A Tensor Form of the Order Parameters

Fluctuating Stripes in Strongly Correlated Electron Systems and the Nematic-Smectic Quantum Phase Transition

Abstract

We discuss the quantum phase transition between a quantum nematic metallic state to an electron metallic smectic state in terms of an order-parameter theory coupled to fermionic quasiparticles. Both commensurate and incommensurate smectic (or stripe) cases are studied. Close to the quantum critical point (QCP), the spectrum of fluctuations of the nematic phase has low-energy “fluctuating stripes”. We study the quantum critical behavior and find evidence that, contrary to the classical case, the gauge-type of coupling between the nematic and smectic is irrelevant at this QCP. The collective modes of the electron smectic (or stripe) phase are also investigated. The effects of the low-energy bosonic modes on the fermionic quasiparticles are studied perturbatively, for both a model with full rotational symmetry and for a system with an underlying lattice, which has a discrete point group symmetry. We find that at the nematic-smectic critical point, due to the critical smectic fluctuations, the dynamics of the fermionic quasiparticles near several points on the Fermi surface, around which it is reconstructed, are not governed by a Landau Fermi liquid theory. On the other hand, the quasiparticles in the smectic phase exhibit Fermi liquid behavior. We also present a detailed analysis of the dynamical susceptibilities in the electron nematic phase close to this QCP (the fluctuating stripe regime) and in the electronic smectic phase.

pacs:
71.10.Hf, 71.45.Lr, 71.10.Ay

I Introduction

The discovery of the high temperature superconductors in the quasi-two-dimensional copper-oxide materials in the late 1980s, and of novel correlated phases in other complex oxides, has brought to the forefront the problem of the physics of strongly correlated electron systems. To this date the understanding of the behavior of these systems remains one of the main open and challenging problems in condensed matter physics. The central conundrum in this field is the fact that these strongly coupled electron systems are best regarded as doped Mott insulators for which both the band theory of metals and the Landau theory of the Fermi liquid (FL) fail.

One characteristic feature of the physics of doped Mott insulators is their inherent tendency to electronic phase separation, frustrated by the effects of Coulomb interactions.Emery and Kivelson (1993); Kivelson and Emery (1994) The ground states resulting from these competing tendencies typically break the translation invariance and/or the point group symmetry of the underlying lattice. From a symmetry point of view, the ground states of doped Mott insulators are charge-ordered phases, which share many similarities with classical liquid crystals, and should be regarded as electronic liquid crystal phases.Kivelson et al. (1998) However, unlike classical liquid crystals, electronic liquid crystals are strongly quantum mechanical states whose transport properties range from insulating to metallic and even superconducting. In contrast with classical liquid crystals, whose ordered phases represent the spontaneous breaking of the continuous translation and rotational symmetry of spacede Gennes and Prost (1993); Chaikin and Lubensky (1998), the electronic liquid crystal phases of strongly correlated systems are sensitive to the effects of the underlying lattice and the symmetry breaking patterns involve the point and space groups, as well as to disorder. More complex ordered states, involving simultaneously charge and spin degrees of freedom, may also arise.Wu et al. (2007)

The sequence of quantum phase transitions described above, electron crystal smectic (stripe) nematic isotropic fluid, representing the progressive restoration of symmetry, is natural from a strong correlation perspective. Indeed, the electron crystal state(s) are naturally insulating (much as in the case of a Wigner crystal), the smectic or stripe phases are either anisotropic metals or superconductors, and the charged isotropic fluids are either metallic or superconducting. While the isotropic metallic phase is essentially a FL (albeit with strongly renormalized parameters), the nematic and smectic metallic phases have a strong tendency to show non-FL character. Indeed, much of the theoretical description of the stripe or smectic phases is usually based on a quasi-one-dimensional analysis, which makes explicit use of this strong correlation physics. Such approaches give a good description of this state deep inside this phase and at energies high compared to a “dimensional crossover” scale below which the state is fully two-dimensional (and strongly anisotropic)Emery et al. (1997); Carlson et al. (2000); Emery et al. (2000); Vishwanath and Carpentier (2001); Granath et al. (2001); Arrigoni et al. (2004); Carlson et al. (2004). Stripe phases (insulating, metallic, and superconducting) have been found in mean-field studies of generalized two-dimensional Hubbard and t-J modelsZaanen and Gunnarsson (1989); Machida (1989); Kato et al. (1990); Poilblanc and Rice (1989); Schulz (1990); Vojta and Sachdev (1999); Vojta et al. (2000); Park and Sachdev (2001); Sachdev (2003); Lorenzana and Seibold (2002); Anisimov et al. (2004); Himeda et al. (2002); Raczkowski et al. (2007).

The same pattern of quantum phase transitions can also be considered in reverse order, with a weak coupling perspective, as a sequence of symmetry breaking phase transitions beginning from the isotropic metal: FL electron nematic electron smectic insulating electron crystal. In this case, one begins with a uniform isotropic metal, well described at low energies by the Landau theory of the FL, with well-defined quasiparticles and a Fermi surface (FS), and considers possible instabilities of the isotropic fluid into a nematic (or hexatic and other such states), as well as phase transitions into various possible charge-density-wave (CDW) phases. The unidirectional CDW-ordered states are the weak coupling analog of the smectic (or stripe) phases, and have the same order parameters as they break the same symmetries. The main difference between a CDW and a smectic resides in the fact that while the CDW arises as a weak coupling (infinitesimal) instability of a FL in which parts of the FS are gappedMcMillan (1975) (which requires the existence of a FS with sharp quasiparticles), the stripe phases do not require such description. While a CDW phase at high energies is essentially a FL, the high-energy regime of a stripe phase is a quasi-one-dimensional Luttinger liquid.Carlson et al. (2000); Emery et al. (2000) A direct quantum phase transition from a FL to a CDW phase is, naturally, possible and this quantum phase transition has been studied in some detail,Altshuler et al. (1995); Chubukov et al. (2005) as well as to a metallic spin-density wave (SDW)Vekhter and Chubukov (2004); v. Löhneysen et al. (2007).

The weak coupling description of an electron nematic phase uses a Pomeranchuk instability of a Fermi liquid statePomeranchuk (1958). Oganesyan, Kivelson, and FradkinOganesyan et al. (2001) showed that the nematic quantum phase transition is a quadrupolar instability of the FS, and gave a characterization of the properties of the nematic Fermi fluid in a continuum model. An electron nematic quantum phase transition has also been found in lattice modelsHalboth and Metzner (2000); Metzner et al. (2003); Dell’Anna and Metzner (2006), which show, however, a strong tendency to exhibit a first-order quantum phase transitionKee et al. (2003); Khavkine et al. (2004); Yamase et al. (2005). Pomeranchuk instabilities in the Landau theory of the FL have also shown the existence of an electron nematic transitionNilsson and Castro Neto (2005); Wölfle and Rosch (2007). Perturbative renormalization group analysis of the stability of the FL in Hubbard-type modelsHonerkamp et al. (2002), as well as high-temperature expansionsPryadko et al. (2004), has also shown that in such models there is a strong tendency to a nematic state. An electron nematic state was shown to be the exact ground state in the strong coupling limit of the Emery model of the copper oxides at low hole dopingKivelson et al. (2004).

