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) = vF|k| − μ, where vF 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 vF. 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 87Rb) 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 = sc. For s > sc, 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 < sc 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).

Figure 2.1
Ground states of bosons on a square lattice with tunneling amplitude between the sites w, and on-site repulsive interactions U. Bosons are condensed in the zero momentum state in the superfluid, and so there are large number flucutations in a typical component of the wavefunction shown above. The Mott insulator is dominated by a configuration with exactly one particle on each site.

Our primary interest here is in the special critical point s = sc, 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 = sc 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 GR 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 sc. This makes ssc 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 ssc in the action

at frequencies just above the mass gap. Note now that a pole has emerged in GR, confirming that quasiparticles are present in the s > sc 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 /(kBT). 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

ω/(kBT) to be fully determined by the fixed-point theory, with no arbitrary scale factors. Note also that for T ξz |ggc|, we can safely set the second argument of to zero. So in this “quantum critical” regime, the finite temperature dynamics is described by the non-quasiparticle dynamics of the fixed point CFT3: see Figure 2.2.

Figure 2.2
Phase diagram of S as a function of s and T. The quantum critical region is bounded by crossovers at T ∼ |ssc| indicated by the dashed lines. Conventional quasiparticle or classical-wave dynamics applies in the non-quantum-critical regimes including a Kosterlitz-Thouless phase transition above which the superfluid density is zero. One of our aims in this review is to develop a theory of the non-quasiparticle dynamics within the quantum critical region.

In contrast, for T ξz |ggc| we have the traditional quasiparticle dynamics associated with excitations similar to those described by (2.8). Our discussion in the next few sections will focus on the quantum-critical region of Figure 2.2.

In the remainder of this subsection, we note extensions to other examples of zero density systems without quasiparticles.

2.1.1 Antiferromagnetism on the honeycomb lattice

We return to electronic models defined on the honeycomb lattice, relevant for systems like graphene, mentioned at the beginning of 2.1. In particular, we consider a Hubbard model for spin S = 1/2 fermions (‘electrons’) on the honeycomb lattice at a density of one fermion per site. As for the boson Hubbard model, the two energy scales are the repulsion energy, U, between two electrons on the same site (which necessarily have opposite spin), and the electron hopping amplitude w between nearest-neighbor sites. For small U/w, the electronic spectrum is that of 4 flavors of massless 2-component Dirac fermions, Ψ, and the short-range interactions only weakly modify the free fermion spectrum. In contrast, for large U/w, the ground state is an insulating antiferromagnet, as illustrated in Figure 2.3. This state is best visualized as one with fermions localized on the sites, with opposite spin orientations on the two sublattices. The excitations of the insulating antiferromagnet come in two varieties, but both are amenable to a quasiparticle description: (i) there are the bosonic Goldstone modes associated with the breaking of spin rotation symmeter, and (ii) there are fermionic quasiparticle excitations with the same quantum numbers as the electron above an energy gap.

Figure 2.3
Phase diagram of the Hubbard model for spin S = 1/2 fermions on the honeycomb lattice at a density of one fermion per site. The large U/w (s < sc) state breaks spin rotation symmetry, and is an insulating antiferromagnet with a gap to all charged excitations. The small U/w (s > sc) state is described at low energies by a CFT3 of free Dirac fermions.

The quantum field theory for the phase transition [617] between the large U/w and small U/w phases is known in the particle-theory literature as the Gross-Neveu model (where it is a model for chiral symmetry breaking). Its degrees of a freedom are a relativistic scalar φa, a = 1, 2, 3 representing the antiferromagnetic order parameter, and the Dirac fermions Ψ. The action for φa is similar to that the superfluid order parameter ψ in (2.1), and this is coupled to the Dirac fermions via a ‘Yukawa’ coupling, leading to the field theory

