REVIEW
Phase transitions in TGFT: Functional renormalization group in the cyclic-melonic potential approximation and equivalence to O$(N)$ models
High Energy Physics - Theory · Submitted 28 Sep 2020 · 114 references
Pith's verdict: All-order cyclic-melonic FRG for U(1) TGFT with a flowing effective dimension and a symmetry-restoration no-go that is solid within the truncation but stronger than what is proven.
T0 · AI review
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Referee report
The paper
The paper claims
The paper establishes that a tensorial group field theory on $U(1)^r$ flows like an O($N$) model with effective dimension falling to zero, so no symmetry-breaking transition occurs on compact domains of fixed volume.
The reading
What carries the argument
The central machinery is the projected renormalization-group flow for the cyclic-melonic effective potential, written as a sum over representation-space lattice points with threshold functions $I^{(s)}_{\gamma}(N_k)$, together with the scale-dependent effective dimension $d_{eff}(k)=\partial \log F^1_r(N_k)/\partial \log N_k$. The non-local combinatorics after projection onto constant fields are encoded in an operator built from Kronecker deltas, and the zero modes singled out by those deltas are what drive $d_{eff}$ to zero at small $k$. This converts the non-autonomous group field theory equation into an O($N$)-model equation that continuously interpolates between the large-volume and zero-dimensional regimes, making the dimensional flow explicit.
Extended reading notes
Core claim
The central discovery is that, in the cyclic-melonic local-potential approximation with anomalous dimension, the non-autonomous renormalization flow of a rank-$r$ complex (or real) tensorial group field theory on $U(1)^r$ is exactly an O($N$) model flow with $N=2$ (or $N=1$) but with a scale-dependent dimension. At large scales the theory behaves like a large-$N$ O($N$) model in $d_r=r-1$ dimensions, with an extra factor $r$ in the mass term; at small scales the equations reduce to the zero-dimensional O($N$) model. The isolated zero modes of the compact spectrum dominate the infrared and drive $d_{eff}$ to zero, so the symmetric phase is the only possible infrared phase. Starting from any potential with a spontaneously broken symmetry, the potential flattens at some finite scale, and the authors argue this restoration extends beyond the cyclic-melonic truncation to any tensor-invariant interactions on any compact group.
Load-bearing premise
The argument transfers the Mermin-Wagner-Hohenberg theorem, proved for local scalar O($N$) models, to a combinatorially non-local TGFT through the flowing effective dimension; the authors state that a rigorous analogue would be desirable but is not provided, so the no-go conclusion depends on that transfer being valid.
Editorial extensions
If this is right
- On a compact group of fixed volume, a rank-$r$ tensorial group field theory has only a symmetric infrared phase, so no global $U(1)$ or $\mathbb{Z}_2$ symmetry-breaking transition exists.
- In the large-volume limit, the large-$N_k$ results apply at all scales, and phase transitions can be described by a mean-field Gaussian fixed point above a critical rank or by a Wilson-Fisher-type fixed point for lower ranks.
- A non-compact configuration space, such as $SL(2,\mathbb{R})$ or a configuration space with additional matter-reference-frame directions, becomes necessary to support a broken or condensate phase.
- The running effective dimension $d_{eff}(k)$ crosses the lower critical dimension $d=2$ at a finite scale, identifying the scale below which the potential flattens and symmetry is restored.
- The identification of the renormalization scale with $N_k=ak$ clarifies the relation between group field theory renormalization and tensor model renormalization in finite volume.
Reading between the lines
- If the zero-mode mechanism is the operative cause, symmetry restoration should occur for any compact group with isolated zero modes, not only $U(1)$; computing the corresponding threshold sums for $SU(2)$ would test the authors' conjecture directly.
- The equivalence to O($N$) models with a flowing dimension suggests a dictionary: O($N$) results at fractional dimension $d$ could be read as statements about a tensorial group field theory at the scale where $d_{eff}(k)=d$, and vice versa.
- For group field theory condensate cosmology, the practical consequence is that the condensate phase must be sought in non-compact or constrained models; a testable next step is to look for a broken phase in a TGFT with a matter reference frame that adds non-compact directions.
- The two branch structure of the non-Gaussian fixed point in the presence of the anomalous dimension may be connected to the multi-critical fixed point structure of O($N$) models; an analytic large-$N$ solution of the rescaled potential equation could determine the exact scaling exponents and the convergence domain.
