A Fourier space coordinate systems

Dual gauge field theory of quantum liquid crystals in two dimensions

Abstract

We present a self-contained review of the theory of dislocation-mediated quantum melting at zero temperature in two spatial dimensions. The theory describes the liquid-crystalline phases with spatial symmetries in between a quantum crystalline solid and an isotropic superfluid: quantum nematics and smectics. It is based on an Abelian-Higgs-type duality mapping of phonons onto gauge bosons (“stress photons”), which encode for the capacity of the crystal to propagate stresses. Dislocations and disclinations, the topological defects of the crystal, are sources for the gauge fields and the melting of the crystal can be understood as the proliferation (condensation) of these defects, giving rise to the Anderson-Higgs mechanism on the dual side. For the liquid crystal phases, the shear sector of the gauge bosons becomes massive signaling that shear rigidity is lost. After providing the necessary background knowledge, including the order parameter theory of two-dimensional quantum liquid crystals and the dual theory of stress gauge bosons in bosonic crystals, the theory of melting is developed step-by-step via the disorder theory of dislocation-mediated melting. Resting on symmetry principles, we derive the phenomenological imaginary time actions of quantum nematics and smectics and analyze the full spectrum of collective modes. The quantum nematic is a superfluid having a true rotational Goldstone mode due to rotational symmetry breaking, and the origin of this ‘deconfined’ mode is traced back to the crystalline phase. The two-dimensional quantum smectic turns out to be a dizzyingly anisotropic phase with the collective modes interpolating between the solid and nematic in a non-trivial way. We also consider electrically charged bosonic crystals and liquid crystals, and carefully analyze the electromagnetic response of the quantum liquid crystal phases. In particular, the quantum nematic is a real superconductor and shows the Meissner effect. Their special properties inherited from spatial symmetry breaking show up mostly at finite momentum, and should be accessible by momentum-sensitive spectroscopy.

keywords:
quantum liquid crystals; quantum phase transitions; Abelian-Higgs duality; superconductivity
1\cortext

[cor1]Corresponding author \cortext[cor2]Correspondence to: Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland

Contents:
Symbol Description
lattice gauge field
dual lattice gauge field
spectral function
electromagnetic (photon) gauge field
Burgers vector
magnetic field
stress gauge field
longitudinal phonon velocity
transverse phonon velocity
compression (sound) velocity
dislocation velocity
rotational velocity
velocity of light
elastic constants
number of dimensions
electric charge
electric field
torque stress gauge field
stress propagator
longitudinal propagator
transverse propagator
electromagnetic propagator
Hamiltonian
Hamiltonian density
compressional source
external source / rotation source
displacement source
dislocation current
electric current density
spacetime electric current density
Lagrangian density
Euclidean Lagrangian density
rotational stiffness length
higher-order compressional length
mass
electric particle density
Burgers vector
momentum absolute value
momentum
spacetime momentum
, discrete point group
particle coordinate
action
Euclidean action
time
imaginary time
imaginary time a velocity
space coordinates
displacement field
partition function
imaginary time convergence factor
dielectric function
dielectric constant
, gauge transformation field
smectic interrogation angle
nematic order parameter field
, disclination current
compression modulus
Lagrange multiplier field
, screening length / penetration depth
London penetration depth
dislocation penetration depth
shear penetration depth
shear modulus
magnetic constant
Poisson ratio
mass density
electric charge density
relativistic stress tensor
conductivity tensor
torque stress
second-gradient torque stress
condensate field
dislocation condensate field
, rotational strain field
frequency
Matsubara frequency
plasma frequency
Higgs mass
Frank vector
deficient rotation
Table 1: List of symbols. Field quantities will be denoted with their spacetime argument .

1 Introduction

How do crystals melt at zero temperature into quantum liquids? This would seem to be a question that was answered a long time ago. The He superfluid solidifies under pressure, through a first-order transition that is regarded as well understood. Similarly, it is widely believed that at low density the Fermi liquid formed from electrons will turn into a Wigner crystal also involving a first-order transition. However, dealing with microscopic constituents which are less simple than helium atoms, in principle zero-temperature phases can be formed which are in between the crystal and the isotropic superfluid: the ‘vestigial’ quantum liquid-crystalline phases.

It appears that such phases are realized in the strongly-interacting electron systems found in iron and copper superconductors, and in recent years this has grown into a sizable research field Ando et al. (2002); Vojta (2009); Oganesyan et al. (2001); Borzi et al. (2007); Fradkin et al. (2010); Fradkin (2012); Chu et al. (2012); Fernandes et al. (2014). This development was started a long time ago, by a seminal paper due to Kivelson, Fradkin and Emery Kivelson et al. (1998) that explained the potential for such vestigial phases to exist in the electron systems. Inspired by this work, one of the authors of this review (J.Z.) in 1997 asked himself the question “what can be learned in general about such quantum liquid crystals?” He decided to focus first on circumstances that simplify the life of a theorist: bosonic matter living in the maximally symmetric Galilean space, and two space and one time dimensions (2+1D). For physical reasons explained below, the interest was particularly focused on the case that the (crystalline) correlations in the quantum liquid states are as pronounced as possible.

As a lucky circumstance it turned out that a rather powerful mathematical methodology was already lying in wait, originating in the field of ‘mathematical elasticity theory’ that was rewritten in a systematic field-theoretical language by Hagen Kleinert in the 1980s Kleinert (1989, 2008). It dealt with the classical statistical physics of crystal melting in three space dimensions, but can be extended to handle the quantum problem in 2+1D. This revolves around a weak–strong duality: it can be viewed as an extension of the well-known Abelian-Higgs or vortex–boson duality used in quantum melting of superfluid order in 2+1D, which is exquisitely able to deal with the strong-correlation aspect given its non-perturbative powers. However, this generalization is far from trivial. Deeply rooted in the intricacies of the symmetries of spacetime itself, the duality reveals that the physics of crystal quantum melting is much richer than in the superfluid case. This is the story that we wish to expose in this review.

The groundwork was laid down in the period 1998–2003 and these results were published in comprehensive form Zaanen et al. (2004). This description was however in a number of regards rather crude and contained several flaws. It turned out that ambitious junior scientists that passed by in Leiden since then were allured by the subject, perfecting gradually the 2003 case and discovering new depths in the problem. Crucial parts were never published Cvetkovic (2006) while the remainder got scattered over the literature Kleinert and Zaanen (2004); Cvetkovic et al. (2006); Cvetkovic and Zaanen (2006a, b); Cvetkovic et al. (2008); Zaanen and Beekman (2012); Beekman et al. (2013); Liu et al. (2015). With the advent of the present third generation of students it appears that the last missing pieces have fallen into place and we believe that we have now an essentially complete theory in our hands at least for 2+1 dimensions. All that remains are some quantitative details such as the effects of crystalline anisotropy that are easy but tedious. We decided to endeavor to present the whole story in a comprehensive and coherent fashion, making it accessible for the larger community beyond the ‘Leiden school’.

More recently we have been concentrating increasingly on the 3+1D case: this is technically considerably harder but rewarding a much richer physical landscape. Several research directions are presently under investigation but a thorough understanding of the easier 2+1D case is a necessity to appreciate fully the vestigial marvels that one discovers in the most physical of all dimensions. This formed an extra motivation for us to write this extensive review.

Before we turn to the mathematical substances described in the bulk of this review let us first present a gross overview of the context, and the main features of this quantum field theory of liquid crystalline matter.

1.1 The prehistory: fluctuating stripes and other high- empiricisms

It all started in the late 1990s during the heydays of the subject of fluctuating order in the copper-oxide high- superconductors. Before the discovery of high- superconductivity in 1986 it was taken for granted that electron systems realized in solids were ruled by the principles of the highly-itinerant electron gas. The essence of such systems is that they are quite featureless: the Fermi liquid is a simple, homogeneous state of matter, ‘equalized’ by the highly delocalized nature of its quasiparticles. Structure can emerge in the form of spontaneous symmetry breaking but this should be of the Bardeen–Cooper–Schrieffer (BCS) kind where it becomes discernible only at the long time and length scales associated with the weak-coupling gap. It came then as a big surprise when inelastic neutron scattering measurements seemed to reveal that at least in the underdoped “pseudogap” regime of the cuprates, the electron quantum liquid is much more textured. Spin fluctuations were observed at energies associated with highly collective physics (0-80 meV) that reveal a high degree of spatial organization, although there is no sign of static translational symmetry breaking (for recent experimental results, see Refs. Kivelson et al. (2003); Vig et al. (2015)).

In the 1990s it was discovered that in the family of doped LaCuO (“214”) superconductors, under specific circumstances, static order can occur in the form of stripes Tranquada et al. (1995); Kivelson et al. (2003). These stripes are a ubiquitous ordering phenomenon found generically in doped Mott insulators other than cuprates (nickelates, cobaltates, manganites …). These are best understood as a lattice of electronic discommensurations (“rivers of charge”) that are formed when the charge-commensurate Mott insulator is doped. For the present purposes these might be viewed as ‘crystals’, likely formed from (preformed) electronic Cooper pairs that break the rotational symmetry (tetragonal, ) of the underlying square lattice of ions in a unidirectional (orthorhombic, ) way. This in turn goes hand-in-hand with an incommensurate antiferromagnetic order. This view was initially received with quite some skepticism. It was argued that this could well be a specialty of the 214-family, being also in other regards atypical (e.g. relatively low superconducting ). This changed with the discovery of charge order in the other underdoped cuprates, at first on the surface by scanning tunneling spectroscopy Hanaguri et al. (2004), followed by a barrage of other experimental observations Comin et al. (2015); da Silva Neto et al. (2014); Comin and Damascelli (2016). This has turned in recent years into a mainstream research subject in the community: see the review Ref. Keimer et al. (2015). There are still fierce debates over the question whether this charge order should be understood as a weak-coupling ‘Peierls-like’ charge density wave (CDW) instability, or as a strongly-coupled affair arising in a doped Mott insulator Zaanen and Gunnarsson (1989). The latest experimental results appear to largely support the strong coupling view Mesaros et al. (2016). Similarly, impressive progress has been made studying doped Mott insulators using several numerical methods. Although the various methods are characterized by multiple a-priori uncontrolled assumptions, it was very recently demonstrated that these invariably predict the ground state of the doped Hubbard model to be of the stripe-ordered kind Zheng et al. (2016).

