Transcript Exact bosonization for interacting fermions in arbitrary dimensions. (New route to numerical and analytical calculations) K.B.

Exact bosonization for interacting fermions in
arbitrary dimensions.
(New route to numerical and analytical calculations)
K.B. Efetov,1 C. Pepin 2, H. Meier 1
1 RUB
Bochum, Germany, 2CEA Saclay, France
Bosonization: mapping of electron models
onto a model describing collective
excitations (charge, spin excitations,
diffusion modes, etc).
Origin of the word: 1D electron systems.
The main idea: writing the electronic operators 

  exp( i ) exp i  dx

A simple Hamiltonian H
H   [ K 2  N   ]dx
2

 ,   i
-Density fluctuations operator
K-compressibility, N-average density
Importance of long wave length
excitations!
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as
Bosonization: for TomonagaLuttinger model (long range
interaction) (Luttinger, Tomonaga
(196?))
The most general form
conjectured by
K.E. & A. Larkin (1975)
Microscopic theory
Haldane (1982)
…………………
2
Formal replacement of electron Green functions by propagators
for collective excitations!
Very often direct expansions
with electronic Green
functions are not efficient
(infrared divergences, high
energy cutoffs respecting
Equivalent representation
symmetries).
Spin excitations
Transformation from electrons to collective excitations: Bosonization
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Difficulties in Monte Carlo simulations for fermionic systems:
negative sign problem
exponential growth of the
computation time with the size of the system.
(From a lecture by M. Troyer)
NP-nondeterministic
polynomial
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Can one bosonize in higher dimensions?
Earlier attempts:
A. Luther 1979: Special form of Fermi surface (square, cube, etc.). Almost 1D.
F.D.M. Haldane 1992: Patching of the Fermi surface, no around corner scattering
Further development of the patching idea:
A. Houghton & Marston 1993; A.H. Castro Neto & E. Fradkin 1994;
P. Kopietz & Schonhammer 1996; Khveshchenko, R. Hlubina, T.M. Rice
1994 et al; C. Castellani, Di Castro, W. Metzner 1994……
Main assumption of all these works: long range interaction.
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I.L. Aleiner & K. B. Efetov 2006, Method of quasiclassical Green
functions supplemented by integration over supervectors
Logarithmic contributions to specific heat and susceptibility
are found. Good agreement with known results in 1D. No
restriction on the range of interaction but not a full account of
effects of the Fermi surface curvature in d>1.
In all the approaches only low energy excitations are considered:
no chance for using in numerics.
Present work: Exact mapping fermion models onto bosonic ones.
New possibilities for both analytical and numerical computations.
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Starting Hamiltonian H
Z  Tr exp(Hˆ / T )
Hˆ  Hˆ 0  Hˆ int
Hˆ 0    tr ,r 'cr cr '    cr cr
r ,r ';
r ,r ';
The interaction, tunneling and
dimensionality are arbitrary!
1
 
Hˆ int 
V
c

r , r ' r cr ' ' cr ' 'cr
2 r ,r '; , '
Small simplification of the formulas
Vr ,r '   r ,r 'V0 ,
V0  0
V
(0)
Hˆ int
 0
2
 c

r , r ,
c
 cr,  cr , 

2
r
'   V / 2
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Hubbard-Stratonovich Transformation with a real field  ( )
 1
Z   Z  exp
 2V0


r   r , d D
0

2

 ˆ f  
Z    exp  Trr , ln  /   ˆr   r ( )   'd  r r
 0

t
 ( )   (   ),
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f
r ,r ' r '
r'
  1/ T
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Another representation for Z  



Z    det r , 1  T exp   ˆr   r     'd 

 0

Basis of auxiliary field Monte Carlo simulations
(Blankenbecler, Scalapino, Sugar (1981))
Procedure: subdividing the interval 0    
and calculation for each slice recursively.
into time slices
However, Z   can in such a procedure be both positive and
negative: sign problem!
Not convenient for analytical calculations either.
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Good news: Z [ ] can exactly be obtained from
a bosonic model!
Derivation of the model (main steps):
 1


Z    Z 0 exp   r  Gr ,r ;  ,  0ddu
0 0
 r ,

  ˆ








u

r


'

Gr ,r ';  , '   r ,r '    '
r
r
 

Gr ,r ';  , ' -fermionic Green function in the external field.
Boundary conditions
G , '  G   ,   G ,   
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Looking for the solution:
(in spirits of a Schwinger Ansatz)
Gr ,r '  , ' 
0 
1




T

G



'
T
 r ,r '' r '',r '''
r ''',r '  '
r '',r '''
1


T

T
 r ,r '' r '',r '   r ,r '
r ''
Bosonic periodic boundary conditions: T    T    
Gr0,r'   ' is the Green function of the ideal Fermi gas
Equations for T and T 1


Tr ,r ' z   ˆr  ˆr ' Tr ,r ' z   ur  Tr ,r ' z   0

 1
Tr ,r ' z   ˆr  ˆr ' Tr,r1' z   ur '  Tr,r1' z   0

Solving these
equations is not
convenient.
Solution is not
unambiguous.
z  { ,  , u}
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New function A:
Ar ,r ' ( z)   Tr ,r '' z nr '',r '''Tr'''1,r ' z   nr ,r '  T z , nˆ T 1 z 
r '',r '''
n p  
nr ,r '-Fourier transform of the Fermi distribution
or
1
e    p   '   1
nr ,r '  Gr0',r '  0
Then, the “partition function” Z  
is
1 


