Recent advances on Hubbard models using quantum cluster methods

Recent advances on Hubbard models using quantum cluster methods David Sénéchal Département de physique Faculté des sciences Université de Sherbrooke 213 CAP Congress, Université de Montréal May 27, 213 David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 1 / 37

Outline 1 Quantum cluster methods 2 Square lattice Hubbard model and superconductivity 3 A spin liquid on the honeycomb lattice? David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 2 / 37

Outline Quantum cluster methods 1 Quantum cluster methods 2 Square lattice Hubbard model and superconductivity 3 A spin liquid on the honeycomb lattice? David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 3 / 37

Quantum cluster methods Section 1 Quantum cluster methods David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 4 / 37

Motivation Quantum cluster methods The one-band Hubbard model: hopping amplitude H = r,r,σ t r,r c rσ creation operator c r σ + U number of spin electrons at r nr n r µ n r,σ r r,σ repulsion Simplified description of the superconducting cuprates Ultra-cold atoms in optical lattices Prototype for interacting fermions on a lattice Multiple band generalizations exist Longer range interactions can be added (extended HM) 3-band model David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 5 / 37

Quantum cluster methods An exponential problem A band structure computation involves the iterated solution of a one-body problem: An eigenvalue problem of dimension D 1 3 1 4. The Hubbard model poses a many-body problem: The Hilbert space dimension increases exponentially with the number L of sites. For a half-filled system: D 4 L 2 πl L dimension D 2 4 4 36 6 4 8 4 9 1 63 54 12 853 776 14 11 778 624 16 165 636 9 Exact diagonalizations limited in practice to small systems (L 16) David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 6 / 37

Cluster kinematics Quantum cluster methods Solve an effective model on a finite cluster of sites Patch up the clusters together (CPT) : extends the cluster self-energy to the whole system: G 1 ( k, ω) = G 1( k, ω) Σ(ω) e 2 e 1 K k k (π, π) (, ) ( π, π) Real space Reciprocal lattice David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 7 / 37

Quantum cluster methods Cluster Dynamical Mean Field Theory H 4 3 H H H 8 7 4 3 H H H 1 2 5 6 H H H 1 2 A cluster s environment is represented by a bath of uncorrelated sites The cluster is the impurity in Anderson s impurity model Bath parameters are determined by a self-consistency procedure Lichtenstein and Katsnelson, PRB 62, R9283 (2). Kotliar et al., PRL 87, 18641 (21). David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 8 / 37

CDMFT (cont.) Quantum cluster methods H = α,β hybridization t αβ c αc β + α,µ Cluster Green function: ( θ αµ bath site annihilation operator ) c αa µ + c.h. + N b µ G 1 (ω) = ω t Γ(ω) Σ(ω) N b θ αµ θβµ Hybridation function: Γ αβ (ω) = µ ω ε µ Self-consistency: G [ (ω) = ω t( k) Σ(ω) ] 1 k ε µ a µa µ + interaction bath energy David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 9 / 37

Quantum cluster methods The CDMFT loop (with an ED solver) 1 Start with a guess value of (θ αµ, ε µ ). 2 Calculate the cluster Green function G (ω) (and Σ(ω)). 3 Calculate the superlattice-averaged Green function Ḡ(ω) = k 1 G 1( k) Σ(ω) and G 1 (ω) = Ḡ 1 (ω) + Σ(ω) 4 Minimize the distance function: d(θ, ε) = W(iω n ) tr G 1 (iω n ) Ḡ 1 (iω n ) ω n over the set of bath parameters. Thus obtain a new set (θ αµ, ε µ ). 5 Go back to step (2) until convergence. 2 David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 1 / 37

Quantum cluster methods Potthoff s variational principle Potthoff s functional: Ω[Σ] = Ω G.S. energy of cluster dω π k G G [ ln det 1 V( k, iω)g cluster Green function (depends on Σ) ] (iω) Exact form of the Baym-Kadanoff functional Expressed in terms of the self-energy Σ Explore a finite space of self-energies: those of cluster Hamiltonians Potthoff, EPJB 36 (23), 335. Potthoff et al., PRL 91, 2642 (23). Michael Potthoff David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 11 / 37

