Second fermionization & Diag.MC for quantum magnetism KITPC 5/12/14 AFOSR MURI Advancing Research in Basic Science and Mathematics N. Prokof’ev In collaboration with B. Svistunov

- 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) - Triangular-lattice Heisenberg model: classical-to-quantum correspondence

Popov-Fedotov trick for S=1/2 Heisenberg model: 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

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

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

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

Dynamics: perfect on physical states: Unphysical empty and doubly occupied sites decouple from physical sites and each other: Proof of is exactly the same: 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

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

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

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

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

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

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

Sign-blessing (cancellation of high-order diagrams) + convergence 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

Correlations reversal with temperature T/J=0.375 but anomalously small. 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!

Square lattice Triangular lattice 0.28 Triangular lattice QCC plot for triangular lattice: 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

Square lattice Triangular lattice 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 0.28 Triangular lattice

QCC, if observed at all temperatures, implies (in 2D): 1.If then the quantum ground state is disordered  spin liquid 2.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

Generalizations: Diagrammatics with expansion on t, not U (i.e. eliminating large expansion parameters!) E.g. for interacting bosons in 3D interesting physics is at ! It means that onsite terms should NOT be projected out  keep them “as is” Physical states still decouple from non-physical ones and non-physical states remain decoupled  can be dealt with in statistics one by one  use Always has a solution for. On-site terms now combine with the chemical potential. Generalizations: Diag. expansions in t, not U (no large expansion parameters!) n=2 state  doublon  2 additional fermions + constraints