Nonperturbative functional renormalization-group approach to the sine-Gordon modeland the Lukyanov-Zamolodchikov conjecture

R. Daviet and N. DupuisSorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France

(Dated: December 5, 2018)

We study the quantum sine-Gordon model within a nonperturbative functional renormalization-group approach (FRG). This approach is benchmarked by comparing our findings for the solitonand lightest breather (soliton-antisoliton bound state) masses to exact results. We then examinethe validity of the Lukyanov-Zamolodchikov conjecture for the expectation value 〈e

i2nβϕ〉 of the

exponential fields in the massive phase (n is integer and 2π/β denotes the periodicity of the potentialin the sine-Gordon model). We find that the minimum of the relative and absolute disagreementsbetween the FRG results and the conjecture is smaller than 0.01.

Introduction. The quantum sine-Gordon model [1]describes many physical systems. In condensed mat-ter it is widely used to understand the phase diagramand the low-energy properties of one-dimensional quan-tum fluids [2–4] and has applications that range fromstrongly correlated electron systems to cold atoms. Inhigh-energy physics it is related to the massive Thirringmodel describing Dirac fermions with a self interac-tion [5]. The sine-Gordon model can also be viewed asa two-dimensional model of classical statistical mechan-ics. In particular it describes the Berezinskii-Kosterlitz-Thouless (BKT) transition which occurs in the XY spinmodel and more generally in two-dimensional systemswith a two-component order parameter with an O(2)symmetry [6–8].

The Hamiltonian of the quantum sine-Gordon modelis defined by

H =

ˆdx

{1

2Π2 +

1

2

(∂ϕ

∂x

)2

− u cos(βϕ)

}, (1)

where Π and ϕ satisfy canonical commutation relations,

[ϕ(x), Π(x′)] = iδ(x − x′). Regularization with a UVmomentum cutoff Λ is implied and u/Λ2, β > 0 are di-mensionless parameters. The phase diagram consists of agapless phase with massless (anti)soliton excitations forβ2 ≥ 8π (and u → 0) and a gapped phase with massive(anti)soliton excitations for β2 ≤ 8π. The soliton and theantisoliton carry the topological charge Q = 1 and −1,respectively [9]. They attract for β2 ≤ 4π and can formbound states, called breathers, with topological chargeQ = 0. The phase transition between the two phases isof BKT type [6–8].

The sine-Gordon model is one of the most studied inte-grable models; its spectrum, thermodynamics and scat-tering properties are well understood [10–14]. Howevernot everything is known and many quantities can beobtained only from nonexact (e.g. perturbative) meth-ods [2–4]. In particular in the massive phase the ampli-tude of the fluctuations about the mean value 〈ϕ〉 = 0 isnot known exactly. It has been conjectured by Lukyanovand Zamolodchikov that [15]

〈ei√

8πaϕ〉 =

[Γ(1−K)

Γ(K)

πu

2(bΛ)2

] a2

1−K

exp

{ˆ ∞0

dt

t

[sinh2(2a

√Kt)

2 sinh(Kt) sinh(t) cosh[(1−K)t]− 2a2e−2t

]}, (2)

where |<(a)| < 1/2√K and K = β2/8π is the “Luttinger

parameter” [4] (the massive phase corresponds toK < 1).Equation (2) is exact for a =

√K, K = 1/2 and in the

semiclassical limit K → 0 [16]. Additional argumentssupporting the conjecture were presented in [17, 18].From the equivalence between the sine-Gordon modeland the massive Thirring model Eq. (2) was shown tobe correct to first order in u [19, 20]. Further evidenceof the correctness of (2), in particular for not too largevalues of a, was provided by a numerical study in a finitevolume [21] and variational perturbation theory [22].

In this Letter, we examine the validity of the

Lukyanov-Zamolodchikov conjecture using a nonper-turbative functional renormalization-group approach(FRG) [23, 24]. We go beyond previous FRG ap-proaches [25–28] and, in order to benchmark our ap-proach, first compute the massMsol of the (anti)soliton aswell as that (M1) of the lightest breather. We then turnto the computation of the expectation value 〈e i2nβϕ〉 =

〈ein√

2πKϕ〉 (n integer) of the exponential fields. We con-firm the Lukyanov-Zamolodchikov conjecture with an ac-curacy, defined as the minimum of the relative and ab-solute disagreements between the FRG results and the

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conjecture, of 0.01.FRG approach. From now on we adopt the point of

view of classical statistical mechanics (or Euclidean fieldtheory) where the sine-Gordon model is defined by thepartition function

Z[J ] =

ˆD[ϕ] e−

´d2r{

12 (∇ϕ)2−u cos(βϕ)−Jϕ

}, (3)

with ϕ(r) a classical field and r a two-dimensional co-ordinate. J is an external source allowing us to obtainthe expectation value φ(r) = 〈ϕ(r)〉 = δ lnZ[J ]/δJ(r) byfunctional derivation. Most physical quantities can beobtained from the free energy − lnZ[J ] or, equivalently,the effective action (or Gibbs free energy)

Γ[φ] = − lnZ[J ] +

ˆd2r Jφ (4)

defined as the Legendre transform of lnZ[J ].We compute Γ[φ] using a Wilsonian nonperturbative

