Magnetic flux induce dissipation effect on the quantum phase diagram of mesoscopic SQUID array
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug Magnetic flux induce dissipation effect on the quantum phasediagram of mesoscopic SQUID array
Sujit Sarkar
PoornaPrajna Institute of Scientific Research,4 Sadashivanagar,Bangalore 5600 80, India. (Dated: November 19, 2018)
Abstract
We present the quantum phase diagram of mesoscopic SQUID array. We predict different quan-tum phases and the phase boundaries along with several special points. We present the results ofmagnetic flux induced dissipation on these points and also on the phase boundaries. The Josephsoncouplings at these points are dependant on the system parameter in a more complicated fashionand differ from the Ambegaokar and Baratoff relation. We derive the analytical relation betweenthe dissipation strength and Luttinger liquid parameter of the system. We observe some interestingbehaviour at the half-integer magnetic flux quantum and also the importance of co-tunneling effect.Keywords: Mesoscopic and nanoscale system, Josephson junction arrays and wire networks,SQUID devicesPACS: 74.78.Na, 74.81.Fa, 82.25.Dq. . INTRODUCTION Josephson junction arrays have attracted considerable interest in the recent years owingto their interesting physical properties like quantum phase transition, quantum critical be-haviour and Coulomb blockade etc [1, 2]. The most universally observed behaviour forthe low-dimensional superconducting system is the superconductor-insulator transition insuperconducting flim, wires, Josephson junction array in one and two dimensional givingrise to intense experimental and theoretical activity [3–17] This superconductor-insulatortransition occurs at low temperature ( mili-Kelvin scale) due the variation of one of the pa-rameter of the system such as the normal state resistance of the flim, thickness of the wire,the Josephson coupling of Josephson junction array and the magnetic flux of the mesoscopicSQUID array.In one dimensional superconducting quantum dot lattice system (mesoscopic SQUID array,Cooperpair box array) one elementary excitation occurs, i.e., the quantum phase slip cen-ters (QPS) [10, 14, 16, 17]. Recently the physics of QPS and the related physics has beenstudied extensively. Quantum phase slip process is a topological excitation, it’s a discreteprocess in space-time domain of a one dimensional superconducting system. In this processthe amplitude of the order parameter destroys temporarily at a particular point that leadsthe phase of the order parameters to change abruptly (in units of 2 π ). This process occursat the time of macroscopic quantum tunneling of the system. As we understand from ourprevious study [10] and also from the existing literature that the QPS process initiate thesuperconductor- insulator transition [16, 17]. It is well known after the work of Calderiaand Leggett [19] the dissipation plays a central role in the macroscopic quantum system.There are several study in the literature based on the Calderia-Leggett formalism to explainthe different physical properties of low-dimensional tunneling junction system [20–26]. Weare mainly motivated from the experimental findings of Chow et al [13]. They have donesome pioneering work to study the effect of magnetic flux on mesoscopic SQUIDs array.They have observed the magnetic flux induced dissipation driven superconductor-insulatorquantum phase transition and the evidence of qunatum critical point. The other interestingpart of this study the length scale dependent superconductor-insulator transition and themagnertic flux induced superconductivity.The goal of this paper is to provide a detailed analysis of the appearance of different quan-2 B FIG. 1: A. Schematic diagram of one dimensional array of small capacitance dc SQUIDs. Eachplaque is a SQUID with two Josephson junctions marked by the cross. Circle with plus signrepresent the applied magnetic flux Φ. B. Equivalent representation of system A, where the dotsare connected through tunnel junctions and the Josephson couplings of this system is tunable dueto presence of magnetic flux Φ. tum phases and phase boundaries in the presence and absence of dissipation for a mesoscopicSQUID array at T = 0. We assume the presence of ohomic dissipation motivated by theexperiment in Ref. ([13] ) and we study the consequence of it in the different quantumphases and phase boundaries and also on the special points like the particle-hole symmetricpoint, charge degeneracy point, and multicritical points. We also derive the relation betweenthe dissipation strength and Luttinger liquid parameter of the system. We are able to findout the relation between the Josephson coupling and the other interaction parameters ofthe system through the analysis of Luttinger liquid parameter of the system , we will noticethat this analytical relation is more complicated and differs from the Ambegaokar-Baratoff[27] relation. To the best of our knowledge, this is the first derivation in the literature. Theplan of the manuscript is as follows. In section 2A, we introduce the basic concept of theappearence of quantum phase slip centers and the source of dissipation for one dimensionalsuperconducting quantum dot lattice. In section 2B, we derive the relation between thedissipation strength and the Luttinger liquid parameter of the system. In section 3, Wepresent the analysis of quantum phases and phase boundaries, in the presence and absenceof magnetic flux induced dissipation. Section IV, is devoted for summary and conclusions.