The upshot of the work on the electron nematic quantum phase transition is that, at the QCP (if the transition is continuous) and in the nematic phase (in the continuum) the electron quasiparticle essentially no longer exists as an asymptotically stable state at low energies, except along symmetry determined directions in the ordered phase. A full solution of this QCP by bosonization methods has confirmed these results, which were gleaned from mean-field theory, and have also provided strong evidence for local quantum criticality at this QCP Lawler et al. (2006); Lawler and Fradkin (2007).

In this paper we will be interested in the quantum phase transition from an electron nematic phase to a charge stripe phase, a unidirectional CDW. For simplicity we will not consider here the spin channel, which plays an important role in many systems. We will only consider the simpler case of unidirectional order. Extensions to the more general case of multidirectional order are straightforward. Here we develop a quantum-mechanical version of the nematic-smectic transition in a metallic system. This is a quantum-mechanical version of the McMillan-deGennes theory for the quantum phase transition from a metallic nematic phase to a metallic smectic (or CDW) phase. The construction of such a generalization of the McMillan-deGennes theory is the main purpose of this paper.

As it is discussed in detail in subsequent sections, here we will follow the “weak-coupling” sequence of quantum phase transitions described above, beginning with the transition from a FL to an electron nematic, and from the latter to a stripe or unidirectional CDW state. The main advantages of this approach are that it allows to address the fate of the electronic quasiparticles and non-Fermi liquid behaviors as the correlations that give rise to these electronic liquid crystal phases develop, as well as to study the quantum critical behavior following the standard Hertz-Millis approach Hertz (1976); Millis (1993); Sachdev (1999). However, the main disadvantage is that this approach does not do justice to the physics of strong correlation. For this reason, in spite of the important insights that are gained through this line of analysis, this approach cannot explain the physics of the “strange metal” regime observed in the “normal state” of high superconductors where non-Fermi liquid effects are widely reported. To do that would require studying this problem as a sequence of quantum melting transitions. An important first step in this direction has been made by Cvetkovic and coworkers Cvetkovic et al. (2006, 2008); Cvetkovic (2007) who have studied a purely bosonic model of such quantum melting. The inclusion of fermionic degrees of freedom in this strong coupling approach is an interesting but challenging open problem.

We have both conceptual and phenomenological motivations for considering this problem. At the conceptual level the main question is to develop a theory of the quantum critical behavior at the electron nematic-smectic phase transition, and of the low-energy physics of both phases near quantum criticality. Although the static properties are the same as in the classical theory (as required by symmetry) the quantum dynamics changes the physics substantially. Thus, physical properties, which determine the transport properties and the fermion spectral function, cannot be gleaned from the classical problem. Provided that the quantum phase transition is continuous or, at most weakly first order, the low-energy fluctuations in one phase (say the nematic metal) must reflect the character of the nearby ordered stripe phase. In other words, under these assumptions, as the quantum phase transition is approached the metallic nematic phase behaves as a state with “fluctuating stripes”. The ample experimental evidence in high temperature superconductors for “fluctuating stripe order” should be interpreted instead as evidence of a nematic phase proximate to a quantum phase transition to a stripe (or smectic)-ordered stateKivelson et al. (2003).

Ii Summary of Results

In this work we follow a phenomenological approach to study the quantum phase transition between an electronic nematic state and electronic smectic state. We postulate the existence of both an electron nematic and a smectic phases with a possible direct phase transition between them. This physics will be represented by an effective field theory involving the nematic and CDW order parameters. The static part of the effective action of the order-parameter theory has the same form as in the classical theory of the nematic-smectic transition, the McMillan-deGennes theory. We will assume that aside from the effects of the coupling to the fermionic quasiparticles, this effective field theory is analytic in the order parameters and their derivatives as this dependence is determined by local physics. As shown below, this assumption implies a dynamical quantum critical exponent .

The fermionic quasiparticles couple to the nematic and smectic (CDW) order parameters in their natural symmetry-dictated way. The fermions will be assumed to be a normal FL, with well-defined quasiparticles and a FS. Thus, we will not attempt to explain why the phase transition exists, which requires a microscopic theory, but rather describe its character. One of our most important results is that this theory gives a description of a phase with fluctuating stripe (smectic) order, of much interest in current experiments. The effective theory that we consider also allows for a possible direct transition between the normal and isotropic FL state and a CDW phase, without going through an intermediate nematic phase, as in the direct transition between a FL and a CDW state, discussed by Altshuler, Ioffe, and MillisAltshuler et al. (1995). Thus, the theory we present here actually describes the behavior of a FL in the vicinity of a possible bicritical point which, as we shall see, is not directly accessible.

Nematic Smectic Mode at the Electronic Nematic-Smectic QCP Smectic
inflection continuous discrete
incommensurate commensurate point rotational rotational
symmetry symmetry
Anisotropic Scaling
Non-analyticity
Gaussian Fixed Point Stable Stable Unstable / First Order Stable Stable Stable Stable
?
or or
Table 1: Summary of results. See the text for a detailed explanation.

The main results of our theory are summarized in Table 1. In Sec. III we discuss the current experimental status of electronic liquid crystal phases in a number of different materials. In Sec. IV we set up the order parameter theory for the electronic liquid crystal phases based on symmetry and analyticity. The static part of this phenomenological theory is (as it should be) similar to its classical counterpart, but we add proper dynamics to describe the quantum fluctuations. We next couple the order parameter theory to the fermionic quasiparticles, in Sec. V. The coupling between the fermionic quasiparticles and the order parameters is completely determined by symmetry. This is a standard approach to study quantum phase transitions in metallic systems Sachdev (1999). It is a consistent scheme for the study of the quantum phase transition provided the effective dimension is close to 4 (here is the dimensionality of space). Several different non-analytic dependences on the order parameters in the effective action appear as a consequence of their coupling to the fermions. We show that these nonanalytic dynamical terms dominate over the dynamics prescribed phenomenologically. Hence, the dynamics of fermionic liquid crystal phase is very different from that of the simple phenomenological theory. We present a detailed analysis of the behavior of the dynamical susceptibilities in both phases and at the QCP.