FRG approach where fluctuation modes are progressivelyintegrated out in the functional integral (3). This de-fines a scale-dependent effective action Γk[φ] which incor-porates fluctuations with momenta between a (running)momentum scale k and the UV scale kin � Λ. The lat-ter condition implies that the initial value Γkin [φ] = S[φ]coincides, as in mean-field theory, with the microscopicaction defined by (3). The effective action of the sine-Gordon model, Γk=0[φ], is obtained when all fluctuationshave been integrated out. The scale-dependent effectiveaction satisfies an exact flow equation which cannot besolved exactly [29]. A common approximation scheme isthe derivative expansion where

Γk[φ] =

ˆd2r

{1

2Zk(φ)(∇φ)2 + Uk(φ)

}(5)

is truncated to second order in derivatives. This leadsto coupled flow equations for the functions Zk(φ) and

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Kk

10-2

10-1

100

u1,k

Figure 1. Flow diagram of the sine-Gordon model projectedonto the plane (Kk, u1,k) where u1,k is the first harmonic ofthe potential Uk(φ) and Kk the running Luttinger parameter.There is an attractive line of fixed points for u1,k = 0 andKk > 1 which terminates at the BKT point (u1,k = 0,Kk =1).

1.0 0.5 0.0 0.5 1.0

(8K/π)12 φ

0.0

0.5

1.0

1.5Zk(φ)

|t|= 0|t|= 11|t|= 12|t|= 13|t|= 14

1.0 0.5 0.0 0.5 1.0

(8K/π)12 φ

0.0

0.5

1.0∆Uk(φ)

Figure 2. Zk(φ) and Uk(φ) for various values of t =

ln(k/kin). ∆Uk(φ) is given by Uk(φ) − Uk(0) normalized sothat ∆Uk(±

√π/8K) = 1. In Figs. 2 and 3, Λ = 1 and

u/Λ2 = 10−3.

Uk(φ), with initial conditions Zkin(φ) = 1 and Ukin(φ) =−u cos(βφ), which can be solved numerically. We referto the Supplemental Material for more detail about theimplementation of the FRG approach [30].

It is convenient to consider the dimensionless functions

Zk(φ) =Zk(φ)

Zk, Uk(φ) =

Uk(φ)

Zkk2, (6)

where Zk = 〈Zk(φ)〉φ is obtained by averaging Zk(φ) on]− β/π, β/π] [31]. The flow diagram, projected onto theplane (Kk, u1,k) is shown in Fig. 1. Here u1,k is the firstharmonic of the potential Uk(φ) = −

∑∞n=0 un,k cos(nβφ)

and Kk = K/Zk can be interpreted as a “running” Lut-tinger parameter. In the massive phase, the flow runs intofixed points characterized by functions Z∗(φ) and U∗(φ)which depend on the parameters u andK (Fig. 2). WhileU∗(φ) slightly deviates from the cosine form of the initialpotential Ukin(φ) = −(u/k2

in) cos(βφ), we see that Z∗(φ)acquires a strong dependence on φ. Zk diverges as k−2

and the running Luttinger parameter Kk ∼ k2 vanishesfor k → 0.

Benchmarking: soliton and breather masses. Thesmallest excitation gap M of the quantum sine-Gordonmodel corresponds to the inverse correlation length of thetwo-dimensional classical model (3),

M2 = limk→0

U ′′k (0)

Zk(0)= limk→0

k2 U′′k (0)

Zk(0). (7)

Since U ′′k (0) converges to a finite value –this property ispreserved even if one retains only the first harmonics ofUk(φ)– Zk(0) must vanish as k2 for M to take a nonzerovalue in the massive phase. Zk(φ) being a normalizedfunction, 〈Zk(φ)〉φ = 1, this is possible only if Zk(φ)strongly varies with φ. Thus only a functional approachwhere the coefficient of (∇φ)2 in the effective action is afunction Zk(φ), and not a mere φ-independent number,can predict the mass of the lowest excitation. Numeri-

3

0.0 0.2 0.4 0.6 0.8 1.0K

0.00

0.01

0.02

0.03

0.04M/Λ

M1

2Msol

M (FRG)

0 0.25 0.5 0.75K

10-410-310-210-1100101

Figure 3. Mass M of the lowest excitation as obtained fromthe FRG approach. The solid and dashed lines show the exactvalues of 2Msol and M1 (the latter being defined only forK ≤ 1/2) [Eqs. (8,9)]. The inset shows the relative (crosses)and absolute (dashed line) errors of the FRG result.

cally we observe a rapid convergence of k2U ′′k (0)/Zk(0)when k → 0 in agreement with a previous study [27].

Only excitations that are in the same topological sectoras the ground state, namely Q = 0, contribute to themass M [1]. The lowest excitation in this sector is asoliton-antisoliton pair with mass

2Msol = bΛ4Γ(

K2−2K

)√πΓ(

12−2K

) [Γ(1−K)

Γ(K)

πu

2(bΛ)2

] 12−2K

, (8)

when 1/2 ≤ K ≤ 1 (Msol is the mass of a single(anti)soliton) and a breather with mass

M1 = 2Msol sin

(π

2

K

1−K

), (9)

when 0 ≤ K < 1/2 [32]. Here b is a scale parameter whichdepends on the precise implementation of the UV cutoffΛ in Eq. (1). Figure 3 shows the value of M obtainedfrom FRG (we refer to the Supplemental Material for adiscussion of the implementation of the UV cutoff Λ andthe determination of the scale factor b). For 0 ≤ K ≤ 0.4our result for the breather mass M ≡ M1 deviates fromthe exact value by at most 2%. The agreement becomesnearly perfect for K � 0.4, which is due to the fact thatthe initial value Γ

(2)kin

(q, φ = 0) = q2 + 8πKu gives theexact breather mass M1,cl =

√8πKu in the semiclassical

limit K → 0 [30]. For 0.4 ≤ K ≤ 1 the agreementbetween M and the exact value 2Msol is not as goodand varies from ∼ 2% for K ' 0.4 to more than 100%for K near 1. Note however that M goes to zero whenK → 1 and the absolute error remains below 10−3 for allvalues of K (see the inset in Fig. 3). In the immediatevicinity of K = 1, the behavior of the mass M differsfrom 2Msol [Eq. (8)] and one recovers the standard BKTscaling characterized by an essential singularity of thecorrelation length [8].