Quantum cluster methods Cluster Dynamical Impurity Approximation (CDIA) Use the same cluster-bath system as in CDMFT The bath parameters (θ αµ, ε µ ) define a space of self-energies. Instead of imposing a self-consistency requirement, find the stationary points of the Potthoff functional Free from the arbitrariness of the distance function d Provides a good estimate of the free energy Ω Ω[Σ ] David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 12 / 37

Outline Extended interactions and superconductivity 1 Quantum cluster methods 2 Square lattice Hubbard model and superconductivity 3 A spin liquid on the honeycomb lattice? David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 13 / 37

Extended interactions and superconductivity Section 2 Extended interactions and superconductivity David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 14 / 37

Extended interactions and superconductivity Early CDMFT results for the cuprate problem.4 ψ.2.8 ψ/j.4 U = 4t U = 8t U = 12t U = 16t U = 24t 1..8.6 M, ψ.4.2 U = 8t t =.3t t =.2t ψ (1 ) M...7.8.9 1..7.8.9 1. 1.1 1.2 1.3 n n Left: t = t =. Right: more realistic model S. Kancharla et al., Phys. Rev. B 77 (28), 184516 phase diagram David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 15 / 37

Extended interactions and superconductivity Early CDMFT results for the cuprate problem (cont.) Local repulsion U favors d-wave superconductivity SC order parameter seems to scale like J = 4t 2 /U Particle-hole asymmetry due to band structure (t = ) Simple model (and small clusters) lead to homogeneous AF-dSC coexistence But: What can we say about the pairing mechanism? David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 16 / 37

Extended interactions and superconductivity Digression: Dynamical spin susceptibility Contains information about spin excitations At zero temperature: ou χ ab site indices (ω) = Ω ground state S z a 1 Sb z ω H + E Ω χ q (ω) = Ω Sq z 1 S ω H + E q Ω z The imaginary part χ q(ω) contains the magnon spectrum. David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 17 / 37

Extended interactions and superconductivity Exchange of magnetic fluctuations Σ an(kf, ω) 1.8.6.4.2 (a) The intensity of the first peak of χ (π,π) is correlated to the presence of superconductivity. 3 (b) δ =.4 δ =.16 χ (ω) 2 δ =.26 1 δ =.29 δ =.37.2.4.6.8 ω Kyung, Sénéchal, Tremblay, Phys. Rev. B 8, 2519 Vignolle et al., Nature Physics 3, 163 (27) La 2 x Sr xcuo 4, x =.16 David Sénéchal (Sherbrooke) (29) quantum cluster methods CAP congress, May 27, 213 18 / 37

Extended interactions and superconductivity Effect of long-range interactions Coulomb interaction (nearest neighbors): H = r,r,σ V is generally larger than J: t r,r c rσc r σ + U n r n r +V r rr J = Can Superconductivity survive V? 4t2 U V V U n r n r µ n r,σ r,σ Yes! The key is the retarded nature of the effective attraction. (like the electron-phonon interaction in conventional superconductivity) David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 19 / 37

Extended interactions and superconductivity Effect of long-range interactions (cont.).4 U = 4 ψ ψ ψ.3.2.1.12.8.4.6 V = V = 1 V = 2 V = 3 U = 8.4 V = V = 1.2 V = 2 V = 4 V = 8.5.1.15.2.25 x V = V =.5 V = 1 V = 1.5 U = 16 V is detrimental to SC in the overdoped region... but not in the underdoped region Two compensating effects: V breaks pairs by itself but V increases J increases pairing Quite different at small coupling (below the Mott transition) Sénéchal, Tremblay, Day, Bouliane, PRB 87 (213), 75123 David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 2 / 37