The nematic-smectic QCP is studied in Sec. VI. In classical liquid crystals, the Goldstone mode of the nematic phase plays a very important role at the nematic-smectic transition. There, this relevant coupling drives the transition weakly first order through a fluctuation-induced first order transitionHalperin et al. (1974). However, in the case of the electronic liquid crystals, we find that the coupling between the nematic Goldstone mode and the smectic field is actually irrelevant at the electronic nematic-smectic QCP. Therefore, these two modes can be treated separately, as they are weakly coupled to each other. Several different nematic-smectic critical theories are studied, depending on the relation between the magnitude of the ordering wave vector of the CDW, , and the Fermi wave vector, . For (Fig. 2(a)), we find that the critical smectic field has a dynamic critical exponent , which will result in a contribution to the heat capacity. This is a correction to the conventional linear behavior of Fermi liquids. These quantum fluctuations lead to the existence of four points on the FS where the assumptions of FL theory are violated (Fig. 2(a)). At these points the imaginary part of the fermion self-energy correction . For (Fig. 2(b)), the system exhibits anisotropic scaling: , and for the incommensurate CDW, while , and for the commensurate case. Besides, a non-analytic term, where is the smectic order parameter, is generated in the action of the low-energy effective theory. This non-analytic term is relevant under the renormalization group (RG) for the incommensurate case, suggesting a weak, fluctuation-induced, first-order transition. This coupling is irrelevant in the commensurate case. Here we also find two points on the FS (Fig. 2(b)), where the system has marginal FL behavior, with a quasiparticle scattering rate , and a low temperature correction to the heat capacity , which is subleading. We also consider the special case of a CDW caused by a nearly nested FS, for which we find that the low-temperature heat capacity correction , which is also subleading, and the fermions form a FL, with . We also calculated the dynamic CDW susceptibility for both cases. The case will not be discussed here. In the presence of a lattice this case is quite trivial (see Sec. VI) while for it to occur in a continuum system, where it is non-trivial, requires unphysical assumptions.

The smectic phase is discussed in Sec. VII. In the smectic phase the anisotropic scaling associated with the Goldstone fluctuations are , . We find that the low-temperature heat capacity correction , which is also subleading. The quasiparticle scattering rate in this case is for much of the FS while at the two special points where the Fermi velocity is parallel to the ordering wave vector. Thus, in this case fermions behave as a FL. We also calculated both the longitudinal and transverse dynamic CDW susceptibilities in the smectic phase.

Lattice effects are also discussed. For the case of an incommensurate smectic phase, we show that there is an unpinned smectic phase close to the nematic-smectic critical point. In this phase, the smectic Goldstone mode has a dynamic critical exponent and the system is a FL with at most of the FS and at some special point on the FS described below. Due to the unpinned smectic ordering, the system receives a correction to the low-temperature heat capacity , and we also computed the dynamic transverse CDW susceptibility. Deep into the smectic phase, an incommensurate CDW may be pinned down by lattice distortion. As expected, the fermions in a pinned smectic are in a conventional FL state.

In Sec. VIII we present a brief discussion of the role of thermal fluctuations for these phases and of the classical-to-quantum crossovers. We conclude with a summary of our main results and a discussion of open questions in Sec. IX. Details of the calculations are presented in several appendices. In Appendix A we discuss the tensor structure of the order parameters. In Appendix B we present details of the nematic-smectic QCP for the case , while the nonanalytic terms induced for the case are presented in Appendix C. In Appendix D we present details of the calculation of the spectrum of Goldstone modes in the smectic phase. In Appendix E we summarize the random phase approximation (RPA) calculation of the fermion self-energy at the nematic-smectic QCP and in the smectic phase.

Iii Experimental Status of Electronic Liquid Crystal Phases

During the past decade or so experimental evidence has been mounting of the existence of electronic liquid crystal phases in a variety of strongly correlated (as well as not as strongly correlated) electronic systems. We will be particularly interested in the experiments in the copper oxide high temperature superconductors, in the ruthenate materials (notably SrRuO), and in two-dimensional electron gases (2DEG) in large magnetic fields. However, as we will discuss below, our results are also relevant to more conventional CDW systems such as the quasi-two-dimensional dichalcogenides.

iii.1 High temperature superconductors

In addition to high temperature superconductivity, the copper oxide materials display a strong tendency to have charge-ordered states, such as stripes. The relation between charge ordered statesKivelson and Fradkin (2007), as well as other proposed ordered statesChakravarty et al. (2001); Varma (2005), and the mechanism(s) of high temperature superconductivity is a subject of intense current research. It is not, however, the focus of this paper.

Stripe phases have been extensively investigated in high temperature superconductors and detailed and recent reviews are available on this subjectKivelson et al. (2003); Tranquada (2007). Stripe phases in high temperature superconductors have unidirectional order in both spin and charge (although not always) and it is typically incommensurate. In general the detected stripe order (by low-energy inelastic neutron scattering) in LaSrCuO, LaBaCuO and YBaCuO (see Refs.Kivelson et al. (2003) and Tranquada (2007) and references therein) is not static but “fluctuating”. As emphasized in Ref.Kivelson et al. (2003), “fluctuating order” means that there is no true long range unidirectional order. Instead, the system is in a (quantum) disordered phase, very close to a quantum phase transition to such an ordered phase, with very low-energy fluctuations that reveal the character of the proximate ordered state. On the other hand, in LaBaCuO near (and in LaNdSrCuO also near ), the order detected by elastic neutron scatteringTranquada et al. (2004), and resonant x-ray scattering in LaBaCuO Abbamonte et al. (2005) also near , becomes true long-range static order.

In the case of LaSrCuO, away from , and particularly on the more underdoped side, the in-plane resistivity has a considerable temperature-dependent anisotropyAndo et al. (2002), which has been interpreted as an indication of electronic nematic order. From these experiments it has been suggested that this phase be identified as an electron nematicAndo et al. (2002). The same series of experiments also showed that very underdoped YBaCuO is an electron nematic as well. The most striking evidence for electronic nematic order in high temperature superconductors are the recent neutron scattering experiments in YBaCuO at Hinkov et al. (2008). In particular, the temperature-dependent anisotropy of the inelastic neutron scattering in YBaCuO shows that there is a critical temperature for nematic order (with ) where the inelastic neutron peaks also become incommensurate. Similar effects were reported by the same groupHinkov et al. (2006) at higher doping levels () who observed that the nematic signal was decreasing in strength suggesting the existence of a nematic-isotropic quantum phase transition closer to optimal doping. Fluctuating stripe order in underdoped YBaCuO has been detected earlier on in inelastic neutron scattering experiments Mook et al. (2000); Stock et al. (2004) which, in hindsight, can be reinterpreted as evidence for nematic order. However, as doping increases the strength of the temperature-independent anisotropic background, due to the increased orthorhombicity of the crystal, also increases thus making this phase transition difficult to observe.

Recent inelastic neutron scattering experiments have found similar effects in LaSrCuO materials where fluctuating stripes where in fact first discoveredTranquada et al. (1995). Matsuda et al Matsuda et al. (2008) have given qualitatively similar evidence for nematic order in underdoped LaSrCuO () which was known to have “fluctuating diagonal stripes”. In the same doping range it has also been found by resonant x-ray scattering experiments that 5% Zn doping stabilizes a static diagonal stripe-ordered state with a very long persistence length which sets in at quite high temperaturesRusydi et al. (2007).

These recent results strongly suggest that the experiments that had previously identified the high temperature superconductors as having “fluctuating stripe order” (both inside and outside the superconducting phase) were most likely detecting an electronic nematic phase, quite close to a state with long-range stripe (smectic) order. In all cases the background anisotropy (due to the orthorhombic distortion of the crystal structure) acts as a symmetry breaking field that couples linearly to the nematic order, thus rounding the putative thermodynamic transition to a state with spontaneously broken point group symmetry. These effects are much more apparent at low doping where the crystal orthorhombicity is significantly weaker.

