$$\def\bG{{\bf G}}\def\br{{\bf r}}\def\bq{{\bf q}}$$


One of my current research topics (in the Physics Department of the Technion Israel Institute of Technology, in collaboration with Daniel Podolsky and Snir Gazit) deals with solid helium-4. We intend to identify and characterize the vibrational excitations (the phonons) in the solid phase with the help of a phenomenological model, that is valid close to the liquid-solid phase transition. We mainly want to give an explanation to the presence of optical phonons in solid helium, that are not well understood up to now.

Introduction : helium

Helium is the lightest element after hydrogen. There are two stable isotopes: helium-3 and helium-4, the most commonly found. They have very different behaviors: helium-4 is a boson while helium-3 is a fermion. Unless stated otherwise, we are dealing here with helium-4.

Figure 1 shows the phase diagram of helium-4.

Phase diagram of helium
Figure 1 : Phase diagram of helium-4
Helium is the only known compound that remains liquid all the way to absolute zero temperature (0K = -275.15°C); there is no triple point (solid-liquid-gas point) in the helium phase diagram. All other substances, including hydrogen, are solid at least up to temperatures of a few kelvins as shown on the typical phase diagram (depending on the specific substance, the solid phase may consist in several different crystallographic phases).
A typical phase diagram
Figure 2 : A typical phase diagram (water, air, oxygen, etc.)

Solid helium-4

However, solid helium can be observed at low temperature and high pressure higher than 2.5 MPa (= 25 bar \(\sim\) 25 atmospheres). Solid helium has a hexagonal close packing structure, except in a small part of the phase diagram, where it shows a body-centered cubic structure, as shown on figure 1.

From fluid to solid state?

Ideal gas: when the temperature is high enough and the pressure low enough (dilute gas), the thermal energy dominates and one can neglect other phenomena (such as interactions, the dimensions of the gas components, etc.). This is the ideal gas model: a gas of free particles (i.e. non interacting particles, apart from the elastic collisions) whose dimensions are negligible compared to the average inter-particle distance.

Classical kinetic energy: according to classical mechanics, the motion of the particles of an ideal gas is directly related to the temperature (equipartition theorem): $$\begin{equation} \frac{1}{2}m v^2=\frac{3}{2}k_B T \label{classkinenergy} \end{equation}$$ where \(k_B\) is the Boltzmann constant. Intuitively, as one lowers the temperature, the atoms progressively slow down and the total kinetic energy decreases.

Interactions: when the temperature decreases, or when pressure increases, other effects than thermal motion become important. In particular, the inter-particle interaction energy becomes comparable to the kinetic energy (\ref{classkinenergy}) that lowers with temperature.

Hard sphere model: When the fluid is not dilute anymore (if the pressure increases), the ideal gas model is not valid: one must take the dimensions of the particles into account. In first approximation, one can consider them as hard (impenetrable) spheres of radius \(r_0\). Figure 3 shows a typical configuration for a set of such hard spheres. This famous simple model allows an intuitive approach of the problem and gives good quantitative results.

Hard sphere model
Figure 3 : typical configuration for a set of hard spheres [Chaikin, Lubensky, 1995]

Correlations: Let us consider an atom at the center of the sample (the black sphere of figure 3); the radial distribution \(g(r)\) is defined as the number of spheres whose center lies inside the ring defined by the two circles of radius \(r - dr\) and \(r + dr\) (the average radius is this \(r\) and the "width" \(2dr\)). Consider a ring of average radius (2r_0\) - twice the sphere radius - it is the closest "shell" to the center; if the fluid is dense enough, as on the figure, the shell contains approximately six atoms (or twelve, in 3D). On the other hand, in a larger ring (\(\sim 3r_0\)), there won't be any sphere due to the exclusion from the inner shell. That is what we call correlations: the presence of a sphere at a given location strongly influences its surroundings.

Away from the black sphere, the correlations quickly decrease. At a large distance, the way the spheres organize is not influenced by the black sphere anymore but only by the closest spheres; the neighbor thus "screen" the influence of remote spheres. Figure 4 shows those correlations; one can observe the \(2r_0\) peak at about \(3.8 Å\) that precisely corresponds to twice the van der Waals radius for argon \(2*1.88 Å\)). When the pressure (and thus the density) increases, one gets closer to the liquid-solid transition, and the peak becomes more and more pronounced. The same phenomenon occurs when the temperature is lowered, although the hard sphere model does not account very well for thermal effects.

