Channeling Quantum Criticality.

We discuss general aspects of renormalization group (RG) flows between two conformal fixed points in 4D with a broken continuous global symmetry in the UV. Every such RG flow can be described in terms of the dynamics of Nambu-Goldstone bosons of broken conformal and global symmetries. We derive the low-energy effective action that describes this class of RG flows from basic symmetry principles. We view the theory of Nambu-Goldstone bosons as a theory in anti--de Sitter space with the flat space limit. This enables an equivalent 3D CFT (${\mathrm{CFT}}_{3}$) formulation of these 4D RG flows in terms of spectral deformations of a generalized free ${\mathrm{CFT}}_{3}$. We utilize this dual description to impose further constraints on the low-energy effective action associated with unitary RG flows in 4D by invoking the chaos bound in 3D. This approach naturally provides a set of independent monotonically decreasing $C$-functions for 4D RG flows with global symmetry breaking by explicitly relating 4D $C$-functions with certain out-of-time-order correlators that diagnose chaos in 3D. We also comment on a more general connection between RG and chaos in QFT.

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this Letter, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional N=2 superconformal field theories (SCFTs) by performing simple calculations involving sets of nonunitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full N=2 superconformal algebra. As a by-product, we find relations between characters in unitary and nonunitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.

We consider line defects in d-dimensional conformal field theories (CFTs). The ambient CFT places nontrivial constraints on renormalization group (RG) flows on such line defects. We show that the flow on line defects is consequently irreversible and furthermore a canonical decreasing entropy function exists. This construction generalizes the g theorem to line defects in arbitrary dimensions. We demonstrate our results in a flow between Wilson loops in four dimensions.

We analyze the conceptual role of background independence in the application of the effective average action to quantum gravity. Insisting on a background independent renormalization group (RG) flow the coarse graining operation must be defined in terms of an unspecified variable metric since no rigid metric of a fixed background spacetime is available. This leads to an extra field dependence in the functional RG equation and a significantly different RG flow in comparison to the standard flow equation with a rigid metric in the mode cutoff. The background independent RG flow can possess a non-Gaussian fixed point, for instance, even though the corresponding standard one does not. We demonstrate the importance of this universal, essentially kinematical effect by computing the RG flow of quantum Einstein gravity in the ``conformally reduced'' Einstein-Hilbert approximation which discards all degrees of freedom contained in the metric except the conformal one. Without the extra field dependence the resulting RG flow is that of a simple ${\ensuremath{\phi}}^{4}$ theory. By including it one obtains a flow with exactly the same qualitative properties as in the full Einstein-Hilbert truncation. In particular it possesses the non-Gaussian fixed point which is necessary for asymptotic safety.

We examine the phase structure of an Abelian Chern-Simons system with relativistic charged matter fields with a sixth-order potential. Using a large $N$ technique, we compute the quantum effective potential and the renormalization group function of the coupling to the next-to-leading order of the $1/N$ expansion in terms of the Chern-Simons coefficient. The model has a phase which exhibits spontaneous breaking of scale symmetry accompanied by a massless dilaton which is a Goldstone mode. We show that the beta function of the sextic coupling exhibits, at the next order in the $1/N$ expansion, nontrivial running that we analyze explicitly in terms of the Chern-Simons coefficient. Viewed as a dynamical system, the renormalization group (RG) flow exhibits a topological normal form of a generic one-dimensional system having a fold bifurcation. We demonstrate that the corresponding IR and UV fixed points, each describing a conformal phase of the theory, approach each other until they merge, giving rise to a scaling behavior similar to Berezinskii-Kosterlitz-Thouless phase transitions. Our study identifies a window in the parameter space of the Chern-Simons coefficient where the renormalization group flow has a stable infrared fixed point and where scale invariance is recovered. We also find that the Chern-Simons interaction modifies the scaling dimension of the operator crossing marginality at the bifurcation points.