The nature of the fluctuating spin order changes substantially as a function of doping: in the very underdoped systems there is no spin gap while inside much of the superconducting dome there is a finite spin gap. In fact in LaBaCuO at there is strong evidence for a complex stripe-ordered state which combines charge, spin and superconducting orderLi et al. (2007); Berg et al. (2007). These experiments have also established that static long-range stripe charge and spin orders do not have the same critical temperature, with static charge order having a higher .

An important caveat to our analysis is that in doped systems there is always quenched disorder, and has different degrees of short range “organization” in different high temperature superconductors. Since disorder also couples linearly to the charge order parameters it ultimately also rounds the transitions and renders the system to a glassy state (as noted in Refs.Kivelson et al. (1998, 2003)). Such effects are evident in scanning tunneling microscopy (STM) experiments in BiSrCaCuO which revealed that the high-energy (local) behavior of the high temperature superconductors has charge order and it is glassyHowald et al. (2003); Kivelson et al. (2003); Hanaguri et al. (2004); Kohsaka et al. (2007); Vershinin et al. (2004).

Finally, we note that in the recently discovered iron pnictides based family of high temperature superconductors, such as La (OF)FeAs Kamihara et al. (2008); Mu et al. (2008), a unidirectional spin-density wave has been found. It has been suggestedFang et al. (2008) that the undoped system LaOFeAs may have a high-temperature nematic phase and that quantum phase transitions also occur as a function of fluorine dopingXu et al. (2008). This suggests that many of the ideas and results that we present here may be relevant to these still poorly understood materials.

iii.2 Other complex oxides

The existence of stripe-ordered phases is well established in other complex oxide materials, particularly the manganites and the nickelates. In general, these materials tend to be “less quantum mechanical” than the cuprates in that they are typically insulating (although with interesting magnetic properties) and the observed charge-ordered phases are very robust. These materials typically have larger electron-phonon interactions and electronic correlations are comparatively less dominant in their physics. For this reason they tend to be “more classical” and less prone to quantum phase transitions. However, at least at the classical level, many of the issues we discussed above, such as the role of phase separation and Coulomb interactions, also play a key roleDagotto et al. (2001). The thermal melting of a stripe state to a nematic has been seen in the manganite material BiCaMnORübhausen et al. (2000).

iii.3 Ruthenates

Recent magneto-transport experiments in the quasi-two-dimensional bilayer ruthenate SrRuO by the St. Andrews groupBorzi et al. (2007) have given strong evidence of a strong temperature-dependent in-plane transport anisotropy in these materials at low temperatures mK and for a window of perpendicular magnetic fields around Tesla. These experiments provide strong evidence that the system is in an electronic nematic phase in that range of magnetic fieldsBorzi et al. (2007); Fradkin et al. (2007). The electronic nematic phase appears to have preempted a metamagnetic QCP in the same range of magnetic fieldsGrigera et al. (2001); Millis et al. (2002); Perry et al. (2004); Green et al. (2005). This suggests that proximity to phase separation may be a possible microscopic mechanism to trigger such quantum phase transitions, consistent with recent ideas on the role of Coulomb-frustrated phase separation in 2EDGsJamei et al. (2005); Lorenzana et al. (2002).

iii.4 2DEGs in large magnetic fields

To this date, the best documented electron nematic state is the anisotropic compressible state observed in 2DEGs in large magnetic fields near the middle of a Landau level, with Landau index Lilly et al. (1999a, b); Du et al. (1999); Pan et al. (1999). In ultrahigh-mobility samples of a 2DEG in AlAs-GaAs heterostructures, transport experiments in the second Landau level (and above) near the center of the Landau level show a pronounced anisotropy of the longitudinal resistance rising sharply below mK, with an anisotropy that increases by orders of magnitude as the temperature is lowered. These experiments were originally interpreted as evidence for a quantum Hall smectic (stripe) phase Koulakov et al. (1996); Moessner and Chalker (1996); Fradkin and Kivelson (1999); MacDonald and M. P. A. Fisher (2000); Barci et al. (2002). Further experimentsCooper et al. (2001, 2002, 2003) did not show any evidence of pinning of this putative unidirectional CDW as the curves are strictly linear at low bias and no broadband noise was detected. In contrast, extremely sharp threshold electric fields and broadband noise in transport was observed in a nearby reentrant integer quantum Hall phase, suggesting a crystallized electronic state. These facts, together with a detailed analysis of the experimental data, suggested that the compressible state is in an electron nematic phaseFradkin and Kivelson (1999); Fradkin et al. (2000); Wexler and Dorsey (2001); Radzihovsky and Dorsey (2002); Doan and Manousakis (2007), which is better understood as a quantum melted stripe phase.

iii.5 Conventional CDW materials

CDWs have been extensively studied since the mid-seventies and there are extensive reviews on their propertiesGrüner (1988, 1994). From the symmetry point of view there is no difference between a CDW and a stripe (or electron smectic). The CDW states are usually observed in systems which are not particularly strongly correlated, such as the quasi-one-dimensional and quasi-two-dimensional dichalcogenides, and the more recently studied tritellurides. These CDW states are reasonably well described as FLs which undergo a CDW transition, commensurate or incommensurate, triggered by a nesting condition of the FSMcMillan (1975, 1976). As a result, a part or all of the FS is gapped in which case the CDW may or may not retain metallic properties. Instead, in a strongly correlated stripe state, which has the same symmetry breaking pattern, at high energy has Luttinger liquid behaviorKivelson et al. (1998); Emery et al. (2000); Carlson et al. (2004).

What will interest us here is that conventional quasi-2D dichalcogenides, the also quasi-2D tritellurides and other similar CDW systems can quantum melt as a function of pressure in TiSeSnow et al. (2003), or by chemical intercalation as in CuTiSeMorosan et al. (2006); Barath et al. (2008) and NbTaSDai et al. (1991). Thus, CDW phases in chalcogenides can serve as a weak-coupling version of the problem of quantum melting of a quantum smectic. Interestingly, there is strong experimental evidence that both TiSeSnow et al. (2003) and NbTaSDai et al. (1991) do not melt directly to an isotropic Fermi fluid but go instead through an intermediate phase, possibly hexatic. (CuTiSe is known to become superconductingMorosan et al. (2006).) Whether or not the intermediate phases are anisotropic is not known as no transport data is available in the relevant regime.

The case of the CDWs in tritellurides is more directly relevant to the theory we present in this paper. Tritellurides are quasi-2D materials which for a broad range of temperatures exhibit a unidirectional CDW (i.e. an electronic smectic phase) and whose anisotropic behavior appears to be primarily of electronic originBrouet et al. (2004); Laverock et al. (2005); Sacchetti et al. (2006, 2007); Fang et al. (2007). However, the quantum melting of this phase has not been observed yet. Theoretical studies have also suggested that it may be possible to have a quantum phase transition to a state with more than one CDW in these materialsYao et al. (2006).

