Yes, we can! Advanced quantum methods for the largest magnetic molecules

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1 Yes, we can! Advanced quantum methods for the largest magnetic molecules ürgen Schnack Department of Physics University of Bielefeld Germany schnack/ French Irish International Workshop on Magnetism and Electronic Structure May 2014, University College Dublin unilogo-m-rot.jpg

2 ? Problem The usual problem unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 1/37

3 ? Problem You have got a molecule! Congratulations! unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 2/37

4 ? Problem You have got an idea about the modeling! H = 2 N ij s (i) s (j) + g µ B B i<j i Heisenberg Zeeman s z (i) unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 3/37

5 ? Matrix In the end it s always a big matrix! ( ) Fe III 10: N = 10, s = 5/2 Dimension=60,466,176. Maybe too big? unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 4/37

6 ? Thank God, we have computers Thank God, we have computers Espresso-doped multi-core 128 cores, 384 GB RAM... but that s not enough! unilogo-m-rot.jpg u rgen Schnack, Advanced quantum methods 5/37

? Contents for you today Contents for you today 3 42 4711 42 0 3.14 4711 3.14 8 17 007 13 1.8 15 081 1. Finite-Temperature Lanczos Method 2.

7 ? Contents for you today Contents for you today Finite-Temperature Lanczos Method 2. Numerical Renormalization Group calculations We are the sledgehammer team of matrix diagonalization. Please send inquiries to jschnack@uni-bielefeld.de! unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 6/37

8 ? FTLM Finite-Temperature Lanczos Method (Good for dimensions up to ) unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 7/37

? Lanczos Lanczos a Krylov space method You do know exact diagonalization. What about diagonalization in reduced basis sets?! Full matrix = small matrix! But which set to choose?

9 ? Lanczos Lanczos a Krylov space method You do know exact diagonalization. What about diagonalization in reduced basis sets?! Full matrix = small matrix! But which set to choose??? Idea: generate the basis set with the operator you want to diagonalize: { φ, H φ, H 2 φ, H 3 φ,... } Hamiltonian creates its own relevant states! But which starting vector to choose??? Cornelius Lanczos ( ) Idea: almost any will do! (1) C. Lanczos,. Res. Nat. Bur. Stand. 45, 255 (1950). unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 8/37

10 ? FTLM Finite-temperature Lanczos Method I Z(T, B) = ν } ν exp { βh ν n } ν exp { βh ν ν n(ν) exp { βɛ n } n(ν) ν Z(T, B) dim(h) R R ν=1 N L n=1 exp { βɛ n } n(ν) ν 2 n(ν) n-th Lanczos eigenvector starting from ν Partition function replaced by a small sum: R = , N L aklič and P. Prelovšek, Phys. Rev. B 49, 5065 (1994). unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 9/37

11 ? FTLM How good is finite-temperature Lanczos? Works very well: compare frustrated cuboctahedron. N = 12, s = 3/2: Considered < 100, 000 states instead of 16,777,216. Exact results: R. Schnalle and. Schnack, Int. Rev. Phys. Chem. 29, (2010). FTLM:. Schnack and O. Wendland, Eur. Phys.. B 78, (2010). unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 10/37

12 ? Icosidodecahedron Icosidodecahedron s = 1/2 Exp. data: A. M. Todea, A. Merca, H. Bögge, T. Glaser, L. Engelhardt, R. Prozorov, M. Luban, A. Müller, Chem. Commun., 3351 (2009). unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 11/37

13 ? Icosidodecahedron Icosidodecahedron s = 1/2 The true spectrum will be much denser. This is miraculously compensated for by the weights. (Exact at low T, coarse grained at high T.) Z(T, B) dim(h) R R ν=1 N L n=1 exp { βɛ n } n(ν, Γ) ν, Γ 2 unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 12/37

14 ? Gd 4 M 8 Gd 4 M 8 Susceptibility T. N. Hooper,. Schnack, St. Piligkos, M. Evangelisti, E. K. Brechin, Angew. Chem. Int. Ed. 51 (2012) unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 13/37

Heesing: Dzyaloshinskii-Moriya & anisotropic

15 ? FTLM Recent developments 12 s = 7/2, dimension = 68,719,476,736 Goal: magnetic properties of anisotropic systems; Oliver Hanebaum: single-ion anisotropy; Christian Heesing: Dzyaloshinskii-Moriya & anisotropic exchange. unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 14/37

16 ? Hamiltonian with single-ion anisotropy Hamiltonian with single-ion anisotropy H ( B) = 2 i<j ij s i s j + i ( ) 2 N d i ei s i + µb B g i s i i [ H, S ] 2 0, [ H, S ] z 0; MAGPACK does not work! You have to diagonalize H ( B) for every field (direction and strength)! Orientational average for powder samples. unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 15/37

17 ? FTLM Glaser-type molecules: Mn III 6 Cr III S 6 θ s = 2, s = 3/2 dim(h) = 62, 500 non-collinear easy axes Hours compared to days, notebook compared to supercomputer! O. Hanebaum,. Schnack, submitted; arxiv: unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 16/37

18 ? FTLM A fictitious Mn III 12 M z vs B z s = 2 dim(h) = 244, 140, 625 collinear easy axes A few days compared to impossible! O. Hanebaum,. Schnack, submitted; arxiv: unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 17/37

19 ? FTLM A fictitious Mn III 12 M x vs B x No other method can deliver these curves! O. Hanebaum,. Schnack, submitted; arxiv: unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 18/37

20 ? Problem The advanced problem unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 19/37

21 ? NRG You deposite a molecule I Molecule with nice properties deposited on metal substrate; Exchange coupled to metal spins; Kondo screening may... unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 20/37

22 ? NRG You deposite a molecule II Kondo screening may improve or worsen the magnetic properties; How does the exchange coupling to the metal influence the magnetic properties? How to calculate such things? unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 21/37

23 ? NRG Numerical Renormalization Group calculations (Good for deposited molecules.) unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 22/37

? NRG Numerical Renormalization Group (Wilson) 1.5 (a) (b) A M [g µ B ] 1 0.5 Free dimer =1.0meV Free trimer =1.0meV Trimer A =0.10eV =1.

