High precision spectral functions from discrete imaginary time

High precision spectral functions from discrete imaginary
time Green's functions
Mats Granath
University of Gothenburg
Sweden-Swiss meeting, Stenungsbaden 26/8 2011
14
Analytic Continuation
Collaborators:
Stellan Östlund
Bo Hellsing
Hugo Strand
Andro Sabashvili
Figure 2.2: Conformal mapping for the exactG( )
Suppose we have data
fn
Monday, April 30, 12
a0 + G̃(i⇥n )
where np is the number of poles and where 0 ⇥ n<N representing an approximate periodized self energy or Greens function at the matsubara frequencies ⇥n . Can we obtain
478
Spectral function
Damascelli, Hussain, and Shen: Photoemission studies of the cuprate superconductors
Fine stucture in ARP
Fermi Surface
like we have often used in the one-body case.
previous
Damascelli, et al. Rev. The
Mod. Phys.
2003results are summarized in Fig.(3-5) which displays the analytic
FIG. 3. Angle-resolved photoemission
spetroscopy:
of anisARPES
experiment
in which
the
theShen
structure
of ! (r" r0 ;(a)
#) $ geometry
This function
analytical
everywhere
except
onemission
the real direction ofZ.X.
photoelectron is specified by theaxis
polar
(-) and
azimuthal
) angles;
(b)tomomentum-resolved
where
it has
a branch(.cut
leading
a jump Eq.(3.128) one-electron
in the value removal
of the and addition
spectra for a noninteracting electron
system
with
a single the
energy
dispersing
E F ; (c)
the same
spectra
for an interacting
function
as we
approach
realband
axis from
eitheracross
the upper
or lower
complex
half! corresponding
Fermi-liquid system (Sawatzky, 1989;
bothfrom
noninteracting
interacting
systems
groundplane.Meinders,
The limit1994).
as weFor
come
the upper and
half-plane
is equal
to !the
(r" r0 ; %)
"
state (T!0 K) momentum distribution
n(k)
is also
shown. (c)
Lower
spectrum
of gaseous hydrogen
whereasfunction
from the
lower
half-plane
it is
equalright,
to !photoelectron
(r" r0 ; %) $ The
Matsubara
and the ARPES spectrum of solid
hydrogen
developed
from
theon
gaseous
one but
(Sawatzky,
Green’s
function
is dened
only
a discrete
innite 1989).
set of points along the
imaginary frequency axis.
Numerical challenge, find
real frequency Green’s
More generally, however, !
should be expressed as
Im(z)
function
from
Matsubara
GF
!
!c ! , where c is the annihilation
operator for
N"1
as we have done for the corresponding energies. This,
i
N"1
N
however, is far from trivial because during the photok i
k
i
G(z) = (i!")
emission process itself the system will relax. The prob-n
an electron with momentum k. This also shows that
†
lemG
simplifies
within
the( sudden
approximation, which is
R of the (N"1) particle Hamil< T (c
! N"1
is not anG(z)
eigenstate
k( ) =
k )ck (0)) >
i
=
G
(!)
extensively used in many-body calculations of phototonian, but is just what remains of the N-particle wave
emission spectra from interacting electron systems and
function after having pulled out one electron. At this
i⇤n ⇥
Gk (i⇥
d eapplicable
Gk ( )only to electrons with
which
is nin) =
principle
point, we can writeRe(z)
the matrix elements in Eq. (4) as
0
high kinetic energy.
In this limit, the photoemission proN"1
cess is assumed to1 be sudden, with no post-collisional
! ! N"1
(8)
# ! fN ! H int ! ! Ni $ ! # " fk! H int ! " ki $# ! m
$,
i
R
interaction
the photoelectron
and the system
Ak (⇥) between
=
ImG
k
k
k
k (⇥)
A
"
!
H
!
"
%M
is the one-electron dipole mawhere
#
$
int
f
i
f,i
left behind (in other words, an electron is instantaG(z) = G (!)
trix element, and the second term is the (N"1)-electron
neously removed and the effective potential of the sysoverlap integral. Note that here we replaced ! fN"1 with
tem changes discontinuously at that instant). The
N"1
3-5 written
Analytical
of an
!(#)
in the complex
frequency
plane. above.
!(#) The total phoeigenstate
!m
, as discussed
N-particle final state ! fN canFigure
then be
as structure A.-M.
Tremblay
reduces to either !! (%) " !" (%) ortoemission
G (&% # ) depending
on the
value of as
thea complex
intensity
measured
function of E kin at a
N
k
N"1
(6)cut along the real axis.
! f !A " f ! f ,
frequency #$ There is a branch
momentum k, namely, I(k,E kin )! & f,i w f,i , is then proportional to
where A is an antisymmetric operator that properly an-!
