# 2

# Zero density matter

The dynamic structure factor at low energies in herbertsmithite, from inelastic neutron scattering. The diffuse continuum of excitations indicates a lack of magnetic ordering and associated quasiparticles, and is consistent with a fractionalized state of matter. Non-quasiparticle physics and fractionalization will be discussed in this chapter. Image adapted from [341] with permission.

# 2.1 Condensed matter systems

The simplest examples of systems without quasiparticle excitations are realized in quantum matter at special densities which are commensurate with an underlying lattice [673]. These are usually modeled by quantum field theories which are particle-hole symmetric, and have a vanishing value of a conserved U(1) charge linked to the particle number of the lattice model — hence ‘zero’ density. Even in the lattice systems, it is conventional to measure the particle density in terms of deviations from such a commensurate density.

In this chapter we will first give some explicit examples of non-quasiparticle zero density field theories arising at quantum critical points in condensed matter systems. This will include an example with emergent gauge fields. We will then discuss how such quantum critical systems can arise holographically. Our discussion in this section will cover quantum critical scaling symmetries as well as deformations away from scaling by temperature and relevant operators. Charge dynamics in these zero density quantum critical theories will be the subject of Chapter 3.

The simplest example of a system modeled by a zero density quantum field theory is graphene. This QFT is furthermore relativistic at low energies.^{1} Graphene consists of a honeycomb lattice of carbon atoms, and the important electronic excitations all reside on a single *π*-orbital on each atom. ‘Zero’ density corresponds here to a density of one electron per atom. A simple computation of the band structure of free electrons in such a configuration yields an electronic dispersion identical to massless 2-component Dirac fermions, Ψ, which have a ‘flavor’ index extending over 4 values. The Dirac ‘spin’ index is actually a sublattice index, while the flavor indices correspond to the physical *S* = 1/2 spin and a ‘valley’ index corresponding to 2 Dirac cones at different points in the Brillouin zone. The density of electrons can be varied by applying a chemical potential, *μ*, by external gates, so that the dispersion is 𝜖(**k**) = *v*_{F}|**k**| − *μ*, where *v*_{F} is the Fermi velocity [126]. Electronic states with 𝜖(**k**) *<* 0 are occupied at *T* = 0, and zero density corresponds to *μ* = 0.

At first glance, it would appear that graphene is not a good candidate for holographic study. The interactions between the electrons are instantaneous, non-local Coulomb interactions, because the velocity of light *c* ≫ *v*_{F}. These interactions are seen to be formally irrelevant at low energies under the renormalization group, so that the electronic quasiparticles are well-defined. However, as we will briefly note later, the interactions become weak only logarithmically slowly [302; 282], and there is a regime with *T* ≳ *μ* where graphene can be modeled as a dissipative quantum plasma. Thus, insight from holographic models can prove quite useful. Experimental evidence for this quantum plasma has emerged recently [159], as we will discuss further in this review. More general electronic models with short-range interactions on the honeycomb lattice provide examples of quantum critical points without quasiparticle excitations even at *T* = 0, and we will discuss these shortly below.

For a more clear-cut realization of quantum matter without quasiparticles we turn to a system of ultracold atoms in optical lattices. Consider a collection of bosonic atoms (such as ^{87}Rb) placed in a periodic potential, with a density such that there is precisely one atom for each minimum of the periodic potential. The dynamics of the atoms can be described by the boson Hubbard model: this is a lattice model describing the interplay of boson tunneling between the minima of the potential (with amplitude *w*), and the repulsion between two bosons on the same site (of energy *U*). For small *U/w*, we can treat the bosons as nearly free, and so at low temperatures *T* they condense into a superfluid with macroscopic occupation of the zero momentum single-particle state, as shown in Figure 2.1. In the opposite limit of large *U/w*, the repulsion between the bosons localizes them into a Mott insulator: this is a state adiabatically connected to the product state with one boson in each potential minimum. We are interested here in the quantum phase transition between these states which occurs at a critical value of *U/w*. Remarkably, it turns out that the low energy physics in the vicinity of this critical point is described by a quantum field theory with an emergent Lorentz invariant structure, but with a velocity of ‘light’, *c*, given by the velocity of second sound in the superfluid [673; 270]. In terms of the long-wavelength boson annihilation operator *ψ*, the Euclidean time (*τ*) action for the field theory is

