Superinsulators, Bose Metals and High-Tc Superconductors

The following sections are included:

• Dedication
• Preface
• Contents

Superinsulation is dual superconductivity. Already in the 70s, Nambu [1], Mandelstam [2] and ‘t Hooft [3] proposed their model of dual superconductivity as a possible explanation of quark confinement (for a review see [4]). The basic idea is that an Abelian magnetic monopole condensate in the U(1)^N−1 subgroup of SU(N) squeezes the lines of the colour-electric field into thin flux tubes, exactly as a charge condensate in a superconductor squeezes magnetic flux into vortices (for a review see [5]). This dual, colour-electric Meissner effect creates strings between the quarks at their endpoints. When quarks are pulled apart, it is energetically favourable to pull out of the vacuum additional quark-antiquark pairs and to form several short strings instead of a long string. As a consequence, colour charge can never be observed at distances above a fundamental length scale, 1/Λ_QCD and quarks are confined. Only colour-neutral hadron jets can be observed in collider events…

Fundamental interactions are mediated by gauge fields. The key ingredient of the gauge principle is that a local action formulation of the interactions, leading to second-order equations of motion, requires the introduction of redundant, non-physical degrees of freedom which have to be accurately “removed” upon quantization. The theory is invariant under local transformations of the non-physical variables while the physical degrees of freedom are left untouched by these transformations. This is the meaning of local gauge invariance, which expresses a redundancy in the local description of physical phenomena rather than a true symmetry. The simplest example of a model with Abelian gauge symmetry is electromagnetism. The physical electric and magnetic fields are expressed via a four-vector gauge potential A_μ = (A₀ = V, −A) as,

In (2+1) dimensions the Maxwell action is power-counting irrelevant, since the coupling constant e² has dimension [mass]. Correspondingly, the model is super-renormalizable and can be expected to be plagued by severe infrared divergences if regulating mass terms are absent. Mass terms, however, typically break gauge invariance. In a series of seminal papers [48], Deser, Jackiw and Templeton showed, on the contrary, that a gauge invariant mass term exists in (2+1) dimensions, if one is willing to pay the price of violating the discrete symmetries of parity 𝒫, with group ${\mathbb{Z}}_2^P$, and time-reversal 𝒯 , with group ${\mathbb{Z}}_2^T$. This is the famous Chern–Simons (CS) term and the augmented massive electrodynamics has the Lagrangian density

In the following chapters we shall consider compact Abelian theories, with gauge group U(1) rather than ℝ. As was pointed out by Polyakov [19], a proper formulation of a compact U(1) gauge theory requires either spontaneous symmetry breaking of a compact non-Abelian group or the introduction of an ultraviolet lattice regularization. We shall choose the latter procedure. Define 𝓁 as the spacing on a hypercubic lattice with sites denoted by {x} and directions indicated by Greek letters and introduce the forward and backward derivatives and shift operators

Statistical mechanics and quantum mechanics both deal with fluctuations, thermal and quantum respectively. Formally, their partition functions,

The Landau theory of second-order phase transitions (for a review see [67]) is one of the pillars of modern statistical mechanics and condensed matter physics. It uses an order parameter spontaneously breaking a given symmetry to distinguish between an ordered and a disordered phase. There are, however, quantum phase transitions (for a review see [68]) between phases that cannot be distinguished by different symmetries, the most notable examples being the transitions between different fractional quantum Hall plateaus at different filling factors (for a review see [69])…

Electric forces are typically much stronger than magnetic ones. Superconductivity (for a comprehensive review see [5]) is made possible only by getting rid of the Coulomb interaction amongst charge carriers by screening it. As a consequence, the elementary bulk excitations (for type II superconductors, those of interest here) are not anymore the electric charge carriers but rather the magnetic vortices, with an energy per unit length $O\left(1/{\lambda }_L^2\right)$, with λ_L the London penetration depth…