0.0 0.2 0.4 0.6 0.8 1.0K

0.0

0.2

0.4

0.6

0.8

1.0

n= 1n= 2n= 3n= 4n= 5

0.00 0.25 0.50 0.75K

10-4

10-3

10-2

10-1

Figure 4. Expectation value 〈ein√

2πKϕ〉 as obtained fromFRG (symbols) vs the Lukyanov-Zamolodchikov conjec-ture (2) valid for K < 1/n (lines). The inset shows therelative (symbols) and absolute (lines) disagreements betweenthe FRG results and the conjecture, respectively εrel and εabs.(Λ = 1 and u/Λ2 = 10−4.)

The Lukyanov-Zamolodchikov conjecture. To obtainthe expectation value of the exponential fields, we con-sider the partition function (3) in the presence of anexternal source term

´d2r(h∗ei

√8πaϕ + c.c.) so that

〈ei√

8πaϕ(r)〉 can be obtained from lnZk[J, h∗, h] by func-tional differentiation wrt h∗(r). To second order of thederivative expansion the effective action now reads

Γk[φ, h∗, h] =

ˆd2r

{1

2Zk(φ, h∗, h)(∇φ)2

+ Uk(φ, h∗, h)

}(10)

and

〈ei√

8πaϕ(r)〉 = −∂Uk(φ = 0, h∗, h)

∂h∗

∣∣∣∣h∗=h=0

. (11)

From the flow equation of Γk[φ, h∗, h] we obtain two cou-pled equations for U (10)

k (φ) ≡ ∂h∗Uk(φ, h∗, h)|h∗=h=0 andZ

(10)k (φ) ≡ ∂h∗Zk(φ, h∗, h)|h∗=h=0 with initial conditions

U(10)kin

(φ) = −ei√

8πaφ and Z(10)kin

(φ) = 0 [30].We have computed the expectation value of the expo-

nential fields ein√

2πKϕ (n integer). These are the nat-ural fields to consider in the sine-Gordon model. Forinstance, in one-dimensional quantum fluids, they arisefrom products of single-particle fields. The FRG resultsfor 1 ≤ n ≤ 5 are shown in Fig. 4.

For n = 1 we find an excellent agreement between theFRG results and the conjecture, with a difference εabs

well below 0.01 for all values of K. The relative disagree-ment εrel is small for K < 0.5 but increases for largervalues of K and becomes of order of 100% for K near1. For these values of K however, the expectation value〈ei√

2πKϕ〉 is very small and what matters is εabs.

4

Note that the Lukyanov-Zamolodchikov conjecturebreaks down in the vicinity of K = 1/n since the expec-tation value 〈ein

√2πKϕ〉 given by Eq. (2) diverges when

K → 1/n [33]. This explains the steep upturn nearK = 1/n of the lines showing the conjecture in Fig. 4.Decreasing the value of u/Λ2 confines the upturn moreand more to the vicinity of 1/n.

For n ≥ 2, εrel behaves similarly to the case n = 1 butεabs is also a monotonously increasing function of K (seethe inset in Fig. 4). εabs remains nevertheless below 0.01up to values of K very close to 1/n; for u/Λ2 = 10−4

this is the case for K = 0.49 (and n = 2), K = 0.33(n = 3) and K = 0.248 (n = 4). Moreover εrel de-creases when u/Λ2 is reduced (which extends the domainof validity of the conjecture to higher values of K, i.e.to values of K closer to 1/n). For instance, for n = 2and K = 0.49, we find εrel = 78/72/66% (while εabs =0.097/0.0097/0.00098) for u/Λ2 = 10−3/10−4/10−5. Wetherefore ascribe the apparent disagreement between theFRG results and the conjecture near K = 1/n to thebreakdown of the latter when K → 1/n. In fact thechange of concavity in the curves showing εabs in the in-set of Fig. 4 suggests that the conjecture might deviatefrom the correct result well beforeK = 1/n (e.g. K ∼ 0.4for n = 2 and u/Λ2 = 10−4). A conservative estimateis that the FRG reproduces Eq. (2), in the domain ofvalidity of the conjecture, to an accuracy (defined as theminimum of εabs and εrel) better than 0.01.

Conclusion. Contrary to the perturbative RG [23, 34,35], which correctly predicts the phase diagram of thequantum sine-Gordon model but fails to describe themassive phase, the nonperturbative FRG allows us tocontinue the flow into the strong-coupling regime andthus compute the low-energy properties of the massivephase. The fact that FRG captures genuinely nonpertur-bative topological excitations, namely (anti)solitons andbreathers, proves its efficiency and is reminiscent of itsability to describe most universal properties of the BKTtransition in the linear O(2) model [36–38] for which it iswidely admitted that topological defects (vortices) playa crucial role.

