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Electronic Liquid Crystal Phases in Strongly Correlated Systems

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Modern Theories of Many-Particle Systems in Condensed Matter Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 843))

Abstract

I discuss the electronic liquid crystal (ELC) phases in correlated electronic systems, what these phases are and in what context they arise. I will go over the strongest experimental evidence for these phases in a variety of systems: the two-dimensional electron gas (2DEG) in magnetic fields, the bilayer material \(\hbox{Sr}_{3}\hbox{Ru}_{2}\hbox{O}_{7}\) (also in magnetic fields), and a set of phenomena in the cuprate superconductors (and more recently in the pnictide materials) that can be most simply understood in terms of ELC phases. Finally we will go over the theory of these phases, focusing on effective field theory descriptions and some of the known mechanisms that may give rise to these phases in specific models.

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Notes

  1. 1.

    You may call the ELC phases the anisotropic states of point particles!

  2. 2.

    On the other hand, in the case of a crystal phase, the expansion is

    $$ \rho({\user2{r}})=\rho_0({\user2{r}})+\sum_{{{\user2{K}}} \in \Upgamma} \rho_{{\user2{K}}}({\user2{r}}) e^{i {{\user2{K}}}\cdot {{\user2{r}}}}+ \hbox{c.c.} $$
    (2.4)

    where \(\Upgamma\) denotes the set of primitive lattice vectors of the crystal phase [3].

  3. 3.

    I will not discuss the case of spiral order here.

  4. 4.

    For a lattice system, rotational symmetries are those of the point (or space) group symmetry of the lattice. Thus, nematic order parameters typically become Ising-like (on a square lattice) or three-state Potts on a triangular lattice (and so forth).

  5. 5.

    Ref. [24] is a recent, complementary, review of the phenomenology of nematic phases in strongly correlated systems.

  6. 6.

    The anisotropy is strongly suppressed by disorder.

  7. 7.

    The 2DEG in a strong magnetic field is inherently a strongly correlated system as the interaction is always much bigger than the (vanishing) kinetic energy.

  8. 8.

    \(\hbox{Cu}_x{\hbox{TiSe}_2}\) is known to become superconducting [89]. The temperature-pressure phase diagram of \(\hbox{TiSe}_2\) exhibits a superconducting dome enclosing the quantum critical point at which the CDW state melts [92].

  9. 9.

    In principle the order parameter of the stripe state may not be pure sinusoidal and will have higher harmonics of the fundamental order parameter.

  10. 10.

    I will ignore here physically correct but more complex orders such as helical phases.

  11. 11.

    This state is a close relative of the resonating valence bond (RVB) state originally proposed as a model system for a high \(T_c\) superconducting state [18, 113], i.e. a (non-resonating) valence bond (VB) state.

  12. 12.

    A difficulty in interpreting the DMRG results lies in the boundary conditions that are used that tend to enhance inhomogeneous, stripe-like, phases.

  13. 13.

    \(\Updelta_{{\rm SC},2}\) is the most relevant operator. For a model with \(\kappa(k_{\perp})=[\kappa_0+\kappa_1\cos(k_{\perp})]^2,\) all perturbations are irrelevant for large \(\kappa_0\) and small \(|\kappa_0-\kappa_1|.\)

  14. 14.

    In Ref. [125] a similar pattern was also considered except that the (say) ‘B′ stripes do no have a spin gap. This patterns was used to show how a crude model with nodal quasiparticles can arise in an inhomogeneous state.

  15. 15.

    Josephson coupling is due to pair tunneling from one stripe to a neighboring one. Josephson processes arise in second order in perturbation theory and involve intermediate states with excitations energies of order J and have an amplitude controlled by \(\delta t.\)

  16. 16.

    While there is some numerical evidence for a state of this type in variational Monte Carlo calculations [115] and in slave particle mean field theory [114, 146] (see, however, Ref.[147, 148]), a consistent and controlled microscopic theory is yet to be developed. Since the difference between the energies of the competing states seen numerically is quite small one must conclude that they are all reasonably likely.

  17. 17.

    A state that is usually described as a pair crystal is commonly known as a pair density wave [149, 150]. However that state cannot be distinguished by symmetry from a (two) CDWs coexisting with a uniform SC.

