General method for calculating the universal conductance of strongly correlated junctions of multiple quantum wires
Armin Rahmani, Chang-Yu Hou, Adrian Feiguin, Masaki Oshikawa, Claudio Chamon, Ian Affleck
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n General method for calculating the universal conductance of strongly correlatedjunctions of multiple quantum wires
Armin Rahmani, Chang-Yu Hou, Adrian Feiguin, Masaki Oshikawa, Claudio Chamon, and Ian Affleck Department of Physics, Boston University, Boston, MA 02215 USA Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands Department of Physics and Astronomy, University of Wyoming, Laramie, Wyoming 82071, USA Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan Department of Physics and Astronomy, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z1 (Dated: June 6, 2018)We develop a method to extract the universal conductance of junctions of multiple quantum wires,a property of systems connected to reservoirs, from static ground-state computations in closed finitesystems. The method is based on a key relationship, derived within the framework of bound-ary conformal field theory, between the conductance tensor and certain ground state correlationfunctions. Our results provide a systematic way of studying quantum transport in the presenceof strong electron-electron interactions using efficient numerical techniques such as the standardtime-independent density-matrix renormalization-group method. We give a step-by-step recipe forapplying the method and present several tests and benchmarks. As an application of the method,we calculate the conductance of the M fixed point of a Y junction of Luttinger liquids for severalvalues of the Luttinger parameter g and conjecture its functional dependence on g . PACS numbers:
I. INTRODUCTION
Advances in molecular electronics can extend the lim-its of device miniaturization to the atomic scales whereentire electronic circuits are made with molecular build-ing blocks.
Single molecule junctions connected to twomacroscopic metallic leads have already been successfullyfabricated, and there are several proposals such as layingquantum wires on top of each other for making junctionsof multiple quantum wires. If we eventually manage to build entire electronic cir-cuits with molecular building blocks, a paramount goalin the field of molecular electronics, junctions of three ormore quantum wires will inevitably be a key ingredient.These junctions are comprised of several quantum wires,i.e., quasi-one-dimensional (1D) metallic structures withatomic scale sizes, that are connected to one another bya given molecular structure as shown schematically inFig. 1. The structure and interactions at the junctiondepend on the particular system under study. What wemean by metallic in the above description of quantumwires is that they are capable of conducting electricitydue to the presence of gapless excitations. A genericdescription for these 1D quantum wires is based on theTomonaga-Luttinger-liquid theory.
Structures involv-ing Luttinger-liquid quantum wires have been the sub-ject of numerous recent studies.
Experimentally, suchquantum wires are realized with carbon nanotubes orthrough the cleaved edge overgrowth technique in GaAsheterostructures.
Electrical current running in thewires can pass through this molecular structure at thejunction.At molecular length scales, quantum mechanics is im-portant and the system represented above must be mod-eled accordingly. The simplest theoretical description of
FIG. 1: (Color online) A schematic illustration of the genericsystem that is the subject of this paper. We have M quantumwires connected to a molecular system of certain structureand interactions. We have currents I α running in wires α =1 . . . M and voltages V α applied to the endpoints of the wires. systems such as this is based on the tight-binding model,with anti-commuting creation and annihilation operatorsintroduced for different atomic sites. An effective Hamil-tonian can then be written in terms of these creationand annihilation operators and generically involves hop-ping terms c † c and density-density interaction terms nn with n = c † c .Suppose we wish to study a rather arbitrary junction,the structure and interactions of which are representedby a tight-binding Hamiltonian. A very basic questionregarding this system is how it conducts electricity. Con-sider a system of M wires. In the presence of voltage bi-ases V α applied to the endpoint of wire α for α = 1 . . . M ,current I α will flow along each wire α . By convention, acurrent I α is positive if it flows toward the junction andnegative if it flows away from it as seen in Fig. 1. In gen-eral, i.e., at arbitrary biases and temperatures, this prob-lem is very complicated and the currents flowing in thequantum wires are nonuniversal functions of the temper-ature, the voltages V α , and the microscopic details of thesystem. In this paper, we are concerned with the linear-response regime where universal behavior can emerge.We work at zero temperature and consider only the limitof infinitesimal biases. The currents in this regime willbe a linear combination of the applied biases as seen inthe following: I α = X β G αβ V β . (1)This linear relationship defines the linear conductancetensor G αβ which is the quantity of interest in this paper.One of the most important ideas of modern physicsis the remarkable universality that emerges near criticalpoints. It turns out that due to the criticality of thebulk of quantum wires, i.e., having divergent correlationlengths, a large degree of universality also emerges inthe behavior of quantum junctions. The universality canbe understood in the framework of the renormalizationgroup (RG). One can argue that in the limit of smallbiases and low temperatures, many of the microscopic de-tails of the junction are irrelevant in the RG sense, whichmeans their contribution to conductance, and other phys-ical observables, decays to zero at large distances and lowenergies.The junctions we are concerned with in this paper fallinto the category of quantum impurity problems. Thejunction, with all the complex structure and interactionsit contains, is localized at the endpoints of the wires. Itcan therefore be thought of as one (rather arbitrary) im-purity inserted into a system, the bulk behavior of whichis given by that of M independent quantum wires. A clas-sic example of quantum impurity problems is the Kondomodel describing the behavior of conduction electronsinteracting with a local magnetic moment. The power-ful methods of boundary conformal field theory (BCFT)have proven useful in a multitude of quantum impurityproblems.