2. BASIC ASPECTS OF QUANTUM PHASE SLIP CENTERS AND ANALYTICALDERIVATIONS OF DISSIPATIVE STRENGTHS IN TERMS OF LUTTINGERLIQUID PARAMETER A. Basic Aspects of Quantum Phase Slip Centers:
It is well known to us that the Josephson coupling of the SQUID is modulated by a facror, E J = E J | cos ( πφφ ) | , where φ is the magnetic flux and φ = hc e is the magnetic flux quantum.Therefore we can consider the the mesoscopic SQUIDs array as a superconducting quantumdot (SQD) lattice with modulated Josephson junction [9, 10]. Fig. 1 shows the equivalancebetween the mesoscopic SQUIDs array and SQD lattice with modulated Josephson junctions.Here we prove the appearance of QPS in SQD array with modulated Josephson couplingthrough an analysis of a minimal model. We consider two SQDs are separated by a Josephsonjunction. These two SQDs are any arbitrary SQDs of the array. Appearance of QPS is anintrinsic phenomenon (at any junction at any instant) of the system. The Hamiltonian ofthe system is H = X i n i C − E J | cos ( π ΦΦ ) | X i cos ( θ i +1 − θ i ) (1)where n i and θ i are respectively the Cooper pair density and the superconducting phase ofthe i’th dot and C is the capacitance of the junction. The first term of the Hamiltonianpresent the Coulomb charging energies between the dots and the second term is nothing butthe Josephson phase only term with modulated coupling, due to the presence of magneticflux. We will see that this model is sufficient to capture the appearance of QPS in SQDsystems. Hence this Hamiltonian has sufficient merit to capture the low temperature dissi-pation physics of SQD array. In the continuum limit, the partition function of the systemis given by, Z = R Dθ ( x, τ ) e − S Q ( x,τ ) , where S Q = R dτ R dx E J [( ∂ τ θ ( x, τ )) + ( ∂ x θ ( x, τ )) ].This action is quadratic in scalar-field θ ( x, τ ), where θ ( x, τ ) is a steady and differentiablefield, so one may think that no phase transition can occur for this case. This situationchanges drastically in the presence of topological excitations for which θ ( x, τ ) is singularat the center of the topological excitations. So for this type of system, we express the θ ( x, τ ) into two components: θ ( x, τ ) = θ ( x, τ ) + θ ( x, τ ), where θ ( x, τ ) is the contributionfrom attractive interaction of the system and θ ( x, τ ) is the singular part from topologicalexcitations. We consider at any arbitrary time τ , a topological excitation with centerat X ( τ ) = ( x ( τ ) , τ ( τ )) . The angle is measured from the center of topologicalexcitations between the spatial coordinate and the x-axis θ ( x, τ ) = tan − ( τ − τx − x ) . Thederivative of the angle is ∇ x θ ( x − X ( τ )) = | x − X ( τ ) | [ − ( τ − τ ) , ( x − x )] which hasa singularity at the center of the topological excitation. Finally we get an interesting4esult when we integrate along an arbitrary curve encircling the topological excitations. R C dx ∇ x θ ( x − X ( τ )) = 2 π. So we conclude from our analysis that when a topologicalexcitation is present in the SQD array, the phase difference , θ , across the junction ofquantum dots jumps by an integer multiple of 2 π . This topological excitation is nothing butthe QPS in the ( x, τ ) plane. According to the phase voltage Josephson relation , V J = − e dθdt ( t is the time), a voltage drop occurs during this phase slip, which is the source of dissipation. Here we derive analytical expression of magnetic flux induced dissipative strength ( α ) interms of the interactions of the system. At first we derive the dissipative action/partitionfunction of a quantum impurity system. In our study, we consider the backscattering pro-cess from a quantum impurity to compare the effective action with the superconductingtunnel junction system. We will see that the analytical structure of this dissipative actionis identical with the mesoscopic SQUIDs array. In a different context, the authors of Ref.