arXiv subjects

Rudrajit Banerjee

Publications and source records attributed to Rudrajit Banerjee.

8 recordsLinked to original sources

arXiv2026

Critical Phenomena on the Bethe Lattice

We investigate the critical behavior of a family of $\mathbb{Z}_2$-symmetric scalar field theories on the Bethe lattice (the tree limit of regular hyperbolic tessellations) using both the non-perturbative Functional Renormalization Group and lattice perturbation theory. The family is indexed by the parameter $\zeta \in (0,1]$, which determines the range of the theory via the kinetic term constructed from the graph Laplacian raised to the power $\zeta$. Specifically, $\zeta=1$ is the short-range theory, while $0<\zeta<1$ defines the long-range model. Due to the hyperbolic nature of Bethe lattices, the Laplacian lacks a zero mode and exhibits a spectral gap. We find that upon closing this spectral gap by a modification of the Laplacian, the scalar field theories exhibit novel critical behavior in the form of non-trivial fixed points with critical exponents governed by $\zeta$ and the spectral dimension $d_s=3$. In particular, our analysis indicates the presence of a Wilson-Fisher fixed point for the short range $\zeta =1$ theory. In contrast, the nearest-neighbor Ising model on the Bethe lattice is known to exhibit mean-field critical exponents. To the best of our knowledge, this work provides the first evidence that a scalar $\phi^4$ theory and the discrete Ising model on the same underlying lattice may lie in distinct universality classes.

hep-th↗

arXiv2025

Analytic semigroups approaching a Schr\"{o}dinger group on real foliated metric manifolds

On real metric manifolds admitting a co-dimension one foliation, sectorial operators are introduced that interpolate between the generalized Laplacian and the d'Alembertian. This is used to construct a one-parameter family of analytic semigroups that remains well-defined into the near Lorentzian regime. In the strict Lorentzian limit we identify a sense in which a well-defined Schr\"{o}dinger evolution group arises. For the analytic semigroups we show in addition that: (i) they act as integral operators with kernels that are jointly smooth in the semigroup time and both spacetime arguments. (ii) the diagonal of the kernels admits an asymptotic expansion in (shifted) powers of the semigroup time whose coefficients are the Seeley-DeWitt coefficients evaluated on the complex metrics.

math-ph↗

arXiv2024

Wick rotation in the lapse, admissible complex metrics, and foliation changing diffeomorphisms

A Wick rotation in the lapse (not in time) is introduced that interpolates between Riemannian and Lorentzian metrics on real manifolds admitting a codimension-one foliation. The definition refers to a fiducial foliation but covariance under foliation changing diffeomorphisms can be rendered explicit in a reformulation as a rank one perturbation. Applied to scalar field theories a Lorentzian signature action develops a positive imaginary part thereby identifying the underlying complex metric as ``admissible''. This admissibility is ensured in non-fiducial foliations in technically distinct ways also for the variation with respect to the metric and for the Hessian. The Hessian of the Wick rotated action is a complex combination of a generalized Laplacian and a d'Alembertian, which is shown to have spectrum contained in a wedge of the upper complex half plane. Specialized to near Minkowski space the induced propagator differs from the one with the Feynman $i\epsilon$ prescription and on Friedmann-Lema\^{i}tre backgrounds the difference to a Wick rotation in time is illustrated.

math-ph↗

arXiv2023

$N$-cutoff regularization for fields on hyperbolic space

We apply a novel background independent regularization scheme, the $N$-cutoffs, to self-consistently quantize scalar and metric fluctuations on the maximally symmetric but non-compact hyperbolic space. For quantum matter fields on a classical background or full Quantum Einstein Gravity (regarded here as an effective field theory) treated in the background field formalism, the $N$-cutoff is an ultraviolet regularization of the fields' mode content that is independent of the background hyperbolic space metric. For each $N > 0$, the regularized system backreacts on the geometry to dynamically determine the self-consistent background metric. The limit in which the regularization is removed then automatically yields the 'physically correct' spacetime on which the resulting quantum field theory lives. When self-consistently quantized with the $N$-cutoff, we find that without any fine-tuning of parameters, the vacuum fluctuations of scalar and (linearized) graviton fields do not lead to the usual cosmological constant problem of a curvature singularity. Instead, the presence of increasingly many field modes tends to reduce the negative curvature of hyperbolic space, leading to vanishing values in the limit of removing the cutoff.

hep-th↗

arXiv2023

States of Low Energy on Bianchi I spacetimes

States of Low Energy are a class of exact Hadamard states for free quantum fields on cosmological spacetimes whose structure is fixed at {\it all} scales by a minimization principle. The original construction was for Friedmann-Lemaître geometries and is here generalized to anisotropic Bianchi I geometries relevant to primordial cosmology. In addition to proving the Hadamard property, systematic series expansions in the infrared and ultraviolet are developed. The infrared expansion is convergent and induces in the massless case a leading spatial long distance decay that is always Minkowski-like but anisotropy modulated. The ultraviolet expansion is shown to be equivalent to the Hadamard property, and a non-recursive formula for its coefficients is presented.

gr-qc↗

arXiv2022

The spatial Functional Renormalization Group and Hadamard states on cosmological spacetimes

A spatial variant of the Functional Renormalization Group (FRG) is introduced on (Lorentzian signature) globally hyperbolic spacetimes. Through its perturbative expansion it is argued that such a FRG must inevitably be state dependent and that it should be based on a Hadamard state. A concrete implementation is presented for scalar quantum fields on flat Friedmann-Lemaître spacetimes. The universal ultraviolet behavior of Hadamard states allows the flow to be matched to the one-loop renormalized flow (where strict removal of the ultraviolet cutoff requires a tower of potentials, one for each power of the Ricci scalar). The state-dependent infrared behavior of the flow is investigated for States of Low Energy, which are Hadamard states deemed to be viable vacua for a pre-inflationary period. A simple time-dependent infrared fixed point equation (resembling that in Minkowski space) arises for any scale factor, with analytically computable corrections coding the non-perturbative ramifications of the Hadamard property in the infrared.

hep-th↗

arXiv2018

Graph rules for the linked cluster expansion of the Legendre effective action

Graph rules for the linked cluster expansion of the Legendre effective action $Γ[ϕ]$ are derived and proven in $D\geq 2$ Euclidean dimensions. A key aspect is the weight assigned to articulation vertices which is itself shown to be computable from labeled tree graphs. The hopping interaction is allowed to be long ranged and scale dependent, thereby producing an in principle exact solution of $Γ[ϕ]$'s functional renormalization group equation.

math-ph↗

arXiv2018

Critical behavior of the hopping expansion from the Functional Renormalization Group

A lattice version of the widely used Functional Renormalization Group (FRG) for the Legendre effective action is solved - in principle exactly - in terms of graph rules for the linked cluster expansion. Conversely, the FRG induces nonlinear flow equations governing suitable resummations of the graph expansion. The (finite) radius of convergence determining criticality can then be efficiently computed as the unstable manifold of a Gaussian or non-Gaussian fixed point of the FRG flow. The correspondence is tested on the critical line of the Lüscher-Weisz solution of the $ϕ^4_4$ theory and its $ϕ_3^4$ counterpart.

hep-lat↗