Coupled Cluster Theories of Quantum Magnetism

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1 Coupled Cluster Theories of Quantum Damian Farnell 1

2 Outline of Talk A very brief overview of the high order CCM Comparison to results of other methods for various Archimedean lattices Spiral model states: example for an anisotropic square ( interpolating square/triangular) model Specific examples of magnetic models: The triangular lattice antiferromagnet in an external magnetic field The spin half 1 2 Model on the square lattice Conclusions and future research 2

3 CCM Overview Ground State $% = S e! S ~ "% ; #$ = #"! ~ S e + " = SICI ; S = 1+ SICI ~ ~! S S. E. $ E g = #" e He % S S "! H ~ =! H! # H $ $ s S # H ~ $ s + S ~ "! CI e He! = 0 "! Se [ H, CI ] e! = # S # S I A similar approach exists for the excited state See Parkinson and Farnell, An Introduction to Quantum Spin Systems LNP 816 Chapter 10. I 0 3

4 Approximation Schemes SUBm: All m body or lower order clusters. LSUBm: All clusters in a locale defined by m. SUBn m: All n body or lower order clusters in a locale defined by m. E.g., SUB4-4 for the s=1/2 and s=1 square-lattice ferrimagnet We extrapolate these results in the limit, m. 4

5 CCM Flowchart 5

6 Archimedean Lattices (a) Square (b) Honeycomb (c) CAVO (d) SrCuBO (e) Triangular (f) Maple-Leaf (g) Kagome 6

7 CCM results for unfrustrated lattices compared to some results of other approximate methods Model Parameter QMC Series ED SWT CCM Square HAF s=1/2 Square HAF s=1 Honeycomb HAF s=1/2 CAVO HAF s=1/2 E G /bond (3) (5) Order Param <s z > (3) 0.307(1) Gap Spin Stiffness 0.175(2) Mag. Suscept. χ (7) (10) (1) (6). SW Velocity E G /bond 2.28(1) (1) (1 st order) Order Param <s z > (4) E (2 nd G /bond ) Order Param <s z > (2 nd ) Gap E G /bond (linear) Order Param <s z > (linear) (SUB2) Spin Stiffness

8 CCM results for frustrated lattices compared to some results of other approximate methods SWT * (1 st order)/ Model Parameter Series ED Swinger Boson CCM Triangle SrCuBO Kagome E * G /bond Order Param <s z > * Spin Stiffness * Mag. Suscept. χ * 0.065(23) E G /bond Order Param <s z > E * G /bond Order Param <s z > ~ 0 Spin Stiffness See Quantum LNP 645 (chapter 2) and An Introduction to Quantum Spin Systems LNP 816 (chapters 10 & 11) for more details about all of these approximate calculations. See also, Series Expansions Methods for Lattice Models by Oitmaa et al. (Cambridge university Press, 2006). < 0 8

9 Spiral model states: the interpolating square/ triangular lattice model (AKA anisotropic square ) (Bishop, Li, et al., PRB 79, (2009)) 1 2 (a) Néel (b) Spiral (c) Striped 9

10 Triangular lattice antiferromagnets in an external magnetic field Co 2+ = spin-1/2 atoms Ni 2+ = spin-1 atoms 10

11 Background Co 2+ atoms in Ba 3 CoSb 2 O 9 and Ni 2+ atoms in Ba 3 NiSb 2 O 9 form s=1/2 and s=1 a triangular lattice. Quantum spins interact via an antiferromagnetic Heisenberg exchance interaction. An interesting effect of spin plateau is seen as one imposes an external magnetic field of strength λ. The relevant Hamiltonian is given by: Triangular lattice for Ba 3 CoSb 2 O 9 and Ba 3 NiSb 2 O 9 H= s s+λ s z 11

12 Spin Plateau (see: Farnell, Richter, & Zinke,. Phys.: Condens. Matt. 21, (2009)) Experiment CCM Co 2+ = spin-1/2 atoms Shirata, Tanaka et al. PRL 108, (2012). arxiv: v1 Ni 2+ = spin-1 atoms Shirata, Tanaka et al.. Phys. Soc. apan 80 (2011)

13 The 1 2 Model on the Square Lattice Nearest and Next Nearest Neighbour Bonds on the square lattice; < > indicates each bond counted once. N # i, j N # i,k H = 1 s i " s j + 2 s i " s k

14 Background to the 1 2 model Various quasi 2D materials are described by the 1 2 model, e.g., Li 2 VOSiO 4 and Li 2 VOGeO 4 2 / 1 in range 5 to 10. See later It is a canonical model used to investigate quantum magnetic systems, especially for case of strong frustration Non magnetic quantum phase (quantum paramagnet) Does this phase survive for T>0? The nature of quantum phase is still not fully resolved Possible deconfined quantum criticality at quantum phase transitions No exact solutions for this 2D model it is a good test of approximate techniques. 14

