Persistent current in a thin superconducting wire
PPersistent current in a thin superconducting wire
Ilya Vilkoviskiyemail: [email protected] Institute of Physics and Technology, 141700 Dolgoprudny, RussiaI.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute,Leninsky ave. 53, 119991 Moscow, RussiaCenter for Advanced Studies, Skolkovo Institute of Science and Technology,143026 Moscow, RussiaJuly 10, 2018
Abstract
In this paper, we explore the persistent current in thin superconducting wiresand accurately examine the effects of the phase slips on that current. The mainresult of the paper is the formula for persistent current in terms of the solutionsof certain (nonlinear) integral equation. This equation allows to find asymptoticsof the current at long(small) length of the wire, in that paper, we interested inthe region in which the system becomes strongly interacting and very few amountsof information can be extracted by perturbation theory. Nevertheless, due to theintegrability, exact results for the current can be obtained. We observe that at thelimit of a long wire, the current becomes exponentially small, we believe that it isthe signal that phase slips may destroy superconductivity for long wires, below BKTphase transition. a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l ontents α and interacting fermions . . . . . . . . . . . . . . . . . . . . . . 13B From Thirring to XXZ spin chain 13C From XXZ Bethe Ansatz to DDV equation 14D From DDV to asymptotics of the current in the limit of big(small) radius ofthe wire 15D.1 Large M L limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16D.2 Small
M L limit (conformal regime) . . . . . . . . . . . . . . . . . . . . . . 16References 172
Introduction
The quantum effects in strongly interacting systems is a very interesting theme. It isknown, that thin superconducting wires may lost two general features of superconductors: zero resistivity and existence of dissipationless (persistent) current in an externalmagnetic flux. This effects was observed at the whole range of temperatures below thecritical T c , even at T → x, t ) = | ∆( x, t ) | e iφ ( x,t ) , note that as φ is a phase, it is defined only modulo2 π : φ ∼ φ + 2 π . If we look at the two dimensional Euclidean plane ( x, τ = it ) , thephase slip could be thought as the following field φ configuration φ ( x, τ ) = arctan( xτ ) (1)While | ∆ | is nontrivial only in a small region near the point x = 0 , τ = 0 : | ∆ | ∼ const for x + τ (cid:29) τ = 0 for small τ , theylooks like a step function, and due to the fact that φ is a phase very close to the config-urations φ = const . In this paper, we examine the phase and wire length dependence ofFigure 1: BKT phase diagram, m - is a phase slips rate. α ∼ √ S , where S is the wirecross sectionthe persistent current at zero temperature, in order to separate the quantum and ther-mal fluctuations. It is known that under varying the thickness of the wire the systemundergoes a BKT phase transition see fig 1. The phase α > α < φ ( x, t ) → φ ( t ) [2].Instead in the insulating phase, system is asymptotically free at the small distances andstrong interacting at large ones, physically it means that we expect that current will havesimple sawtooth-like behaviour for the rings of small circumference and experience someunknown behaviour at the limit of large (compared to the only dimensional parameterof the