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arXiv:0812.0883v1 [nucl-th] 4 Dec 2008

December

Proceeding˙Kazimierz˙2008

4,20089:15WSPC/INSTRUCTIONFILE

International Journal of Modern Physics E

c ? World Scientific Publishing Company

QUANTUM MONTE CARLO METHOD APPLIED TO STRONGLY

CORRELATED DILUTE FERMI GASES WITH FINITE

EFFECTIVE RANGE.

GABRIEL WLAZ? LOWSKI and PIOTR MAGIERSKI

Faculty of Physics, Warsaw University of Technology, ul. Koszykowa 75

00-662 Warsaw, Poland

e-mail: gabrielw@if.pw.edu.pl, magiersk@if.pw.edu.pl

Received (received date)

Revised (revised date)

We discuss the Auxiliary Field Quantum Monte Carlo (AFQMC) method applied to

dilute neutron matter at finite temperatures. We formulate the discrete Hubbard-

Stratonovich transformation for the interaction with finite effective range which is free

from the sign problem. The AFQMC results are compared with those obtained from

exact diagonalization for a toy model. Preliminary calculations of energy and chemical

potential as a function of temperature are presented.

1. Introduction

Properties of dilute and degenerate Fermi gases with large scattering lengths are of

particular interest in the context of cold fermionic atoms and dilute neutron matter.

In the case of trapped fermionic atoms, due to very low density, the scattering

length is the only parameter which determines an atom-atom interaction. Taking

advantage of Feshbach resonances it can be tuned, using external magnetic field,

to the values which greatly exceed the average interparticle distance. This limit

which is dubbed the unitary regime exhibits universal properties and has been the

subject of intensive theoretical effort in the last couple of years (see1and references

therein). Similarly, neutron matter at densities corresponding to kF ? 0.6 fm−1is

a dilute system of fermions in a sense that the average distance between particles

is much larger than the range of the neutron-neutron interaction. It is superfluid

and its physics is determined by two parameters: scattering length aSand effective

range reffin1S0neutron-neutron channel. The influence of other channels as well

as of three-body forces is marginal and can be neglected in this density range2.

The properties of dilute neutron matter are crucial for understanding the structure

and thermal evolution of neutron star crust3. They can also provide constraints for

parameters of nuclear energy density functional.

Despite of these similarities the important qualitative difference between dilute

neutron matter and cold atomic gases arises from the fact that in the former case

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G. Wlaz? lowski, P. Magierski

the nonzero effective range cannot be neglected. Therefore contrary to cold atomic

gases, physics of dilute neutron matter cannot be captured by simple contact, delta-

like interaction (except for very low densities kF? 0.06fm−1).

Since Fermi gas with long scattering length is an example of strongly correlated

system, non-perturbative methods based on Monte Carlo approaches are often used

to determine its properties. There exist various calculations at zero temperature

4,5,6,7,8,9,10,11,12,13,14. Here we present selected aspects of AFQMC approach

at finite temperatures, which is able to generate fully non-perturbative results for

dilute neutron matter without sign problem.

2. Theoretical approach

Neutron-neutron interaction in dilute neutron matter can be described by small

number of parameters characterizing the low energy physics of two-body collisions.

In this limit the main contribution to the scattering amplitude comes from s-wave

scattering and is determined by two parameters: scattering length aSand effective

range reffi.e.:

f(p) =

1

2reffp2− ip,

−1/aS+1

(1)

where aS≈ −18.5fm and reff≈ 2.8fm.

To capture low energy physics of dilute neutron matter we consider the Hamil-

tonian of the form:

?

+1

2

λ,λ′=±

ˆH =

?

λ=±

d3− →rˆψ†

λ(− →r )

?−∇2

2m

?

ˆψλ(− →r )

?

?

d3− →r d3− →r′V (− →r −− →r′)ˆψ†

λ(− →r )ˆψ†

λ′(− →r′)ˆψλ′(− →r′)ˆψλ(− →r ), (2)

where

{ˆψ†

Throughout this work we shall use units in which Planck’s and Boltzmann’s con-

stants are equal to one. The interaction has the form:

6g,− →r −− →r′= 0

g,

0, otherwise.

thefieldoperatorsobey thefermionic anticommutationrelations

λ(− →r ),ˆψλ′(− →r′)} = δλλ′δ(− →r −− →r′) and λ denotes the spin degree of freedom.