Radial distribution for liquid argon
Figure 4: the radial distribution \(g(r)\) for liquid argon. This curve is the result of Monte Carlo simulations based on a model that includes additional corrections (three-body terms, quantum corrections, etc.) [J.A. Barker et al. (IBM, San Jose, California, Diffusion Research Unit, Australian National University, Canberra, 1971)]. The circles correspond to experimental measurements on liquid argon [J.A. Yarnell et al. (Los Alamos laboratory, New Mexico and IBM New York, 1973)].

Transition: The interactions combined with the effects of finite dimensions drive the transition from the liquid to the solid state. The presence of the peak in the radial distribution shows the fluid tendency to periodicity, that becomes more and more pronounced as one approaches the transition: the fluid develops an instability towards the solid phase.

Symmetries: from the point of view of symmetries, the liquid phase and gaseous phase are identical. One can differentiate between the two only when they coexist and one observes the meniscus resulting from the difference in their refraction indices, that separated a denser liquid phase from a less dense gaseous phase. Hence one rather speaks of fluid, when one cannot distinguish between the two states. Fluids are homogeneous and isotropic; the set of translation, reflection or rotation operation (euclidean group) leave the fluid unchanged. The local density \(\rho(x,y,z)\) for instance, is uniform and equal to the average density of the system (total number of atoms divided by the total volume): $$\begin{equation} \rho(x_1,y_1,z_1) = \rho(x_2,y_2,z_2)= \rho_0 \label{translationsymmetry} \end{equation}$$
On the other hand, solids are very different. The symmetric group that characterizes a solid phase is a lot smaller than the group of a fluid. There is a symmetry breaking when one crosses the transition from the fluid to the solid (the whole concept of symmetry - along with the phase transition and the associated symmetry breaking - is one of the cornerstones of modern physics). The translation symmetry for example is broken: indeed, the equation (\ref{translationsymmetry}) isn't valid anymore, but only its discrete version: $$\begin{equation} \rho(x,y,z) = \rho(x+a_x,y+a_y,z+a_z) \label{translationsymmetry2} \end{equation}$$ where \(a_x,a_y,a_z\) are characteristic constants of the solid; for a cubic lattice (see figure 5), \(a_x=a_y=a_z=a_0\).

Solid helium

The phase that we are interested in here is the body-centered cubic phase which occurs close to the liquid-solid transition (see figure 1). The corresponding symmetry group is \(O_h\) and it consists of 48 symmetry operations.

BCC Lattice
Figure 5: the body-centered cubic conventional cell which corresponds to the crystallographic configuration of solid helium close to the transition (see figure 1). The atoms are shown is black; the lines are just a guide for the eyes.

Motion of atoms in the solid phase: the phonons

In the solid phase, the atoms are not completely frozen: they have a fixed equilibrium position, or average position, but they can move about this equilibrium position, partly because of temperature. Due to (repulsive) interactions with its neighbors, an atom will see a potential well. For a typical solid, the equilibrium position corresponds to the center of the well and its lower point. The atom can thus oscillate around this position, but the equilibrium position is the most favorable in energy (see the dash gray curve on figure 6).
Solid helium is very peculiar since the potential well has two minima on either side of the equilibrium position, as shown by the blue curve on figure 6. Since the minima do not coincide with the equilibrium position, it is easier, energetically speaking, for the helium atom to explore the surroundings of its equilibrium position.