[28] have predicted the dissipation driven quantum phase transition to occur. They havealso considered the presence of backscattering events originated in the LL under the effectof dynamically screened Coulomb interaction.Here we consider that the impurity is present at the origin where the fermions scatter fromthe left to the right and vice versa. The Hamiltonian describing this process is H = V ( R † (0) L (0) + h.c ) = V Z dxδ ( x ) cosθ ( x ) . The total Hamiltonian of the system is H = H + V Z dxδ ( x ) cosθ ( x, τ ) (2) H = 12 π Z uK ( ∂ x θ ( x, τ )) + uK ( ∂ x φ ( x, τ )) , (3)The corresponding Lagrangian of the system is L = 12 πK Z u ( ∂ τ φ ( x, τ )) + u ( ∂φ ( x, τ )) + V Z dxδ ( x ) cos ( θ ( x, τ )) = L + L (4)5here L and L are the non-interacting and the interacting part of the Lagrangian and K isthe Luttinger liquid parameter of the system. The only non-linear term in this Lagrangianis expressed by the field θ ( x = 0). We would like to express the action of the system as aneffective action by integrating the field θ ( x = 0). Therefore one may consider θ ( x = 0) as aheat bath, which yields the source of dissipation in the system. The constraint condition forthe integration is θ ( τ ) = θ ( x = 0 , τ ). We can write the partition function of the quantumimpurity system. Z = R Dθ ( x, τ ) Dθ ( τ ) δ ( θ ( τ ) − θ (0 , τ )) e − R β Ldτ
Now we use the standardtrick of introducing the Lagrange multiplier with auxiliary field λ ( τ ). Z = Z Dθ ( τ ) e − R β L dτ Z Dλ ( τ ) e − iλ ( τ ) θ ( τ ) Z Dθ ( x, τ ) e R β ( − L + iλ ( τ ) θ (0 ,τ )) dτ (5)The Fourier transform of the first term of Eq.(3) is L = P q P iω n ω n + v q πKv θ ( q, iω n ) θ ( − q, − iω n ). At first we would like to calculatethe integral: R β dτ [ L − iλ ( τ ) θ (0 , τ )], we can write this term as X q X iω n ω n + v q πKv θ ( q, iω n ) θ ( − q, − iω n ) − √ L ( λ ( iω n ) θ ( − q, − iω n ) + λ ( − iω n ) θ ( q, iω n ) . (6)This integral appears in the integral θ ( x, τ ). This integral is quadratic in θ . Now we wouldlike to perform the Gaussian integration by completing the square. We can write the resultas − L P iω n ,q πKvω n + v q . In the infinite length limit one can write, L P q πKvω n + v q = R dq π πKvω n + v q = πK ω n .Now we would like to append this result of integration in the second integral of Z , i.e., theintegral over λ . One can write the integrand as X iω n ( − πK ω n λ ( iω n ) λ ( − iω n ) + i λ ( iω n ) θ ( − q, − iω n )+ λ ( − iω n ) θ ( q, iω n ) . (7)This integral is again the quadratic integral of λ , therefore, the Gaussian integral canbe performed by completing the square. We perform the integration and finally obtain P iω n ω n πK θ ( iω n ) θ ( − iω n ). From this analytical expression, we obtain the effect of bath on θ ( τ ). Appearance of the factor ω n signifies the dissipation. Therefore, the effective actionreduces to S = X iω n ω n πK θ ( iω n ) θ ( − iω n ) + Z dτ V