Magnetism and Superconductivity on Depleted Lattices 1. Square Lattice Hubbard Hamiltonian: AF and Mott Transition 2. Quantum Monte Carlo 3. The 1/4 depleted (Lieb) lattice and Flat Bands 4. The 1/5 depleted lattice model of CaV 4 O 9. 5. The 1/3 depleted lattice model of La 4 Ni 3 O 8 6. Superconductivity and Bose-Einstein condensation in Flat Bands 7. Summary (and Future Directions) Funding: DOE DE-SC14671
. Cast of Characters Natanael Tiago Ehsan Huaiming Costa Mendes Khatami Guo Warren Thereza Raimundo Rajiv Pickett Paiva Dos Santos Singh
1. Square Lattice Hubbard Hamiltonian: AF and Mott Transition 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. Band structure : ǫ(k x,k y ) = 2t(cosk x +cosk y ) Chemical potential µ controls filling. Half-filling (ρ = 1) at µ =. On-site repulsion U sufficiently large Mott Insulator Antiferromagnetic exchange interaction J t 2 /U Antiferromagnetism Consider first 2D square lattice geometry (Cu atoms in CuO 2 sheets of cuprates).
Special Features of square lattice hopping Hamiltonian (U = ).4 4 2.3 N(E).2 k y (π,π).1-2 -4-2 2 4 E -4-4 -2 2 4 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.)
2. Quantum Monte Carlo Compute operator expectation values  = Z 1 Tr [Âe βĥ ] Z = Tr [e βĥ ] Discretize inverse temperature β = L τ. Express Z and A as path integrals. Hubbard-Stratonovich fields S iτ decouples interaction. Quadratic Form in fermion operators: Do trace analytically Z = detm ({S iτ })detm ({S iτ }) {S iτ } dim(m σ ) is the number of spatial sites/orbitals. Sample HS field stochastically. Algorithm is order N 3 L. N 1 2 1 3 lattice sites/electrons Measurements: c iσ c jσ [M 1 σ ] ij = [G σ ] ij
QMC results- Fermi distribution n(k x,k y ) U = 2 Fermi function: 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 1.8.6.4.2 n(k) U = 2 Gradient of Fermi function: π 2.5 π/2 -π/2 2 1.5 1.5 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
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 1.8.6.4.2 n(k) U = 4 Gradient of Fermi function: π π/2 -π/2 1.5 1.5 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
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 = 12 -.15 (,) (1,) (1,1) (,) Hirsch: Proof of AFLRO in ground state of half-filled Hubbard Hamiltonian.
Density of States.4.3 U=2 U=4 U=6 N(ω=).2.1..1 1 T N(ω = ) vanishes Insulating Mott/Slater Gap.
3. The 1/4 depleted (Lieb) lattice and Flat Bands Divides into A and B sites All neighbors of A belong to B and vice-versa. Bipartite doesn t mean N A = N B! Lieb Lattice CuO 2 sheets of cuprate superconductors La 2 x Sr x CuO 4, Y 1 Ba 3 Cu 3 O 7+δ,
Theorem (Lieb) Near neighbor hopping Hamiltonian has spectrum N A - N B eigenvalues E = (Lieb: N A = 2N B.) Flat band! 2N B eigenvalues in ± pairs. Topologically localized states (Sutherland): Provide simple picture of E = modes. n 3 n n 4 2 ψ = n 1. +1 1 +1 1. n 1 n 2 n 3 n 4 Getting Hψ: tψ n1 tψ n2 = t(1) t( 1) = Can do this construction around any missing site. These are the N A N B = N B states with E =!
Unlike previous delocalized plane wave states, these ψ have only four nonzero entries (at locations n 1,n 2,n 3,n 4 ). The electron is spread out over only four sites. BUT can form linear combinations of degenerate (all E = ) eigenvectors ψ. These states can be viewed as having momenta K. with the very curious property that E(K) = is completely independent of K. This is a flat band! What happens to Magnetism on the Lieb Lattice (flat band)? Lieb rigorously proved ferromagnetic order. (Ground state non-zero spin.) What happens to SC (pairing of K with K) if E(K) is independent of K? Other depleted lattices...
Magnetic correlations (resolved by orbital).2.1 c αβ (r). -.1 -.2.2. (a) U/t pd = 2 L = 8 T/t pd =.42 -.2 -.4 (b) U/t pd = 4..5 1. 1.5 2. 2.5 3. 3.5 4. 4.5 5. 5.5 6. 6.5 r
Ferromagnetic Structure Factor Nonzero extrapolation of S(q = ) to thermodynamic limit. Validates (and quantifies!) Lieb theorem.
Extending Lieb s result further. Ground state phase diagram U p U d. Ferromagnetic order parameter is not significantly reduced if U d =. Ferromagnetic is destroyed if U p = (metallic phase). Interpret this as excessive dilution of interactions.
