Advancements in Quantum Magnetism: Second Fermionization and Diag MC

1. Second fermionization & Diag.MC for quantum magnetism N. Prokof’ev In collaboration with B. Svistunov AFOSR MURI Advancing Research in Basic Science and Mathematics KWANT, 6/2/15

2. (same for correlation functions) - Popov-Fedotov trick for spin-1/2 Heisienberg model: - Generalization to arbitrary spin & interaction type; SU(N) case - Projected Hilbert spaces (tJ-model) & elimination of large expansion parameters ( U in the Fermi-Hubbard model) - Heisenberg models in D=2,3: classical-to-quantum correspondence

3. Popov-Fedotov trick for S=1/2 Heisenberg model: spin-1/2 f-fermions spin-1/2 f-fermions • Dynamics: perfect on physical states: - Unphysical empty and doubly occupied sites decouple from physical sites and each other: - Need to project unphysical Hilbert space out in statistics in the GC ensemble because

4. Popov-Fedotov trick for S=1/2 with complex Now Flat band Hamiltonian to begin with: + interactions Standard Feynman diagrams for two-body interactions

5. Partition function of physical sites in the presence of unphysical ones (K blocked sites) Proof of Number of unphysical sites with n=2 or n=0 configuration of unphysical sites Partition function of the unphysical site

6. Arbitrary spin (or lattice boson system with n < 2S+1): Mapping to (2S+1) fermions: … Onsite fermionic operator in the projected subspace converting fermion to fermion . For example, Matrix element, same as for SU(N) magnetism: a particular symmetric choice of

7. Proof of is exactly the same: Dynamics: perfect on physical states: Unphysical empty and multiply-occupied sites decouple from physical sites and each other: Partition function of the unphysical site Always has a solution for (fundamental theorem of algebra)

8. Projected Hilbert spaces; t-J model: Dynamics: perfect on physical states: Unphysical empty and doubly occupied sites decouple from physical sites and each other: as before, but C=1! previous trick cannot be applied

9. Solution: add a term For we still have but , so Zero! Feynman diagrams with two- and three-body interactions Also, Diag. expansions in t, not U, to avoid large expansion parameters: n=2 state  doublon  2 additional fermions + constraints + this trick

10. The bottom line: Standard diagrammatic expansion but with multi-particle vertexes: If nothing else, definitely good for Nature cover !

11. First diagrammatic results for frustrated quantum magnets Triangular lattice spin-1/2 Heisenberg model: Magnetism was frustrated but this group was not Oleg Starykh Univ. of Utah Boris Svistunov Umass, Amherst Sergey Kulagin Umass, Amherst Chris N. Varney Umass, Amherst

12. Frustrated magnets perturbative `order’ Cooperative paramagnet High-T expansions: sites, clusters. … T=0 lmit: Exact diag. DMRG (1D,2D) Variational Projection Strong coupling … Experiments: CM and cold atoms Skeleton Feynman diagrams with broken symmetry

13. standard diagrammatics for interacting fermions starting from the flat band. Main quantity of interest is magnetic susceptibility

14. How we do it Configuration space = (diagram order, topology and types of lines, internal variables) Diagram order MC update MC update Diagram topology This is NOT: write diagram after diagram, compute its value, sum

15. Standard Monte Carlo setup: (depends on the model and it’s representation) - configuration space - each cnf. has a weight factor - quantity of interest Monte Carlo configurations generated from the prob. distribution

16. TRIANGULAR LATTICE HEISENBERG ANTI-FERROMAGNET (expected order in the ground state)

17. Sign-blessing (cancellation of high-order diagrams) + convergence 113824 7-th order diagrams cancel out! High-temperature series expansions (sites or clusters) vs BDMC

18. Uniform susceptibility Full response function even for n=0 cannot be done by other methods

19. Correlations reversal with temperature Anti-ferro @ T/J=0.375 but anomalously small. Ferro @ T/J=0.5 Quantum effect? No, the same happens in the classical Heisenberg model : (unit vector)

20. Quantum-to-classical correspondence (QCC) for static response: Quantum has the same shape (numerically) as classical for some at the level of error-bars of ~1% at all temperatures and distances!

21. QCC plot for triangular lattice: Triangular lattice Triangular lattice Square lattice 0.28 Naïve extrapolation of data   spin liquid ground state! (a) (b) 0.28 is a singular point in the classical model!

22. Gvozdikova, Melchy, and Zhitomirsky ‘10 Kawamura, Yamamoto, and Okubo ‘84-‘09

23. Triangular lattice Triangular lattice Square lattice 0.28 QCC for static response also takes place on the square lattice at any T and r ! [Not exact! relative accuracy of 0.003]. QCC fails in 1D

24. QCC for Heisenberg model on pyrochlore lattice Yuan Huang Kun Chen Spin-ice state

25. QCC, if observed at all temperatures, implies (in 2D): • If then the quantum ground state is disordered spin liquid • If the classical ground state is disordered (macro degeneracy) then the quantum ground state is a spin liquid • Possible example: Kagome antiferromagnet • 3. Phase transitions in classical models have their counterpatrs in quantum • models on the correspondence interval

26. Conclusions/perspectives Arbitrary spin/Bose/Fermi system on a lattice can be “fermionized” and dealt with using Feynman diagrams without large parameters The crucial ingredient, the sign blessing phenomenon, is present in models of quantum magnetism Accurate description of the cooperative paramagnet regime (any property) QCC puzzle: accurate mapping of quantum static response to

27. Theory vs experiment (cold atoms solve neutron stars) MIT group: Mrtin Zwierlein, Mark Ku, Ariel Sommer, Lawrence Cheuk, Andre Schirotzek Uncertainty due to location of the resonance BDMC results virial expansion (3d order) Ideal Fermi gas