Extended interactions and superconductivity Finite-frequency contribution to SC order.25.2.15.1.5.15.1.5.1.8.6.4.2 U = 8,x =.5 V = V = 1 U = 8,x =.2 U = 16,x =.5 V = 2 V = 3 1 2 3 4 5 ω Nambu representation: ( G G(k, ω) = (ω) F (ω) I F (ω) = ω dω 2π F(k, ω ) ) F(ω) G ( ω) Gorkov function k SC order parameter ψ = I F ( ) Low ω: magnetic fluctuations ( J) High ω: pair-breaking effect of V. Sénéchal et al. PRB 87 (213), 75123 David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 21 / 37

Extended interactions and superconductivity High-T c part: Conclusions Quantum cluster methods support a pairing mechanism via magnetic fluctuations The spectral features of the order parameter are correlated with the AF susceptibility This magnetic interaction is retarded, which makes SC robust against longer-range Coulomb repulsion At low doping: J increases with V, which compensates V s pair breaking effect. At higher doping: V hinders superconductivity David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 22 / 37

Outline A spin liquid on the honeycomb lattice? 1 Quantum cluster methods 2 Square lattice Hubbard model and superconductivity 3 A spin liquid on the honeycomb lattice? David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 23 / 37

A spin liquid on the honeycomb lattice? Section 3 A spin liquid on the honeycomb lattice? David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 24 / 37

A spin liquid on the honeycomb lattice? The graphene lattice to et al.: The electronic properties of graphene Miao et al., mb interacgeometries, reira et al., ade effects to magnetic e transport n a plethora le detection spin injeci et al., 27; ctronic flexnd/or strucof metal atal., 28 or et al., 28; as done in haus et al., selhaus and δ 3 a1 a 2 A δ 1 δ 2 B k y Of particular importance for the physics of graphene are the two points K and K at the corners of the graphene Brillouin FIG. 3. Color zone online BZ. Electronic These are dispersion named in the Dirac honeycomb Castro-Neto et al, Rev. Mod. Phys. 81 (29) 19 points for lattice. Left: energy spectrum in units of t for finite values of reasons that will become clear later. Their positions in Γ b1 K M K k x b2 Castro Neto et al.: The electronic properties of graphene FIG. 2. Color online Honeycomb lattice and its Brillouin zone. Left: lattice structure of graphene, made out of two interpenetrating triangular lattices a 1 and a 2 are the lattice unit vectors, and i, i=1,2,3 are the nearest-neighbor vectors. Right: corresponding Brillouin zone. The Dirac cones are located at the K and K points. David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 25 / 37

A spin liquid on the honeycomb lattice? What is a spin liquid? A non magnetic, insulating state Spin-spin correlations decay exponentially... or algebraically (algebraic spin liquid) May be classified according to their excitations Many types, according to theory Here: same as a nonmagnetic, Mott insulator David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 26 / 37

