Vinod Ashokan Department of Physics, National Institute of Technology

Jalandhar, India Email: ashokanv@nitj.ac.in

GROUND-STATE PROPERTIES OF 1-D ELECTRON GAS

Contents • What is 1D electron gas and where it can be

found ?

• How we expect it to behave

• Methods of study of its properties (QMC,

simulation)

• High density theory

• Results

In 1D, particles cannot avoid each other

The interesting physics comes from the reduced dimensionality and the Coulomb interaction i.e. a consequences of both

The electrons in 1D do not obey the conventional Fermi-liquid theory, and an appropriate description is given by exactly solvable Tomonaga-Luttinger Liquid model

Electron gas can be describe in presence of neutralizing background, with ions and electrons equal charge density to have a neutral system

Jellium Model

1D

2D

Electron gas in 3D, 2D and 1D Pair excitations

( )

−+= 22

2

2kqk

mE

Fk20 q

)(qEk

Fk20 q

)(qEk

Momentum distribution n(k)

n(k)

n(k) n(k) Non interacting Interacting

Experimentalists look for power law behaviour in various quantities and spin-charge separation as a signature of 1d behaviour

|)(| Fkk −Θ= (Non interacting)

Experimental realization of 1D systems

Carbon Nanotube

Energy-momentum dispersion relation of graphene, the material used to make carbon nanotube, is linear over a large range

Bechgaard salts and Semiconductor Wire

(left) charge-transfer salts (e.g. (BEDT-TTF)2X (right) semiconductor devices

Nanowires of atoms

The existence 1D electronic states between self-organized Pt nanowires spaced1.6 or 2.4 nm apart on a Ge(001) surface is observed by low-temperature scanning tunneling microscopy

The system of gas of electron with neutralizing background can be described by single parameter well known by , which is average distance between two electrons, defined as

Parameter necessary to describe electron gas systems (in unit of Bohr radius) (T=0)

=

.1,2

1

,2,1

,3,4

3

2/1

3/1

Dn

Dn

Dn

ar Bs π

π

sr

sr

eF mk

re

EK

EP

2/

/

.

.22

02

=

3

41 30r

n

π=

In 3D

=sr

Hamiltonian

Infinitely-thin wire

Details of 1D HEG simulation

The Ewald interaction for the infinitely thin wire may be written as

,2

)(2

1ˆ1

2

2

Madji

ij

N

i i

VN

xVx

H ++∂∂

−= ∑∑<=

∑ ∫∞

−∞=−

−+−

+=

n

L

Lijij

ij ynLx

dy

LnLxxV ,

||

1

||

1)(

2/

2/

0

1lim ( )

| |Mad xV V x

x→

= −

(1)

(2)

(3)

Harmonic wire

More sophisticated model has a wire of finite width – the confinement comes from a harmonic potential

If the confinement is strong enough, we can factorize the wave function

..and we take as a Gaussian

x

y

z

),()exp(),( 2/1 RiqxLRxq φψ −=

)(Rφ

(4)

We can work out the interaction as a function of x by integrating over the transverse part of

The electron wave function of two dimensional harmonic oscillators in cylindrical co-ordinates is given as

This leads to interaction potential

and its Fourier transform

),( Rxqψ

V

∫∞

∞−

− ∫−+−

−=2/122

22)'(

0

2

])'()'[(

'|)'(||)(|)'()(

RRxx

dRdRRRexxd

eqV xxiq φφ

ε

)4/exp()2()( 222/12 bRbR −= −πφ

),4

||()(

2

2

4

b

xerfce

bxV b

xπ=

).()exp()( 221

22

0

2

bqEbqe

qVε

=

(5)

(6)

(7)

(8)

Using Ewald summation this can be written as

,)cos(])[(2

2

||

||

1

2

||

2)(

1

21

)4/()( 22

∑

∑

∞

=

∞

−∞=

−

+

−

−−

−=

nij

ij

ij

m

ijbmLxij

GnxbGnEL

b

mLxerfc

mLx

b

mLxerfce

bxV ijπ

−=→ b

xVVx

Mad 2)(lim

0

π

It possesses a long range Coulomb tail and is finite at 0=ijx

(9)

(10)

Where and x’ is related to x by a backflow transformation. The Jastrow factor is,

The wave function

Quantum Monte Carlo (QMC) techniques provide a practical method for solving the many body Schrödinger equation

)'(...)'()'(

....

....

)'(..)'()'(

)'(..)'()'(

)](exp[)(

21

11211

11211

nnnn

n

n

xxx

xxx

xxx

RJR

φφφ

φφφφφφ

ψ =

)exp()( xikx nn =φ

−Θ−+

=

∑

∑ ∑

=

≠ =

u

p

N

r

rijrijuuij

N

ji

N

AijA

xxLLx

xL

AaRJ

0

2

1

)()(

2cos)(

α

π

(11)

Variational Monte Carlo (VMC) The ground state energy E0 of a quantum system may be written as

Using a trial wave function • (R), a variational estimate E is given by

formulate it as a Monte Carlo integral

Method of calculations (QMC)

CASINO code (University of Cambridge, UK )

(12)

(13)

(14)

Diffusion Monte Carlo (DMC)

A. Ground State Energy at high density Results

Fig 1. VMC energy plotted against the reciprocal of the square of the system size for the infinitely thin wire. The energies per particle were extrapolated to the thermodynamic limit using the form .

