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arXiv:1605.04641 · Physical review. B./Physical review. B

Effective Hamiltonian based Monte Carlo for the BCS to BEC crossover in the attractive Hubbard model

Kanika Pasrija, Prabuddha Chakraborty, Sanjeev Kumar

Published Oct 21, 2016• 5 citations
arXivHTMLPDF

Abstract

We present an effective Hamiltonian based real-space approach for studying the weak-coupling BCS to the strong-coupling Bose-Einstein condensate crossover in the two-dimensional attractive Hubbard model at finite temperatures. We introduce and justify an effective classical Hamiltonian to describe the thermal fluctuations of the relevant auxiliary fields. Our results for ${T}_{c}$ and phase diagrams compare very well with those obtained from more sophisticated and CPU-intensive numerical methods. We demonstrate that the method works in the presence of disorder and can be a powerful tool for a real-space description of the effect of disorder on superconductivity. From a combined analysis of the superconducting order parameter, the distribution of auxiliary fields, and the quasiparticle density of states, we identify the regions of metallic, insulating, superconducting, and pseudogapped behavior. Our finding of the importance of phase fluctuations for the pseudogap behavior is consistent with the conclusions drawn from recent experiments on NbN superconductors. The method can be generalized to study superconductors with nontrivial order-parameter symmetries by identifying the relevant auxiliary variables.

DOI
10.1103/physrevb.94.165150

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We present an effective Hamiltonian based real-space approach for studying the weak-coupling BCS to the strong-coupling Bose-Einstein condensate crossover in the two-dimensional attractive Hubbard model at finite temperatures. We introduce and justify an effective classical Hamiltonian to describe the thermal fluctuations of the relevant auxiliary fields. Our results for Tc{T}_{c}Tc​ and phase diagrams compare very well with those obtained from more sophisticated and CPU-intensive numerical methods. We demonstrate that the method works in the presence of disorder and can be a powerful tool for a real-space description of the effect of disorder on superconductivity. From a combined analysis of the superconducting order parameter, the distribution of auxiliary fields, and the quasiparticle density of states, we identify the regions of metallic, insulating, superconducting, and pseudogapped behavior. Our finding of the importance of phase fluctuations for the pseudogap behavior is consistent with the conclusions drawn from recent experiments on NbN superconductors. The method can be generalized to study superconductors with nontrivial order-parameter symmetries by identifying the relevant auxiliary variables.
A Fermi liquid with weak attractive interaction undergoes a BCS transition to a superconductor with reducing temperature. With increasing interaction strength, the thermal transition is progressively modified as the high temperature `metallic' phase develops a pseudogap due to pairing fluctuations and the resistivity above T_c shows insulating behaviour. The crossover to insulating character occurs much before the system can be considered to be in the BEC regime of preformed fermion pairs. We use a new Monte Carlo tool to map out the BCS-BEC crossover in the attractive Hubbard model on large two dimensional lattices and explicitly compute the resistivity to demonstrate how the metal to superconductor (MS) thermal transition at weak coupling crosses over to an insulator to superconductor (IS) transition at intermediate coupling. Our high resolution access to the single particle and optical spectrum at finite temperature allows us to completely describe the transport crossover in this longstanding problem.
Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak [Bardeen-Cooper-Schrieffer, (BCS)] and strong [Bose-Einstein condensation (BEC)] couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice from which several different quantities have been calculated and should be useful as a road map to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature Tc{T}_{c}Tc​ in terms of the band filling $\ensuremath{\langle}n\ensuremath{\rangle}$ and interaction strength UUU from which we pinpoint a somewhat wide region $|U|/t\ensuremath{\approx}5\ifmmode\pm\else\textpm\fi{}1$ (ttt is the hopping amplitude) and $\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath{\approx}0.79\ifmmode\pm\else\textpm\fi{}0.09$ leading to a maximum ${T}_{c}\ensuremath{\approx}0.16t$. Two additional temperature scales, namely, pairing Tp{T}_{p}Tp​ and degeneracy Td{T}_{d}Td​ have been highlighted: The former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), whereas the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.
We report a systematic study of the superconducting ground state in the two-dimensional attractive Hubbard model, employing the {\it numerically exact} auxiliary-field quantum Monte Carlo method. We focus primarily on the pairing order and condensate fraction across the BCS-BEC crossover. Our numerical calculations cover a broad and physically relevant parameter regime, with the interaction strength U/t=2∼12U/t=2\sim 12U/t=2∼12 (where ttt is the nearest-neighbor hopping amplitude) and fermion density n=0.25∼1.0n=0.25\sim 1.0n=0.25∼1.0. We systematically characterize the dependence of the pairing order and condensate fraction on both the interaction strength and fermion density, and perform finite-size extrapolations to obtain their thermodynamic-limit values. As complementary quantities, we also provide high-precision numerical results for the double occupancy and total energy. Furthermore, we investigate the effects of next-nearest-neighbor hopping on the superconducting properties of the model. Our results offer a comprehensive set of unbiased numerical data for the attractive Hubbard model, establishing quantitative benchmarks for understanding superconductivity across the BCS-BEC crossover and for validating future theoretical and numerical approaches.
Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak (Bardeen-Cooper-Schrieffer, BCS) and strong (Bose-Einstein condensation, BEC) couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice, from which several different quantities have been calculated and should be useful as a roadmap to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature, TcT_cTc​, in terms of the band filling, $\ave{n}$, and interaction strength, UUU, from which we pinpoint a somewhat wide region ∣U∣/t≈5±1|U|/t \approx 5 \pm 1∣U∣/t≈5±1 (ttt is the hopping amplitude) and $\ave{n} \approx 0.79 \pm 0.09$ leading to a maximum Tc≈0.16tT_c \approx 0.16 tTc​≈0.16t. Two additional temperature scales, namely pairing, TpT_pTp​, and degeneracy, TdT_dTd​, have been highlighted: the former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), while the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites, for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.
In this work we use functional integral techniques to examine the nearest-neighbor attractive Hubbard model on a quasi-two-dimensional lattice. It is a simple phenomenological model for the high-Tc{T}_{c}Tc​ cuprates that allows both extended (nonlocal) s- and d-wave singlet superconductivity as well as mixed symmetry states. The Hartree-Gor'kov mean-field theory of the model has a finite-temperature phase diagram which shows a transition from pure s-wave to pure d-wave superconductivity, via a mixed symmetry s+ids+ids+id state, as a function of doping. Including Gaussian fluctuations we examine the crossover from weak-coupling BCS superconductivity to the strong-coupling Bose-Einstein condensation of composite s- or d-wave bosons and comment on the origin and symmetry of the pseudogap.
We use the variational cluster approximation to study the superconducting ground state in the two-dimensional attractive Hubbard model, putting particular emphasis on the significance of quantum fluctuations of the system. We first show that the order parameter is suppressed in comparison with that obtained using the mean-field theory owing to the effects of spatial fluctuations in two-dimensional systems. We then show that the calculated Bogoliubov quasiparticle spectra and condensation amplitude clearly exhibit the character of Cooper pairs in momentum space and that the pair coherence length ξξξ evaluated from the condensation amplitude demonstrates a smooth crossover in real space from a weakly paired BCS state (ξ≫aξ\gg aξ≫a) to a BEC state of tightly bound pairs (ξ≪aξ\ll aξ≪a), where aaa is the lattice constant. The calculated kinetic and potential energies in the superconducting and normal ground states indicate that the superconducting state in the weak-coupling region is driven by the gain in potential energy, while that in the strongcoupling region is driven by the gain in kinetic energy.