cosθ ( τ ) (8)6he above action implies that a single particle moves in the potential V cosθ ( τ ) subject todissipation with friction constant , πK .Now we calculate the dissipative action of mesoscopic SQUIDs array. Here, we calculatethe effective partition function of our system. Our starting point is the Calderia-Legget [19]formalism. Following this reference we write the action as S = S + α πT X m ω m | θ m | . (9) S is the action for non-dissipative part and S is the standard action for the system withtilted wash-board potential [1, 20, 21] to describe the dissipative physics for low dimensionalsuperconducting tunnel junctions, α is the dissipative strength of the system, α = R Q R s cos | πφφ | (10)(the extra cosine factor which we consider in α is entirely new which probes the effect ofan external magnetic flux and is also consistent physically), ω m = πβ m is the Matsubarafrequency and R Q (= 6 . k Ω) is the quantum resistance, R s is the tunnel junction resis-tance, β is the inverse temperature. In one of our previous work, we have explained fewexperimental findings by using the above expression of α . In the strong potential, tunnelingbetween the minima of the potential is very small. In the imaginary time path integral for-malism, tunneling effect in the strong coupling limit can be described in terms of instantonphysics. In this formalism, it is convenient to characterize the profile of θ in terms of itstime derivative, dθ ( τ ) dτ = X i e i h ( τ − τ i ) , (11)where h ( τ − τ i ) is the time derivative at time τ of one instanton configuration. τ i is thelocation of the i-th instanton, e i = 1 and − h from −∞ to ∞ , we get R ∞−∞ dτ h ( τ ) = θ ( ∞ ) − θ ( −∞ ) = 2 π. It is well known that the instanton (anti-instanton) is almost universallyconstant except for a very small region of time variation. In the QPS process the amplitudeof the superconducting order parameter is zero only in a very small region of space as afunction of time and the phase changes by ± π . So our system reduces to a neutral systemconsisting of equal number of instanton and anti-instanton. One can find the expression for θ ( ω ) after the Fourier transform applied to both the sides of Eq. 11 which yields θ ( ω ) = iω P i e i h ( iω ) e iωτ . Now we substitute this expression for θ ( ω ) in the second term of Eq. 97nd finally we get this term as P ij F ( τ i − τ j ) e i e j , where F ( τ i − τ j ) = παβ P m | ω m | e iω ( τ i − τ j ) ≃ ln ( τ i − τ j ). We obtain this expression for very small values of ω ( → F ( τ i − τ j )effectively represents the Coulomb interaction between the instanton and anti-instanton.This term is the main source of dissipation of SQUID array system. Following the standardprescription of imaginary time path integral formalism, we can write the partition functionof the system as [8, 10, 18, 24–26]. Z = ∞ X n =0 n ! z n X e i Z β dτ n Z τ n − dτ n − ... Z τ dτ e − F ( τ i − τ j ) e i e j . (12)We would like to express the partition function in terms of integration over auxiliary field, q ( τ ). After some extensive analytical calculations, we get Z = Z Dq ( τ ) e ( − P iωn | ωn | πα q ( iω n ) q ( − iω n ))+(2 z R β dτcosq ( τ )) . (13)Thus by comparing the first term of the action of Eq. (8) and the first term of exponentialof Eq. (13), we conclude that the dissipative strength α and the Luttinger liquid parameterof the system are related by the relation, K = 4 α .
