Truncated Unity Functional renormalization group (TUfRG) for 2D - PowerPoint PPT Presentation

Truncated Unity Functional renormalization group (TUfRG) for 2D lattices: getting more quantitative 1. fRG: quantitative issues 2. TUfRG in momentum space: recent results 3. TUfRG for frquency dependence: outlook Carsten Honerkamp Institute for Theoretical Solid State Physics RWTH Aachen University Support DFG FOR 723, FOR 912, SPP1458/9 & Ho2422/x-x, RTG 1995

Functional RG for Hubbard-type models Model bandwidth Λ Interaction at scale bandwidth ~ eV (few eV s) not much structure, mean-field decoupling ambiguous/impossible functional Intermediate energy scales: particle-hole pairs, renormalization group (fRG): particle-particle loop corrections generate structure lower Λ in effective low-energy interaction V Λ (k 1 , k 2 , k 3 ) k 1 k 3 H e ff = 1 q ) c † q,s c † X p ⇥ , � V ( � p + � p, � 10-100 meV q,s 0 c � p 0 ,s 0 c � p + � � p 0 � � � p,s 2 ≥ T c k 2 k 4 p, � p 0 , � � q s,s 0 ⇒ e.g. guided mean-field decoupling 0 Functional Renormalization Group (fRG): Provides low-energy effective action & momentum structure V Λ ( k,k’,k+q) ! Removes ambiguities of mean-field decouplings.

Functional RG fRG captures all one-loop contributions: Cooper Peierls unbiased description of competing orders Vertices at scale Λ = d/d Λ Vertex- d/d Λ interaction Corrections Λ -derivative of V Λ (k 1 , k 2 , k 3 ) Screening 1-loop diagram k 1 k 3 Brillouin zone k = wavevector, band, frequency k 2 k 4 patch k Fermi surface Keep track of wavevector structure: N -patch wave vector k n Discretize Brillouin zone into N patches n More recently: channel decomposition & form factor expansion Often neglected: self-energy, higher-order interactions, frequency dependence

Flow to strong coupling G 0 Standard cases without self-energy feedback: = Flow to strong coupling G 0 Initial condition V ( k 1 , k 2 , k 3 ) = U Leading low-energy correlations Flow Energy scales è ‘ Weather forecast ’ Λ c = estimate for gaps in electronic spectrum Metzner, Salmhofer et al. RMP 2012

1. Quantitative issues: testing fRG for materials Take model Hamiltonian with parameters given, e.g., by DFT & cRPA Single-particle parameters, fit or Wannier matrix elements target bands Interaction parameters, e.g., Wannier matrix elements, cRPA n Can fRG become quantitative low-energy frontend of ab-initio theory? n Besides groundstate: Energy scales for phase transitions & relevant excitations? Trends within material families?

Trends in 1111 iron arsenide superconductors metallic antiferromagnet (AF-SDW)

La-1111 versus Sm-1111 Why is T c in La-1111 much lower than in Sm-1111? La-1111 RE- OFeAs ‘ RE -1111’ RE =La,Sm, … As Fe FeAs-Tetrahedra Sm-1111 elongate for Sm-1111 Andersen & Boeri 2011

Trends in 1111 iron arsenides fRG for 8-band model reproduces sizable T c -difference for pairing, while keeping AF-SDW scale unchanged Sm-1111 La-1111 Experimental trend reproduced Superconducting scale differs ~ factor 3 Overall energy scale ok, or a little too small … AF-SDW scale comparable Lichtenstein, Maier, Platt, Thomale, CH, Boeri, Andersen PRB 2014

Gaps in bi- & trilayer graphene (b) Phys. Rev. Lett. 108, See also: 076602 (2012) PRL 2012 B. E. Feldman et al., Nature Phys. 2009, A. S. Mayorov et al., Science 2011 Trilayer gaps: Bao et al., Clean current-annealed Nature Phys. suspendend BLG 2011. Proc. Nat. Acad. Sci., 109, 10802 (2012) Gap scale ≈ 2-3meV Trilayer: Nature Physics, 7, 948 (2011), T c ≈ 5K in bilayer, Lee, C.N. Lau et al. 2014 Even larger in trilayer (40meV) Also: Nijmegen (Maan) group