The FRG result for the expectation value 〈ein√

2πKϕ〉of the exponential fields is in very good agreement withthe conjecture proposed by Lukyanov and Zamolod-chikov [15]. The minimum of the relative and absolutedisagreements is smaller than 0.01 for all values of n ex-cept in the immediate vicinity of K = 1/n where the con-jecture breaks down. This undoubtedly provides us witha very strong support of the Lukyanov-Zamolodchikovconjecture. We also stress that FRG allows one to ob-tain 〈ein

√2πKϕ〉 for all values ofK whereas the conjecture

is limited to the range K < 1/n.Finally we would like to point out that the nonper-

turbative FRG approach presented in this Letter opensup the possibility to study various nonintegrable exten-

sions of the quantum sine-Gordon model where both per-turbative RG and exacts methods are inoperative in thestrong-coupling phase.

Acknowledgment. We thank P. Azaria for enlighten-ing discussions and a critical reading of the manuscript.

[1] For an introduction to the quantum sine-Gordon model seeR. Rajaraman, Solitons and instantons (North-Holland,Amsterdam, 1989).

[2] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik,Bosonization and strongly correlated systems (CambridgeUniversity Press, 1998).

[3] Alexei M. Tsvelik, Quantum field theory in condensed mat-ter physics, 2nd ed. (Cambridge University Press, 2007).

[4] T. Giamarchi, Quantum physics in one dimension (OxfordUniversity Press, Oxford, 2004).

[5] Sidney Coleman, “Quantum sine-Gordon equation as themassive Thirring model,” Phys. Rev. D 11, 2088–2097(1975).

[6] V. L. Berezinskii, “Destruction of long-range order in one-dimensional and two-dimensional systems having a con-tinuous symmetry group i. classical systems,” Sov. Phys.JETP 32, 493 (1971); “Destruction of long-range order inone-dimensional and two-dimensional systems possessinga continuous symmetry group. ii. quantum systems,” Sov.Phys. JETP 34, 610 (1972).

[7] J. M. Kosterlitz and D. J. Thouless, “Ordering, metasta-bility and phase transitions in two-dimensional systems,”J. of Phys. C 6, 1181 (1973).

[8] J. M. Kosterlitz and D. J. Thouless, “The critical proper-ties of the two-dimensional XY model,” J. Phys. C 7, 1046(1974).

[9] Classically the topological charge of a static field con-figuration is defined by Q = (β/2π)

´∞−∞ dx ∂xϕ =

(β/2π)[ϕ(∞)−ϕ(−∞)]. In the quantum theoryQ becomesa topological quantum number [1].

[10] L. D. Faddeev and V. E. Korepin, “Quantum theory ofsolitons,” Phys. Rep. 42, 1–87 (1978).

[11] E. K. Sklyanin, L. A. Takhtajan, and L. D. Faddeev,“The Quantum Inverse Problem Method. 1,” Theor. Math.Phys. 40, 688–706 (1980), [Teor. Mat. Fiz.40,194(1979)].

[12] A. B. Zamolodchikov, “Exact two-particle S-matrix ofquantum sine-Gordon solitons,” Commun. Math. Phys.55, 183–186 (1977).

[13] A. B. Zamolodchikov and Al. B. Zamolodchikov, “Factor-ized S-matrices in two dimensions as the exact solutionsof certain relativistic quantum field theory models,” Ann.Phys. 120, 253–291 (1979).

[14] F. A. Smirnov, Form Factors in Completely IntegrableModels of Quantum Field Theory, Advanced series inmathematical physics, v. 14 (World Scientific, 1992).

[15] S. Lukyanov and A. B. Zamolodchikov, “Exact expec-tation values of local fields in the quantum sine-Gordonmodel,” Nucl. Phys. B 493, 571 – 587 (1997).

[16] For a =√K the expectation value 〈ei

√8πaϕ〉 is simply

related to the free energy which is exactly known. ForK = 1/2 the exact value is deduced from the mappingonto the massless Thirring model [15].

5

[17] V. Fateev, S. Lukyanov, A. B. Zamolodchikov, and Al. B.Zamolodchikov, “Expectation values of boundary fields inthe boundary sine-Gordon model,” Phys. Lett. B 406, 83–88 (1997).

[18] V. Fateev, S. Lukyanov, A. B. Zamolodchikov, and Al. B.Zamolodchikov, “Expectation values of local fields in theBullough-Dodd model and integrable perturbed conformalfield theories,” Nucl. Phys. B 516, 652–674 (1998).

[19] R. H. Poghossian, “Perturbation theory in angular quan-tization approach and the expectation values of exponen-tial fields in sine-Gordon model,” Nucl. Phys. B 570, 506–522 (2000).

[20] V. V. Mkhitaryan, R. H. Poghossian, and T. A. Se-drakyan, “Perturbation theory in the radial quantizationapproach and the expectation values of exponential fieldsin the sine-Gordon model,” J. Phys. A 33, 3335–3346(2000).

[21] Z. Bajnok, L. Palla, G. Takács, and F. Wágner, “Thek-folded sine-Gordon model in finite volume,” Nucl. Phys.B 587, 585–618 (2000).

[22] W.-F. Lu, “Sine-Gordon expectation values of exponen-tial fields with variational perturbation theory,” Phys.Lett. B 602, 261–268 (2004).

[23] The perturbative renormalization group is a popularmethod to study the sine-Gordon model [2–4]. In the mas-sive phase however, perturbative scaling equations lead toan unphysical divergence of the coupling constant u. Thephase diagram can be determined but the understandingof the physical properties of the massive phase is out ofreach of the perturbative approach.

