arXiv subjects

Johannes Thürigen

Publications and source records attributed to Johannes Thürigen.

At least 19 recordsLinked to original sources

arXiv2026

Primitive asymptotics in $ϕ^4$ vector theory

A longstanding conjecture in $ϕ^4_4$ theory is that primitive graphs dominate the beta function asymptotically at large loop order in the minimal-subtraction scheme. Here we investigate this issue by exploiting additional combinatorial structure coming from an extension to vectors with $O(N)$ symmetry. For the 0-dimensional case, we calculate the $N$-dependent generating function of primitive graphs and its asymptotics, including arbitrarily many subleading corrections. We find that the leading asymptotic growth rate becomes visible only above $\approx 25$ loops, while data at lower order is suggestive of a wrong asymptotics. Our results also yield the symmetry-factor weighted sum of 3-connected cubic graphs, and the exact asymptotics of Martin invariants. For individual Feynman graphs, we give bounds on their degree in $N$ depending on their coradical degree, and construct the primitive graphs of highest degree explicitly. We calculate the 4D primitive beta function numerically up to 17 loops, and find its behaviour to be qualitatively similar to the 0D case. The locations of zeros quickly approach their large-loop asymptotics at negative integer $N$, while the growth rate of the beta function differs from the asymptotic prediction even at 17 loops.

hep-th↗

arXiv2025

Statistics and asymptotics of subdivergence-free Feynman integrals in $ϕ^4$ theory

Recent algorithmic improvements have made it possible to evaluate subdivergence-free (=primitive=skeleton) Feynman integrals in $ϕ^4$ theory numerically up to 18 loops. By now, all such integrals up to 13 loops and several hundred thousand at higher loop order have been computed. This data enables a statistical analysis of the typical behaviour of Feynman integrals at large loop order. We find that the average value grows exponentially, but the observed growth rate is accurately described by its leading asymptotics only upwards of 25 loops. This is in contrast with the $N$-dependence of the $ON(N)$-symmetric $ϕ^4$ theory, which is close to its large-order asymptotics already around 10 loops. Secondly, the distribution of integrals has a largely continuous inner part but a few extreme outliers. This makes uniform random sampling inefficient. We find that the value of the integral is correlated with many features of the graph, which can be used for importance sampling. With a naive test implementation we obtained an approximately 1000-fold speedup compared with uniform sampling. This suggests that in future work, Feynman amplitudes at large loop order might be computed numerically with statistical methods, rather than through enumerating and evaluating every individual integral.

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arXiv2025

Combinatorial Dyson-Schwinger Equations of Quartic Matrix Field Theory

Matrix field theory is a combinatorially non-local field theory which has recently been found to be a non-trivial but solvable QFT example. To generalize such non-perturbative structures to other models, a more combinatorial understanding of Dyson-Schwinger equations and their solutions is of high interest. To this end we consider combinatorial Dyson-Schwinger equations manifestly relying on the Hopf-algebraic structure of perturbative renormalization. We find that these equations are fully compatible with renormalization, relying only on the superficially divergent diagrams which are planar ribbon graphs, i.e. decompleted dual combinatorial maps. Still, they are of a similar kind as in realistic models of local QFT, featuring in particular an infinite number of primitive diagrams as well as graph-dependent combinatorial factors.

math-ph↗

arXiv2024

Quantum Gravity, Hydrodynamics and Emergent Cosmology: A Collection of Perspectives

This collection of perspective pieces captures recent advancements and reflections from a dynamic research community dedicated to bridging quantum gravity, hydrodynamics, and emergent cosmology. It explores four key research areas: (a) the interplay between hydrodynamics and cosmology, including analog gravity systems; (b) phase transitions, continuum limits and emergent geometry in quantum gravity; (c) relational perspectives in gravity and quantum gravity; and (d) the emergence of cosmological models rooted in quantum gravity frameworks. Each contribution presents the distinct perspectives of its respective authors. Additionally, the introduction by the editors proposes an integrative view, suggesting how these thematic units could serve as foundational pillars for a novel theoretical cosmology framework termed "hydrodynamics on superspace".

gr-qc↗

arXiv2024

New Fixed Points from Melonic Interactions

Generalizations of vector field theories to tensors allow to similarly apply large-$N$ techniques but find a richer though often still tractable structure. However, the potential of such tensor theories has not been fully exploited since only a symmetry-reduced ``isotropic'' part of their phase space has been studied so far. Here we present for the first time the richness of the tensorial phase space applying the functional renormalization group to tensor fields of rank $r$ in the cyclic-melonic potential approximation including the flow of anomalous dimensions. Due to a decoupling of the flow equations of the $r$ couplings at given order, we find non-Gaussian fixed points in any regime of $0<s\le r$ non-vanishing coupling types. Each of these regimes contains isotropic fixed points of Wilson-Fisher type as well as new anisotropic fixed points. This new classification reveals a rich structure of renormalization group dynamics including candidates for asymptotic safe fixed points even at critical dimension. Considering the tensorial interactions as generating discrete geometries, the various fixed points correspond to continuum limits of distinguished ensembles of triangulations raising hope to find new classes of continuum geometry in this way.