system - the phase slips rate) circumference. This fact makes the insulating regioninteresting both for theoretical and experimental physicists.It is very hard to analyze the current in the whole range of lengths because there isno any simple model for wires of a large circumference, instead, one can use underlyingintegrability in order to derive an exact equation for the current, which then can besolved numerically or analysed exactly in the limits of large or small wires, see appendixD for details.The paper is organized as follows: at the section 2 we discuss the physical system andmake a review of the effective model for 1-dimensional superconducting rings. At thesection 3 we argue what interacting terms could be added to the Lagrangian, and arguethat all physics is dominated by the one term - which bring us to the Sine-Gordon theory.At the section 3 we present the results of the paper. The ground state energy and thecurrent could be written in terms of solutions of Destri-De Vega equation (DDV) , (25),the details explained in the Appendix. In the Appendix A we review dualities betweenbosons and fermions and relations between their statistical sums (54) .In Appendix B wesketchy explain the relations between XXZ spin chains and Thirring model. Relationsbetween XXZ Bethe Ansatz (59) and nonlinear equations (69) which we finally use asthe main computational tool explained in the Appendix C. Finally in the Appendix D ,we show how to calculate persistent current , using the DDV equation. We consider a superconducting ring of the circumference L, at the limit of small tem-perature T →
0. Low energy physics of the wire is described by the order parameter∆ = | ∆( x, t ) | e iφ ( x,t ) , moreover as fluctuations of the | ∆( x, t ) | described by a massivefield we assume that at low energies | ∆( x, t ) | is an independent of { x, t } constant.Effective theory described be the Lagrangian , see [1] for details: L = α π (cid:18)
12 ( v∂ t φ ( x, t )) −
12 ( v − ∂ x φ ( x, t )) (cid:19) + m V slips (2)4igure 2: Superconducting ring of the circumference L, in the external magnetic fluxAs was argued in [1] : for the standard dirty metals the only dimensionless parameterwhich control the physics of the system is α π = e (cid:113) C w L kin , v = πσSC w where C w -is thegeometric wire capacitance per length and L kin = πσ N S - is the kinetic wire inductance , σ N - is the normal state Drude conductance of the wire and S is the wire cross section.The term M V slips is introduced to take phase slips into account, and we will inves-tigate it carefully below. As usual , the current is the gradient of an order parameterphase J = σ e ∂ x φ , and the charge density ρ = V C w = − Ce ∂ t φ , so that the continuityequation ∂ t ρ + ∂ x J = 0 turns to an equation of motion for the field φv ∂ t φ − ∂ x φ = 0 (3)In order to couple the external electro-magnetic field we should analyze the transforma-tion law of field φ under the gauge transformations A µ → A µ + ∂ µ f (4)∆ → ∆ e if (5)The gauge invariant Lagrangian is L = α π (cid:18)
12 ( ∂ t φ ( x, t ) − A t ) −
12 ( ∂ x φ ( x, t ) − A x ) (cid:19) (6)In order to introduce external magnetic flux, one can choose A µ in the form A t = 0 (7) A x = Φ Lα (8)5ere we choose the normalization of the flux Φ in such a way that it is 2 π periodic:statistical sum is periodic function of Φ with the period 2 π .To the end of this section, let us note that velocity in the Lagrangian [6] can be scaledout by redefinition of coordinate x → vx and time t → vt , later, for simplicity, we willuse this convention. We will study the Sine-Gordon theory S = β ∼∞ (cid:90) L (cid:90) [ 12 πα (cid:18)
