Quantum Simulation Studies of Charge Patterns in Fermi-Bose Systems

Transcription

1 Quantum Simulation Studies of Charge Patterns in Fermi-Bose Systems 1. Hubbard to Holstein 2. Peierls Picture of CDW Transition 3. Phonons with Dispersion 4. Holstein Model on Honeycomb Lattice 5. A New Langevin-Based QMC Algorithm 6. Conclusions- Potential Applications Funding: Department of Energy DE-SC and DE-NA

2 Natanael Costa Bo Xiao WeiTing Chiu YuXi Zhang Thomas Blommel George Batrouni Ehsan Khatami ZiYang Meng Phonon dispersion and the competition between pairing and charge order, N.C. Costa, T. Blommel, W.-T. Chiu, G.G. Batrouni, and RTS, Phys. Rev. Lett. 120, (2018). Charge Order in the Holstein Model on a Honeycomb Lattice, Y.-X. Zhang, W.-T. Chiu, N.C. Costa, G.G. Batrouni, and RTS, arxiv: Localization, Charge Ordering, and Superfluidity in the Disordered Holstein Model, B. Xiao, E. Khatami, N. Costa, G. Batrouni, and RTS, in preparation.

3 Theorists have a reputation for oversimplification: In that sense the Holstein model is an unfortunate name. This talk: More realistic cows! Like the Hubbard Model, the Holstein Model has a considerable lineage: The Hubbard model: Electron Correlations in Narrow Energy Bands, J. Hubbard, Proc. Royal Soc. London 276, 238 (1963). The Holstein model: Studies of polaron motion: Part II. The small polaron, Annals of Physics 8, 343 (1959).

4 Noninteracting electron kinetic energy: 1. Hubbard to Holstein Ĥ el ke = t ij σ (ĉ iσĉjσ +ĉ jσĉiσ Hubbard: Spin, electrons on same site i interact with each other: ) Ĥ el el = U i ˆn i ˆn i Holstein: Spin, electrons interact with boson displacement on site i Ĥ el ph = λ ) ˆX i (ˆni + ˆn i H boson = 1 2 ω2 0 i i ˆX 2 i i ˆP 2 i Bosons local energy independent of momentum (dispersionless) ω(q) = ω 0. Similarly, electron-boson coupling is local independent of momentum. Dimensionless coupling: λ D = λ 2 /(ω 2 0 W) where W = electronic bandwidth.

5 Slightly technical: Understanding Hubbard-Holstein Connection Quantum Monte Carlo at finite temperature- Partition function: Z = Tr[e βĥ ] = Tr[e τĥe τĥ e τĥ ] Path integral: Insert complete sets of boson coordinate states. 1 2 ω2 0 ˆX 2 i 1 2 ω2 0 X 2 i ˆP 2 i [ Xi (τ + τ) X i (τ) τ ] 2 Adiabatic approximation (ignore boson kinetic energy). Complete square on every site i and time slice τ: 1 2 ω2 0 X 2 +λx (ˆn + ˆn ) = 1 2 ω2 0 ( λ X + (ˆn ω0 2 + ˆn ) )2 λ2 ) 2 (ˆn 2ω0 2 + ˆn Integrate out bosons: effective on-site interaction between electrons. λ2 ) 2 (ˆn 2ω0 2 + ˆn = λ2 (ˆn 2ω0 2 + ˆn ) λ2 ω0 2 ˆn ˆn Holstein: Close connection to attractive Hubbard model. U eff = λ 2 /ω 2 0.

6 Hubbard Model in momentum space: Ĥ el el = U i ˆn i ˆn i = U N ĉ p q ĉ k+q ĉp ĉk k k+q kpq p U p q Holstein Model in momentum space: λ N λ N ˆX q ĉ p qσĉ pσ = pq (â )ĉ q +â q pq p qσĉ pσ p p q λ q k λ q k+q Let s tie the boson lines together...