Iv Order-Parameter Theory

In this section we will construct, using phenomenological arguments, an effective order parameter theory that will describe both the electron nematic and the electron smectic (or unidirectional CDW) phases. Although by symmetry the order parameter theory must be very similar to the ones used in classical liquid crystal phases, we will go through the construction of the phenomenological theory in some detail for several reasons. In 2D the rotation group is Abelian which allows for a significant simplification of the formulas by using a complex order parameter for the nematic phase, instead of a tensor expressions commonly used for 3D classical liquid crystals. Proper dynamical terms now need to be included to describe the quantum fluctuations at zero-temperature. Besides, in order to provide a clear relation between this paper and earlier studies of the CDW state of fermions, we would like to discuss also the relation between the smectic phase and the CDW state.

iv.1 The normal-electronic nematic transition

The nematic order parameter in 2D is a representation of the rotational group Oganesyan et al. (2001). It is defined as a symmetric traceless tensor of rank two.

(1)

The 2D rotational group is isomorphic to . Hence, we define instead the complex order-parameter field

(2)

where and are the space and time coordinates. We will use this complex order parameter field in this paper to take the advantage of the Abelian nature of .

The conjugate field is . Under a global rotation by an angle , the fields and transform, respectively, as and . Hence, and carry the angular momentum quantum numbers and , respectively.

This complex order parameter can be generalized easily to other angular momentum channels , but not to higher dimensions , since it relies heavily on the special property of the 2D rotational group . In higher dimensions, the rotational group will no longer be Abelian, so one will need to use the tensor formula as in the classical liquid crystal theories. In Appendix A, formulas using the complex order parameter are translated into the conventional tensor form for comparison.

The order-parameter field we just defined is invariant under spatial-inversion and time-reversal

(3)
(4)

In even spatial dimensions, including 2D in which our system lives, a chiral transformation is different from a space inversion. To change the chirality in 2D, we can reverse the direction and keep the direction unchanged. Under this chiral transformation, the nematic field will be changed into the conjugate field

(5)

Here, is the chiral transformation operator.

The effective action must preserve the symmetries of the system, both continuous, as the translational and rotational symmetries, and discrete, as the time reversal, spatial inversion and chiral symmetries. With the assumption of analyticity, the action must be

Here the dynamical term is quadratic in time derivatives. This is because the term linear in time derivatives is not allowed by the chiral symmetry. It is the imaginary part of . It corresponds to a pseudoscalar, and is not allowed.

In 2D, cubic terms in the nematic field are not allowed. Hence, if , the normal-nematic transition is second order, instead of a first-order transition as in the 3D case de Gennes and Prost (1993); Chaikin and Lubensky (1998). For and , the rotational invariant ground state will be stable. When becomes negative, will develop an expectation value with module , which breaks the rotation symmetry. The residual rotational symmetry would be . The argument of determines the direction of the nematic order parameter.

The action of Eq. (LABEL:eq:N_action) has an internal symmetry associated with the phase of the complex field , which is not physical. By symmetry, terms of the form

(7)

are allowed Oganesyan et al. (2001); chi (). This kind of terms are irrelevant at the QCP and in the isotropic phase, which leads to the existence of an “emergent” internal symmetry at quantum criticality. But it will be important in the nematic phase, as it makes the two Frank constants to attain different values. (This effect is formally analogous to the role of spin-orbit interactions in the Schrödinger equation: in their absence spin is an internal degree of freedom.) This emergent symmetry of the normal phase and at the critical point is very important for the classical normal-nematic transition, especially in the study about the fluctuation effects Priest and Lubensky (1976); Korzhenevskii and Shalaev (1979); Nelson and Toner (1981).

iv.2 The electronic nematic phase

In the nematic phase, the rotational symmetry is broken. Hence, we expect the fluctuations of the amplitude of the nematic order parameter, , to correspond to a massive mode with an energy gap of (), and the fluctuations of the phase, , constitute the gapless Goldstone mode. Without loss of generality, throughout this paper, we assume that is real and positive. This state corresponds to a nematic order in the main axis direction. In this state, the action of is

where and are the two Frank constants. This action is only valid for small nematic fluctuations. It cannot be used to study topological defects of the nematic phase, known as disclinationsde Gennes and Prost (1993). The field has dynamic critical exponent . This makes the effective dimension of this system , which is above the lower critical dimension of the theory , and nematic order will not be destroyed by fluctuations.

iv.3 CDW multi-critical point

The smectic order is a unidirectional CDW, described by a single complex order parameter field. If we assume analyticity, the effective low-energy theory of the bosonic field can be determined as:

(9)

where is the term in the quadratic order of , and are the cubic and quartic terms respectively.

The term in the momentum space is,

(10)

The function has the physical meaning of the inverse of the CDW susceptibility. If we assume the ordering wave-vector of the CDW is (with its magnitude), will have the form

(11)

where is the energy gap of the CDW excitations Brazovskii (1975), and is a positive constant.

When the energy gap decreases to zero, all the density wave modes with will become soft and critical when . This is very different from an ordinary - or -theory, where we only need to consider one mode (or two modes for a complex field) at small momentum. Here, we need to consider all the modes with the wave vector whose magnitude is close to . In other words, the point is not a critical point but a multi-critical point with an infinite number of critical modes. Even if a lattice background is present, the point may still be a multi-critical point of critical modes, if the lattice has a -fold rotational symmetry for . For a multi-critical point, higher-order terms become important. Without a detailed knowledge of these higher-order terms, it is not possible to determine whether the transition is first or second order, or how many CDWs will be formed in the ordered phase.

Brazovskii Brazovskii (1975) studied the classical version of this problem, considering only the isotropic interactions. Chubukov and co-workers Chubukov et al. (2005) studied the quantum problem in a fermionic system in the high-density regime where the cubic and quartic terms of can be ignored.

In general, depending on the non-Gaussian terms, the ordered phase may have only one or several CDWs Brazovskii (1975). For a rotational invariant system, it is often assumed that CDWs form a triangular lattice to minimize the breaking of the rotational symmetry, as the 2D Wigner crystal state Wigner (1934). For systems with a strong lattice potential, the system is often assumed to become an electron crystal state which preserves the point group rotational symmetry of the background lattice, e.g. the rare-earth tritelluridesYao et al. (2006).

For isotropic systems, outside the nematic phase, the term in Eq. (9) favors that three CDWs form by a first-order transition Brazovskii (1975). However, inside a nematic phase, as we will show below, the nematic order parameter, which is coupled to , favors only one CDW and will compete with the term. For a continuous quantum phase transition, will be a subleading perturbation compared to , at least close enough to the transition. Hence, the smectic phase, a unidirectional CDW, will be energetically favorable. On the other hand, in the case of a first-order transition, depending on microscopic details either the smectic phase or the state with three CDWs would be preferred. We represent these different possibilities in the schematic phase diagram shown in Fig. 1.

On a square lattice, due to the point group symmetry of the lattice, the electron crystal phase usually consists of CDWs perpendicular to each other. The phase transition between this phase and the FL may be second order due to the absence of the cubic term which, in contrast to isotropic systems, is prohibited by momentum conservation. We have confirmed this structure of the phase diagram in a microscopic mean-field calculation. However, at the multi-critical point where both the CDW modes and the nematic mode are critical (the point in Fig. 1), the coupling between CDWs and the nematic order parameter (Eq. (12)) is relevant. This suggests a fluctuation driven first-order transition near the multi-critical point. Hence, this multi-critical point is essentially unreachable.