Dealing with a weak-coupling CDW associated with a Fermi surface instability the ‘solid-like’ correlations should rapidly disappear when the charge order melts, be it as function of temperature or by quantum fluctuations in the zero-temperature state. The physical ramification of “strongly coupled” in this context is that such correlations should remain quite strong even in the liquid state. The energy scale associated with the formation of the charge order on the microscope scale is by definition assumed to be large, and the melting process is driven by highly collective degrees of freedom — the main theme of this review. Observation of the consequences of such “fluctuating order” in electron systems is not easy Zaanen et al. (2001); Kivelson et al. (2003). One has to have experimental access to the dynamical responses of the electron system in a large window of relevant energies and momenta. Until very recently only spin fluctuations could be measured, and it was early on pointed out that these should be able to reveal information about such fluctuations in the case that the charge order is accompanied by stripe antiferromagnetism Zaanen et al. (1996). Elastic neutron scattering then revealed the surprise that at somewhat higher energies the spin fluctuations in superconducting, underdoped cuprates, which lack any sign of static stripe order, look very similar to those of the striped cuprates Tranquada et al. (2004). The difference is that in the former a gap is opening up at small energies in the spin-wave spectrum of the latter. On basis of these observations the idea of dynamical or fluctuating stripes was born: at mesoscopic distances ( nanometers) and energies ( meV) the electron liquid approaches closely a striped state but eventually quantum fluctuations take over, turning it into a featureless superconducting state at macroscopic distances.

It proved very difficult to make this notion more precise, a main obstacle being the absence of experimental means of directly observing fluctuating charge order. The spin fluctuations represent an inherently indirect measure and one would like to measure instead the charge fluctuations. Roughly twenty years after the idea of “fluctuating stripes” emerged, this appears to be now on the verge of happening due to the arrival of the high resolution RIXS beam lines and of a novel EELS spectrometer Vig et al. (2015). Also in this context the computational progress is adding urgency to this affair by the very recent Quantum Monte Carlo results for a three-band model signaling strong stripe fluctuations at elevated temperatures Huang et al. (2016).

However, at first sight it sounds like a tall marching order to interpret such results. This is about the quantum physics of strongly-interacting forms of matter and without the help of powerful mathematics it may well be that no sense can be made of the observations. The theory presented in this review is precisely aiming at making a difference in this regard. It is very useful in physics to know what happens in the limits. The established paradigm dealing with order in electron systems is heavily resting on the weak-coupling limit: one starts from a free Fermi gas, to find out how this is modified by interactions. But this is inherently perturbative and when the interactions become strong one loses mathematical control. By mobilizing some big guns of quantum field theory (gauge theory, weak–strong duality) we will demonstrate here that exactly the opposite limit describing in a mathematically precise way the ‘maximal solid-like’ quantum fluid becomes also easy to compute, at least once one has mastered the use of the field-theoretical toolbox. The only restriction is that this works solely for bosons, preformed Cooper pairs in the present empirical context. This is powerful mathematics and it predicts a ‘universe’ of novel phenomena, that we will outline in the remainder of this introduction.

Returning to the historical development, the main source of inspiration for this work has been all along the seminal 1998 paper by Kivelson, Fradkin and Emery Kivelson et al. (1998). These authors argued that the fluctuating stripe physics forms a natural stage for the formation of new zero-temperature phases of matter: the quantum liquid crystals. In most general terms it follows a wisdom which is well tested in the realms of the physics of classical, finite-temperature matter de Gennes and Prost (1995). Typically the system forms a fully symmetric liquid at high temperatures, while it breaks the translations and rotations of Euclidean space at low temperatures, forming a solid. However, given particular microscopic conditions (e.g. ‘rod-like molecules’) one finds the partially ordered or vestigial phases. One manifestation corresponds to the nematic-type liquid crystalline order where translational symmetry is restored — the liquid aspect — while rotational symmetry is still broken (“the rods are lined up”). There are also smectic-type phases which break translations in one direction while the system remains fluid in the other directions (“stack of liquid layers”), see Fig. 1. A priori, the same hierarchy of symmetry breakings can occur at zero temperature, with the difference that the liquids are now identified as quantum liquids. Crudely speaking, one can now envisage that the ‘stripiness’ takes the role of the rod-like molecules on the microscopic scale. Subsequently one can picture that a quantum smectic is formed which behaves like a zero-temperature metal or superconductor in one spatial direction, while it insulates in other directions. Similarly, metallic or superconducting zero-temperature states can be imagined which are anisotropic because of the spontaneous breaking of spatial rotations: the quantum nematics. The notion of quantum liquid crystals appeared to be a fruitful idea. Not long thereafter evidences were found for the occurrence of such quantum nematic order in part of the underdoped regime of YBaCuO and BiSrCaCuO cuprate superconductors Ando et al. (2002); Hinkov et al. (2008); Daou et al. (2010); Howald et al. (2003); Kohsaka et al. (2007).

However, Kivelson et al. Kivelson et al. (1998) took it a step further by conceptualizing it in the language of the celebrated Kosterlitz–Thouless–Nelson–Halperin–Young (KTNHY) theory of topological melting in two ‘classical’ dimensions Kosterlitz and Thouless (1972, 1973); Halperin and Nelson (1978); Nelson and Halperin (1979); Young (1979). In this framework the liquid is not understood as the state where the constituents of the solid are liberated, freely moving around in a gaseous state. Instead it is asserted that the solid stays locally fully intact, and instead the ‘isolated’ topological excitations associated with the restoration of translational invariance (dislocations, Sec. 4) proliferate. Such a liquid still breaks the rotational symmetry since rotational-symmetry restoration requires different defects: disclinations. Therefore states of matter where the dislocations are ‘condensed’ while the disclinations are still ‘massive’ are symmetry-wise identical to the smectics and nematics formed from the rods of Fig. 1. In fact, the theory we will present here starts out from this basic notion: is just the generalization of the KTNHY theory to the zero-temperature quantum realms in 2+1D, showing that in this quantum setting there is a lot more going on.

(a) solid –
(b) smectic –
(c) nematic –
(d) liquid –
Figure 1: Sketch of classical melting of a 2D crystalline solid. The thick dashed lines denote crystal axes along which the ‘particles’ are ordered while the thin dashed lines are for reference with the previous situation only. Denoted in the figure labels is the symmetry group as a subgroup of the 2D Euclidean group , where denotes the discrete point group of the lattice. \subreffig:classical melting solid Solid: regularly ordered breaking both translations and rotations completely to a discrete space group. \subreffig:classical melting nematic Smectic: translational symmetry is restored in the horizontal direction, but these liquid layers are still periodically stacked in the vertical direction. Rotations are broken. \subreffig:classical melting nematic Nematic: all translational symmetry is now restored like a liquid, but the constituents maintain orientational order. \subreffig:classical melting liquid. Liquid: completely disordered and all symmetry is restored. For the complementary dual picture, see Fig. 6

Also in other areas the concept of quantum liquid crystals flourished. It became clear that the stripe phases formed in high-Landau-level quantum Hall systems turn into quantum nematic phases. The theme became particular prominent in the iron-pnictide superconductors Chuang et al. (2010) where such nematic order appears to be very pronounced although the debate about its precise microscopic origin as well as its relation to the superconductivity is still raging Fernandes et al. (2014). For completeness we will shortly review these matters in Sec. 1.4. They are interesting subjects by themselves, revealing physics of a different kind than we are addressing. It is questionable whether the material in this paper is of any consequence in these realms. The quantum Hall nematics may be ‘sufficiently orderly’, but the dynamical information which is our main output cannot possibly be measured in two-dimensional electron gases. The pnictides are almost surely situated on the weak-coupling side: no evidence of any kind emerged for strong charge-order correlations in their electron systems.

With regard to the potential empirical relevance of the theory the only obvious theater of which we are presently aware are the underdoped cuprates. Even in this context it remains to be seen whether any of the phenomena that the theory predicts will occur in a literal fashion in nature. It hinges after all on an extreme limit, and it depends on whether the microscopic conditions in real electron systems permit getting near enough to this “maximal solid behavior” such that the remnants of its physics are discernible in experiment. At present this work is therefore in first instance of a general theoretical interest. However, anybody who will take the effort to master this affair will be rewarded by the striking elegance and beauty of the physics of the maximally-correlated quantum fluid, making one wonder whether nature can ignore this opportunity.

