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Quantum Criticality

Amongst the plethora of strongly correlated systems there are some hints of general underlying principles. The idea quantum criticality and the quantum phase transition have proved particularly useful in understanding strongly correlated condensed matter systems.

A continuous classical phase transition occurs due to a competition between thermal fluctuations and interactions. One familiar example is the evaporation of water to form steam. This occurs as a competition between hydrogen bonds between water molecules and the thermal kinetic energy of the water molecules. In fact, this transition is first order with a steplike jump in density. As one increases pressure, however, there is a critical pressure at which the transition becomes continuous; there is a continuous change in the density from water to steam. At higher pressures, there is no transition at all. At the critical pressure and near to the critical temperature, the water/steam system shows interesting scaling relations in its correlations. Correlations of fluctuations in pressure at one point with those at other points scale in a particular way with the distance between the two points and with temperature and pressure. Such scaling relations are known as universal, because they are independent of the microscopic details of the system. Other systems with different underlying microscopic details can show the same scaling relations and are then said to fall into the same universality class. For a wonderful introduction to these ideas see the book by Prof. J. C. Cardy[1]

Quantum Phase Transitions occur due to a competition between quantum fluctuations and interactions. They are found at zero temperature (rather than at finite temperature as in the classical case) as some system parameter is varried. In real systems this may involve the application of hydrostatic pressure, changing chemical doping or the concentration of disorder, or applying a magnetic field. In all cases, changing the system conditions changes the effective strength of interactions and quantum fluctuations. In the same way as classical systems in the vicinity of continuous phase transitions display scaling relations in their response to external probes, so do quantum systems in the vicinity of quantum phase transitions (It turns out that quantum phase transitions in D-dimensions are related, by analytical continuation to imaginary time, to classical phase transitions in D+1-dimensions). A very nice summary of these scaling relations may be found in the review by Shivaji Sondhi[2] or the book of Subir Sachdev[3]. Many many strongly correlated condensed matter systems display these types of scaling relations, even when a microscopic theory for them is not available. Examples include high temperature superconductors, the transitions between quantum Hall states, superconductor to insulator transitions and a whole class of magnetic phase transitions. The research carried out here in St. Andrew's in Prof. MacKenzie's group is a fine example of the latter type of system.

Quantum vs Thermal Effects One of the features of quantum critical systems that I find most intriguing, and which ties them to the other main aspect of my research, is the unusual way in which they combine classical/thermal effects and quantum effects. This is outlined in a very nice review article written by Subir Sachdev and Matthias Vojta[4]. I paraphrase some aspects of this review here.

Perhaps the easiset system through which to see this intertwining of quantum and classical effects is the antiferromagnet. A phase diagram of this system is shown in the figure below. As the antiferromagnetic coupling between the spins is varied at zero temperature, the system undergoes a quantum phase transition from a phase with long-range antiferromagnetic order to a paramagnetic phase.

At low temperatures above either the anti-ferromagnetic or para-magnetic phase the low energy behaviour of the system may be described by effective classical theories. In the anti-ferromagnetic phase, the low energy fluctuations are anti-ferromagnetic spinwaves. The low energy behaviour of the system is dominated by spinwaves whose wavelength is near to the correlation lenght of the system. It turns out that the thermal occupation of these modes is very large. In the same way as the electromagnetic field obeys the classical Maxwell equations when there are large numbers of photons, these anti-ferromagnetic spinwaves with wavelength near to the correlation length behave classically. In the paramagnetic phase, the low energy excitations have a particle like character. It requires a finite energy to produce these excitations and at low temperatures, only an exponentially small number are produced (from the relevant Boltzman factor). The separation of these thermally generated quasi-particles is exponentially large. In particular, the separation greater than the de Broglie wavelength of the quasi-particles which, therefore, behave semiclassically.

As the temperature is increased in either the anti-ferromagnetic phase or in the para-magnetic phase, the underlying quantum nature of the system is revealed. This is the opposite of the usual situation where thermal fluctuations tend to dephase a quantum system and generate classical behaviour as the temperature is increased. In the anti-ferromagnetic phase this effect occurs due to interactions between anti-ferromagnetic spinwaves. At low temperatures, these interactions are not particularly strong and the classical picture holds. As the temperature is increased, the total number of excited spinwaves becomes larger and larger and the interaction energy increases accordingly. At a certain point, the interaction energy becomes comparable to the energy of a single spinwave quantum and the quantum nature of the occupation of the spinwave modes is revealed. In the para-magnetic phase, as the temperature is increased, the number of quasi-particle excitations increases and their separation decreases until eventually it becomes comparable to their de Broglie wavelength. At this point, interference effects become important and the quantum nature of the quasi-particle excitations.

Whether one starts from the anti-ferromagnetic or para-magnetic phase, at high temperatures, it becomes impossible to disentangle thermal and quantum effects. The behaviour in this regime is universal and independent of the zero-temperature state from which one started. This is known as the Quantum Critical phase. The Quantum Critical phase is finely balanced between the quantum and classical worlds.

[1] Scaling and Renormalization in Statistical Physics, John Cardy, CUP, (1996). (Homepage)

[2] Continuous Quantum Phase Transitions, S. Sondhi, S. M. Girvin, J. P. Carini and D. Shahar, Rev. Mod. Phys.69 , 315 (1997). (Homepage)

[3] Quantum Phase Transitions, Subir Sachdev CUP (1999). (Homepage-contains links to a variety of review articles)

[4] Quantum Phase Transitions in Antiferromagnets and Superfluids, Subir Sachdev and Matthias Vojta Physica B280, 333 (2000). (cond-mat link)