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A single renormalization-group framework produces the full phase diagram of an interacting Dirac semimetal, including emergent Lorentz symmetry, the quantum critical fan, and a classical Ising transition.

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2026-08-01 15:31 UTC pith:3NZG7D3E

load-bearing objection Solid fRG study with genuinely new velocity-flow results, but the finite-T Ising 'confirmation' is partly built into the projection choice and the non-universal phase diagram needs more truncation control. the 4 major comments →

arxiv 2607.18396 v1 pith:3NZG7D3E submitted 2026-07-20 cond-mat.str-el cond-mat.stat-mechhep-ph

Quantum critical fan and emergent relativistic symmetry of two-dimensional Dirac semimetals

Friederike Ihssen , Bilal Hawashin , Mireia Tolosa-Sime\'on , Michael M. Scherer This is my paper
classification cond-mat.str-el cond-mat.stat-mechhep-ph
keywords chiral Ising modelGross-Neveu-Yukawafunctional renormalization groupDirac semimetalquantum critical fanemergent Lorentz symmetryprecondensationquasiparticle weight
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tries to establish that a single non-perturbative functional renormalization group (fRG) framework, with separately flowing fermion and order-parameter velocities and a fully field-dependent potential, quantitatively captures the entire phase diagram of the chiral Ising Gross–Neveu–Yukawa model in 2+1 dimensions. At zero temperature, it finds a quantum critical point with exponents close to the chiral Ising universality class and shows that relativistic symmetry (equal fermion and boson velocities) emerges as the RG flow approaches the critical point, on both sides of the transition. At finite temperature, the symmetry-broken phase survives to a finite Tc, and the thermal transition belongs to the two-dimensional Ising universality class. The paper also fixes the extent and scaling of the quantum critical fan, where the thermal correlation length behaves as xi_T ~ T^{-1/z} with z ~ 1, and describes precondensation and quasiparticle-weight suppression inside the fan. If correct, this provides a controlled benchmark for interacting Dirac semimetals and a pathway toward material-specific phase diagrams, for example in moire materials near a semimetal-to-insulator transition.

Core claim

The central claim is that the chiral Ising Gross–Neveu–Yukawa model, even when starting with distinct bare fermion and order-parameter velocities (v_psi/v_phi = 1.4), flows to a relativistically symmetric fixed point at the zero-temperature quantum critical point: the velocity ratio v_psi/v_phi approaches unity both from the symmetric side and from the side with spontaneously broken Z_2 symmetry. At finite temperature, the transition into the ordered phase is governed by static bosonic fluctuations and falls into the two-dimensional Ising universality class, with bosonic anomalous dimension eta_phi ~ 0.236 (exact: 1/4) and correlation-length exponent nu ~ 1.02–1.1 (exact: 1). Around the quan

What carries the argument

The argument is carried by a leading-order derivative expansion of the flowing effective action, keeping separate temporal and spatial wave-function renormalizations Z_phi/psi,|| and Z_phi/psi,perp, a scale-dependent Yukawa coupling, and a fully field-dependent local potential U(rho). The potential is resolved by rewriting its flow as a convection-diffusion partial differential equation and solving it with high-order fluid-dynamical numerical methods, which is necessary to capture the flat (convex) region near the broken phase and the classical critical regime. A spatial Litim-type regulator allows all Matsubara sums to be performed analytically. The key identity is the beta function for v_p

Load-bearing premise

The entire quantitative phase diagram rests on the leading-order derivative expansion with only uniform temporal/spatial wave-function renormalizations and a local field-dependent potential, and the paper itself flags that this truncation produces an artificial IR fixed point on the symmetric side (where the scalar mass does not freeze), slightly violates the relativistic scaling relation via eta_psi,|| = 0.037 vs eta_psi,perp = 0.031, requires discarding a logarithmic loop d

