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.
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.
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 , ].
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 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.
Quasi-one-dimensional (Q1D) systems, i.e., three- and two-dimensional (3D/2D) arrays composed of weakly coupled one-dimensional lattices of interacting quantum particles, exhibit rich and fascinating physics. They are studied across various areas of condensed matter and ultracold atomic lattice-gas physics, and are often marked by dimensional crossover as the coupling between one-dimensional systems is increased or temperature decreased, i.e., the Q1D system goes from appearing largely 1D to largely 3D. Phase transitions occurring along the crossover can strongly enhance this effect. Understanding these crossovers and associated phase transitions can be challenging due to the very different elementary excitations of 1D systems compared to higher-dimensional ones. In the present work, we combine numerical matrix product state (MPS) methods with mean-field (MF) theory to study paradigmatic cases of dimensional crossovers and the associated phase transitions in systems of both hard-core and soft-core lattice bosons, with relevance to both condensed matter physics and ultracold atomic gases. We show that the superfluid-to-insulator transition is a first order one, as opposed to the isotropic cases, and calculate transition temperatures for the superfluid states, finding excellent agreement with analytical theory. At the same time, our MPS+MF approach keeps functioning well where the current analytical framework cannot be applied. We further confirm the qualitative and quantitative reliability of our approach by comparison to exact quantum Monte Carlo calculations for the full 3D arrays.
Abstract The study of ground-state properties of the Fermi-Hubbard model is a long-lasting task in the research of strongly correlated systems, and is crucial for the understanding of notable quantum phenomena including superconductivity and magnetism. Owing to the exponentially growing complexity of the system, a quantitative analysis usually demands high computational cost and is restricted to small samples, especially in two or higher dimensions. Here, we introduce a variational method in the frame of fermionic Gaussian states, and obtain the ground states of one- and two-dimensional attractive Hubbard models via imaginary-time evolution. We calculate the total energy and benchmark the results in a wide range of interaction strength and filling factor with those obtained via exact two-body results, the density matrix renormalization group based on matrix product states (MPS), and projector Quantum Monte Carlo (QMC) method. For both 1D and 2D cases, the Gaussian variational method presents accurate results for total energy with a maximum systematic error ∼ 4 % in the intermediate interaction region. The accuracy of these results has negligible dependence on the system size. We further calculate the double occupancy and find excellent agreement with MPS and QMC, as well as the experimental results of cold quantum gases in optical lattices. The results suggest that the Gaussian pairing state is a good approximation to the ground states of attractive Hubbard model, in particular in the strong and weak coupling limits. Moreover, we generalize the method to the attractive Hubbard model with a finite spin-polarization, which can be mapped to the repulsive interaction case via particle-hole transformation, and obtain accurate results of ground state energy and double occupancy. Our work demonstrates the ability of the Gaussian variational method to extract ground state properties of strongly correlated many-body systems with negligible computational cost, especially of large size and in higher dimensions.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.