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

(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

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

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

Proof of 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

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

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)

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

Feynman diagrams with two- and three-body interactions 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

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

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

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

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

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

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

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

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

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

No, the same happens in the classical Heisenberg model : (unit vector) 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)

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!

QCC plot for 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!

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

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

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

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

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

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