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Research Projects

Non-linear Response Near to Quantum Phase Transitions

Prof. Shivaji L. Sondhi1& Dr Andrew G. Green

How does a quantum-critical system behave when driven far from thermal equilibrium, and is this behaviour universal? My collaboration with Prof. S. L. Sondhi (Princeton) has recently begun to shed some light on this matter. It was shown in [1] that a Hamiltonian quantum critical system, in particular the superfluid/insulator transition, can indeed show a universal non-linear response when driven far from thermal equilibrium. This is an important extension to the quantum critical paradigm; although scaling arguments might lead one to anticipate such universal, non-linear response, there was previously no controlled calculation which demonstrated this.

These ideas have direct application in solid-state systems, for example to the understanding of non-linear IV characteristics near to criticality. Perhaps the most exciting new experiments to which these ideas apply are those on quantum phase transitions of atomic condensates in optical lattices. Since the energy scales are so low and the perturbations e.g. changes in the lattice potential or translation of the optical lattice) so rapid and so large, these systems are easily driven into the strongly non-linear regime.

I am currently involved in several projects related to this work:

Non-linear Response within the epsilon-expansion Our original work[1] used a 1/N-expansion within a Boltzmann equation approach and was able to sidestep a direct treatment of the coupling to the heat bath. The alternative non-perturbative technique of epsilon- expansion will be used to supplement these results. This will further elucidate the role of the heat bath in determining the universal response. The nature of the heatbath is a crucial distinction between solid-state systems and atomic condensates in optical lattices. Despite having formally identical quantum groundstates and critical points, the response and particularly the non-linear response of the two types of system may be markedly different due to this distinction.

Universal Current-Fluctuations Near to Quantum Criticallity In collaboration with Prof. J. E. Moore (UC Berkeley), I have recently embarked upon a consideration of the possibility of universal conductance fluctuations in the non-linear response of quantum critical systems (and indeed in the thermal equilibrium system- this does not appear to have been calculated in the literature). Preliminary results suggest an unusual scaling of these fluctuations with electric field.

Universal Nernst Effect at the Super-fluid/Insulator Transition. The Boltzmann transport equations developed in previous studies of electrical conductivity[2] will be extended to describe the Nernst effect[3], i.e. the transverse current response of a system to a combination of thermal gradient and magnetic field. The response near to the quantum critical point is expected to be universal. This calculation is particularly topical because of a recent resurgence of experimental interest in the Nernst effect in strongly correlated electron systems[3]. Previous analyses of this effect have considered the contribution of small amplitude, Gaussian fluctuations to the Nernst effect using a Kubo formula approach. Such approaches do not work near to the critical point, since fluctuations are very large. The combination of a Boltzmann transport equation and non-perturbative (1/N or epsilon expansion) technique should be able to capture the key physics.

Non-linear Transport of Quantum Critical Itinerant Magnets. Scaling arguments suggest that there will be universal non-linear response in quantum critical systems where the critical modes are not charged, for example in itinerant magnets. On the other hand, the fact that these modes are not charged appears to suggest that they will not be affected by the application of an electric field. Together with Prof. C. Pepin (Saclay) I have obtained preliminary results that show how the naive expecations of scaling can be seen in a microscopic calculation.

[1] Tail States in a Superconductor with Magnetic Impurities, A Lamacraft and B D Simons, Phys. Rev. Lett. 85 , 4783 (2000). (Gzipped Postscript)

[2] "Nonzero-temperature transport near quantum critical points", K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997)

[3] N. P. Ong, et al, Annalen der Physik 13, 9-14 (2004);
P. N. Ong and Yayu Wang, Physica C 408-410C , 11-15 (2004);
R. Bel et al, Phys. Rev. B 70, 220501 (2004); Phys. Rev. Lett. 92, 217002 (2004); Phys. Rev. Lett. 91, 066602 (2003).

1Department of Physics, Princeton University

New Physics at the Breakdown of Quantum Criticality

Prof Ben Simons1 & Dr. Santiago Grigera & Dr Andrew G. Green

There is growing experimental evidence that as one approaches an apparent quantum critical point (by varying temperature and other control parameters), eventually quantum criticality breaks down[1]. This is often due to the formation of new electronic phases such as superconductivity[2]. From the theoretical side, some of the basic descriptions[3] of quantum criticality can break down as one approaches the transition[4] indicating a transition to new behaviour.

These observations suggest a broader principle: that ``Nature abhors a quantum critical point'', a suggestion which was discussed in detail by Lonzarich, Laughlin and Montoux in their Quantum Conundrum[1]. The idea is that, at a quantum critical point, there is an enormous degeneracy of the quantum groundstate due to the existence of many low(zero) energy ways of distorting the critical wavefunction. In nature, this degeneracy will be broken either spontaneously or by a vanishingly small symmetry breaking field, leading to the formation of a new phase.

Intruiging contributions to this debate have been made by experiments performed in Prof. A. Mackenzie's group in St Andrews. Transport and susceptibility measurements of very pure Sr3 Ru2 O7 [6] showed it initially to be an apparent textbook example of a new type of quantum criticality[7]; since it was associated with a metamagnetic critical end point, the divergent fluctuations were not associated with a change of symmetry at the phase transition.

