c Copyright by Michael J. Lawler. All rights reserved.

Transcription

1 c Copyright by Michael J. Lawler. All rights reserved.

2 QUANTUM ELECTRONIC LIQUID CRYSTALS BY MICHAEL J. LAWLER B.Eng., Queen s University, 1999 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2006 Urbana, Illinois

3 QUANTUM ELECTRONIC LIQUID CRYSTALS Michael J. Lawler, Ph.D. Department of Physics University of Illinois at Urbana-Champaign, 2005 Eduardo Fradkin, Adviser This thesis is devoted to an investigation of quantum electronic liquid crystal phases of matter. With the recent discovery of electronic phases that are neither isotropic liquids, nor crystalline solids, it is now vital to gain a comprehensive understanding of these intermediate electronic liquid crystal phases. Examples of these new phases include the highly anisotropic phases in quantum Hall systems, static stripes in cuprate superconductors and Fermi surface distortions in some heavy fermion and ruthenate systems. Owing to the strong correlations involved, such an investigation requires both the use of non-perturbative methods as well as their refinement as they are pushed to these new regimes of interest. Here we shall focus in particular on two such electronic liquid crystal phases representing two paradigms: one which focus on the establishment of orientational order (the nematic phase) and the other on translational order (the smectic phase) both of which are intermediately ordered in comparison to a crystal. We begin by discussing the quantum nematic phase which arises due to shape (Pomeranchuk) instabilities of the Fermi surface of a Fermi liquid. We choose to do so through a simple model that captures the essence of this quantum phase and quantum phase transition. Much has been understood about the order parameter theory of this phase transition using the random phase approximation (RPA). Unfortunately, this order parameter field couples poorly to electromagnetic fields suggesting that unambiguous signatures of this phase may be found only by a direct measurement of quasi-particle properties. However, perturbative methods fail miserably to describe quasi-particle properties owing to a surprisingly strong coupling between these quasi-particles and this order parameter field. We face this challenge by employing the controlled non-perturbative method of bosonization to compute fermion (quasi-particle) correlators and demonstrate precisely how they couple to the order parameter. Our results show explicitly the vanishing of the fermion residue while maintaining a finite local density of states. These non-perturbative results demonstrate unambiguously the non-fermi liquid nature of both the nematic quantum critical point and the nematic phase. iii

4 At finite temperatures, we then discuss properties of the fermion correlation function near this fixed point theory. We show that though the fixed point is above its upper critical dimension, the equal time fermion correlation function takes on a universal scaling form in the vicinity of the QCP. We find that in the quantum critical regime, this equal-time correlation function has an ultra local behavior in space, while the equal-position auto correlation function exhibits scaling and is mildly behaved. This behavior should also apply to other quantum phase transitions of metallic Fermi systems. Lastly, we discuss the implication of the existence of a sliding symmetry, equivalent to the absence of a shear modulus, on the low-energy theory of the quantum hall smectic (QHS) state. This phase has received much attention in the literature owing to its intuitive appeal of the electrons spontaneously forming stripes, or rivers of charge. However, development of it led to differing opinions about the continuum limit in the presence of this sliding symmetry. We show, through renormalization group calculations, that such a symmetry causes the naive continuum approximation in the direction perpendicular to the stripes to break down through infrared divergent contributions originating from operators labeled irrelevant by power counting. In particular, we show that the correct fixed point has the form of an array of sliding Luttinger liquids making sliding symmetry a dramatic features of this liquid crystalline phase. Similar considerations apply to all theories with sliding symmetries. iv

5 I dedicate this thesis to my wife who has encouraged me to heights I never thought I would reach and to my family both here in North America and South Korea. v

6 Acknowledgment First, I would like to thank my adviser Eduardo Fradkin for providing stability and allowing me to grow, for all the times I made major mistakes and he said such is the life of a theorist! and for providing me with the fascinating topic to which I devoted my thesis research. Secondly, I would like to thank my wife / office-mate / collaborator Eun-Ah Kim for listening to all of my ideas, hopes and dreams, for supporting and encouraging me through all the difficult times while also being there to celebrate all the wonderful times. I would like to thank the UIUC faculty for all the efforts they have placed into my education and for all the stimulating discussions that we have had over the years. In particular, I would like to thank Gordon Baym, Nigel Goldenfeld, Tony Leggett, Rob Leigh, Philip Phillips, Mike Stone, John Stack and Smith Vishveshwara. My thesis research was largely motivated by the previous work by Vadim Oganesyan, Steve Kivelson and Daniel Barci without which, I would never have made nearly as much progress as I have made and with whom I have had many stimulating conversations and discussions. I would like to thank my group-mates over the years: Congjun Wu, Vincent Liu, Daniel Barci, Eddy Ardonne, Victoria Fernández, Kai Sun, Stephanos Papanikolaou, Akbar Jafari and Kumar Raman for providing an inspiring and intense community. I would also like to thank many of my fellow classmates, to many to mention here, for providing a very dynamic and wonderful environment at UIUC. Of course, without the support of my family in Chicago: Judy, Dennis, Kevin, Tom, Laurel, and Grandma Lawler to name a few, I never would have felt at home here in Illinois. I am very happy that we got to know each other so much better as a result of living this close together. Family in Canada: Adrian, Helene, James, Sharon, Chris, David, Grandma Doreen and Grandpa Ken, and my family in Korea: Kim Oak-Bae, Dong Yeon-Soon, Kim Kyoung-Ae, Kim Jonghwa vi