2 along the real-time axis from the knowlThe problem of nding ! (r" r0 ; %)
tisymmetrizes the N-electronedge
wave
function
so
that
the
of the Matsubara Green’s function is a problem of analytical continuation.
k 2
N"1
N
Monday,Pauli
April 30,
12
is the wave function
principle
is satisfied, " fkUnfortunately,
!
! c m,i ! 2 ' ( Econtinuation
"E
(9)
! (# = &'of)the
does not have !aMunique
analytical
be- "h ) * ,
kin #E
&
&
Example: non-interacting Anderson t=1
x x
impurity
function as we approach the real axis from either the upper or lower complex ha
plane. The limit as we come from the upper half-plane is equal to !! (r" r0 ; %
whereas from the lower half-plane it is equal to !" (r" r0 ; %) $ The Matsuba
Green’s function is dened only on a discrete but innite set of points along t
imaginary frequency axis.
⇤
1
4 ⌅2
(|⌅|
A(r = 0, ⌅) =
⇤ 4 ⌅ 2 + ⇥2
G(0, i
⇥
2) + ⇤
(⌅ +
2
4+⇥
n) =
d
t
o x x x
t
G(z) =
t
(i!")
n
Im(z)
R
G(z) = G (!)
4 + ⇥2 )
Re(z)
A(0, )
i n
A
G(z) = G (!)
P (z)
a0 + a1 z + ... + am 1 z m
=
Q(z)
1 + b1 + ... + bm z m
Figure 3-5 Analytical structure of !(#) in the complex frequency plane. !(#)
reduces to either !! (%) " !" (%) or G (&% # ) depending on the value of the compl
frequency #$ There is a branch cut along the real axis.
P (i
G(i n ) =
Q(i
1
= 25, ⇤n =
0.25
n)
2⇥Theorem 81If
(n
+ ), nmax
= 40
1. ! (#) is analytical
in the upper half-plane
2
2. ! (#) = G (&' ) for all Matsubara frequencies
3. lim | #! (#) = )*+ /
condition number
max
%!"
0.20
n)
The problem of nding !! (r" r0 ; %) along the real-time axis from the know
edge of the Matsubara Green’s function is a problem of analytical continuatio
Unfortunately, ! (# = &'# ) does not have a unique analytical continuation b
cause there is an innite number of analytical functions that have the same val
along ¡this discrete
set of points. For example, suppose we know ! (# = &'# ) " th
¡
¢¢
! (#) 1 + ($% + 1 has the same value as ! (#) for all points # = &'# becau
(&'! $ + 1 = 0$ Baym and Mermin[13], using results from the theory of compl
functions, have obtained the following result.
#
min |
1050
then the analytical continuation is unique and
!! (r" r0 ; %) =
0.15
lim
&'! !(+&)
G (r" r0 ; &'# )
(3.13
Double precision cannot capture the peak
40 digit precision does (width 10^(-10))
The key point is the third one on the asymptotic behavior at high frequenc
That this is the correct asymptotic behavior at high frequency follows trivially fro
the spectral representation Eq.(3.125) as long as we remember that the spectr
weight is bounded in frequency. The non-trivial statement is that this asymptot
behavior su!ces to make the analytical continuation unique. In practice this rare
poses a problem. The simple replacement &'# ! % + &, su!ces. Nevertheless, t
asymptotic behavior reects a very fundamental property of the physical system
namely the anticommutation relations! It is thus crucial to check that it is satise
More on the meaning of the asymptotic behavior in subsection (3.6.1).
0.10
Out[475]=
t
0.05
see also, Karrasch, Meden, Schönhammer, PRB 2010
-3
x
-2
-1
x
x
1
-0.05
2
x
x
x
3
Monday, April 30, 12
3
MATSUBARA GREEN’S FUNCTION AND ITS RELATION TO USUAL GREEN’S FUN
TIONS. (THE CASE OF FERMIONS)
High precision self-consistent Green’s function calculations
High precision calculations requires a finite basis (?)
Can we work with a finite set of imaginary frequencies?
We discretize imaginary time: the GF of the diagrammatic expansion, not the path integral
directly.
Advantage: non-interacting GF and spectral function can be captured exactly.
Spectral weight conservation can be build into the formalism.
G(
) = G0 (
⇥j =
G(⇥i
N
)+
d 1 d 2 G0 (
(
1
2 )G0 ( 2
j
N 1
⇥j ) = G0 (⇥i
⇥j ) + (
N
)2
G0 (⇥i
⇥l ) (⇥l
⇥l )G0 (⇥l
l,l =0
(ambiguity at
i
j
4
Monday, April 30, 12
1)
= 0)
⇥j ) + ...
) + ...