V (− →r −− →r′) =

− →r −− →r′∈ Nb

, (3)

where Nb= {(b,0,0),(−b,0,0),(0,b,0),(0,−b,0),(0,0,b),(0,0,−b)}.

This choice has been motivated, as will become clear later, by the special form

of discrete Hubbard-Stratonovich (H-S) transformation. It imposes limitations on

the density range of neutron matter which can be described by this particular

interaction. Clearly the interaction depends on two parameters: g and b. Solving

the equation for T-matrix elements:

d3− →

(2π)3V−

T−→

p−→

p′ = V−→

p−→

p′ +

?

k

→

p−→

kG−→

p′−→

kT−→

k−→

p′,(4)

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where G is the free particle propagator and using the relation:

−4π

mT−1

−→

p−→

p≈ −1

aS

+1

2reffp2− ip + O(p3)(5)

we can adjust parameters g and b to reproduce scattering length aS and effective

range reffof the neutron-neutron interaction. Still however the regularization pro-

cedure is required, which can be determined by introducing the momentum cut-off

pcut. This prescription sets to zero all two-body matrix elements, if the relative

momentum of two particles exceeds a given momentum cut-off.

In order to compute thermal properties of dilute neutron matter we place the

system on a 3D spatial lattice. The lattice spacing b and size L = Nsb introduce

natural ultraviolet (UV) and infrared (IR) momentum cut-offs given by pcut= π/b

and p0= 2π/L, respectively. The momentum space has the shape of a cubic lattice,

with size 2π/b and spacing 2π/L. To simplify the analysis, however, we place a

spherically symmetric UV cut-off, including only momenta satisfying p ≤ pcut.

To evaluate expectation values of observables we have used a path integral

representation of partition function:

Z(β,µ) = Tr

?Nτ

k=1

?

exp[−τ(ˆH − µˆ N)]

?

, (6)

O(β,µ) =

1

Z(β,µ)Tr

?

ˆO

Nτ

?

k=1

exp[−τ(ˆH − µˆ N)]

?

,(7)

where β = 1/T = Nττ, T is the temperature and µ - the chemical potential,ˆ N

- the particle number operator andˆO is a quantity of interest. The propagator

e−τ(ˆ H−µˆ

N)is decomposed using the second order expansion15,16:

exp[−τ(ˆ H − µˆ N)] ≈ exp

?

−τ(ˆT − µˆ N)

2

?

exp(−τˆV )exp

?

−τ(ˆT − µˆ N)

2

?

, (8)

whereˆT is the kinetic energy operator. To express the many body operator e−τˆV

by a sum of one body operators we have extended the discrete H-S transformation

introduced by Hirsch17. Namely, one may rewrite the interaction in the form:

ˆV =g

2

−→

r −−

→

r′∈Nb

+ˆ n↓(− →r )ˆ n↓(− →r′) + ˆ n↑(− →r )ˆ n↓(− →r ) + ˆ n↑(− →r′)ˆ n↓(− →r′)),

where ˆ nλ(− →r ) =ˆψ†

element of the above sum can take the following values:

iff n↑(− →r ) + n↓(− →r ) + n↑(− →r′) + n↓(− →r′) = 0

0, iff n↑(− →r ) + n↓(− →r ) + n↑(− →r′) + n↓(− →r′) = 1

g/2, iff n↑(− →r ) + n↓(− →r ) + n↑(− →r′) + n↓(− →r′) = 2

3g/2, iff n↑(− →r ) + n↓(− →r ) + n↑(− →r′) + n↓(− →r′) = 3

6g/2, iff n↑(− →r ) + n↓(− →r ) + n↑(− →r′) + n↓(− →r′) = 4

?

(ˆ n↑(− →r )ˆ n↑(− →r′) + ˆ n↑(− →r )ˆ n↓(− →r′) + ˆ n↓(− →r )ˆ n↑(− →r′) +

(9)

λ(− →r )ˆψλ(− →r ). Note that in the coordinate representation each

0,

(10)

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G. Wlaz? lowski, P. Magierski

where nλ(− →r ) denotes eigenvalue of density operator. This allows us to introduce

the H-S transformation of the form:

e−τbV=

?

−→

r −−

→

r′∈Nb

1

k

k

?

i=1

eσi(−→

r ,−→

r′)[ˆ n↑(−→

r )+ˆ n↓(−→

r )+ˆ n↑(−→

r′)+ˆ n↓(−→

r′)]

(11)

where values of σiand k have to fulfil the equations:

1

k

1

k

1

k

1

k

?k

?k

i=1eσi= 1

?k

?k

i=1e2σi= e−τg

i=1e3σi= e−3τg

i=1e4σi= e−6τg

2

2

2 .

