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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{4}\textrm{He}} atoms and of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{3}\textrm{He}} impurities in solid Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{4}\textrm{He}} .The study suggested the affinity of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{3}\textrm{He}} impurities to bind to defects and promote Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^{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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{a_{i}^{\ast }}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{a_{i}^{}}} which creates and annihilates an atom at the i-th lattice point. We assume the commutation relations,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\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}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{v}} betwwen two atoms so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{v}=\infty} for two atoms in same lattice points.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{v}=-\mathit{v_{o}}} otherwise.

Then, the total potential energy of the system Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi =-\mathit{v_{o}}\sum_{<ij>}\mathit{a_{i}^{\ast }}\mathit{a_{i}}\mathit{a_{j}^{\ast }}\mathit{a_{j}}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{i}\mathit{a_{i}^{\ast }}\mathit{a_{i}}=N_{0}} Making Fourier transforms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{a_{i}^{\ast }}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{a_{i}}} as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{a_{\mathbf{k}}^{\ast }}=N^{-\frac{1}{2}}\sum_{j}e^{(-i\mathbf{kr}_{j})}\mathit{a_{j}^{\ast }}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K\cong \sum_{\mathbf{k}}(\frac{\hbar^{2}\mathbf{k^{2}}}{2m})\mathit{a_{\mathbf{k}}^{\ast }}\mathit{a_{\mathbf{k}}}} provided Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle kd\ll 1} . Putting together both the kinetic energy and potential energy terms, the total Hamiltonian for the lattice liquid

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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}}}

Information about the thermodynamic properties of our lattice liquid can be obtained by constructing the Grand canonical partition function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Xi_{L}=Tr[e^{(-\alpha\sum_{i}\mathit{a_{i}^{\ast }}\mathit{a_{i}}-\beta H_{L})}]}

Thus, defining the characteristic function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{\psi }= log\; \Xi _{L}} ,

the number of atoms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{0}=-(\frac{\partial\bar{\psi } }{\partial \alpha })_{\beta }}

internal energy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = -(\frac{\partial\bar{\psi } }{\partial \beta })_{\alpha }}

pressure Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle pV/kT = \bar{\psi }} and so on.

Spin System eqivalent to the lattice model

Now we consider a set of N spins each of which is localized at each lattice point and has a magnitude of 1/2. Then, the spin operators, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{j\pm }= S_{jx}\pm iS_{jy}} satisfy similar commutation relations:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [S_{i+},S_{j+}]_{-}= [S_{i-},S_{j-}]_{-}=[S_{i-},S_{j+}]_{-}=0\; \; (i\neq j)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [S_{i+},S_{i+}]_{+}= [S_{i-},S_{i-}]_{+}=0,\; \; [S_{i-},S_{i+}]_{+}=1\; \; (i=j)}

Besides, the z component of spin Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{iz}} can be expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{iz}=S_{i+}S_{i-}-\frac{1}{2}} .So, there should exist a system of spins equivalent to the lattice liquid with the lattice Hamiltonian by the correspondence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{a_{i}^{\ast }}\leftrightarrow S_{i+}, \mathit{a_{i}}\leftrightarrow S_{i-}} .

We consider the spin system whose Hamiltonian is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{F}=-J\sum_{<ij>}{(S_{ix}S_{jx}+S_{iy}S_{jy})}-J^{'}\sum_{<ij>}{S_{iz}S_{jz}}-H\sum_{i}{S_{iz}}}

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{F}} represents the energy of spin which interact with an anisotropic exchange coupling and are subjected to an external magnetic field. Using the commutation relations, the Hamiltonian can easily be written as: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{F}=\frac{J}{2}\sum_{<ij>}(S_{i+}-S_{j+})(S_{i-}-S_{j-})-J^{'}\sum_{<ij>}S_{i+}S_{i-}S_{j+}S_{j-}-(H+\frac{z}{2}J-\frac{z}{2}J^{'}){\sum_{i}S_{i+}S_{i-}}+\frac{N}{2}(H-\frac{1}{4}zJ^{'})} where z is the number of nearest neighbours.Now the partition function of this spin system is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Xi _{F}=Tr[e^{(-\beta H_{F})}]}

Then comparing with our lattice gas model, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Xi _{L}=C\Xi _{F}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\hbar^{2}}{2md^{2}}=\frac{J}{2}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathit{v_{o}}=J^{'}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha =-\beta[ {H+(\frac{z}{2})J-(\frac{z}{2})J^{'}}]\; ,\; C=e^{[(\frac{N}{2})\beta [H-(\frac{z}{4})J^{'}]]}}

Therefore, if we know the thermodynamic properties of our spin system, we can presume the behaviours of our lattice liquid by simple translation, and vice versa.