Non-interacting GF
G0,k (⇤ ) = e
⇥j =
N
G0,k (⇥j ) =
⇥
k
[nf ( k )⇥( ⇤ + 0+ ) + (nf ( k )
1)⇥(⇤
0+ )]
j
1
N 1
e
i⇥n
j
G0,k (i⇤n )
n=0
1
2
G0,k (⇥0 ) = nf ( k )
0
N 1
G0,k (i⌅n ) =
N
Periodic:
ei⇥n j G0,k (⇤j ) =
j=0
i
n
i
n
+i
N
5
Monday, April 30, 12
2N
N
coth
=
2⇥
2N
N
(i⌅n
⇥k )
Dyson equation
Gk (i⇥n ) = (i⇥n
Gk (i⇤n ) =
2N
Coth
2N
k (i⇥n ))
k
(i⇤n
⇥k
1
k (i⇤n ))
G±
0 = G0 ±
+
G = G0 + G+
G
+
G
0
0
0 G0 G0 + ...
is 1PI diagrams of periodized Green’s functions
G(i⇤n ) is periodic G(i⇤n ) = G(i⇤n ± i
6
Monday, April 30, 12
N)
with
N
=
2⇥
N
2N
Analytic Structure/Analytic continuation
Discretize the
Gk ( ) =
spectral
representation
Gk (z) =
1
Z
e
En ⇥ (En Em )
e
m,n
d⇥A(k, ⇥)
2N
|⇥m|c†k |n⇤|2
Coth
2N
A(k, ⇥) =
(z
Periodic
N
=
m,n
|⇥m|c†k |n⇤|2 e
En
(1 + e
⇥
) (Em
En
a
Im z⇥
3
3
2
2
1
1
0
0
b
N
⇥
7
Monday, April 30, 12
⇥)
⇥)
Im z⇥
2⇥
1
Z
4
2
0
2
4
⇥
4
2
0
2
4
Represent spectral function in terms of a (set of) generalized Lorentzians
L⇤,⇥ (⇤) =
”G (z)”
R
G(z)
i 2N
sinh 2N (⇤
1
2
=
=
⇤
C
4N
i 2N
⇥+i )
sinh 2N (⇤
dz
L⇥, (z )
coth
(z
2⌅
2N
2N
coth
Reduces to Lorentzian
for large N:
4N
(z
⇤ + i⇥) + tanh
⇥
i )
(z
N
C
z
N
⇥
N
i +i
⇥
d⇥
L(⇥) = 1
2
⇥
Im z⇥
⇥+i +i
N
⇥
i⇥)
⇤
2
Spectral weight
conserving:
0 < Imz <
Im
⇥
(⇤
z)
4N
2
⇥)2 +
a
Im z⇥
3
3
2
2
1
1
0
0
b
⇥
⇥+i
⇥
i
Re
⇥
8
Monday, April 30, 12
4
2
0
2
4
⇥
4
2
0
2
4
Map periodically repeated poles to single poles:
G(z) =
[a coth
4N
4N
(z
⇤ + i⇥ ) + a tanh
Analytic Continuation
2
1
(a + a ) = 1
2
14
2⇥N
2⇥N
⇥N
⇥N
0
⇤
i⇥ )]
Single Poles from conformal mapping:
13
1 weight 0
Spectral
1
conservation: 2
4N
(z
z = coth(
4N
z
⇥
i )
4
Analytic Continuation
0
⇥N
⇥N
2
1
0
Fit G(z (i
Figure 2.1: The function G(i⇤)
1
n )) to P (z )/Q(z )
2
The function G̃n differs only from Gn in that we use the tanh and coth functions instead9of
the digamma functions to define our periodized greens function data.
It isApril
useful
1/⇥. The result is in Fig. 2.2.
Monday,
30, to
12make one more conformal mapping; let ⇥
Figure 2.2: Conformal mapping for the exactG( )
Suppose we have data
fn
a0 + G̃(i⇥n )
where np is the number of poles and where 0 ⇥ n<N representing an approximate periodized self energy or Greens function at the matsubara frequencies ⇥n . Can we obtain
{a } and { } from {fn } ?
How many poles are in principle necessary to fit arbitrary values of fn ?. First of all, there
⇥
Applied to DMFT (Iterated perturbation theory) on
Hubbard model
T=W/25
25 frequencies
40 digits precision
U =2
U =4
a
⇥k
b
⇥k
A( k , ⇥)
T=W/500
501 frequencies
700 digits precision
⇤
2
1
0
1
ImG(z )
10
Monday, April 30, 12
⇤
2
4
2
0
2
4
1.
2.
3.
Do calculations to high precision at finite N
Make “numerically exact” analytic continuation
Take N to infinity to find full analytic Green’s
function and spectral function
M.Granath, A. Sabashvili, H.U.R. Strand, S. Östlund, arXiv: 1103.3516
H.U.R. Strand, A. Sabshvili, M. Granath, B. Hellsing, S. Östlund, PRB 2011
Formalism looking for more applications...
There is also a consistent functional formulation
= ⇤({G})
T r(G ⇥) + T r log
+
⇥
k (i⇥n ) =
Gk (i⇥n )
⇥
G /(2⇥) ⇥ ⌅
Gives exact free energy in the non-interacting limit for any N
11
Monday, April 30, 12