Here *s* is the coupling which tunes the system across the quantum phase transition at some *s* = *s*_{c}. For *s > s*_{c}, the field theory has a mass gap and no symmetry is broken: this corresponds to the Mott insulator. The gapped particle and anti-particle states associated with the field operator *ψ* correspond to the ‘particle’ and ‘hole’ excitations of the Mott insulator. These can be used as a starting point for a quasiparticle theory of the dynamics of the Mott insulator via the Boltzmann equation. The other phase with *s < s*_{c} corresponds to the superfluid: here the global U(1) symmetry of *S* is broken, and the massless Goldstone modes correspond to the second sound excitations. Again, these Goldstone excitations are well-defined quasiparticles, and a quasiclassical theory of the superfluid phase is possible (and is discussed in classic condensed matter texts).

Our primary interest here is in the special critical point *s* = *s*_{c}, which provides the first example of a many body quantum state without quasiparticle excitations. And as we shall discuss momentarily, while this is only an isolated point at *T* = 0, the influence of the non-quasiparticle description widens to a finite range of couplings at non-zero temperatures. Renormalization group analyses show that the *s* = *s*_{c} critical point is described by a fixed point where *u → u*^{*}, known as the Wilson-Fisher fixed point [673; 754]. The value of *u*^{*} is small in a vector large *M* expansion in which *ψ* is generalized to an *M*-component vector, or for small 𝜖 in a field theory generalized to *d* = 3 − 𝜖 spatial dimensions. Given only the assumption of scale invariance at such a fixed point, the two-point *ψ* Green’s function obeys

at the quantum critical point. Here *η* is defined to be the anomalous dimension of the field *ψ* which has scaling dimension

in *d* spatial dimensions. The exponent *z* determines the relative scalings of time and space, and is known as the dynamic critical exponent: [*t*] = *z*[*x*]. The function *F* is a scaling function which depends upon the particular critical point under study. In the present situation, we know from the structure of the underlying field theory in (2.1) that

Indeed, the Wilson-Fisher fixed point is not only Lorentz and scale invariant, it is also conformally invariant and realizes a conformal field theory (CFT), a property we will exploit below.

A crucial feature of *G*^{R} in (2.4) is that for *η ≠* 0 its imaginary part does not have any poles in the *ω* complex plane, only branch cuts at *ω* = *±ck*. This is an indication of the absence of quasiparticles at the quantum critical point. However, it is not a proof of absence, because there could be quasiparticles which have zero overlap with the state created by the *ψ* operator, and which appear only in suitably defined observables. For the case of the Wilson-Fisher fixed point, no such observable has ever been found, and it seems unlikely that such an ‘integrable’ structure is present in this strongly-coupled theory. Certainly, the absence of quasiparticles can be established at all orders in the 𝜖 or vector 1*/M* expansions. In the remaining discussion we will assume that no quasiparticle excitations are present, and develop a framework to describe the physical properties of such systems. We also note in passing that CFTs in 1+1 spacetime dimensions (*i.e.* CFT2s) are examples of theories in which there are in fact quasiparticles present, but whose presence is not evident in the correlators of most field variables [391]. This is a consequence of integrability properties in CFT2s which do not generalize to higher dimensions.

We can also use a scaling framework to address the ground state properties away from the quantum critical point. In the field theory *S* we tune away from the quantum critical point by varying the value of *s* away from *s*_{c}. This makes *s* − *s*_{c} a relevant perturbation at the fixed point, and its scaling dimension is denoted as

We can also express these scaling results in terms of the dimension of the operator conjugate to *s* − *s*_{c} in the action

at frequencies just above the mass gap. Note now that a pole has emerged in *G*^{R}, confirming that quasiparticles are present in the *s > s*_{c} Mott insulator.