7 and Baek Yoon-Kyoung who as always, have been my support structure that has felt solidly behind me in this lengthy and difficult endeavor. Lastly, I would like to thank the entire committee for making time to oversee this thesis. This acknowledgment can go on indefinitely but I should stop for the sake of length. Praise be to God. I was supported in part by the University of Illinois Fellowship for the summers of 2000 and 2005 as well as Fall I was supported by the University of Illinois as a teaching assistant between the year 1999 and 2005 during Fall and Spring semisters. Lastly, I was supported by the University of Illinois as a research assistant for the summers of 2001 through 2006 as well as for the spring of Research for this thesis was supported in part by the National Science Foundation through Grants Nos. DMR and DMR , by an A. O. Beckmann Award of the Research Board of the University of Illinois and by the DOE Award No. DEFG02-91ER45439 (A78) at the Materials Research Laboratory of the University of Illinois. vii

8 Table of Contents List of Tables x List of Figures xi List of Abbreviations xiii Chapter 1 Introduction Quantum Hall liquid crystals Solid state liquid crystals Strong interactions and bosonization Pomeranchuk Instabilities and orientational ordering The Quantum Hall smectic phase In this thesis Chapter 2 Bosonization of D-dimensional Fermi Systems Chapter 3 Nonperturbative behavior of the quantum phase transition to a nematic Fermi fluid Introduction The nematic quantum phase transition and the order parameter Saddle point expansion near the nematic QCP Theory of the Quadrupole Moment Density Order parameter theory of the nematic QCP Fermions in the critical regime Diagrammatic expansion for the bosonized theory Perturbative results Non-perturbative results Conclusion Chapter 4 Local Quantum Criticality at the Nematic Quantum Phase Transition 63 Chapter 5 Quantum Hall smectics, sliding symmetry and the renormalization group Introduction Hydrodynamic Theory of Quantum Hall Smectics Scaling Theories of the Quantum Hall Smectic: Perturbation Theory, RG and Dangerous Irrelevant Operators Conclusions viii

9 Chapter 6 Conclusions Appendix A Analysis of V 2 + and V A.1 Partial fraction expansion of V 2 + (q, s) A.2 Partial fraction expansion of V2 (q, s) Appendix B Inclusion of F 0 (q) Appendix C Equal-Position Boson Propagator Appendix D The Quantum Lifshitz Model Appendix E Anisotropic scaling and the real space renormalization group References List of Publications Author s Biography ix

10 List of Tables C.1 Low frequency / long wavelength limit of the required integrals x

11 List of Figures 1.1 (a), (b) and (c): freezing process of nematogens introducing two new intermediate phases, the nematic and smectic, before the eventual formation of a crystal. (d), (e) and (f): freezing process for fermions (here in two-dimensions) introducing two new intermediate phases, the quantum nematic and quantum smectic phase, before the eventual formation of a Wigner crystal Gadenken phase diagram of electronic liquid crystal phases of the electron gas in two-dimensions. Here the stripe or smectic state exists only at zero temperature valid when considering electronic liquid crystal phases in the absense of a lattice. Similar reasoning and Gadenken phase diagrams were has been presented in [5, 6] (a) GaAs/AlGaAs heterojunction setup used to construct most quantum Hall systems. Electrons from the doped silicon fall into the potential well created by the heterojunction. (b) Free fermions in a magnetic field form Landau Levels labeled by N that are split by the Zeeman effect into polarized levels whose filling is labeled by ν Formation of charge ordered phases from Hartree-Fock calculations in quantum Hall systems[21]. (a) Hartree and Fock potentials and (b) Stripes, bubbles and crystal phases New phases of Quantum Hall Systems (taken from[3, 24]) Quantum Hall liquid crystal Phase diagram based on generic considerations (see text) from our knowledge of quantum Hall physics and classical liquid crystals[5] Candidates for surface (Pomeranchuk) instabilities in lattice systems. In Sr 3 Ru 2 O 7, a possible phase transition to a nematic phase is seen at finite magnetic fields[25]. In URu 2 Si 2, a hidden order is seen in zero magnetic field below 17.5 K and may be a nematic spin nematic (invariant under rotating both spins and space by π) also called helicity ordered[26] Suppression of T C in some cuprate superconductors. In La 2 x Ba x CuO 4, T C (black dots) is suppressed to very small temperatures near x 1/8 with maxima on both sides at about 25 K[30]. This suppression has been shown to be due to static stripe ordering[1, 31] with similar behavior in La 1.6 x Nd 0.4 Sr x CuO 4 [2]. In YBa 2 Cu 3 O 6+x, more direct measurements of hole doping by relating it to unit cell lengths in the c direction have revealed similar suppression of T C [32], at hole doping near 1/8, though much weaker than for La 2 x Ba x CuO 4. The origin of this suppression is unknown for certain, but the authors propose that it may also be related to a tendency towards static stripe ordering Charge ordering revealed through STM expermiments on Bi 2 Sr 2 CaCu 2 O 8+x in the very underdoped regime[33]. (A) Topography (inset magnification: 2), with data in (B) 12 mev LDOS, (C) 16 mev LDOS, and (D) 22 mev LDOS revealing charge ordering in a checker board or stripe pattern xi