Thus, BCFT is the main analytical tech-nique used in this paper.Determining the conductance of quantum junctionsin the presence of strong electron-electron interactionsis a long-sought and challenging goal. The Landauer-B¨uttiker’s formalism, which is the method of choice in thecalculation of quantum conductance, does not accountfor these interactions, which indeed play a key role inlow dimensions. Functional renormalization-group meth-ods have been helpful in studying the interaction effectsin the vicinity of the junction, but their applicability isalso dependent upon the presence of large noninteractingleads.
In recent years, efficient numerical methods, such asthe density-matrix renormalization group (DMRG), have been developed for studying strongly correlatedquasi-1D quantum problems. Since the quantum junc-tions described above can be thought of as quasi-1D (byfolding all the wires to one side so they run parallel to oneanother), these numerical methods could potentially beefficient tools for computing the conductance of junctionswith an arbitrary number of wires and in the presence ofstrong interactions. In fact, DMRG has already been applied to the studyof quantum junctions. However, when it comes to cal-culating the conductance of strongly correlated junc-tions, there are fundamental difficulties even when we arearmed with powerful tools such as DMRG. One such dif-ficulty arises from the fact that conductance is a propertyof an open quantum system. We define the conductancein terms of the current passing through the system andthe underlying assumption is that we have reservoirs thatcan act as sources and drains for electrons. To study con-ductance, we either need to model the reservoirs carefullyor send them to infinity. The latter is a simpler and moreelegant way of formally dealing with quantum transport,but has the downside that for a numerical calculation ofconductance, we would need to model large enough sys-tems that faithfully approximate the semi-infinite ones.Another difficulty with calculating the linear conduc-tance is that, within the linear-response framework, con-ductance is formally related to dynamical correlationfunctions. It may then appear that one needs to use themuch more computationally demanding time-dependentnumerical methods such as time-dependent DMRG tocalculate the conductance.For junctions of two quantum wires, time-dependentDMRG has already been used for conductance calcula-tions.
A brute force calculation with time-dependentmethods in large systems is not, however, currently fea-sible for strongly correlated junctions of more than twoquantum wires.It is the objective of this paper to make such calcula-tions possible with a combination of analytical and nu-merical techniques. More specifically, the main objectiveof this paper is to develop a formalism that would allowus to apply numerical methods such as time-independentDMRG and the related matrix product states to calculatethe linear-response conductance of strongly-correlatedjunctions of an arbitrary number of quantum wires withrather generic structures and interactions in the junc-tion. In this paper, we focus on the systems with spin-less electrons, but our method can also be extended tosystems with spin-1/2 electrons.One particular application for the formalism we seekto develop in this paper is the problem of the M fixedpoint in a Y junction of spinless Luttinger liquids. Theexistence of this nontrivial fixed point was conjecturedmany years ago, but its nature, and more specifically itsconductance, had remained an open question in quantumimpurity problems.