hep-th↗

arXiv2023

Curvature effects in the spectral dimension of spin foams

It has been shown in [1] that a class of restricted spin foam models can feature a reduced spectral dimension of space-time. However, it is still an open question how curvature affects the flow of the spectral dimension. To answer this question, we consider another class of restricted spin foam models, so called spin foam frusta, which naturally exhibit oscillating amplitudes induced by curvature, as well as an extension of the parameter space by a cosmological constant. Numerically computing the spectral dimension of $1$-periodic frusta geometries using extrapolated quantum amplitudes, we find that quantum effects lead to a small change of spectral dimension at small scales and an agreement to semi-classical results at larger scales. Adding a cosmological constant $Λ$, we find additive corrections to the non-oscillating result at the diffusion scale $τ\sim 1/\sqrtΛ$. Extending to $2$-periodic configurations, we observe a reduced effective dimension, the form of which sensitively depends on the values of the gravitational constant $G$ and the cosmological constant $Λ$. We provide an intuition for our results based on an analytical estimate of the spectral dimension. Furthermore, we present a simplified integrable model with oscillating measure that qualitatively explains the features found numerically. We argue that there exists a phase transition in the thermodynamic limit which crucially depends on the parameters $G$ and $Λ$. The dependence on $G$ and $Λ$ presents an exciting opportunity to infer phenomenological insights about quantum geometry from measurement of the spectral dimension, in principle. [1]: S. Steinhaus and J. Thürigen, Emergence of Spacetime in a restricted Spin-foam model, Phys. Rev. D 98 (2018) 026013

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arXiv2023

QFT with Tensorial and Local Degrees of Freedom: Phase Structure from Functional Renormalization

Field theories with combinatorial non-local interactions such as tensor invariants are interesting candidates for describing a phase transition from discrete quantum-gravitational to continuum geometry. In the so-called cyclic-melonic potential approximation of a tensorial field theory on the $r$-dimensional torus it was recently shown using functional renormalization group techniques that no such phase transition to a condensate phase with a tentative continuum geometric interpretation is possible. Here, keeping the same approximation, we show how to overcome this limitation amending the theory by local degrees freedom on $\mathbb{R}^d$. We find that the effective $r-1$ dimensions of the torus part dynamically vanish along the renormalization group flow while the $d$ local dimensions persist up to small momentum scales. Consequently, for $d>2$ one can find a phase structure allowing also for phase transitions.

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arXiv2023

Mean-field phase transitions in TGFT quantum gravity

Controlling the continuum limit and extracting effective gravitational physics are shared challenges for quantum gravity approaches based on quantum discrete structures. The description of quantum gravity in terms of tensorial group field theory (TGFT) has recently led to much progress in its application to phenomenology, in particular cosmology. This application relies on the assumption of a phase transition to a nontrivial vacuum (condensate) state describable by mean-field theory, an assumption that is difficult to corroborate by a full renormalization group flow analysis due to the complexity of the relevant TGFT models. Here we demonstrate that this assumption is justified due to the specific ingredients of realistic quantum geometric TGFT models: combinatorially non-local interactions, matter degrees of freedom and Lorentz group data together with the encoding of micro-causality. This greatly strengthens the evidence for the existence of a meaningful continuum gravitational regime in group-field and spin-foam quantum gravity, the phenomenology of which is amenable to explicit computations in a mean-field approximation.

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arXiv2023

Phase transitions in TGFT: a Landau-Ginzburg analysis of Lorentzian quantum geometric models

In the tensorial group field theory (TGFT) approach to quantum gravity, the basic quanta of the theory correspond to discrete building blocks of geometry. It is expected that their collective dynamics gives rise to continuum spacetime at a coarse grained level, via a process involving a phase transition. In this work we show for the first time how phase transitions for realistic TGFT models can be realized using Landau-Ginzburg mean-field theory. More precisely, we consider models generating 4-dimensional Lorentzian triangulations formed by spacelike tetrahedra whose quantum geometry is encoded in non-local degrees of freedom on the non-compact group $\mathrm{SL}(2,\mathbb{C})$ and subject to gauge and simplicity constraints. Further we include $\mathbb{R}$-valued variables which may be interpreted as discretized scalar fields typically employed as a matter reference frame. We apply the Ginzburg criterion finding that fluctuations around the non-vanishing mean-field vacuum remain small at large correlation lengths regardless of the combinatorics of the non-local interaction validating the mean-field theory description of the phase transition. This work represents a first crucial step to understand phase transitions in compelling TGFT models for quantum gravity and paves the way for a more complete analysis via functional renormalization group techniques. Moreover, it supports the recent extraction of effective cosmological dynamics from TGFTs in the context of a mean-field approximation.