4. The 1/5th depleted Square Lattice: Magnetism in CaV 4 O 9 CaV 2 O 5 : A one-fifth depleted lattice: t t 1 2 3 1 Intra-plaquette t and inter-plaquette t hoppings differ.
t t 1 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?
Anisotropy of NN Spin Correlations σ i σ i+a t - σ i σ i+a t.8.6.4.2 (a) β=2 U=1 U=2 U=3 U=4 U=5 U=6 U=7 U=8 U=12.2.4.6.8 1.8.6.4.2 t /t t/t 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.)
Antiferromagnetic Structure Factor S AF 3 25 2 15 ( 12) U=6, β=2 64 sites 144 sites m(a t )-m(a t ) ( 12) 1 5 Plaquettes.2.4.6.8 1 t /t.8.6.4 t/t Dimers.2 Region with AFLRO in vicinity of t t for U = 6. (Blue bar is Heisenberg limit.)
Quantum Monte Carlo Phase Diagram (Half-Filling)
RPA Phase Diagram (general density) 1/5 depleted structure first seen in CaV 4 O 9 also observed in iron selenide family. Luo etal, PRB 84 14596 (211) Magnetic ground state of K.8 Fe 1.6 Se 2 π,π π 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
5. The 1/3 depleted lattice model of La 4 Ni 3 O 8 Layered Nickelates: analogs of cuprates if Ni 1+ can be stabilized like Cu 2+. Trilayer La 4 Ni 3 O 8 insulating transition at 15K, likely AF order. Real space ordering of charge to 1/3 depleted lattice: Zhang etal. PNAS 113, 8945 (216); Botana etal. PRB 94, 8115 (216) Begin by studying Heisenberg model for single layer. 1/5 depleted (CaV 4 O 9 ): decoupled clusters in J J and J J limits. Here: Extended linear chains remain when J J.
Strong coupling (Heisenberg) limit: (g = J /J) Presence of extended chain structure leads to AF down to small J. QMC for Hubbard model in progress.
6. Superconductivity and Bose-Einstein condensation in Flat Bands 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.
1 4 2 3 cdw order at ρ = 2/3 and ρ = 4/3? Pairing (superconductivity) when doped away from these values? ρ = 1 is in between these favored cdw densities, right in middle of flat band. Will pairing occur at ρ = 1? Related (?) question: System of bosons in a flat band: no k which minimizes ǫ(k) Would Bose-Einstein condensation still occur? (Stamper-Kurn)
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] 1.5 1..5 n d,u = n px,u =..5 n d,u = 4 n px,u = 4 n d,u = 8 n px,u = 8 1. 1.2 1.4 1.6 1.8 2. ρ[t] As total ρ increases past ρ = 2/3 minority orbitals depopulate. (Strange?)
Near-neighbor density-density correlations. 4 N =18, β =36 n i n j [t] 3 2 1 1 n d n px,u = n d n px,u = 4 n d n px,u = 8 n d n px,u = 1 n px n py,u = n px n py,u = 4 n px n py,u = 8 n px n py,u = 1 1. 1.2 1.4 1.6 1.8 2. 2.2 ρ[t] Also show clear signature of flat band between ρ = 2/3 and ρ = 4/3.
Local moment.25.2 U = 4, β =36 N =27 N =48 N =75 N =18 m 2 [t].15.1.5. 1. 1.2 1.4 1.6 1.8 ρ[t] Constant within the flat band.
Pairing structure factor P s = (1/N 2 ) ij i j j = c i c i P s [t].2.15.1.5 U = 4..15 mean field N =27 N =48 N =75 N =18 (a).8. (b).8 U = 4 U = 8...2 P s [t].1.5 U = 8. 1. 1.2 1.4 1.6 1.8 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).
7. Summary (and Future Directions) Extension of QMC simulations to depleted lattices: Quantify nature of ferromagnetic order on Lieb lattice. Effect of strong correlations in a flat band. Regime of AF ground state in model of CaV 4 O 9. LRAFO stabilized in isotropic region between PM dimer and plaquette phases. Regime of AF ground state in model of La 4 Ni 3 O 8. Nature of superconducting order in a flat band. Whither do pairs condense if E(k) is k-independent? Tianxing Ma, Yueqi Li: Disordered Lieb Lattice Huaiming Guo, Chunhan Feng: Decorated Honeycomb Lattice
Lieb Construction: How to get localized states! [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 4 [1] [2] [3] Square plaquette lattice! ˆT (c 1 c 2 +c 3 c 4) vac =.
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 6 2 5 3 4 [1] [2] [3] Kagome Lattice! ˆT (c 1 c 2 +c 3 c 4 +c 5 c 6) vac =.