12 1.10 The fermion occupation number at zero temperature for weakly interacting Fermions in (a) one and (b) two and greater dimensions. In one dimension there is no discontinuous jump across the Fermi surface, while in two dimensions there is a jump of magnitude Z which can be thought of as a surface tension stabilizing the phase Simple shape distortions of the free fermion ground state utilized by Pomeranchuk to understand the stability of a Fermi liquid (a) Landau level buckling and the formation of Chiral Luttinger Liquids in the quantum Hall smectic phase. (b) A DNA Lipid complex. In the interpretation of the quantum Hall smectic as a DNA lipid complex, the z-axis is equivalent to the time axis and the y-axis that labels the different lipid layers is equivalent to the labeling of the stripes (y = λi) The (normalized) quadrupole density spectral function S 2 + (q, s) = S+ 2 (q, s)/n(0), Eq. (3.26) and Eq.(3.29) for q 0, very close to the nematic quantum phase transition from the Fermi liquid phase, F 2 (0) 1 +. As the QCP is approached, there is a large increase of spectral weight in the the z = 3 overdamped quadrupolar mode at very low frequency. Notice the delta function contribution of the propagating mode with z = 1 discussed in the text. The propagating mode with z = 2 gives a similar delta function contribution (with smaller spectral weight) to S2 (q, s) Bosonization s Feynman diagram series Phase diagram near the nematic quantum phase transition of a Fermi fluid. The crossover between Fermi liquid behavior and quantum critical regime is defined by the correlation length ξ 1/2 ; KT is a Kosterlitz-Thouless transition The cutoff surface Interesting ultraviolet-infrared mixing Cutoff shapes for ω q = q x qy 2r + qx 4 = Λ D.1 Phase diagram of the quantum dimer model on bipartite lattices as a function of potential energy vs kinetic energy V/t taken from[103] E.1 Effect of Anisotropic scaling on the Kosterlitz Thouless real space RG picture xii

13 List of Abbreviations 2DEG Two dimensional electron gas ARPES Angle resolved photo emission spectroscopy BCS Bardeen Cooper Schrieffer (theory of superconductivity) CDW Charge density wave DOS Density of states FS Fermi surface IQHE Integer quantum Hall effect OPE Operator product expansion QCP Quantum critical point QHN Quantum Hall nematic QHS Quantum Hall smectic RG Renormalization group RIQHE Re-entrant integer quantum Hall effect RK Rokhsar Kivelson (point) RPA Random phase approximation RSXS Resonant soft X-ray scattering STM Scanning tunneling microscopy xiii

14 Chapter 1 Introduction Recent discoveries of electronic liquid crystal phases, such as the static stripe ordered phases in some cuprate superconductors[1, 2] as well as highly anisotropic phases in quantum Hall systems[3, 4], have called into question[5, 6] the simple picture of a direct first order quantum phase transition of an electron gas to an electron crystal, a phase proposed by Wigner[7] many years ago. Wigner discovered that if one considers the strongly interacting (correlated) limit, the ground state of a collection of electrons is actually that of a crystal (a prediction which was verified experimentally 25 years ago[8]). However, his analysis neglects quantum fluctuations which by playing the role of temperature, as argued by Kivelson, Fradkin and Emery[6], may melt such a crystal in unusual ways depending on the nature of such fluctuations. These recent discoveries of intermediate phases, therefore, expose our lack of understanding of the process by which a Fermi liquid freezes. At the same time, they raise the exciting possibility that by understanding them, we may gain deep insight into the physics of high temperature superconductivity[9], quantum Hall liquids and other strongly correlated phases. One intriguing aspect of some electronic liquid crystal phases is that they are non-fermi liquids. While in a broad sense these phases may be called non-fermi liquids in that they result from the breakdown of a Fermi liquid, such an instability alone does not guarantee that a description within the framework of a Fermi liquid theory governed by gapless fermionic excitations is not possible. However, in the presence of Goldstone modes arising from the spontaneous symmetry breaking of a continuous symmetry, a Fermi liquid description turns out to indeed fail despite the wisdom that such modes tend to couple only weakly to observable quantities. Thus a successful description of these phases may bring insights into phases that involve electrons living in a hostile environment 1

15 as well as shed light on the role of these newly discovered liquid crystal phases in the context of many strongly correlated systems. Quantum phase transitions involving liquid crystal phases as one of the two phases involved, may also be quite intriguing. For example, should one of the two phases be a Fermi liquid, what happens to the gapless fermions? While an order parameter theory may be constructed and understood within the framework of a renormalization group study following methods introduced by Hertz[10], how this order parameter couples to the fermions is far from being well understood. Do the fermions simply decouple from the critical physics? Can we ignore them like we can ignore gapped degrees of freedom? Indeed, while this type of phase transition was the first quantum phase transition studied, it is not simply the imaginary time equivalent of a well known d + 1-dimensional classical phase transition. Instead, it is quite exotic and many open questions remain. We may seek to gain a qualitative definition of electronic liquid crystal phases from their classical counter-parts. Classical liquid crystal phases arising from the freezing of either a collection of disks or of ellipsoidal shaped objects known as nematogens, have been studied for many years[11]. A simple description of this freezing process is as follows (see Fig. 1.1a-c). Beginning with a gas of nematogens at high temperatures (Fig. 1.1a), cooling may cause the nematogens to maximize their entropy by all of them aligning in one direction (Fig. 1.1b). This nematic phase breaks the rotational invariance inherent in the gas phase while leaving the translational invariance intact. Upon further cooling, again entropy may be maximized by the formation of layers in which the nematogens are free to move within, but not between (Fig. 1.1c). This layered phase, called a smectic, then breaks the translational invariance of the gas phase, but only in one of the three directions. Further cooling must eventually lead to a crystal phase. Liquid crystal phases may therefore be understood and classified by the spontaneous symmetry breaking of the various symmetries inherent in the gas / liquid phase. Electronic liquid crystals may be classified in a similar way. However, it is important to keep in mind that electronic liquid crystals, as quantum phases of matter, are not fully dictated by symmetry principles alone and one needs to understand of their quantum dynamics before a complete description can be achieved. As a result, we cannot extend our understanding of the classical problem directly and must re-think the problem starting from our understanding of Fermi liquid 2