15 Ground State Energy CCM. R. Darradi et al. PRB 78, (2008) ED: N=40. Richter & Schulenburg. Eur. Phys.. B 73, (2010) Series Expansions, Sirker et al., PRB B 73,

16 Order Parameters CCM. R. Darradi et al. PRB 78, (2008) ED: N=40. Richter & Schulenburg. Eur. Phys.. B 73, (2010) Series Expansions, Oitmaa et al. PRB (1996) 16

17 The Spin Stiffness the response to a twist in local axes of spins, which so indicates stability of ground state CCM. R. Darradi et al. PRB 78, (2008) ED: N=40. Richter & Schulenburg. Eur. Phys.. B 73, (2010) ED + LSWT, Einarsson, PRB 51, 6151 (1995) 17

18 Excitation Energy Gap CCM: DF ED: N=40. Richter & Schulenburg. Eur. Phys.. B 73, (2010) Series: Kotov et al. PRB (1999) Gap opens at 2/ But is this result too large... Higher LSUBm please!! 18

19 Initial Results for the Quantum Fidelity 19 ) / / ( ) / ( ) / ( ) / ( ) / / ( ) / ( F!!! + " + " " " + " " = δ=0.001 above

20 Sign Rule for Small 2 / 1 A M A z &( 1) ci I M A & si ci 0 I A B A B! = " = # $ I = 1 Sites on A go over one sublattice i N A %%%%%%% A B A A B A Hard sign rule: rule above can be proven exactly at 2 =0 by noting that the above formulation results in a negative definite matrix formulation of the Schrodinger equation and so for which c I >0. Soft sign rule: rule can t be proven exactly (often due to frustration), although it is seen up to some value of 2 numerically via ED and CCM results. Weight of states: Sum over states I for which the rule holds such that: weight=σ I c I 2 20

21 Marshall Peierls Sign Rule CCM: DF ED: N=40. Richter & Schulenburg. Eur. Phys.. B 73, (2010) SWT & ED: Ivanov & richter,. Phys.: Condens. Matter 6 (1994) N N f 2 2 M =!! weight c / c I i= 1 i= 1 I weight! N M = " i= 1 c I 2 21

22 Range of Paramagnetic Regime Method 2 M 2 c1 2 c2 ED Series Dimer Boson Model CCM (±0.01) (±0.01) 5 1 Richter and. Schulenburg, Eur. Phys.. B 73, (2010) 2 Sushkov et al. PRB 63, (2001) 3 Kotov et al. PRB (1999) 4 Bishop, Farnell, Parkinson PRB (1998). 5 Darradi et al. PRB 78, (2008) 22

23 Nature of the paramagnetic regime Sushkov et al. PRB 63, (2001). 23

24 Inverse Magnetic Susceptibilities CCM + ED: Darradi, PRB 78, (2008) Series: Sirker PRB 73, (2006)? Taken from Sushkov, PRB 63, (2001) 24

25 Results for 1 <0 CCM ED Richter et al. PRB 81, (2010) 25

26 Spin Spin Correlation Functions CCM ED 1 <0: Results (thicker lines) indicate single PT point at (probably first order ) 1 >0: Results (thinner lines) indicate PT points at (probably first order) See: Richter et al. PRB 81, (2010) 26

27 Quantum Phase Diagram at T= c Gapless Gapped 2 c Gapless Gapless 1 2 M to Phase diagram from a synthesis all of the results presented here. Phase diagram from R. Nath, A. Tsirlin, H. Rosner, and C. Geibel, Phys. Rev. B 78, (2008). 27

28 Conclusions The CCM provides a useful tool in tackling (especially 2D) problems in quantum magnetism. It is accurate, reliable, and flexible. Can be used to detect phase transition point points, and possibly even their order. 28

29 Afterword and Thanks Final words in The Theory Of Made Simple by DC Mattis: I cannot resist pointing out the obvious: independently of the technical importance of magnetic substances which cannot be denied the theory of magnetism continues to lie at the core of practically all of contemporary theoretical physics. Its study continues to be a perpetual delight and inspiration. Thanks to: WE Hereaus Stiftung. All at the Physikszentrum, Bad Honnef. Also to: Prof. Raymond F. Bishop (Manchester), Prof. ohannes Richter (Magdeburg, Germany), Dr. Peggy Li (Manchester), Dr. ohn B. Parkinson (Manchester), Dr. oerg Schulenburg (Magdeburg), Dr. Sven Krueger (Magdeburg), Dr. Klaus Gernoth (Manchester), Dr. Ronald Zinke (Magdeburg), Dr. Rachid Darradi (Technische Universität Braunschweig, Germany), and Prof. Charles Campbell (Minnesota, USA), et (many more) al. 29

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