1.2 Platonic perfection and the big guns of quantum field theory

Quantum field theory as it comes alive in condensed matter physics is precisely tied to the universal long-wavelength physics associated with zero-temperature matter. Inspired by the empirical developments described above we became aware that actually the general description of the quantum liquid crystals is among the remaining open problems that can be tackled at least in principle by the established machinery of quantum field theory. More generally, this is about quantum many-body systems that spontaneously break spatial symmetries. This is what we set out to explore some 15 years ago. This program is not quite completed yet. Dimensionality is a particularly important factor and quite serious complications arise in 3+1 and higher dimensions. However, in two space dimensions the theory is brought under complete control, which this review is intended to present in a comprehensive and coherent fashion.

In order to get anywhere we consider matter formed from bosons: there are surely some very deep questions related to fermionic quantum liquid crystals but there is just no controlled mathematical technology available that can tackle the fermion sign problem (see also Sec. 1.4). As related to the empirical context of the previous paragraphs, at zero temperature one is invariably dealing with nematic (or smectic) superconductors formed from Cooper pairs which are bosons. Therefore, insofar as any of our findings can be of direct relevance in this empirical context, it is natural to explore what the bosonic theory has to tell.

The next crucial assumption is that we start out with a system living in Galilean-invariant space. There is no ionic background lattice and our bosonic system has to break the spatial symmetries all by itself. This assumption detaches our theoretical work from a literal application to the empirical electron systems. This is however the natural stage for the elegant physics associated with the field theory and it is just useful to know what happens in this limit, as we hope to demonstrate. After all, there are signs that the strength of the ‘anisotropy’ coming from the lattice might not be at all that large: the case in point is that the scanning tunneling spectroscopy (STS) images of cuprate stripes are littered with rather smooth dislocation textures of a type that would not occur when the effective lattice potential would be dominant Mesaros et al. (2011). We will later present several results following from the continuum theory that might be still of relevance to the lattice incarnation when the pinning energy of the lattice is sufficiently weak. Of course, the experimentalists should take up the challenge to engineer such a continuum bosonic quantum liquid crystal, for instance by exploiting cold atoms etc.

The experienced condensed matter physicist might now be tempted to stop reading: what new is to be learned about a system of bosons in the Galilean continuum? This realm of physics is supposed to be completely charted: dealing with bosonic particles like He atoms these are well known to form either close packed (in 2+1D, triangular) crystals or superfluids at zero temperature. Dealing with ‘rod-like bosons’ there is surely room to have an intermediate quantum nematic phase corresponding to a superfluid breaking space rotations in addition. Resting on generic wisdoms of order parameter theory it is obvious that a Goldstone boson will be present in this phase associated with the rotational symmetry breaking that can be sorted out in a couple of lines of algebra. What is the big deal?

This industry standard paradigm is based on a weak-interaction, ‘gaseous’ perspective. To describe the superfluid one takes the free boson Bose–Einstein condensate perspective dressed by weak interactions (Bogoliubov theory). In helium one typically finds a strong first-order transition to the crystal phase, which can be well understood as a classical crystal dressed by mild zero-point motions. The reason, however, for this review to be quite long is that we will mobilize the ‘big gun’ machinery of quantum field theory. This is actually geared to deal with a physics regime that might be described as a ‘maximally strongly-interacting’ regime of the microscopic bosons. In fact, the reader will find out that these bosons have disappeared altogether from the mathematical description that is entirely concerned with the emergent collective degrees of freedom that are formed from a near infinity of microscopic degrees of freedom.

We will find out that the zero-temperature liquids are invariably superfluids or superconductors. However, these are now characterized by transient crystalline correlations extending on length scales that are large compared to the lattice constant. From this non-perturbative starting point, it is rather natural for the field theory to describe the kind of physics that is envisioned by the fluctuating stripes hypothesis, where the superconductor is locally, at the smallest length scales, still behaving as a crystal.

The big gun machinery that we will employ is weak–strong or Kramers–Wannier duality Kramers and Wannier (1941). The immediate predecessor of the present pursuit is the intense activity in the 1970s revolving around the Berezinskii–Kosterlitz–Thouless (BKT) topological melting theory Berezinskii (1970); Kosterlitz and Thouless (1972, 1973), and the particular implementation in the form of the Kosterlitz–Thouless–Nelson–Halperin–Young (KTNHY) theory of finite-temperature melting of a crystal in two dimensions, involving the hexatic vestigial phase Kosterlitz and Thouless (1972, 1973); Halperin and Nelson (1978); Nelson and Halperin (1979); Young (1979). The central notion is that the destruction of the ordered state can be best understood in terms of the unbinding (proliferation) of the topological defects associated with the broken symmetry. The topological defects of the superfluid are vortices. The vortex is thereby the unique agent associated with the destruction of the order: in a strict sense a single delocalized vortex suffices to destroy the order parameter of the whole system. In the ordered state the excitations can therefore be divided in smooth configurations corresponding to the Goldstone bosons, whose existence is tied to the presence of order, and the singular or multivalued configurations characterized by topological quantum numbers. As long as the latter occur only as neutral combinations (e.g. bound vortex–antivortex pairs) the order parameter cannot be destroyed. Conversely, when the topological excitations unbind and proliferate, the system turns automatically into the disordered state, which now can be seen as a condensate, a ‘dually ordered state’ formed out of topological defects. This is in essence the basic principle of field-theoretical weak–strong or Kramers–Wannier dualities. Many weak–strong mappings have been developed since, ranging all the way to the fanciful dualities discovered in string theory such as the AdS/CFT correspondence Zaanen et al. (2015).

If this principle applies universally (which is not at all clear) it will lead to the staggering consequence that, away from the critical state, all field-theoretical systems are always to be regarded as ordered states. It is just pending the access of the observer to order operators or disorder operators whether he/she perceives the disordered state as ordered or the other way around. The benefit for theorists is that the mathematical description of the weakly-coupled ordered/symmetry broken state is very well controlled (Goldstone bosons and so forth) while strongly-coupled disordered states are typically much harder to describe. Now in the dual description the latter are yet again of the tranquil, ordered kind. This review explores in detail the workings of the weak–strong duality as applied to zero-temperature quantum crystals and its duals in 2+1 spacetime dimensions.

Turning to crystals, the symmetry that is broken is the Euclidean group associated with space itself which is a much richer affair than the internal global symmetry of -spin systems/superfluids. In crystals this can be broken to the smallest possible subgroups as classified in terms of the space groups. Solids are of course overly familiar from daily life but to a degree this familiarity is deceptive. The symmetry principles which are involved are much more intricate than the usual internal symmetries. The Euclidean group in space dimensions involves independent translations, , forming an infinite group, in semidirect relation with the orthogonal group including rotations and reflections. Semidirect here means that rotating, translating and rotating back is in general not the same as simply translating (the rotation group acts on the translation group). This is denoted as . Crystals are described by space groups which are comprised of lattice translations , again in semidirect relation to discrete point group symmetries , augmented by non-symmorphic symmetries such as glide reflections. This will be the underlying theme throughout this review: the surprising richness of the physics is an expression of this intricate symmetry structure.

The point of departure will be the theory describing the nature of the maximally ordered phase (the solid): this is the 19th century theory of elasticity, promoted to the Lagrangian of the quantum theory by adding a kinetic term. Although it has been around for years, as a field theory it is quite involved given its tensor structure. The topological content is also remarkably rich, despite the fact that the basics have been identified a long time ago, and much of it has been exported to engineering departments. The topological defect associated with the restoration of translational invariance is the dislocation identified by Burgers in the 1930s Burgers (1939). Its topological invariant is the Burgers vector associated with the discrete lattice translation symmetry of the crystal; this translational defect does not affect the rotational symmetry and its charge has therefore to keep track of this information. The rotational symmetry is restored by a separate disclination defect with the Frank vector (or Frank scalar in 2D) as topological charge Frank (1958). The dislocation can be in turn viewed as a bound disclination–antidisclination pair while the disclination also corresponds to a bound state of an infinite number of dislocations with parallel Burgers vector, see Fig. c below.

In a solid, dislocations and disclinations are topologically distinct defects with an explicit hierarchy: the deconfined (in the solid) dislocations are intrinsically easier to produce than the confined disclinations, although a priori one cannot exclude the possibility that the disclinations will proliferate together with the dislocations giving rise to the first-order transition directly from the solid to the isotropic liquid. This depends on the details of the ‘UV’ (the ‘chemistry’ on the molecular scale).

We now insist that the disclinations stay massive and thereby the breaking of the rotational symmetry of the solid is maintained. However, when the dislocations proliferate, translational symmetry is restored and the system turns into a fluid. Given the vectorial nature of the Burgers vectors this can be accomplished in different ways. When free dislocations occur with precisely equal probability the translational symmetry is restored in all possible directions while the rotational symmetry is still broken as characterized by the point group of the ‘parent’ crystal. These form the family of ‘nematic-like’ liquid crystals. There is actually an ambiguity in the vocabulary that is not settled: in the soft-matter community it is convention to reserve nematic for the uniaxial, -symmetric variety (ordered states of ’rod-like’ molecules), while for instance the nomenclature -atics has been suggested for 2D nematics characterized by a -fold axis Park and Lubensky (1996). In full generality, these substances are classified by their point group symmetries. Because we are mainly interested in long-distance hydrodynamic properties which do not really differ between the different point groups, and by lack of a generally accepted convention, we will call all these substances nematics, with a point group prefix when needed. See Sec. 5 for a more nuanced view.

