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)

system fermionsH H Feynman diagrams

system fermionsH HTr e Tr e

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

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

N

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 doubly 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

{ , , }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

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

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

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

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”

/ 30U t

† ( ) ( ) ( ) . .ij i jF

i ij

H E f f Q Q h c

Physical states still decouple from non-physical ones and non-physical states remain decoupled can be dealt with in statistics one by one use ln( )T z

1 2 1 2

/ ( 1)/ 1

, ,

1 ( , ) 0tot tot

tot

tot

n E T N T Ntot

n n N n n

C e e F N T z

Always has a solution for . On-site terms now combine with the chemical potential.z

Generalizations: Diag. expansions in t, not U (no large expansion parameters!) n=2 state doublon 2 additional fermions + constraints

&

†