Potential well for the helium atom
Figure 6: Potential well for an atom in a solid (along a given direction): the equilibrium position is at the center, at \(x=0\); the dash gray curve shows the typical case, while the blue curve the potential for an atom of solid helium. [After Glyde (1994)]

Zero point energy or vacuum energy:

There is an additional effect to take into account to understand solid helium. According to classical mechanics and equation (\ref{classkinenergy}), as one gets closer to the absolute zero (\(T=0 K=-273.15°C\)), the atoms must move less and less and finally end up completely frozen when the system reaches zero energy. However, one of the fundamental principles of quantum mechanics, the Heisenberg uncertainty principle, states that a physical system must have a zero point energy, or vacuum energy, which is higher than the classical energy minimum. (According to the Heisenberg principle, it is impossible to specify with absolute precision both position and velocity of a system; thus a frozen system is ruled out.) The zero point energy is the total energy of the system minus its thermal energy - equation (\ref{classkinenergy}). This implies a motion of the system, even at zero temperature.

This is precisely the reason why helium does not freeze at atmospheric pressure: its zero-point energy is too high for that, and the atoms motion is too large to allow for solidification. One must increase the pressure, thus the density, in order to reach the solid phase. And even in the solid phase, the fluctuations around the equilibrium positions are enormous: of the order of 30% away from the equilibrium position - to be compared to the 10% of the Lindemann criterion for solid fusion (see Polturak et Gov, 2003). This number of 30% strongly disagrees with the usual crystallographic standards and shows how much solid helium is an unusual material, for which one must adapt the models and think differently.

Open questions and motivations

In the bcc phase, pure solid helium-4 has only one atom per elementary cell. In principle, standard approaches allow to immediately conclude that the solid has three acoustic phonons (energetic excitation mode with no energy gap) and no optical phonon (gapped energetic excitation).

The observation of optical modes in neutron scattering experiments on solid helium-4 (see Gov 1999 and Markovich 2002) contradicts our common understanding of this phase.

Despite several attempts to explain the presence of these phonons, the question remains open. Our work suggests an explanation based on a mechanism that can take place in solids where the quantum fluctuations are important, which is the case for helium-4. Our approach rests on a phenomenological model based on the notion of density wave. In this sense, the optical phonons would be the analogues of Higgs modes in density wave systems.

Our approach is also relevant to density wave systems such as the dichalcolgenide compound 2H-NbSe\(_2\) (see below).

Density waves:

We just saw that in the solid phase, the atoms can move around their equilibrium position: they are not fixed at one point. Hence, instead of seeing a solid as a set of atoms fixed at specific location, one can think of it as a dense environment where some points (the lattice points) gather more matter than other. By choosing this point of view, on can describe the solid by defining the density function \(\rho(\br)\) that gives the value of the density of matter at point \(\br\).

Moreover, the solid is periodic in space, as stated by equation (\ref{translationsymmetry2}). Now, any periodic function of position \(\bf{r}\), like the density \(\rho(\br)\), can be decomposed into Fourier series: $$\begin{equation} \rho(\br)= \rho_0 + \sum_{\bG_i} \rho_{i} e^{i \bG_i \cdot \br} \label{CDWFourierGS} \end{equation}$$ where the coefficients \(\rho_{i}\) are real and \(\rho_0\) is the average density of the system (the total number of atoms divided by the total volume). The vectors \(\bG_i\) are called the reciprocal lattice vectors; they account for the periodicity of the system. In a pure solid where all atoms are identical, the coefficients \(\rho_{i}\) must all be equal, independent of the reciprocal vector: \(\rho_{i}=\bar\rho\), \(\forall i\).

In one dimension: One intuitively understands that by considering a one-dimensional lattice with constant \(a_0\): on the \(x\)-axis, an atom sits at every \(a_0\). The system density is thus: $$\begin{equation} \rho(x)= \bar\rho e^{i\frac{2 \pi}{a_0} x} + \bar\rho e^{-i\frac{2 \pi}{a_0}x} =2 \bar\rho \cos \left(\frac{2 \pi}{a_0} x\right) \label{CDW1D} \end{equation}$$ Here, there are two opposite reciprocal vectors: \(\bG_\pm=\pm 2 \pi / a_0 \hat x\) (where \(\hat x\) is the unit vector along the \(x\)-axis). The density is periodic with period \(a_0\) as shown on figure 7.

Figure 7: A one-dimensional charge density wave models a one-dimensional lattice of atoms with constant \(a_0\); the atoms mark the density peaks.