The topology allows for yet another possibility Ostlund and Halperin (1981), which is sometimes overlooked. It is a topological requirement for the nematic order that the Burgers vectors in the dislocation condensate are locally anti-parallel since a net ‘Burgers uniform magnetization’ corresponds to a finite disclination density, which we excluded from the start. However, there is no requirement to populate all Burgers directions equally as happens in the nematics. Instead, one can just preferentially populate the Burgers vectors in one direction. The effect is that in this direction the system turns into a fluid while translations are still broken in the orthogonal direction: this is the topological description of the smectic as the state that can occur in between the crystal and the nematic. Obviously, when the disclinations proliferate one will eventually end up in an isotropic liquid although still other phases are possible with a higher point group symmetry associated with a preferential population of certain Frank vectors.

In the present context of crystal quantum melting in two spatial dimensions, the crucial ingredient is that the dislocations are ‘quite like’ vortices with regard to their status in the duality, as was already realized by KTNHY in the 2+0D case. It was then asserted that an unbinding BKT transition can take place involving only the dislocations (keeping the disclinations massive) into a hexatic state (in our terminology: -nematic). Our pursuit is in essence just the next logical generalization of this affair: how does this topological melting work out in the 2+1D bosonic quantum theory realized at zero temperature?

The quantum version is however richer in a number of regards, and in a way more closely approaching a platonic perfected incarnation of liquid crystalline order. As we will see, the fluids associated with the quantum disordered crystal can also be viewed as ‘dual superconductors’ but now the “gauge bosons that acquire a mass” are associated with shear forces while the ‘dual matter’ corresponds to the Bose condensate formed out of dislocations. The similarity between dislocations and vortices is rooted in the fact that, like the vortices, the dislocations restore an Abelian symmetry: the translational symmetry of space. For this reason, the crystal duality is still governed by the general rules of Abelian dualities. However, there are also fundamental differences: in constructing the duality the richness associated with field-theoretical elasticity comes to life. For instance, in the dual dislocation superconductors the information associated with the rotational symmetry breaking in the liquid crystals is carried by the Burgers vectors of the dislocations. In the gauge-theoretical description these take the role of additional ‘flavors’ which in turn determine the couplings to the ‘stress photons’ mediating the interactions between the dislocations.

As we will explain at length in Secs. 79, this has the net effect that shear stress has a similar fate in the liquid as the magnetic field in a normal superconductor: the capacity to propagate shear forces is expelled from the dual stress superconductor at distances larger than the shear penetration depth. By just populating the Burgers charges equally or preferentially in a particular orientational direction these dislocation condensates describe equally well nematic- and smectic-type phases where now the solid-like behavior of the latter is captured naturally by the incapacity of the dislocations to ‘Higgs’ shear stress in a direction perpendicular to their Burgers vector.

In fact, one could nonchalantly anticipate that this general recipe applies a priori equally well to the classical, finite-temperature liquid crystals in three space dimensions as to the 2+1D quantum case in the usual guise of thermal field theory. In equilibrium, one can compute matters first in a spacetime with Euclidean signature and Wick rotate to Minkowski time afterwards. Where is, then, the difference between 3D classical and 2+1D quantum ‘elastic matter’? The quantum matter is formed from conserved constituents (like electrons, atoms) at finite density, and we are interested in phenomena occurring at energies which are small compared to the thermodynamic potential. Under these conditions Lorentz invariance is badly broken: the ‘crystal’ formed in spacetime is made from worldlines and although these do displace in space directions they are incompressible in the time direction. Compared to 3D crystals this ‘spacetime crystal’ is singularly anisotropic; as realized by Nelson and coworkers its only sibling in the classical world is the Abrikosov lattice formed from flux lines in superconductors Marchetti and Nelson (1990).

Nevertheless, one can take the bold step to postulate the existence of a Lorentz-invariant ‘world crystal’ corresponding to a spacetime as an isotropic elastic medium. This is characterized by stress tensors that are symmetric in spacetime labels and it is very easy to demonstrate that the nematic-type quantum liquid crystal which is dual to this medium is the vacuum of strictly linearized gravity where the disclinations are the exclusive sources of curvature Kleinert and Zaanen (2004); Zaanen and Beekman (2012). As an intriguing consequence, since gravity is incompressible in 3D there are no massless propagating modes in 2+1D while in 3+1D one just finds the two spin-2 gravitons. As we will see, this is very distinct from the mode spectrum of the real life non-relativistic quantum nematics.

However, a significant simplification is associated with the fact that the symmetry breaking only affects the 2D space in the non-relativistic case. In the soft-matter tradition it is well understood that the classification of nematic-type orders is in terms of the point groups, and in 2D this is a rather simple affair given that the 2D rotational groups are all Abelian. We will discuss the precise nature of these orientational order parameters in Sec. 5, actually making the case that these are most conveniently approached in the language of discrete -gauge theory. In three space dimensions hell breaks loose since the point groups turn non-Abelian with the effect that the order parameters acquire a highly non-trivial tensor structure. This can be also brought under control employing discrete non-Abelian gauge theory; this will be subject of a separate publication. In fact, in the duality construction we will close our eyes for the intricacies associated with particular nematic symmetries and concentrate instead of the maximally symmetric ‘spherical cow’ cases descending from isotropic elasticity; the lower symmetry cases just invoke adding details like anisotropic velocities which do not play any interesting role in the duality per se.

There is yet another aspect that is special to the non-relativistic quantum liquid crystals, which in turn plays a crucial role for their physics. In the classical setting, dynamics does not affect the thermodynamics, but this is different in the quantum incarnation since quantum physics ‘entangles’ space and time. From the study of the motions of classical dislocations in solids it is well known that these are subjected to a special principle rooted in topology: dislocations can move ballistically in the direction of their Burgers vector (called glide motion) while in the absence of interstitial and/or substitutional defects climb motion perpendicular to the Burgers vector is completely impeded. In addition, the inertial mass associated with the climb motion is identical to the mass of the constituents of the solid, but dislocations do not fall in the gravitational field of the earth, the reason being that the dislocation “does not carry volume”. It can only accelerate by applying shear forces to the medium. This glide principle will play a remarkable role in the quantum problem. As we will see that it is responsible for the capacity of the zero-temperature quantum fluid, that is ultimately a dislocation condensate, to propagate sound.

1.3 The warped dual view on quantum liquid crystals

This completes the exposition of the basic ingredients for the dual quantum theory of elasticity for bosonic matter in 2+1D. These in turn form the building blocks for a quantum field theory with remarkable mathematical qualities. By just blind computation one obtains results that shed a different, often surprising light on a seemingly very classical physics topic. This will be the substance of the remainder of this review but to whet the appetite of the reader let us present a list of these surprises, roughly in the same order as they appear below.

  1. Phonons are gauge bosons. The theory of quantum elasticity is just the 19th century theory of elasticity with a kinetic energy term added to its Lagrangian. This is nothing else than the long-wavelength theory of acoustic phonons. Using Kleinert’s way of employing the stress–strain duality Kleinert (1989, 2008) we show how to rewrite this in terms of -gauge fields. In contrast to textbook wisdoms, phonons can be regarded as ‘photons’ when the question is asked how the medium propagates forces. The elastic medium is richer than the vacuum of electromagnetism in the regard that the crystal directions enter as ‘flavors’ in the gauge theory. These stress photons are sourced by external shear and compressional stresses but also by the dislocations, which in turn have the same status as charged particles in electromagnetism, with the same complication that they carry the Burgers vector charge as a ’flavor’ (Sec. 6).

  2. The disordered solid is a stress superconductor. Since individual dislocations are ‘charged particles’ interacting via ‘stress photons’, when the dislocations proliferate and condense the resulting quantum fluid can be viewed as a stress superconductor. Shear stress is the rigidity exclusively associated with translational symmetry breaking, and it is this form of stress that falls prey to the analogue of the Meissner effect. Shear stress is “expelled from the liquid” in the same way that magnetic fields cannot enter the superconductor. One can identify a shear penetration depth having the meaning that at short distances the medium remembers its elastic nature with the effect that shear forces propagate. At a length scale larger than the average distance between (anti-)dislocations, shear stress becomes perfectly ‘screened’ by the response of the Bose-condensed dislocations (Sec. 8).

  3. The disordered solid is a real superfluid. The fact that dislocations “do not occupy volume” is in the duality encapsulated by the glide principle. After incorporating this glide constraint, one finds that the dislocation condensate decouples from the purely compressional stress photons: the quantum liquid carries massless sound, which in turn can be viewed as the longitudinal phonon of the disordered crystal that “lost its shear contributions” (Sec. 8). The mechanism involves a mode coupling between the longitudinal phonon and a condensate mode having surprising ramifications for experiment. Besides the specialties associated with the orientational symmetry breaking, we find a bosonic quantum fluid that just carries sound. By studying the response of its EM charged version to external magnetic fields (item 9.) we prove that this liquid is actually a superfluid! At first sight this might sound alarming since we have constructed it from ingredients (phonons, dislocations) that have no knowledge of the constituent bosonic particles forming the crystal. It seems to violate the principle that superfluidity is governed by the off-diagonal long range order (ODLRO) of the (constituent) bosonic fields. This is less dramatic than it appears at first sight: the braiding of the dislocations will give rise to the braiding of the worldlines of the constituent bosons. In fact, it amounts to a reformulation of the usual ODLRO principle to the limit of maximal ‘crystalline correlations’ in the fluid: “a bosonic crystal that has lost its shear rigidity is a superfluid”.

