Quantum s=1/2 antiferromagnet on the Bethe lattice at percolation I. Low-energy states, DMRG, and diagnostics

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1 Quantum s=1/2 antiferromagnet on the Bethe lattice at percolation I. Low-energy states, DMRG, and diagnostics Hitesh J. Changlani, Shivam Ghosh, Sumiran Pujari, Christopher L. Henley Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca NY , USA American Physical Society (APS) March Meeting, 2011 H.J.Changlani, APS 2011 Diluted Quantum Antiferromagnet on the Bethe Lattice

2 The Coordination 3 Bethe Lattice Percolation on the Bethe Lattice similar to that in higher dimensions Yet, so much like a 1D chain! To disconnect one segment from other need to cut exactly 1 bond Lack of loops conducive for Density Matrix Renormalization Group (DMRG) calculations. Tractable analytically The z=3 Bethe Lattice Bipartite lattice. Sign problem free Quantum Monte Carlo calcs! 2

3 Heisenberg Antiferromagnet (HAF) On Percolation Clusters (PC) H Heisen = J S i S j i, j Probability of connected bond p Probability of broken bond 1-p (Quenched disorder) Senthil and Sachdev (PRL 96) : PC 1D like, so probably disordered... But long range order within a cluster! (A.Sandvik PRL 2002) Lowest excitations are not linear spin waves Could be rotor modes with uniform susceptibility χ (refs. Sachdev's book, T. Vojta and J. Schmalian PRL '05) SW ~N 3 / 4 rot = M2 2 N ~N 1 N is no. of sites in percolation cluster 3

4 Question Does that mean that lowest excitation for a percolation cluster (a fractal object with Hausdorff dimension dh ) is, 1 d H rot ~N ~L?? L. Wang & A. Sandvik (PRL 2006) : No for square lattice PC Our Work : No for Bethe lattice PC 4

5 Balanced vs Unbalanced Clusters Balanced number of even = number of odd Singlet (S=0) ground state and triplet (S=1) first excited state We'll deal with balanced clusters from here on... Nred = Ngreen BALANCED 5

6 HAF On Square Lattice At Percolation threshold L. Wang, A. Sandvik (PRL 2006 and PRB 2010) studied square lattice PC's with Quantum Monte Carlo (QMC). Both balanced and unbalanced clusters. Balanced case : Computed energy gap indirectly for large clusters. Gap scales as L-z z=2dh not dh! Excitations lower than the expected rotor states! Result : Proposed scenario of dangling spins arising out of local sublattice imbalance 6

7 Dangling Spins Scenario A balanced cluster, is either, (a) locally balanced (i.e. each spin can be paired up with neighboring spin on Big Gap 0.33 J opposite sublattice) (b) or not Has dangling spins Small Gap 0.03 J Are the the 'unusual' low energy states due to interaction of separated dangling spins? (YES!) 7

8 Our Results Low energy spectra of balanced clusters Spatial profile of the dangling degrees of freedom Techniques Exact diagonalization for cluster sizes 22 sites Density Matrix Renormalization Group (DMRG) for larger clusters ( 32 sites at present) Successful technique (S.R. White PRL 92, PRB 93 ) for ground and low energy excited states of 1-d and small 2-d systems. Generalizes for lattice without loops. Prev. work on undiluted Bethe lattice by Otsuka (PRB '96), Lepetit et. al ('98), B.Friedman ('98), V.Murg et. al (PRB '10) 8

9 Our Results : Dangling degrees of freedom have spin ½ Evidence from the Quasi Degenerate (QD) States n = 2 dangling n = 4 dangling n = 6 dangling 26 = 64 E (3,1,1) (1) 4 QD QD 22 = 4 QD Important Observation (3) (1) 2 = 16 (1) (5,3,3) (3) (1) No. of QD states is 2n (n is the no. of dangling d.o.f.) QD (3) (3,3,1) (5,5) (5) (3,5,7) (3,3) (3) (5) (3) (1) Dangling d.o.f a spin ½ object! 9

10 Our Results- Lowest gap for each cluster (balanced clusters at p=pc=1/2) Our work (Bethe lattice) Wang and Sandvik (square lattice) Locally balanced With dangling N = 16 sites Frequency Log (Δ) 10

11 Question Where are the dangling spins (on each cluster)? 11

12 Locating the dangling spins The Simple Heuristic Wang and Sandvik : Try to dimerize as much as possible... what's remaining is dangling. 12

13 Locating the Dangling Spins - I Consider magnitude of the matrix element, (the spin flip diagnostic ) c i= triplet S +i singlet Measure of how decoupled or dangling a spin is, from its environment Dangling degree of freedom Area = ci 2 13

14 Locating The Dangling Spins - II Define the inter-site correlation matrix ij = Si S j expectation is calculated in ground state. Look for modes with 0 or near 0 eigenvalues. Modes give combination of spins which team up when interacting with other spins. For systems inaccessible by Exact Diag., we use linear spin wave theory. red is positive green is negative 14

15 Effective Hamiltonians What about higher excited states? We have gone beyond just the singlet-triplet states and explained the other quasi degenerate states numerically. We write an effective model in terms of quasi- spins (we call them ti ) Simplest guess respecting spin symmetries H eff = J ij t i t j i, j Turns out to be pretty good! That is, we can fit the quasi degenerate spectrum and find good approximations to these states. More details in Shivam Ghosh's talk (the one after this!) 15

16 Summary Emerging picture : Interaction of far off dangling spins lead to excitations lower than expected. L. Wang, A. Sandvik (PRL '06 & PRB '10) We have gone beyond just the singlet-triplet states and explained the other quasi degenerate states giving further credibility to the dangling spin picture Dangling degrees of freedom can be detected using the spin flip diagnostic and inter-site correlation matrix 16

17 Additional information (if time permits...and I'm sure it wont!) 17

18 N=20 sites N=14 sites Freq. Log (Δ) rot ~N 0.54 dangling ~N 0.33 Log (Δavg) Log N Log N

19 Area = ci 2 A dangling degree of freedom could be spread out over a few sites! 19

20 The Renormalization Step Form the density matrix (DM) of the systems+site by tracing over the environment Keep the most important states (ones with the highest eigenvalues in DM) 20

21 Step 1 Attack Inwards Perform Wilson's Numerical RG (warm up procedure) i.e. perform an energy based truncation. Attack the cluster from all open sides (like an army marching in!) 21

22 Step 2 Form The Environments Form the systems and environment blocks for the central site ( root ) On the way out, the renormalization step forms the environments for every site in the system 22

23 Step 3 Sweep In-Out-In Repeat Step 1 (march in, but now with environments!) and Step 2 till numerical convergence is attained. Typically 4-5 sweeps needed till convergence. For our purpose, modest number of states per block were kept (6 to 24) We target multiple states (by weighting them in the DM) or obtain lowest energy states in each spin sector to get the low energy states 23