12 ( ∂ t χ ( x, t )) −
12 ( ∂ x χ ( x, t )) + m cos( χ ( x, t )) (cid:19) − Φ2 πL ∂ t χ ( x, t )] dxdt (9)There are several ways to arrive to this effective theory, which takes phase slips intoaccount, in that paper we try to write it down from general principles, but it also can bederived as an effective theory of phase slips - which are vortices in the imaginary timeformalism [1] .Starting from the free theory, it is natural to ask in which ways this theory can bedeformed to an interacting one and what is the physical meaning of that deformation.As φ is a phase, it is natural to consider following operators V m ( x, t ) = e i ( mφ ( x,t ) , m ∈ Z (10)This operators has fixed conformal dimension ∆ = m α , and mutual local to each other,that means that the product V m ( x, t ) V m ( y, τ ) = (( x − y ) + ( t − τ ) ) m m α : V m ( x, t ) V m ( y, τ ) : (11)is single valued operator in the complex plane : z = x + it , ¯ z = x − it , here we assumethat : V m ( x, t ) V m ( y, τ ) : means the normal ordered product, which is meromorphic in z , ¯ z . It is possible to enlarge the algebra of local operators, introducing the dual field ∂ x χ ( x, t ) = ∂ t φ ( x, t ) (12) ∂ t χ ( x, t ) = − ∂ x φ ( x, t ) (13)Consider the following operators [11] : V m,n ( z, ¯ z ) = e i ( mφ ( z, ¯ z )+ nα χ ( z, ¯ z )) (14) V m ,n ( z, ¯ z ) V m ,n (0 ,
0) = | z | ( m m α + n n α ) z m n n m ¯ z − m n n m : V m ,n ( z, ¯ z ) V m ,n (0 ,
0) :(15)Which are mutual local to each other at the integers n, it is imply that χ is compactifyedon a circle of radius α , χ ∼ χ + πα
6e also want our Lagrangian to be invariant under the global gauge transformations: φ → φ + f, χ → χ , general Lagrangian satisfying this restrictions contains only operators V ,n , obviously it can be written down in terms of field χ only (also it is useful to makea little redefinition χ → α χ : L = 12 πα (cid:32)
12 ( ∂ t χ ( x, t )) + 12 ( ∂ x χ ( x, t )) + (cid:88) n m n cos( nχ ( x, t )) (cid:33) (16)Note that operators e ± i ( nχ ( z, ¯ z ) can be identified with operators of creation/annihilation ofvortexes (phase slips) with vorticity n, see [2] for more details and physical interpretation.Here the masses m n is the dimensional parameters, with the dimension[ m n ] = 2(1 − ( nα ) m n m scales tozero as L (1 − n ) α , so, it is natural to leave only first of them. This is a standard way toget a Sine-Gordon Lagrangian. L = 12 πα (cid:18)
12 ( ∂ t χ ( x, t )) + 12 ( ∂ x χ ( x, t )) + M cos( χ ( x, t )) (cid:19) (18)It is important to note that this theory have two qualitatively different regimes : α > , α < α < T = β → S = β ∼∞ (cid:90) L (cid:90) [ 12 πα (cid:18)
12 ( ∂ t χ ( x, t )) −
12 ( ∂ x χ ( x, t )) + m cos( χ ( x, t )) (cid:19) − Φ2 πL ∂ t χ ( x, t )] dxdt (19) J = L πβ ∂ log( Z (Φ)) ∂ Φ = L π ∂E vac (Φ) ∂ Φ (20)The best way to solve this problem was provided by Destri and De-Vega [5]. Below, Ireview their results, and apply them to our problem:7
Results