  4. The rotational Goldstone mode deconfines in the quantum nematic. By insisting that the disclinations stay massive while the dislocations proliferate, the quantum liquids we describe are automatically quantum liquid crystals, where we just learned that these are actually superfluids in so far their ‘liquid’ aspect is concerned. Given that the isotropy of space is still spontaneously broken there should be a rigidity present, including the associated Goldstone boson: the ‘rotational phonon’ and the reactive response to torque stress. But now we face a next conundrum: the phonons of the crystal are purely translational modes, how does this rotational sector “appear out of thin air” when the shear rigidity is destroyed? Why is there no separate mode in the crystal associated with the rotational symmetry breaking? The reason is that translations and rotations are in semidirect relation in the space groups describing the crystal: one cannot break translations without breaking rotations. To do full justice to this symmetry principle, in Sec. 6.5 we introduce a more fanciful ‘dynamical Ehrenfest constraint’ duality construction, where the role of rotations and the associated disclination sources is made explicit already in the crystal. We find, elegantly, that in this description the torque photon as well as the associated disclination sources are literally confined in the crystal, in the same physical (although not mathematical) sense as quarks are confined in QCD. When the dislocations condense the torque stresses and the disclinations deconfine becoming the objects encapsulating the ‘rotational physics’ of the quantum nematics (Sec. 8.3). As compared to the classical (finite-temperature) nematics there is one striking difference. It is well known that the rotational Goldstone mode couples to the dissipative circulation in the normal hydrodynamical fluid, and is overdamped. The superfluid is however irrotational, thereby protecting the rotational Goldstone mode as a propagating excitation.

  5. Partial translational melting is a quantum smectic. The dislocation condensate consists in essence of -fields in space dimensions. By preferentially condensing one Burgers orientation in the dislocation condensate we construct the quantum smectic in Sec. 9. Although one anticipates the conventional picture of “stacks of liquid layers”, the zero temperature quantum case defies such intuitions. It cannot be viewed as simply a ’solid liquid’, and the transverse and longitudinal characters mix in the mode spectrum, depending on the interrogation angle of the linear response (see Fig. 17 below). Surprisingly, the ‘most isotropic’ response is found with field propagating at 45 to the layers. Nevertheless, for fields propagating along the layers (in the ‘liquid direction’), the duality construction flawlessly reproduces the transverse “undulation mode” in the solid direction with a quadratic () dispersion, that is known in classical smectics de Gennes and Prost (1995); Chaikin and Lubensky (2000).

  6. Order parameter theory of 2+1-dimensional nematics. In Sec. 5 we develop a completely general theory of order parameters arising due to broken rotational symmetry in 2+1 dimensions. The conventional ‘uniaxial’ nematic is just one example of a host of possible -atic orders, which derive directly from the space group of the crystal that undergoes dislocation-mediated melting. This is extended to a gauge theory formulation, where the topological defects in the nematic phase (disclinations) are represented by -fluxes. At zero temperature, this leads to the prediction of a new phase, the -deconfined phase, where these gauge fluxes are frozen but not condensed.

  7. Transverse phonons become massive shear modes in the quantum liquid crystal. Up to this point we have highlighted universal features of the long-wavelength limit. These are in fact not depending on the assumption of maximal crystalline correlations intrinsic to the duality description. By adiabatic continuation they are smoothly connected to the outcomes of the weakly-interacting, gaseous description. The difference between the two descriptions becomes manifest considering the spectrum of finite-energy excitations. The dual stress superconductor description yields a plethora of propagating massive modes in the quantum liquid crystals which depend critically on the assumption that the crystalline correlation length/shear penetration depth is large compared to the lattice constant. Their origin is easily understood in terms of the dual relativistic superconductor description. In the Higgs phase of a real superconductor the photon becomes massive; in the stress superconductor the stress photons (the phonons) become massive, and they propagate shear forces only over a short range. In other words, these new modes in the liquid phases correspond to massive shear photons. A simple example is the transverse phonons of the crystal that just acquire a ‘Higgs mass’ in the liquid. However, the stress superconductor is more intricate than just the Abelian-Higgs condensate. The case in point is the way that the longitudinal phonon of the crystal, sensitive to both the compression and the shear modulus, turns into the sound mode of the superfluid, rendering the sound mode of the quantum nematic to be of a purely compressional nature. The existence of these massive modes is critically dependent on the assumption that interstitials (the constituent bosons) are absent. When the crystalline correlation length shrinks towards the lattice constant these modes will get damped to eventually disappear in the gaseous limit where they are completely absent at small momenta. We believe that the roton of e.g. He can be viewed as a remnant of such a shear photon in the regime where the crystalline correlation length has become a few lattice constants.

  8. Elasticity and the charged bosonic Wigner crystal. Up to here we have dealt with electromagnetically neutral systems, but as we will show in Secs. 10,11 it is straightforward to extend the description to electrically charged systems. The first step is to derive the elastic theory of the charged bosonic Wigner crystal. We shall obtain the spectrum of (coupled) stress and electromagnetic (EM) photons, considering the coupling of the 2+1D matter to 2+1D electrodynamics. The unpinned Wigner crystal behaves as a perfect conductor, where the plasmon now propagates with the longitudinal phonon velocity. In the transverse optical response however, next to the expected plasmon there is also a weak massless mode, with quadratic dispersion at low energies, extrapolating to the transverse phonon at high energies.

  9. The charged quantum nematic shows the Meissner effect. Since a dislocation does not carry volume it does not carry charge either. Accordingly, in the dislocation condensate the longitudinal EM response is characterized by a ‘true’ plasmon, a sound wave that has acquired a plasmon energy. The surprise is in the transverse EM response: the EM photon acquires a mass and the system expels magnetic fields according to the Meissner effect of a superconductor. This proves the earlier assertion that the bosonic quantum nematic, described in terms of a dual stress superconductor, is indeed a genuine superfluid that turns into a superconductor when it is gauged with electromagnetic fields. The mechanism is fascinating: the Meissner effect is in a way hiding in the Wigner crystal where it is killed by a term arising from the massless shear photons. When the latter acquire a mass this compensation is no longer complete with the outcome that EM photons are expelled (Sec. 11.4).

  10. The charged quantum smectic shows strongly anisotropic properties. The quantum nematic is just an isotropic superconductor but the charged quantum smectic is equally intriguing as the neutral counterpart: in the ‘liquid’ direction it is characterized by a finite superfluid density and the capacity to expel EM fields, but at an angle of 45 with respect to the ‘solid’ direction its EM response is identical to that of the Wigner crystal. For momenta along the ‘solid’ direction, there is a massive transverse plasma polariton and massive mode from arising from mode coupling with the shear photon (i.e. phonon). At intermediate angles, the plasma polariton persists but the spectral weights of the coupled modes interpolate between the ‘magic’ angles with massless and massive modes at finite momenta. For transverse fields propagating at finite momenta near but not exactly in the liquid direction, magnetic screening at finite frequencies (the skin effect) is enhanced with respect to the Wigner crystal (Sec. 11.5).

  11. The massive shear mode is detectable by finite-momentum spectroscopy. The charged case is the one of greatest potential relevance to the physics of the electron systems. Given the very small mass of electrons the only way to exert external forces on the system probing the physical properties of the liquid crystalline states is by its electromagnetic responses. A key prediction is that the massive shear photons are in principle observable by electromagnetic means, albeit with the practical difficulty that these carry finite optical weight only at finite momenta. This runs into the usual difficulty that because of the large mismatch between the ‘material velocity’ and the speed of light the experimentalists can only easily interrogate the zero-momentum limit. However, using modern techniques, such as electron energy-loss spectroscopy or inelastic X-ray scattering, this regime becomes accessible and in Sec. 11.4 we will present a very precise prediction regarding the massive shear photon that should become visible in the longitudinal EM channel.

  12. Directly probing the liquid crystalline order parameter. Last but not least, is it possible to measure the order parameters of the quantum-liquid crystals directly using electromagnetic means? Given the presence of the pinning energy in the real solids, an even more pressing issue is whether there is any way to couple into the rotational Goldstone boson of the nematic that is expected to be characterized in any case by a finite ‘anisotropy gap’ caused by the pinning. In Sec. 11.4 we shall see that the rotational Goldstone mode leaves its signature in the transverse conductivity at low but non-zero momenta.

1.4 Quantum liquid crystals: the full landscape

We will now shortly discuss the relation of our ‘maximally-correlated’ quantum liquid crystals to other manifestations of quantum liquid-crystalline order.

Electrons in solids are a natural theater to look for quantum fluids such as our present quantum liquid crystals. The greatly complicating circumstance is that electrons are fermions. One is facing a monumental obstruction attempting to describe systems of interacting fermions at any finite density in general mathematical terms: the fermion sign problem. In constructing the theory we employ the general methodology of quantum field theory: mapping the quantum problem onto an equivalent statistical physics problem in Euclidean spacetime, to then mobilize the powerful probabilistic machinery of statistical physics to solve the problem Sachdev (2011). As we emphasized before, we just generalize the classical KTNHY story to 2+1 Euclidean dimensions and after Wick rotation a quantum story unfolds. But this does not work for fermions since the fermionic path integral does not lead to valid statistical physics, since fermion signs correspond to “negative probabilities”. It is just not known how to generalize deeply non-perturbative operations like the weak–strong duality behind the present theory to a fermionic setting at any finite density.

To circumvent this trouble we assume that the electrons are subjected to very strong interactions that first bind them in ‘local pairs’ which subsequently form a tightly-bound crystal that can only melt by topological means. Given the sign problem, the only other option is to start from the opposite end: depart from the non-interacting Fermi gas to find out what happens when interactions are switched on. There is surely interesting physics to be found here which is however rather tangential to the theme of this review. For completeness let us present here a short sketch of these other approaches (see also Ref. Fradkin et al. (2010)).

