We demonstrate that summation of connected diagrams to high order starting from a BCS hamiltonian is a viable generic unbiased approach for strongly correlated fermions in superconducting or superfluid phases. For the 3D attractive Hubbard model in a strongly correlated regime, we observe convergence of the diagrammatic series, evaluated up to 12 loops thanks to the connected determinant diagrammatic Monte Carlo algorithm. Our study includes the polarized regime, where conventional quantum Monte Carlo methods suffer from the fermion sign problem. Upon increasing the Zeeman field, we observe the first-order superconducting-to-normal phase transition at low temperature, and a thermally activated polarization of the superconducting phase well described by quasiparticle theory.
We study the three-dimensional (3D) attractive Hubbard model by means of the determinant quantum Monte Carlo method. This model is a prototype for the description of the smooth crossover between BCS superconductivity and Bose-Einstein condensation. By detailed finite-size scaling we extract the finite-temperature phase diagram of the model. In particular, we interpret the observed behavior according to a scenario of two fundamental temperature scales;
T∗ associated with Cooper pair formation and
Tc with condensation (giving rise to long-range superconducting order). Our results also indicate the presence of a recently conjectured phase transition hidden by the superconducting state. A comparison with the 2D case is briefly discussed, given its relevance for the physics of high-
Tc cuprate superconductors.
The dual fermion approach provides a formally exact prescription for calculating properties of a correlated electron system in terms of a diagrammatic expansion around dynamical mean-field theory (DMFT). Most practical implementations, however, neglect higher-order interaction vertices beyond two-particle scattering in the dual effective action and further truncate the diagrammatic expansion in the two-particle scattering vertex to a leading-order or ladder-type approximation. In this work, we compute the dual fermion expansion for the two-dimensional Hubbard model including all diagram topologies with two-particle interactions to high orders by means of a stochastic diagrammatic Monte Carlo algorithm. We benchmark the obtained self-energy against numerically exact diagrammatic determinant Monte Carlo simulations to systematically assess convergence of the dual fermion series and the validity of these approximations. We observe that, from high temperatures down to the vicinity of the DMFT N\'eel transition, the dual fermion series converges very quickly to the exact solution in the whole range of Hubbard interactions considered ($4\ensuremath{\le}U/t\ensuremath{\le}12$), implying that contributions from higher-order vertices are small. As the temperature is lowered further, we observe slower series convergence, convergence to incorrect solutions, and ultimately divergence. This happens in a regime where magnetic correlations become significant. We find, however, that the self-consistent particle-hole ladder approximation yields reasonable and often even highly accurate results in this regime.
In non-perturbative regimes, the superfluid instability in the
2d Hubbard model can be described by an emergent BCS theory with small effective pairing constants. We compute the effective couplings using a controlled bold-line diagrammatic Monte Carlo approach, which stochastically sums all skeleton Feynman diagrams dressed in the one- and two-particle channels to high expansion orders, and map out the resulting superfluid ground-state phase diagram in a range of next-nearest-neighbor hopping
0≤t′≤0.3t, interaction strength
0≤U≤3t, and lattice filling
0≤n≤2. The phase diagram is dramatically transformed in the hole-doped region and becomes particularly rich at larger doping and
t′. At
t′=0.3, the weak-coupling picture with the dominant triplet pairing sharply peaked at
n≈0.73 due to the Van Hove singularity is replaced by a plateau of the singlet
dx2−y2 paring, while for
U≳3t the effective couplings are consistent with the
dx2−y2 high-temperature superconductivity in the hole-doped region near cuprates' optimal doping.
Using a dual representation of lattice fermion models that is based on spin-charge transformation and fermionization of the original description, I derive an algorithm for diagrammatic Monte Carlo simulation of strongly correlated systems. This scheme allows eliminating large expansion parameters, as well as large corrections to the density matrix that generally prevent diagrammatic methods from being efficient in this regime. As an example, I compute the filling factor for the Hubbard model at infinite on-site repulsion and compare the results to controllable data obtained from numerical linked-cluster expansion. I find excellent agreement between the two methods, as well as rapid convergence of the diagrammatic series. I also report results for the momentum distribution and kinetic energy of the electrons.
In nonperturbative regimes, the superfluid instability in the two-dimensional (2D) Hubbard model can be described by an emergent BCS theory with small effective pairing constants. We compute the effective couplings using a controlled bold-line diagrammatic Monte Carlo approach, which stochastically sums all skeleton Feynman diagrams dressed in the one- and two-particle channels to high expansion orders, and map out the resulting superfluid ground-state phase diagram in a range of next-nearest-neighbor hopping $0\ensuremath{\le}{t}^{\ensuremath{'}}\ensuremath{\le}0.3t$, interaction strength $0\ensuremath{\le}U\ensuremath{\le}3t$, and lattice filling $0\ensuremath{\le}n\ensuremath{\le}2$. The phase diagram is dramatically transformed in the hole-doped region and becomes particularly rich at larger doping and ${t}^{\ensuremath{'}}$. At ${t}^{\ensuremath{'}}=0.3$, the weak-coupling picture with the dominant triplet pairing sharply peaked at $n\ensuremath{\approx}0.73$ due to the Van Hove singularity is replaced by a plateau of the singlet ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ paring, while for $U\ensuremath{\gtrsim}3t$ the effective couplings are consistent with the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ high-temperature superconductivity in the hole-doped region near cuprates' optimal doping.
We consider a lattice version of the ``negative-
U′′ Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse--grained quantum
XY model the phase fluctuation algebra between number and phase operators (given by the Euclidean group
E2), we map the ``phase-only'' action onto a solvable quantum spherical model. We establish the self--consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature
Tc as a function of the coupling strength
∣U∣, exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature
Tg due to a state with bound pairs but without long--range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{g}\ensuremath{\sim}4$ is almost independent on the pairing strength
∣U∣/t, whereas $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{c}$ increases with increasing
∣U∣---in striking analogy with the phenomenology of the high--temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.