Our aim is to compute statistical sum, ground state energy and current at the limit ofzero temperature, here and later we will work in the imaginary time formalism: Z = (cid:90) Dφ e − S cl [ φ ] ∼ e − βE vac (21)Where S cl is given by the formula (19) J = L πβ ∂ log( Z (Φ)) ∂ Φ = L π ∂E vac (Φ) ∂ Φ (22)Classical result (at m = 0) tells that E vac (Φ) = − π L (1 − α π ) (23) J (Φ) = α Φ4 π (24)For the case m (cid:54) = 0, we will write down an intregral equation for a ground state energy(69) , [6], see appendix for the details. (cid:15) ( θ ) = − M L cosh( θ ) − i Φ − G (cid:63) log(1 + e (cid:15) ) + G c (cid:63) log(1 + e ¯ (cid:15) ) (25)Where (cid:63) is a convolution, and G c , G are known functions (68) G ( θ ) = ∞ (cid:90) dk π sinh( πk ( π γ − πk ( πγ − πk ) cos( kθ ) (26) G c ( λ ) = G ( λ + iπ ) (27) E = − M ∞ (cid:90) −∞ cosh( θ ) (cid:0) log(1 + e (cid:15) ) + log(1 + e ¯ (cid:15) ) (cid:1) dθ π (28)It is simple to get from this equation that in the limit of large M L → ∞ E = − M (cid:18) π (cid:19) / e − ( ML − π ) cos(Φ) + o ( e − ML ) (29) J = M L √ π e − ( ML − π ) sin(Φ) + o ( e − ML ) (30)Here M is the soliton mass, it is related to the bare mass m through the formula [9] M = πm Γ (cid:16) − α (cid:17) Γ (cid:16) α (cid:17) α − (cid:16) α − α ) (cid:17) √ π Γ (cid:16) − α (cid:17) (31)Higher corrections could be also computed order by order. The technique allows oneto compute the current numerically, here is the plots of the current J (Φ), for differentvalues of M L
M L , Current J (Φ) - interpolates between Sawtoothand Sine In this paper we started investigation of the so-called ‘insulating’ region of thin su-perconducting nanowires. Following the paper of A.Semenov and A.Zaikin we usedSine-Gordon theory as an effective low energy theory for our system. In order to exploresuperconducting properties of the ring we computed the simplest quantity of supercon-ductors : the persistent current dependence on the external magnetic flux and length ofthe ring. We found that for the ring of large lengths the current is exponentially sup-presed J ∼ e − ML . This result can not be obtained in pertrubation theory, instead weused integrability of the theory in order to find asymptotics of the current exactly. Ourresults agrees with the results of [1] , which was obtained with the help of renorm groupargument, which are not very confident at the regime of strong coupling. Although,our exact computation shows that heuristic argument of renorm group leads to a qual-itatively correct results. Also our method reproduce exponential and pre-exponentialfactors of current dependence as well as systematical computation of higher order ofpertrubation series in ( e − ML ).Though the current is suppressed in the bulk of superconductor, we can’t confi-dently claim that superconductivity is broken, for example, it would be interesting tocompute resistivity of the wire connected to the resistor, which is yet a nontrivial prob-lem. We believe that it will be very interesting to discover the region near, and below,the BKT phase transition experimentally. Not only because it may have applicationsin nanoelectronics, but because it can boost our understanding of Sine-Gordon theory. Acknowledgements
I thank Andrew G. Semenov for introducing me to the theme, and for interesting dis-cussions during the work. I also thank Andey Marshakov for permanent support and his9nterest to this study. Finally I would like to thank Andrei D. Zaikin for his commentson earlier versions of the paper. 10
Free field theory and boson-fermion correspondence, from Sine-Gordonto Thirring