How to describe nematic order departing from a free Fermi gas? The object to work with is the Fermi surface, an isotropic sphere in momentum space (when working in the Galilean continuum). When the Fermi surface turns into an ellipsoid, the isotropy is lost and symmetry-wise it corresponds to a uniaxial nematic deformation. This can be accomplished by switching on an electron–electron interaction of a quadrupolar nature (associated with the Landau Fermi liquid parameter Oganesyan et al. (2001); the surprise is that this is perturbatively unstable! Upon inspecting the leading order perturbative corrections one discovers an extremely bad IR divergence. As it turns out, the rotational Goldstone mode does not decouple from the quasiparticle excitations in the deep infrared and just as in the classical nematic it is overdamped. The quasiparticles pick up an IR divergence as well. This is perhaps the most profound problem in this field: although the interactions are weak the nematic Fermi fluid cannot be a Fermi liquid, but the fermion sign problem is in the way of finding out what it is instead!

In real electron systems the anisotropy of the underlying lattice will render the rotational symmetry to become discrete: in pnictides and cuprates one is typically dealing with a square lattice with a fourfold () symmetry axis that turns into a twofold () axis in the nematic state. This anisotropy gap of this “Ising nematic” protects the physical systems from this divergence. It is still debated whether the nematic order found in pnictides Fernandes et al. (2014) is of this ‘near Fermi-liquid’ kind or rather of a strongly-coupled “spin nematic” nature, where also the complications of orbital degeneracy Krüger et al. (2009) may play a crucial role.

One can subsequently ask the question what happens in such a metallic nematic when the order disappears at a zero-temperature quantum phase transition. This is typically approached from the Hertz–Millis perspective Hertz (1976); Millis (1993). One assumes that the order is governed by a bosonic order-parameter theory such that the quantum phase transition is equivalent to a thermal phase transition in Euclidean space time Sachdev (2011). However, the critical order-parameter fluctuations are perturbatively coupled to the electron–hole excitations around the Fermi surface of a free fermion metal. The latter can in turn give rise to IR singularities changing the nature of the universality class: the (Ising) nematic transition in 2+1D is a case in point Metlitski and Sachdev (2010).

The most recent results are associated with a special model characterized by sign cancellations, making it possible to unleash the powers of Monte Carlo simulations; these indicate that the perturbative assumptions wired in the Hertz–Millis approach break down, instead showing a non-Fermi-liquid behavior in the metallic state and a strong tendency towards superconductivity at the quantum critical point Schattner et al. (2016); Lederer et al. (2016).

There is yet another series of ideas that are more closely related to the strong-coupling bosonic perspective of this review. One can arrive at a notion of a quantum smectic which is quite different from the quantum smectics we will highlight, identified by Emery et al. Emery et al. (2000). One starts out from static stripes assuming that metallic 1+1D Luttinger liquids are formed on every stripe, which are subsequently coupled into a 2+1D system. One can now demonstrate that for particular forward-scattering-dominated interactions the inter-stripe interactions become irrelevant with the effect that the system continues to behave like a Luttinger liquid in the direction parallel to the stripes, while becoming, in the scaling limit, insulating in the perpendicular direction. These ideas were taken up and further elaborated on in the context of quantum Hall (QH) physics. Upon going to high filling fractions in the QH two-dimensional electron gases, it is easy to show that at some point one will form the QH “stripes” Koulakov et al. (1996); Moessner and Chalker (1996). These are microscopically very different from the doped-Mott-insulator stripes: they consist of linear arrays of filling fraction and incompressible QH fluids, while chiral edge states that propagate on the boundaries are quite literal incarnations of the Luttinger liquids of Kivelson et al. Kivelson et al. (1998). Besides this ‘edge mode’ quantum smectic, one can also contemplate it be subjected to dislocation melting Fradkin and Kivelson (1999) and there is experimental evidence for the formation of nematic states as well, observed in terms of anisotropic QH transport Lilly et al. (1999); Xia et al. (2011). The transport properties in such quantum nematics are tied to the chiral QH edge states and, for the topologically-ordered quantum liquid crystal QH phases, these are governed by Chern–Simons topological field theory. This adds an extra layer of physics since the associated topological order communicates with the nematic order parameter Balents (1996); Mulligan et al. (2010, 2011); Maciejko et al. (2013); You et al. (2014). The general hydrodynamics of QH liquid crystals is in turn deeply rooted in the effectively non-commutative geometry associated with 2+1D matter in magnetic fields Radzihovsky and Dorsey (2002).

Finally, there is a set of new topological-order phenomena associated with the effects of topological crystal melting dealing with more complicated “intertwined” orders Cho et al. (2015). Perhaps the simplest way to understand this is to see how topological order associated with the deconfining states of discrete gauge theories can arise as an emergent phenomenon related to stripes. This was born in the context of the magnetic stripes, where it was named “stripe fractionalization” Zaanen et al. (2001); Z. Nussinov and J. Zaanen (2002); Zhang et al. (2002). The charge stripes are at the same time domain walls in the antiferromagnet. Imagine now that the charge order is subjected to dislocation quantum melting. Insisting that the antiferromagnetic order persists, the magnet domain walls should stay intact and as a consequence only double dislocations proliferate. Since the dislocations are Bose condensed, the staggered antiferromagnetic order parameter becomes identified with its opposite: the system turns into a spin nematic which breaks magnetic rotation symmetry but has vanishing staggered magnetization (“headless arrows”). One can now unbind a double-charge dislocation pair into an isolated dislocation but this implies a frustration in the spin background that is identified as the -disclination of the spin nematic. When these proliferate as well, one enters a quantum-disordered magnetic phase. Apart from this it is also possible that the antiferromagnet quantum disorders by itself but since locally the antiferromagnet correlations are still strong, the double dislocations continue to be bound, and this is somehow a different state than the fully disordered one. As it turns out, this is precisely described by “” lattice gauge theory, where the spin nematic and fully disordered phase correspond to the Higgs and the confining phase of the gauge theory. The topologically ordered deconfining phase of the Ising gauge theory has now the simple interpretation of the otherwise featureless condensate of the double charge dislocations! Putting this on a torus, this phase would boast an either even or odd number of charge stripes when traversing the circumferences of the torus, although one cannot hope to detect the stripes since they form a nematic dislocation condensate.

Recently evidence emerged that the superconductivity in static stripe systems may behave similar to the antiferromagnetism, with the phase of the order parameter reversing from stripe to stripe: the “pair density waves” Fradkin et al. (2015); Hamidian et al. (2016). One can imagine that similar “stripe fractionalization” topological orders may occur, now involving the superconducting order parameter Berg et al. (2009). The jury is still out on whether this is of relevance to cuprates Mross and Senthil (2012, 2015). Another context where this could be relevant are the ‘unbalanced condensates’ formed by cold atoms. Conventionally one expects here the so-called FFLO states Fulde and Ferrell (1964); Larkin and Ovchinnikov (1965) which are symmetry-wise of the same kind as pair density waves. One can envisage in this context similarly partially-melted phases Radzihovsky and Vishwanath (2009).

1.5 Organization of this paper

This is a review paper and we have therefore striven to present a reasonably self-contained exposition. Understanding of basic condensed matter physics (phase transitions, Green’s functions) as well as field theory and path integrals, is assumed. This article supersedes our earlier works Zaanen et al. (2004); Kleinert and Zaanen (2004); Cvetkovic (2006); Cvetkovic et al. (2006); Cvetkovic and Zaanen (2006a, b); Cvetkovic et al. (2008); Zaanen and Beekman (2012); Beekman et al. (2013); Liu et al. (2015) wherever results are contradictory. We warm up in Sec. 2 by treating a simpler and well-studied problem: vortex–boson or Abelian-Higgs duality for interacting bosons. Here the principles of the duality construction are laid out, and we will refer to it often in the remainder of the text. Classical elasticity is restated in field theory language in Sec. 3, up to the derivation of the phonon propagators. The topological defects of spatially ordered systems, dislocations and disclinations, are introduced in Sec. 4, including field-theoretic defect currents leading to a compact form for the glide constraint for dislocations. Up to this point, nothing essentially new has been offered. In Sec. 5 an independent, gauge-theoretic treatment of the order parameters of the nematic–isotropic liquid is included, which also contains a justification for our use of the word nematic for any state with complete translational symmetry but broken rotational symmetry. The remainder of the review concerns the main topic: dislocation-mediated quantum melting of two-dimensional crystals. In Sec. 6 we perform the duality transformation by going from the strain variables of Sec. 3 to stress variables expressed as gauge fields. Here we rederive the phonon propagators in this dual language. Higher-order elasticity is considered as well, leading to the introduction of torque stress gauge fields. Sec. 7 deals with the condensate of dislocation worldlines, culminating in a Higgs term for the dual stress gauge fields. This is then used to derive the hydrodynamics of the quantum nematic in Sec. 8 and the quantum smectic in Sec. 9. These are the main results presented in this review. In the quantum nematic, where translational symmetry is restored, a Goldstone mode related to rotational symmetry breaking emerges. This ‘deconfinement’ is discussed as the release of a constraint in the latter half of Sec. 8. Finally, it is straightforward to incorporate electrically charged media into the melting program, which is the topic of Sec. 10. In Sec. 11 we derive and compare several experimental signatures such as conductivity, spectral functions and the electron energy-loss function, as well as the Meissner effect, indicating superconductivity. Together with a summary, an outlook for future developments is presented in Sec. 12. A details the Fourier space coordinate systems which are employed throughout this text, while B, C contain details about derivations in electrically charged elastic media.