.18 can be observed in photoemission A spin liquid experiments. on the honeycomb As shown in Fig. lattice? 1, D sp(k) 5 for values of U/t below about 3.5, as expected for a semimetal. For larger values of U/t, the system enters an insulating phase.4.4.8.12.12 as a result of interactions. The values of the gap are obtained by S AF /N U/t = 4.5 extrapolation of the QMC data to the TDL as shown in Fig. 2a. U/t = 4.3 U/t = 4. From previous analysis of the model, long-range antiferromagnetic correlations are expected when the Mott insulator appears. U/t = 3.6.6 U/t = 3.8 We therefore measured the antiferromagnetic spin structure factor, U/t = 3.5 SAF (Supplementary Information), which indicates long-range antiferromagnetic order if m 2 s 5 limnr (SAF/N).. Figure 2b shows.5.1.15.2.25.3.35 the QMC results together with a finite-size extrapolation. The results c.16 of this extrapolation are also presented in the phase diagram in Fig. 1. Antiferromagnetic order appears for U/t. 4.3, a value that is consistent with previous estimates for the onset of long-range antiferro- U/t = 3.4 U/t = 4. magnetic order 26,27 U/t = 4.3. This leaves an extended window, 3.5, U/t, 4.3, finite.12 size scaling within which the system is neither a semimetal nor an antiferromagnetic Mott insulator..12 L = 6 The controversy Meng et al. assert spin liquid phase between the AF and the semi-metal.5.8.8 a 1 b1 K a M 2 9 Γ K.4.4 12 y k y b 2.4 15 k x x.3 TDL 1 2 3 4 5 6 7 8 Δ sp (K)/t U/t.2 m s.2.5.1.15.2.25.3.35.4.45 1/L Δ s ( 6).1.1 Figure 2 Finite-size extrapolations of the excitation gaps and the antiferromagnetic structure factor. a, The single-particle gap at the Dirac SM SL AFMI point, Dsp(K), shown here for different values of U/t, is linear in 1/L. Dsp(K)is obtained by fitting the tail of the Green s function, G(K, t) (inset), to the 2 2.5 3 3.5 4 4.5 5 5.5 6 form e {tdsp(k). b, Antiferromagnetic structure factor, SAF, for various values U/t of U/t, fitted using third-order polynomials in 1/L. Antiferromagnetic order appears for U/t. 4.3, as seen in the histogram P(S Figure 1 Phase diagram for the Hubbard model on the honeycomb lattice AF/N) from a Monte Carlo bootstrapping analysis (inset). a.u., arbitrary units. c, Spin gap, D at half-filling. The semimetal (SM) and the antiferromagnetic Mott s, for ZY Meng et al, Nature 464 (21), 847851. different values of U/t, fitted using second-order polynomials in 1/L. Inset, insulator (AFMI) are separated by a gapped spin-liquid (SL) phase in an D intermediate-coupling regime. D sp(k) denotes the single-particle gap and D s for L 5 6, 9, 12 and 15, and the extrapolated values (TDL), as functions of s U/t. Error bars, s.e.m. denotes the spin gap; m s denotes the staggered magnetization, whose FIG. 4: Finite size S. Sorella scaling et al, of Sci. the Rep. spin 2, gap 992 and (212) charge-charge c saturation value is 1/2. Error bars, s.e.m. Inset, the honeycomb lattice withthe honeycomb lattice at half-filling. a, Spingap primitive vectors a 1 and a 2, and the reciprocal lattice with primitive vectors Further details on the nature of this intermediate region are s = E(S =1) energy b1 andb2. Openandfilledsitesrespectivelyindicatetwodifferentsublattices. obtained for by a examining given spin the S. spin Solid excitation curves gap, which are quadratic is extracted from fits of data. T The David DiracSénéchal points K and(sherbrooke) K9 the M and C points are marked. quantum Error cluster thebars long-time methods of the behaviour extrapolated of the CAP imaginary-time values congress, are computed displaced May 27, spin spin 213 with the 27 bootstra / 37 Δ s /t Δ sp /t, ms S AF /N Δ s /t P(S Sorella et al. assert the contrary. The argument revolves around inital states in QMC and proper Δ s /t

A spin liquid on the honeycomb lattice? Strategy Assume that the semi-metal to SL transition is the Mott transition Detect this with CDIA, with a few cluster-bath systems Check the semi-metal to AFM transition as well Hypothesis: beyond the smallest systems, U c does not depend strongly on system size. Conclusion: the AFM transition always pre-empts the Mott transition. Therefore: no spin liquid for the Hubbard model on the honeycomb lattice. David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 28 / 37

A spin liquid on the honeycomb lattice? The Mott transition on a square lattice s4-4b s4-8b θ 1..8.6.4.2 θ N() s4-4b.1.8.6.4.2 N(ω = ) s4-4b-l First order transition between a metal (red) and an insulator (blue) M. Balzer et al., Europhys. Lett. 85 (29), 172. θ. 3 4 5 6 7 8. U 1..1 N() s4-8b.8.8.6.4 θ.2 U c1 U c2.6.4.2.. 4.4 4.6 4.8 5. 5.2 5.4 5.6 5.8 U N(ω = ) David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 29 / 37