3. QUANTUM FIELD THEORETICAL STUDY OF MODEL HAMILTONIAN OFTHE SYSTEM AND EXPLICIT DERIVATION OF DISSIPATIVE STRENGTH
In the previous section and also from our previous study [10], we have shown explicitlythat the mesoscopic SQUIDs array is equivalent to the array of superconducting quantumdots (SQD) array with modulated Josephson coupling. We first write the model Hamiltonianof SQD with nearest neighbor (NN) Josephson coupling Hamiltonian ( H J ) and also withthe presence of the on-site ( H EC ) and NN charging energy ( H EC ) between SQD as, H = H J + H EC + H EC . (14)We would like to recast our model Hamiltonian in the magnetic flux induced Coulombblocked regime ( E C >> E J ). In this regime, one can recast the Hamiltonian in spinoperators. It is also observed from the experiments that the quantum critical point existsfor larger values of the magnetic field, when the magnetic field induced Coulomb blockade8 Eg K=1A A n=1n=0 E E EQ (K =1/2)B CCK=1/4 DD P1 (K=2)P2 J FIG. 2: Quantum phase diagram ( E J vs. V g ) of the SQD array. We have depicted the differentphases of the model Hamiltonians by: A. Mott insulating phase, B. Charge-density wave (CDW);C. First kind of Repulsive Luttinger liquid (RL1); D. Second kind of Repulsive Luttinger liquid(RL2); E. Superconductivity. Q and P2 are the multi-critical points (please see the text). K is 1 atthe phase boundary between Luttinger liquid (RL2) and superconductivity and also at the phaseboundary between Mott phase and superconductivity. K = 1 / K = 2 and 1 / P Q point respectively. Q and P Q is the charge degeneracy point and P phase is more prominent than the E J induced SC phase. Thus our theoretical model isconsistent with the experimental findings. During this mapping process we follow Ref. ([8–10]), so that. H J = − E J X i ( S i † S i +1 − + h.c ), E J = E J | cos ( π ΦΦ ) | , H EC = E C X i ( S iZ − h ) . are the Hamiltonian H EC accounts for the influence of gate voltage ( eN ∼ V g ), where eN is the average dot charge induced by the gate voltage. When the ratio E J E C →
0, the SQDarray is in the insulating state having a gap of the width ∼ E C , since it costs an energy ∼ E C to change the number of pairs at any dot. The exceptions are the discrete points at N = (2 n + 1), where a dot with charge 2 ne and 2( n + 1) e has the same energy because thegate charge compensates the charges of extra Cooper pair in the dot. On this degeneracy9 ∆ E J AB0 0.5 1 1.5 2050100150200 Magnetic Flux ∆ E J AB FIG. 3: Shift of the Josephson coupling (∆ E J ) due to the magnetic flux induced dissipation effectwith magnetic flux (measured with respect to magnetic flux quantum, φ = hc e ). The red lineis for the shifting of the Josephson coupling for the phase boundary between the RL2 and thesuperconducting phase. Blue line is for the shift of the phase boundary between the CDW andRL1 phase. Inset shows the shift of the Josephson coupling for the particle-hole symmetric point(green line) and the Charge degeneracy point (yellow line) in the figure. point, a small amount of Josephson coupling leads the system to the superconducting state.Here h = N − n − allows the tuning of the system around the degeneracy point by meansof gate voltage. H EC = 4 E Z P i S iZ S i +1 Z . At the Coulomb blocked regime, the higherorder expansion leads to the virtual state with energies exceeding E C . In this second orderprocess, the effective Hamiltonian reduces to the subspace of charges 0 and 2, and takes theform [8–10], H C = − E J E C X i S iZ S i +1 Z − E J E C X i ( S i +2 † S i − + h.c ) . (15)With this corrections H EC become H EC ≃ (4 E Z − E J E C ) X i S iZ S i +1 Z . In this analytical expression, we only consider the nearest -neighbor contribution of the inter-action. There is no evidence of next-nearest-neighbour interaction for mesoscopic SQUID ar-ray system [13]. One can express spin chain systems to as spinless fermions systems throughthe application of Jordan-Wigner