0meV A 0 0 1 2 3 4 5 6 7 8 B [10-3 W/(g µ B ) 8.

24 ? NRG Numerical Renormalization Group (Wilson) 1.5 (a) (b) A M [g µ B ] Free dimer =1.0meV Free trimer =1.0meV Trimer A =0.10eV =1.0meV Trimer A =0.15eV =1.0meV Trimer A =0.17eV =1.0meV Trimer A =0.20eV =1.0meV Trimer A =0.50eV =1.0meV Trimer A =1.00eV =1.0meV A B [10-3 W/(g µ B ) 8.64 T] Magnetic properties of deposited spin systems; Martin Höck (until 07/2013): anisotropic single spins (PRB 87, (2013)); Henning-Timm Langwald: deposited Heisenberg systems. unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 23/37

25 ? NRG NRG minimal model H = H electrons + H coupling + H impurity H electrons = i j, σ t ijd iσ d jσ + g e µ B BS z H coupling = 2 A S s 0; H impurity = Hamiltonian of your molecule! s 0 spin density at contact; A NRG construction of a small (!) effective model in order to evaluate properties of the deposited cluster, the impurity (3). (1) K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975) (2) M. Höck,. Schnack, Phys. Rev. B 87, (2013) (3) Impurity is a technical term in this context and not an insult to chemists. unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 24/37

26 ? NRG NRG in a cartoon A A t1 t2 t3 t Metallic surface is replaced by semi-infinite Hubbard chain; Parameters of the chain: hopping matrix elements and on-site energies; Stepwise enlargement of the chain (t 1 > t 2 > t 3... ); Truncation of basis set when necessary. unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 25/37

27 ? NRG Physical example A X. Chen et al., Phys. Rev. Lett. 101, (2008). unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 26/37

28 ? NRG Increasing coupling to the substrate A H.-T. Langwald and. Schnack, submitted; arxiv: unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 27/37

29 ? NRG Increasing coupling to the substrate A H.-T. Langwald and. Schnack, submitted; arxiv: unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 28/37

30 ? NRG Increasing coupling to the substrate A H.-T. Langwald and. Schnack, submitted; arxiv: unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 29/37

31 ? NRG Increasing coupling to the substrate A H.-T. Langwald and. Schnack, submitted; arxiv: unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 30/37

32 ? NRG Increasing coupling to the substrate A H.-T. Langwald and. Schnack, submitted; arxiv: unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 31/37

33 ? NRG Increasing coupling to the substrate A H.-T. Langwald and. Schnack, submitted; arxiv: unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 32/37

34 ? NRG Weak vs. strong coupling A=0 weak coupling limit: unperturbed trimer A 0.1W no impact of the substrate A >> strong coupling limit: effective dimer A 0.5W partial screening; effective remainder Inbetween: no simple characterization unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 33/37

? Summary Summary Exact diagonalization is great but limited. Finite-Temperature Lanczos is a good approximate method for Hilbert space dimensions smaller than 1010.

35 ? Summary Summary Exact diagonalization is great but limited. Finite-Temperature Lanczos is a good approximate method for Hilbert space dimensions smaller than Magnetic molecules change their properties on metallic surfaces. Question: appropriate model? NRG deals with molecules that are exchange-coupled to the substrate. unilogo-m-rot.jpg u rgen Schnack, Advanced quantum methods 34/37

36 ? Collaboration Many thanks to my collaborators worldwide M. Czopnik, T. Glaser, O. Hanebaum, Chr. Heesing, N.B. Ivanov, F. Kaiser, H.-T. Langwald, A. Müller, Chr. Schröder (Bielefeld) K. Bärwinkel, H.-. Schmidt, M. Neumann (Osnabrück) M. Luban (Ames Lab, USA); P. Kögerler (Aachen, ülich, Ames); R.E.P. Winpenny, E..L. McInnes (Man U, UK); L. Cronin, M. Murrie (Glasgow, UK); E. Brechin (Edinburgh, UK); H. Nojiri (Sendai, apan); A. Postnikov (Metz, France); M. Evangelisti (Zaragoza), A. Tennant (Oak Ridge, USA). Richter,. Schulenburg (Magdeburg); A. Honecker (Göttingen); U. Kortz (Bremen); B. Lake (HMI Berlin); B. Büchner, V. Kataev, H.-H. Klauß (Dresden); P. Chaudhuri (Mühlheim);. Wosnitza (Dresden-Rossendorf);. van Slageren (Stuttgart); R. Klingeler (Heidelberg); O. Waldmann (Freiburg) unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 35/37

37 ? The end Thank you very much for your attention. The end. unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 36/37

38 ? Information Molecular Magnetism Web Highlights. Tutorials. Who is who. Conferences. unilogo-m-rot.jpg ürgen Schnack, Advanced quantum methods 37/37

Extreme magnetocaloric properties of Gd 10 Fe 10 sawtooth spin rings: a case study using the finite-temperature Lanczos method

Extreme magnetocaloric properties of Gd 10 Fe 10 sawtooth spin rings: a case study using the finite-temperature Lanczos method Jürgen Schnack Department of Physics University of Bielefeld Germany http://obelix.physik.uni-bielefeld.de/