Lecture 1

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Crήstos Panagòpoulos

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Matter is the substance of which all physical objects are composed. The density of matter is a measure of the composition of matter and the compactness of the constituent entities in it. Dense matter physics is the study of the physical properties of material substance compressed to high density. The density range begins with hundreds of grams per cubic centimeter and extends to values ten to fifteen orders of magnitudes higher.

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Condensed matter physics is concerned with the "condensed" phases that appear whenever the number of constituents in a system is extremely large and the interactions between the constituents are strong

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“… at each new level of complexity, entirely new properties appear, and the understanding of this behavior requires research which I think is as fundamental in its nature as any other” Philip W. Anderson 1972

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One of the reasons for calling the field "condensed matter physics" is that many of the concepts and techniques developed for studying solids actually apply to fluid systems: For instance, the conduction electrons in a conductor form a type of quantum fluid with essentially the same properties as fluids made up of atoms:

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Under high pressures and low temperatures electrons may condense into a quantum fluid: A quantum fluid can refer to a cluster of valence electrons (the electrons located within the outermost energy level of an atom) moving together after they undergo fermionic condensation (fermions are particles with half-integer spin) 

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Quantum fluids exhibit the remarkable property of remaining liquid at absolute zero temperature and zero pressure. This effect arises from their large zero-point energy and the small inter-atomic forces, both of which prevent the formation of a solid phase. A quantum fluid can also refer to a superfluid (made up of atoms).

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What exactly is a superfluid? As the name suggests, a superfluid possesses fluid properties similar to those possessed by ordinary liquids and gases, such as the lack of a definite shape and the ability to flow in response to applied forces. A superfluid phase is a phase of matter characterised by the complete absence of viscosity - formed by fermionic particles (fermions are particles with half-integer spin) at low temperatures. It is the phase or state of matter in which it loses all its resistance to change its shape: Resistance to changing the shape of an object is its intrinsic property. In liquids, this property is manifested through its stickiness or internal resistance to flow. But superfluid is totally devoid of viscosity.

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Superfluid has another bizarre property. It cannot be made to rotate like water in a pot. - water if stirred with a stick in a container swirls round imaginary axes. Superfluids do not show this property. When stirred, it will create lots of vortices. Strangely though, the superfluid loses it superfluidity at those vortices, whereas it retains its quality elsewhere.

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Imagine that a (tightly sealed!) bucket of superfluid rotates.  A vortex can form in the middle, with fluid moving around in a circle, much like a water vortex in a draining bathtub.  The amazing difference is that, at a given distance from the vortex center, only certain fluid velocities are allowed!   There is a minimum velocity, then twice that minimum, then three times the minimum, etc.  No in-between values can occur, so the vortices are said to be quantized.

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Therefore, in contrast to the example of a glass of water above, the rotation in superfluids is always inhomogeneous. The fluid circulates around quantized vortex lines. The vortex lines are shown as yellow in the figure, and the circulating flow around them is indicated by arrows. There is no vorticity outside of the lines because the velocity near each line is larger than further away. (curl v = 0, where v(r) is the velocity field.)

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The velocity around each vortex line is determined by h/m, where h is the Planck's constant, and m the mass of one atom. The presence of the Planck's constant means that quantized vorticity is a consequence of quantum mechanics. h is very small, but so is m, so the ratio h/m is quite macroscopic. Therefore, superfluidity is a quantum phenomenon on a macroscopic scale. The number of vortex lines depends on the constant h/m. There are approximately 1000 vortex lines in a container of radius 1 cm that is rotating 1 round per minute.

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Where could we find superfluidity? n p p He - 3 He - 4 Helium Helium - 4 atoms are bosons  particles with integer spin. Helium - 3 atoms are fermions  particles with half integer spin.

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As the temperature drops, so does the pressure and/or the volume & the reverse. Hence, we can cool He gas in liquid or reduce its volume. However, reducing its volume is equivalent to applying pressure. Onnes

1911 (-269 C): 

1911 (-269 C) Onnes

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1938 Kapitza and Allen discover superfluidity in He-4

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For T > 2.4Κ (-271 C) When it is heated up it boils like water For T < 2.4Κ Perfect thermal conductor

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For T < 2.4Κ – gravity ... If the bottle containing helium rotates for a while and then stops, helium will continue to rotate for ever – there is no internal friction (for as long as He is at T = -269 C or lower

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Helium-4 atoms are bosons  particles with integer spin. Superfluidity had been found in helium-4 at about 2 degrees kelvin, but because helium-4 has integer spin, it can form a condensed phase without the need for a pairing mechanism: Due to integer-spin, bosons obey Bose–Einstein statistics, one consequence of which is the Bose–Einstein condensation of particles — in such case a number of bosons can share the same quantum state, and their superfluidity can be understood in terms of the Bose statistics (which determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium) that they obey.

