In
many-body physics , the problem of analytic continuation is that of numerically extracting the spectral density of a
Green function given its values on the imaginary axis. It is a necessary post-processing step for calculating
dynamical properties of physical systems from
Quantum Monte Carlo simulations, which often compute Green function values only at
imaginary times or
Matsubara frequencies .
Mathematically, the problem reduces to solving a
Fredholm integral equation of the first kind with an ill-conditioned kernel. As a result, it is an ill-posed inverse problem with no unique solution and where a small noise on the input leads to large errors in the
unregularized solution . There are different methods for solving this problem including the maximum entropy method,
[1]
[2]
[3]
[4] the average spectrum method
[5]
[6]
[7]
[8] and Pade approximation methods.
[9]
[10]
Examples
A common analytic continuation problem is obtaining the spectral function
A
(
ω
)
{\textstyle A(\omega )}
at real frequencies
ω
{\textstyle \omega }
from the Green function values
G
(
i
ω
n
)
{\textstyle {\mathcal {G}}(i\omega _{n})}
at
Matsubara frequencies
ω
n
{\textstyle \omega _{n}}
by numerically inverting the integral equation
G
(
i
ω
n
)
=
∫
−
∞
∞
d
ω
2
π
1
i
ω
n
−
ω
A
(
ω
)
{\displaystyle {\mathcal {G}}(i\omega _{n})=\int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}{\frac {1}{i\omega _{n}-\omega }}\;A(\omega )}
where
ω
n
=
(
2
n
+
1
)
π
/
β
{\textstyle \omega _{n}=(2n+1)\pi /\beta }
for
fermionic systems or
ω
n
=
2
n
π
/
β
{\textstyle \omega _{n}=2n\pi /\beta }
for
bosonic ones and
β
=
1
/
T
{\textstyle \beta =1/T}
is the inverse temperature. This relation is an example of
Kramers-Kronig relation .
The spectral function can also be related to the
imaginary-time Green function
G
(
τ
)
{\textstyle {\mathcal {G}}(\tau )}
be applying the
inverse Fourier transform to the above equation
G
(
τ
)
:
=
1
β
∑
ω
n
e
−
i
ω
n
τ
g
(
i
ω
n
)
=
∫
−
∞
∞
d
ω
2
π
A
(
ω
)
1
β
∑
ω
n
e
−
i
ω
n
τ
i
ω
n
−
ω
{\displaystyle {\mathcal {G}}(\tau )\ \colon ={\frac {1}{\beta }}\sum _{\omega _{n}}e^{-i\omega _{n}\tau }{\mathcal {g}}(i\omega _{n})=\int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}A(\omega ){\frac {1}{\beta }}\sum _{\omega _{n}}{\frac {e^{-i\omega _{n}\tau }}{i\omega _{n}-\omega }}}
with
τ
∈
0
,
β
{\textstyle \tau \in [0,\beta ]}
. Evaluating the
summation over Matsubara frequencies gives the desired relation
G
(
τ
)
=
∫
−
∞
∞
d
ω
2
π
−
e
−
τ
ω
1
±
e
−
β
ω
A
(
ω
)
{\displaystyle {\mathcal {G}}(\tau )=\int _{-\infty }^{\infty }{\frac {d\omega }{2\pi }}{\frac {-e^{-\tau \omega }}{1\pm e^{-\beta \omega }}}A(\omega )}
where the upper sign is for fermionic systems and the lower sign is for bosonic ones.
Another example of the analytic continuation is calculating the optical conductivity
σ
(
ω
)
{\displaystyle \sigma (\omega )}
from the current-current correlation function values
Π
(
i
ω
n
)
{\displaystyle \Pi (i\omega _{n})}
at Matsubara frequencies. The two are related as following
Π
(
i
ω
n
)
=
∫
0
∞
d
ω
π
2
ω
2
ω
n
2
+
ω
2
A
(
ω
)
{\displaystyle \Pi (i\omega _{n})=\int _{0}^{\infty }{\frac {d\omega }{\pi }}{\frac {2\omega ^{2}}{\omega _{n}^{2}+\omega ^{2}}}\;A(\omega )}
Software
The Maxent Project : Open source utility for performing analytic continuation using the maximum entropy method.