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arXiv2021

Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom

In the tensorial group field theory approach to quantum gravity, the theory is based on discrete building blocks and continuum spacetime is expected to emerge from their collective dynamics, possibly at criticality, via a phase transition. On a compact group of fixed volume this can be expected to be only possible in a large-volume or thermodynamic limit. Here we show how phase transitions are possible in TGFTs in two cases: a) considering the non-local group degrees of freedom on a non-compact Lie group instead of a compact one (or taking a large-volume limit of a compact group); b) in models including $\mathbb{R}$-valued local degrees of freedom (that can be interpreted as discrete scalar fields, often used in this context to provide a matter reference frame). After adapting the Landau-Ginzburg approach to this setting of mixed local/non-local degrees of freedom, we determine the critical dimension beyond which there is a Gaussian fixed point and a continuous phase transition which can be described by mean-field theory. This is an important step towards the realization of a phase transition to continuum spacetime in realistic TGFT models for quantum gravity.

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arXiv2021

Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme

Various combinatorially non-local field theories are known to be renormalizable. Still, explicit calculations of amplitudes are very rare and restricted to matrix field theory. In this contribution I want to demonstrate how the BPHZ momentum scheme in terms of the Connes-Kreimer Hopf algebra applies to any combinatorially non-local field theory which is renormalizable. This algebraic method improves the understanding of known results in noncommutative field theory in its matrix formulation. Furthermore, I use it to provide new explicit perturbative calculations of amplitudes in tensorial field theories of rank $r>2$.

hep-th↗

arXiv2021

One-matrix differential reformulation of two-matrix models

Differential reformulations of field theories are often used for explicit computations. We derive a one-matrix differential formulation of two-matrix models, with the help of which it is possible to diagonalize the one- and two-matrix models using a formula by Itzykson and Zuber that allows diagonalizing differential operators with respect to matrix elements of Hermitian matrices. We detail the equivalence between the expressions obtained by diagonalizing the partition function in differential or integral formulation, which is not manifest at first glance. For one-matrix models, this requires transforming certain derivatives to variables. In the case of two-matrix models, the same computation leads to a new determinant formulation of the partition function, and we discuss potential applications to new orthogonal polynomials methods.

math-ph↗

arXiv2021

Renormalization in combinatorially non-local field theories: the Hopf algebra of 2-graphs

It is well known that the mathematical structure underlying renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality of the field theory. Consequently, one might suspect that non-local field theories such as matrix or tensor field theories cannot benefit from a similar algebraic understanding. Here I show that, on the contrary, the renormalization and perturbative diagramatics of a broad class of such field theories is based in the same way on a Hopf algebra. These theories are characterized by interaction vertices with graphs as external structure leading to Feynman diagrams which can be summed up under the concept of "2-graphs". From the renormalization perspective, such graph-like interactions are as much local as point-like interactions. They differ in combinatorial details as I exemplify with the central identity for the perturbative series of combinatorial correlation functions. This sets the stage for a systematic study of perturbative renormalization as well as non-perturbative aspects, e.g. Dyson-Schwinger equations, for a number of combinatorially non-local field theories with possible applications to quantum gravity, statistical models and more.

math-ph↗

arXiv2020

Phase transitions in TGFT: Functional renormalization group in the cyclic-melonic potential approximation and equivalence to O$(N)$ models

In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. We derive the flow equation for the effective potential at any order restricting to a subclass of tensorial interactions called cyclic melonic and projecting to a constant field in group space. For a tensor field of rank $r$ on U$(1)$ we explicitly calculate beta functions and find equivalence with those of O$(N)$ models but with an effective dimension flowing from $r-1$ to zero. In the $r-1$ dimensional regime, the equivalence to O$(N)$ models is modified by a tensor specific flow of the anomalous dimension with the consequence that the Wilson-Fisher type fixed point solution has two branches. However, due to the flow to dimension zero, fixed points describing a transition between a broken and unbroken phase do not persist and we find universal symmetry restoration. To overcome this limitation, it is necessary to go beyond compact configuration space.

