Cluster Functional Renormalization Group

Transcription

1 Cluster Functional Renormalization Group Johannes Reuther Free University Berlin Helmholtz-Center Berlin California Institute of Technology (Pasadena) Lefkada, September 26, 2014 Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

2 Collaborators Ronny Thomale Würzburg Peter Wölfle Karlsruhe Christian Platt Würzburg JR and Ronny Thomale, Phys. Rev. B 89, (2014) JR and P. Wölfle, Phys. Rev. B 81, (2010) JR, R. Thomale, and S. Trebst, Phys. Rev. B 84, (2011) Y. Singh, S. Manni, JR, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, (2012) JR, R. Thomale, S. Rachel, arxiv: (2014) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

3 Cluster spin models J ij S i S j c J ij S i S j Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

4 Cluster spin models c J ij S i S j Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

5 Cluster spin models c J ij S i S j Use functional renomoralization-group method: Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

6 Outline 1 Pseudofermion FRG 2 Cluster implementation 3 Application to the bilayer Heisenberg model 4 Conclusion Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

7 Pseudofermion FRG Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

8 Pseudo fermions Introduce two fermionic operators f i, f i for each lattice site i. Then: S µ i = 1 2 f i σ µ f i with f i = ( fi f i ) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

9 Pseudo fermions Introduce two fermionic operators f i, f i for each lattice site i. Then: S µ i = 1 2 f i σ µ f i with f i = ( fi Spin space is two-dimensional (, ) while two fermions define a four-dimensional space ( 0, 0, 0, 1, 1, 0, 1, 1 ). f i ) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

10 Pseudo fermions Introduce two fermionic operators f i, f i for each lattice site i. Then: S µ i = 1 2 f i σ µ f i with f i = ( fi Spin space is two-dimensional (, ) while two fermions define a four-dimensional space ( 0, 0, 0, 1, 1, 0, 1, 1 ). A spin 1/2 is only realized in the subspace with f i f i + f i f i = 1 (one fermion per site)! f i ) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

11 Pseudo fermions Introduce two fermionic operators f i, f i for each lattice site i. Then: S µ i = 1 2 f i σ µ f i with f i = ( fi Spin space is two-dimensional (, ) while two fermions define a four-dimensional space ( 0, 0, 0, 1, 1, 0, 1, 1 ). A spin 1/2 is only realized in the subspace with f i f i + f i f i = 1 (one fermion per site)! Empty and doubly occupied sites carry spin zero and act like vacancies in the lattice = excitation energy of order J = unphysical occupations are naturally suppressed at zero temperature f i ) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

12 H = ij Fermionic Hamiltonian J ij S i S j 1 4 ( ) ( ) J ij f i σ µ f i f j σµ f j ij µ Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

13 Fermionic Hamiltonian H = ij J ij S i S j 1 4 ( ) ( ) J ij f i σ µ f i f j σµ f j ij µ Diagrammatics in the fermions: bare progagator: G 0 (iω) = 1 iω = bare self energy: Σ 0 = 0 interaction vertex: Γ 0 = J ij Propagator is strictly local (no fermion hopping). Interactions are non-local. Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

14 Fermionic Hamiltonian H = ij J ij S i S j 1 4 ( ) ( ) J ij f i σ µ f i f j σµ f j ij µ Diagrammatics in the fermions: bare progagator: G 0 (iω) = 1 iω = bare self energy: Σ 0 = 0 interaction vertex: Γ 0 = J ij Propagator is strictly local (no fermion hopping). Interactions are non-local. FRG may be applied (relatively) straightforwardly: Introduce infrared frequency cutoff in the propagator: G 0 (iω) = 1 iω G0 Λ Θ( ω Λ) (iω) = iω Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

15 Pseudofermion FRG Define a flowing self energy Σ Λ and a two-particle vertex Γ Λ (1, 2 ; 1, 2), 1 = {ω 1, i 1, α 1 }. = FRG equations:... Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

16 Pseudofermion FRG Define a flowing self energy Σ Λ and a two-particle vertex Γ Λ (1, 2 ; 1, 2), 1 = {ω 1, i 1, α 1 }. = FRG equations:... Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

17 Pseudofermion FRG Define a flowing self energy Σ Λ and a two-particle vertex Γ Λ (1, 2 ; 1, 2), 1 = {ω 1, i 1, α 1 }. = FRG equations:... Initial conditions: Σ Λ = 0 Γ Λ (1, 2 ; 1, 2) = 1 4 J i 1,i 2 σ µ α 1 α 1 σ µ α 2 α 2 δ i1 i 1 δ i2 i 2 (i 1 i 2, α 1 α 2 ) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

18 Pseudofermion FRG This type of FRG approach works surprisingly well for 2D spin models JR and P. Wölfle, Phys. Rev. B 81, (2010) JR, R. Thomale, and S. Trebst, Phys. Rev. B 84, (2011) Y. Singh, S. Manni, JR, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, (2012) JR, R. Thomale, S. Rachel, arxiv: (2014) even though: there is no well defined point of expansion (however, certain mean-field limits describing magnetic order and disorder are included exactly). 0D spin clusters may not be treated accurately. Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