[24] Nonperturbative flow equations have been obtained fromcontinuous unitary transformations but the determinationof the the spectrum (soliton and breather’s masses) andthe validity of the Lukyanov-Zamolodchikov conjecturehave not been addressed. See S. Kehrein, “Flow Equa-tion Solution for the Weak- to Strong-Coupling Crossoverin the Sine-Gordon Model,” Phys. Rev. Lett. 83, 4914–4917 (1999); “Flow equation approach to the sine-Gordonmodel,” Nucl. Phys. B 592, 512 – 562 (2001).

[25] S. Nagy, I. Nándori, J. Polonyi, and K. Sailer, “Func-tional Renormalization Group Approach to the Sine-Gordon Model,” Phys. Rev. Lett. 102, 241603 (2009).

[26] V. Pangon, “Structure of the broken phase of the sine-Gordon model using functional renormalisation,” Int. J.Mod. Phys. A 27, 1250014 (2012).

[27] V. Pangon, “Generating the mass gap of the sine-Gordonmodel,” (2011), arXiv:1111.6425.

[28] V. Bacsó, N. Defenu, A. Trombettoni, and I. Nándori,“c-function and central charge of the sine-Gordon modelfrom the non-perturbative renormalization group flow ,”Nucl. Phys. B 901, 444 – 460 (2015).

[29] For reviews on the nonperturbative functional renormal-ization group, see J. Berges, N. Tetradis, and C. Wet-terich, “Non-perturbative renormalization flow in quan-tum field theory and statistical physics,” Phys. Rep. 363,223 (2002); B. Delamotte, “An Introduction to the Non-perturbative Renormalization Group,” in RenormalizationGroup and Effective Field Theory Approaches to Many-Body Systems, Lecture Notes in Physics, Vol. 852, editedby A. Schwenk and J. Polonyi (Springer Berlin Heidel-berg, 2012) pp. 49–132; P. Kopietz, L. Bartosch, andF. Schütz, Introduction to the Functional RenormalizationGroup (Springer, Berlin, 2010).

[30] See Supplemental Material.

[31] Note that we do not rescale the field in order to preservethe 2π/β periodicity of Zk(φ) and Uk(φ).

[32] Al. B. Zamolodchikov, “Mass scale in the sine-Gordonmodel and its reductions,” Int. J. Mod. Phys. A 10, 1125–1150 (1995).

[33] Near K = 1 and u = 0 the behavior of the system iscontrolled by the BKT fixed point, which explains whythe conjecture cannot be valid arbitrary close to K = 1.

[34] P. B. Wiegmann, “One-dimensional Fermi system andplane XY model,” J. Phys. C. 11, 1583–1598 (1978).

[35] D. J. Amit, Y. Y. Goldschmidt, and S. Grinstein,“Renormalisation group analysis of the phase transition inthe 2D Coulomb gas, Sine-Gordon theory and XY-model,”J. Phys. A 13, 585 (1980).

[36] M. Gräter and C. Wetterich, “Kosterlitz-Thouless PhaseTransition in the Two Dimensional Linear σ Model,” Phys.Rev. Lett. 75, 378–381 (1995).

[37] G. v. Gersdorff and C. Wetterich, “Nonperturba-tive renormalization flow and essential scaling for theKosterlitz-Thouless transition,” Phys. Rev. B 64, 054513(2001).

[38] P. Jakubczyk, N. Dupuis, and B. Delamotte, “Reexam-ination of the nonperturbative renormalization-group ap-proach to the Kosterlitz-Thouless transition,” Phys. Rev.E 90, 062105 (2014).

6

Supplemental MaterialWe consider the sine-Gordon model defined by the Eu-

clidean action

S[ϕ] =

ˆd2r

{1

2(∇ϕ)2 − u cos(βϕ)

}, (1)

where regularization with a UV momentum cutoff Λ isimplied and u/Λ2, β > 0 are dimensionless parameters.In the following, instead of the parameter β, we will oftenuse the Luttinger parameter K = β2/8π.

I. EXACT FLOW EQUATION

To implement the functional renormalization-group(FRG) approach we add to the action the infrared regu-lator term

∆Sk[ϕ] =1

2

∑q

ϕ(−q)Rk(q)ϕ(q) (2)

such that fluctuations are smoothly taken into accountas k is lowered from the microscopic scale kin down to0 [1–3]. The regulator function in (2) is defined by

Rk(q) = Zkq2r

(q2

k2

), (3)

where the function r(y) satisfies r(0) =∞ and r(∞) = 0,and Zk is a field renormalization factor defined below.Thus ∆Sk suppresses fluctuations with momenta |q| � kbut leaves unaffected those with |q| � k. Various choicesfor r(y) are discussed below.

The partition function

Zk[J ] =

ˆD[ϕ] e−S[ϕ]−∆Sk[ϕ]+

´d2r Jϕ (4)

is k dependent. The scale-dependent effective action

Γk[φ] = − lnZk[J ] +

ˆd2r Jφ−∆Sk[φ] (5)

is defined as a modified Legendre transform of − lnZk[J ]which includes the subtraction of ∆Sk[φ]. Here φ(r) =〈ϕ(r)〉 is the order parameter (in the presence of the ex-ternal source J). Provided that the initial value kin ofk is sufficiently large wrt the UV momentum cutoff Λof the sine-Gordon model, all fluctuations are completelyfrozen by ∆Skin [ϕ] and Γkin [φ] = S[φ]. On the other handthe effective action of the sine-Gordon model is given byΓk=0 since ∆Sk=0 = 0. The FRG approach aims at de-termining Γk=0 from ΓΛ using Wetterich’s equation [4–6]

∂kΓk[φ] =1

2Tr{∂kRk

(Γ

(2)k [φ] +Rk

)−1}, (6)

where Γ(2)k [φ] denotes the second-order functional deriva-

tive of Γk[φ].