  18. 18.

    A charge 4e SC order parameter is an expectation value of a four fermion operator.

  19. 19.

    A more elaborate version of this phase diagram, based on a one-loop Kosterlitz RG calculation for physically very different systems with the same RG structure system, was given in Refs. [154–156].

  20. 20.

    Although the Pomeranchuk argument is standard and reproduced in all the textbooks on Fermi liquid theory (see, e.g. Ref.[157]) the consequences of this instability were not pursued until quite recently.

  21. 21.

    In fact, perturbative renormalization group calculations [168, 169] have found a runaway flow in the \(d_{x^2-y^2}\) particle-hole channel, which is a nematic instability, but it was not recognized as such. See, however, Ref.[170].

  22. 22.

    Another class of nematic state can occur inside a \(d_{x^2-y^2}\) superconductor. This quantum phase transition involves primarily the nodal quasiparticles of the superconductor and it is tractable within large N type approximations [188, 189].

  23. 23.

    There are other collective modes at higher energies. In particular there is an underdamped longitudinal collective mode with z = 2 [39]. These higher energy modes contribute to various crossover effects [192], but decouple in the asymptotic quantum critical regime.

  24. 24.

    A similar behavior was found in the quantum Lifshitz model at its QCP [207].

  25. 25.

    In 3D the situation is more complex and the possible are more subtle. In particular, in 3D there are three vector order parameters involved [17].

  26. 26.

    The \(\ell=0\) case is, of course, just the conventional Stoner ferromagnetic instability.

  27. 27.

    The term “nematic-spin-nematic′′ is a poor terminology. A spin nematic is a state with a magnetic order parameter that is a traceless symmetric tensor, which this state does not.

  28. 28.

    The p-wave \((\ell=1)\;\beta\) phase has the same physics as the ‘spin-split′ metal of Ref. [221]. A similar state was proposed in Ref. [222] as an explanation of the “hidden order′′ phase of \(\hbox{URu}_2\hbox{Si}_2.\)

  29. 29.

    This is consistent with the general arguments of Ref. [224].

  30. 30.

    It also turns out that in the (so far physically unrealizable) case of x = 1, the ground state is a nematic insulator as each row is now full. However, for \(x \to 1\) the ground state is again a nematic metal.

  31. 31.

    Important work with a similar approach has been done on the problem of the quantum melting of the stripe state in quantum Hall systems [36, 37].

  32. 32.

    For a different perspective see Ref. [230].

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Acknowledgments

I am deeply indebted to Steve Kivelson with whom we developed many of the ideas that are presented here. Many of these results were obtained also in collaboration with my former students Michael Lawler and Kai Sun, as well as to John Tranquada, Vadim Oganesyan, Erez Berg, Daniel Barci, Congjun Wu, Benjamin Fregoso, Siddhartha Lal and Akbar Jaefari, and many other collaborators. I would like to thank Daniel Cabra, Andreas Honecker and Pierre Pujol for inviting me to this very stimulating Les Houches Summer School on “Modern theories of correlated electron systems” (Les Houches, May 2009). This work was supported in part by the National Science Foundation, under grant DMR 0758462 at the University of Illinois, and by the Office of Science, U.S. Department of Energy, under Contracts DE-FG02-91ER45439 and DE-FG02-07ER46453 through the Frederick Seitz Materials Research Laboratory of the University of Illinois.

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  1. Department of Physics, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL, 61801-3080, USA

    Eduardo Fradkin

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  1. , Departamento de Física/IFLP, Universidad Nacional de La Plata, La Plata, 1900, Argentina

    Daniel C. Cabra

  2. , Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, Göttingen, 37077, Germany

    Andreas Honecker

  3. , Laboratoire de Physique Théorique, Université Paul Sabatier, Route de Narbonne 118, Toulouse Cedex 4, 31062, France

    Pierre Pujol

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Fradkin, E. (2012). Electronic Liquid Crystal Phases in Strongly Correlated Systems. In: Cabra, D., Honecker, A., Pujol, P. (eds) Modern Theories of Many-Particle Systems in Condensed Matter Physics. Lecture Notes in Physics, vol 843. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10449-7_2

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