Figure 1: (online color) Schematic phase diagram at as a function of and defined in Eqs.(LABEL:eq:N_action) and (11). The cross point of the two dash lines is the multi-critical point . The red thick lines stand for first order phase boundaries. Other phase boundaries may be first or second order. More complex electron crystal phases are possible, for example, an anisotropic electron crystal phase where more than one CDWs and nematic coexist, but they are beyond the discussion of this paper.

In this paper, we study the nematic-smectic phase transition and the smectic phase using a weak coupling approach by perturbing about a FL state. This approach is consistent provided the nematic phase is narrow enough in coupling constant space so that the nematic-smectic transition is not too far from the FL phase.

It is useful to compare to the classical version of this problem. The theory of classical (thermal) melting in two dimensions, the Kosterlitz-Thouless-Halperin-Nelson-Young theoryNelson and Halperin (1979); Young (1979) (see Ref.Chaikin and Lubensky (1998)), is a theory of a phase transition driven by the proliferation of topological defects: a dislocation unbinding transition in the case of melting of a 2D Wigner crystal (a triangular lattice) into a hexatic phase, and disclination unbinding transition in the hexatic-isotropic phase transition. (The case of the square lattice was discussed only recently in Ref.DelMaestro and Sachdev (2005)). The reason for the success of the classical theory of melting in two dimensions is that, as in all Kosterlitz-Thouless phase transitionsKosterlitz and Thouless (1973); Chaikin and Lubensky (1998), at finite temperatures the classical ordered state with a spontaneously broken continuous symmetry is not possible in two dimensions. Instead, there is a line (or region) of classical critical behavior with exactly marginal operators. The defect-unbinding phase transition appears as an irrelevant operator becoming marginally relevant.

In the case of the quantum phase transitions in two dimensions that we are interested in, there are no such exact marginal operators available at zero temperature, and hence, no lines of fixed points available. Thus, the quantum phase transition is not triggered by a defect-unbinding operator becoming marginal, but instead by making the coupling constant of an irrelevant operator large (as in standard continuous phase transitions, classical or quantum). Instead, the quantum phase transition is closer to Landau-type (or, rather, Hertz-Millis like) description in that it is governed (as we will see) by a quantum-mechanical analog of the celebrated McMillan-deGennes theory for a nematic-smectic phase transition in classical liquid crystals in three dimensionsde Gennes and Prost (1993); Chaikin and Lubensky (1998). The approach that we will pursue here does not contain much of the physics of strong correlations as it begins with a state with well-defined fermionic quasiparticles. It also does not treat correctly the tendency of strongly correlated systems to exhibit inhomogeneous states and phase separation. The only way to account for this physics correctly is to use the opposite approach, a strong coupling theory of quantum melting of the crystal and stripe phases, as advocated in Ref.Kivelson et al. (1998). So far, this theory only treats the physics deep inside a stripe phases, and the theory of their quantum melting to a nematic phase does not yet exist. Thus, although from a strong-coupling perspective it would be highly desirable to have such a defect unbinding theory of this quantum phase transition (such a description does exist for an insulating systemZaanen et al. (2004) but its extension to a metallic state is not available and it is highly non-trivial), we will pursue instead a Hertz-Millis approach Hertz (1976); Millis (1993); Sachdev (1999) to this quantum phase transition.

iv.4 The electronic nematic-smectic transition

Nematic order will remove the degeneracy of CDW modes in different directions and select one CDW. As a result, the Brazovskii CDW multi-critical point becomes just a critical point. For simplicity, we assume that the nematic order parameter is small enough so that a Landau-type expansion still makes sense, which is equivalent to assuming that the system is still “close enough” to the nematic-isotropic QCP. However, as we will show later, the critical theory we get using these assumptions has the only form allowed by symmetry, assuming analyticity.

By symmetry, the coupling between the CDW and the nematic field is

(12)

whose tensor form is shown in Appendix A. Here, is the polar angle of . This term is irrelevant in the isotropic phase, but in the nematic phase, where gets the expectation value ; this term will be of the same order as , which was defined in Eq. (10), and hence it becomes important.

Inside the nematic phase the amplitude fluctuations of the nematic order parameter are gapped while the orientational fluctuations, the nematic Goldstone modes, are gapless, at least strictly in the absence of a lattice and other orientational symmetry breaking couplings. Thus, deep enough in the nematic phase it is possible to integrate out the gapped nematic amplitude fluctuations and derive an effective theory involving the gapless nematic Goldstone mode. However, as the nematic-smectic phase transition is approached, the gap of the fluctuations of the smectic order parameter will get smaller and will approach zero at the QCP. Thus, in this regime, the nematic phase has low-energy “fluctuating stripes”. This regime is the analog of that in conventional liquid crystals where the McMillan-deGennes classical theory applies . We will now see how this theory arises in the quantum case.

The leading term in of Eq. (12) will be

(13)

This term will stabilize the density wave in either or direction and destabilize the other, depending on the sign of . As a result, the nematic order will select a special direction along which only one CDW will form. Past this phase transition the system will be in a smectic state, a unidirectional CDW. For simplicity, we assume , which selects in the direction.

Only the density fluctuations close to matter for the low-energy theory. We define a complex field , describing the density fluctuations around as

(14)

where is small. The real part of measures the density fluctuations.

Under a spatial inversion, will become its conjugate field . Hence, the term is not allowed in the Lagrangian, and the dynamical term for is at least quadratic in time derivatives.

The cubic term of the field in the isotropic-CDW transition vanishes in the nematic-smectic transition, due to momentum conservation. By expanding Eq. (12) around , and , we obtain

Here is the action of the nematic Goldstone mode defined in Eq. (LABEL:eq:nematic_goldstone).

The action of Eq. (LABEL:eq:ns_action) is just a 2D version McMillan-de Gennes theory of the nematic-smectic transition in the classical liquid crystals but with quantum dynamics. The constants in Eq.(LABEL:eq:ns_action) are

(16)

Here is the energy gap of field, which mainly comes from the CDW gap defined in Eq. (11). The correction term comes from the nematic ordering. The term comes from the interactions between CDWs and it gets a correction from the amplitude fluctuations of the nematic order, which has been integrated out. The nematic Goldstone field couples to the CDW field as a gauge field with a “charge” . Here the two in the denominator comes from the fact that the nematic order parameter has an angular momentum . This gauge-like coupling is required by the rotational symmetry since, under spatial rotation by a small angle , the fields transform as and (for the angular momentum channel ). In fact, with the symmetry constrain and the assumption of analyticity, the action we show in Eq. (LABEL:eq:ns_action) is the only allowed form for the effective low-energy theory, provided the topological excitations of are ignored Renn and Lubensky (1988). Therefore, although we only keep linear terms of in our calculations above, which is valid close to the normal-nematic critical point, the action in Eq. (LABEL:eq:ns_action) will have the same form even deep inside the nematic phase.