What would settle it

A high-precision quantum Monte Carlo simulation of the chiral Ising Gross–Neveu–Yukawa model with a bare velocity ratio v_psi/v_phi = 1.4 could measure the velocity ratio at the quantum critical point from both sides of the transition; if v_psi/v_phi does not flow to 1, or if the finite-temperature correlation-length exponent deviates distinctly from the two-dimensional Ising value, the central claims would be refuted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The phase diagram of the chiral Ising model in 2+1 dimensions can now be computed in a single framework, covering both universal quantum/classical critical behavior and non-universal regions near the QCP.
  • Quantum critical fan boundaries, precondensation scales, and domain sizes can be attached to physical scales (moire lattice constant, sample size) for materials such as twisted tetralayer WSe2.
  • The predicted increase of the Fermi velocity along the precondensation line is testable via cyclotron-mass measurements in Dirac semimetals.
  • The vanishing of the quasiparticle weight at the QCP and its angularly anisotropic recovery at finite temperature set expectations for spectral-function and transport calculations in correlated Dirac materials.
  • Higher-derivative extensions of the truncation could remove the symmetric-side artifacts and further sharpen the non-universal predictions, with universal exponents expected to remain stable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the symmetric-side artifacts (non-freezing scalar mass and the artificial IR fixed point eta_phi -> 1) were resolved by including momentum-dependent wave-function renormalizations, the quantitative extent of the quantum critical fan and precondensation region could shift, even though the universal exponents would likely survive.
  • The emergent relativistic symmetry found on the symmetry-broken side of the chiral Ising transition suggests that analogous velocity-ratio flows occur in the chiral Heisenberg Gross–Neveu–Yukawa model relevant to twisted WSe2, where a continuous order parameter would also test the role of Goldstone modes.
  • The discarded logarithmic divergence in the fermionic loop contributing to eta_phi, described as spurious for the spatial regulator, invites a cross-check with a covariant regulator scheme to confirm regulator independence of the quoted anomalous dimension.
  • The velocity-ratio beta function, proven only to linear order in perturbations, could be tested at second order; if the O(r^2) term changed sign sufficiently far from the QCP, it might create a basin boundary that alters the emergence region at finite couplings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript presents a functional renormalization group study of the (2+1)-dimensional chiral Ising Gross–Neveu–Yukawa model, using a leading-order derivative expansion with a fully field-dependent effective potential and separate wave-function renormalizations for bosonic and fermionic fields in temporal and spatial directions. It reports a T=0 quantum critical point with exponents close to the chiral Ising universality class, emergent relativistic symmetry via velocity flows, a finite-temperature transition in the 2D Ising universality class, the extent and scaling of the quantum critical fan with ξ_T ~ T^{-1/z} and z≈1, precondensation, and suppression of the fermionic quasiparticle weight. The results are benchmarked against previous fRG studies, perturbative RG, conformal bootstrap, and QMC.

Significance. If correct, the work would provide a unified non-perturbative description of the phase diagram of a Dirac semimetal near a quantum critical point, including non-universal regions relevant to moiré materials such as twisted tetralayer WSe2. The use of a full field-dependent potential via DiFfRG, the explicit derivation of velocity flows, and the transparent discussion of regulator artifacts are strengths. The central quantitative claims, however, rest on a truncation whose known limitations (symmetric-side mass flow, projection dependence of ηφ, discarded log-divergent loop) are acknowledged in the text but not fully resolved; these limitations are directly relevant to the strength of the benchmarks.