As we have discussed in Chapter 7, the degrees of freedom in the vicinity of the SIT are Cooper pairs localized on granules of minimum size 𝓁 and Josephson-like vortices in between. This information is sufficient to establish the entire phase diagram. Charges and vortices acquire Aharonov– Bohm [44] and Aharonov–Casher [54] phases when one type of excitation encircles the other. These mutual statistics interactions (for a review see [79]), moreover, are the infrared (IR)-dominant ones since they have infinite range. In the Euclidean partition function describing point charges and vortices they must be accounted for by the topological Gauss linking number of their trajectories (3.27), with the value k = 1 to guarantee the Dirac quantization condition [17]. Unfortunately, this topological invariant is non-local. It was, however, noted by Wilczek [55] that it can be made local by coupling the point charge and vortex currents Q_μ and M_μ to two fictitious gauge fields a_μ and b_μ with mixed Chern–Simons term, giving rise to the Euclidean partition function

In what follows we shall compute the electromagnetic response in the various quantum phases of the system at T = 0 in order to establish their nature. To accomplish this we minimally couple the current j_μ = (1/2π)k_μνb_ν to the physical electromagnetic gauge field A_μ in the Euclidean action,

Since Fisher first pointed out that the direct SIT transition point is a universal metal with exactly one quantum of sheet resistance R_□ = R_Q = h/4e² = 2π/4e² (in natural units) [9], it was predicted that, under certain conditions, this single point could open up to form a fully fledged metallic intermediate phase between superconductor and superinsulator [6]. This metal made out of bosons was subsequently called Bose metal [25, 26] and is known today also as bosonic topological insulator [97]. It was observed in numerous experiments on different materials [28–30] (for a review see [27]). In this chapter we will work in the continuum formulation in Minkowski space-time and we will show how the granular structure of the SIT [35] is typical of a bosonic topological insulator and self-organizes from the competition of two quantum BKT transitions…

Until now we have focused on effective field theories, based exclusively on symmetry considerations and topology. The question arises, if there exists a concrete system that realizes the SIT effective field theory at the microscopic level. The answer is affirmative, such a system exists and it is provided by Josephson junction arrays (JJA) [7]…

Second-order phase transitions are characterized by scale-invariant correlation functions. The lack of a scale implies that the lattice spacing can be removed at these critical points, giving rise to a continuum field theory. It turns out that critical field theories actually have a larger invariance under the full conformal group. In 3D this has no far reaching consequences. In 2D, however, the conformal group is infinite-dimensional [120]. This implies a huge number of constraints which, ultimately, are sufficient to completely classify all possible critical phenomena in two dimensions [73]…

In the preceding chapter we have learned about the power of generic symmetry considerations to establish the quantum numbers of admitted excitations and, thereby, to provide an exhaustive classification of possible Bose metal ground states in presence of frustration. This is an example of a dynamical symmetry. A dynamical symmetry is an organizational principle for the classical configuration space and the quantum mechanical Hilbert space based on a fundamental symmetry of the problem [127]. The admitted states in the model are the representations of the symmetry algebra and are characterized by quantum numbers corresponding to the values of the Cartan subalgebra generators in these representations. The Hamiltonian is expanded in the Casimir operators of the symmetry algebra so that each representation has the same energy for all its states [127]. Excitations can be combined by the representation combination rules, called fusion rules…

The following sections are included:

• The 3D SIT
• Topologically ordered (Higgsless) superconductivity
• 3D superinsulators
• 3D bosonic insulators
• Finite-temperature phase transitions

The infrared-dominant term in the effective gauge action of the 3D SIT is the topological BF term (14.1). In this chapter we will ask the question of the microscopic origin of this term. In 2D, the topological Chern–Simons mass term is radiatively induced at one-loop level by coupling the gauge field to massive fermions, the induced coupling being determined by the sign of the fermion mass [147, 148]. As we now show, in 3D the BF term is correspondingly radiatively induced by gauging the spin of electrons [149]…

We have established that, under certain conditions, the electromagnetic response of insulating materials can become the compact version of QED [18, 19], which admits magnetic monopoles. When these Bose condense, superinsulation is induced and electric charge is confined. In this chapter we shall, instead, consider the physics of non-condensed magnetic monopoles, in particular their interaction with non-confined electric charge…