Ginzburg-Landau theory: the ground state

When examining the symmetries, we said above that the density is uniform on the liquid side of the transition \(\rho(\br)=\rho_0\). On the solid side, it is periodic and can be described by equation (\ref{CDWFourierGS}), with \(\rho_{i}=\bar\rho\), \(\forall i\) for a pure solid like helium-4.

Following a proposal by Alexander and McTague, we describe the liquid-solid transition with a Ginzburg-Landau energy: $$\begin{equation} F=\int d{\bf r}_1d{\bf r}_2 \rho({\bf r}_1) \chi_0^{-1}({\bf r}_1,{\bf r}_2)\rho({\bf r}_2) -B\int d{\bf r} \rho({\bf r})^3 +C\int d{\bf r} \rho({\bf r})^4 \label{GinzburgLandau} \end{equation}$$ with : $$\begin{equation} \chi_0^{-1}({\bf r}_1,{\bf r}_2)=\left[r+c (k_0^2+\nabla^2)^2\right] \delta({\bf r}_1-{\bf r}_2) \end{equation}$$ and: \(r=a(T-T^*)\) is the parameter that drives the transition; \(T^*\) is the mean field transition temperature, and \(a>0\).

This phenomenological model allows to account for the fluid instability towards the solid phase and for the character of the transition (first order); the density is given by: $$\begin{equation} \rho(\br)= \rho_0 + \bar\rho\sum_{\bG_i} e^{i \bG_i \cdot \br} \label{rhoGLGS} \end{equation}$$ with \(\bar\rho=0\) on the liquid side of the transition and \(\bar\rho\neq 0\) on the solid side; \(\bar\rho\) is the order parameter of the transition.

On the solid side, we can ascertain what the most favorable crystallographic configuration is by identifying which reciprocal vectors \(\bG_i\) minimize the energy (\ref{GinzburgLandau}).

In three dimensions:

The most favorable configuration is that on the body-centered cubic lattice (bcc), that indeed corresponds to the solid helium close to the transition (figure 1). There are twelve reciprocal vectors forming an octahedron shown on figure 8.

Figure 8: The set of the reciprocal vectors of the bcc lattice. Their origin is the center of the octahedron and their tips are at the middle of the edges. We label six of the twelve vectors with a positive index \(\bG_{i}\), \(i=1,\ldots,6\); the six other are their opposite and are denoted as \(\bG_{-i}\).

Now that the reciprocal vectors have been identified, the value of the order parameter \(\bar\rho\) corresponds to the absolute minimum in energy and can be computed as a function of the phenomenological parameters \(r,B,C\). In the liquid phase, the minimum is \(\bar\rho=0\) (the black curve at \(r=0.2\) on figure 9, where one observes an inflexion point at \(\bar\rho\sim 0.2\) that indicates the apparition of the instability towards the solid phase). In the solid phase, the energy minimum is at \(\bar\rho\neq 0\) (the red curve at \(r=-0.2\) on figure 9). At the transition, the two minima are degenerate (blue curve, \(r=0\)).

Ginzburg-Landau energy
Figure 9: Ginzburg-Landau energy (\ref{GinzburgLandau}) in the ground state (no phonon) as a function of the order parameter \(\bar\rho\) for \(B=5\) and \(C=1\).

In two dimensions:

The configuration that minimizes the energy has six reciprocal vectors organized on a hexagon.

Figure 10: The set of six reciprocal vectors of the two-dimensional case: the vectors' origin is at the center of the hexagon and their tips are at the corners.
The set of six reciprocal vectors (\(\bG_1,\bG_2,\bG_3\) and their three opposite) form three density wave, each described by equation (\ref{CDW1D}). Their superposition according to equation (\ref{rhoGLGS}) produces a triangular lattice; figure 11 shows the density function \(\rho(\br)\): the white zones correspond to maximum density (where an atom sits) and the dark blue zones are the lowest density zones (vacuum).
Triangular lattice in the 2D case
Figure 11: The triangular lattice that minimizes the Ginzburg-Landau energy in two dimensions.

Density waves in two dimensions are not pure theoretical objects: the phenomenon has for instance been observed by scanning tunneling microscopy (STM) in the dichalcolgenide compound 2H-NbSe\(_2\) [Giambattista et al. (1988)]. The recent experiments by Jennifer E. Hoffmann's team at Harvard show the wealth of such density wave phenomena; the technical progress in imaging (see figure 12) allow for a better understanding.