II. DERIVATIVE EXPANSION

To solve the exact flow equation (6) we use a derivativeexpansion of the scale-dependent effective action. Suchan expansion is made possible by the regulator term ∆Sk,which ensures that all vertices Γ

(n)k (q1 · · ·qn) are smooth

functions of momenta qi and can be expanded in powersof q2

i /max(k,M)2 when |qi| � max(k,M) with M thesmallest mass in the spectrum. Thus the derivative ex-pansion is sufficient to compute the soliton and breathermasses in the massive phase as well as the expectationvalue of the exponential fields.

To second order of the derivative expansion, the effec-tive action

Γk[φ] =

ˆd2r

{1

2Zk(φ)(∇φ)2 + Uk(φ)

}(7)

is fully determined by two functions of the field, Zk(φ)and Uk(φ), which are periodic on the interval ] −π/β, π/β]. In practice we consider the dimensionlessfunctions

Uk(φ) =Uk(φ)

Zkk2, Zk(φ) =

Zk(φ)

Zk, (8)

where Zk is defined from the condition

β

2π

ˆ π/β

−π/βdφ Zk(φ) = 1. (9)

The flow equations for the functions Uk(φ) and Zk(φ),obtained by inserting (7) into (6), are given in Appendix .

III. CHOICE OF THE REGULATOR FUNCTIONRk

The initial value of the running momentum scale mustsatisfy kin � Λ in order to ensure that all fluctuationsare frozen and Γkin [φ] = S[φ]. In practice we take kin ∼100Λ. Figure 5 shows the mass M obtained from FRGwith different regulator functions:

r(y) =

α

ey − 1

α(1− y)2

yΘ(1− y)

αe−y

y(1 + γy)

(10a)

(10b)

(10c)

(α, γ are free parameters). In the range K ≤ 0.4, theaccuracy of the FRG result is better than 2% regardlessof the choice of the regulator. For K > 0.4 it deterioratesand the results become strongly dependent on the regu-lator. Note however that although the relative error canbe larger than 100%, the absolute error remains smallsince the mass M goes to zero as K → 1.

7

0.0 0.2 0.4 0.6 0.8K

0.00

0.01

0.02

0.03

0.04M/Λ

(10a) α= 4(10a) α= 20(10b) α= 4(10b) α= 20

M1

2Msol

0.25 0.50 0.75K

10-3

10-2

10-1

100

101

Figure 5. Mass M as obtained from the FRG approach withthe regulator functions (10a) (+) and (10b) (×) and α = 4.The solid and dashed lines show the exact values of 2Msol

and M1 (the latter being defined only for K ≤ 1/2). Theinset shows the relative error of the FRG result for the sameregulator functions and α = 4 or α = 20. The regulator func-tion (10c) gives results (not shown) similar to (10a). (Hereand in the following figures: Λ = 1 and u/Λ2 = 10−3.)

IV. THE NONUNIVERSAL SCALE FACTOR b

In a theory with a UV momentum cutoff Λ the solitonmass is given by [7]

Msol =2Γ(

K2−2K

)√πΓ(

12−2K

) [Γ(1−K)

Γ(K)

π

2uΛ−2KR

] 12−2K

. (11)

The value of ΛR (which depends on Λ) can be deter-mined by considering the one-loop correction to the massM1,cl =

√8πKu of the lightest breather in the semiclas-

sical limit K → 0 [7], i.e.

M21

M21,cl

= 1 + 2K ln

(M1,cle

C

2ΛR

), (12)

where C is the Euler constant. An elementary calculationbased on the action (1) gives [1]

M21

M21,cl

= 1− 2K

ˆ ∞0

dqq

q2 +M21,cl

fΛ(q), (13)

where fΛ(q) is a momentum cutoff function (e.g. fΛ(q) =Θ(Λ−q) for a hard cutoff). By comparing (12) and (13),we find that Λ and ΛR are related by

ln

(2ΛRe

−C

M1,cl

)=

ˆ ∞0

dqq

q2 +M21,cl

fΛ(q). (14)

For M1,cl � Λ, the rhs gives ln(Λ/M1,cl) + const andEq. (14) allows us to determine the scale factor b = ΛR/Λ

0.0 0.2 0.4 0.6 0.8K

10-7

10-6

10-5

10-4

10-3

10-2

10-1

|Mi/M

0(bi/b 0

)K/(

1−K

)−

1| f1(q) = exp(−q2/Λ2)f2(q) = exp(−q4/Λ4)

Figure 6. |(Mi/M0)(bi/b0)K/(1−K) − 1| for fixed values of Λand u/Λ2 and two different cutoff functions, f1(q) and f2(q).M0 is the mass obtained with the hard cutoff fΛ(q) = Θ(Λ−q). The scale parameter b = ΛR/Λ is obtained from (14).

for a given cutoff function fΛ(q) (for a hard cutoff b =eC/2). The soliton mass in the model defined with amomentum cutoff Λ can therefore be written as

Msol = bΛ2Γ(

K2−2K

)√πΓ(

12−2K

) [Γ(1−K)

Γ(K)

πu

2(bΛ)2

] 12−2K

, (15)

where b = ΛR/Λ is obtained from (14).

