Phy5670/QLG

Introduction

A lattice model first proposed in 1956 has been introduced to explain the peculiar properties and phase transitions in liquid helium.In this model, the liquid is regarded as a lattice composed of atoms and holes, where each atom can migrate by exchanging its position with an adjacent hole.The kinetic and potential energies are defined in terms of creation and annihilation operators for the atom sin each lattice site such that they boil down to the appropriate form in the limit of vanishing lattice spacing.An analogy has been drawn between the lattice liquid and a system of vector spins subjected to an external magnetic field, with identification of corresponding physical quantities.Finally,the quantum lattice gas model has been implemented to describe essential aspects of the motion of $^{4}{\textrm {He}}$ atoms and of $^{3}{\textrm {He}}$ impurities in solid $^{4}{\textrm {He}}$.The study suggested the affinity of $^{3}{\textrm {He}}$ impurities to bind to defects and promote $^{4}{\textrm {He}}$ atoms to interstitial sites which can turn the bosonic quantum disordered crystal into a metastable supersolid.It is further proposed that such defects can form a glass phase during the $^{4}{\textrm {He}}$ solid growth by rapid cooling.

The Lattice Gas Model

The lattice model for classical liquid is well known for providing an adequate approximation for the purpose of taking into consideration the large atomic correlation in liquid phase.Each atom is assumed to occupy only discrete lattice points and the configurations of liquid are described by the distributions of atoms and holes among the lattice points.By choosing a proper magnitude of lattice constant, and excluding the multiple occupation of atoms on each lattice point, one can account for the effect of the strong repulsion between atoms. In extending this idea of lattice model to quantum liquid such as liquid helium, it is of importance to take into account of the effect of the zero point motion of the atoms.Since, considerable increment in kinetic energy will be caused by localizing each atom in the lattice configuration, it must be reduced by mixing various configurations.Thus, in the lattice representation on which our lattice model rests, the kinetic energy has large non-diagonal elements such as to produce transitions among various lattice configurations.Taking consideration of this effect of the kinetic energy on one hand, and of the excluding effect of the strong repulsion between atoms on the other hand, we shall construct a lattice model for liquid helium. For simplicity, we assume a simple cubic lattice of lattice constant d. Adopting the scheme of second quantization, we define operators ${\mathit {a_{i}^{\ast }}}$ and ${\mathit {a_{i}^{}}}$ which creates and annihilates an atom at the i-th lattice point. We assume the commutation relations,

$[{\mathit {a_{i}^{\ast }}},{\mathit {a_{j}^{\ast }}}]_{-}=[{\mathit {a_{i}}},{\mathit {a_{j}}}]_{-}=[{\mathit {a_{i}}},{\mathit {a_{j}^{\ast }}}]_{-}=0\;\;for\;\;i\neq j$

$[{\mathit {a_{i}^{\ast }}},{\mathit {a_{i}^{\ast }}}]_{+}=[{\mathit {a_{i}}},{\mathit {a_{i}}}]_{+}=0,[{\mathit {a_{i}}},{\mathit {a_{i}^{\ast }}}]_{+}=1\;\;for\;\;i=j$ In other words, the operators with different lattice indices are commutable to each other and with respect to the same lattice point they Fermionic character. Now, we set up the potential energy of the lattice liquid.We consider the potential energy ${\mathit {v}}$ betwwen two atoms so that

${\mathit {v}}=\infty$ for two atoms in same lattice points.

${\mathit {v}}=-{\mathit {v_{o}}}$ otherwise.

Then, the total potential energy of the system $\phi =-{\mathit {v_{o}}}\sum _{<ij>}{\mathit {a_{i}^{\ast }}}{\mathit {a_{i}}}{\mathit {a_{j}^{\ast }}}{\mathit {a_{j}}}$ where $<ij>$ means summation over nearest neighbouring pair points.

As to the kinetic energy, we assume that each atom can make transition only to one of the nearest neighbour sites when it is vacant.So, the proposed kinetic energy of the system $K=({\frac {\hbar ^{2}}{2md^{2}}})\sum _{<ij>}({\mathit {a_{i}^{\ast }}}-{\mathit {a_{j}^{\ast }}})({\mathit {a_{i}}}-{\mathit {a_{j}}})$ The total number of atoms is given by $\sum _{i}{\mathit {a_{i}^{\ast }}}{\mathit {a_{i}}}=N_{0}$ Making Fourier transforms of ${\mathit {a_{i}^{\ast }}}$ and ${\mathit {a_{i}}}$ as

${\mathit {a_{\mathbf {k} }^{\ast }}}=N^{-{\frac {1}{2}}}\sum _{j}e^{(-i\mathbf {kr} _{j})}{\mathit {a_{j}^{\ast }}}$ ${\mathit {a_{\mathbf {k} }}}=N^{-{\frac {1}{2}}}\sum _{j}e^{(i\mathbf {kr} _{j})}{\mathit {a_{j}}}$

Then, the kinetic energy can be written in the form

$K\cong \sum _{\mathbf {k} }({\frac {\hbar ^{2}\mathbf {k^{2}} }{2m}}){\mathit {a_{\mathbf {k} }^{\ast }}}{\mathit {a_{\mathbf {k} }}}$ provided $kd\ll 1$. Putting together both the kinetic energy and potential energy terms, the total Hamiltonian for the lattice liquid

<math>H_{L}=(\frac{\hbar^{2}}{2md^{2}})\sum_{<ij>}(\mathit{a_{i}^{\ast }}-\mathit{a_{j}^{\ast }})(\mathit{a_{i}}-\mathit{a_{j}})-\mathit{v_{o}}\sum_{<ij>}\mathit{a_{i}^{\ast }}\mathit{a_{i}}\mathit{a_{j}^{\ast }}\mathit{a_{j}}<math>