This thesis is dedicated to certain nonlocal aspects of conformal quantum field theories (CFTs). Two main directions are the study of CFTs on a particular globally-nontrivial spacetime corresponding to finite temperature, and the study of particular nonlocal CFT observables localized on light-rays. Specifically, we introduce bootstrap techniques for determining finite-temperature data of CFTs, and make novel predictions for the 2+1-dimensional Ising model. We propose the “stringy equivalence principle”, stating that coincident gravitational shocks commute, as a generalization of the strong equivalence principle of Einstein’s General Relativity that must hold in all consistent theories of gravity. We prove it in Anti-de Sitter (AdS) spacetimes by studying light-ray operators in the holographically dual CFT. We also derive an operator product expansion (OPE) for light-ray operators in CFT, by which two light-ray operators on the same light-sheet can be expanded as a sum of single light-ray operators. Light-ray operators model detectors — such as calorimeters. We use the light-ray OPE to compute energy event shape observables suitable for conformal collider physics. An additional part of this thesis determines the low-energy vacua of two-dimensional maximal super-Yang-Mills theory, which describes the dynamics of stacks of D-strings in Type IIB string theory. By computing an invariant of the renormalization group (RG) flow from high to low energy — a modified thermal partition function named the refined elliptic genus — we prove the existence of multiple vacua, and identify the superconformal field theories capturing their dynamics. The vacua correspond to bound states of (p,q)-strings in Type IIB string theory. Our computation serves as a check of the strong-weak S-duality of the Type IIB string.

We present a new type of generalization of the Rényi entropy that follows naturally from its representation as a thermodynamic quantity. We apply it to the case of [Formula: see text]-dimensional conformal field theories (CFTs) reduced on a region bounded by a sphere. It is known how to compute their Rényi entropy as an integral of the thermal entropy of hyperbolic black holes in [Formula: see text]-dimensional anti-de Sitter spacetime. We show how this integral fits into the framework of extended gravitational thermodynamics, and then point out the natural generalization of the Rényi entropy that suggests itself in that light. In the field theory terms, the new generalization employs aspects of the physics of Renormalization Group (RG) flow to define a refined version of the reduced vacuum density matrix. For [Formula: see text], it can be derived directly in terms of twist operators in field theory. The framework presented here may have applications beyond this context, perhaps in studies of both quantum and classical information theoretic properties of a variety of systems.

On higher order geometric and renormalization group flows

We discuss a two-parameter renormalization group (RG) flow when parameters are organized in a single complex variable, τ, with modular properties. Throughout the work we consider a special limit when the imaginary part of τ characterizing the disorder strength tends to zero. We argue that generalized Riemann-Thomae (gRT) function and the corresponding generalized Devil's staircase emerge naturally in a variety of physical models providing a universal behavior. In 1D we study the Anderson-like probe hopping in a weakly disordered lattice, recognize the origin of the gRT function in the spectral density of the probe and formulate specific RG procedure which gets mapped onto the discrete flow in the fundamental domain of the modular group SL(2,Z). In 2D we consider the generalization of the phyllotaxis crystal model proposed by L. Levitov and suggest the explicit form of the effective potential for the probe particle propagating in the symmetric and asymmetric 2D lattice of defects. Analyzing the structure of RG flow equations in the vicinity of saddle points we claim emergence of BKT-like transitions at Imτ→0. We show that the RG-like dynamics in the fundamental domain of SL(2,Z) for asymmetric lattices asymptotically approaches the “Silver ratio”. For a Hubbard model of particles on a ring interacting via long-ranged potentials we investigate the dependence of the ground state energy on the potential and demonstrate by combining numerical and analytical tools the emergence of the generalized Devil's staircase. Also we conjecture a bridge between a Hubbard model and a phyllotaxis.

The renormalization group (RG) flow on the space of couplings of a simple model with two couplings is examined. The model considered is that of a single component scalar field with φ4 self interaction coupled, via Yukawa coupling, to a fermion in flat four-dimensional space. The RG flow on the two-dimensional space of couplings, [Formula: see text], is shown to be derivable from a potential to sixth order in the couplings, which requires two-loop calculations of the β-functions. The identification of a potential requires the introduction of a metric on [Formula: see text] and it is shown that the metric defined by Zamolodchikov, in terms of two-point correlation functions of composite operators, gives potential flow to this order.

Symmetries and their anomalies give strong constraints on renormalization group (RG) flows of quantum field theories. Recently, the identification of a theory’s global symmetries with its topological sector has provided additional constraints on RG flows to symmetry preserving gapped phases due to mathematical results in category and topological quantum field theory. In this paper, we derive constraints on RG flows from ℤ2-valued pure- and mixed-gravitational anomalies that can only be activated on non-spin manifolds. We show that such anomalies cannot be matched by a unitary, symmetry preserving gapped phase without symmetry fractionalization. In particular, we discuss examples that commonly arise in 4d gauge theories with fermions.