transformation. In Jordan-Wigner transformation the re-lation between the spin and the electron creation and annihilation operators are S z ( x ) = ψ † ( x ) ψ ( x ) − / S − ( x ) = ψ ( x ) exp[ iπ P x − j = −∞ n j ], where S + = ( S − ) + , n ( x ) = ψ † ( x ) ψ ( x )10 g EJ α =1/4 EA n=1 α P1 ( = 1/2)B α =1/ Q ( =1/8) α n=0EA CC EDDP2 FIG. 4: Dissipative quantum phase diagram ( E J vs. V g ) of the SQD array. We have depicted thedifferent phases of the model Hamiltonians by: A. Mott insulating phase, B. Charge-density wave(CDW); C. First kind of Repulsive Luttinger liquid (RL1); D. Second kind of Repulsive Luttingerliquid (RL2); E. Superconductivity. Q and P2 are the multi-critical points (please see the text). K is 1 at the phase boundary between Luttinger liquid (RL2) and superconductivity and also atthe phase boundary between Mott phase and superconductivity. K = 1 / K = 2 and 1 / P Q point respectively. Q and P Q is the charge degeneracy point and P is the fermion number at the site x . We have transformed all Hamiltonians in spinlessfermions as follows: H J = − E J P i ( ψ i † ψ i +1 − + h.c ), H EC = 2 hE C P i ( ψ i † ψ i − / .H EC ≃ (4 E Z − E J E C ) P i ( ψ i † ψ i − / ψ i +1 † ψ i +1 − / . In order to study the continuum field theory of these Hamiltonians, we recast the spinlessfermions operators in terms of field operators by a relation [30]. ψ ( x ) = [ e ik F x ψ R ( x ) + e − ik F x ψ L ( x )] (16)where ψ R ( x ) and ψ L ( x ) describe the second-quantized fields of right- and the left-moving11ermions respectively. We would like to express the fermionic fields in terms of bosonic fieldby the relation ψ r ( x ) = U r √ πα e − i ( rφ ( x ) − θ ( x )) , (17)where r denotes the chirality of the fermionic fields, right (1) or left movers (-1). Theoperators U r preserve the anti-commutivity of fermionic fields. φ field corresponds to thequantum fluctuations (bosonic) of spin and θ is the dual field of φ . They are related by therelations φ R = θ − φ and φ L = θ + φ . The Hamiltonians without ( H ) and withco-tunneling ( H ) effect are the following H = H + 4 E Z (2 πα ) Z dx : cos (4 √ Kφ ( x )) :+ E C πα Z ( ∂ x φ ( x )) dx (18) H = H + (4 E Z − E J E C )(2 πα ) Z dx : cos (4 √ Kφ ( x )) :+ E C πα Z ( ∂ x φ ( x )) dx (19)Where, H is the non-interacting part of the Hamiltonian. The Luttinger liquid param-eters of the Hamiltonian H and H are K and K are respectively. K = ππ + 2 sin − ∆ (20) K = ππ + 2 sin − ∆ (21)∆ = E Z E J ∆ = E Z E J − E J E C . We calculate the dissipation strength by calculating Kfor both cases and then we use the relation K = 4 α . Therefore the dissipative strength inabsence ( α ) and presence ( α ) of co-tunneling effect are α = 14 ππ + 2 sin − ∆ (22) α = 14 ππ + 2 sin − ∆ (23)This is the first analytical derivation of flux induced dissipation strength in terms of theinteractions of the system. In the derivation of K and α , we only consider the nearest-neighbour hopping consideration when we consider the effect co-tunneling. We consider thesetwo processes to emphasis the importance of co-tunneling effect for this system. Before we12roceed further for the analysis of the Hamiltonian , H and H , we want to explain indetail for the different values of K at the different points (like P1, Q and P2) of phasediagram and also at the phase boundaries. The physical analysis of the phases and cleardistinction of the phase boundaries will depend on the values of the K. Point Q is thecharge degeneracy point for the low Cooper-pair density ( n i = 1 / K will beevaluated from the relevance of sine-Gordon terms. At the point Q, system is in the secondorder commensurability, sine-Gordon coupling terms ,Eq. 18 and Eq. 19, will becomerelevant for K = 1 /