In this section, we rigorously show that the theory of one bosonic field φ ( x, t ) - compact-ified on a circle : φ ( x, t ) ∼ φ ( x, t ) + 2 π is equivalent to a theory of Dirac fermions.Consider , for the moment pure Sine Gordon theory: S = β (cid:90) L (cid:90) πα (cid:18)
12 ( ∂ t χ ( x, t )) −
12 ( ∂ x χ ( x, t )) + m cos( χ ( x, t )) (cid:19) dxdt (32)Field φ defined to take values in R , but (at least at the classical level) we should specifya boundary conditions, for the applications to a superconductors we should consider allfields of the form : χ ( x + L, t ) = χ ( x, t ) + 2 πn (33) χ ( x, t + β ) = χ ( x, t ) + 2 πm (34)The best way to deal with different sectors is to introduce chemical potentials Φ , and µ in the action S m,n = β (cid:90) L (cid:90) πα (cid:18)
12 ( ∂ t χ ( x, t )) −
12 ( ∂ x χ ( x, t )) + m cos( χ ( x, t )) (cid:19) dxdt + m Φ + nµ (35) Z (Φ , µ ) = (cid:88) m,n ∈ Z Z m,n = (cid:88) m,n ∈ Z (cid:90) m,n Dφe iS m,n (36)Where integration (cid:82) m,n performed under the fields satisfying conditions (33,34). It iseasy to understand that Z (Φ , µ ) = (cid:90) Dφe iS e i Φ2 πL β (cid:82) L (cid:82) ∂ t χ ( x,t ) dxdt + i µ πβ β (cid:82) L (cid:82) ∂ x χ ( x,t ) dxdt (37)Where integration performed in all sectors. Alternatively Z m,n = π (cid:90) π (cid:90) Z (Φ , µ ) e inφ + imµ dµdφ (38)We state that the same statistical sum Z (Φ , µ ) can be obtained from the dual fermionicmodel (Thirring model) with the fields ψ ( x, t ) = (cid:18) ψ L ( x, t ) ψ R ( x, t ) (cid:19) and Hamiltonian H T h = (cid:90) (cid:18) ψ † σ ( − i∂ x + Φ L ) ψ + µβ ψ † ψ + M ψ † σ ψ + 2 gψ † R ψ R ψ † L ψ L (cid:19) dx (39)This statement can be easily proved by direct computation at the free (and massles)fermions point g = M = 0, and than one can follow the Coleman’s standard argumentto understand it holds for any orders of g and M in perturbation theory.11 .1 Free boson S = β (cid:90) L (cid:90) πα (cid:18)
12 ( ∂ t χ ( x, t )) −
12 ( ∂ x χ ( x, t )) + Φ L ∂ t χ ( x, t ) + µβ ∂ x χ ( x, t ) (cid:19) dxdt (40)Classical solutions have the form χ ( x, t ) = 2 πnxL + 2 πmtβ + (cid:88) k ∈ Z / { } χ k ( t ) e πikxL (41) φ k ( t ) just the oscillators, each of them make a contribution equal to a e πik βL − e − πik βL ,and we will get Z α (Φ , µ ) = (cid:89) k (cid:54) =0 e πik βL − e − πik βL (cid:88) m,n e iπ ( βL M α − Lβ n α )+ i Φ n + iµm (42)Or, after Poisson resummation Z α (Φ , µ ) = (cid:89) p (cid:54) =0 e πip βL − e − πip βL (cid:88) m,n e iπ ( βα L M + α βL ( n + Φ2 π ) )+ iµm (43)At the limit of zero temperature ( β → i T → i ∞ ) Z ∼ e − βE vac (Φ) (44) E vac (Φ) = − π L (1 − α π ) (45)Which reproduce classical answer for the current of the wire, in the presence of externalflux J (Φ) = α Φ4 π (46) A.2 Free fermions
The method of Destri and De-Vega is applied to Thirring model, so it is worth to showdirectly that , at least at the free level, fermionic theory reproduce the same answer asbosonic.We begin with the free and massles theory of Dirac fermions, and show that it’s statisticalsum is identical to bosonic one at the point α = √ H ff = (cid:90) (cid:18) ψ † σ ( − i∂ x + Φ L ) ψ + µβ ψ † ψ (cid:19) dx (47) ψ α,r = c α,r e πiL r (48) H ff = (cid:88) r ∈ Z +1 / πL c † R, − r c R,r ( r + Φ2 π + µL πβ ) + (cid:88) s ∈ Z +1 / πL c † L,s c L, − s ( s − Φ2 π + µL πβ ) (49)12 f (Φ , µ ) = tr (cid:16) e iβH ff (cid:17) (50) Z f (Φ , µ ) = (cid:89) p (cid:54) =0 e πip