The outline of this paper is as follows. In Sec. II, wesummarize the main results of this paper. We presenta key relationship [Eq. (2)] between the junction con-ductance and certain static correlation functions in a fi-nite system. This relationship serves as the basis of themethod developed here. We also present in Sec. II astep-by-step recipe for applying the method in practiceas well as a summary of the new results on the Y junc-tion, which were obtained with this method. In Sec. III,we present some explicit calculations in a noninteract-ing lattice model that help motivate the derivation ofEq. (2) and clarify the connection of the continuum re-sults to lattice calculations. The results derived are infact some special cases, which can be obtained with el-ementary methods, of the general relation Eq. (2), thederivation of which requires the machinery of BCFT. InSec. IV, we briefly review the main analytical techniques,namely, bosonization and boundary conformal field the-ory, used in this paper and set up the notation. Sec-tion V is devoted to deriving Eq. (2) in the BCFT frame-work. In Sec. VI, we discuss in detail the method pro-posed in this work for conductance calculations and clar-ify practical issues regarding a lattice-model implementa-tion. In Sec. VII, we present numerical benchmarks withDMRG for interacting systems and exact diagonalizationfor noninteracting systems to verify the correctness of themethod. In Sec. VIII, we study a Y junction of quantumwires and obtain the previously unknown conductance ofits M fixed point as a function of the Luttinger parameter g . Finally, we conclude in Sec. IX by outlining the out-look for future applications and the impact of the resultsobtained in this paper. Some of the results of this paperhave been briefly reported in Ref. 42. II. MAIN RESULTS
We developed a method to extract the universal lin-ear conductance G αβ of quantum multi-wire junctionsdefined in Eq. (1) from a calculation of the ground stateexpectation values of operators involving currents anddensities in an appropriately constructed finite system.At the core of the method lies an important generalrelationship, which we recently derived in Ref. 47 usingthe machinery of BCFT. The relationship is derived inSec. V and simply states thatlim x →∞ h J αR ( x ) J βL ( x ) i GS h ℓ sin (cid:16) πℓ x (cid:17)i e h = G αβ , (2)where x is the distance from the boundary on the left ina system of length ℓ , with ℓ → ∞ and finite x/ℓ , con-structed from the junction of interest and an appropriatemirror image placed on the right endpoints of the wiresas seen in Fig. 2. Here J αR ( x ) ( J βL ( x )) is the right-moving(left-moving) current on wire α ( β ). Note that althoughEq. (2) holds asymptotically ( x → ∞ ), it can be used toextract the conductance G αβ even with finite but largeenough x .The relationship above is the key ingredient of ourmethod for calculating the conductance. If we can com-pute the quantity h J αR ( x ) J βL ( x ) i GS , which is a ground-state expectation value in a finite system of length ℓ , asa function of x , then we can multiply it by a universalfunction to get the left-hand side of Eq. (2) above. Thisquantity will then saturate to the universal conductanceof the junction for large x .Apart from the derivation of the key Eq. (2), we pro-vide a recipe for applying this continuum result to a lat- FIG. 2: (Color online) A schematic illustration of the fi-nite system, constructed from the junction of interest and anappropriate mirror image, which we use in our method to ex-tract the conductance G αβ . The conductance is related tothe ground state h J αR ( x ) J βL ( x ) i GS correlation function of chi-ral currents in this finite system through Eq. (2). tice calculation. This requires specifying the procedurefor constructing the lattice (tight-binding) Hamiltonianof the aforementioned finite system from the Hamilto-nian of the junction of interest (that couples infinitelylong wires). It also requires specifying lattice operators,i.e., in terms of the tight-binding fermionic creation andannihilation operators, the correlation functions of whichare a good approximation to h J αR ( x ) J βL ( x ) i GS . This is im-portant because the chiral current operators are definedfor the continuum theory, and chiral creation and anni-hilation operators can not be directly modeled on thelattice. The recipe for applying the key relation Eq. (2)to a lattice computation is given in Sec. VI.For quick reference and an illustration of the method,here we give a simple example and explain a step-by-stepapplication of the method to the well-known problem ofa weak link in the Luttinger liquid. The starting pointfor applying our method is always a tight-binding latticeHamiltonian of spinless electrons for bulk wires and theirconnection at the junction, namely, H = H boundary + H bulk . For a weak link in a Luttinger liquid, we canwrite H boundary = − tc † , c , − tc † , c , and the following Hamiltonian for the bulk of the wires: H bulk = X α =1 ∞ X j =0 (cid:2) − c † α,j c α,j +1 − c † α,j +1 c α,j (3)+ V ( n α,j −
12 )( n α,j +1 −
12 ) (cid:3) . Note that there is some arbitrariness in dividing the sys-tem into the junction and wires. The boundary Hamilto-nian above is a minimal choice, but including more sitesin H boundary does not affect the results as long as thesystem is large enough and the correlation functions dis-cussed below are computed far away from the boundary.Given the system Hamiltonian, our method consists ofthe following steps.1. Construct the finite system as in Fig. 2. For thiswe need to construct a Hamiltonian H ′ = H + H ′ bulk + H ℓ where H and H ℓ respectively describethe junction on the left side of the system ( x = 0)in Fig. 2 and the mirror image at x = ℓ . The recipefor constructing these Hamiltonians is simple. Theleft boundary Hamiltonian H is simply equal to H boundary and the bulk Hamiltonian H ′ bulk has ex-actly the same form as the bulk Hamiltonian of thesemi-infinite system but a finite number of terms,i.e., P ∞ j =0 → P N − j =0 . The construction of the rightHamiltonian goes as follows. First we consider thesame Hamiltonian as H boundary but acting on theother endpoint, namely − tc † ,N c ,N − tc † ,N c ,N andthen we apply two transformations K and C on thisHamiltonian. K simply takes the complex conju-gate and C changes c → c † . In this case, assuminga real hopping amplitude t , we have H ℓ = C ( − tc † ,N c ,N − tc † ,N c ,N )= tc † ,N c ,N + tc † ,N c ,N .