We now also mention the influence of a non-zero temperature, *T*, on the quantum critical point, and we will have much more to say about this later. In the imaginary time path integral for the field theory, *T* appears only in boundary conditions for the temporal direction, *τ*: bosonic (fermionic) fields are periodic (anti-periodic) with period ℏ/(*k*_{B}*T*). At the fixed point, this immediately implies that *T* should scale as a frequency. So we can generalize the scaling form (2.2) to include a finite *ξ* and a non-zero *T* by

where Δ_{α} is a discrete lattice derivative, and *ρ*_{i} *≡* (−1)^{ix+iy} is a background ‘charge’ density. (2.13) is analogous to Gauss’s Law in electrodynamics, and a key indication that the physics of resonating valence bonds is described by an emergent compact U(1) gauge theory.

**Number operators**, , counting the number of singlet valence bonds on each link of the square lattice; here we have numbered links by integers, but in the text we label them by the sites they connect. Modulo a phase factor, these operators realize the electric field operator of a compact U(1) lattice gauge theory.

CFTs with emergent gauge fields appear when we consider a quantum phase transition out of the resonating valence bond state into an ordered antiferromagnet with broken spin rotation symmetry. To obtain antiferromagnetic order, we need to consider a state in which some on the electrons are unpaired, as illustrated in Figure 2.4b. For an appropriate spin liquid, these excitations behave like relativistic bosons, *z*_{α}, at low energy; here *α* =↑, ↓ is the ‘spin’ index, but note here that *α* does not correspond to spacetime spin, and behaves instead like a ‘flavor’ index associated with the global SU(2) symmetry of the underlying lattice antiferromagnet. The quantum transition from the resonating valence bond state to the antiferromagnet is described by the condensation of the *z*_{α} bosons [651; 652], and the quantum critical point is proposed to be a CFT3 [685]; see Figure 2.6. Such a transition is best established for a set of quantum antiferromagnetics with a global SU(*M*) symmetry, as reviewed recently by Kaul and Block [466]. These models are expressed in terms of SU(*M*) spins, , residing on the sites of various bipartite lattices, with bilinear and biquadratic exchange interactions between nearest neighbor sites. The low energy physics in the vicinity of the critical point is described by a field theory with an emergent U(1) gauge field *A* and the relativistic scalar *z*_{α} (now *α* = 1…*M*)

where *F* = d*A*. The order parameter for the broken symmetry in the antiferromagnet involves a gauge-invariant bilinear of the *z*_{α}. In the present case, it turns out that the resonating valence bond phase where the *z*_{α} are gapped is unstable to the proliferation of monopoles in the U(1) gauge field, which leads ultimately to the appearance of a broken lattice symmetry with ‘valence bond solid’ (VBS) order (see Figure 2.6) [651; 652]. However, the monopoles are suppressed at the quantum critical point at *s* = *s*_{c}, and then *S*_{z} realizes a conformal gauge theory [685]: numerical studies of the lattice Hamiltonian are consistent with many features of the 1*/N* expansion of the critical properties of the gauge theories.

**Phases of a square lattice antiferromagnet.**The vicinity of the critical point is described by the theory

*S*

_{z}in (2.14) (compare Figure 2.3). The phase with Néel order is the Higgs phase of the gauge theory, while the Coulomb phase of

*S*

_{z}is destabilized by monopoles, leading to valence bond solid order.

Our interest here is primarily on the non-quasiparticle dynamics of the *T >* 0 quantum critical region near *s* = *s*_{c}. Here, the scaling structure is very similar to our discussion of the Wilson-Fisher critical theory, with the main difference being that the associated CFT is now also a gauge theory.