Reducing “sign problem” by the determinant propagators

“Sign problem” in fermionic QMC

The path-integral Monte Carlo (PIMC) method remains problematic for solving many-fermion systems due to the“sign” problem.

The antisymmetric free-fermion propagator is not positive-definite, only itsmagnitude can be sampled by the Monte Carlo method (as used in simulations of Bose particles)and observables must then be weighed by the overall sign.

We work in the ensemble of the coordinate space-imaginary time trajectories[=β/P]

X^s= (R^s₁, . . . ,R^s_P), R^s= (r^s_1,, . . . ,r^s_Ns,), {s=↑,↓}

To evaluate the fermionic partition function we sum up overN! permutations with different sign: sgn(σs) =±1

ZF= 1 N^↑!N^↓!

X

σ_s∈P

Z

dX^↑dX^↓

P

Y

i=1

sgn(σ↑) sgn(σ↓)·|hRi−1|e^−Hˆ|πˆσ_↑πˆσ_↓Rii|

Physical expectation values are evaluated in QMC with the statistical error:δA∼1/hsi

hAi=

DAˆ·sgn(σ)E

hsi ±δA, s=hsgn(σ)i=ZF

ZB

=e^{−βN(fF−fB)}

! Average sign decays exponentially with particle numberN, inverse temperatureβand the free energy difference of Bose and Fermi system

A.Filinov (1 Christian-Albrechts-Universit¨at Kiel, ITAP, 2 JIHT RAS, Izhorskaya Str., 13, 125412 Moscow, Russia )Simulations of Hydrogen Plasma with Fermionic Propagator Path Integral Monte Carlo41 st International Workshop on High-Energy-Density Physics with Intense Ion and Laser Beams 7 / 25

Ideal gas: Fermi repulsion / Bose attraction

PIMC: Simulations of ideal gas with Boltzmann / Bose / Fermi statistics

! Sampling of particle trajectories with the bosonic density matrix will require a strong cancellation between different permutation classesσs(i)∈P,i= (N1,N2..,N_N):

r→0limg^F(r)≈ hsi ·g^B(r), hsi ∼10⁻⁵

0 0.5 1 1.5 2

0 1 2 3 4

Paircorrelationfunction

Distance,r/aB

Boltzmann Bose Fermi

Simulation parameters:

Chemical potential:

µ=−0.086[Ha]

Density parameter:

rs= 3, ρa³_B= ⁴₃πr_s³−1

Temperature:

θ=T/TF = 1 Average particle number:

hNi^Fermi= 29.49 hNi^Boltz= 29.97 hNi^Bose= 30.43

A.Filinov (1 Christian-Albrechts-Universit¨at Kiel, ITAP, 2 JIHT RAS, Izhorskaya Str., 13, 125412 Moscow, Russia )Simulations of Hydrogen Plasma with Fermionic Propagator Path Integral Monte Carlo41 st International Workshop on High-Energy-Density Physics with Intense Ion and Laser Beams 8 / 25

Reducing “sign problem” by the determinant propagators

Use theAntisymmetric Propagators (determinants)

Takahashi and Imada (1984); V.Filinov (2004); Lyubartsev, Vorontsov-Velyaminov (2005) Modified Partition function:

Z_F = 1 N^↑!N^↓!

Z

dX^↑dX^↓

P

Y

i=1

sgn(i)·e^{−SA(i−1,i)}

sgn(i) = sgnM^↑(i−1,i)·sgnM^↓(i−1,i),

e^{−SA(i−1,i)}=e^{−W(R↑i,Ri↓)}e^{ln|det M↑}(i−1,i)|+ln|det M^↓(i−1,i)|

⇒The composite sign of different permutations P

σ_↑(σ_↓)

reduces to a sign of determinants M^↑(↓) Fermionic propagatorpartially reduces thesign-problem. The exchange matrix gives a repulsive (Pauli-blocking like) contribution to the interactions of the spin-like fermions

D_i−1,i^s =X

σs

hRi−1|e^−Kˆ|πˆσ_sRii= 1

λ^DNsdetM^s(i−1,i), s=↑(↓) where M^sis theN^s×N^santisymmetric diffusion matrix

M^s=||m_kl(i−1,i)||=exp(−π λ²

[r^s_l(i)−r^s_k(i−1)]²), (k,l= 1, . . .N^s)

A.Filinov (1 Christian-Albrechts-Universit¨at Kiel, ITAP, 2 JIHT RAS, Izhorskaya Str., 13, 125412 Moscow, Russia )Simulations of Hydrogen Plasma with Fermionic Propagator Path Integral Monte Carlo41 st International Workshop on High-Energy-Density Physics with Intense Ion and Laser Beams 9 / 25

Improved high-temperature density matrix

Pauli-blockingbecomes effective only for few “determinant” propagators. This originally prevents to use this idea at low-temperatures to study degenerate systems with the

“primitive” propagators: e^{−( ˆK+ ˆV)}≈e^−Kˆe^−Vˆ+O(²) The fourth-order factorization:

S.A. Chin et al., J.Chem.Phys. 117, 1409 (2002); K.Sakkos et al., J.Chem.Phys. 130, 204109 (2009)

e^−βHˆ=

P

Y

i=1

e^{−( ˆK+ ˆV)}≈

P

Y

i=1

e^−Wˆ1e^−t1Kˆe^−Wˆ2e^−t1Kˆe^Wˆ1e^−t0Kˆ+O(⁴⁽⁶⁾) (=β/P, (2t1+t0= 1), (2v1+v2= 1), 0≤t0≤1−1/√

3)

Combination of two ideas allowed for a first time to perform ab-initio fermionic simulations of up toNe∼66at high degeneracy! (nλ³_T.3)

–Ensembles of electrons in 3D quantum dots, S.A. Chin et al., J.Chem.Phys. 117, 1409 (2002)

–The uniform electron gas at warm dense matter conditions(2015-2020), T.Dornheim, S.Groth, M. Bonitz, Phys.Rep. 744 (2018)

! All present numerical simulations have been restricted to the canonical ensemble (CE) and the thermodynamic observables related with the properties of the diagonal density matrix (and derivatives of the partition function)

Present method: all advantages of the GCE and additional thermodynamic functions

A.Filinov (1 Christian-Albrechts-Universit¨at Kiel, ITAP, 2 JIHT RAS, Izhorskaya Str., 13, 125412 Moscow, Russia )Simulations of Hydrogen Plasma with Fermionic Propagator Path Integral Monte Carlo41 st International Workshop on High-Energy-Density Physics with Intense Ion and Laser Beams 10 / 25