Here γμ are suitable 2 × 2 Dirac matrices, while the Γa are ‘flavor matrices’ which act on a combination of the valley and sublattice indices of the fermions. A Dirac fermion ‘mass’ term is forbidden by the symmetry of the honeycomb lattice. We have chosen units to that the Fermi velocity, vF = 1, and then the boson velocity, c, cannot be adjusted. The γμ and the Γa turn out to commute with each other, and so when the velocity c = vF = 1, the action Sφ becomes invariant under Lorentz transformations. The equal velocities are dynamically generated under the renormalization group flow, and so we find an emergent Lorentz invariance near the quantum critical point, similar to that found above for the superfluid-insulator transition for bosons. The tuning parameter across the quantum critical point, s, is now a function of U/w. The phase with broken spin-rotation symmetry (s < sc) has φa ≠ 0, we then see that the Yukawa term, λ, endows the Dirac fermions with a ‘mass’ i.e. there is a gap to fermionic excitations; this ‘mass’ is analogous to the Higgs boson endowing the fermions with a mass in the Weinberg-Salam model. The quantum critical point at s = sc can be analyzed by renormalization group and other methods, as for the boson-only theory S in (2.1). In this manner, we find that the quantum critical theory is a CFT, described by a renormalization group fixed at u = u* and λ = λ*. This is a state without quasiparticle excitations, with Green’s functions for φa and Ψ similar to those of the other CFT3s noted above.

2.1.2 Quadratic band-touching and z ≠ 1

This subsection briefly presents an example of a system at commensurate density with dynamic critical exponent z ≠ 1; all other models described in the current 2.1 have an emergent relativistic structure and z = 1. We consider a semiconductor in which the parabolic conduction and valence bands touch quadratically at a particular momentum in the Brillouin zone: such band-touching can arise generically in materials with strong spin-orbit coupling [549]. Without doping, the electron chemical potential is precisely at the touching point, and so the free fermion Hamiltonian has a scale invariant structure with z = 2. In d = 3, the long-range Coulomb interaction is a relevant perturbation to the free fermion fixed point [6; 5], and its effects can be described by the action [584]

contains the quadratic dispersion of both bands, and the scalar field mediates the Coulomb interaction. A renormalization group analysis [584] shows that the coupling e flows to a non-trivial fixed point with z smaller than 2. A recent experiment [492] on the pyrochlore iridate Pr2Ir2O7 displays evidence for this non-trivial critical behavior.

2.1.3 Emergent gauge fields

Condensed matter models also lead to field theories which have fluctuating dynamic gauge fields, in contrast to those in (2.1) and (2.10). These arise most frequently in theories of quantum antiferromagnets, and can be illustrated most directly using the ‘resonating valence bond’ (RVB) model [623; 39]. Figure 2.4a shows a component, |Di, of a wavefunction of an antiferromagnet on the square lattice, in which nearby electrons pair up in to singlet valence bonds—the complete wavefunction is a superposition of numerous components with different choices of pairing between the electrons:

Figure 2.4
RVB states and excitations. (a) Sketch of a component of a resonating valence bond wavefunction on the square lattice. (b) Excited state with neutral excitations carrying spin S = 1/2. In the field theory 𝒮z each excitation is a quantum of the zα particle, while the gauge field A represents the fluctuations of the valence bonds (see Figure 2.5).

where the ci are unspecified complex numbers. In a modern language, such a resonating valence bond wavefunction was the first to realize topological order and long-range quantum entanglement [753]. In more practical terms, the topological order is reflected in the fact that an emergent gauge fields is required to describe the low energy dynamics of such a state [77; 277]. A brief argument illustrating the origin of emergent gauge fields is illustrated in Figure 2.5. We label each configuration by a set of integers

where Δα is a discrete lattice derivative, and ρi (−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.

Figure 2.5
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, zα, 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 zα 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 zα (now α = 1…M)

where F = dA. The order parameter for the broken symmetry in the antiferromagnet involves a gauge-invariant bilinear of the zα. In the present case, it turns out that the resonating valence bond phase where the zα 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 = sc, and then Sz 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.

Figure 2.6
Phases of a square lattice antiferromagnet. The vicinity of the critical point is described by the theory Sz 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 Sz 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 = sc. 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.