1. 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

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. 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

5. 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

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. 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

10. 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 !

11. 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

12. 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

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
Diagram topology
MC update
This is NOT: write diagram after diagram, compute its value, sum

15. 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

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

17. 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

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

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

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:
Square lattice
0.28
Naïve extrapolation of data   spin liquid ground state!
(a) (b) 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. Kun Chen
Yuan Huang
QCC for Heisenberg model on pyrochlore lattice
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