Original abstract
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.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. | background 1.00 | 0.05 | 0.03 | hedged possibility 0.91 |
| s1 | However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. | gap 0.93 | 0.04 | 0.02 | asserted 1.00 |
| s2 | In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. | contribution 0.81 | 0.89 | 0.03 | asserted 0.96 |
| s3 | 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. | method 0.84 | 0.97 | 0.06 | asserted 1.00 |
| s4 | 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. | result 0.99 | 0.96 | 0.26 | asserted 1.00 |
| s5 | 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. | result 0.96 | 0.88 | 0.13 | asserted 1.00 |
| s6 | 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. | result 0.81 | 0.95 | 0.41 | asserted 1.00 |
| s7 | To overcome this limitation, it is necessary to go beyond compact configuration space. | future work 0.87 | 0.34 | 0.19 | asserted 0.99 |
What was examined, and what was not
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References
How the literature uses it
20 arXiv papers cite this in 88 sentences
Papers citing it · 56 sentences without a confident label
Builds on it 3 papers
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In this way, the present work directly builds on previous ones on the application of mean-field theory to TGFT [ref] as well as the application of the FRG methodology to the same class of quantum gravity models [ref] , and it extends this line of research by considering the influence of non-compact local directions of the domain of the group field.
Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom · Discussion and conclusion · 1.00
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Here we have generalized the functional RG equations for tensorial field theory of arbitrary rank $$ in the cyclic-melonic potential approximation from the isotropic [ref] to the full anisotropic case.
New Fixed Points from Melonic Interactions · Conclusions · 1.00
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For fields on $^×G^$ the calculation of the Hessian $^(2)$ is a straightforward generalization of the purely non-local case $=0$ treated in [ref] .
QFT with Tensorial and Local Degrees of Freedom: Phase Structure from Functional Renormalization · FRG equation for a cyclic-melonic potential · 0.93
Uses its method or tool 3 papers
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Using the common optimized regulator function [ref] [equation] where $θ$ is the Heaviside function, we can perform the trace in the momentum space of $U(1)$ representations and find for the effective potential $U_k(ρ)=μ_kρ+V_k(ρ)$ [ref] [equation] where the particular threshold functions $s^s$ in the dimensionless variable $= k$ result from an integral approximation of the $s$-dimensional traces with quadratic cutoff [We have also explicitly calculated the trace as a sum with simplex and box cutoffs taking into account subleading orders in $$ leading to the same qualitative results [ref] .], thus including the usual unit volume $s=π^s/2/Γ(1+s/2)$.
(No) phase transition in tensorial group field theory · Functional renormalization group in the melonic potential approximation · 0.98
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We will therefore consider here a uniform-field projection on the truncation of $$ to cyclic-melonic interactions, called the cyclic-melonic potential approximation [ref] .
QFT with Tensorial and Local Degrees of Freedom: Phase Structure from Functional Renormalization · LPA for non-local interactions · 0.83
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We calculate the RG flow using the non-perturbative method of the functional RG [ref] in the cyclic-melonic potential approximation [ref] .
New Fixed Points from Melonic Interactions · Introduction · 0.99
Compares with it 5 papers
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Beyond the Gaussian regime we find a Wilson-Fisher type point for $<<4$, similar to the purely tensorial theory [ref] .
QFT with Tensorial and Local Degrees of Freedom: Phase Structure from Functional Renormalization · Introduction · 0.93
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Similar to previous results in the isotropic case [ref] , we find that all these fixed points exist even for the critical theory at $-1==4$ due to the tensor-specific flow of the anomalous dimension.
New Fixed Points from Melonic Interactions · Introduction · 0.83
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Notably, in spite of the assumptions of Landau-Ginzburg theory, most importantly the selection of the uniform field configuration, the truncation of the action functional and symmetries, our results agree with results from more involved renormalization studies of TGFTs [ref] .
Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom · Discussion and conclusion · 0.93
Cites it as support 3 papers
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However, similar to local theories, a phase of broken global symmetry, i.e. a condensate phase, can only be generated when the domain of the group field is non-compact (or its volume is sent to infinity in a thermodynamic limit), since otherwise the symmetry will be restored upon flow towards the IR [ref] .
Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom · Introduction · 0.98
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This hypothesis, while remaining such, is increasingly supported by growing number of mean-field [ref] and renormalization group studies [ref] , from the more formal perspective.
Emergent Cosmology from Quantum Gravity in the Lorentzian Barrett-Crane Tensorial Group Field Theory Model · Introduction · 0.99
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This expectation is also supported by prior studies employing Landau-Ginzburg mean-field theory [ref] and the FRG methodology [ref] to TGFT.
QFT with Tensorial and Local Degrees of Freedom: Phase Structure from Functional Renormalization · Renormalization group flow in the non-autonomous case · 1.00
Cites it as background 9 papers
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The phase structure of TGFTs for Abelian groups $G=$U$(1),$ with and without local arguments has been studied in [ref] and [ref] , using Landau-Ginzburg mean-field theory and the FRG methodology, respectively.
