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Second fermionization & Diag.MC for quantum magnetism
KWANT, 6/2/15 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)
spins fermionsH H Feynman diagrams
system fermionsH HTr e Tr e
- Heisenberg models in D=2,3: classical-to-quantum correspondence
(same for correlation functions)
Popov-Fedotov trick for S=1/2
Heisenberg model:
† †i ifermi ij
ijj jf f fH fJ
spin-1/2 f-fermionsspin-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
/ /spin fermiH T H TspinZ Tr e Tr e
'fermi spinH phys phys H phys
0fermiH unphys
spin ij i jij
H J S S
† † 1f ij i i j ji
jjj
H J f f f f n
/ 2i T with complex
Flat band Hamiltonian to begin with: + interactions 1f jj
H n
Popov-Fedotov trick for S=1/2
Now/ /spin fermiH T H T
spinZ Tr e Tr e
Standard Feynman diagramsfor two-body interactions
/fH TspinZ Tr e Proof of
/ ( )
1
f K
K
NH T K
f spin spinK
Tr e Z Z C
Number of unphysical sites with n=2 or n=0
Partition function of the unphysical site
configuration of unphysical sites
Partition function of physicalsites in the presence of unphysical ones (K blocked sites)
( 1)/ /2 /2
0,2
0n T i i
n
C e e e
/fH Tf spinTr e Z
( )a a aspin ij i j
ija
H J S S Arbitrary spin (or lattice boson system with n < 2S+1):
Mapping to (2S+1) fermions: 1,0,...,0 zS S
0,0,...,1 zS S
0,1,...,0 1zS S …
( ) ( ) ( ) . . ( 1)ij i jfermi i
ij i
H Q Q h c n
Matrix element,same as for
Onsite fermionic operator in the projected subspaceconverting fermion to fermion . For example,
1
1N
n n
† †
1,
(1 )nQ f f P f f n
( )a a aij i jJ S S
SU(N) magnetism: a particular symmetric choice of
( )ij
Dynamics: perfect on physical states:
Unphysical empty and multiply-occupied sites decouple from physical sites and each other:
'fermi spinH phys phys H phys
0fermiH unphys
/fH TspinZ Tr e Proof of is exactly the same:
/ ( )
1
f K
K
NH T K
f spin spinK
Tr e Z Z C
Partition function of the unphysical site
( 1)/ 1,1
0,1 1
(1 ) 0; ( )n T nn n
n n
C e N z n n
Always has a solution for (fundamental theorem of algebra)
/Tz e
Projected Hilbert spaces; t-J model:
†(1 ) (1 )t J i j j s js is i sij ij
H J S S t n f f n
† † †(1 ) (1 )fermi i i j j j s js is i sij ij
H J f f f f t n f f n
Dynamics: perfect on physical states:
Unphysical empty and doubly occupied sites decouple from physical sites and each other:
'fermi t JH phys phys H phys
0fermiH unphys
as before, but C=1! / ( )
1
f K
K
NH T K
f t J t JK
Tr e Z Z C
previous trick cannot be applied
Solution: add a term 3 3 3unphys i i ii i
H i T n P i T n n n
3
3
/
0,1
( )
1
2 2f K
K
KN
H T N N Kf t J J
i
K nt
nTr e Z Z C e
For we still have
but , so
'fermi t JH phys phys H phys
3i nfermiH unphys e unphys
3fermi fermiH H H
Zero!(0)2 3fermi t body bodyH H V V
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
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 KulaginUmass, Amherst
Chris N. Varney Umass, Amherst
Magnetism was frustrated but this group was not
Oleg Starykh Univ. of Utah
spin i jij
H J S S
Triangular lattice spin-1/2 Heisenberg model:
T
Frustrated magnets
J or 1CT
perturbative `order’
High-T expansions:sites, clusters. …
T=0 lmit:Exact diag.DMRG (1D,2D)VariationalProjectionStrong coupling …
Cooperative paramagnet
Experiments: CM and cold atoms
with broken symmetry
Skeleton Feynman diagrams
(0) (0)G G G G ˆ ˆ ˆ ˆˆU J J U
(1 )J
standard diagrammatics for interacting fermions starting from the flat band.
Main quantity of interest is magnetic susceptibility
†
† †'0
'n
z zi j i i j jS S f f f f
G
Jˆ ˆU J
{ , , }i i iq p
Diagram order
Diagram topology
MC update
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
Standard Monte Carlo setup:
- each cnf. has a weight factorcnfW
- quantity of interest
cnf cnfcnf
cnfcnf
A W
AW
- configuration space (depends on the model and it’s representation)
/cnfE Te
Monte Carlo
cnf
MCi
cnfcnf
A e configurations generated from the prob. distribution
cnfW
TRIANGULAR LATTICE HEISENBERG ANTI-FERROMAGNET
(expected order in the ground state)
Sign-blessing (cancellation of high-order diagrams) + convergence
1138247-th order diagrams cancel out!
High-temperature series expansions (sites or clusters)
vs BDMC
Uniform susceptibility ( , )nq i ( 0)q Full response function even
for n=0 cannot be done by other methods
Correlations reversal with temperature
Anti-ferro @ T/J=0.375but 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!
( , )q T ( , )clq T ( )clT T
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!
(0) 0.28clT
(0.28) ~ 1000cl
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
Kun Chen Yuan Huang
QCC for Heisenberg model on pyrochlore lattice
Spin-ice state
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
( 0) 0clT T
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
classical
MIT group: Mrtin Zwierlein, Mark Ku, Ariel Sommer, Lawrence Cheuk, Andre Schirotzek
Theory vs experiment (cold atoms solve neutron stars)
Uncertainty due to location of the resonance sa
virial expansion (3d order)
Ideal Fermi gas
BDMC results 834 1.5B G