The electromagnetic response of Mott insulators is the usual Maxwell action, augmented by the θ-term for strong topological insulators, that of superconductors is the Proca action, including a photon mass term ${\lambda }_L^{-2}{A}_{\mu }{A}^{\mu }$ which, in the physical Weyl gauge A₀ = 0, is equivalent to the London equations $j_{\text{i}\text{n}\text{d} }={\lambda }_L^{-2}A$. What is the corresponding electromagnetic response of superinsulators? Until now we have derived that, in the superinsulating phase, positive and negative charges are linearly bound by an electric flux string, which causes the infinite resistance. In this chapter we will focus on the full electromagnetic response and we will derive the field theory for the confining strings…

In this chapter we will focus again on 2D superinsulating films. We have shown that the electromagnetic response of such materials is compact QED. Compact QED in 2D, however, does not have a phase transition, it is always in a confining phase, since monopole instantons are always in a plasma phase. Things are different, however, when compact QED is coupled to matter. Dynamical matter can induce a deconfinement phase transition [165]…

In the last part of the book we are going to discuss the possible relevance of magnetic monopoles for the physics of high-T_c superconductivity. Here we shall construct an effective long-distance field theory that reproduces the observed universal aspects of high-T_c superconductors [41]. In the next chapter we will then focus on a new microscopic mechanism of pairing. Contrary to the material presented in previous chapters, there is as yet no experimental evidence for these ideas, which have to be considered, for the moment, just one possible model, albeit one that seems to explain most, if not all, the present observations…

Electron pairing in high-T_c superconductors (for a review see [42, 172]) is due to a different mechanism than the standard s-wave Bardeen–Cooper–Schrieffer (BCS) phonon-mediated attraction (for a review see [5]). Typical existing proposals for a different mechanism, however, still focus on BCS-type pairing, either of different objects or mediated by different massless fields. In Anderson’s famed resonating valence bond (RVB) model [188] and variants thereof [189] the BCS wave function is a superposition of neutral couples of spin-singlet electrons that become charged and superconduct upon doping (for a review see [190]). Otherwise, BCS-type pairing by spin density waves [191] and by bosonic collective excitations of the fermions that become massless at a quantum critical point have been considered [192,193]. Whatever the correct mechanism, however, it must guarantee that electron pairs in high-T_c materials are localized on scales of a few nm by very strong interactions, a situation very different from the weak BCS pairing in standard superconductors [194], as illustrated in Fig. 20.1. The phenomenology of high-T_c materials points to the formation of localized electron pairs that subsequently Bose condense at a lower temperature [195]…

In Chapter 20 we have seen that magnetic monopoles can provide local seeds for strong-coupling electron pairing in higher angular momentum states. Here we will ask the question in which quantum materials can such monopoles appear. We have already mentioned effective monopoles due to curvature defects in graphene and graphite and to hedgehog defects in magnetically ordered materials, like cuprates. However, there is also another possibility, which we now discuss. To this end we shall consider Mott insulators, materials in which strong Coulomb interactions suppress charge transport. The main model to describe this physics is the Hubbard model, either in its fermionic or bosonic versions (for a review see [206]). The Hamiltonian H_Hubbard of this model is formulated on a lattice and contains two terms: a kinetic term with parameter t, describing the hopping of charges from one site to an adjacent one; and an on-site potential U modelling the Coulomb repulsion…

As we have stressed repeatedly in previous chapters, in the superconductor-to-insulator (SIT) transition, exogenous disorder plays an important role as a tuning parameter, since it can influence the strength of the Coulomb interaction [95], but is irrelevant in the renormalization group sense, i.e. it does not influence the infrared character of the emerging phases. In this chapter we will focus on the interplay between disorder, topology, and interactions and explain in more detail why this is the case…

To conclude we would like to sum up in a compact form the main points of this book…

The following section is included:

• Bibliography