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arXiv:1408.2088 · Europhysics Letters (EPL)

Emergent BCS regime of the two-dimensional fermionic Hubbard model: Ground-state phase diagram

Youjin Deng, Evgeny Kozik, Nikolay V. Prokof'ev, Boris V. Svistunov

Published Jun 1, 2015• 97 citations
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Abstract

For over half a century, the Hubbard model has played a paradigmatic role in attempts to understand quantum phenomena exhibited by correlated electrons in solids. Despite a substantial effort and the apparent simplicity of the model, its behavior in many important regimes has remained unknown. Here we study superfluidity in the two-dimensional Hubbard model with controlled error bars up to the coupling strength U = 4 and filling factor . We show, by means of unbiased diagrammatic Monte Carlo simulations, that in this regime the superfluid transition is governed by Fermi liquid physics with an emergent weak BCS-type coupling driving the instability. The corresponding ground-state phase diagram in the plane describes the competition between the superfluid states of p - and d -wave symmetry. We also report dimensionless coupling constants in this effective BCS regime.

DOI
10.1209/0295-5075/110/57001

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For over half a century, the Hubbard model has played a paradigmatic role in attempts to understand quantum phenomena exhibited by correlated electrons in solids. Despite a substantial effort and the apparent simplicity of the model, its behavior in many important regimes has remained unknown. Here we study superfluidity in the two-dimensional Hubbard model with controlled error bars up to the coupling strength U = 4 and filling factor . We show, by means of unbiased diagrammatic Monte Carlo simulations, that in this regime the superfluid transition is governed by Fermi liquid physics with an emergent weak BCS-type coupling driving the instability. The corresponding ground-state phase diagram in the plane describes the competition between the superfluid states of p - and d -wave symmetry. We also report dimensionless coupling constants in this effective BCS regime.
Superfluid ground-state phase diagram of the 2d2d2d Hubbard Model in the emergent BCS regime
In non-perturbative regimes, the superfluid instability in the 2d2d2d 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′≤0.3t0 \leq t^{\prime} \leq 0.3t0≤t′≤0.3t, interaction strength 0≤U≤3t0 \leq U \leq 3t0≤U≤3t, and lattice filling 0≤n≤20 \leq n \leq 20≤n≤2. The phase diagram is dramatically transformed in the hole-doped region and becomes particularly rich at larger doping and t′t't′. At t′=0.3t'=0.3t′=0.3, the weak-coupling picture with the dominant triplet pairing sharply peaked at n≈0.73n \approx 0.73n≈0.73 due to the Van Hove singularity is replaced by a plateau of the singlet dx2−y2d_{x^2-y^2}dx2−y2​ paring, while for U≳3tU \gtrsim 3tU≳3t the effective couplings are consistent with the dx2−y2d_{x^2-y^2}dx2−y2​ high-temperature superconductivity in the hole-doped region near cuprates' optimal doping.
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 investigate the ground-state phase diagram of the two-dimensional Hubbard model based on the optimization variational Monte Carlo method. We use a wave function that is an off-diagonal type given as ψ=exp⁡(−λK)PGψ0ψ=\exp(-λK)P_Gψ_0ψ=exp(−λK)PG​ψ0​, where ψ0ψ_0ψ0​ is a one-particle state, PGP_GPG​ is the Gutzwiller operator, KKK is the kinetic operator, and λλλ is a variational parameter. The many-body effect plays an important role as an origin of spin correlation and superconductivity in correlated electron systems. We examine the competition between the antiferromagnetic state and superconducting state by varying the Coulomb repulsion UUU, the band parameter t′t't′ and the electron density nen_ene​. We show a phase diagram that includes superconducting and antiferromagnetic phases and that t′=0t'=0t′=0 is most favorable for superconductivity.
We establish the full ground state phase diagram of the disordered Bose-Hubbard model in two dimensions at a unity filling factor via quantum Monte Carlo simulations. Similarly to the three-dimensional case we observe extended superfluid regions persisting up to extremely large values of disorder and interaction strength which, however, have small superfluid fractions and thus low transition temperatures. In the vicinity of the superfluid-insulator transition of the pure system, we observe an unexpectedly weak--almost not resolvable--sensitivity of the critical interaction to the strength of (weak) disorder.
We analyze the ground-state properties of mixtures consisting of scalar bosons and spin-12\frac{1}{2}21​ fermions using a mean-field treatment of the local boson-fermion interaction on a simple cubic lattice. In the deep superfluid limit of the boson sector and the BCS regime of the fermion sector, we derive BCS-type equations to determine the phase diagram of the system. We find a competition between a charge density wave and a superconducting phase. In the opposite limit, we study the Mott-insulator-to-superfluid transition of the boson sector in the presence of a staggered density-induced alternating potential provided by the fermions, and determine the mean-field transition line. In the two-superfluids phase of the mixture, we restrict to nearest-neighbor-induced interactions between the fermions and consider the extended Hubbard model. We perform a mean-field analysis of the critical temperature for the formation of boson-assisted sss-, extended ${s}^{\ensuremath{-}}$-, ddd-, and ppp-wave pairs at fermionic half-filling. We compare our results with a recent dynamical mean-field study [P. Anders et al., Phys. Rev. Lett. 109, 206401 (2012)].
The superfluid state in the attractive Hubbard is studied for the stacked square lattices to simulate the two- to three-dimensional crossover effects. We use the variation cluster approximation to calculate the order parameter and pair coherence length from the weak-coupling BCS state to the strong-coupling BEC state at T = 0 K. We show that in the weak-coupling BCS region the density of states at Fermi level plays an important role in the dimensional crossover of the superfluid state but in the strong-coupling BEC region the quantum fluctuations depending on the dimensionality of the system play an essential role.