major comments (4)
  1. [Sec. IVC, App. A3, Eqs. (A10)–(A13)] The finite-temperature Ising claim is not an independent confirmation. In Eqs. (A10)–(A13), ηφ is projected onto the Goldstone mode of an O(N) theory and only afterwards the N→1 limit is taken; the authors explicitly state that in d=2 this prescription is chosen because it reproduces the exact Ising value ηφ=0.25, whereas the direct radial projection gives 0.436. The reported ηφ,c≈0.236 is therefore largely encoded in the projection choice, and since ηφ,⊥ enters the potential flow (Eq. (31)), the quoted ν values are not independent of this choice. To support the conclusion 'we confirm that this transition lies in the two-dimensional Ising universality class', the authors should either show a projection-independent check (e.g., a different regulator or a direct radial projection with error estimates) or clearly rephrase the statement as consistency of the projection-improved LPA'.
  2. [Sec. IVB, App. D] The T=0 correlation-length exponent is extracted only on the symmetry-broken side of the transition. As the authors state in Section IVB and Appendix D, on the symmetric side the scalar mass does not freeze out and ηφ,⊥ flows to an artificial infrared fixed point, which is attributed to the insufficient momentum dependence of LPA'. This is a load-bearing issue for the claim of a 'quantitatively reliable unified framework': the QCP is not controlled from both sides, and ν=1.008(9) could be biased by the asymmetric extraction. Please provide a robustness test (e.g., a different regulator or an extended truncation such as NLO) showing that the extracted ν is unchanged within errors when the symmetric side is treated differently.
  3. [Sec. IIIB, Eq. (29), Table I] The consistency relation Eq. (29) is violated by the reported fermionic anomalous dimensions: with ηψ,∥=0.037 and ηψ,⊥=0.031, the right-hand side gives z≈1.006 rather than z=1. The authors call this 'slightly violated' (Section IVB), but the QCF analysis in Section IVC2 assumes z=1 and uses the exponent a≈1 as evidence; the velocity-flow argument for emergent Lorentz symmetry also relies on these projections. Please quantify the uncertainty this violation introduces in the QCF scaling and in the velocity-ratio statement, or explain why it is within the truncation error.
  4. [App. B2] In Appendix B2, a logarithmically divergent fermionic loop contributing to ηφ,⊥ is dropped as spurious, with the justification that it is an artifact of the non-analytic momentum dependence of the Litim regulator. Since ηφ,⊥ is one of the key quantities entering the finite-temperature exponents and the QCF, a more quantitative justification is needed. For instance, compare with a smooth regulator where the divergence is regulated, or show explicitly that the term is absent in the covariant scheme and does not affect the extracted exponents.
minor comments (5)
  1. [Fig. 1] The horizontal axis label appears as 'h §'; it should read 'hΛ'.
  2. [Eq. (48)] The expression for the anisotropy ratio is ambiguous; please add parentheses to clarify the numerator and denominator.
  3. [Table I] The entry 'LPA′4 [35]' contains a stray subscript. Also, the caption should state that the ηψ value from this work is ηψ,⊥, with the parallel value deferred to Appendix E.
  4. [Sec. IVC2] The QCF boundary prefactors A± are defined by a ±10% deviation criterion. This is a convention-dependent definition and should be flagged as such in the figure caption as well as in the text.
  5. [App. E, Fig. 9] The right panel shows the velocity ratio at k=πT; the heuristic identification |k|≈|p| is only mentioned in the caption. Please add a corresponding caveat in the main text.

Circularity Check

1 steps flagged

Finite-T Ising 'confirmation' is partially built into the ηφ projection prescription; zero-T exponents and QCF scaling are otherwise independent.

specific steps
  1. other [Appendix A3 (projection prescription) and Section IVC (finite-temperature transition)]
    "For the computation of the bosonic two-point function, we use a projection onto the Goldstone mode of a O(N) type theory and subsequently take the limit N→1. This trick is commonly applied in Z2-type theories, as it yields quantitatively better results for the scaling exponents: For example in d=2, the direct projection onto the scalar mode yields a critical anomalous dimension of ηϕ,c = 0.436 [98], whereas the modified projection procedure captures the exact solution ηϕ = 0.25 much better, see e.g., Figure5 for the result of the present work. The value of ηϕ,c ≈ 0.236 is approached asymptotic"

    The finite-T bosonic anomalous dimension used to 'confirm' 2D Ising universality is computed with a projection (Goldstone mode, N→1) that was adopted precisely because it reproduces the exact Ising η=0.25 in d=2; the paper even cites its own Figure 5 as evidence. Since the finite-T transition is dimensionally reduced to two spatial dimensions, obtaining η≈0.236 is largely a consequence of the preselected projection rather than an independent test of the LPA' truncation. The correlation-length exponent ν≈1.02/1.1 is not preselected this way and supplies some independent support, so the circularity is partial rather than total.

full rationale

The zero-temperature critical exponents are not circularly obtained: h_c is determined by fine-tuning the flow, and ν and ηψ/ηϕ are read from power-law fits and flow plateaus (Figs. 3, 4), then compared with external bootstrap/QMC results. The velocity-ratio argument that vψ/vϕ=1 is a closed, attractive subspace is a genuine one-loop computation with an external fixed-point check; self-citations such as [35] supply context and benchmarks rather than the load-bearing result. The finite-temperature Ising claim is the only partially circular element: the bosonic anomalous dimension is obtained with a Goldstone-mode projection preselected to reproduce exact Ising η=0.25 in d=2, and the paper points to its own Fig. 5 as evidence. Thus the 'confirmation' of η≈0.236 is not independent. The correlation-length exponent ν≈1.02/1.1 is not preselected, so the overall circularity is partial. The paper's own caveats (symmetric-side mass non-freeze-out, Eq. (29) violation, dropped spurious divergence, limited momentum resolution in App. E) are correctness risks, not circularity.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

