Magnetism and Superconductivity in Decorated Lattices

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1 Magnetism and Superconductivity in Decorated Lattices Mott Insulators and Antiferromagnetism- The Hubbard Hamiltonian Illustration: The Square Lattice Bipartite doesn t mean N A = N B : The Lieb Lattice Construction of Decorated Lattices Superconductivity on the Lieb Lattice The 1/5th depleted Square Lattice: Magnetism in CaV 4 O 9 Conclusion Funding: NSF (Physics at the Information Frontier) DARPA (Optical Lattice Emulation Program) XXXVIII Meeting of the Brazilian Physical Society, May 28, 215

2 . Cast of Characters Vlad Benoit Ehsan George Fred Iglovikov Gremaud Khatami Batrouni Hebert

3 (Single band) Hubbard Hamiltonian Ĥ = t ij σ (c iσ c jσ +c jσ c iσ )+U i (n i 1 2 )(n i 1 2 ) µ iσ (n iσ +n iσ ) Two spin species σ =,. Kinetic energy t describes hopping between near-neighbor sites ij. On-site repulsion U discourages double occupancy Chemical potential µ controls filling. Half-filling (ρ = 1) at µ =. Cuprate materials (LaSrCuO, YBaCuO,...) drive interest in 2D square lattice: Cu atoms in CuO 2 sheets are in that geometry. Ignore bridging O atoms. Ignore La, Sr, Y, Ba in between layers.

4 Mott Insulating Behavior in Hubbard Model Half-filled lattice : Average number of fermions is one per site. Kinetic energy and entropy both favor particles moving around lattice. Metal: odd number (one) particle per cell/site. (Like cuprate superconductors.)

5 Large repulsive interaction U between fermions on the same site. +U Mott Insulator forms. Basic physics of parent compounds of cuprate superconductors! Also transition metal monoxides: FeO, CoO, MnO.

6 Two ways to destroy Mott Insulator: Decrease U/t: By applying pressure (MnO) Shift n 1: Dope chemically (cuprate superconductors) What is optimal spin arrangement? Hopping of neighboring parallel spins forbidden by Pauli. Antiparallel arrangement lower in second order perturbation theory. x t t E (2) = E (2) t 2 /U = J Mott insulating behavior and antiferromagnetism go hand-in-hand. Qualitative picture of cuprate physics before doping. Still do not really understand why cuprates become superconducting after doping.

7 Special Features of square lattice hopping Hamiltonian (U = ) N(E).2 k y (π,π) E k x Left: Van Hove singularity of density of states at E = (half-filling) Right: Nesting : At ρ = 1, wavevector (π, π) connects big sections of Fermi surface. Favors ordering at (π, π) like antiferromagnetism.

8 Determinant Quantum Monte Carlo Map Hamiltonian with electron-electron interactions electrons coupled to space-(imaginary) time dependent field S iτ. Trace out fermions analytically determinants detm σ (S iτ ). Dimension of M is number of sites in lattice. (Classical) Monte Carlo sampling of S iτ Algorithm is order N 3 L. L is number of (imaginary) time slices. N lattice sites/electrons L 1 2 to reach low T (large β). Measurements (any single or two particle Green s function): c iσ c jσ [M 1 σ ] ij = [G σ ] ij Sign Problem: At low temperature detm σ can go negative. At special symmetry points (half-filling ρ = 1) detm = detm. No Sign Problem!

9 DQMC Results- Density of States.4.3 U=2 U=4 U=6 N(ω=) T N(ω = ) vanishes Insulating Mott Gap.

10 DQMC results- Antiferromagnetic spin correlations c(l) = (n j+l n j+l )(n j n j ) βt = 2 T = t/2 = W/16 (bandwidth W = 8t).1.5 (a) 2 x 2 U = 2. C(lx,ly) -.5 (1,1) -.1 β = 32 β = 2 (,) (1,) β = (,) (1,) (1,1) (,) Hirsch/Hirsch-Tang: Proof of AFLRO in ground state of half-filled Hubbard Hamiltonian.

11 U = 2 Fermi function: DQMC results- Fermi distribution n(k x,k y ) ρ =.2 ρ =.4 ρ =.6 ρ =.8 ρ = 1. π π/2 -π/2 -π -π -π/2 π/2 π -π/2 π/2 π -π/2 π/2 π -π/2 π/2 π -π/2 π/2 π n(k) U = 2 Gradient of Fermi function: π 2.5 π/2 -π/ n(k) -π -π -π/2 π/2 π -π/2 π/2 π -π/2 π/2 π -π/2 π/2 π -π/2 π/2 π

12 Fermi surface is smeared further by increasing interaction strength. U = 4 Fermi function: π π/2 -π/2 -π ρ =.2 β = 8 -π -π/2 π/2 π ρ =.4 β = 8 -π/2 π/2 π ρ =.6 β = 6 -π/2 π/2 π ρ =.8 β = 4 -π/2 π/2 π ρ = 1. β = 8 -π/2 π/2 π n(k) U = 4 Gradient of Fermi function: π π/2 -π/ n(k) -π -π -π/2 π/2 π -π/2 π/2 π -π/2 π/2 π -π/2 π/2 π -π/2 π/2 π

13 Square lattice is bipartite Divides into A and B sites such that all neighbors of A belong to B and vice-versa. Antiferromagnetism is unfrustrated. Hubbard model has particle-hole symmetry: c iσ c iσ c jσ ( 1) i c iσ ( 1) i+j c iσ c jσ = c jσ c iσ n iσ 1/2 1 n iσ 1/2 = 1/2 n iσ Ĥ Ĥ (at µ = ) Implications for DQMC: No sign problem at half-filling.

