Phase diagram of a quantum Coulomb wire

IntroductionSystemMethodResults

Summary

Phase diagram of a quantum Coulomb wire

Guillem Ferré, Grigori Astrakharchik, Jordi Boronat

Barcelona Quantum Monte Carlo groupDepartment de Física i Enginyeria Nuclear

Campus Nord B4-B5, Universitat Politècnica de Catalunya, Barcelona, Spain


Summary

Outline

1 Introduction

2 System

3 Method

4 Results


Summary

Outline

1 Introduction

2 System

3 Method

4 Results


Summary

Electron gas

Electron gas is a long-standing physical problem whose studystarted a long time ago. Few systems are more universal.

Mainly studied by means of Monte Carlo calculations.

Phase diagrams in 2D & 3D are well known.

Landau’s Fermi-liquid theory works for electron gas in 2D& 3D, while the appropriate framework for 1D isTomonaga-Luttinger theory.

Numerical treatment of large fermionic systems is morecomplicated than bosons due to the "sign problem".

Girardeau’s Fermi-bose mapping for 1D systems reducesfermionic complexity to a bosonic one.


Summary

Phase diagram for 2D & 3D electron gas


Summary

Motivation

Theoretical knowledge of an electron gas in 1D geometryis more scarce than its counterpart at higher dimensions.

There is no full determination of the density-temperaturephase diagram of a 1D Coulomb gas.

Coulomb wire is different from other Tomonaga-Luttigersystems.

Ground-state properties have already been studied, themost accurate results obtained using DMC.


Summary

Outline

1 Introduction

2 System

3 Method

4 Results


Summary

Hamiltonian

System composed by N particles with charge e and mass m ina 1D box of length L with periodic boundary conditions.

H = −12

N∑

i=1

∂2

∂x2i

+

N∑

i<j

1∣

∣xi − xj∣

∣

Atomic units: Bohr radius a0 = ~2/(me2) for length,

Hartree Ha = e2/a0 for energy.

Potential diverges at contact.


Summary

Sign problem

Arises when performing numerical simulations with fermions

Fixed-node approximation gives an upper-bound energyfor fermions.

Bad scaling with number of particles: N3 for fermionsagainst N2 for bosons.

But...

Nodal position is known.

Girardeau’s mapping is applicable: the many-body wavefunction ΨB of charged bosons on a line with e2/rinteraction can be mapped onto a many-body wavefunction ΨF of fermions as ΨB = |ΨF |

Results are exact in our case.


Summary

Outline

1 Introduction

2 System

3 Method

4 Results


Summary

Monte Carlo Methods

Ground-state properties have already been studied usingdiffusion Monte Carlo (DMC).

Energy can be obtained exactly.

Static structure factor can be obtained using pureestimators.

At finite temperature, we use path-integral Monte Carlo (PIMC).

The knowledge of the partition function gives access to amicroscopic description of the properties of the system,Z = Tr e−βH = Tr e−β(K+V )

The main goal is the calculation of the phase diagram, thuswe mainly want to determine energetic and structureproperties.


Summary

PIMC method

Noncommutativity of quantum operators K and V makesimpractical direct calculations.

Approximations are needed.

e−βH =(

e−τ(K+V ))M

τ =β

M

The exact result for M → ∞ is warranted by Trotter formula.

e−β(K+V ) = limM→∞

(

e−τ K e−τ V)M


Summary

PIMC method

The partition function can be written as a multidimensionalintegral with a distribution law that resembles a classicalpolymer.

Z =

∫

d~R1...d~RM

M∏

α=1

ρ(

~Rα, ~Rα+1; τ)

The statistical distribution law can be interpreted as aprobability distribution.

As temperature decreases, M needs to increase to reachconvergence.


Summary

PIMC method

Possibility to map the quantum system onto a classicalmodel of interacting polymers.

Worm algorithm: Permutations between particles at lowtemperature.


Summary

Action

We use a fourth-order approximation of ρ(

~Rα, ~Rα+1; τ)

.

Chin Action

e−τ H ≃ e−v1τWa1 e−t1τ K e−v2τW1−2a1 e−t1τ K e−v1τWa1 e−2t0τ K

Based on the symplectic developments of Chin.Two free parameters (t0,a1) that need to be adjusted forevery system.Already proved its efficiency in the study of systems suchas liquid 4He.Can be effectively sixth order with a good choice of theparameters.


