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Quantum Many-Body Systems in 1D: One Dimensional Hydrogen

V E A Arias, A. Jose

Published Jun 22, 2015• 0 citations
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Abstract

Out of all atoms, hydrogen possesses the smallest mass, which results in very strong quantum effects at low temperatures. The hydrogen isotopes have different masses (hydrogen H, deuterium D and tritium T) and follow as significantly different behavior. The role of quantum fluctuations is further increased in low-dimensional geometry. A precise description of the equation of state and correlation functions of one-dimensional hydrogen is of a large fundamental and experimental importance.

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Out of all atoms, hydrogen possesses the smallest mass, which results in very strong quantum effects at low temperatures. The hydrogen isotopes have different masses (hydrogen H, deuterium D and tritium T) and follow as significantly different behavior. The role of quantum fluctuations is further increased in low-dimensional geometry. A precise description of the equation of state and correlation functions of one-dimensional hydrogen is of a large fundamental and experimental importance.
We review the recent activity in the theoretical description of spin-polarized atomic hydrogen and its isotopes at very low temperatures. Spin-polarized hydrogen is the only system in nature that remains stable in the gas phase even in the zero temperature limit due to its small mass and weak interatomic interaction. Hydrogen and its heavier isotope tritium are bosons, the heavier mass of tritium producing a self-bound (liquid) system at zero temperature. The other isotope, deuterium, is a fermion with nuclear spin one making possible the study of three different quantum systems depending on the population of the three degenerate spin states. From the theoretical point of view, spin-polarized hydrogen is specially appealing because its interatomic potential is very accurately known making possible its precise quantum many-body study. The experimental study of atomic hydrogen has been very difficult due to its high recombination rate, but it finally led to its Bose-Einstein condensate state in 1998. Degeneracy has also been observed in thin films of hydrogen adsorbed on the 4He surface allowing for the possibility of observing the Berezinskii–Kosterlitz–Thouless superfluid transition.
The hydrogen atom is the simplest of all atoms consisting of proton of charge e and the electron of charge -e. It is a two particle system. A two-body problem can always be reduced to a one-body problem by choosing the centre of mass coordinate and the relative coordinates. The existence of degenerate energy eigenvalues implies that linear combination of the corresponding eigenfunctions are solutions of the wave equation with the same energy. The isotopes of atomic hydrogen, deuterium and tritium also provide examples of hydrogenic systems. Since the mass of deuterium is md˜2mp and that of tritium is mt˜3mp, the reduced masses of them will be slightly different from that of hydrogen. In addition to deuterium and tritium, there exists other hydrogenic systems in which the role of proton is played by another positive particle.
The ground-state properties of one-dimensional electron-spin-polarized hydrogen 1 H, deuterium 2 H, and tritium 3 H are obtained by means of quantum Monte Carlo methods. The equations of state of the three isotopes are calculated for a wide range of linear densities. The pair correlation function and the static structure factor are obtained and interpreted within the framework of the Luttinger liquid theory. We report the density dependence of the Luttinger parameter and use it to identify different physical regimes: Bogoliubov Bose gas, super-Tonks–Girardeau gas, and quasi-crystal regimes for bosons; repulsive, attractive Fermi gas, and quasi-crystal regimes for fermions. We find that the tritium isotope is the one with the richest behavior. Our results show unambiguously the relevant role of the isotope mass in the properties of this quantum system.
Systems of spin-aligned hydrogen isotopes are studied. They are shown to exhibit even more extreme "quantum" behavior than the helium isotopes. Spin-aligned hydrogen is predicted to be a gas at all temperatures and its Bose-Einstein condensation and possible superfluidity are discussed. Spin-aligned deuterium is predicted to show critical behavior strongly influenced by quantum mechanics. The preparation of spin-aligned hydrogen (in the presence of large magnetic fields) is also discussed.
Motivated by emerging experimental possibilities to confine atoms and molecules in quasi-one-dimensional geometries, we analyze ground-state properties of strictly one-dimensional molecular matter comprised of identical particles of mass m. Such a class of systems can be described by an additive two-body potential whose functional form is common to all substances which only differ in the energy $\ensuremath{\epsilon}$ and range l scales of the potential. With this choice De Boer's quantum theorem of corresponding states holds and the ground-state properties expressed in appropriate reduced form are only determined by the dimensionless parameter ${\ensuremath{\lambda}}_{0}^{2}\ensuremath{\sim}{\ensuremath{\Elzxh}}^{2}{/ml}^{2}\ensuremath{\epsilon},$ measuring the strength of zero-point motion in the system. The presence of a minimum in the two-body interaction potential leads to a many-body bound state which is a Luttinger liquid stable for not very large ${\ensuremath{\lambda}}_{0}.$ As ${\ensuremath{\lambda}}_{0}$ increases, the asymmetry of the two-body potential causes quantum expansion, softening, and eventual evaporation of the Luttinger liquid into a gas phase. Selecting the pair interaction potential in the Morse form we analytically compute the properties of the Luttinger liquid and its range of existence. We find that as ${\ensuremath{\lambda}}_{0}$ increases, the system first undergoes a discontinuous evaporation transition into a diatomic gas followed by a continuous dissociation transition into a monoatomic gas. In particular we find that spin-polarized isotopes of hydrogen and 3He{}^{3}\mathrm{He}3He are monoatomic gases, 4He{}^{4}\mathrm{He}4He is a diatomic gas, while molecular hydrogen and heavier substances are Luttinger liquids. We also investigate the effect of finite pressure on the properties of the liquid- and monoatomic gas phases. In particular we estimate a pressure at which molecular hydrogen undergoes an inverse Peierls transition into a metallic state which is a one-dimensional analog of the transition predicted by Wigner and Huntington in 1935 [E. Wigner and H.B. Huntington, J. Chem Phys. 3, 764 (1935)].