Upon closer investigation, however, this quantum critical behaviour was found to break down very close to the critical point. Even more dramatically, in the very purest crystals of Sr3 Ru2 O7, quantum critical behaviour is circumvented by the appearance of bifurcated structure in the magnetic phase diagram. In our early work[8] we were able to capture this rich phase diagram using a minimal extension of the Landau theory of the metamagnet. This Landau theory suggests a microscopic mechanism that, we believe, may have general validity. We are currently working in close collaboration with the experementalists in St Andrews and elsewhere to pin down this microscopic mechanism and to test its generality.

[1] R. B. Laughlin, G. G. Lonzarich, P. Monthoux and D. Pines, Adv. Phys. 50, 361 (2001);

[2] N. D. Mathur et al, Nature 394, 39 (1998);
C. Petrovic et al, Europhys. Lett. 53, 354 (2001).

[3] J. A. Hertz, Phys. Rev. B 14, 1165 (1976)
A. J. Millis, Phys. Rev. B 48, 7183 (1993)
T. Moriya, Solid State Science 56 (Springer, Berlin, Heidelberg, 1985).

[4] T. R. Kirkpatrick and D. Belitz, Phys. Rev. B 67, 024419 (2003)
D. Belitz, T. R. Kirkpatrick and J. Rollbühler, Phys. Rev. Lett 93, 155701 (2004)
D. Belitz, T. R. Kirkpatrick and Thomas Vojta, Phys. Rev. B 65, 165112 (2002)

[5] S. A. Grigera et al., Science 294, 329 (2001)
S. A. Grigera et al., Phys. Rev. B 67, 214427 (2003)
R. S. Perry et al., Phys. Rev. Lett. 92, 166602 (2004)
R. S. Perry, et al., Phys. Rev. Lett. 86, 2661 (2001)

[6] A. J. Millis, et al. Phys. Rev. Lett. 88, 217204 (2002).

[7] A. G. Green, S. A. Grigera, R. A. Borzi, A. P. Mackenzie R. S. Perry and B. D. Simons, Phys. Rev. Lett. 95, 086402 (2005).

1Theory of Condensed Matter, Cavendish Laboratory, Cambridge University

Errors in Reading Out the State of a Superconducting Quantum Bit

Prof Steven M. Girvin1, Dr Krishnendu Sengupta1 & Dr Andrew G. Green

The question of how the classical world emerges from the underlying quantum mechanical world presents science with some of its deepest problems. The attempt to produce quantum bits has generated a new fervour in the investigation of quantum measurement in solid-state systems.

I have been motivated in particular by experiments carried out in Saclay and Yale[2,3]. These use Josphson junctions in novel ways to create quantum bits and to read out their state. They involve operating Josphson junctions in regimes where non-linear classical effect, quantum tunneling effects and the effects of thermal and quantum fluctuations in the environment are all important. Current analytical techniques struggle to describe all of these effects in concert. I have recently developed an extension of the Quantum Langevin equation with addresses this problem in the regime of incoherent hopping[1].

[1] "Extensions to the Quantum Langevin Equation in the Non-Linear Hopping Regime'' A.G. Green, cond-mat/0508261

[2] D. Vion, et al., Science, April (2002).

[3] I. Siddiqi et al, Phys. Rev. Lett. 94, 027005 (2005)
I. Siddiqi et al, Phys. Rev. Lett. 93, 207002 (2004).

1Department of Physics, Yale University

The Information Theory of Non-linear Optical Systems

Prof. Peter B. Littlewood1, Dr Partha P. Mitra2, Dr Lorenz G. L. Wegener3 & Dr Andrew G. Green

An important sideline to my work over the past few years has been the application of ideas from condensed matter and many-body physics to the propagation of information in non-linear media. The efficiency with which information may be transmitted or stored is limited by signal distortion. These distortions may be due either to amplifier noise or to characteristics of the medium in which the signals propagate. The ideas of information theory, first developed by Shannon, give a quantitative measure of storage and transmission efficiency. Many of the ideas in this field are very similar to those of statistical mechanics; the measure of the information content of a signal is precisely the same as its entropy for example.

The transmission of information along an optical fibre is limited both by amplifier noise and by the non-linear propagation of electromagnetic waves in the fibre. In fact, the propagation of the low frequency envelope of the electric field is described by the familiar non-linear Schrodinger equation with the roles of space and time interchanged. Together with my collaborators, I have been interested in signal distortions in frequency division multiplexed(FDM) systems; systems in which the transparent bandwidth is divided into a number of smaller sub-bands which are used independently. Significant distortion of the optical signal in FDM systems occurs due to non-linear interaction with the signal in the other channels. I have used Feynman path integral and diagrammatic techniques in order to estimate the errors due to this[1,2,3]. My PhD student, Andrew Berridge, has recently made further advances in this direction.

Another development has been our analysis of the effect of non-linearities upon phase noise. By including the effect of chromatic dispersion, we have shown that phase noise may be much smaller than previously anticipated. This opens up the possibility of frequency modulation rather than amplitude modulation in optical communication[4]. Above all, a theoretical understanding of channel capacity will enable the prediction of variations in capacity with technologically relevant parameters and motivate the development of new coding schemes to improve transmission rates.

[1] Mitra and Stark, Nature 411, 1027 (2001)

[2] Green, Littlewood, Mitra and Wegener, Phys. Rev. E66, 046627 (2002)

[3] Mitra, Stark and Green, Optics and Photonics News 13, S22 (2002)

[4] Green, Littlewood, Mitra and Wegener, Optics Letters 28, 2455 (2003)

1Theory of Condensed Matter, Cavendish Laboratory, Cambridge University 2Cold Stream Harbour Laboratory 3No longer in Physics