16 (a) gas phase (b) Nematic phase (c) Smectic phase BZ (d) Fermi liquid (e) Nematic phase (f) Smectic phase Figure 1.1: (a), (b) and (c): freezing process of nematogens introducing two new intermediate phases, the nematic and smectic, before the eventual formation of a crystal. (d), (e) and (f): freezing process for fermions (here in two-dimensions) introducing two new intermediate phases, the quantum nematic and quantum smectic phase, before the eventual formation of a Wigner crystal theory. Naturally, electrons are not disk or ellipsoidal shaped objects. So at a first glance, it appears unlikely that such liquid crystal phases are stable. However, we are also not interested in a transition that is driven entropically, but we are after energetically favorable phases at zero temperature. To this end, we begin by constructing a Fermi sea using Pauli exclusion and forming the well known Fermi liquid ground state (see Fig. 1.1d). Then, we may turn up the strength of interactions, following the path first outlined by Wigner, and consider possible instabilities of this state using the shape of the Fermi surface itself as a guide for different possible liquid crystal-like ground states. The simplest possible such state is the instability of the circular Fermi surface towards becoming an ellipse (see Fig. 1.1e). Stretching this ellipse may lead to an instability to forming a unidirectional 3

17 T Fermi Liquid g replacements Wigner Crystal Hexatic Nematic Xtals Smectic es and Adams (1979) Quantum Fluctuations r 1 s Figure 1.2: Gadenken phase diagram of electronic liquid crystal phases of the electron gas in twodimensions. Here the stripe or smectic state exists only at zero temperature valid when considering electronic liquid crystal phases in the absense of a lattice. Similar reasoning and Gadenken phase diagrams were has been presented in [5, 6] charge density wave or stripe state which can be thought of as introducing a Brillouin zone in only one direction (see Fig. 1.1f). Finally introducing the Brillouin zone also in the other direction then brings us to a crystal phase. Hence, formation of various Fermi surface shapes through balancing Pauli exclusion and interactions, while not as intuitive as the nematogen example, is perfectly allowed for by the rules of quantum mechanics. In the absense of a lattice, the relative strength of interactions may be tuned simply by tuning density: interactions become weak in the high density limit and strong in the low density limit. Hence, loosely speaking we may think of density as playing a role similar to that of temperature and use it to tune through various electronic liquid crystal phases. The phase diagram we present in Fig. 1.2, is the result of this kind of reasoning, guided by what is known from the classical liquid crystal literature. In two dimensions, there is no stripe or smectic phase at finite temperature due 4

18 to the finite energy of dislocations in a stripe lattice[12] so this phase is placed only at T = 0 in the phase diagram. However, both the nematic (elliptic Fermi surface) and hexatic (hexagonal Fermi surface) phases, have topological defects whose energy diverges logarithmically in the system size and allow for them to have stable quasi-long range order at finite temperatures (for a review of this type of reasoning, see [13]). The various rotationally order phases occur in the phase diagram since they are the natural intermediate phase one enters upon melting of specific lattices. For example, a triangular lattice may melt into the hexatic phase and a rectangular lattice may melt into a nematic phase. A beautiful illustration of this general picture in the absence of a lattice occurs in the higher Landau levels of very clean quantum Hall systems[3, 4] and we shall turn to a detailed discussion of this important example in section 1.1. In the presence of a lattice, the above reasoning must be modified as a lattice introduces an explicit spatial symmetry breaking field. However, electronic liquid crystal phases can again be defined in a similar way, only they now involve the spontaneous symmetry breaking of the lattice symmetry. For example, one can define a nematic phase in the presence of a square lattice by the elongation of the Fermi surface along one of the lattice directions (see for example, Yamase et. al[14]) or along a lattice diagonal[15]. Furthermore, a stripe phase may be defined by the spontaneous breaking of lattice translational symmetry in one direction introducing rivers of charge separated by a lattice spacing, doubling the unit cell in one direction[6]. Since such symmetry breaking involves only discrete symmetries, all of these phases may exist at finite temperature with long range order and there are now a number of exciting possible experimental realizations of these ideas which we will review in section 1.2. In the remaining sections of this chapter (Sec ), we will outline how to gain a basic understanding of quantum liquid crystal phases from a theoretical point of view and present a framework by which we may begin to understand these new experiments. In Sec. 1.3, we will look at fermions living in the presence of strong correlations by reviewing the case of the one-dimensional electron gas where this is known to occur. In Sec. 1.4 we then turn to orientational ordering of a Fermi liquid by considering shape fluctuations of the Fermi surface. Lastly, in Sec. 1.5 we look at quantum stripe formation in the context of the quantum Hall smectic phase and introduce the sliding symmetry that distinguishes this quantum phase from its classical counter part. 5