14 Bipartite doesn t mean N A = N B! For example, the Lieb Lattice (CuO 2 sheets of cuprates) Some theorems (Lieb and Sutherland) Near neighbor hopping Hamiltonian has spectrum N A - N B eigenvalues E = (In above, N A = 2N B.) Flat band! 2N B eigenvalues in ± pairs. U > ground state has spin S nonzero (suggested from AF picture!)

15 Topologically localized states (Sutherland): Provide simple picture of E = modes Ψ = (c 1 c 2 +c 3 c 4) vac ˆT Ψ = Can do this construction about any missing site. These are the N A N B = N B states with E =. These are the N A N B = N B states with E =. Note: Can view Lieb as 1/4-depleted square lattice.

16 Construction of Decorated Lattices [1] Begin with any bipartite lattice. [2] Select midpoints of all bonds. [3] Connect any two midpoints that share a vertex. Theorem: The resulting lattice has localized states [1] [2] [3] Square plaquette lattice! ˆT (c 1 c 2 +c 3 c 4) vac =.

17 Lieb Construction: How to get localized states (continued)! [1] Begin with any bipartite lattice. [2] Select midpoints of all bonds. [3] Connect any two midpoints that share a vertex. Theorem: The resulting lattice has localized states [1] [2] [3] Kagome Lattice! ˆT (c 1 c 2 +c 3 c 4 +c 5 c 6) vac =.

18 Superconductivity on the Lieb Lattice Ĥ = t ij σ (c iσ c jσ +c jσ c iσ ) U i (n i 1 2 )(n i 1 2 ) µ iσ (n iσ +n iσ ) (Partial) Particle-Hole Transformation c i c i c iσ c j n i 1/2 ( 1) i c i c i ( 1) i+j c i c j = c j c i 1 n i 1/2 = 1 2 n i n i 1/2 n i 1/2 Ĥ(U > ) Ĥ(U < ) (at µ = ) Repulsive (U > ) Hubbard model: antiferromagnetism. Attractive (U < ) Hubbard model: charge density wave (CDW) and pairing.

19 a 1 b c Lower dispersing band < ρ < 2/3 Flat band 2/3 < ρ < 4/3 Upper dispersing band 4/3 < ρ < 2 CDW order at ρ = 2/3 and ρ = 4/3? Pairing (superconductivity) when doped away from these values? Related (?) question: System of bosons in a flat band: no k which minimizes ǫ(k) Would Bose-Einstein condensation still occur? (Stamper-Kurn)

20 DQMC simulations of Attractive Hubbard Model on Lieb Lattice Occupations of minority ( copper ) and majority ( oxygen ) orbitals: 2. N =18, β =36 n i [t] n d,u = n px,u =..5 n d,u = 4 n px,u = 4 n d,u = 8 n px,u = ρ[t] As total ρ increases past ρ = 2/3 minority orbitals depopulate. (Strange?) Local moment

21 .25.2 U = 4, β =36 N =27 N =48 N =75 N =18 m 2 [t] ρ[t] Constant within the flat band. For square lattice, m 2 falls immediately with doping ρ = 1+δ.

22 Pairing structure factor P s = (1/N 2 ) ij i j j = c i c i P s [t] U = mean field N =27 N =48 N =75 N =18 (a).8. (b).8 U = 4 U = P s [t].1.5 U = ρ[t] P s has minimum at ρ = 4/3 (and 2/3) where CDW competes strongly. U = 4 and half-filling, extrapolate to non-zero order parameter for N. U = 8 and half-filling: no LRO at β = 36. (LRO on square lattice at β = 12). Large U: bosonic limit : No BEC in a flat band. (No k with ǫ(k) smallest.)

23 The 1/5th depleted Square Lattice: Magnetism in CaV 4 O 9 CaV 2 O 5 : A one-fifth depleted lattice: t t a 1 a 2 V V O3 V V O1 Intra-plaquette t and inter-plaquette t hoppings differ.

24 t t a 1 a 2 t t t >> t: singlets form on dimers and no long range AF order. t >> t : singlets form on plaquettes and no long range AF order. Troyer: Heisenberg limit determined region of AF order around J J. What about U > Hubbard?

25 Antiferromagnetic Structure Factor S AF ( 12) U=6, β=2 64 sites 144 sites m(a t )-m(a t ) ( 12) 1 5 Plaquettes t /t t/t Dimers.2 Region with AFLRO in vicinity of t t for U = 6. (Blue bar is Heisenberg limit.)

26 Anisotropy of NN Spin Correlations σ i σ i+a t - σ i σ i+a t (a) β=2 U=1 U=2 U=3 U=4 U=5 U=6 U=7 U=8 U=12 Difference between near-neighbor spin correlations on intra-plaquette (t) and inter-plaquette (t ) bonds is small in region of AF ordering. High symmetry point (HSP): nn spin correlations on 2 types of bond are equal. (Alternate way of determining quantum critical point qualitatively.)

27 Quantum Monte Carlo Phase Diagram (Half-Filling)

28 RPA Phase Diagram (general density) Because of sign problem must do RPA for densities away from half-filling. Q=( π,π) FM Q=(,) AF Q=(,) Stripe Q=( π,) Stripe 6 4 U c 2 1 DOS 2 1 t =t=1-2 2 ω.8.6 ρ.4 Q=(,) FM Q=(π,π) FM Q=(,) AF Q=(π,π) AF Q=(,) Stripe Q=(π,) Stripe.2

29 Conclusions Hubbard model describes Mott insulators (and antiferromagnetism). Interesting lattices exist where wave functions are localized. These are often associated with flat bands. Putting together exceptional lattices and Hubbard physics. unusual charge order and magnetic order. and unusual superconductivity.