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Specifically, the superfluidity of helium-4 can be regarded as a consequence of Bose-Einstein condensation - a phase of matter formed by bosons cooled to temperatures very near to absolute zero where a large fraction of the atoms collapse into the lowest quantum state, at which point quantum effects become apparent on a macroscopic scale in an interacting system ----- just like in Bose-Einstein condensates The primary difference between superfluid helium and a Bose-Einstein condensate is that the former is condensed from a liquid while the latter is condensed from a gas.

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Fermions such as helium-3 follow Fermi-Dirac statistics and should not actually be condensable in the lowest energy state. For this reason superfluidity should not be possible in helium-3 which, like helium-4, can be liquidised at a temperature of some degrees above absolute zero. But fermions can in fact be condensed, but in a more complicated manner. This was proposed in the BCS theory for superconductivity in metals, formulated by John Bardeen, Leon Cooper and Robert Schrieffer (Nobel Prize in Physics 1972). The theory is based on the fact that electrons are fermions (they consist of one particle only, an odd number) and therefore follow Fermi-Dirac statistics just as helium-3 atoms do. But electrons in greatly cooled metals can form pairs and then behave as bosons.

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Because of the analogy with electrons and BCS it was expected that He-3 would also become a superfluid. Although many research groups had worked with the problem for years, particularly during the 1960s, none had succeeded and many considered that it would never be possible to achieve superfluidity in helium-3. Early 1970s

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In 1972 Leggett reported new states where all of the pairs' spins line up spontaneously, like a row of bar magnets.

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1938 Pyotr L. Kapitsa discovered the superfluidity of liquid Helium 4 Nobel Prize in 1978 1941-47 Lev Landau formulated the theory of quantum Bose liquid - 4He superfluid liquid.  1956-58 he further formulated the theory of quantum Fermi liquid. Nobel Prize in 1962 Early 1970s David M. Lee, Douglas D. Osheroff, and Robert C. Richardson discovered the superfluidity of liquid Helium 3. Nobel Prize in 1996 Anthony Leggett first formulated the theory of superfluidity in liquid 3He in 1965. Nobel Prize in 2003

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In fact, the phenomenon of superconductivity, in which the electrons condense into a new fluid phase in which they can flow without dissipation, is analogous to the superfluid phase (Just as atoms can move without viscosity in a superfluid - the atoms formed up into pairs, electrons can flow without resistance in a superconductor). Now let us see how one cools an experiment:

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Cooling to 0.3K

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3He/4He

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In matter at temperatures close to absolute zero, the thermal, electric, and magnetic properties undergo great change, and the behaviour of matter may seem strange when compared with that at room temperature. Thermal fluctuations are greatly reduced and effects of interactions at the quantum-mechanical level can be observed. As the temperature is lowered, order sets in (either in space or in motion), and quantum-mechanical phenomena can be observed on a macroscopic scale. Why bother measure at low temperatures?

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Considerable attention has been addressed to the general problem of ordering in disordered systems leading to studies of spin glasses, localization, and lower dimensionality. Quantum statistics are investigated in atomic hydrogen and deuterium, stabilized in states known as spin-polarized hydrogen (H↓) and spin-polarized deuterium (D↓). Many practical applications have emerged, including the use of superconductivity for large magnets, ultra-fast electronics for computers, and low-noise and high-sensitivity instrumentation. This type of instrumentation has opened new areas of research in biophysics, and in fundamental problems such as the search for magnetic monopoles, gravity waves, and quarks.

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The development of low-temperature techniques has revealed a wide range of other phenomena: The behaviour of oriented nuclei is studied by observing the distribution of gamma-ray emission of radioactive nuclei oriented in a magnetic field. Other areas of study include surfaces of liquid 3He and liquid 4He, 3He–4He mixtures, cryogenics, acoustic microscopy, phonon spectroscopy, monolayer helium films, molecular hydrogen, determination of the voltage standard, and phase transitions.