βL − e − πip βL ( (cid:88) m ∈ Z ,n ∈ Z e iπ ( β L m + βL ( n + Φ2 π ) )+ iµm ++ (cid:88) m ∈ Z +1 ,n ∈ Z +1 / e iπ ( β L m + βL ( n + Φ2 π ) )+ iµm ) (51)If we in addition project to a even m (fermion charge) we obtain Z evenf (Φ , µ ) = (cid:89) p (cid:54) =0 e πip βL − e − πip βL (cid:88) m ∈ Z ,n ∈ Z e iπ ( β L m + βL ( n + Φ2 π ) )+ iµm (52)analogically Z oddf (Φ + π, µ ) = (cid:89) p (cid:54) =0 e πip βL − e − πip βL (cid:88) m ∈ Z +1 ,n ∈ Z e iπ ( β L m + βL ( n + Φ2 π ) )+ iµm (53)Finally, we obtain Z bosonα = √ (Φ , µ ) = Z evenf (Φ , µ ) + Z oddf (Φ + π, µ ) (54)Which simply means that we have to impose antiperiodic boundary conditions in evencharge sector and periodic in odd charge sector. Of course, as the bosonic and fermionicstatistical sums are identical, fermionic models reproduce the same answers for theground state energy and current as the bosonic ones (45,46). A.3 General α and interacting fermions We assme that for a general α , formula analogical to (54) holds Z bosonα (Φ , µ ) = Z evenT h,g (Φ , µ ) + Z oddT h,g (Φ + π, µ ) (55)Where Z even/oddT h,g is the statistical sum of the Thirring Hamiltonian with the antiperi-odic/periodic boundary conditions in even/odd fermionic charge sector H T h,g = (cid:90) (cid:18) ψ † σ ( − i∂ x + Φ L ) ψ + µβ ψ † ψ + 2 gψ † R ψ R ψ † L ψ L (cid:19) dx (56) B From Thirring to XXZ spin chain
This section based on the work of Destri and De-Vega [5], see also [7]Here I just briefly review their results:1) Sine-Gordon can be realised as a certain limit of XXZ spin chain with a number ofinhomogeneities 13) Spectrum of energy can be analysed with the help of Bethe-Ansatz equations (BAE) N (cid:89) n =1 sinh( λ + iθ n + iγ/ λ + iθ n − iγ/
2) = − e − i Φ N (cid:89) j =1 sinh( λ − λ j + iγ )sinh( λ − λ j − iγ ) (57)With the limit θ i = ( − i − Θ / , πγ Θ = log( NML ) → ∞ as N → ∞ , and γ relates to theSine-Gordon coupling constant as γ = π (1 − α ) ∈ [0 , π ] , the external magnetic flux corresponds to a twist e − i Φ in theBAE .The energy and momentum of the { λ j } , configuration given by the formula: e − i E ± P N = N (cid:89) n =1 sinh( ± λ j + Θ / iγ/ ∓ λ j − Θ / − iγ/
2) (58)3) This XXZ regularisation of Sine-Gordon gives the answer for the ground state energyin the presence of external magnetic flux, which (in the massless limit) agrees with theconformal one (instead of coordinate Bethe Ansatz and cut-off regularisation).
C From XXZ Bethe Ansatz to DDV equation
Here we review connection between the XXZ Bethe Ansatz and nonlinear integral equa-tion. [6].Our starting point will be the Bethe-Ansatz equations for XXZ model with inhomo-geneities N (cid:89) n =1 sinh( λ + iθ n + iγ/ λ + iθ n − iγ/
2) = − e − i Φ N (cid:89) j =1 sinh( λ − λ j + iγ )sinh( λ − λ j − iγ ) (59)Taking the logarithm of both sides , and introducing new functions Z ( λ ) = log − e − i Φ N (cid:89) j =1 sinh( λ − λ j + iγ )sinh( λ − λ j − iγ ) − log (cid:32) N (cid:89) n =1 sinh( λ + iθ n + iγ/ λ + iθ n − iγ/ (cid:33) (60) φ ( λ, x ) = i log (cid:18) sinh( ix + λ )sinh( ix − λ ) (cid:19) (61)We arrive to the following equations Z ( λ ) = N [ φ ( λ + Θ , γ/
2) + φ ( λ − Θ , γ/ − N/ (cid:88) j =1 φ ( λ − λ k , γ ) −
2Φ (62)As usual vacuum state characterises by the condition Z vac ( λ j ) = ( − N − j ) π (63)14sing it, we can rewrite the sum over roots - λ k as a contour integral N/ (cid:88) j =1 φ ( λ − λ k , γ ) = (cid:73) Γ φ ( λ − µ, γ ) ddµ log(1 + e iZ ( µ ) ) dµ πi (64)Where contour Γ - encircles all roots λ k , this is the way how nonlinear integral equationappears Z ( λ ) = N [ φ ( λ + Θ , γ/