V. UNIVERSALITY

Equation (15) implies universality in the sense thatMsol/Λ is a universal function of K and u/Λ2 up to thescale factor b. This universality is due to the flow beingcontrolled by the Gaussian fixed point u = 0 for Msol �k � Λ. In this momentum range all correlation func-tions exhibit the same scaling, e.g. 〈eiβϕ(r)e−iβϕ(r′)〉 ∼(|r − r′|Λ)−4K , up to a nonuniversal prefactor that de-pends on the cutoff function fΛ(q). Various theories, cor-responding to different implementations of the cutoff (or,equivalently, different normalizations of the correlationfunctions), are simply related by the scale factor b.

In Fig. 6 we show the ratio

Mi

M0

(bib0

) K1−K

, (16)

where M0 is the mass obtained with the hard cutofffΛ(q) = Θ(Λ − q) and Mi that obtained with the cutofff1(q) = e−q

2/Λ2

or f2(q) = e−q4/Λ4

. The scale parameterb = ΛR/Λ is obtained from (14). Universality impliesthat the ratio (16) is equal to one. This property is sat-isfied with a very high accuracy (the ratio differs fromone by less than 10−3) in the range K ≤ 0.4 where themass M is obtained with high precision. Not surpris-ingly, deviations from universality are more pronouncedforK > 0.4 due to the lesser accuracy of the FRG results.

8

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35K

0.00

0.01

0.02

0.03

0.04

0.05

0.06

M1

M1

M1, cl

M1, corr

M (FRG)

Figure 7. Mass M1 of the lightest breather for K � 1.The FRG result is compared to the leading-order (semi-classical) value M1,cl =

√8πKu and the next-to-leading-

order value M1,corr given by (12) as well as the exact resultM1 = 2Msol sin

(πK

2(1−K)

)given by (15).

VI. SEMICLASSICAL LIMIT

In the limit K � 1, the leading correction to the massM1,cl of the lightest breather is given by Eq. (12) withΛR = bΛ. When K → 0 the partition function can becomputed by a saddle-point approximation and the ini-tial value Γkin [φ] = S[φ] of the effective action gives theexact result M1,cl =

√8πKu (the flow does not bring

any correction to Γkin [φ] = S[φ] when K → 0). In Fig. 7we compare the FRG result for M1 with the exact value,the semiclassical limit K → 0 and the perturbative re-sult (12). For K . 0.1, the agreement between FRG andthe exact solution is nearly perfect.

VII. EXPECTATION VALUE OF EXPONENTIALFIELDS

To compute the expectation value of the exponentialfields, we add to the action the source term

Sh[ϕ] = −ˆd2r(h∗ein

√2πKϕ + c.c.) (17)

so that 〈ein√

2πKϕ(r)〉 can be obtained from lnZk[J, h∗, h]by functional differentiation wrt h∗(r). The scale-dependent effective action reads

Γk[φ, h∗, h] = − lnZk[J, h∗, h]+

ˆd2r Jφ−∆Sk[φ] (18)

and satisfies the flow equation

∂tΓk[φ, h∗, h] =1

2Tr{∂tRk

(Γ(2)[φ, h∗, h]+Rk

)−1}

(19)

with initial conditions Γkin [φ, h∗, h] = S[φ] + Sh[φ].We consider the ansatz

Γk[φ, h∗, h] =

ˆd2r

{1

2Zk(φ, h∗, h)(∇φ)2

2 1 0 1 2

(8K/π)12 φ

200

0

200Z

(10)k (φ)

|t|= 0|t|= 8.7|t|= 9

2 1 0 1 2

(8K/π)12 φ

1

0

1U

(10)k (φ)

|t|= 0|t|= 6.5|t|= 10

Figure 8. Z(10)k (φ) and U

(10)k (φ) for various values of t =

ln(k/kin) and n = 1.

+ Uk(φ, h∗, h)

}. (20)

The expectation value of interest reads

〈ein√

2πKϕ(r)〉 = −∂Uk(φ, h∗, h)

∂h∗

∣∣∣∣h∗=h=0φ=0

≡ −U (10)k (φ = 0),

(21)noting that φ = 0 corresponds to the minimum of thepotential U(φ, h∗, h) when h∗ = h = 0. The RG equationfor U (10)

k (φ) involves Z(10)k (φ) ≡ ∂h∗Zk(φ, h∗, h)|h∗=h=0.

In practice we consider the dimensionless functions

U(10)k (φ) =

U(10)k (φ)

Zkk2, Z

(10)k (φ) =

Z(10)k (φ)

Zk. (22)

The flow equations satisfied by U (10)k (φ) and Z(10)

k (φ) aregiven in Appendix . The results obtained in the massivephase are shown in Fig. 8 for n = 1.