Model for layered graphene E.g. AB (bernal) stacked bilayer: Four bands, 2 quadratic band crossing points @ K , K ’ Take ab-initio-derived interaction parameters (‘constrained RPA’), interpolate between mono-layer and graphite values Wehling et al. PRL 2011

N-Layer graphene @ charge neutrality Single layer: Raghu, Scherer 0 , CH et al., PRL 2008 Quantum spin Hall AA bilayer , charge density Sanchez de la Pena, spin density wave ungapped Scherer, CH, 2014 wave semimetal ABC trilayer AB bilayer AB bilayer, ABC trilayer Scherer (N-1) , Uebelacker, CH, 2012

The ‘scale challenge’ fRG scales for gaps in layered graphene seems far too large compared to experiment, even with ‘realistic’ model parameters Sources of error: v N-patch fRG (in-)sufficient approximation? fRG Th. Lang et al. PRL 2012, compares QMC gaps with fRG scale, pure onsite Hubbard U v Model incorrect? Other interactions? Long-range Coulomb! v Model parameters incorrect? cfRG instead of cRPA?

Resolve patching ambiguities Choice of representative wavevectors for patches matters: Daniel D. Scherer, Michael M. Scherer, C. Honerkamp, Phys. Rev. B 92 (2015) Yanick Volpez, Daniel D. Scherer, and Michael M. Scherer Convergence requires several patch rings Phys. Rev. B 94 , 165107 (2016) QSH phase replaced by charge-modulated phase

2. Truncated unity fRG in momentum space n Builds on channel decomposition à la Salmhofer et al. (Husemann, Salmhofer, PRB 2009) n Incorporates numerical advantages of singular-mode (SM-)fRG , Q.H. Wang et al. PRB 21012 n Idea: insert resolutions of unity in momentum space factor basis into one-loop RG eqns δ kk 0 = 1 X e i ( k � k 0 ) x N G 0 x = J. Lichtenstein, D. Sanchez dlP, D. Rohe, CH, S.A. Maier Computer Physics Communications 2017 G 0 n Truncation of basis provides physically transparent approximation & high momentum resolution n Parallelizes nicely on high-performance architectures (= headroom for attacking frequency-dependence, selfenergies, … )

Channel decomposition Husemann, Salmhofer, Giering, Eberlein & Metzner , Maier &CH ... Karrasch et al. Instead of one function of three variables, use three functions P,D, C of one ’strong/bosonic‘ variable s = k 1 + k 2 , t = k 3 − k 1 , u = k 4 − k 1 V Λ ( k 1 , k 2 , k 3 ) = V 0 ( k 1 , k 2 , k 3 ) + P Λ ( k 1 , k 3 ; s ) + D Λ ( k 1 , k 4 ; t ) + C Λ ( k 1 , k 3 ; u ) ‘weak/fermionic’ variables, smooth dependence µ = − 0.7t, 〈 n 〉 ≈ 0.99 µ = − t, 〈 n 〉 ≈ 0.86 µ = − 1.2t, 〈 n 〉 ≈ 0.72 1 1 1 25 25 25 23 23 23 27 27 0.5 0.5 0.5 27 21 21 21 31 29 17 29 29 19 19 17 k y a / π k y a / π k y a / π 19 17 31 31 0 0 0 15 1 3 3 3 1 15 1 15 13 13 13 5 5 − 0.5 5 − 0.5 − 0.5 11 11 11 7 7 7 9 9 9 − 1 − 1 − 1 − 1 0 1 − 1 0 1 − 1 0 1 k x a / π k x a / π k x a / π 12 15 10 µ = − 0.7t, T=0.04t µ = − t, T=0.04t µ = − 1.2t, T=0.01t 10 30 30 30 Data for 8 10 25 25 25 5 V Λ ( k 1 , k 2 , k 3 ) k 1 + k 3 = const . 6 20 20 20 k 3 − k 1 = const . k 1 k 1 k 1 15 15 4 15 from 2D 5 0 10 10 10 2 Hubbard model, 5 5 5 0 CH (2000) 0 10 20 30 10 20 30 10 20 30 − 5 − 2 k 2 k 2 k 2 k 2 − k 3 = k 4 − k 1 = const .

