Quantum critical point

A quantum critical point is a special class of continuous phase transition that takes place at absolute zero, typically in a material where the phase transition temperature has been driven to zero by the application of a pressure, field or through doping. Conventional phase transitions occur at finite temperature, when the growth of random thermal fluctuations leads to a change in the physical state of a system. Condensed matter physics research over the past few decades has revealed a new class of phase transitions, called a quantum phase transition, which take place at absolute zero, and which are driven by the zero point quantum fluctuations associated with Heisenberg's uncertainty principle.

Overview

Within the class of phase transitions, there are two main categories: At a first-order phase transition, the properties shift discontinuously, as in the melting of solid, whereas at a second order phase transition, the state of the system changes in a continuous fashion. Second-order phase transitions are marked by the growth of fluctuations on ever-longer length-scales. These fluctuations are called "critical fluctuations". At the critical point where a second-order transition occurs the critical fluctuations are scale invariant and extend over the entire system. At a finite temperature phase transition, the fluctuations that develop at a critical point are governed by classical physics, because the characteristic energy of quantum fluctuations is always smaller than the characteristic Boltzmann thermal energy $k_B T$.

At a quantum critical point, the critical fluctuations are quantum mechanical in nature, exhibiting scale invariance in both space and in time. Unlike classical critical points, where the critical fluctuations are limited to a narrow region around the phase transition, the influence of a quantum critical point is felt over a wide range of temperatures above the quantum critical point, so the effect of quantum criticality is felt without ever reaching absolute zero. Quantum criticality was first observed in ferroelectrics, in which the ferroelectric transition temperature is suppressed to zero.

A wide variety of metallic ferromagnets and antiferromagnets have been observed to develop quantum critical behavior when their magnetic transition temperature is driven to zero through the application of pressure, chemical doping or magnetic fields. In these cases, the properties of the metal are radically transformed by the critical fluctuations, departing qualitatively from the standard Fermi liquid behavior, to form a metallic state sometimes called a non-Fermi liquid or a "strange metal". There is particular interest in these unusual metallic states, which are believed to exhibit a marked preponderance towards the development of superconductivity.

Quantum critical endpoints

Quantum critical points arise when a susceptibility diverges at zero temperature. There are a number of materials (such as CeNi₂Ge₂^[1]) where this occurs serendipitously. More frequently a material has to be tuned to a quantum critical point. Most commonly this is done by taking a system with a second-order phase transition which occurs at finite temperature and tuning it—for example by applying pressure or magnetic field or changing its chemical composition. CePd₂Si₂ is such an example,^[2] where the antiferromagnetic transition which occurs at about 10K under ambient pressure can be tuned to zero temperature by applying a pressure of 28,000 atmospheres.^[3] Less commonly a first-order transition can be made quantum critical. First-order transitions do not normally show critical fluctuations as the material moves discontinuously from one phase into another. However if the first order phase transition does not involve a change of symmetry then the phase diagram can contain a critical endpoint where the first-order phase transition terminates. Such an endpoint has a divergent susceptibility. The transition between the liquid and gas phases is an example of a first-order transition without a change of symmetry and the critical endpoint is characterized by critical fluctuations known as critical opalescence.

A quantum critical endpoint arises when a finite temperature critical point is tuned to zero temperature. One of the best studied examples occurs in the layered ruthenate metal, Sr₃Ru₂O₇ in a magnetic field.^[4] This material shows metamagnetism with a low-temperature first-order metamagnetic transition where the magnetization jumps when a magnetic field is applied within the directions of the layers. The first-order jump terminates in a critical endpoint at about 1 kelvin. By switching the direction of the magnetic field so that it points almost perpendicular to the layers, the critical endpoint is tuned to zero temperature at a field of about 8 teslas. The resulting quantum critical fluctuations dominate the physical properties of this material at nonzero temperatures and way from the critical field. The resistivity shows a non-Fermi liquid response, the effective mass of the electron grows and the magnetothermal expansion of the material is modified all in response to the quantum critical fluctuations.

Non-equilibrium quantum phase transition

What happens when a quantum critical point is affected by noise? An intuitive guess would be that the external noise defines an effective temperature. This effective temperature would introduce a well defined energy scale in the problem and break the scale invariance of the quantum critical point. On the contrary, it was recently found that certain types of noise can induce a non-equilibrium quantum critical state. This state is out-of-equilibrium because of the continuous energy flow introduced by the noise, but it still retains the scale invariant behavior typical of critical points.

Notes

1. Non-Fermi-Liquid Effects at Ambient Pressure in a Stoichiometric Heavy-Fermion Compound with Very Low Disorder: CeNi₂Ge₂, P. Gegenwart, F. Kromer, M. Lang, G. Sparn, C. Geibel and F. Steglich. Phys. Rev. Lett. 82, #6, 1293-1296 (1999).Abstract
2. The normal states of magnetic d and f transition metals, S R Julian, C Pfleiderer, F M Grosche, N D Mathur, G J McMullan, A J Diver, I R Walker and G G Lon zarich, J. Phys.:condens. matt. 8, #48, 9675-9688 (1996) Abstract
3. N.D. Mathur, F.M. Grosche, S.R. Julian, I.R. Walker, D.M. Freye, R.K.W. Haselwimmer, G.G. Lonzarich (1998). "Magnetically mediated superconductivity in heavy fermion compounds". Nature (London) 394: 39–43. Bibcode:1998Natur.394...39M. doi:10.1038/27838.
4. Magnetic Field-Tuned Quantum Criticality in the Metallic Ruthenate Sr₃Ru₂O₇, S. A. Grigera, R. S. Perry, A. J. Schofield, M. Chiao, S. R. Julian, G. G. Lonzarich, S. I. Ikeda, Y. Maeno, A. J. Millis, A. P. Mackenzie, Science 294, #5541, 329-332 (2001). Abstract

References

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• Subir Sachdev (1999). Quantum Phase Transitions. Cambridge University Press.
• M.A. Continentino (2001). Quantum Scaling in Many-Body Systems. World Scientific.
• P. Coleman, A. J. Schofield (2005). "Quantum criticality". Nature 433 (7023): 226–229. arXiv:cond-mat/0503002. Bibcode:2005Natur.433..226C. doi:10.1038/nature03279. PMID 15662409.
• E.G. Dalla Torre; et al. (2010). "Quantum critical states and phase transitions in the presence of non-equilibrium noise". Nature Physics 6 (10): 806–810. arXiv:0908.0868. Bibcode:2010NatPh...6..806D. doi:10.1038/nphys1754.
• Carr, Lincoln D. (2010). Understanding Quantum Phase Transitions. CRC Press. ISBN 978-1-4398-0251-9.
• Amusia, M., Popov, K., Shaginyan, V., Stephanovich, V. (2014). Theory of Heavy-Fermion Compounds - Theory of Strongly Correlated Fermi-Systems. Springer. ISBN 978-3-319-10825-4.