19 1.1 Quantum Hall liquid crystals Now let us return to the quantum hall systems, that is the two-dimensional electron gas in a magnetic field. The experimental setup is described in Fig. 1.3(a). The GaAs and AlGaAs layers are undoped neutral semiconductors with different band gaps. When placed next to each other, a potential well is created between them and after doping the AlGaAs with Si electron donors some distance from the junction, the added electrons fall into the well leaving behind a frozen sheet of positive charge. The electrons in the well form a high mobility two-dimensional electron gas (2DEG) in a charge neutral lattice. Usually the mobilities are on the order of 10 6 to 10 7 cm 2 /V s, the record being cm 2 /V s or a mean free path of order 0.1 mm! Since the experimental findings below are only for the highest mobility samples, we will therefore make the assumption throughout this paper that disorder plays a minor role. Furthermore, Since the roughness of the junction interface has little inflouence and the lattice symmetry governing the effective mass tensor is nearly ideal (m ij = m δ ij ), this setup is essentially an ideal isotropic 2DEG with a uniform positive neutralizing background charge[16, 17]. Now, if we solve the non-interacting case of a two-dimensional electron gas in a magnetic field, we will find that the electrons fall into a series of highly degenerate energy levels called Landau Levels. The conventional notation is that the lowest Landau level is labeled N = 0, the second Landau level, N = 1 and so on. These energy levels naturally split due to the Zeeman energy. In order to characterize the state of the system we define its filling fraction, ν = # of electrons degenarcy of a Landau Level = N BA/(h/e) = n sh eb (1.1) where n s = N/A is the areal density of electrons. ν = 1 thus fills half a the first landau level but completly fills the lowest energy level due to spin (as shown in Fig. 1.3(b)). We can therefore change the state of the system by either changing the density of electrons or by changing the magnetic field (both can actually be done). To make some progress at understanding this system from a theoretical or computational point of view, we begin by making the single Landau level approximation. This approximation involves defining a new Hamiltonian that describes only the electrons in the top Landau level. Aleiner and 6

20 Bz Bz N= Si Doped AlGaAs Positive charge from Si dopant e 2DEG GaAs 10nm Conduction Band Fermi Surface 75nm ε F N=2 N=1 hωc N=0 E Zeeman ν=1 E 2DEG B 0 B 0, g 0 B 0, g 0 (a) GaAs/AlGaAs heterojunction (b) Landau levels Figure 1.3: (a) GaAs/AlGaAs heterojunction setup used to construct most quantum Hall systems. Electrons from the doped silicon fall into the potential well created by the heterojunction. (b) Free fermions in a magnetic field form Landau Levels labeled by N that are split by the Zeeman effect into polarized levels whose filling is labeled by ν. Glazman took this approach[18] and defined an effective hamiltonian by: Z = T r N e βh eff = T r N [ T r N Ne βh ] (1.2) Clearly, in the process of confining the electrons to a single layer, the kinetic energy has disappeared! So, H eff is just a potential energy. Keep in mind, though, that this does not mean the electrons have no dynamics. Indeed, the electrons are strongly governed by the Lorentz force. Unfortunately, this process cannot be carried out exactly. However, at the Hartree-Fock level, it can indeed be understood and the case N = 5 is shown in Fig. 1.4(a) (u HF = u H u F ). Notice the zeros in the Hartree potential! Notice also that for the most part, the potential energy is about a fifth of the Landau level energy spacing ( ω c ) justifying our assumption that little Landau level mixing is occurring. Therefore, we can qualitatively view this interaction as a competition between a long ranged screened Coulomb interaction, q q 0 and a short ranged attractive interaction, q q 0. In 1979, the lowest Landau level was studied in Hartree-Fock by Fukuyama, Platzman and 7

21 Potential (h c ) ~ u ex ~ u H ~ u HF q q (a 1 B ) 2k F 2.7R c (a) Hartree and Fock potentials (b) Stripes and bubbles Figure 1.4: Formation of charge ordered phases from Hartree-Fock calculations in quantum Hall systems[21]. (a) Hartree and Fock potentials and (b) Stripes, bubbles and crystal phases. Anderson[19]. They found that below some reasonable temperature, the electrons will form a Wigner Crystal, that is they form a hexagonal lattice. Three years later, the fractional quantum hall effect was discovered[20] and proved that Hartree-Fock calculation was completely wrong. Since then, Hartree-Fock has had an untrustworthy reputation in this field. In 1996, Koulakov et. al.[21] and Moessner and Chalker[22] studied the higher Landau levels in Hartree-Fock despite its failure in the lowest Landau level. They present the following picture, shown in Fig 1.4(b). Because of the nodes in the interaction potential, the electron wavefunction likes to form rings, of radius R c = 2N + 1l, l the magnetic length, around the center of mass of the electron. In the low density regime, that is the classical limit, ν 1/N and the electrons form a Wigner crystal, as Hartree-Fock predicts in the lowest Landau level. As the filling fraction is increased past 1/N, the average distance between particles becomes about R c and quantum effects begin to occur. When two rings overlap, the result is the formation of a larger ring with two particles in the center and the system forms a super Wigner Crystal called the bubble phase. As the density is further increased, more electrons fall into each bubble keeping the lattice spacing about two are three R c. Surprisingly, when ν reaches about 0.4, the bubble phase collapses into a stripe pattern the stability of which is attributed to the competition between the short range attractive force and the long range coulomb force. Modulations of the stripe turn out to lower the energy so that in Hartree-Fock, the stripes actually lock next to each other[23]. Because of this shear force, this phase is called a stripe crystal 8

