applied to Sr2CuO3, SrBaCuO3, and Ba2CuO3, this purely one-dimensional treatment yields no long-range magnetic order, in contrast to the magnetic ordering observed experimentally. To account for this behavior, we extend the multi-step approach by incorporating interchain couplings through a self-consistent mean-field scheme. This combined approach stabilizes finite staggered magnetizations, providing a consistent description of magnetic order in agreement with experiment. For Sr2CuO3.5 and SrCuO2, we also tested an approach proposed for ladder materials, however, we find that these materials are not well suited for this approach due to the small magnitude of the intraladder hopping parameters.
Predicting critical temperature in quantum simulators for high-Tc superconductivity: the matrix product state plus mean field approach
Quantum simulation based on ultra cold atomic lattice gases is one of the most promising platforms to investigate high-Tc superconductivity beyond the limited capabilities of quantum many body numerics on classical computers. Yet, despite enormous progress since the field's inception, realizing a high-Tc superconducting state still remains out of reach. The present work lays the groundwork to purpose the recently proposed, and already partly realized, mixed-dimensional (mixD) models, towards this end. These systems offer the proven capability to realize very high pairing energies while retaining appreciable mobility of pairs. We specifically investigate the potential of 2D mixD-models with anisotropic tunneling, using the matrix product state plus mean field theory (MPS+MF) for fermions, and show that these models may enter a high-Tc superconducting phase. These simulations in turn are based on a comprehensive characterization of the 1D mixD-systems, which are the sub-units of which the 2D system is comprized. In this, we cover the range of currently experimentally relevant system sizes, and establish practical heuristics to determine when finite\hyp size effects preclude the use of a 1D mixD-system to build the 2D ones.
We provide a quasi-one-dimensional model which can support a pair-density-wave (PDW) state, in which the superconducting (SC) order parameter modulates periodically in space, with gapless Bogoliubov quasiparticle excitations. The model consists of an array of strongly interacting one-dimensional systems, where the one-dimensional systems are coupled to each other by local interactions and tunneling of the electrons and Cooper pairs between them. Within the interchain mean-field theory, we find several SC states from the model, including a conventional uniform SC state, PDW SC state, and a coexisting phase of the uniform SC and PDW states. In this quasi-1D regime we can treat the strong correlation physics essentially exactly using bosonization methods and the crossover to the 2D system by means of interchain mean-field theory. The resulting critical temperatures of the SC phases generically exhibit a power-law scaling with the coupling constants of the array, instead of the essential singularity found in weak-coupling BCS-type theories. Electronic excitations with an open Fermi surface, which emerge from the electronic Luttinger liquid systems below their crossover temperature to the Fermi liquid, are then coupled to the SC order parameters via the proximity effect. From the Fermi surface thus coupled to the SC order parameters, we calculate the quasiparticle spectrum in detail. We show that the quasiparticle spectrum can be fully gapped or nodal in the uniform SC phase and also in the coexisting phase of the uniform SC and PDW parameters. In the pure PDW state, the excitation spectrum has a reconstructed Fermi surface in the form of Fermi pockets of Bogoliubov quasiparticles.
Superconductivity in its various manifestations has been stimulating both experimental and theoretical progress in condensed-matter physics for more than a hundred years. The remarkable property of electrons to pair up and form quasi-particles gives rise to a plethora of phenomena featuring important practical applications not only in science, but, for instance, also in medicine and metrology. Recently, new directions in investigating this fascinating subject emerged, such as superconductivity out-of equilibrium and topological superconductors. Providing experimental evidence for enhanced superconducting correlations in optically pumped copper oxides at temperatures far above the equilibrium transition temperature, the first issue caused considerable excitement. On the other hand, topological superconductors are believed to provide realizations of highly fault-tolerant qubits by means of hosting non-Abelian quasi-particles, which can be the building blocks of scalable quantum computers. Experimentally verifying the emergence of these Majorana edge modes, exotic quasi-particles in heterostructures consisting of a conventional superconductor and semiconductors or topological insulators, is one of the most urgent questions to be answered right now. Both subjects cannot be accounted for with analytically solvable approximations only, and also provide very challenging numerical problems. We implemented a matrix-product state (MPS) based toolkit exploiting U(1) symmetries, providing a flexible and efficient platform to study these complex systems. In order to efficiently simulate out-of equilibrium setups we studied, compared, and developed time-evolution algorithms for MPS enabling us to choose the most suitable method for a given task. We also developed a new framework to represent operators in an enlarged Hilbert space so that benefits from conserving U(1) symmetries can also be exploited in systems that originally break such symmetries (projected purification). Using this method we could efficiently model mesoscopic phenomena such as a charging energy controlled by a gate electrode without further approximations. Equipped with this techniques we studied out-of equilibrium spectral functions to explore how to identify superconducting correlations more reliably on ultra-short timescales. We found conclusive evidence that in particular two-particle spectral functions yield excellent probes for the formation of a (quasi-)condensate out-of equilibrium. Furthermore, we also investigated the question whether in a particular model system there is the possibility of true long-range order out-of equilibrium by studying correlation matrices and the scaling of their eigenvalues. Here, we observe a change in the algebraic decay of the correlations, even though the extrapolated order parameter is still zero within the error bounds. Furthermore, we also investigated the effects of coupling a superconductor-semiconductor heterostructure, which is subject to an in-plane magnetic field and a charging energy controlled by a gate voltage, to normal leads. In the context of experimentally verifying the existence of Majorana edge modes, such systems are believed to be the most promising and recent studies seem to underline this expectation. However, in order to consistently analyze the experimental data, the effects of quantum fluctuations caused by hybridization of the heterostructure with the leads have to be understood. Here, only perturbative limits are available so far, i.e., the weak and strong tunneling limit, while the experimentally relevant regime is expected to be somewhere inbetween. We aimed to fill this gap using the projected purification method to calculate the ground state phase diagram over a wide parameter regime. Our results indicate that the experimental situation is much more involved than what is predicted from perturbative analysis.