Density wave in 2H-NbSe2 observed by STM
Figure 12: atomically resolved scanning tunneling microscopy (STM) topographies of doped NbSe\(_2\). The temperature is lower than the transition temperature to the charge density wave state (33K). Jennifer E. Hoffmann's team, Harvard University.

Ginzburg-Landau theory: the phonons

Model (\ref{GinzburgLandau}) also allow to consider phonons and characterize them. In order to take them into account, the density function must be modified in the following way: $$\begin{equation} \rho(\br)=\rho_0+\sum_i \int_{\bq}\left[ \bar\rho+\delta_i(\bq) e^{i(\bq \cdot \br-\omega(\bq) t)}\right] e^{i \bG_i \cdot \br} \end{equation} $$ Here: \(\delta_i(\bq)\) represents the possible amplitude variations of the density waves, and \(e^{i\bq \cdot \br}\) the modulations of the wave vectors - or reciprocal vectors - \(\bG_i\) (expansion or contraction of the wave) ; the exponential in \(\omega(\bq) t\) accounts for the (time) oscillation of the phonon; \(\omega(\bq)\) is the phonon energy. The phonon is thus characterized by its vector \(\bq\) and the amplitudes \(\delta_i\).

The phonons must be studied as low energy excitations. The amplitudes \(\delta_i\) are of course small compared to \(\bar\rho\). In order to examine and characterize the phonons, one can simply linearize the Euler-Lagrange equation obtained from the Ginzburg-Landau energy, with respect to these amplitudes. The linearized Euler-Lagrange equations translate into a square matrix of dimensions the total number of reciprocal vectors \(\bG_i\) (12 by 12 in 3D ; 6 by 6 in 2D). This matrix can be diagonalized, and even analytically with Mathematica for instance, at least for \(\bq=0\), i.e. at low energy. This enables us to identify all the phonons allowed by the model (\ref{GinzburgLandau}) and that can appear in the system.

In two dimensions:

In this case, as one can anticipate beforehand, one finds two acoustic phonons, respectively oscillating along \(x\) and \(y\). Those have complex amplitudes \(\delta_i\), whose sign and modulus are shown on figure 13. These acoustic phonons have zero energy at \(\bq=0\). Furthermore, there are four less intuitive optical phonons; we tag two of them as "breather" and "anti-breather". These phonons have real amplitudes \(\delta_i\). The "breather" corresponds to the amplitude modulation of the three density wave in phase: at each lattice site, the atoms concentrate and then spread all in phase (the amplitudes \(\delta_i\) are all equal - see figure 13), whence the term "breather" by analogy with respiration. As for the anti-breather, the atoms of the two sublattices "breath" out of phase (the amplitudes \(\delta_i\) have opposite sign). Lastly, there remain two optical phonons (a and b) with different amplitude modulations (see figure 13).

Phonons in two dimensions
Figure 13: The six phonons obtained from the Ginzburg-Landau energy (\ref{GinzburgLandau}) in two dimensions: there are two acoustic phonons and four optical ones.

In three dimensions

One finds the three acoustic phonons oscillating along the three spatial directions (see figure 14) and with zero energy for \(\bq=0\); we also find nine optical phonons with non zero energy, including a "breather".

Phonons in three dimensions
Figure 14: The twelve phonons obtained from the Ginzburg-Landau energy (\ref{GinzburgLandau}) in three dimensions: there are three acoustic phonons and nine optical ones.

Ongoing research

The description of solid helium (2D and 3D) with the help of the Ginzburg-Landau model (\ref{GinzburgLandau}) in the vicinity of the liquid-solid transition allow to consider phonons as density wave modulations. This lead us to unveil many optical phonons that the usual common description (in terms of atoms motion) fails to reveal.

Now, we have to characterize them further: symmetries, fundamental representation, lifetime. We then must determine to which experimental probe they may couple: neutron inelastic scattering or Raman spectroscopy for example.

We also hope to be able to complete our approach and analysis with Monte-Carlo simulations.