22 since it is not distinguishable from a highly rectangular crystal as will be shown below. This shear force can be very small, however, so that this phase may still be thought of as a layered system not unlike that of the smectic liquid crystal discussed above. Please note that all crystal phases are expected to be pinned by disorder so that these Hartree-Fock states should be at least mildly insulating. It should also be noted why the Laughlin wavefunction, which is marvelously successful at describing the fractional hall effect at N = 0, loses to Hartree-Fock for larger N. Moessner and Chalker, making the short range interaction approximation as well as assuming no inter-landau level mixing, showed that Hartree-Fock should actually be exact in the large N limit. Therefore, it makes sense that these predictions should begin to come true at some N. Further, because of the zeros of the Hartree-potential, electrons can make use of these zeroes to lower there energy by forming ring-like shapes which minimize any short ranged interactions. In the lowest Landau level, the Laughlin wavefunctions have multiple zeros when two particles collide and therfore also minimize a short ranged interaction. These two forms compete and it turns out that Laughlin state generally wins at the lowest Landau level while Hartree-Fock wins in the higher Landau levels though sometimes the competition is subtle. Due to its infamous history, the above Hartree-Fock picture received little attention until experiments in early 1999 published surprising results. All earlier transport experiments showed, using lower mobility samples, data similar to that in Fig. 1.5a. The wealth of structure shown in the lowest Landau level is the usual fractional quantum hall effect. At the shown filling fractions, ν = 5/3 and ν = 4/3 and others, as well as at all integer filling fractions, ρ xx is zero, (i.e. dissapationless) and ρ xy (not shown) is quantized at a value of h/νe 2. In the higher Landau levels, it was expected that the transition region from the zeros of each integer filling would shrink with temperature creating broader and broader IQHE plateaus. In 1999, two experimental groups[3, 4] discovered that in very high mobility samples, instead of a shrinking region between integer zeros, a sudden giant anisotropy occurs in the resistance matrix at filling fractions ν = {9/2, 11/2,...} shown in Fig. 1.5b) peaks whose width do not change with temperature below about 50mK. A closer look, Fig. 1.5c, shows that beside each of these large peaks, there is also a deep valley where R xx 0 as in the quantum hall effect phases[24]. A glance at R xy 9

23 (a) 0.25 h/e (b) ρ xx (Ω) N=2 Landau level 11/2 9/ N=1 5/2 ν=2 T=150mK 5/3 4/3 N= B (Tesla) ρ xx (Ω) 1000 T=25mK 9/2 V 750 V I I 11/ /2 7/2 5/ N= B (Tesla) R xx and R yy R xy h/e Ω ν=9/2 RIQHE RIQHE ν=5 ν= magnetic field (T) R xy R xx and R yy (c) magnetic field (T) (a) Lower mobility behaviour (b) Anisotropic transport at filling fractions ν = {9/2, 11/2,...} (c) Re-entrant integer quantum hall plateaus Figure 1.5: New phases of Quantum Hall Systems (taken from[3, 24]) shows that it is quantized (in units of h/e 2 ) to the same value as the nearby IQHE plateau. This is the first example of the IQHE occurring at fractional filling! As such, it is called the re-entrant IQHE or RIQHE. Comparing these experimental findings and the qualitative picture presented by Hartree-Fock, we find tremendous agreement! The anisotropic phase can be thought of as a stripe phase as indeed such a phase will have anisotropic transport. Viewing the RIQHE phase as a bubble phase, the electrons in the top Landau level should be trapped inside bubbles and with the super lattice pinned by disorder, the phase should be insulating and behave like an empty Landau level. Should the voltage be increased, at some point the crystal should depin from the disorder and a sudden increase in the current should be observed. This indeed happens[24]. As the voltage is lowered, the current is found to persist beyond the initial jump-up value forming a Hysteresis loop. One major disagreement between the Hartree-Fock picture, however, is that the stripe phase will have highly anisotropic transport to such an extent that the resistivity should be infinite in one direction and zero in the other. Experiment, however, shows that the anisotropy in this resistivity is finite in both directions. That is, the anisotropic phase is metallic! Further, from a theoretical point of view, the lower critical dimension of a classical stripe-like state (a smectic liquid crystal) is three and at finite temperatures, topological defects should proliferate and destroy the stripes[12]. For these and other reasons, a broader picture than Hartree-Fock can provide is necessary. Following the initial experiments, Fradkin and Kivelson, having already thought much about 10

24 "1/N" Isotropic Wigner Nematic Crystal Stripe Smectic Crystal (M+1/2) (filling factor) M Figure 1.6: Quantum Hall liquid crystal Phase diagram based on generic considerations (see text) from our knowledge of quantum Hall physics and classical liquid crystals[5]. stripe ordering in high temperature superconductors, presented the general phase diagram shown in Fig. (1.6) based on the ideas of classical liquid crystals[5]. The top of this diagram is essentially the lowest Landau level, with the fractional quantum hall states and other phases grouped under the label isotropic. As the magnetic field decreases and the filling fraction increases we move down from the top and encounter new phases distinguished by their spacial symmetries. In particular, they propose that the highly anisotropic metallic phase is actually a nematic, a phase not mentioned in the previous Hartree Fock calculations. Further, it should be noted that the stripe crystal phase is a label that includes both the modulation locked stripe phase, briefly discussed earlier, and other crystal phases such as the bubble phases found in Hartree-Fock. The picture of these new phases found in quantum Hall systems is therefore very similar to the general phase diagram discussed earlier (see Fig. 1.2). 1.2 Solid state liquid crystals While an atomic lattice plays only a minor role in quantum Hall settings, electronic liquid crystal phases have also been found when its presence is very strong. In deed, there is evidence for 11