19 Pseudofermion FRG This type of FRG approach works surprisingly well for 2D spin models JR and P. Wölfle, Phys. Rev. B 81, (2010) JR, R. Thomale, and S. Trebst, Phys. Rev. B 84, (2011) Y. Singh, S. Manni, JR, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, (2012) JR, R. Thomale, S. Rachel, arxiv: (2014) even though: there is no well defined point of expansion (however, certain mean-field limits describing magnetic order and disorder are included exactly). 0D spin clusters may not be treated accurately. Can this be resolved within an cluster FRG approach which: uses the isolated cluster limit as well defined expansion point? treats finite spin clusters exactly? Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

20 Cluster implementation See also: A. Rançon and N. Dupuis, Phys. Rev. B 83, (2011) A. Rançon and N. Dupuis, Phys. Rev. B 84, (2011) M. Kinza, J. Ortloff, J. Bauer, and C. Honerkamp, Phys. Rev. B 87, (2013) C. Taranto, S. Andergassen, J. Bauer, K. Held, A. Katanin, W. Metzner, G. Rohringer, and A. Toschi, Phys. Rev. Lett. 112, (2014) Talk by A. Toschi, 17:50 Session VIII Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

21 General idea: Replace Exact cluster vertices Γ Λ (1, 2 ; 1, 2) = 1 4 J i 1,i 2 σ µ α 1 α 1 σ µ α 2 α 2 δ i1 i 1 δ i2 i 2 (i 1 i 2, α 1 α 2 ) in the initial conditions by the exact cluster vertex Γ ex (1, 2 ; 1, 2) if J i1,i 2 is a coupling within a cluster. Then the isolated clusters are treated exactly! J S S i i 1 2 i 1 i 2 Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

22 General idea: Replace Exact cluster vertices Γ Λ (1, 2 ; 1, 2) = 1 4 J i 1,i 2 σ µ α 1 α 1 σ µ α 2 α 2 δ i1 i 1 δ i2 i 2 (i 1 i 2, α 1 α 2 ) in the initial conditions by the exact cluster vertex Γ ex (1, 2 ; 1, 2) if J i1,i 2 is a coupling within a cluster. Then the isolated clusters are treated exactly! J S S i i 1 2 i 1 i 2 Is this allowed? Which consequences does it have on the FRG scheme? One severe difficulty indeed arises: Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

23 FRG equations First parametrize: Γ Λ (1, 2 ; 1, 2) = γ Λ (1, 2 ; 1, 2) γ Λ (1, 2 ; 2, 1) with γ Λ (1, 2 ; 1, 2) = γ Λ (2, 1 ; 2, 1) and γ Λ (1, 2 2 2' ; 1, 2) = δi1 1 1' i 1 δ i2 i 2 FRG equation for γ Λ (1, 2 ; 1, 2): Λ γλ (1, 2 ; 1, 2) = 1 2π [ γ Λ (1, 2 ; 3, 4)γ Λ (3, 4; 1, 2) 3,4 + γ Λ (2, 4; 3, 1)γ Λ (3, 1 ; 2, 4) γ Λ (1, 4; 1, 3)γ Λ (3, 2 ; 4, 2) + γ Λ (1, 4; 1, 3)γ Λ (3, 2 ; 2, 4) + γ Λ (1, 4; 3, 1)γ Λ (3, 2 ; 4, 2) ] (G Λ (iω 3 )S Λ (iω 4 ) + G Λ (iω 4 )S Λ (iω 3 )) single scale propagator S Λ = G Λ [ Λ (G Λ 0 ) 1 ]G Λ Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

24 FRG equations First parametrize: Γ Λ (1, 2 ; 1, 2) = γ Λ (1, 2 ; 1, 2) γ Λ (1, 2 ; 2, 1) with γ Λ (1, 2 ; 1, 2) = γ Λ (2, 1 ; 2, 1) and γ Λ (1, 2 2 2' ; 1, 2) = δi1 1 1' i 1 δ i2 i 2 FRG equation for γ Λ (1, 2 ; 1, 2): Integrated up: γ Λ (1, 2 ; 1, 2) = γ (1, 2 ; 1, 2) + Λ dλ 1 2π 3,4 [ γ Λ (1, 2 ; 3, 4)γ Λ (3, 4; 1, 2) + γ Λ (2, 4; 3, 1)γ Λ (3, 1 ; 2, 4) γ Λ (1, 4; 1, 3)γ Λ (3, 2 ; 4, 2) + γ Λ (1, 4; 1, 3)γ Λ (3, 2 ; 2, 4) + γ Λ (1, 4; 3, 1)γ Λ (3, 2 ; 4, 2) ] P Λ (iω 3, iω 4 ) with P Λ (iω 3, iω 4 ) = G Λ (iω 3 )S Λ (iω 4 ) + G Λ (iω 4 )S Λ (iω 3 ) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