2. Having constructed the Hamiltonian H = H + H ′ bulk + H ℓ of the finite system, measure, by a nu-merical DMRG calculation, the following groundstate expectation value: h J R ( m ) J L ( m ) i = − v h J ( m ) J ( m ) i , where J α ( m ) = i (cid:16) c † α,m c α,m +1 − c † α,m +1 c α,m (cid:17) issimply the current operator and v is the charge car-rier velocity for the Luttinger liquids described by H bulk . The above equation is valid for time-reversalsymmetric systems like the example at hand. Thegeneral construction of the operator h J R ( m ) J L ( m ) i in terms of the lattice creation and annihilation op-erators is given in section VI.3. Fit the data for h J R ( m ) J L ( m ) i to the asymptoticfunctional form from Eq. (2), i.e., h J R ( m ) J L ( m ) i ∝ (cid:2) N sin (cid:0) πN m (cid:1)(cid:3) − , and obtain G from the overallcoefficient.Note that if G = 0, the fitting is tricky. These detailswill be discussed later.The other important results of this paper concern aconcrete application of the method to a previously un-solved quantum impurity problem. These results arepresented in Sec. VIII. The main problem solved in thatsection by an explicit numerical DMRG calculation isdetermining the conductance of a nontrivial fixed pointin an interacting Y junction of three quantum wires de-picted in Fig. 3. The existence of this fixed point knownas the M fixed point was conjectured in Ref. 45,46, butthe properties of the fixed point including its conduc-tance remained unknown. In this paper, by combiningthe method developed in Secs. V and VI and numericalcomputations with time-independent DMRG, we calcu-late the universal conductance of this fixed point as a function of the Luttinger parameter g (a parameter thatquantifies the strength of electron-electron interactionsin the wires). Based on the numerical results, we claimthat the conductance of the M fixed point depends onthe Luttinger parameter as G ( g ) = G ( g ) = − gγ g + 3 γ − gγ e h , (4)where γ ≈ .