No new entities are introduced. The main burden the reader pays for is the fRG truncation and projection scheme: it is benchmarked against known exponents, then used to produce the non-universal phase diagram, so the non-universal quantitative claims inherit the truncation's uncertainty. The free parameters listed are non-universal inputs or explicitly fitted quantities; the universal exponents themselves are extracted rather than imposed.

free parameters (5)
  • initial Yukawa coupling h_Lambda = h_c ≈ 2.45306(4)
    Tuning parameter used to locate the QCP; all phase-diagram axes are measured relative to h_c.
  • initial velocity ratio v_psi,Lambda / v_phi,Lambda = 1.4
    Chosen as 'somewhat arbitrary' but guided by estimates for WSe2 from Ref. [41]; controls non-universal QCF extent and terminal QCP velocities.
  • initial boson mass m_Lambda^2 = 1
    Sets the UV cutoff scale Lambda=1; an arbitrary normalization that affects non-universal temperatures such as T_c.
  • initial quartic scalar coupling lambda_Lambda = 0
    Chosen so the UV potential is non-interacting; assumed justified because order-parameter fluctuations build up near the transition.
  • QCF boundary prefactors A_+, A_- = A_+ = 1.96, A_- = -2.17
    Fitted to where the numerically determined exponent a deviates by ±10% from z=1; these are not predicted from first principles.
axioms (6)
  • domain assumption The flowing action is well approximated by the leading-order derivative expansion with uniform wave-function renormalizations and a local field-dependent potential U_k(rho), Eq. (14).
    All quantitative results depend on this truncation; the paper's own symmetric-phase artifacts (Appendix D) show its limits.
  • ad hoc to paper The Litim/sharp regulator shape functions in Eqs. (19)-(20) and (21)-(22) provide controlled RG flows, and the logarithmically divergent fermionic loop in Eq. (B8) can be discarded as spurious.
    The divergence is regulator-induced, but dropping it is a prescription, not a systematically derived scheme; it directly affects the bosonic anomalous dimension.
  • domain assumption The low-energy physics of the Dirac semimetal is captured by the chiral Ising GNY action Eq. (1) with eight-component fermions and a single Z2 scalar order parameter, ignoring O(phi^6, psi^4, psi^2 phi^2) and competing orders.
    Valid for one particular order-parameter channel; not the chiral Heisenberg model believed relevant to WSe2, so material connection is indirect.
  • ad hoc to paper Initial conditions at Lambda: m_Lambda^2=1, Z_phi=Z_psi=1, lambda_Lambda=0, v_phi,Lambda=1, v_psi,Lambda=1.4, with h_Lambda tuned to approximately h_c.
    Non-universal values chosen for convenience and guided by WSe2 estimates; they affect T_c, QCF prefactors, and velocity flows.
  • domain assumption At finite temperature, fermions are gapped by odd Matsubara frequencies and do not contribute to the critical singularities, so the transition is in the classical 2D Ising universality class.
    Standard dimensional-reduction argument used to interpret the finite-T numerical results.
  • standard math The Wetterich equation (12) and the analytic Matsubara-summed threshold functions in Appendix B are correct within the stated conventions.
    Formal foundation of the fRG method; assumed correct without machine checking.

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read the original abstract

Two-dimensional Dirac semimetals near a quantum critical point can be described by Gross--Neveu--Yukawa models. In view of recent experimental advances exhibiting a transition from Dirac semimetal to insulator in highly-tunable van-der-Waals heterostructures, a better understanding of finite-temperature effects is mandatory. Here, we study the Gross--Neveu--Yukawa phase diagram of the chiral Ising model with a non-perturbative field-theory approach at zero and finite temperature, both in the semimetallic phase and in the insulating phase with spontaneously broken $\mathbb{Z}_2$ symmetry. At zero temperature, we find a quantum critical point with critical exponents that are close to the ones of the chiral Ising universality class, and show that relativistic symmetry is emergent close to the quantum critical point. At finite temperature, the ordered phase survives up to a finite critical temperature, at which we observe a classical phase transition into the disordered phase. We confirm that this transition lies in the two-dimensional Ising universality class. Finally, we determine the extent and scaling properties of the quantum critical fan, and the behavior of the quasiparticle weight, therein. In summary, we present a unified field-theoretical framework for the phase diagram of the chiral Ising model in the surroundings of its quantum critical point.