The theory with the effective action given in Eq.(LABEL:eq:ns_action) has a critical field and gapless Goldstone boson . A naive mean-field theory would suggest that this is a continuous phase transition. In the case of the theory of classical liquid crystals, where the same naive argument also holds, Halperin, Lubensky and Ma Halperin et al. (1974) used the expansion to show that there is a run-away behavior in the renormalization group flows, similar to that of superconducting transition coupled to a fluctuating electromagnetic field. They concluded that in both cases the transition is probably weakly first order, a fluctuation-induced first-order transition. In other terms, in the classical theory the coupling of the smectic to the nematic Goldstone mode (which has the same form as a coupling to a gauge field) is relevant. To ascertain what happens in the case of the metallic nematic-smectic QCP we will also need to take into account the effects of the fermionic degrees of freedom. We will see that the fermionic fluctuations change the critical behavior in an essential way.

iv.5 The electronic smectic phase: a unidirectional CDW

In the smectic phase, the amplitude fluctuations of the order parameter, , are gapped but the phase fluctuations, , are gapless, as required by the Ward-identity. This happens in systems for which lattice effects can be neglected, and hence are described formally in a continuum, or if the smectic order is sufficiently incommensurate. Therefore, upon integrating out the gapped amplitude fluctuations , the effective low-energy theory of the Goldstone mode becomes

(17)

When we are close to the nematic-smectic critical point, the coefficients of this effective action are

(18)

with being the expectation value of the CDW order parameter. The vanishing of the stiffness term Peierls (1935); Landau (1937) is required by the Ward identity of rotational invariance. Thus, an underlying lattice, which will break the continuous rotational symmetry down to its discrete point group, will lead to a non-vanishing stiffness. Nevertheless, in many cases and particularly away from situations in which the FS is strongly nested, the breaking of rotational invariance can be parametrically small enough that at low temperatures its effects to a first approximation can be neglected and treated perturbatively afterward.

A simple scaling analysis of the effective action of Eq. (17) shows that, at the tree level, the scaling dimensions of space and time , and , are , and , respectively. Although the time direction and the direction scales in the same way, the and directions now scale differently. This is a typical phenomenon for anisotropic states. Now the effective dimensions of this quantum theory is . Hence, our theory is above its (upper) critical dimension. So the higher-order interactions of the smectic Goldstone mode will be irrelevant, if we don’t consider topological defects. The fact that we are above the critical dimension also tells us that the quantum fluctuations in the quantum smectic phase will not destroy the long-range order.

This scaling is very different from the classical smectic phase of 3D, where and , if the modulation is on the direction. This classical theory is at its lower critical dimension, and long-range order is destroyed by fluctuations Peierls (1935); Landau (1937), resulting in a power-law quasi-long-range order. This system has a line of critical points, so the higher order terms of the action that need to be considered were found to lead to logarithmic corrections to the power-law behavior Grinstein and Pelcovits (1981).

The above analysis implies that our quantum problem is above the lower critical dimension. Therefore all these effects of the 3D classical smectic phase will not be present in the 2D quantum case. The scaling behavior of a 2D quantum system is similar to the columnar state of the classical liquid crystals, instead of that of classical smectics. The classical columnar state has two density waves so that it is a solid in two directions but a liquid in the third direction. The Goldstone fluctuations of this state scale as and de Gennes and Prost (1993), which is the same as in the present case, if we consider the time direction in our problem as the direction. The difference between the classical columnar state and the 2D quantum smectic state is that in the 3D columnar state, the Goldstone mode is a planar vector but in the present case it is a scalar.

V Coupling the Order Parameter Theory to Fermions

(a)  
(b)  
(c)  
(d)  
Figure 2: (online color) The FS of the nematic phase (a and b) and the reconstructed FS in the smectic phase (c and d). (a) and (c) are for at the QCP and in the smectic phase respectively, while (b) and (d) are , also at the QCP and in the smectic phase respectively. In (a) and (b), the black dots marked the non-FL points on the FS caused by the smectic mode fluctuations at the nematic-smectic QCP. The relevance of the points in (c) is explained in Sec. VII. In (c) we have show the case of to be comparable to so as to keep the FS reconstruction simple. Here we show the effective Brillouin Zone with an open orbit and a closed pocket. The reconstructed FS of case (d) is partially gapped and the FS has an open orbit.

We will now proceed to couple the phenomenological theory of the nematic and smectic phases to a system of a priori well-defined fermionic quasiparticles described by the Landau theory of the FL. In a fermionic liquid crystal state, the bosonic order-parameter fields, defined above, will couple to the fermions.

Let us define and to be the fermion creation and annihilation operators of a FL. We will assume that the FL has a well-defined FS, which for simplicity we will assume is circular. (For lattice systems the FS will have the symmetry of the point group of the lattice.) The Fermi wave vector is . The Fermi velocity is set to so that the energy and momentum have the same units. Consistent with the assumptions of the Landau theory of the FLBaym and Pethick (1991) the effective Hamiltonian of the fermionic quasiparticles will be taken to be that of a free Fermi system, with a well-defined FS, and a set of quasiparticle interactions parametrized by the Landau parameters. These interactions are irrelevant in the low-energy limit of the FL but play an important role in the physics of electronic liquid crystal phases Oganesyan et al. (2001). In any case in our discussion it will be unnecessary to include the Landau parameters explicitly since their effects will already be taken into account through the coupling to the liquid crystal order parameters.

By symmetry, the nematic order-parameter field, , couples to the fermion density quadrupole Oganesyan et al. (2001)

(19)

In 2D, since the rotational group is , the density quadrupole can be defined in terms of a two-component real director field (i.e. a headless vector) or, in terms of complex field

(20)

Same as the nematic order parameter, is also invariant under rotations by .

The coupling between and is

(21)

Here is a coupling constant. Again, the chiral symmetry of the system requires that the effective action depends only on the real part of , and that there is no dependence on the imaginary part, since it is a pseudo scalar. The tensor form of this coupling is shown in Appendix A. In what follows we choose the sign of to be negative, so that a positive expectation value of the nematic order parameter means a FS stretched along the direction and compressed in the direction, as shown in Figs. 2(a) and (b).

The sign of alone is not important. What matters is the relative sign between and the coupling constant defined in Eq. (12). Under a redefinition of becoming , both and change sign. If , prefers the direction in which the FS is stretched, but when , it prefers the direction where the FS is compressed. In general, the sign of is determined by microscopic details of the system to which this model may apply.

If , very close to a nesting condition the curvature of the FS controls the CDW instability as it controls how singular the charge susceptibility is near the nesting wave vector. In this case one finds that it leads to the condition , when connects two points on the Fermi surface where the curvature is smallest, as shown in Fig.2(b). In general, far from a nesting condition, the curvature of the FS alone is not the dominant factor, and the sign of may be positive or negative, depending on the microscopic details.

The smectic order-parameter field should be coupled to the CDW of the fermions. The CDW operator of the fermions, close to the ordering wave vector , is

(22)

where . The smectic order-parameter field couples to this fermion density wave as

(23)

Integrating out the bosons, attractive four-fermion interactions are generated of the form

(24)

Hence, the order-parameter fields can be regarded as Hubbard-Stratonovich fields used to decouple four-fermion interactions. In this picture, the couplings between the order-parameter fields and fermions are measuring the strength of the attractive four-fermion term.