2, which is depicted in the Fig.1. Point P1 is the particle hole symmetricpoint, i.e., there is one particle in each site. At this point system is in the first ordercommensurability, i.e., the sine-Gordon coupling term is cos (2 √ Kφ ( x )). So this term willbecome relevant for K=2, which is depicted in the Fig.1. We are understanding from Eq. 18and Eq. 19 that the applied gate voltage acts as a chemical potential. So the proper tuningof gate voltage will drive the system from the insulating state to the other quantum phasesof the system. We are now interested in finding the value of K at the phase boundaries,hence analysis is the following: We follow the Luther-Emery [29] trick during the analysis.One can write the sine-Gordon Hamiltonian for arbitrary commensurability as H = H + λ Z dx cos (2 n √ Kφ ( x )) , (24)where n is the commensurability and λ is the coupling strength. H isthe free part of the Hamiltonian. We know that for the spinless fermions, ψ R † ψ L + ψ L † ψ R = πa R dxcos (2 √ Kφ ( x )), which is similar to the analytical expres-sion of sine-Gordon coupling term but with the wrong coefficient inside the cosine. One canset ˜ φ ( x ) = 2 √ ˜ Kφ ( x ) then the Eq. 24 become H = H + λ Z dxcos (2 ˜ φ ( x )) . (25) K and ˜ K are related by the relation, K = ˜ Kn .At the phase boundary, ˜ K = 1 that implies K = 1 /n . So for the first and second ordercommensurability the value of K at the phase boundary are 1 and 1/4 respectively whichis depicted in Fig. 1. The point to be noticed that if we start from an initial model with K = 1 /n , i.e., in general a strongly interacting model, the resulting spin-less fermionsmodel corresponds in the boson language to K = 1 which means that it is non-interacting.For this particular value of K the spin-less fermions whose bosonized form is Eq. 25 are just13ree particle with backscattering. This special value of K is known as the Luther-Emery[29] line, the importance of the Luther-Emery solution is to provide a solution for themassive phase on the whole line K = 1 /n for arbitrary λ .Here we do the analysis for Hamiltonian H . In the limit ∆ = ∆ , for E J < E Z and relatively small field, the anti-ferromagnetic Ising interaction dominate the physics ofanisotropic Heisenberg chain. When the field is large, i.e., the applied gate voltage is large,the chain state is in the ferromagnetic state. In the language of interacting bosons, TheNeel phase is the commensurate charge density wave phase with period 2, i.e., there isonly one boson in every two sites. In Fig. 1 this phase region is described by region B.The ferromagnetic state is the Mott insulating state, this is the phase A of our quantumphase diagram (Fig. 1). H is the Heisenberg XXZ model Hamiltonian in a magneticfield. The emergence of two Luttinger liquid phases for the following reasons: RL1 andRL2 respectively occur due to commensurate-incommensurate transition and the criticalityof Heisenberg XY model. For the intermediate values of the field, system is either in thefirst kind of repulsive Luttinger liquid (RL1) phase for K < / K > /
2. The physical significance of RL1phase is that the coupling term is relevant but the applied magnetic field, i.e., the appliedgate voltage on the dot, breaks the gapped phase whereas in the RL2 phase non of thecoupling term is relevant due to the larger values of K ( > / and ∆ = ∆ . So we predict theexistence of two RL from two different sources. In previous studies this clarification wasabsent and they had reported only one RL [8]. The value of K = 1 at the phase boundarybetween the MI and SC phase and also between the RL2 and SC phase. From the analysisof K at the phase boundary we obtain E Z = 0, according to our theory and also from theexperimental findings this condition is unphysical. So the interaction space of Hamiltonian H is not sufficient to produce the whole phase diagram of Fig. 2. It indicates that weshall have to consider more extended interaction space to get the correct phase diagram.If we consider the co-tunneling effect in this Hamiltonian system, i.e., the Hamiltonian H . The phase boundary analysis at the MI and SC phase and also for the RL2 and SCphase implies that we get the