Figures

Figures reproduced from arXiv: 2607.18396 by Bilal Hawashin, Friederike Ihssen, Michael M. Scherer, Mireia Tolosa-Sime\'on.

Figure 1
Figure 1. Figure 1: Our developments prepare the calculation of thermodynamic observables and spectral functions, which we aim for in future work [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: , we follow the fRG flow trajectories of the order￾parameter anomalous dimension, approaching Tc. The value of ηϕ,c ≈ 0.236 is approached asymptotically when T → Tc from both sides of the transition. The correlation length, shown in the lower panel of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: , where Z −1 ψ,⊥ clearly dips along the preconden￾sation line. Together, these observations lead to an in￾crease of the fermionic velocity vψ ∝ Zψ,⊥/Zψ,∥ along the precondensation line. Such behavior of the Fermi velocity in the precondensation region of a pertinent finite￾temperature ordering transition could be experimentally scrutinized with measurements of the Dirac fermion’s cyclotron mass m∗ = ℏ √ πn… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗

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Reference graph

Works this paper leans on

114 extracted references · 16 linked inside Pith

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    Initial conditions We need to choose a set of initial conditions at the UV scaleΛ, which we identify to be close to the pertinent QCP. For this purpose, we choose the initial Yukawa coupling hΛ as tuning parameter and fix the remaining quantities as follows: The fully field-dependent order-parameter potential is chosen to be non-interacting, i.e., UΛ(ρ) =...

  2. [2]

    In the present work, we solve this entire system of differential equations using the DiFfRG framework [52], which builds on the use of fluid dynamical methods in the fRG

    Numerical implementation of the potential While the RG scale evolution of the field-independent couplings h, Zϕ,⊥, Zϕ,∥, Zψ,⊥, and Zψ,∥ corresponds to solving a set of algebraic and simple coupled ordinary differential equations, the resolution of the fully field- dependent potential requires solving a highly nonlinear partial differential equation. In th...

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    Temperatures for which a finite expectation value vanishes at a finite scalekprecond

    Phase diagram and precondensation To gain further insight on the phase diagram, we scan a range of Yukawa interactions and identify the order- parameter expectation value as well as the fermion mass gap, Eq.(35), see the left and middle panels of Figure 6, respectively. Temperatures for which a finite expectation value vanishes at a finite scalekprecond. ...

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    Quantum critical fan Quantum critical points induce a zero-temperature power-law scaling of the correlation length with critical exponent ν. Accordingly, the characteristic energy scale∆ associated with the ground state, which, for example, can be the gap to excited states, scales as [2] ∆∝ |hΛ −h c|zν ,(44) with the dynamical critical exponentz of the QC...

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    Its anisotropic dependence on the temporal and spatial direction is resolved min- imally by projecting onto Zψ,∥ and Zψ,⊥, respectively, and evaluating at p = ( πT, ⃗0 ), cf

    Fermionic quasiparticle weight The present setup allows us to study the fermionic wave-function renormalizationZψ which acts as a proxy for the quasiparticle weight. Its anisotropic dependence on the temporal and spatial direction is resolved min- imally by projecting onto Zψ,∥ and Zψ,⊥, respectively, and evaluating at p = ( πT, ⃗0 ), cf. also Eq.(23). Th...

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    Alternatively, one can evaluate the Yukawa coupling from the fermionic two-point function as ∂th= 1 dγNf ϕ Tr " − →δ δ ¯ψ(−p) ∂tΓk ← −δ δψ(p) # p=p ex Φ(x) = Φ0

    Flow of the Yukawa coupling The flow of the Yukawa coupling can be evaluated from the projection of the Wetterich flow onto the three point vertexΓ ϕ ¯ψψ at the minimal external fermionic momentum pex = (pex,0,⃗0 )with pex,0 = πT and vanishing bosonic external momentum ∂th= 1 dγNf Tr " δ δϕ(p−q) − →δ δ ¯ψ(−p) ∂tΓk ← −δ δψ(q) # p=q=p ex Φ(x) = Φ0 (A4a) whe...