A spin liquid on the honeycomb lattice? Cluster-bath systems h6-6b h2-4b h6-4b h4-6b h6b-4b S. Hassan et DS, PRL 11 (213) 9642 A. Liebsch & Ishida, J. Phys. Condensed Matter 24 (212), no. 5, 5321. David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 3 / 37

A spin liquid on the honeycomb lattice? The Mott transition ǫ 2.2 1.8 2 1.6 1.4 1.2.8 1.6.4.2 CDIA CDMFT M h2-4b U c 3 3.5 4 4.5 5 5.5 6 6.5 7 U.8.7.6.5.4 M.3.2.1 ǫ 1.8 1.6 1.4 1.2 1.8.6.4.2 CDIA CDMFT M (CDMFT) h6b-4b U c 2 3 4 5 6 7 8 U.8.7.6.5.4 M.3.2.1 ǫ 3 2.5 2 1.5 1.5 CDIA CDMFT M h4-6b.8.7.6.5.4.3.2 3 4 5 6 7 8 U U c M double occupancy.14.12.1.8.6.4.2 h2-4b h4-6b h6b-4b 5 5.5 6 6.5 7 7.5 8 U David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 31 / 37

A spin liquid on the honeycomb lattice? Dependence on systems size in one dimension PRL 94, 2646 (25) PHYSICAL REVIEW LETTERS FIG. 3. (a) jz c j as a function of U=t for t =t :75 and different sizes. (b) The same for the dimer order parameter D d. The v i;j s are independently minimized; very similar results are obtained with the parametrization of Eq. (5). Inset: The size scaling of jz c j for U=t 5; the dashed line is a three parameter fit. The variational Capello phase et al, PRL diagram 94 (25), for the 2646 half-filled t t Hubbard model is finally shown in Fig. 4. In the region of t =t & :5 and U>, we find no evidence of a phase transition, apart from finite-size effects at small U=t. The FIG. 4. Variational phase diagr model. The error bars take into a dashed lines are guides to the eye lating compounds. We also be great potential promise for fut ducting phases that result by d This work was partially sup (COFIN 23 and 24), by F by FIRB RBAU1LX5H. We sions with C. Castellani, A. Pa David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 32 / 37

Naughty clusters A spin liquid on the honeycomb lattice? Ring cluster systems h6-6b and h6-4b show no sign of Mott transition. The reason is not clear. But even a bath-less cluster can be used to look for the AFM transition (using the variational cluster approximation) gap 1.8 1.6 1.4 1.2 1..8 h6 (CPT) (VCA) M.6.4.2. 1 2 3 4 5 6 7 8 U.7.6.5.4.3.2.1. M David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 33 / 37

A spin liquid on the honeycomb lattice? Spectral properties semi-metal (U = ) Fermi surface plot, semi-metal (U = 7) semi-metal (U = 7) insulator (U = 7) 4 ω 2 2 4 4 ω 2 2 4 Γ David Se ne chal (Sherbrooke) M K Γ Γ quantum cluster methods M K Γ CAP congress, May 27, 213 34 / 37

A spin liquid on the honeycomb lattice? Spin-liquid part : Conclusions Quantum cluster methods predict that the Mott transition is pre-empted by AF order on the honeycomb lattice However: care must be take to consider cluster systems that allow the Mott transition to occur (exclude the ring cluster) Clusters are small, but the Mott transition is a rather local phenomenon. In any case: U c will generally increase with size. The critical U for AF is roughly the same as found in large-scale QMC calculations. David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 35 / 37

A spin liquid on the honeycomb lattice? Acknowledgements Collaborators on these projects: V. Bouliane, A. Day, S. Hassan, A.-M. Tremblay David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 36 / 37

A spin liquid on the honeycomb lattice? QUESTIONS? David Sénéchal (Sherbrooke) quantum cluster methods CAP congress, May 27, 213 37 / 37