condition E J = q E Z E C , which is consistent physically.Now we discuss the effect of magnetic field induced dissipation on the quantum phase14iagram of superconducting quantum dot lattice. We have already proven in the previoussection the analytical relation between the LL parameter ( K ) and the dissipation strengthin presence of magnetic flux. Here we use the analytical expression for K to study theeffect of magnetic flux on the quantum phases and quantum phase boundaries. Thereforethe modified quantum phase diagram in presence of magnetic flux include the magneticflux induce dissipation effect. In the previous paragraph we emphasis the importance ofco-tunneling effect. Therefore we consider the Eq.(21) and Eq.(23) when we consider theshift of the Josephson couplings due to the presence of magnetic flux induced dissipation. Itis very clear from Eq. 21 and Eq. 23 that the analytical relation of the Josephson couplingwith the system interactions parameters at different phase boundaries and different pointsare more complicated than the Ambegaokar and Baratoff relation [27]. The analyticalexpression for the Josephson couplings at the different phase boundaries and special pointsare the following:Particle-Hole symmetric point ( K = 2), E J = 0 . E C ( s E Z E C + 1) (26)Charge-Degeneracy point ( K = 1 / E J = 0 . E C ( s E Z E C −
1) (27)Phase boundary between the RL2 and SC phase ( K = 1), E J = 4( s E Z E C K = 1 / E J = 0 . E C ( s E Z E C + 1) . (29)Our main intension is to study the effect of magnetic flux on the particle hole symmetricpoint ( K = 2 ), charge degeneracy point ( K = 1), multicritical point and also for the phaseboundaries between the different quantum phases.The analytical expressions for the shift of Josephson couplings (∆ E J ) due to the presenceof magnetic flux are the following: 15article-Hole symmetric point ( α = 1 / E J = 0 . E C ( s E Z E C + 1)(1 / | cos ( πφφ ) | −
1) (30)Charge-Degeneracy point ( α = 1 / E J = 0 . E C ( s E Z E C − / | cos ( πφφ ) | −
1) (31)Phase boundary between the RL2 and SC phase ( α = 1 / E J = 4( s E Z E C / | cos ( πφφ ) | −
1) (32)Phase boundary between CDW state RL1 phase ( α = 1 / E J = 0 . E C ( s E Z E C + 1)(1 / | cos ( πφφ ) | −
1) (33)In Fig. 3, we present our results of deviation of Josephson coupling as a function ofmagnetic flux. We observe that this deviation is maximum at the half-integers values ofmagnetic flux quantum. It reveals from our study that the effect of magnetic flux inducedissipation for the half-integer values of magnetic flux quantum is slightly prominant forthe special points compare to the qunatum phase boundaries. Inset shows the shift ofJosephson coupling for the particle-hole symmetric point and the charge degeneracy point.It is also clear from the inset that the magnetic flux induce dissipation is more prominantfor the particle-hole symmetric point compare to the charge degeneracy point.It is very clear from the analytical expression from Eq. 30 to Eq. 33 and also from theFig.3 from our study that there is no appreciable changes in the quantum phase diagramfor small magnetic flux in the system. As the applied magnetic fluxes changes from 0 . φ to 0 . φ , quantum phase diagram shows some appreciable change and it is robust for thehalf-integer magnetic flux quantum.In Fig. 4, we present the magnetic flux induce dissipative quantum phase diagram of oursystem. This schematic phase diagram is for values of magnetic flux which appreciably effectthe phase boundaries and special points as we have discussed in the previous paragraph.We present the phase boundaries and special points in terms of the dissipative strength ofthe system. We observe from our study that magnetic flux induce dissipation favour theinsulating phase and the gapless LL phase over the superconducting phase of the system16hich is consistent with the experimental findings [13]. SUMMARY AND CONCLUSIONS
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