25 (a) Sr 3Ru 2O 7 (b) URu 2Si 2 Figure 1.7: Candidates for surface (Pomeranchuk) instabilities in lattice systems. In Sr 3 Ru 2 O 7, a possible phase transition to a nematic phase is seen at finite magnetic fields[25]. In URu 2 Si 2, a hidden order is seen in zero magnetic field below 17.5 K and may be a nematic spin nematic (invariant under rotating both spins and space by π) also called helicity ordered[26]. their existence in a variety of systems including some ruthenates, heavy fermions and cuprate superconductors. In the ruthenates, there has been a proposal that Fermi surface distortions may occur in Sr 3 Ru 2 O 7, at finite magnetic fields. In Sr 3 Ru 2 O 7, a c-axis magnetic field induces a phase transition below 1.3 K at H 7.8 T and again at H 8.1 T which is first order at low temperatures but becomes second order above about 0.6 K[25]. The authors propose that this is due to a distortion of the Fermi surface as seen in Fig. 1.7a (though the Fermi surface of the spin down electrons should also distort exhibiting a similar spontaneous symmetry breaking). This picture is also supported by studies of lattice nematics[14] where a similar first order transition becoming second order at some finite temperature also occurs. In the absence of a magnetic field, URu 2 Si 2, a heavy fermion system, undergoes a thermodynamic phase transition below 17.5 K whose order parameter has yet to be identified despite the 20 years of development since its original discovery[27, 28]. Recently, this phase transition has been proposed to be that of a Fermi liquid to a nematic spin nematic or helicity ordered state[26] (see 12

26 (a) La 2 xba xcuo 4 (b) YBa 2Cu 3O 6+x Figure 1.8: Suppression of T C in some cuprate superconductors. In La 2 x Ba x CuO 4, T C (black dots) is suppressed to very small temperatures near x 1/8 with maxima on both sides at about 25 K[30]. This suppression has been shown to be due to static stripe ordering[1, 31] with similar behavior in La 1.6 x Nd 0.4 Sr x CuO 4 [2]. In YBa 2 Cu 3 O 6+x, more direct measurements of hole doping by relating it to unit cell lengths in the c direction have revealed similar suppression of T C [32], at hole doping near 1/8, though much weaker than for La 2 x Ba x CuO 4. The origin of this suppression is unknown for certain, but the authors propose that it may also be related to a tendency towards static stripe ordering. also Wu and Zhang[29]), in which the Fermi surface of the spin up electrons moves up while the Fermi surface of the spin down electrons move down as depicted in Fig. 1.7b. This has nematic spin nematic symmetry since rotating the spins by π and further rotating space by π leaves the ground state invariant. Varma and Zhu were able to quantitatively understand discontinuities in thermodynamic functions based on this model and it is hoped that further experiments will continue to strength this proposal. In some cuprate superconductors, including La Ba x Sr x CuO 4 for x = and La 1.48 Nd 0.4 Sr 0.12 CuO 4, there is strong evidence from neutron scattering and resonant soft X-ray scattering (RSXS) of static stripe ordering[1, 2, 31]. In all cases, this ordering is doping dependent, occurs above the superconducting dome and is accompanied by a strong suppression of T C near x = 1/8 doping. In Fig. 1.8a we have shown an example of this type of behavior. Also of note, recently Liang et. al.[32] have made careful measurements on YBa 2 Cu 3 O 6+x of hole doping via unit cell length measurements in the c-direction and found a weak suppression near hole doping of about 1/8. While the cause of this suppression is unknown, the authors propose that it may be 13

27 Figure 1.9: Charge ordering revealed through STM expermiments on Bi 2 Sr 2 CaCu 2 O 8+x in the very underdoped regime[33]. (A) Topography (inset magnification: 2), with data in (B) 12 mev LDOS, (C) 16 mev LDOS, and (D) 22 mev LDOS revealing charge ordering in a checker board or stripe pattern. due to static stripe ordering similar to that found in La 2 x Ba x CuO 4. Within our classification of electronic liquid crystals, these stripe phases are best thought of as a pinned smectics on a lattice and it appears that the one-dimensional nature of this phase competes with the nearby d-wave superconductivity. Based in part on some of these experimental results, Kivelson, Fradkin and Emery proposed that a nematic phase, which has d-wave symmetry and may live in proximity to such static stripes, may provide a mechanism for the large values of T C found in cuprate superconductors. While this may or may not be the case, it is clear from these experimental results that charge inhomogenaity certainly affects T C qualitatively. That electronic liquid crystal phases may actually help raise T C in the cuprates, it is noteworthy to mention that in Bi 2 Sr 2 CaCu 2 O 8+x, scanning tunneling microscope (STM) experiments (for example, Hoffman et. al[33].) measuring the local density of states (DOS) have revealed definitive charge ordering of about 4 lattice spacings with a kind of checker board or stripe pattern well below T C, as shown in Fig. 1.9B-D). Unfortunately, due to the large amount of disorder here, it is difficult to predict what will happen in cleaner samples. Nevertheless, it appears to suggest a 14