1.6 Conventions and notation

  • For the temporal components we use the following notation. Italic denotes real time; Greek denotes imaginary time, and fraktur denotes an imaginary temporal component with dimensions of length, rescaled with a factor of (shear) velocity via (cf. Eqs. (9), (141)). Most of the calculations are performed in imaginary time , for which the partition function , where is the Euclidean action. Note that the Euclidean Lagrangian differs by a sign compared to the real-time Lagrangian in the non-kinetic components due to the choice . We will suppress the subscript when there is no confusion. Consequently there is no distinction between contravariant and covariant (upper and lower) indices.

  • Greek indices refer to spacetime indices, while Roman indices refer to spatial indices only. Spatial Fourier directions (see below) use capital indices . For quantum elasticity fields in Euclidean time, like the stress tensor , the lower index is the space-time vector index while the upper index is a purely spatial index related to the Burgers charge. This is important since these tensors are not fully symmetric unlike the ‘classical’ stress tensor . However, there is no essential difference between upper and lower indices since the metric is Euclidean.

  • The fully antisymmetric symbols in 3+1, 3, 2+1 and 2 dimensions respectively obey .

  • The Fourier transforms are:

  • In Fourier space we use Matsubara frequencies . Explicitly and . The momentum is and , where is an appropriate velocity to be defined in the text. The analytic continuation is where is a convergence factor, such that the Fourier transforms are calculated using .

  • In position space, all fields will be real-valued; in Fourier space we therefore use the notation .

  • Next to the standard -coordinate system, we employ two other coordinate systems in Fourier-Matsubara space. The -system has spatial components parallel () and orthogonal () to the spatial momentum . The -system has components parallel () and orthogonal (, ) to the spacetime momentum , where . These systems are detailed in A, which we encourage the reader to study before repeating any calculations. We shall often employ longitudinal and transverse projectors () for spatial coordinates,

    (1)
    (2)

    Obviously .

  • Planck’s constant is set , energies are equivalent to frequencies via .

2 Vortex–boson duality

There are two reasons to start with a different problem: vortex–boson or Abelian-Higgs or -duality in the superfluid–Bose-Mott insulator phase transition Fisher et al. (1989); Fisher and Lee (1989); Lee and Fisher (1991); Kleinert (1989); Nguyen and Sudbø (1999); Hove and Sudbø (2000); Herbut and Tešanović (1996). First, it is an easy warm up for the more complicated physics and dualities relevant to elasticity later. And second, it is the simplest duality that yet contains the most important ingredients we need, namely the nature of the condensate of topological defects and the status of dual gauge fields. Because the interpretation will, from time to time, differ from that found in some of the literature, we advise even researchers well-versed in this field to read through this section.

The idea behind vortex–boson duality is the following. An ordered state is eventually destroyed by the proliferation (unbinding) of vortices or topological defects, which in two spatial dimensions are point-like, and quantum mechanically it is the Bose–Einstein condensation thereof. The defects are topological singularities in the order parameter field and disturbances in the order parameter field are communicated by the Goldstone modes. This neatly comes together in the dual formulation, where it turns out that the Goldstone modes are in fact dual gauge fields, that precisely mediate the interaction between topological defect sources. Thus we shall adopt the view that the topological defects are ordinary bosonic particles that interact with each other by exchanging gauge fields, like electric charges interact by exchanging photons. Next, the proliferation of topological defects introduces a collective bosonic condensate field to which the gauge fields are minimally coupled. This is exactly as in a superconductor where photons are minimally coupled to the Cooper-pair condensate. We know what will happen: the photons acquire a mass (gap) through the Anderson–Higgs mechanism such that magnetic fields are expelled. Thus the dual gauge fields correspond to massless Goldstone modes in the ordered phase and massive excitations in the disordered phase.

This general framework is virtually literally realized in the superfluid–Bose-Mott insulator phase transition, which we will detail in this section. It will also form the basis of the dislocation-mediated melting transitions in the remainder of the paper. More details can be found in Refs. Cvetkovic and Zaanen (2006a); Beekman et al. (2011).

2.1 Bose–Hubbard model

The Bose-Hubbard model Fisher et al. (1989) describes bosons, created by operators on lattice sites with commutation relations , with Hamiltonian

(3)

Here the first sum is over nearest-neighbor sites, is the hopping parameter and is an on-site repulsion term. A chemical potential was fixed to ensure a large, integer average number of bosons per site such that . Now we can write where is the number operator and is the phase operator. They obey and are canonically conjugate. Then the Hamiltonian reads

(4)

Here we have rescaled with the number of bosons per site . The physics of this model is as follows. For we have a well-defined phase variable nearly constant in space, and the first term denotes the fluctuations from that phase. This is a superfluid with a spontaneously broken -symmetry and phase rigidity. In this limit Eq. (4) is also called the or quantum rotor model. For the repulsion dominates, and the boson number is fixed at each site to the average value, with a large gap to the particle and hole excitations. This is the Mott insulator of bosonic particles (there are no spin degrees of freedom). The parameter tunes the phase transition. In the following, we start out deep in the weak-coupling limit , the superfluid phase, where we can neglect the quantized bosonic degrees of freedom in the second term. Furthermore we take the continuum limit, and approximate since deviations from the preferred phase value are costly. The continuum limit Hamiltonian of the system is now

(5)

where is the lattice constant and a constant term has been dropped. Following from the commutation relation, the number excitations are canonical momenta conjugate to . The quantum partition sum of this Hamiltonian becomes the Euclidean imaginary time path integral with the following action Sachdev (2011),

(6)

where , and is the phase velocity of the superfluid. In this section and the rest of the review, we shall work exclusively in the imaginary time path integral formalism as is standard for quantum phase transitions Sachdev (2011). In this convention we have , the canonical momenta being , in order to have the partition function , where is the Euclidean action and we have dropped the subscript . While the Euclidean theory of the phase field in Eq. (6) is emergently relativistic with ‘speed of light’ , we must keep in mind that the phase-disordered side in fact corresponds to the Bose-Mott insulator.

2.2 The superfluid as a Coulomb gas of vortices

When weakly-interacting bosons condense they form a superfluid, spontaneously breaking global internal -symmetry. The resulting Goldstone mode is the zero-sound mode of the superfluid, and it is a single free massless mode described by a scalar field, as derived from the Bose-Hubbard model above. The partition function for the relativistic zero-sound mode is

(7)
(8)
(9)

Here is the inverse temperature; by we denote the phase velocity of the superfluid condensate, and we have defined . All components have dimension of inverse length via . Furthermore, is the coupling constant: for small , deviations from the spontaneously chosen value of the superfluid phase are very costly and thus strongly suppressed, stabilizing the superfluid regime. When is large the phase can fluctuate wildly. Now we need to remember that is in fact a compact variable originating from in Eq. (4), and a change by brings it back to its original value. Such windings of the phase variable correspond to vortices in the superfluid, and at large can be created easily. To incorporate the vortices, we must treat the phase as a multivalued field Kleinert (1989, 2008); Zee (2010), having both smooth and singular contributions

(10)

Here obeys

(11)

for any contour that encloses the singularity, and is the winding number of the quantized vorticity. By definition, the same contour integral for the smooth part is zero. We shall restrict ourselves to 2+1 dimensions, the generalization to higher dimensions can be found in Ref. Beekman et al. (2011). For a vortex of winding number we have by Stokes’ theorem

(12)

Here is a surface pierced by the vortex, and is its boundary. Note that derivatives operating on a multivalued field do not commute. We have defined the vortex current

(13)

We can also conveniently write it as,

(14)

Here is a delta function that is non-zero only on the vortex worldline pointing in direction  Kleinert (1989, 2008); , if and only if the worldline pierces the surface (see also Sec. 4.3). We shall from now on treat topological defects as ‘ordinary’, bosonic particles in 2+1 dimensions encoded by these currents. They obey an integrability condition,

(15)

implying that vortex worldlines cannot begin or end but must either appear as closed spacetime loops, or extend to the boundary of the system.

To see how vortices interact in the superfluid, we perform the duality operation, which is in fact a Legendre transformation to the relativistic canonical momentum associated with the field-theoretic velocity field . The canonical momentum is

(16)

The factor is used in the imaginary time formalism and is consistent with Refs. Kleinert (1989); Zaanen et al. (2004); Cvetkovic and Zaanen (2006a); Cvetkovic (2006). The field is related to the supercurrent of the superfluid by analytic continuation to real time. Now we can go from a Lagrangian in terms of the velocity to a Hamiltonian density in terms of the momentum ,

(17)

Here we used Eq. (9) and Eq. (16). Again, this form of the Hamiltonian density stems from the convention for imaginary time . The partition function is given by . As we are interested in a quantum field-theoretic description, we actually prefer to work with Lagrangians. Therefore we will formally reobtain a Lagrangian, but keep the as the principal field, thus . Hence

(18)

Note that we could have also directly obtained this form by a Hubbard–Stratonovich transformation applied to Eq. (9), but we would not have the interpretation of as the canonical momentum. Also note that the dual coupling constant is inversely proportional to the original coupling constant . This is therefore a weak–strong duality, in the sense of Kramers and Wannier Kramers and Wannier (1941).