Gapless fermions will introduce nonanalytic terms to the low-energy effective theory of the nematics and smectics. For the case of the nematic order parameter, it was shown by Oganesyan and co-workers Oganesyan et al. (2001) that the fermions generate nonanalytic Landau damping termsHertz (1976); Millis (1993), so the theory of the isotropic-nematic metallic QCP becomes

(25)

The nematic susceptibility at this FL-nematic QCP isOganesyan et al. (2001)

(26)

The phase mode of the nematic order-parameter field in the nematic phase, the nematic Goldstone mode, has an effective action of the form

where is the expectation value of the nematic order parameter and is the angle between and the main axis direction of the nematic ordering. The stiffnesses and (the Frank constants) are given in Ref.Oganesyan et al. (2001). With this action, it follows that the transverse nematic susceptibility in the electron nematic phase isOganesyan et al. (2001)

(28)

For the case of a nematic order parameter aligned along the -axis, the angular factor becomes .

For the case of a charged smectic, a unidirectional CDW, a similar effect will be observed. Besides, if connects to points on the FS which have just the opposite Fermi velocity as shown in Fig. 2(b), the discontinuity leads to another type of nonanalytic terms as will be shown in Sec. VI.2.

Vi the Nematic-Smectic Metallic Quantum Critical Point

In this section, we study the metallic nematic-smectic QCP. The two cases shown in Figs. 2(a) , and (b) , are studied separately.

Deep in the nematic phase, the amplitude fluctuations of the nematic order parameter are gapped, and the low-energy fluctuations are due to the nematic Goldstone mode, , whose action is given in Eq. (LABEL:eq:quantum_nematic_goldstone). However, as the nematic-smectic QCP is approached (from the nematic side) the fluctuations of the smectic order parameter become progressively softer and, provided the quantum phase transition is continuous, become gapless at this QCP. In this scenario, the nematic phase looks like a “fluctuating stripe” phase qualitatively similar to the phenomenology of the cuprate superconductors, as discussed in Sec. III.1.

The case will not be discussed here. The reason is that since now the CDW fluctuations with cannot decay into particle-hole pairs, in this case fermions only renormalize the coefficients of the smectic effective action, while the nematic fluctuations will still be Landau damped. For isotropic systems and for , the CDW (Lindhard) susceptibility in general decreases faster than linear as , where is a constant. This implies that a CDW with is unlikely to be realized as it would require an anomalously attractive interaction at a large . However, for a lattice system the phase fluctuations of the nematic mode get gapped by lattice anisotropies and in this case the fermions only yield the trivial effect of renormalizing the coefficients of the effective action at the CDW transition.

vi.1

For , the leading contribution to the effective action of the order parameter field, resulting from integrating out the fermions, has the form

(29)

Here is the CDW susceptibility of the fermions, given by the fermion loop integral (bubble)

(30)

where is the Fermi-Dirac distribution function.

The static part of the fermion CDW susceptibility depends on the details of the dispersion relation from way above the FS to the bottom of the band. However, since is analytic for , the static part, , will not change the analytic structure of Eq. (LABEL:eq:ns_action), but just renormalize the coefficients, in particular the critical value of the coupling constant. The important contribution comes from the dynamical part, . The singular contributions to this integral are dominated by the behavior of the integrand around the four points on the FS, which are connected by the ordering wave vector , as marked with black dots on Fig. 2(a). If we expand the dispersion relation of the fermions around these four points, , to leading order we get a Landau damping contribution

(31)

which is linear in . The formula above can be checked by taking the limit of or . In these two regimes, the fermion loop integral can be computed by RPA without expanding the dispersion relations around the four points. After setting , for , one finds with being the density of states and for , , which can be reached by expanding Eq. (37). Both of them agree with the general formula given above.

The term linear in in the effective action for the smectic field of Eq. (31), which is due to the contributions of the fermions, dominates over the “naive” dynamical term proportional to of the phenomenological theory. We can thus write an effective action for the electron nematic-smectic quantum phase transition of the form

where , and . The point and is the nematic-smectic critical point. With the nonanalytic dynamical term, the dynamic critical exponent of the field becomes , instead of as it would generally be in the absence of fermions (or, if the fermions were gapped as in the case of an insulator).

The nematic Goldstone mode has a dynamic critical exponent Oganesyan et al. (2001), larger than the exponent for the smectic fluctuations. Thus, the Goldstone mode of the nematic order parameter and the smectic fluctuate on very different energy scales, with being the low-energy mode. If we only focus on the asymptotic low-energy theory, we should integrate out the high-energy mode . This process will lead to an effective theory of . In turn, the low-energy mode will mediate interactions of the field . However, we will show by a scaling argument that in the case of the quantum metallic system the coupling between the smectic field and the nematic Goldstone mode is irrelevant.

The action of Eq.(LABEL:eq:quantum_mcmillan) is invariant under a rescaling parametrized by a factor

(33)

where , , , , and are the stiffness in Eq. (LABEL:eq:quantum_mcmillan).

When , at the tree level and in the long-wavelength regime, both the gauge-like “coupling constant” and scale to infinite, but the ratio scales to as a function of . This implies that the gauge-like coupling is irrelevant. Quantum fluctuations may change the tree-level scaling behavior as we include loop corrections. However, for large enough or small enough , the irrelevancy of the gauge-like coupling will not be changed. As a byproduct, we notice that and scale to zero in the long-wavelength regime, which means that these two terms are irrelevant also. However, we should keep in mind that these operators are actually dangerous irrelevant, in the sense that is necessary to find the proper equal-time correlation function for and is necessary for stability in the ordered phase, and they are only irrelevant at this QCP.

Notice that at the QCP there are two critical modes: the amplitude of the CDW order parameter, which has , and the transverse (Goldstone) mode of the nematic phase, which has (and it is clearly dominant at low enough energies). Thus, we also need to check the scaling behavior of and for the high-energy mode. Under this rescaling,

(34)

At the critical point where , it can be seen that also scales to as in the long-wavelength limit, which means, for the mode, the gauge-like coupling is still irrelevant.

These conclusions are confirmed by one-loop perturbation theory calculations, presented in Appendix B, where we show that integrating out (or ) does not change the action of (or ). This is one of our main results.

Figure 3: (color online) The spectral density of the smectic susceptibility at the nematic-smectic QCP, , as a function of and for . The spectral density is singular near the origin (lower right corner) and decays monotonically away from there. Here we show contour plots at constant spectral density with values from up to . The red line, , marks the peak of the spectral density as a function of momentum parallel to the nematic orientation. The inset is the energy dependence of at a fixed small momentum (along the dashed vertical line).

In conclusion, there are two essentially decoupled soft modes at the nematic-smectic QCP. The nematic Goldstone mode is governed by the same action as in the nematic phase, Eq. (LABEL:eq:quantum_nematic_goldstone). Since as the nematic-smectic QCP is approached from the nematic side, the nematic Goldstone mode and the smectic order parameters effectively decouple; the effective action for the smectic field in this limit reduces to

(35)

which implies that the dynamic smectic susceptibility is