Comment on Absence versus Presence of Dissipative Quantum Phase Transition in Josephson Junctions Th eo S epulcre1Serge Florens2and Izak Snyman3

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Comment on “Absence versus Presence of Dissipative Quantum Phase Transition in

Josephson Junctions”

Th´eo S´epulcre,1Serge Florens,2and Izak Snyman3

1Wallenberg Centre for Quantum Technology, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

2Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France

3Mandelstam Institute for Theoretical Physics, School of Physics,

University of the Witwatersrand, Johannesburg, South Africa∗

In Ref. [1], a Josephson junction shunted by an ohmic

transmission line is studied. The authors present a phase

diagram with features not anticipated in the established

literature [2]. We show that their Numerical Renor-

malization Group (NRG) calculation suﬀers from several

ﬂaws, and cannot be trusted to substantiate their claims.

FIG. 1. Top: Low energy spectrum v. NRG step N, scaled

with ΛN. Results of the NRG scheme in [1] for the cosine

and quadratic potential are compared to exact results for the

quadratic potential. We took nS= 50 kept states, nB= 300

bosonic states for N= 0 and nB= 15 for N > 0, Λ = 2.0,

α= 10, EC= 0.01W,EJ/EC= 10. Bottom:⟨cos(φ)⟩v. α,

for EJ/EC= 0.15, like the triangles in the top panel of Fig. 4

of [1]. The blue dots reproduce the result of [1] with the same

truncation parameter nB= 15 for N > 0. The yellow squares

and green diamonds were obtained by increasing nBto 29

and 43 respectively. The inset zooms in on the two smallest

values of α, which are still unconverged at n= 43, showing a

downward trend.

NRG captures low energy physics by building recur-

sive Hamiltonians, HN+1 =HN+ ∆HN+1, that are it-

eratively diagonalized. Scale separation is required for

∗izak.snyman@wits.ac.za

NRG to work, i.e. ∆HN+1 should decrease exponentially

with N[3]. For the NRG scheme in Ref. [1], ∆HN+1 is

of the same order as H0[See Eqs. (S51) and (S52) in the

supplementary material to [1].]. This is a known problem

that can only be cured by introducing an infrared cut-

oﬀ [4]. As a result, the NRG fails to ﬂow to the correct

infrared ﬁxed point. To demonstrate this, we considered

large conductance αand large EJ/EC, where the system

studied in [1] is nearly harmonic, allowing us to expand

−EJcos(Ξ) ≃EJ(Ξ2/2−1). We compared low energy

spectra obtained with the NRG scheme of [1] for the co-

sine and quadratic potentials, to the exact spectrum ob-

tained for the latter. As the top panel of Fig. 1shows,

the NRG results diverge from the exact spectrum after

the seventh RG step. Thus the NRG scheme proposed

in [1] is unreliable and cannot be trusted to predict the

phase diagram. (See Appendix Afor discussion of the

RG ﬂow of mobility µ10.)

The phase diagram in [1] is ﬂawed in another way.

Even if one trusted the employed NRG scheme, the

re-entrant superconductivity seen at small αand small

EJ/ECis a numerical artefact. The blue dots in the bot-

tom panel of Fig. 1reproduce the result for ⟨cos(φ)⟩v.

αat EJ/EC= 0.15 in the upper panel of Fig. 4 of [1],

obtained with the truncation parameter nB= 15 in each

mode for N > 0. For this result to be correct, it must not

change when nBis increased. Instead we see that the re-

gion where ⟨cos(φ)⟩vanishes, grows to include the inter-

val α∈[0,0.2] when nBis increased. Thus, the apparent

re-entrant superconductivity in the phase diagram in [1]

stems from unconverged data. In [1] it is argued that su-

perconductivity makes common sense when the junction

is shunted by a suﬃciently large impedance. We stress

that taking the thermodynamic limit N→ ∞ before

α→0, couples the junction to divergent φ-ﬂuctuations

that render the junction’s zero-frequnecy response non-

trivial. The object Letter also contains a brief functional

Renormalization Group (fRG) argument in support of

superconductivity at α < 1 and large EJ/EC. The ap-

proximations involved are not controlled by any obvious

small parameter. It is still not known whether fRG can

reproduce infrared Luttinger exponents for 1 < α < 2 [4],

where phase-slips aﬀect results non-trivially. Until this

is settled, fRG’s validity in the more challenging α < 1

regime remains unclear.

arXiv:2210.00742v2 [cond-mat.supr-con] 24 Oct 2023

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Commenton“AbsenceversusPresenceofDissipativeQuantumPhaseTransitioninJosephsonJunctions”Th´eoS´epulcre,1SergeFlorens,2andIzakSnyman31WallenbergCentreforQuantumTechnology,ChalmersUniversityofTechnology,SE-41296Gothenburg,Sweden2Univ.GrenobleAlpes,CNRS,GrenobleINP,InstitutN´eel,38000Grenoble,France3Man...

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Comment on Absence versus Presence of Dissipative Quantum Phase Transition in Josephson Junctions Th eo S epulcre1Serge Florens2and Izak Snyman3

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