Spektra : Free online tool for performing analytic continuation using the average spectrum Method.
SpM : Sparse modeling tool for analytic continuation of imaginary-time Green’s function.
See also
References
^ Silver, R. N.; Sivia, D. S.; Gubernatis, J. E. (1990-02-01).
"Maximum-entropy method for analytic continuation of quantum Monte Carlo data" . Physical Review B . 41 (4): 2380–2389.
Bibcode :
1990PhRvB..41.2380S .
doi :
10.1103/PhysRevB.41.2380 .
PMID
9993975 .
^ Jarrell, Mark; Gubernatis, J. E. (1996-05-01).
"Bayesian inference and the analytic continuation of imaginary-time quantum Monte Carlo data" . Physics Reports . 269 (3): 133–195.
Bibcode :
1996PhR...269..133J .
doi :
10.1016/0370-1573(95)00074-7 .
ISSN
0370-1573 .
^ Reymbaut, A.; Bergeron, D.; Tremblay, A.-M. S. (2015-08-27).
"Maximum entropy analytic continuation for spectral functions with nonpositive spectral weight" . Physical Review B . 92 (6): 060509.
arXiv :
1507.01956 .
Bibcode :
2015PhRvB..92f0509R .
doi :
10.1103/PhysRevB.92.060509 .
S2CID
56385057 .
^ Burnier, Yannis; Rothkopf, Alexander (2013-10-31).
"Bayesian Approach to Spectral Function Reconstruction for Euclidean Quantum Field Theories" . Physical Review Letters . 111 (18): 182003.
arXiv :
1307.6106 .
Bibcode :
2013PhRvL.111r2003B .
doi :
10.1103/PhysRevLett.111.182003 .
PMID
24237510 .
^ White, S. R. (1991).
"The Average Spectrum Method for the Analytic Continuation of Imaginary-Time Data" . In Landau, David P.; Mon, K. K.; Schüttler, Heinz-Bernd (eds.). Computer Simulation Studies in Condensed Matter Physics III . Springer Proceedings in Physics. Vol. 53. Berlin, Heidelberg: Springer. pp. 145–153.
doi :
10.1007/978-3-642-76382-3_13 .
ISBN
978-3-642-76382-3 .
^ Sandvik, Anders W. (1998-05-01).
"Stochastic method for analytic continuation of quantum Monte Carlo data" . Physical Review B . 57 (17): 10287–10290.
Bibcode :
1998PhRvB..5710287S .
doi :
10.1103/PhysRevB.57.10287 .
^ Ghanem, Khaldoon; Koch, Erik (2020-02-10).
"Average spectrum method for analytic continuation: Efficient blocked-mode sampling and dependence on the discretization grid" . Physical Review B . 101 (8): 085111.
arXiv :
1912.01379 .
Bibcode :
2020PhRvB.101h5111G .
doi :
10.1103/PhysRevB.101.085111 .
S2CID
208548627 .
^ Ghanem, Khaldoon; Koch, Erik (2020-07-06).
"Extending the average spectrum method: Grid point sampling and density averaging" . Physical Review B . 102 (3): 035114.
arXiv :
2004.01155 .
Bibcode :
2020PhRvB.102c5114G .
doi :
10.1103/PhysRevB.102.035114 .
S2CID
214775183 .
^ Beach, K. S. D.; Gooding, R. J.; Marsiglio, F. (2000-02-15).
"Reliable Pad\'e analytical continuation method based on a high-accuracy symbolic computation algorithm" . Physical Review B . 61 (8): 5147–5157.
arXiv :
cond-mat/9908477 .
Bibcode :
2000PhRvB..61.5147B .
doi :
10.1103/PhysRevB.61.5147 .
S2CID
17880539 .
^ Östlin, A.; Chioncel, L.; Vitos, L. (2012-12-06).
"One-particle spectral function and analytic continuation for many-body implementation in the exact muffin-tin orbitals method" . Physical Review B . 86 (23): 235107.
arXiv :
1209.5283 .
Bibcode :
2012PhRvB..86w5107O .
doi :
10.1103/PhysRevB.86.235107 .
S2CID
8434964 .