Complex Langevin study of an attractively
interacting two-component Fermi gas in 1Dwith population imbalance
Shoichiro Tsutsui(RIKEN Nishina Center for Accelerator-Based Science)
In collaboration withTakahiro M. Doi (RCNP Osaka Univ.)
Hiroyuki Tajima (Kochi Univ.)1

My research interest : QCD at finite density

J. Phys.: Conf. Ser. 706 022004

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Fermi gas

PRL 124, 203402 (2020)3

Common feature: sign problem

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Common feature: sign problem

 What is the sign problem ?
 Sign problem in cold atom (and QCD)
 Complex Langevin (theory and application)
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Sign problem: an intuitive picture

Numerical evaluation of highly oscillatory integrals
is difficult6

Sign problem: precise statement

Monte Carlo evaluation of highly oscillatory integrals
is difficult
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Monte Carlo integration

Positive semi-definite

Random number

is viewed as a probability density
function if
8Monte Carlo integration for complex P(x)Non positive semi-definite

is not viewed as a probability density
function if
9Monte Carlo integration for complex P(x)10Monte Carlo integration for complex P(x)This procedure is known as reweighting.
11Monte Carlo integration for complex P(x)Positive semi-definite

Evaluate the numerator and denominator separately

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Sign problem: more precise statement

Statistical error

Signal-to-noise ratio is exponentially small
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Sign problem in ultracold Fermi gas
Grand partition function

attractive contact interaction

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Sign problem in ultracold Fermi gas
Grand partition function

Hubbard-Stratonovich transformation

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Sign problem in ultracold Fermi gas
Grand partition function

Non positive semi-definite

Reweighting

Sign problem
Except for16

Sign problem in other systems

Fermion determinant is non positive semi-definite when
• Even species of fermions with imbalance (↑≠↓)• Odd species of fermions• Repulsive interaction

Related topics:
polaron, FFLO, High-Tc superconductor, Effimov effect, bose-fermi mixture, …

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Sign problem in QCD

Fermion determinant is non positive semi-definite when
• Chemical potential is nonzero

Condition of positivity is different from that in non-rela. system

18Complex LangevinParisi, Phys. Lett. 131B (1983) 393, Klauder PRA 29 (1984) 2036

19Complex Langevin

Drift term

White noise

Parisi, Phys. Lett. 131B (1983) 393, Klauder PRA 29 (1984) 2036

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Complex Langevin

Reach equilibrium

Parisi, Phys. Lett. 131B (1983) 393, Klauder PRA 29 (1984) 2036

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Justification of complex LangevinIfor

has “good” properties,

Obtained by complex Langevin

Original path integral

Aarts, Seiler, Stamatescu, PRD 81 (2010) 054608
Aarts, James, Seiler, Stamatescu, EPJ C71 (2011) 1756
Nagata, Nishimura, Shimasaki, PRD 92 (2015) 011501, PTEP 2016 013B0122

Practically useful criterion
Distribution of the drift term should decay exponentially.

Exponential decay

Power-law. decay

Nagata, Nishimura, Shimasaki, PRD 92 (2015) 011501, PTEP 2016 013B01

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Application
Our setup:• Two-component Fermion• Attractive contact interaction• 1D

Corresponding Hamiltonian:24

Application
Our setup:• Two-component Fermion• Attractive contact interaction• 1D• Lattice regularization25

Application
Our setup:• Two-component Fermion• Attractive contact interaction• 1D• Lattice regularization(thermal de Broglie length)26

Application
Our setup:• Two-component Fermion• Attractive contact interaction• 1D• Lattice regularization

Continuum limit:

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Dimensionless parameters

We set** This is not the natural unit, where c=1 !

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What is expected ?
Poralon (inpurity dressed by medium)
Pseudogap
https://physics.aps.org/articles/v9/86

FFLO-like state

Orso, PRL 98 (2007) 070402

Tajima, ST, Doi, arXiv:2005.12124

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What is expected ?
Poralon ← Today’s topic
Pseudogap
https://physics.aps.org/articles/v9/86

FFLO-like state

Orso, PRL 98 (2007) 070402

Tajima, ST, Doi, arXiv:2005.12124

30Complex Langevin works !

Exponential decay

31Extracting the polaron energy32Dispersion relation of polaron

Momentum in lattice unit

Fitting function:

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Dispersion relation of polaron: Polaron energy

Momentum in lattice unit

Fitting function:

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Polaron energy

Temperature: scattering length: determined by35

Exact result at T=0 limit obtainedby thermodynamic Bethe ansatz
J. B. McGuire, J. Math. Phys. 7, 123 (1966).

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Exact result at T=0 limit obtained
by TBAJ. B. McGuire, J. Math. Phys. 7, 123 (1966).

Continuum limit37

Exact result at T=0 limit obtained
by TBAJ. B. McGuire, J. Math. Phys. 7, 123 (1966).

Continuum limit
T=0 limit

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Exact result at T=0 limit obtained
by TBAJ. B. McGuire, J. Math. Phys. 7, 123 (1966).

Continuum limit
T=0 limit

Complex Langevin agrees
with TBA

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What about T-dependence?
TMA = T-matrix approach(self-consistent diagrammatic calc.)

CL = Complex Langevin• TMA agrees with TBA in thistemperature region.
• Lattice artifact may not be
negligible in strong couplingregime.

40Summary What is the sign problem ?
• Exponentially small signal-to-noise ratio in
Monte Carlo simulations

 Sign problem in cold atom
• Non positive definite fermion determinantcauses the sign problem.

 Complex Langevin (theory and application)• In our setup (1D, attractive, βh≠0), complex Langevin is reliable.
• We obtain polaron energy at T ≠0
• Consistent with TBA

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Appendix42

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Flow of the drift term

constant

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Two point function

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One particle energy

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