The Discrete Gaussian model, I. Renormalisation group flow at high temperature

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.

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.

On higher order geometric and renormalization group flows

We analyze the effect of decoherence, modeled by local quantum channels, on quantum critical states and we find universal properties of the resulting mixed state's entanglement, both between system and environment and within the system. Renyi entropies exhibit volume law scaling with a subleading constant governed by a "g function" in conformal field theory, allowing us to define a notion of renormalization group (RG) flow (or "phase transitions") between quantum channels. We also find that the entropy of a subsystem in the decohered state has a subleading logarithmic scaling with subsystem size, and we relate it to correlation functions of boundary condition changing operators in the conformal field theory. Finally, we find that the subsystem entanglement negativity, a measure of quantum correlations within mixed states, can exhibit log scaling or area law based on the RG flow. When the channel corresponds to a marginal perturbation, the coefficient of the log scaling can change continuously with decoherence strength. We illustrate all these possibilities for the critical ground state of the transverse-field Ising model, in which we identify four RG fixed points of dephasing channels and verify the RG flow numerically. Our results are relevant to quantum critical states realized on noisy quantum simulators, in which our predicted entanglement scaling can be probed via shadow tomography methods.

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.

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.

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.

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.

The 2D Discrete Gaussian model gives each height function \({\eta : {\mathbb{Z}^2\to\mathbb{Z}}}\) a probability proportional to \({\exp(-\beta \mathcal{H}(\eta))}\), where \({\beta}\) is the inverse-temperature and \({\mathcal{H}(\eta) = \sum_{x\sim y}(\eta_x-\eta_y)^2}\) sums over nearest-neighbor bonds. We consider the model at large fixed \({\beta}\), where it is flat unlike its continuous analog (the Discrete Gaussian Free Field). We first establish that the maximum height in an \({L\times L}\) box with 0 boundary conditions concentrates on two integers M, M + 1 with \({M\sim \sqrt{(1/2\pi\beta)\log L\log\log L}}\). The key is a large deviation estimate for the height at the origin in \({\mathbb{Z}^{2}}\), dominated by “harmonic pinnacles”, integer approximations of a harmonic variational problem. Second, in this model conditioned on \({\eta\geq 0}\) (a floor), the average height rises, and in fact the height of almost all sites concentrates on levels H, H + 1 where \({H\sim M/\sqrt{2}}\). This in particular pins down the asymptotics, and corrects the order, in results of Bricmont et al. (J. Stat. Phys. 42(5–6):743–798, 1986), where it was argued that the maximum and the height of the surface above a floor are both of order \({\sqrt{\log L}}\). Finally, our methods extend to other classical surface models (e.g., restricted SOS), featuring connections to p-harmonic analysis and alternating sign matrices.

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 construct a new duality for two-dimensional discrete Gaussian models. It is based on a known one-dimensional duality and on a mapping, implied by the Chinese remainder theorem, between the sites of an N × M torus and those of a ring of NM sites. The duality holds for an arbitrary translation-invariant interaction potential v(r) between the height variables on the torus. It leads to pairs (v,v˜) of mutually dual potentials and to a temperature inversion according to β˜=π2/β . When v(r) is isotropic, duality renders an anisotropic v˜ . This is the case, in particular, for the potential that is dual to an isotropic nearest-neighbor potential. In the thermodynamic limit, this dual potential is shown to decay with distance according to an inverse square law with a quadrupolar angular dependence. There is a single pair of self-dual potentials v⋆=v⋆˜ . At the self-dual temperature β⋆=β⋆˜=π the height–height correlation can be calculated explicitly; it is anisotropic and diverges logarithmically with distance.

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.