hep-th↗

arXiv2020

(No) phase transition in tensorial group field theory

Continuum spacetime is expected to emerge via phase transition in discrete approaches to quantum gravity. A promising example is tensorial group field theory but its phase diagram remains an open issue. The results of recent attempts in terms of the functional renormalization group method remain inconclusive since they are restricted to truncations of low order. We overcome this barrier with a local-potential approximation for $\textrm{U}(1)$ tensor fields at arbitrary rank $r$ focusing on a specific class of so-called cyclic-melonic interactions. Projecting onto constant field configurations, we obtain the full set of renormalization-group flow equations. At large cut-offs we find equivalence with $r-1$ dimensional $\textrm{O}(N)$ scalar field theory in the large-$N$ limit, modified by a tensor-specific, relatively large anomalous dimension. However, on small length scales there is equivalence with the corresponding scalar field theory with vanishing dimension and, thus, no phase transition. This is confirmed by numerical analysis of the full non-autonomous equations where we always find symmetry restoration. The essential reason for this effect are isolated zero modes. This result should therefore be true for tensor field theories on any compact domain and including any tensor-invariant interactions. Thus, group field theories with non-compact degrees of freedom will be necessary to describe a phase transition to continuum spacetime.

hep-th↗

arXiv2019

Multi-critical behaviour of 4-dimensional tensor models up to order 6

Tensor models generalize the matrix-model approach to 2-dimensional quantum gravity to higher dimensions. Some models allowing a $1/N$ expansion have been explored, most of them generating branched-polymer geometries. Recently, enhancements yielding an additional 2d quantum-gravity (planar) phase and an intermediate regime of proliferating baby-universes have been found. It remains an open issue to find models escaping these lower dimensionality universality classes. Here we analyse the dominant regime and critical behaviour of a range of new models which are candidates for such effective geometries, in particular interactions based on the utility graph $K_{3,3}$. We find that, upon proper enhancement, the two-phase structure of a branched-polymer and a 2d gravity regime is the common case in $U(N)$-invariant rank $D=4$ tensor models of small orders. Not only the well known so-called necklace interactions but also $K_{3,3}$-type interactions turn out as the source for the planar regime. We give a systematic account of the enhancement scaling, the counting of leading-order diagrams and the multi-critical behaviour of a wide range of interactions, in particular for all order-6 interactions of rank 3 and 4. These findings support the claim of universality of such mixtures of branched-polymer and planar diagrams at criticality. In particular, this hints at the necessity to consider new ingredients, or interactions of higher order and rank, in order to obtain higher dimensional continuum geometry from tensor models.

hep-th↗

arXiv2018

Phase transitions in group field theory: The Landau perspective

In various approaches to quantum gravity continuum spacetime is expected to emerge from discrete geometries through a phase transition. In group field theory, various indications for such a transition have recently been found but a complete understanding of such a phenomenon remains an open issue. In this work, we investigate the critical behavior of different group field theory models in the Gaussian approximation. Applying the Ginzburg criterion to quantify field fluctuations, we find that this approximation breaks down in the case of three-dimensional Euclidean quantum gravity as described by the dynamical Boulatov model on the compact group $\text{SU}(2)$. This result is independent of the peculiar gauge symmetry and specific form of nonlocality of the model. On the contrary, we find that the Gaussian approximation is valid for a rank-$1$ GFT on the noncompact sector of fields on $\text{SL}(2,\mathbb{R})$ related to Lorentzian models. Though a nonperturbative analysis is needed to settle the question of phase transitions for compact groups, the results may also indicate the necessity to consider group field theory on noncompact domains for phase transitions to occur.

gr-qc↗

arXiv2018

Emergence of Spacetime in a restricted Spin-foam model

The spectral dimension has proven to be a very informative observable to understand the properties of quantum geometries in approaches to quantum gravity. In loop quantum gravity and its spin foam description, it has not been possible so far to calculate the spectral dimension of spacetime. As a first step towards this goal, here we determine the spacetime spectral dimension in the simplified spin foam model restricted to hypercuboids. Using Monte Carlo methods we compute the spectral dimension for state sums over periodic spin foam configurations on infinite lattices. For given periodicity, i.e. number of degrees of freedom, we find a range of scale where an intermediate spectral dimension between 0 and 4 can be found, continuously depending on the parameter of the model. Under an assumption on the statistical behaviour of the Laplacian we can explain these results analytically. This allows us to take the thermodynamic limit of large periodicity and find a phase transition from a regime of effectively 0-dimensional to 4-dimensional spacetime. At the point of phase transition, dynamics of the model are scale invariant which can be seen as restoration of diffeomorphism invariance of flat space. Considering the spectral dimension as an order parameter for renormalization we find a renormalization group flow to this point as well. Being the first instance of an emergence of 4-dimensional spacetime in a spin foam model, the properties responsible for this result seem to be rather generic. We thus expect similar results for more general, less restricted spin foam models.

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