(12)

This set of equations has a solution if k ? 7 and g < 0 (attraction).

The partition function can be expressed in the form:

Z(β,µ) =

??

τj

?

−→

r −−

→

r′∈Nb

Dσ(− →r ,− →r′,τj)TrˆU({σ}),

ˆU({σ}) = Tτexp{−τ[ˆh({σ}) − µ]}, (13)

where Tτdefines the time-ordered product andˆh({σ}) denotes the one-body Hamil-

tonian. Consequently the expectation value of an arbitrary operator takes the

form15,16:

? ?Dσ(− →r ,− →r′,τj)TrˆU({σ})

The many-fermion problem has thus been reduced to the typical AFQMC problem.

Consequently the standard Metropolis algorithm can be applied, using TrˆU({σ})

as a measure18.

To prove that the measure is positive it is sufficient to note that the operator

ˆU({σ}) is invariant with respect to time reversal. This is due to the special form

of expression (11) and the fact that the sigma field is real. Therefore the single

particle spectrum ofˆU({σ}) consists of pairs which are conjugate to each other.

Clearly TrˆU({σ}) = det(1+ˆU({σ})) =?

to spin-up and spin-down particles then the matrix representation ofˆU takes the

form:

O(β,µ) = ?ˆO? =

Z(β,µ)

TrˆOˆU({σ})

TrˆU({σ})

. (14)

i|(1+ui)|2? 0. In particular if the single

particle basis is chosen in such a way that the time reversed partners correspond

U =

?U↑↑ 0

0 U↓↓

?

(15)

where U↑↑= U↓↓∗. Matrix elements between states with different spin projections

are equal to zero. Finally one gets

TrˆU = det[1 + U] = det[1 + U↑↑]det[1 + U∗

↑↑] = |det[1 + U↑↑]|2? 0. (16)

Hence the lack of the sign problem is a direct consequence of the time-reversal

symmetry which is preserved by each term in the sum (11).

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Table 1. Comparison of AFQMC results with

exact diagonalization in a restricted space

(see text for details). T/ǫF = 0.58, µ/ǫF =

0.41.

ObservableAFQMC result

(1000 samples)

Exact result

?ˆT?

?ˆV ?

?ˆ H?

?ˆ N?

0.32

-0.088

0.23

5.58

0.31

-0.086

0.23

5.49

3. Numerical results

In order to verify the correctness of our code we have performed several tests. As a

first check the thermodynamics of a free gas was reproduced for g = 0.

To check our results for an interacting system we have diagonalized the Hamil-

tonian (2) exactly, restricting the single-particle Hilbert space to the lowest 7 states.

It corresponds to the Fock space of 214states. The test provided an estimate for

the size of the imaginary time step. Comparison of AFQMC results with the exact

ones for selected observables at fixed temperature is given in the table 1.

To perform calculations for a neutron matter we have used the box with Ns= 8

and lattice constant b = 3.21fm. The neutron mass was fixed at 939 MeV. The

chemical potential was chosen in such a way to keep the total number of particles

equal to 55. This corresponds to the density of neutron matter ρ∼= 0.02ρ0where

ρ0 is the saturation density of nuclear matter. For this density we have varied

temperatures from 0.06ǫF (0.26 MeV) to 1.0ǫF (3.8 MeV) where ǫF is the Fermi

energy. For the lowest temperature we have used 2360 imaginary time steps while

for the highest temperature only 216. In all runs the single-particle occupation

probabilities for the highest energy states were below 0.01 at all temperatures. At

low temperatures the Singular Value Decomposition technique was used to avoid

instabilities of the algorithm.

Figure 1 presents an example of the results for the energy and chemical potential

as a function of temperature. The energy and chemical potential versus temperature

for the free Fermi gas at the same particle density has also been plotted (dotted

line). The free Fermi gas results were shifted by constant values −0.54 and −0.60

for the energy and the chemical potential, respectively.

For our lowest temperature 0.06ǫF(0.26 MeV) we obtained the energy (relative

to the energy of the free Fermi gas EFFG) E/EFFG= 0.48 ± 0.02 (E/N = 1.28

MeV). This value is in good agreement with recent fixed-node quantum Monte

Carlo calculations13. The chemical potential at this temperature was calculated to

be µ/ǫF= 0.40 (1.77 MeV).