Z    Z 0 exp    r  Arr z dud 
 r , 0 0

What is good in that?
How to find A without solving equation for T?
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“Free lunch”:
a)   T z   ˆ  ˆ T z   u  T z   0
r ,r '
r
r ' r ,r '
r
r ,r '
b)

 1
Tr ,r ' z   ˆr  ˆr ' Tr,r1' z   ur '  Tr,r1' z   0

an operator ...,nˆ  f r   f r , nˆ 
1. Act with
...,nˆ 
on a) and multiply by T 1 from the right.
2. Multiply b) by T , nˆ  from the left.
3. Subtract the obtained equations from each other.
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Final equation for A
  ˆ ˆ













u






 Ar ,r ' z   unr ,r ' r    r '   z  { ,  , u}
r
r'
r
r'
 

Partition function
 1  2

Z   Z  exp
 r , D

 2V0 0

1 


Z    Z 0 exp    r  Arr z dud 
 r , 0 0

Boundary conditions
r    r    ,
Ar ,r '    Ar ,r '    
Purely bosonic problem! Linear (almost separable) real equation for A
New possibilities for both numerical and analytical studies!
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How can one calculate Z   numerically ?
(Non-expert’s view)
Introduce the Green function g for the linear equation and
represent it in the form:      ' 0


1
ˆ
ˆ
ˆ
g r ,r ';r1 ,r1 '  , 1   Pr ,r '  , 1  1  Pr ,r '  1 ,0Pr ,r '  , 1   r ,r1 r ',r1 '

where the operator P is
 

ˆ

Pr ,r '  , 1   T exp   hr ,r '  ' ,  , u d ' 
 

 1

hˆr ,r '  ˆr  ˆr '  ur    r '  
1  



Z    Z 0 exp     g r ,r ;r1 ,r1 '  , 1 r   r1  '  r1 '  ' dd ' du
  ,r ,r1 ,r1 ' 0 0 0



Possibility to do recursion in time (as usual)!
Compensation of the Bose-denominator
no singularities in g.
In the absence of singularities, the exponent remains real for any subdivision of
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Z   is positive: no sign problem!15
the interval 0,   into slices
Perturbation theory:
expanding in r   and subsequent integration over this field.
Main order:
 T
ddk 
n p  k / 2   n p  k / 2  d d p  
Z  Z 0 exp  
ln 1  V0 

d
i    p  k / 2    p  k / 2 2 d  
 2  2  
RPA-like formula
(first order in the expansion in collective excitations).
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Analytical calculations:
BRST (Becchi, Rouet, Stora, Tyutin)
-possibility of integration over the auxiliary
field before doing approximations
How to calculate BA0  if A0 is the solution of the
equation F  A  0?
A well known trick: BA0    Ba  F a  det F  da
 a 
Next step:
 F a   C  exp ifF a df
 F 
det
   exp F / a  dd
 a 
 ,  -Grassmann variables
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Description with a supersymmetric action and superfields
Introducing new Grassmann variables  , * and superfields
r ,r ' R  ar ,r ' z   f rT,r ' z  * r ,r ' z  r,r ' z  *



R   ,  , u,  ,  *

 is anticommuting (!)
The “partition function” Z   as the functional integral over 
Z    Z0 expSss   Ssb 
i
  ˆ 
S ss      r ',r   hr ,r ' r ,r ' dR
2 r ,r '
 

hˆr ,r '  ˆr  ˆr '  ur    r '  
Ssb     r  r ,r ' *dR  i  nr ,r ' r    r '  r ,r 'udR
r
Gaussian averaging over
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r ,r '

can immediately be performed!
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Final superfield theory (still exact).
S   S0   SB   SI 
Z  Z 0  exp  S [ ]D
Z 0-partition function of the ideal Fermi gas


i
 

S0     r ',r   ˆr  ˆr ' r ,r ' dR
2 r ,r ' 
 


S0  is the bare action
(fully supersymmetric)


V0
S B          1 r ,r R  * r ,r R1 1*  2i r R1   1dRdR1
2
V0
SI 
2
       R  R  dRdR
1
r
r'
1
1
1
r
 r R   u r ',r R   nr ,r ' r ,r ' R   nr ,r ' 
r'
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The terms S0 and S I are invariant
under the transformation of the fields  :




r ,r '  , *  r ,r '    , *   *  nr ,r '
 , *
(Almost) supersymmetry
transformation.
-anticommuting variables
What to calculate?
Logarithmic contributions exist in any dimensions and they can be
studied by RG. Reduction of the exact model to a low energy one is
convenient. Variety of phenomena for, e.g., cuprates and other strongly
correlated systems can be attacked in this way.
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Conclusions.
The model of interacting fermions can be bosonized in any
dimension for any reasonable interaction: free lunch but…..
What about a free dinner?
Can the bosonization be the key to the ultimate solution of
the sign problem?
Can one solve non-trivial models (e.g., models for high
temperature cuprates or for quantum phase transitions)
using the supersymetric model for collective excitations?
To be answered
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