Summary

Chin Action

Density matrix

ρ(

~Rα, ~Rα+1

)

=

(

m

2π~2τ

)3dN/2(

1

2t21 t0

)dN/2 ∫

d~RαAd~RαB exp[

−m

2~2τ

N∑

i=1

(

1

t1

(

~rα,i −~rαA,i)2

+1

t1

(

~rαA,i −~rαB,i)2

+1

2t0

(

~rαB,i −~rα+1,i)2

)

− τ

N∑

i<j

(

v1

2V (rα,ij ) + v2V (rαA,ij) + v1V (rαB,ij) +

v1

2V (rα+1,ij )

)

− τ3u0

~2

m

N∑

i=1

(

a1

2|~Fα,i |

2+ (1 − 2a1)|

~FαA,i |2+ a1|

~FαB,i |2+

a1

2|~Fα+1,i |

2)]

K. Sakkos, J. Casulleras, and J. Boronat. “High order Chin actions in Path Integral Monte Carlo”. In:

The Journal of Chemical Physics 130 (2009), p. 204109

http://dx.doi.org/10.1063/1.3143522


Summary

Outline

1 Introduction

2 System

3 Method

4 Results


Summary

Ground-state energies

For ground-state 1D Coulomb1 :

At low density, energy’s leading term is the potentialenergy for a set of particles at the fixed positions of aWigner crystal.

EW

N= e2n ln N

At high density, the kinetic energy dominates.

EIFG

N=

~2k2

F

6m, kF = πn

1G. E. Astrakharchik and M. D. Girardeau. “Exact ground-state propertiesof a one-dimensional Coulomb gas”. In: Phys. Rev. B 83 (2011), p. 153303.


Summary

Energies at finite temperature

At low temperature PIMCrecovers DMC ground-stateenergy.

For a fixed finite temperatureT . 1 Ha, low densitycorresponds to a classicalgas with EC = T/2.

For termperatures T & 1 Ha,Wigner crystal behavior is nolonger observed.

G. Ferré, G. E. Astrakharchik, and J. Boronat. “Phase diagram of a quantum Coulomb wire”. In: Phys. Rev. B 92

(2015), p. 245305


Summary

Static structure factor at T = 0.01 Ha

At lowest densityn = 0.01a−1

0 , quantumPIMC results are nearlyindistinguishable fromclassical S(k).

Bragg peak emerges atk/kF = 2 when increasingdensity, forming a Wignercrystal.

Increasing further, thesystem evolves to an idealFermi gas.

Classical gas - Wigner crystal - ideal Fermi gas


Summary

Static structure factor at n = 10−3 a−10

At low temperatures, Braggpeaks are identified. AtT = 10−5Ha a quantumcrystal is observed while athigher T = 10−4Ha aclassical one is found.

Once in classical regime thecrystal "melts" and becomesa gas.

Quantum Wigner crystal - classical Wigner crystal - classical gas


Summary

Phase diagram

Regimes

Classical Coulombgas regime.

Wigner crystal, withcrossover betweenquantum and classicalregimes.

Ideal Fermi gas, withcrossover betweenquantum and classicalregimes.


Summary

Regime transitions changing T

Wigner crystal meltingtransition followsT ∼ EW → T ∼ n.

Quantum Wignercrystal becomesclassical followingT ∼ EBZ → T ∼ n3/2.

For high densities,ideal Fermi gasbecomes classical gasfollowingT ∼ EF → T ∼ n2.


Summary

Regime transitions changing density

For T . 10−2Ha,classical Coulomb gasbecomes Wignercrystal.

For T . 10−2Ha,Wigner crystalbecomes ideal Fermigas at n ≈ 1.

For T > 10−2Ha,classical Coulomb gasbecomes ideal Fermigas at n ≈ 1.


Summary

Conclusions

Complete PIMC study of the density-temperature phasediagram of a 1D quantum Coulomb wire.

The singularity of the Coulomb interaction at x = 0 allowsus to solve the sign problem.

Analysis focused on energetic and structural properties

Enhanced stability of the Fermi gas.

Double crossing of gas-crystal-gas regime.


Summary

Thank You!

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Phase diagram of a quantum Coulomb wire