9

APPENDIX A: FLOW EQUATIONS

The flow equations read

∂tUk = (ηk − 2)Uk +1

4πZkl20,0, (23)

∂tZk = ηkZk −1

4πZk

[2U ′′′k l

60,2Z

′k − 2U ′′′k l

22,0Z

′k + U ′′′k

2l40,2 + l21,0Z′′k + l80,2Z

′k

2 − 5

2l42,0Z

′k

2]

(24)

and

∂tU(10)k = (ηk − 2)U

(10)k − 1

4πZk

[l21,0U

(10)k′′ + l41,0Z

(10)k

], (25)

∂tZ(10)k = ηkZ

(10)k − 1

4πZk

{2l60,2

[U

(10)k′′′Z ′k + U ′′′k Z

(10)k′]

+ 2U(10)k′′′U ′′′k l

40,2 − 8U ′′′k l

61,2U

(10)k′′Z ′k

+ 4U ′′′k l23,0U

(10)k′′Z ′k − 4U ′′′k

2l41,2U(10)k′′ − 4l81,2U

(10)k′′Z ′k

2 + 5l43,0U(10)k′′Z ′k

2 − l22,0[2U

(10)k′′′Z ′k

+ U(10)k′′Z ′′k + 2U ′′′k Z

(10)k′]− 8U ′′′k Z

(10)k l81,2Z

′k + 2U ′′′k Z

(10)k l62,1Z

′k + 8U ′′′k Z

(10)k l43,0Z

′k − 4U ′′′k

2Z(10)k l61,2

+ U ′′′k2Z

(10)k l42,1 + U ′′′k

2Z(10)k l23,0 + l21,0Z

(10)k′′ + 2l80,2Z

(10)k′Z ′k − l42,0

[5Z

(10)k′Z ′k + Z

(10)k Z ′′k

]− 4Z

(10)k l10

1,2Z′k

2 + Z(10)k l82,1Z

′k

2 + 8Z(10)k l63,0Z

′k

2

}, (26)

where t = ln(k/kin) and ηk = −∂t lnZk. The equation for ηk is simply derived from (9). To alleviate the notationswe do not write explicitly the φ dependence of Uk(φ), Zk(φ), U (10)

k (φ) and Z(10)k (φ). The initial conditions are

Ukin(φ) = −(u/k2in) cos(βφ), Zkin(φ) = 1, U

(10)kin

(φ) = −ein√

2πKφ/k2in, Z

(10)kin

(φ) = 0 (27)

and Zkin = 1. The threshold function ldn,m is defined in Appendix .

Weak-coupling limit

The perturbative flow equations are recovered by retaining a single harmonic of the potential, i.e. Uk(φ) =−uk cos(βφ), and neglecting the flow of Zk(φ). Using

l20,0(U ′′, 1, 0) = l20,0(0, 1, 0)− l21,0(0, 1, 0)U ′′ +O(U ′′2), (28)

where l21,0(0, 1, 0) = 1 is universal (i.e. independent of the regulator function r), one obtains

∂tuk = −2uk(1−Kk) +O(u2k),

∂tKk = u2k(8πK2)2l40,2(0, 1, 0) +O(u3

k),(29)

where Kk = K/Zk is a “running” Luttinger parameter. Sufficiently far away from the BKT point (Kk = 1, uk = 0),one can neglect the renormalization of Kk when uk → 0 and the flow equations become ∂tuk = −2uk(1 − K) toleading order. For K < 1, uk grows as k2K−2. This is the massive phase studied in the Letter. When 1−Kk and ukare of the same order, both equations in (29) must be solved simultaneously and one recovers the scaling behavior ofthe BKT phase transition [9].

APPENDIX B: THRESHOLD FUNCTIONS

The threshold function ldn1,n2≡ ldn1,n2

(U ′′k , Zk, ηk) is defined by

ld0,0 =

ˆ ∞0

dq qd−1 RkZkk2

G,

ldn1,n2= −∂t

ˆ ∞0

dq qd−1Gn1G′n2 if (n1, n2) 6= (0, 0),

(30)

10

where

G = [(Zk + r)y + U ′′k ]−1, ∂tG = − RkZkk2

G2,

G′ = −G2[Zk + r + yr′], ∂tG′ = 2

RkZkk2

G3[Zk + r + yr′]− R′kZk

G2,

(31)

and

Rk = Zkk2yr, Rk = −Zkk2y(ηkr + 2yr′),

R′k = Zk(r + yr′), R′k = −Zk[ηk(r + yr′) + 2y(2r′ + yr′′)],(32)

with y = q2 = q2/k2, r ≡ r(y), r′ ≡ r′(y), etc. The prime denotes a derivative wrt y, the dot a derivative wrt t and∂t acts only on the k dependence of Rk.

[1] J. Berges, N. Tetradis, and C. Wetterich, “Non-perturbative renormalization flow in quantum field theoryand statistical physics,” Phys. Rep. 363, 223 (2002).

[2] B. Delamotte, “An Introduction to the NonperturbativeRenormalization Group,” in Renormalization Group andEffective Field Theory Approaches to Many-Body Systems,Lecture Notes in Physics, Vol. 852, edited by A. Schwenkand J. Polonyi (Springer Berlin Heidelberg, 2012) pp. 49–132.

[3] P. Kopietz, L. Bartosch, and F. Schütz, Introduction tothe Functional Renormalization Group (Springer, Berlin,2010).

[4] C. Wetterich, “Exact evolution equation for the effectivepotential,” Phys. Lett. B 301, 90 (1993).

[5] Ulrich Ellwanger, “Flow equations for n point functionsand bound states,” Z. Phys. C 62, 503–510 (1994).

[6] T. R. Morris, “The exact renormalization group and ap-proximate solutions,” Int. J. Mod. Phys. A 09, 2411–2449(1994).

[7] Al. B. Zamolodchikov, “Mass scale in the sine-Gordonmodel and its reductions,” Int. J. Mod. Phys. A 10, 1125–1150 (1995).

[8] R. Rajaraman, Solitons and instantons (North-Holland,Amsterdam, 1989).

[9] See, e.g., P. M. Chaikin and T. C. Lubensky, Principles ofCondensed Matter Physics (Cambridge University Press,1995).