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    Thus in the projection procedure, we take derivatives with respect to the corresponding external momentump= (p 0, ⃗ p)

    Flow of the fermionic anomalous dimensions The spatial momentum and frequency dependence of the fermionic two-point function is described by the wave- functions parallel and perpendicular to the heat-bath as defined in Eq.(23). Thus in the projection procedure, we take derivatives with respect to the corresponding external momentump= (p 0, ⃗ p). Then the ...

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    Flow of the bosonic anomalous dimensions For the computation of the bosonic two-point function, we use a projection onto the Goldstone mode of aO(N ) type theory and subsequently take the limitN→ 1. This trick is commonly applied inZ2-type theories, as it yields quantitatively better results for the scaling exponents: For example in d = 2, the direct proj...

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    To retain meaningful expressions it is useful to cast the flows in a standard form

    Matsubara summation The summation over all frequencies in Eq.(14) can be done analytically when using a spatial regulator. To retain meaningful expressions it is useful to cast the flows in a standard form. To this aim, we consider the scalar parts of the propagators GB,n(M 2(⃗ p)) = 1 ω2 ϕ,n +M 2(⃗ p), GF,n(M 2(⃗ p)) = 1 ω2 ψ,n +M 2(⃗ p),(B1) with the re...

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    With this choice all mo- mentum loops can be performed analytically as the mo- mentum dependence in the dispersion relations Eq.(B2) drops out

    Loop integration In this work, we use the spatial flat (or Litim) regu- lator [100, 101], defined in the main text by the shape functions in Eqs.(19) and (20). With this choice all mo- mentum loops can be performed analytically as the mo- mentum dependence in the dispersion relations Eq.(B2) drops out. Consequently, the momentum dependence in the threshol...

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    To evaluate these projections, we expand the Wetterich equation(12) around the background fieldΦ = Φ 0 + δΦwith fluc- tuations δΦ = ( ξ, ψ,¯ψT )T

    RG flow of the bosonic and Fermi velocities The RG flow of the bosonic and Fermi velocities are defined in terms of the scale-dependent wave-function renormalizations as ∂tv2 ϕ = 1 Zϕ ∂t(Zϕv2 ϕ)−v 2 ϕ∂tZϕ , =v 2 ϕ(ηϕ,⊥ −η ϕ,∥)(C1) ∂tvψ = 1 Zψ [∂t(Zψvψ)−v ψ∂tZψ] =v ψ(ηψ,⊥ −η ψ,∥).(C2) The projection prescriptions forηψ,⊥, ηψ,∥, ηϕ,⊥, andηϕ,∥ are given in E...

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    Forvψ/vϕ = 1, the microscopic action features full rotational (Lorentz) in- variance after rescaling of spacetime

    Covariant regulator scheme Beforeevaluatingtheseexpressionsexplicitly, itisworth noting a general structural feature. Forvψ/vϕ = 1, the microscopic action features full rotational (Lorentz) in- variance after rescaling of spacetime. If the regulator insertion respects this symmetry, the theory space defined by vψ/vϕ = 1is a symmetry-protected subspace, wh...

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    Evaluation of the loop integrals To evaluate the loop integrals for the velocity flows, we introduce the bosonic and fermionic shape functionsrB andr F via Rϕ k (p, q) =Zϕp2 BrB(p2 B/k2)δ(p−q),(C19) Rψ k (p, q) =iZψ/pF rF (p2 F /k2)δ(p−q),(C20) and introduce the dressed propagator functions G−1 ϕ (p2) =Z ψp2 1 +r B(p2/k2) +m 2 ϕ ,(C21) G−1 ψ (p2) =Z 2 ψp2...

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    Covariant Litim regulator To explicitly evaluate the radial integral in Eq.(C31), we choose the covariant Litim regulator, defined in the main text by the shape functions in Eqs.(19) and (20). We find ˜∂t Z ∞ 0 dx x d 2 +1G′ ϕ(x)∂x[(1 +r F (x/k2))Gψ(x)] = kd−4 (d+ 1)Z ϕZ 2 ψ −2(d+ 1) + (1 +m 2 ψ)ηϕ,∥ (1 +m 2 ϕ)2(1 +m 2 ψ)2 .(C33) Hence, we find for the fl...

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    Velocity flow at finite temperature We show the different components of the bosonic wave- function renormalization in Figure 8. In comparison to both fermionic wave-function renormalizations, the bosonic ones grow large, highlighting the change from a system that is dominated by fermions atk = Λto a system that is mainly determined by bosons atk→0. We fin...

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