We study the phase diagram of the Hubbard model in the limit where U, the onsite repulsive interaction, is much smaller than the bandwidth. We present an asymptotically exact expression for T$_c$, the superconducting transition temperature, in terms of the correlation functions of the non-interacting system which is valid for arbitrary densities so long as the interactions are sufficiently small. Our strategy for computing T$_c$ involves first integrating out all degrees of freedom having energy higher than an unphysical initial cutoff $\Omega_0$. Then, the renormalization group (RG) flows of the resulting effective action are computed and T$_c$ is obtained by determining the scale below which the RG flows in the Cooper channel diverge. We prove that T$_c$ is independent of $\Omega_0$. Using this method, we find a variety of unconventional superconducting ground states in two and three dimensional lattice systems and present explicit results for T$_c$ and pairing symmetries as a function of the electron concentration.

Haros graphs have been recently introduced as a set of graphs bijectively related to real numbers in the unit interval. Here we consider the iterated dynamics of a graph operator R over the set of Haros graphs. This operator was previously defined in the realm of graph-theoretical characterization of low-dimensional nonlinear dynamics and has a renormalization group (RG) structure. We find that the dynamics of R over Haros graphs is complex and includes unstable periodic orbits of arbitrary period and nonmixing aperiodic orbits, overall portraiting a chaotic RG flow. We identify a single RG stable fixed point whose basin of attraction is associated with the set of rational numbers, and find periodic RG orbits that relate to (pure) quadratic irrationals and aperiodic RG orbits, related with (nonmixing) families of nonquadratic algebraic irrationals and transcendental numbers. Finally, we show that the graph entropy of Haros graphs is globally decreasing as the RG flows towards its stable fixed point, albeit in a strictly nonmonotonic way, and that such graph entropy remains constant inside the periodic RG orbit associated to a subset of irrationals, the so-called metallic ratios. We discuss the possible physical interpretation of such chaotic RG flow and put results regarding entropy gradients along RG flow in the context of c-theorems.

In this article, we propose the realization of conformal manifolds, which are smooth manifolds of scale-conformal invariant interacting Hamiltonians in two-dimensional quantum many-body systems. Such phenomena can occur in various interacting systems, including topological surfaces or 2D bulks. Building on previous observations, we demonstrate that a conformal manifold can emerge as an exact solution when the number of fermion colors, \(N_c\), becomes infinite. We identify distinct exact marginal deformation operators uniquely associated with the conformal manifolds. By considering \(N_c\) as finite but large, we show that quantum fluctuations induce a fermion field renormalization that results in mildly infrared relevant or irrelevant renormalization-group (RG) flow within a conformal manifold, producing standard isolated infrared stable Wilson-Fisher fixed points. These can be grouped with ultraviolet stable fixed points into a discrete manifold due to the spontaneous symmetry breaking of an emergent \(SO(\mathcal{N})\) dynamical symmetry in the RG flow as \(N_c \rightarrow \infty\). Additionally, we find a correlation between the direction of the RG flow within the manifold and an increase in EPR-like entanglement entropy. The infrared-stable Wilson-Fisher fixed points, induced by quantum fluctuations, are linked to theories on the conformal manifold where interaction operators are maximally entangled in flavor space. Our studies provide an effective framework for addressing topological quantum critical points with high-dimensional interaction parameter spaces, potentially housing many fixed points of various stabilities. They also highlight the central role of entangled conformal operators and their entropy in shaping universality classes of surface topological quantum phase transitions. We conclude with open questions and possible future directions.

We investigate phase transitions in scalar field theories using the functional renormalization group (RG) equation. We analyze a system with $U(2)\times U(2)$ symmetry, in which there is a parameter $\lambda_2$ that controls the strength of the first-order phase transition driven by fluctuations. In the limit of $\lambda_2\to0$, the $U(2)\times U(2)$ theory is reduced to an $O(8)$ scalar theory that exhibits a second-order phase transition in three dimensions. We develop a new insight for the understanding of the fluctuation-induced first-order phase transition as a smooth continuation from the standard RG flow in the $O(8)$ system. In our view from the RG flow diagram on coupling parameter space, the region that favors the first-order transition emerges from the unphysical region to the physical one as $\lambda_2$ increases from zero. We give this interpretation based on the Taylor expansion of the functional RG equations up to the fourth order in terms of the field, which encompasses the $\epsilon$-expansion results. We compare results from the expansion and from the full numerical calculation and find that the fourth-order expansion is only of qualitative use and that the sixth-order expansion improves the quantitative agreement.