Channel decomposition Husemann, Salmhofer, Giering, Eberlein & Metzner , Maier & CH ... Karrasch et al. Instead of one function of three variables, use three functions P,D, C of one ’strong/bosonic‘ variable s = k 1 + k 2 , t = k 3 − k 1 , u = k 4 − k 1 V Λ ( k 1 , k 2 , k 3 ) = V 0 ( k 1 , k 2 , k 3 ) + P Λ ( k 1 , k 3 ; s ) + D Λ ( k 1 , k 4 ; t ) + C Λ ( k 1 , k 3 ; u ) ‘weak/fermionic variables‘, captured by smooth form factors f x ( k ), form factor expansion: X P Λ ( k 1 , k 3 ; s ) = f x 1 ( k 1 ) f ∗ x 3 ( k 3 ) P Λ ( x 1 , x 3 ; s ) x 1 ,x 3 X ˙ ˙ ˙ C Λ ( x 1 , x 3 ; s ) = P Λ ( x 1 , x 3 ; s ) = D Λ ( x 1 , x 3 ; s ) = P Λ ( k 1 , k 3 ; s ) = T ˙ X V Λ ( k 1 , k ; s ) L PP ( k ; s ) V Λ ( k, k 3 ; s ) N L k

Form factor basis: X P Λ ( k 1 , k 3 ; s ) = f x 1 ( k 1 ) f ∗ x 3 ( k 3 ) P Λ ( x 1 , x 3 ; s ) x 1 ,x 3 bonds on real space lattice x i = ~ b i X Form factors/basis functions f n ( k ) most easily organized on real space Bravais lattice spanned by bond vectors ~ b = b 1 ~ e 1 + b 2 ~ e 2 real lattice bond functions b ( ~ r ) = � ~ f ~ r, ~ b Symmetrize wrt IRREPs of reciprocal lattice point group G k ) = e i ~ k · ~ b ( ~ b bond exponentials f ~ X f l ( ~ r ) = a l ( R ) � ~ r,R ~ b R ∈ G e.g. - + - ‘ d -wave‘ f l ( ~ b ( ~ X k ) = a l ( R ) f R ~ k ) R ∈ G f d x 2 − y 2 ( ~ k ) ∝ cos k x − cos k y For most cases: Short bonds b most important <=> form factors f n ( k ) smooth C. Platt, W. Hanke, R. Thomale, Adv. Phys. 2013

a l ( ~ X f l ( ~ r ) = b ) � ~ Fermion bilinear interaction r, ~ b ~ b e.g. - + - ‘ d -wave‘ In real space, P -interaction becomes pair-pair scattering: 2 3 ! 2 3 ~ b 1 − ~ H V = 1 b 3 X 4X l 3 ( ~ b 3 ) c † r 3 ,s c † 4X a l 1 ( ~ 5 V P a ∗ ~ r 1 − ~ r 3 + b 1 ) c ~ b , s 0 c ~ r 1 + ~ r 1 ,s l 1 ,l 3 5 ~ r 3 + ~ 2 2 ~ b 3 ,s 0 l 1 ,l 3 ~ ~ b 3 b 1 outgoing pair, s,s 0 incoming pair, ~ short ranged short ranged r ~ pair distance, can get long r 3 ~ exchange boson ~ b 3 ~ r Channel decomposition is way of rewriting full interaction as sum ~ b 1 of interactions between all possible/necessary fermion bilinears! ~ r 1 particle-hole-pairs, no spin flip particle-hole-pairs, particle-particle- spin flip pairs V Λ ( k 1 , k 2 , k 3 ) = V 0 ( k 1 , k 2 , k 3 ) + P Λ ( k 1 , k 3 ; s ) + D Λ ( k 1 , k 4 ; t ) + C Λ ( k 1 , k 3 ; u ) Intuitive representation with meaningful truncations