28 possible harmony between some kinds of charge ordering and superconductivity which presents a kind of counter-balance to the effect of static stripe ordering. These experiments therefore call for a careful study of the relationship between electronic liquid crystals and superconductivity. 1.3 Strong interactions and bosonization Electronic liquid crystal phases and phase transitions in the systems discussed so far, typically occur in the presense of strong interactions. This makes these phases challenging to study, but fortunately in modern times, there are tools that are naturally suited for this purpose. In particular, we shall find the method of bosonization, which was first developed in one-dimension (for a review, see Sénéchal[34]) and then later extended to higher dimensions by a number of authors[35 38], invaluable. It naturally describes the system in terms of bosons while allowing a reconstruction of the fermion operator for the purposes of studying fermion correlations. Indeed, in the regime where interactions are strong enough to destabolize the Fermi liquid fixed point, a description in terms of bosons is often particularly well suited. As an example of how bosonization may be utilized to understand fermions in the presence of strong correlations, consider the one-dimensional electron gas (which is a non-fermi liquid). In one dimension, interactions are so strong that even in the weakly interacting regime, there are no fermionic low lying excitations of the Fermi sea. More precisely, if we define the existence of low lying fermionic excitations through the scaling limit of the fermion correlator G F (q, ω) = Z ω v + reg, (1.3) q + iɛ sign ω where the fermion residue Z takes values 0 < Z 1, then in one dimension Z = 0. Indeed, perhaps the best definition of a non-fermi liquid is a phase in which Z = 0 so that Z plays a role similar to an order parameter for the Fermi liquid phase. It may also be helpful to think of Z as a surface tension[36, 37]. Since there is no Fermi surface in one-dimension, only Fermi points, there is no surface tension to stabilize the phase. However, in higher dimensions, there is always a surface and the Fermi liquid has a sizable basin of stability with a finite Z. This view can perhaps be understood more intuitively from the role Z plays in the 15

29 n k n k PSfrag replacements Singularity PSfrag replacements Z k k (a) Luttinger Liquid n k (b) Fermi Liquid n k Figure 1.10: The fermion occupation number at zero temperature for weakly interacting Fermions in (a) one and (b) two and greater dimensions. In one dimension there is no discontinuous jump across the Fermi surface, while in two dimensions there is a jump of magnitude Z which can be thought of as a surface tension stabilizing the phase. fermion occupation number n k shown in Fig In this function Z is manifested as a distinct jump across the Fermi surface in two and greater dimensions demonstrating the existence (or effect) of a surface tension that keeps the fermions below the Fermi surface. Now lets look at how the residue vanishes in 1D more carefully using the method of bosonization. Consider a simple model of spinless fermions interacting via a forward scattering density-density interaction with an energy dispersion that is linearized about the left(l) and right(r) Fermi points: S F = η={r,l} ( ) dx [i ψ t η +v η x ψ η πv 4 η ={R,L} ( g + sign (vη v η )h ) ψη ψ η ψη ψ η ] (1.4) where v L = v R = v and in terms of the standard bosonization notation, we have defined g = 2(g 2 + g 4 )/π and h = 2(g 2 g 4 )/π. On the other hand, the equivilant bosonic description of this system is S B = 1 2πv η={r,l} [ dxdt t ϕ η v η x ϕ η (v η x ϕ η ) η ={R,L} ( g + sign (vη v η )h ) v η x ϕ η v η x ϕ η (1.5) ] where by comparison with Eq. 1.4 v η x ϕ η ψ η ψ e ta. Notice that in terms of the bosons, the theory is entirely quadratic. It turns out that this has happened because through the bosonization 16

30 process, the kinetic energy has been approximated by the linearization of the energy dispersion and lifted up to be placed on par with the potential energy. Non-linear terms in the energy dispersion then introduce cubic and quartic interactions in the boson theory. Bosonization can therefore be thought of as somewhat the opposite to mean field theory where the potential energy is approximated through the introduction of an average field and reduced to the level of the kinetic energy. The boson and fermion theories are connected via the operater identity ψ η = 1 2πa e iϕη, (1.6) where a is a short distance cutoff, so that through this relationship, fermion propagators of (1.4) may be computed exactly. In particular, the equal time propagator for right movers, ψ R, is simply: G F (R) (r, 0) = 1 ( a ) 1 2 (K+K 1 ) 2πa r (1.7) where K = (1 + h)/(1 + g) is the Luttinger parameter which takes the value of unity for free fermions (g = h = 0). Hence, we may define an anomalous exponent, η = 1 2 (K + K 1 2) which is generically non-zero in the presence of any finite interaction. Due to this power-law structure, we conclude that the fermion residue vanishes in the presence of even weak interactions. However, it was necessary to employ a non-perturbative method to obtain this result but due to the simple nature of the bosonic description, the loss of the fermionic quasiparticles was not difficult to capture. For the purposes of studying electronic liquid crystals, this type of analysis will prove fruitful since, utilizing the concept that Z is a surface tension, spontaneous Fermi surface distortions may naturally produce a weakening or collapse of Z. Indeed the action S B can be considered a useful toy model, or benchmark model, for the purposes of understanding aspects of a full two or three dimensional problem in which a Fermi liquid description breaks down. 17