Scale invariance beyond criticality within the mean-field analysis of tensorial field theories · Model setup · 0.97
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What is the physical interpretation and what are the physical signatures of a geometrogenesis phase transition [ref] [ref] [ref] , leading from a non-geometric phase to a geometric one or viceversa?
Tensorial Group Field Theory condensate cosmology as an example of spacetime emergence in quantum gravity · Spacetime emergence and its many levels · 0.81
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While the investigation of such theories' phase structure, has attracted considerable attention in recent years [ref] , much remains to be understood concerning the general conditions under which critical behaviour occurs, and even more about whether this leads to well-defined continuum spacetime geometries.
Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom · Introduction · 0.88
What it cites · 125 sentences without a confident label
Builds on 2 papers
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The following discussion follows and slightly generalizes the arguments in the Appendix of Ref. [ref] .
Functional Renormalisation Group analysis of Tensorial Group Field Theories on $\mathbb{R}^d$ · Scaling dimensions · 0.86
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Indeed, our equations will agree with those results [ref] in quartic truncation but generalize them to arbitrary order.
Functional Renormalisation Group analysis of a Tensorial Group Field Theory on $\mathbb{R}^3$ · The large-$$ regime · 0.84
Uses the method or tool of 1 paper
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As a regulator function [ref] which satisfies the usual properties we choose the optimized regulator [ref] [equation] where $θ$ is the Heaviside function.
Optimised Renormalisation Group Flows · FRG equation for Abelian group with generalized propagator · 0.98
Uses data or values from 2 papers
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The scaling dimensions necessary to rescale the FRG equation in the large-$$ limit can be taken from results on the renormalizability of TGFTs [ref] .
Tensorial methods and renormalization in Group Field Theories · Scaling dimensions · 0.91
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The scaling dimensions necessary to rescale the FRG equation in the large-$$ limit can be taken from results on the renormalizability of TGFTs [ref] .
Renormalizable Models in Rank $d\geq 2$ Tensorial Group Field Theory · Scaling dimensions · 0.91
Compares with 6 papers
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Alternatively, the flow equation of the anomalous dimension solved for $η_k$ is [equation] These large-$$ equations are consistent with earlier results in quartic ($=2$) truncation [In Ref. [ref] , Eq. (4.5–4.7), the same equations are found for rank $=3$, real field $=1$ and linear kinetic term $ζ=1/2$ setting $=3$.
Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model · Rescaled flow equations · 0.90
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It is the same kind of non-local operator as found for the connected two-point correlation function using Landau-Ginzburg mean field theory [ref] .
Phase transitions in group field theory: The Landau perspective · FRG equation for projected fields · 0.94
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A similar result has been already discussed in Ref. [ref] for a theory with closure constraint, thus for $= + 2 = 6$ (cf. App. [ref]) in quartic $= 2$ truncation [Also in [ref] both NGFP$_1$ and NGFP$_2$ are found in the quartic truncation ($=2$) and discussed for rank $=3$, that is $=2$.
Functional Renormalisation Group analysis of a Tensorial Group Field Theory on $\mathbb{R}^3$ · Results with anomalous dimension · 0.88
Cites as support 7 papers
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This theory space is approximately stable under the FRG in the large $k$ regime as shown in the $=3$ case with an additional closure constraint [ref] .
Asymptotic safety in three-dimensional SU(2) Group Field Theory: evidence in the local potential approximation · Restriction to cyclic-melonic interactions · 0.96
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Previous research using mean-field arguments [ref] as well as FRG studies applied to TGFTs [ref] also nourish this expectation.
Phase transitions in group field theory: The Landau perspective · Symmetry restoration · 1.00
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Previous research using mean-field arguments [ref] as well as FRG studies applied to TGFTs [ref] also nourish this expectation.
Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model · Symmetry restoration · 1.00
Cites as background 27 papers
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The phase structure has been studied in low-order truncations of the theory space [ref] and in an autonomous limit [ref] .
Functional Renormalization Group analysis of rank 3 tensorial group field theory: The full quartic invariant truncation · Introduction · 0.92
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The phase structure has been studied in low-order truncations of the theory space [ref] and in an autonomous limit [ref] .
Functional Renormalisation Group Approach for Tensorial Group Field Theory: a Rank-3 Model · Introduction · 0.92
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The phase structure has been studied in low-order truncations of the theory space [ref] and in an autonomous limit [ref] .
Functional Renormalisation Group analysis of a Tensorial Group Field Theory on $\mathbb{R}^3$ · Introduction · 0.92
Discussion
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