arXiv:cond-mat/0011334 · Journal of Physics A Mathematical and General

Non-uniform ground state for the Bose gas

András Sütö

Published Dec 19, 2000• 2 citations

Abstract

We study the ground state Ψ 0 = ∑ a X | X ⟩ of N hard-core bosons on a finite lattice in configuration space, X = { x 1 ,..., x N }. All a X being positive, the ratios a X /∑ a Y can be interpreted as probabilities P a ( X ). Let E 0 denote the energy of the ground state and |∂ X | the number of nearest-neighbour particle-hole pairs in the configuration X . We prove the concentration of P a onto X with |∂ X | in a (| E 0 |) 1/2 -neighbourhood of | E 0 |, show that the average of a X over configurations with |∂ X | = n increases exponentially with n , discuss fluctuations about this average, derive upper and lower bounds on E 0 and give an argument for off-diagonal long-range order in the ground state.

DOI: 10.1088/0305-4470/34/1/303

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arXiv:cond-mat/0011334 · Journal of Physics A Mathematical and General

Non-uniform ground state for the Bose gas

András Sütö

Published Dec 19, 2000• 2 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/0305-4470/34/1/303

Related papers

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Motivated by some inconsistencies in the way quantum fluctuations are included beyond the classical treatment of hard-core bosons on a lattice in the recent literature, we revisit the large-S semi-classical approach to hard-core bosons on the square lattice at T=0. First of all, we show that, if one stays at the purely harmonic level, the only correct way to get the 1/S correction to the density is to extract it from the derivative of the ground state energy with respect to the chemical potential, and that to extract it from a calculation of the ground state expectation value of the particle number operator, it is necessary to include 1/\sqrt{S} corrections to the harmonic ground state. Building on this alternative approach to get 1/S corrections, we provide the first semiclassical derivation of the momentum distribution, and we revisit the calculation of the condensate density. The results of these as well as other physically relevant quantities such as the superfluid density are systematically compared to quantum Monte Carlo simulations. This comparison shows that the logarithmic corrections in the dilute Bose gas limit are only captured by the semi-classical approach if the 1/S corrections are properly calculated, and that the semi-classical approach is able to reproduce the 1/k divergence of the momentum distribution at k=0. Finally, the effect of 1/S^2 corrections is briefly discussed.Motivated by some inconsistencies in the way quantum fluctuations are included beyond the classical treatment of hard-core bosons on a lattice in the recent literature, we revisit the large-S semi-classical approach to hard-core bosons on the square lattice at T=0. First of all, we show that, if one stays at the purely harmonic level, the only correct way to get the 1/S correction to the density is to extract it from the derivative of the ground state energy with respect to the chemical potential, and that to extract it from a calculation of the ground state expectation value of the particle number operator, it is necessary to include 1/\sqrt{S} corrections to the harmonic ground state. Building on this alternative approach to get 1/S corrections, we provide the first semiclassical derivation of the momentum distribution, and we revisit the calculation of the condensate density. The results of these as well as other physically relevant quantities such as the superfluid density are systematically compared to quantum Monte Carlo simulations. This comparison shows that the logarithmic corrections in the dilute Bose gas limit are only captured by the semi-classical approach if the 1/S corrections are properly calculated, and that the semi-classical approach is able to reproduce the 1/k divergence of the momentum distribution at k=0. Finally, the effect of 1/S^2 corrections is briefly discussed.

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The ground state of a system of

N

impenetrable hard core quantum particles in a 1-D box is analyzed by using a new scheme applied recently to study a similar system of two such particles {\it [Centl. Eur. J. Phys., 2(4), 709 (2004)]}. Accordingly, each particle of the system behaves like an independent entity represented by a {\it macro-orbital}, -a kind of pair waveform identical to that of a pair of particles moving with (

q

-q

) momenta at their {\it center of mass} which may have any momentum

K

in the laboratory frame. It concludes: (i) $ = 0$, (ii) $ \ge λ/2$ and (iii)

q \ge q_o (= π/d)

(with

d = L/N

being the average nearest neighbour distance), {\it etc.} While all bosons in their ground state have

q = q_o

and

K = 0

, fermions have

q= q_o

with different

K

ranging between 0 and

K = K_F

(the Fermi wave vector). Independent of their bosonic or fermionic nature, all particles in the ground state define a close packed arrangement of their equal size wave packets representing an ordered state in phase (

ϕ-

)space with

Δϕ= 2nπ

(with

n

= 1,2,3, ...), $ = λ/2 = d$, and

q = q_o

. As such our approach uses greatly simplified mathematical formulation and renders a visibly clear picture of the low energy states of the systems and its results supplement earlier studies in providing their complete understanding.

We consider a gas of bosons interacting through a hard-sphere potential with radius

\frak{a}

in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4πρ\frak{a}

and shows that corrections are smaller than

C ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, for a sufficiently large constant

C > 0

. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order

ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, in agreement with the Lee-Huang-Yang prediction.

Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius

\mathfrak {a}

a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4\pi \rho \mathfrak {a}

4 π ρ a and shows that corrections are smaller than

C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}

C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant

C > 0

C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order

\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}

ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

We investigate the properties of the ground state of a system of interacting bosons on regular lattices with coordination number

k\geq 2

. The interaction is a pure, infinite, on-site repulsion. Our concern is to give an improved upper bound on the ground state energy per site. For a density

ρ

a trivial upper bound is known to be

-kρ(1-ρ)

. We obtain a smaller variational bound within a reasonably large family of trial functions. The estimates make use of a large deviation principle for the energy of the Ising model on the same lattice.