Now comes the important step in the duality construction. We again separate the phase field in smooth and singular parts as in Eq. (10):

(19)

On the smooth part, one is allowed to perform integration by parts , and to then integrate out in the path integral as a Lagrange multiplier field, producing the constraint

(20)

It follows that is the conserved supercurrent of the superfluid, and the conservation law (continuity equation) arises because the phase fluctuations have to be smooth. The constraint Eq. (20) can be enforced explicitly by expressing it as the curl of another vector field, the dual gauge potential or dual gauge field ,

(21)

The physical field is invariant under the addition of the gradient of any smooth scalar field

(22)

so is a gauge potential for the ‘field strength’ . We substitute the definition Eq. (21) into Eq. (18) to find

(23)

Here we performed integration by parts on and substituted the definition of the vortex current Eq. (12). This form is identical to the Maxwell Lagrangian for charged particles interacting by exchanging photon fields. Vortices in a 2+1D superfluid are identical to a gas of charged particles, interacting by the electromagnetic-like dual gauge fields . Therefore we refer to this as the Coulomb phase for vortex particles. Recall that encodes for the supercurrent which encodes for the superfluid sound mode . We conclude that the Goldstone modes that communicate the rigidity of the superfluid medium also mediate interactions between the topological defects of that medium: the vortices. The gauge freedom Eq. (22) can in turn be viewed as enforcing vortex conservation Eq. (15).

2.3 The phase-disordered superfluid as a vortex condensate

The real power of duality is that it reveals how to take the analogy with electrodynamics one step further. When the free photon encounters a superconductor, which is a condensate of bosons (Cooper pairs), it minimally couples to the condensate and acquires a mass through the Anderson–Higgs mechanism. The interaction mediated by massive gauge bosons is not long- but short-ranged. The rigidity communicated by the dual gauge fields persists over a short distance only, signaling that long-range order has been lost. In other words, a hallmark of the disordered phase is that the dual gauge fields become massive.

In the dual language such a disordered state corresponds to a condensate of vortices. When the coupling constant increases above a critical value, vortices can be freely created and annihilated and undergo an unbinding transition, which is very similar to the Berezinskii–Kosterlitz–Thouless transition in 2+0 dimensions. This statement can be made very precise on a lattice by considering a grand canonical ensemble of meandering vortex worldlines Kleinert (1989) and has been confirmed by numerical calculations Nguyen and Sudbø (1999); Hove and Sudbø (2000), see also Sec. 7. The vortex condensate is described by a collective complex scalar field to which the dual gauge fields are minimally coupled.

Since the flow around the vortex core is superflow, the vortex condensate is governed by the same velocity as the superfluid phase fluctuations. And as we are working with a relativistic superfluid, the vortices also have relativistic dynamics. (In other words, we assume there is no ‘normal component’ of the fluid, which is valid near the zero-temperature quantum phase transition. Therefore the vortices do not carry entropy and move relativistically.) It will however be useful to keep track of the influence of the condensate degrees of freedom by assigning to it a hypothetical velocity , which could be different from . This turns out to be even more useful in the dual elasticity context. The minimal coupling term then becomes

(24)

The tildes indicates rescaling with the different velocity . Recall from Eqs. (16), (20) and (21) that , so we have . The action for the dual superconductor is

(25)

Here and are dual Ginzburg–Landau parameters. When the condensate field obtains a finite vacuum expectation value and its phase (not to be confused with the superfluid phase ) obtains a spontaneously chosen value . We will always work in the London limit or strong type-II limit, where fluctuations in the condensate amplitude are neglected. The dual gauge field acquires a Higgs mass with units of energy, defined by . The interactions mediated by have turned short-ranged: the dual Meissner effect. In addition to the zero-sound mode, which has turned massive, there is a second mode provided by the condensate itself. By gauge transformation this can also be interpreted as the longitudinal polarization of the dual gauge field. In the physical case where , the two modes are degenerate. In the Bose-Hubbard model, these two degenerate gapped modes correspond to the particle and hole excitations of the Bose-Mott insulator.

At this point we would like to dispel one common misconception regarding vortex–boson duality, namely that instead of the massive dual gauge field, the dual vortices in the Mott insulator phase (singularities in ) would correspond to the particle and hole (doublon/holon) excitations. As the insulator is a condensate of vortices which has the action of a type-II superconductor Eq. (25), there are the dual incarnations of Abrikosov vortices, sometimes called vortex pancakes in two dimensions. In several works, following the ideas laid out in Refs. Fisher and Lee (1989); Lee and Fisher (1991), it is surmised that these dual vortex and antivortex excitations be the particle and hole excitation in the Mott insulator. As we mentioned, that role is played by the dual gauge fields, coding for the Goldstone modes in the superfluid but acquiring a mass through the Anderson–Higgs mechanism in the Mott insulator Cvetkovic and Zaanen (2006a). There are several ways to see this. First, vortices are topological defects as opposed to localized particle excitations. The energy of a single vortex grows with the system size whereas the particle/hole excitations are local and at rest have a well-defined energy, the Mott gap. Second, the particle and hole have a well-defined boson number. Conversely, a vortex in the Mott insulator is precisely that state where boson number is no longer well-defined, since number and phase are canonically conjugate operators, and according to the duality the core of the Mott vortex recovers the superfluid phase. Similarly, since the Mott insulator is a dual Ginzburg–Landau superconductor, if the charged particle (hole) were a vortex (antivortex), this would imply that an Abrikosov vortex in a real superconductor would carry an elementary charge , which to the best of our knowledge and observations is not the case. Third, in higher dimensions the vortices become extended objects, for instance vortex lines in three dimensions. But the particle and hole excitations in the Mott insulator are always particle-like in any dimension. Instead, the dual vortices are just that: topological defects that carry a quantized value of the current whose rigidity was lost in the disordered state Zaanen and Beekman (2012). This statement was proven by the derivation in Ref. Cvetkovic and Zaanen (2006a) that the propagating modes on each side of the phase transition are represented by the dual gauge fields. Will we repeat this argument below.

For more details on the duality construction see for instance Refs. Kleinert (2008); Beekman et al. (2011). One concludes that the disordered superfluid is dual to a superconductor in 2+1D, where vortices no longer have long-range correlations. Such a disordering transition is reflected by the gapping out of the dual gauge field representing the original Goldstone mode, a theme that will feature prominently in quantum elasticity.

2.4 Propagators and duality

Although we can read off most of the physics directly from the Lagrangian for the superfluid Eq. (9), more rigor is acquired by studying the Green’s functions or propagators. The phase–phase propagator (two-point correlation function) is easy to compute in momentum space,

(26)

as one expects for a massless scalar field. However, the phase variables are not well defined across the phase transition and therefore we cannot compare the correlator Eq. (26) with anything on the disordered side. This was called “dual censorship” in Ref. Cvetkovic and Zaanen (2006a), and conveys the fact that one cannot interrogate the disordered phase with ordered means. In this light, it is better to consider the velocity–velocity correlator, where , and this is precisely the point of the duality construction. By definition , but we can also calculate it directly from the generating functional with external sources :

(27)

Integrating out the phase field will leave a Lagrangian , leading to

(28)

Obviously, since the duality is a mathematical identity, calculating the same quantity using dual variables one should obtain the same results. This is a more subtle issue than it seems at face value. With the external source in place, the dual current is redefined through the left-hand side of Eq. (16), so that

(29)

Thus, the above equation relates the original velocity fields to the dual variables. The dual Lagrangian becomes

(30)

It follows directly from that

(31)

This relation was first derived in Ref. Zaanen et al. (2004), relating propagators on the original and the dual sides. The propagator can be calculated directly from (18) but only after implementing the constraint . Another route automatically taking care of this constraint is to substitute the dual gauge field Eq. (21), via

(32)

Since the dual gauge fields couple to the vortex sources, their propagators have a meaningful interpretation by themselves. Since their action is just that of massless free vector fields, transforming to the -system (see Sec. 1.6 and A) and using relation Eq. (398), we find

(33)

Recall from Sec. 1.6 that , and is a shorthand for . The propagators can be obtained as etc. One should integrate out the fluctuating gauge fields, which we do for and then to find,

(34)

In the last line, we see that is to be integrated out as a Lagrange multiplier for the constraint , the conservation of vortex current. This is a consequence of the gauge invariance Eq. (22). This ascertains that in future calculations, is it allowed to impose the Coulomb gauge , or any other gauge fix at will, right at the Lagrangian level Eq. (33). We find two propagators in the Coulomb gauge,

(35)
(36)

Next to the expected massless propagating mode corresponding to the transverse polarization , we also discern the static Coulomb force between ‘vortex charges’ , with propagator . It is interesting that the single scalar field through the duality transformation also immediately yields these static forces. This feature will become more revealing in the elasticity context. With these results one finds the supercurrent correlators,

(37)
(38)

Here we used the longitudinal and transverse projectors Eqs. (1) and (2). Given these equations and the generating functional relation Eq. (31), one can reobtain Eq. (28). Note that the propagating mode shows up in and while the transverse current correlator is constant: a sound mode has no transverse component.

Eq. (31) holds both in the ordered (Coulomb) and the disordered (Higgs) phase. In the disordered phase, we add the Higgs term as in Eq. (25). We can now choose the unitary gauge , simplifying the Higgs term to . The gauge freedom is absorbed, and we find the Lagrangian

(39)

Here is the Higgs mass. Note the condensate velocity in the top-left matrix element. Employing external sources to find the propagators, we must integrate out the fluctuating gauge fields. Starting with , we obtain

(40)

From here on, one can integrate out the remaining gauge field components. In the end, we are simply inverting the matrix in Eq. (39). We can read off the propagators

(41)