2) + φ ( λ − Θ , γ/ − − (cid:73) Γ φ ( λ − µ, γ ) ddµ log(1 + e iZ ( µ ) ) dµ πi (65)After some manipulations [6] this equation could be transformed to a more convenientform: Z ( λ ) = M L sinh( πλγ ) − π Φ π − γ − i ∞ (cid:90) −∞ G ( λ − µ − iη + ) log(1 + e iZ ( µ + iη + ) ) dµ ++ i ∞ (cid:90) −∞ G ( λ − µ + iη − , γ ) log(1 + e − iZ ( µ − iη − ) ) dµ (66) G ( µ ) = K K = ∞ (cid:90) dk π sinh( k ( π − γ ))sinh( k π − γ ) cosh( γk ) cos( kµ ) (67)This equation (or its simple modification) is called Destri-De Vega (DDV) equation. D From DDV to asymptotics of the current in the limit of big(small)radius of the wire
For now we will restrict ourself to a repulsive regime only ( γ < π and no breathers).For this case Destri and De Vega suggest a prescription η + = η − = η = min ( γ ; π − γ ).and the change of an argument : Z ( γλπ + iη ) = i(cid:15) ( λ ) , G ( λ ) → γπ G ( γπ λ ) we will get G ( θ ) = ∞ (cid:90) dk π sinh( πk ( π γ − πk ( πγ − πk ) cos( kθ ) (68) (cid:15) ( θ ) = − M L cosh( θ ) − i Φ − G (cid:63) log(1 + e (cid:15) ) + G c (cid:63) log(1 + e ¯ (cid:15) ) (69)Where (cid:63) is a convolution, and G c is crossed G : G c ( λ ) = G ( λ + iπ ) , remind that energyis eqal to E = − M ∞ (cid:90) −∞ cosh( θ ) (cid:0) log(1 + e (cid:15) ) + log(1 + e ¯ (cid:15) ) (cid:1) dθ π (70)15urprisingly, G coincides with the logarithmic derivative of soliton-soliton scatteringmatrix G ( θ ) = πi ∂ θ log( S ( θ )), and G c coincides with matrix of crossed process. SoDDV equation looks similar to a TBA equation, despite the fact that it is much moresimple, because in general TBA for Sine-Gordon is a system of a number of couplednonlinear equations. D.1 Large
M L limit
Recall that equations 69, 70 contain full information about the energy of the groundstate of Sine-Gordon (on finite ring with length L ) in the external magnetic flux Φ.Having this equation it is easily to analyze the large M L limit, namely we can solve thisequation recursively (cid:15) = − M L cosh( λ ) − i Φ (71) (cid:15) = G (cid:63) log(1 + e (cid:15) ) + G c (cid:63) log(1 + e ¯ (cid:15) ) (72) ... Let us make just zero order approximation, E = − M ∞ (cid:90) −∞ cosh( θ ) Re (cid:16) log(1 + e − ML cosh( πλγ ) − i Φ ) (cid:17) dθ π (73)Expanding logarithm in Taylor series E = − Mπ (cid:88) n ≥ ( − n − n K ( nM L ) cos( n Φ) (74)For the free fermions point ( γ = π ), nonlinear part just vanishes, and it becomes anexact answer, for example we can make a conformal limit M L →
0, and reproduce theclassical answer (45,46) E = − Lπ (cid:88) n ≥ ( − n − n cos( n Φ) = − π L (cid:18) − π Φ (cid:19) (75)Away from the free fermions point , we can leave only the firs term from the infinitesum E = − Mπ K ( M L ) cos(Φ) + o ( e − mL ) (76) D.2 Small
M L limit (conformal regime)
The analisys of small
M L regime is more complicated, it was done in [6], with the resultsin agreement with the conformal ones (45,46). E = − π L (cid:18) − π − γ )2 π Φ (cid:19) + ... (77)16 eferences [1] Andrew G. Semenov, Andrei D. Zaikin Persistent currents in quantum phaseslip rings , Phys. Rev. B 88, 054505 (2013) ,DOI: 10.1103/PhysRevB.88.054505.arXiv:1306.5456[2] K.Yu. Arutyunov, D.S. Golubev, A.D. Zaikin :
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