UC Berkeley

In this thesis, we examine the question of technical naturalness from the point of view of nonrelativistic quantum field theories of Lifshitz type.

Lifshitz field theories are distinguished from standard relativistic quantum field theories by the spacetime scaling symmetries that they enjoy in the vicinity of their renormalization group fixed points. These scaling symmetries are parametrized by the dynamical critical exponent, which measures the degree of scaling of time relative to spatial coordinates. Whereas in relativistic theories, space and time scale equally with each other, in Lifshitz field theories, the dynamical critical exponent may differ from unity. Furthermore, Lifshitz field theories live in spacetimes that possess a foliation structure by spatial leaves of constant time. Therefore, in contrast to relativistic theories, Lifshitz field theories are not invariant under the full diffeomorphism group of spacetime, but rather only those diffeomorphisms that preserve the built-in foliation structure. In the flat spacetime, this would simply exclude the usual spacetime boost symmetries. Since time is treated on a fundamentally different footing as is space, these theories are often referred to as anisotropic. In addition, these theories often require the tuning of multiple parameters in order to approach their fixed points under renormalization group flow, and are consequently called multicritical.

We will explore some new and interesting lessons that nonrelativistic theories have to teach us about technical naturalness. We begin this study at the modest level of Lifshitz scalar field theories. We examine the nature of Nambu-Goldstone (NG) bosons that arise from spontaneous symmetry breaking in multricritical systems described by Lifshitz scalar field theories. The NG modes in such nonrelativistic theories were previously classified into two types: (1) Type-A, which disperses linearly, and which derives its kinetic energy from a term which is quadratic in time derivatives; and (2) Type-B, which disperses quadratically, and which is described by a pair of fields with kinetic terms linear in time derivatives. In principle, Type-A modes dispersing by a power different from unity, and Type-B modes dispersing by a power different from two, can exist. However, the naive expectation from relativistic quantum field theory is that these would require fine tuning and are therefore technically unnatural. We discover that this is not the case. Instead of fine-tuning, all one needs is a new type of symmetry by which the fields transform by a polynomial function of some appropriate degree in the spatial coordinates. This polynomial shift symmetry protects the naturalness of the corresponding NG bosons. This leads to a refinement of the classification of technically natural NG modes in nonrelativistic theories.

Having discovered these polynomial shift symmetries, we turn our attention to the classification of Lagrangians that are invariant (up to total derivatives) with respect to these symmetries. We develop a novel graph-theoretical technique in order to address this problem. In this language, the invariants display beautiful patterns that otherwise remain obscured. For example, linear-shift invariants are presented as equal-weight sums over all labeled trees with some fixed number of vertices. Furthermore, we develop a graph-theoretical method for constructing invariants under polynomial shifts of high degree from invariants under polynomial shifts of lower degree. In this way, one no longer needs to repeat the entire classification process for each degree of the polynomial shift symmetry.

The third part of the thesis uses some of the theories, which are built out of invariants constructed in the second part of the thesis, to study a novel feature that these Lifshitz theories possess as one changes the energy scale at which the systems are examined. We find that these systems can flow from one fixed point described by one value of the dynamical exponent, to another fixed point described by a different value of the dynamical exponent. Furthermore, the system can explore any number of fixed points between the extreme high energy regime and the extreme low energy regime. We refer to this behavior as a cascade. Not only can Type-A modes cascade into other Type-A modes (or similarly for Type-B modes), but Type-A modes can flow towards Type-B at low energies as well. Both mechanisms are protected by symmetries. The purely Type-A or Type-B cascade is protected by the polynomial shift symmetries in space. The Type-A to Type-B cascade can be protected by various symmetries, including a linear shift symmetry in time. Furthermore, we re-examine the Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem, which prohibits the spontaneous breaking of global internal continuous symmetries in relativistic theories in two spacetime dimensions. Naively, this theorem would prevent the existence of a Type-A mode unless its dynamical exponent is strictly less than the spatial dimension, which is when the theory is in its lower critical dimension. The cascade represents a mechanism by which this result can be circumvented.

Next, we examine the renormalization group flow of one particular Lifshitz scalar field theory. We perform the analysis explicitly using three different standard techniques of renormalization and show that they are all mutually consistent. Furthermore, we demonstrate that the RG flow of Lifshitz theories can be interpreted physically in many different, but consistent, ways due to the additional freedom of renormalizing the dynamical exponent.

The lessons in Lifshitz field theories discussed in this thesis are most readily applied in the area of condensed matter physics, where systems often display a richer spectrum of behavior than is described by relativistic physics. We perform a preliminary study of the effects of coupling these Lifshitz theories with other systems. In particular, we study the naturalness problem of the linear dependence on temperature of the resistivity of so-called strange metals, which are high-temperature superconductors above their critical temperature. We show that this behavior is reproduced by the standard electron-phonon interaction picture of superconductivity, if the phonons are allowed to be multicritical and at their lower critical dimension. We also examine the impact that this model has on the heat capacity of the system.