7 To second order in λ, interaction is U eff = λ 2 D 0 (q,w) = λ 2 1 ω 2 ω 2 q D 0 (q,ω) is free boson propagator. k λ q k+q p λ p q Suppressing ω dependence, and, for Holstein, setting ω q = ω 0 : U eff = λ 2 D 0 (q,w) = λ2 ω 2 0 Effective el-el coupling (reproducing previous argument). Repulsive interaction (+U Hubbard): Local moments form. up/down spin alternation favored by J: Antiferromagnetism. Attractive interaction ( U Hubbard; Holstein): Local pairs form. Double occupied/empty alternation favored: Charge Density Wave. x t x t t t

8 Local order can become long range if thermal/quantum fluctuations reduced. Repulsive (AF) Attractive (CDW) Holstein Model: Charge order at half-filling (bipartite lattice). Superconducting order when doped. CDW transition at finite T in 2D (Ising universality class). Contrast to Hubbard: AF order only at T = 0 in 2D (Heisenberg universality). Quantitative values for T c obtained for square lattice only recently! Weber and Hohenadler, Phys. Rev. B 98, (2018).

9 Square Lattice Holstein model Structure factor S cdw (π,π) = 1 n i n j ( 1) i+j S af (π,π) = 1 N N i,j SiS z j ( 1) z i+j i,j High T: n i n j e i j /ξ S(π,π) independent of N. Low T: n i n j constant S(π,π) N. Finite size scaling collapse determines T c = 1/β c.

10 Order at q = (π, π): Fermi-Surface Nesting (FSN) for half-filled 2D square lattice. 2k F In 1D, always have FSN at q = 2k F. Lattice distortion lowers electronic energy more than bond stretching energy cost.

11 2. Peierls Picture of CDW Transition Peierls Instability: FSN in one dimensional system creates very high susceptibility to lattice distortion at 2k F. χ(q) = k f(ǫ k ) f(ǫ k+q ) ǫ k ǫ k+q Widely used and useful picture! However: CDW in NbSe 2 : no sign of FSN. Attributed instead to momentum dependent electron-boson coupling. Charge order (e.g. stripes etc) in cuprates. Attributed instead to electron-electron interactions.

12 3. Phonons with Dispersion We explored momentum dependence of boson dispersion. Easy to implement in Determinant QMC. A close cousin of momentum dependent coupling. Σ g (k,ω) dqdν g(q) 2 1 ω ν ǫ(k q) Σ ω (k,ω) dqdν g ω ν ǫ(k q) 2ω 0 ν 2 ω 2 0 2ω(q) ν 2 ω(q) 2 Σ g : momentum dependent electron-boson coupling. Σ ω : momentum dependent boson dispersion. ν 0 (boson carries no energy): Σ g = Σ ω if g(q) 2 /ω 0 = g 2 /ω(q). (For nonzero ν the two self-energies are not equal.)

13 Conventional Holstein Model: Ĥ = t (ĉ ij σ iσĉjσ +ĉ jσĉiσ ) +λ i ˆX i (ˆni + ˆn i ) ω2 0 i ˆX 2 i i ˆP 2 i An intersite boson coupling introduces q dependence in boson energy. Ĥ 2 = 1 2 ω2 2 ( ˆX i ± ˆX j ) 2 i,j (q) - y x 0-0 (a) (b) (c) (q) (q) q q y x q q x q q y x y y y x x a. ˆXi + ˆX j : lowers ω(q = (π,π)) (Checkerboard CDW). b. Mixed signs in x,y directions: lowers ω ( q = (0,π) ) (Stripe CDW). c. ˆXi ˆX j : raises ω(q = (π,π)) (no CDW, superconductivity?) Phonon bandwidth: ω ω max ω min.