( )E E∞−

( )E E∞−

( ) 2E N E BN−∞= +

Table: The VMC energies for the infinitely thin and harmonic wire extrapolated to the thermodynamic limit

Fig 2. Correlation energy as a function of rs for an infinitely thin wire (and, in the inset, for a harmonic wire of width b = 0.5). The solid line is a quadratic fit as a function of rs .

Infinitely thin wire

Harmonic wire

Exchange Energy

2( ) 0.027431(3) 0.00791(1) 0.00196(1)c s s sr r rε = − + −

( ) 0.0274156 0.00845 ...c s sr rε = − + +

Fitted Quadratic Function (Infinitely thin wire b=0)

Conventional Perturbation Theory Result (Infinitely thin wire b=0)

Fitted Quadratic Function (Harmonic wire b=0.5)

2( ) 0.00079(1) 0.00581(4) 0.00062(3)c s s sr r rε = − + −

B. Pair Correlation Function

parallel-spin

Fig 3. PCF of an infinitely thin wire at several densities. The data shown are for N = 99.

Fig 4. PCF of a harmonic wire of width b = 0.5 at several densities. The data shown are for N = 99.

Fig 5. PCFs g(r ) of 1D HEGs in infinitely thin wires. Results obtained from VMC simulation are compared with recent high density theory at several densities. The inset shows the oscillations of g(r ) at larger distances in greater detail.

C: Static Structure Factor

Fig 6. SSF of an infinitely thin wire at several system sizes for rs = 0.8. The main plot shows the behavior at the peak, and the inset shows a zoomed-out view.

Fig 7. VMC SSF of an infinitely thin wire with N = 99, compared with the high-density theory (solid line). The main plot shows the SSF for rs = 0.5, and the inset is for rs = 0.6

Fig 8. SSF peak height at k = 2kF plotted against system size N for infinitely thin wires with different coupling parameters rs .

D: Momentum Density

Fig 9 MD of an infinitely thin wire at several densities

Fig 10. MD of a harmonic wire of width b = 0..5 at several densities. The data shown are for N = 99. The statistical error bars are much smaller than the symbols and have therefore been omitted for clarity. The inset shows a zoomed-out view.

Tomonaga-Luttinger Liquid Parameters

In TL liquid theory the exponent • is related to the TL liquid parameter by

FIG 11 Tomonaga-Luttinger liquid exponent • extracted from our MDs against the fitting range of data (|k − kF | <• kF ) for a ferromagnetic, infinitely thin wire and a harmonic wire of width b = 0.5. The extracted exponent is linearly fitted with the solid line in the region > 0.035 and extrapolated to • = 0.

Fig.12 Exponent • , found by fitting to the MDs of ferromagnetic 1D electron gases and extrapolating to •= 0, plotted against rs . The error bars on the data points approximately account for the random error due to the Monte Carlo evaluation of the momentum density of the trial wave function (approximate because the data points in the MD are correlated); however, the random noise on the exponents is clearly larger than these error bars. There is an additional uncertainty in the MD and hence • due to the stochastic optimization of the trial wave function, and this may be responsible for the larger noise. Nevertheless, the noise in the exponent • as a function of rs is at least an order of magnitude smaller than the systematic behavior.

Fig 13. Exponent • found by fitting to the MDs of ferromagnetic systems for rs = 0.5 and different harmonic wire widths b

Infinitely thin wire

For repulsive interactions

Fig 14.Tomonaga-Luttinger parameter K• plotted against rs and, in the inset, plotted as a function of exponent • for an infinitely thin wire. The low-density DMC data are adopted from Lee and Drummond

We also derived expression for static structure factor in high density limit for finite width wire, which for infinitely wire reduces to

)()()()( 10 qSqSqSqS v++=

Where

and for x>1 an additional part . These are exact to order rs.

Pair correlation function g(r) as a Fourier transform of S(q) is,

High density theory

and

>

<=

F

FF

kq

kqk

q

qS

2,1

2,2)(0

,]|1|

ln2[|1|ln|1|

]1

ln2)[1ln()1(

2

2

2

2

1

−+−−+

++++=+

x

xxx

x

xxx

x

grSS ss

v π

]ln2[ln2 xxx +

x<1

∫∞

∞−−−= )](1[

2

11)( qSedq

nrg iqr

π

K. Morawetz, V. Ashokan, Renu Bala and KN Pathak, Phys. Rev. B. 97, 155147 (2018)

Summary We have presented calculations of the ground-state energy, PCF, SSF, and MD of the infinitely thin 1D HEG model using VMC and DMC. We observe the development of peaks at increasingly large even-integer multiples of kF in the SSF as the density is increased We have also calculated the structure factor in high density limit exactly and compared with our simulation data, which are in exact agreement in this limit. SSF peak height at k=kF plotted against system size N for infinitely thin wire for different coupling parameter rs From simulated ground state energy correlation energy is obtained as discussed and are in good agreement with conventional perturbation theory The Luttinger liquid behavior in high density is clearly observed in our simulation, and the Luttinger liquid parameter is extracted from MDs data and our high density Luttinger liquid parameter is smoothly goes over to the low density data.

Collaborators Prof. K. N, Pathak, Prof. Klaus Morawetz, Prof. N. D. Drummond Dr. Renu Bala

THANKS