25 FRG equations First parametrize: Γ Λ (1, 2 ; 1, 2) = γ Λ (1, 2 ; 1, 2) γ Λ (1, 2 ; 2, 1) with γ Λ (1, 2 ; 1, 2) = γ Λ (2, 1 ; 2, 1) and γ Λ (1, 2 2 2' ; 1, 2) = δi1 1 1' i 1 δ i2 i 2 FRG equation for γ Λ (1, 2 ; 1, 2): Iterative solution: Set γ 0 = γ and insert the solutions successively, yielding γ 1, γ 2,... γ Λ n+1(1, 2 ; 1, 2) = γ (1, 2 ; 1, 2) + Λ dλ 1 2π 3,4 [ γ Λ n (1, 2 ; 3, 4)γ Λ n (3, 4; 1, 2) + γ Λ n (2, 4; 3, 1)γ Λ n (3, 1 ; 2, 4) γ Λ n (1, 4; 1, 3)γ Λ n (3, 2 ; 4, 2) + γ Λ n (1, 4; 1, 3)γ Λ n (3, 2 ; 2, 4) + γ Λ n (1, 4; 3, 1)γ Λ n (3, 2 ; 4, 2) ] P Λ (iω 3, iω 4 ) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

26 Iterative solution Example for a diagrammatic contribution to γ Λ 3 (1,2 ;1,2): Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

27 Iterative solution Example for a diagrammatic contribution to γ Λ 3 (1,2 ;1,2): Assume that the propagator lines (sites) 4,5,6 and 3,7,8,9,10 are located on the same cluster, respectively. Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

28 Iterative solution Example for a diagrammatic contribution to γ Λ 3 (1,2 ;1,2): Using the new initial conditions, the bare interaction needs to be replaced by the exact cluster vertex for all intra-cluster couplings,. Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

29 Iterative solution Example for a diagrammatic contribution to γ Λ 3 (1,2 ;1,2): Using the new initial conditions, the bare interaction needs to be replaced by the exact cluster vertex for all intra-cluster couplings,. Over-counting of sub diagrams! Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

30 New classes of vertices How can this over-counting of terms be avoided? Define classes of vertices γ Λ 1,n, γλ 2,n,..., γλ 10,n with γλ n = 10 x=1 γλ x,n + γ ex Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

31 New classes of vertices Express the FRG flow equations in terms of these new classes. Example: particle-particle channel {Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

32 New classes of vertices Express the FRG flow equations in terms of these new classes. Example: particle-particle channel {Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

33 [ [ [ [ [ [ [ [ Introduce counter terms Counter terms cancel the redundant diagrams in each iteration step separately! (works similarly for the other interaction channels) = Well defined cluster-expansion scheme. Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

34 Application to the bilayer Heisenberg model Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

35 Bilayer Heisenberg model H =J S ia S ja ij a=1,2 + J S i1 S i2 i Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

36 Bilayer Heisenberg model Phase diagram for J, J > 0 and g = J /J : H =J S ia S ja ij a=1,2 + J S i1 S i2 i ~ 8g Numerical value from Monte Carlo: g c = L. Wang, K. S. D. Beach, and A. W. Sandvik, Phys. Rev. B 73, (2006) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

37 Bilayer Heisenberg model Calculate the frequency dependent susceptibility via χ Λ=0 (k, ω) = + k = (k x, k y, k z ) with k x, k y [ π, π] and k z = 0, π. In the following k z = π. For a stable numerical implementation of the cluster FRG, some approximations and modifications are necessary. (for details, see JR and Ronny Thomale, Phys. Rev. B 89, (2014)) Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

38 Bilayer Heisenberg model Results for the static susceptibility χ(k) = χ(k, ω = 0): Sharp peaks at k = (±π, ±π) indicate a transition into the antiferromagnetic Néel phase at g Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

39 Bilayer Heisenberg model Calculate the spin-excitation spectrum A(k, ω) = 1 π Im χ(k, ω + i0+ ) via analytical continuation (Padé approximation): Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

40 Bilayer Heisenberg model Calculate the spin-excitation spectrum A(k, ω) = 1 π Im χ(k, ω + i0+ ) via analytical continuation (Padé approximation): Fit a bosonic Green s function with a single excitation to χ(k, ω) ( 1 χ(k, z) = W k ) 1 z+e k +iδ k z E k +iδ k to obtain the dispersion of spin excitations E k the quasiparticle weight W k the damping (inverse lifetime) δ k Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

41 Bilayer Heisenberg model Dispersion E k in comparison to dimer expansion: (Z. Weihong, Phys. Rev. B 55, (1997)) g= Quasiparticle weight W k and damping δ k : ~ ~ Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

42 Conclusion Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

43 Conclusion Within a cluster FRG approach the spin clusters are treated exactly while the inter-cluster couplings are addressed via RG. In order to circumvent an over-counting of diagrams, counter terms need to be introduced. A simplified numerical implementation yields qualitatively correct results for the bilayer Heisenberg model. (Improvements concerning g c are possible.) The exact cluster vertices enter the initial conditions and only need to be evaluated once = Large spin clusters are possible (as long as the exact cluster vertices can be calculated)! Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25

44 Thank you for your attention! Johannes Reuther Cluster Functional Renormalization Group () Lefkada, September 26, / 25