Download Triangular-lattice Heisenberg model

Second fermionization & Diag.MC for quantum magnetism
N. Prokof’ev
In collaboration with B. Svistunov
KITPC 5/12/14
AFOSR
MURI
Advancing Research in Basic Science and Mathematics
H system  H fermions  Feynman diagrams
  Hsystem
Tre
  H fermions
 Tre
- 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:
H spin   J ij Si S j
ij


H fermi  J ij  fi†  fi  f j†  f j

ij
spin-1/2 f-fermions
-
Dynamics: perfect on physical states:
H fermi phys  phys '  H spin phys
- Unphysical empty and doubly occupied sites decouple from physical sites
and each other:
H fermi unphys  0
- Need to project unphysical Hilbert space out in statistics in the GC ensemble because
Z spin  Tre
 H spin /T
 Tre
 H fermi /T
Popov-Fedotov trick for S=1/2





H f   J ij  fi†  fi  f j†  f j    n j  1 
j
ij
with complex
Now
Z spin  Tre
 H spin /T
Flat band Hamiltonian to begin with
 Tre
  i T / 2
 H fermi /T


H 
   n j  1 
f
j
+ interactions
Standard Feynman diagrams
for two-body interactions
Proof of
Z spin  Tre
Trf e
 H f /T
 H f /T
Partition function of physical
sites in the presence of
unphysical ones (K blocked sites)
 Z spin

 K
( K )
    Z spin C
K 1   K

Number of unphysical
sites with n=2 or n=0
C

N
configuration of
unphysical sites
e  ( n 1)/T  e  i / 2  ei / 2  0
n  0,2
Trf e
Partition function of
the unphysical site
 H f /T
 Z spin
Arbitrary spin (or lattice boson
system with n < 2S+1):
Mapping to (2S+1) fermions:
H spin   J ij( a ) Sia S aj
ija
1,0,...,0  Sz  S
0,1,...,0  Sz  S  1
…
0,0,...,1  Sz  S
( ij )
H fermi     
Q(i)  Q( j )  h.c.    (ni  1)
ij

i
N
Matrix element,
same as for
J S S
(a)
ij
a
i
a
j
Onsite fermionic operator in the projected subspace
converting fermion  to fermion  . For example,
Q  f † f Pn1  f † f
(1  n )

  
 ,
SU(N) magnetism: a particular symmetric choice of
( ij )

n   n
 1
Proof of
Z spin  Tre
 H f /T
is exactly the same:
Dynamics: perfect on physical states:
H fermi phys  phys '  H spin phys
Unphysical empty and doubly occupied sites
decouple from physical sites and each other:
Trf e
 H f /T
 Z spin
H fermi unphys  0

 K
( K )
    Z spin C
K 1   K

N
Partition function of the unphysical site

C    
  n 0,1
  ( n 1)/T
(1   n ,1 )   N n z n 1 0;
 e
n 1

z  e  /T
Always has a solution for
(fundamental theorem of algebra)
( n   n )

Projected Hilbert spaces; t-J model:
H t  J   J Si S j  t  (1  n j  s ) f js† fis (1  ni  s )
ij 
ij 
H fermi    J     fi† fi f j† f j  t  (1  n j s ) f js† fis (1  ni s )
ij 
ij 
Dynamics: perfect on physical states:
H fermi phys  phys '  Ht  J phys
Unphysical empty and doubly occupied sites
decouple from physical sites and each other:
Trf e
 H f /T
 Zt  J
H fermi unphys  0

 K
( K )
    Z t  J C
K 1   K

N
as before, but C=1!
previous trick cannot be applied
Solution: add a term
H 3  i T  n3 Punphys i T  ni 3ni ni
i
i
For
H fermi  H fermi  H3
but
H fermi unphys  ei n3 unphys
Trf e
 H f /T
we still have
H fermi phys  phys '  Ht  J phys
, so



i n3
( K ) N  K
 Zt  J 2    Zt  J 2
C  e 
K 1   K
  n3 0,1

N
K
N
H fermi  H
(0)
t
 V2body  V3body
Zero!
Feynman diagrams with twoand 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
How we do it
Configuration space = (diagram order, topology and types of lines, internal variables)
Diagram order
{qi , i , pi }
Diagram topology
This is NOT: write diagram after diagram, compute its value, sum
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:
H spin  J  Si S j
ij 
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
Frustrated magnets
`order’
Cooperative paramagnet
TC or   1
T=0 lmit:
Exact diag.
DMRG (1D,2D)
Variational
Projection
Strong coupling
…
J
T
High-T expansions:
sites, clusters. …
Experiments: CM and cold atoms
Skeleton Feynman diagrams
with broken
symmetry
perturbative
standard diagrammatics for interacting fermions starting from the flat band.
 
 
Uˆ  Jˆ

 
 †
G  G  G  G
(0)
Main quantity of interest is magnetic susceptibility
S S
 
G
(0)
z
i
Ĵ
z
j  0
n
 fi fi f f j '  '
†
†
j
ˆ Uˆ
Uˆ  Jˆ  Jˆ 


(1  J  )
TRIANGULAR LATTICE HEISENBERG ANTI-FERROMAGNET
(expected order in the ground state)
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
 (q  0)
Full response function
 (q, in ) even
for n=0 cannot be done by other methods
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
 (q, T ) has the same shape (numerically) as classical  (q, Tcl )
at the level of error-bars of ~1% at all temperatures and distances!
for some Tcl (T )
QCC plot for triangular lattice:
Triangular lattice
Triangular lattice
Square lattice
0.28
Naïve extrapolation of data  Tcl (0)  0.28  spin liquid ground state!
(a) cl (0.28) ~ 1000
(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
QCC, if observed at all temperatures, implies (in 2D):
1. If Tcl (T  0)  0 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
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 U / t 30 ! It means
that onsite terms should NOT be projected out  keep them “as is”
( ij )
H F   E f† f    
Q( i)  Q( j )  h.c.
i
ij

Physical states still decouple from non-physical ones and non-physical states remain
decoupled  can be dealt with in statistics one by one  use   T ln( z )
C


Ntot 1
1   tot    e  n E /T  e  ( Ntot 1)/T 
F
(
N
,
T
)

z
0

tot
Ntot  n1  n2 
 


n1 ,n2 ,

Always has a solution for
potential.
z.
On-site terms now combine with the chemical
Generalizations: Diag. expansions in t, not U (no large expansion parameters!)
n=2 state  doublon  2 additional fermions + constraints
 
&

 †
 