14 No dispersion (H 2 = 0) we found β c = 6.0±0.1. Initial effect of H 2, checkerboard CDW still dominant, but shifted T c. Top: ˆXi ˆX j : lowers ω ( q = (0,0) ) Bottom: ˆXi + ˆX j : lowers ω ( q = (π,π) ) β c for checkerboard CDW increases. β c for checkerboard CDW decreases.

15 For sufficiently large mixed H 2, which favors stripe CDW, Checkerboard to stripe transition: Key observation here: No alteration to electron band structure. Half-filled square lattice. Fermi surface nesting remains at (π, π). But CDW ordering vector can be elsewhere. CDW transition outside of canonical Peierls picture.

16 Can also examine these phenomena via CDW gap. Plateau in ρ(µ) shrinks/expands with boson dispersion. Density of states has a gap at Fermi Energy. Like AF-Slater gap in +U Hubbard.

17 Suppress checkerboard CDW without replacing it with stripes. Superconducting phase at commensurate filling. 2D superconducting transition (Kosterlitz-Thouless universality).

18 4. Holstein Model on Honeycomb Lattice Dirac spectrum for fermions. Quantum critical point for Hubbard Model: Minimal U c /t 3.87 to induce antiferromagnetic order. Effect of electron-boson interactions on Dirac fermions and charge order? (a) (b) Long range real space charge correlations develop as β increases.

19 CDW structure o(n) when charge correlations long range (β > β c ). Data collapse/crossing yield critical temperature.

20 Phase Diagram of Holstein Model on Honeycomb Lattice Charge Order in the Holstein Model on a Honeycomb Lattice, Y.-X. Zhang, W.-T. Chiu, N.C. Costa, G.G. Batrouni, and RTS, arxiv: Charge-Density-Wave Transitions of Dirac Fermions Coupled to Phonons, C. Chen, X.Y. Xu, Z.Y. Meng, and M. Hohenadler, arxiv:

21 5. A New Langevin-Based QMC Algorithm Many QMC approaches propose local moves. Inexpensive to evaluate change in action. In the case of DQMC: o(n 2 ) to update boson coordinate. Nβ boson degrees of freedom: Complete scaling is o(n 3 β). Alternate approach in Lattice Gauge Theory QMC: Update all bosonic field variables at same time via Langevin equation. In principle complete scaling is o(n β). Assumes number of (conjugate gradient) iterations to compute M 1 v is independent of system size N and inverse temperature β. Does not work for Hubbard Model! Matrix M is much more ill-conditioned than LGT. Number of conjugate gradient iterations grows catastrophically with β. [ ] 2 Key observation: ˆP2 i Xi (τ+ τ) X i (τ) moderates eigenspectrum of M. (No such term for auxiliary bosonic field in Hubbard model.) τ

22 CPU time in hours τ=1/8 λ=2 1/2 ω=2 µ=-1 Langevin time step: dt=0.05 DQMC, β=4, CPU N 3 Langevin, β=4, CPU N 1.2 Langevin, β=8, CPU N N=L x XL y Left: Langevin CPU time is indeed nearly linear in N. Below: Increase in accessible system size improves scaling collapse. DQMC (left) vs. Langevin (right). S(π,π)/L 7/ X6 8X8 10X10 12X12 14X14 DQMC, ω=1, λ=2 1/2, β c =6 (0.1) (β-β c )L S(π,π)/L 7/ X12 14X14 16X16 18X18 20X20 24X24 Langevin, ω=1, λ=2 1/2, β c = (β-β c )L

23 6. Conclusions- Potential Applications Phonon dispersion alters CDW wavevector in square lattice Holstein model. Non-Peierls CDW Mechanism: order decoupled from Fermi Surface Nesting. Honeycomb lattice Holstein model: Semi-metal to CDW quantum critical point. Langevin approach to updating boson degrees of freedom: Greater efficiency for generalizations of Holstein model. * Momentum dependent fermion-boson coupling. * Interactions of form n B i n F i. Both make conventional DQMC o(n 4 ), i.e. unfeasible.