The relation between the physical quantities of both systems can be found straightforwardly. The density of the lattice liquid is connected with the magnitude of the magnetization of the spin systemalong z axis.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_{0}=-\frac{\partial (log \Xi _{L})}{\partial \alpha }}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{\partial(logC+log\Xi _{F})}{\partial (\beta H)}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{N}{2}+<\sum S_{iz}>}

Denoting the density of lattice liquid by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{N_{0}}{N}=\rho } and the mean value of the z component of spin by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <S_{z}>=\zeta } , then it holds that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\rho =1+2\zeta } It is natural to assume Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\rho =1+2\zeta } for liquid phase and so we may confine ourselves to the case $\zeta >0$.The chemical potential of the lattice liquid $\mu =-{\frac {\alpha }{\beta }}$ corresponds to the external magnetic field apart from an adiditive constant.The pressure of the lattice liquid is identified with a linear combination of free energy per one spin f and external magnetic field:

$pd^{3}=-f+{\frac {1}{2}}(H-{\frac {z}{4}}J^{'})$

where $f=-({\frac {1}{\beta N}})log\Xi _{F}$ and the volume $V=Nd^{3}$

The kinetic energy of the lattice liquid corresponds to the exchange energy of spins which prefer to direct perpendicularly to the z axis, if externel magnetic field is absent, on account of anisotropic exchange coupling.In other words, to maximize the negative of exchange energy by making as many spins as possible point in one direction in the xy plane corresponds to minimize the kinetic energy of the lattice liquid by making the localization of each atom as small as possible. And when the spin system magnetizes spontaneously at sufficiently low temperatures, there appears a long range order by which the direction of any two spins, however distant, are correlated to each other.Corresponding to this phenomena of long range order in ferromagnets, there would appear in the lattice liquid a kind of 'long range order of momentum', which might be responsible for superfluidity in helium.

Phase Transition in Helium

In the preceding section, we found that there exists an intimate connection between the thermodynamical properties of the system of ferromagnets and that of the lattice liquid. Therefore, one can expect that the appearance of ferromagnetic ordering in one system will be related to the occurrence of the "--transition" or of superfluidity in the other. In order to examine thus relation, it is more convenient to start with the ferromagnetic system because the physical phenomenon

Possible Role of $^{3}{\textrm {He}}$ impurities in solid $^{4}{\textrm {He}}$

In this section, we discuss the role of very low density of impurities in a bosonic hard core solid.Our model that describes the impurity motion in an otherwise ideal quantum bosonic crystal, maps to a quantum spin model with antiferromagnetic coupling and antiferromagnetic order in one direction and ferromagnetic coupling in the perpendicular direction with impurities moving through the lattice.The impurity motion between sub-lattices couples to quantum fluctuations of these pseudo-spin degreesof freedom which correspond to the boson hopping from an occupied site of the solid to an empty interstitial site.We find that for the limit which describes the case of solid $^{4}{\textrm {He}}$, interstitial atoms are well-defined delocalized excitations. Let us consider a lattice gas model to describe the bosonic solid and the added impurities.In such a model,we need to consider the interstitial sites as part of the lattice and, thus, the ideal quantum solid containing no vacancies and no impurities, corresponds to a fractionally occupied lattice.For example, the ideal triangular solid corresponds to the case of 1/3 filling, namely to a ${\sqrt {3}}\times {\sqrt {3}}$ ordered solid and the ideal square lattice solid corresponds to the ${\sqrt {2}}\times {\sqrt {2}}$ checkerboard solid, i.e., 1/2 filling of the lattice with bosons. Our model Hamiltonian describing a bosonic quantum solid (such as solid $^{4}{\textrm {He}}$ ) and a small concentration of impurities (such as $^{3}{\textrm {He}}$ atoms) may be written as follows:

${\hat {H}}={\hat {H}}_{B}+{\hat {H}}^{\ast }+{\hat {H}}_{int}-\mu \sum _{i}{\hat {N}}_{i}-\mu ^{\ast }\sum _{i}{\hat {n}}_{i}$