42. The above expression is a universal re-sult which, for any nonvanishing hopping amplitude t ,holds independently of the value of t . We also explicitlyverify that the conductance exhibits universal behavior.Namely we find that at large length scales, the conduc-tance is independent of the hopping amplitude t at thejunction. FIG. 3: A schematic illustration of the Y junction numeri-cally studied in section VIII. The M fixed point correspondsto the time reversal symmetric case φ = 0 , π . We also present in Sec. VIII a numerical verification ofa theoretical prediction for the conductance of the chi-ral fixed point. The chiral fixed point is stable in therange 1 < g < φ = 0 , π . III. CONDUCTANCE AND CORRELATIONS INA NONINTERACTING LATTICE MODEL
There is a well-established framework, namely theLandauer-B¨uttiker formalism (see for example Ref. 48),for calculating conductances of multi-wire quantum junc-tions in the absence of electron-electron interactions.This framework is applicable both in the continuum andto lattice models. The key quantities in this frameworkare the incoming and outgoing scattering states Ψ in andΨ out , which for a junction of M wires can be representedby an M × M ×
1) column vector for spinless (spinful)electrons. The scattering states are labeled by momen-tum quantum number k , and the effect of the junction isencoded in an M × M (2 M × M ) unitary scattering ma-trix for spinless (spinful) electrons. The scattering matrixrelates the incoming and outgoing scattering states asΨ out ( k ) = S ( k )Ψ in ( k ) . (5)In the Landauer-B¨uttiker formalism, conductance issimply related to the elements of the scattering matrix S ( k F ) at the Fermi level. Because including spin is asimple extension of the method developed in this paper,we work with spinless electrons here. The conductancebetween two different wires G αβ is then given by G αβ = | S αβ ( k F ) | e h . (6)Let us now consider the simplest junction of two latticewires. Each wire has hopping amplitude set to unity inthe bulk and the two wires are connected by a hoppingamplitude t . More generically, one can consider a nonin-teracting junction of M wires that are coupled quadrat-ically as described below. Consider a lattice system of M wires with electron annihilation operators c α,j , where α = 1 , ..., M is the wire index and j = −∞ , ..., † j = (cid:16) c † ,j c † ,j · · · c † M,j (cid:17) : H = − − X j = −∞ (cid:16) Ψ † j Ψ j +1 + Ψ † j +1 Ψ j (cid:17) + Ψ † Γ Ψ , (7)where Γ is a Hermitian M × M matrix with diagonal ele-ments Γ αα = µ α (endpoint chemical potentials) and off-diagonal elements Γ αβ = Γ ∗ βα = t αβ (hopping betweenendpoints). In the bulk of each wire, the nearest-neighborhopping amplitude is set to unity. The scattering eigen-states are | ψ i k = X j Ψ † j (cid:0) A + e ikj + A − e − ikj (cid:1) | i , (8)where Ψ † j is a row vector of operators c † α,j and A +( − ) is a column vector of scattering amplitudes A +( − ) α for α = 1 . . . M , and the standard matrix multiplicationconvention applies. By plugging the above states inthe Schr¨odinger equation H| ψ i k = ǫ k | ψ i k , we obtain ǫ k = − k and S ( k ) = − (Γ + e − ik ) − (Γ + e ik ) . (9)The scattering matrix relates the incoming and outgoingstates Ψ in = A + and Ψ out = A − as Ψ out = S Ψ in . Forthe simple case of just two wires connected with hoppingamplitude t , we haveΓ = (cid:18) − t − t (cid:19) , S = t − t − e − ik t ( e ik − e ik t − e − ik t ( e ik − e ik t − e − ik t − t − e − ik ! . (10)The above scattering matrix at half-filling then yields thefollowing conductance : G = 4 t (1 + t ) e h . (11)As discussed in the Introduction, the key result of thispaper is a general relationship between the conductance and certain static correlation functions in a finite sys-tem. The purpose of this section is to illustrate thisrelationship in the very special case of a simple nonin-teracting system where exact calculations can be donewith elementary methods. Consider the following finitesystem shown in Fig. 4, which consists of our junctionwith hopping t on the left. At a finite length away fromthe junction, the other endpoints of the wires are coupledwith a hopping amplitude − t . Let us now calculate thecurrent-current correlation function h J ( x ) J ( x ) i . FIG. 4: A simple noninteracting system of two wires con-structed with a junction with hopping t and mirror imagewith hopping − t . The quantity of interest is the expectationvalue h J ( x ) J ( x ) i with J α the current operator on wire α . To begin with, let us assume a simple special casewhere t = 1. In this case, our system is a loop, withno impurity, that is threaded with a flux π . This givesrise to anti-periodic BC. In this special case, the Hamil-tonian can be simply diagonalized and we get H = X k − k c † k c k , k = 2 n − N π (12)where c k is the Fourier transform of c j with j = 1 , . . . N and n = − N + 1 , . . . N . We focus on the half-filled caseand assume N is even. The ground state is then givenby a Slater determinant | GS i = c † k . . . c † k N | i and thefermionic correlation function C ( a, b ) ≡ h GS | c † a c b | GS i , where a and b are the site index as in Fig. 5, can bewritten as C ( a, b ) = 12 N N X n =1 h e iπ n − N ( a − b ) + c.c. i . (13)Each term in the above sum corresponds to one filledmomentum level where the contributions of k and − k levels are complex conjugates. The sum is just a geomet-ric series, which can be calculated exactly and the resultis C ( a, b ) = 12 N sin π ( a − b )2 sin π ( a − b )2 N . (14)If J ( m ) (on wire 1) is the current operator betweensites m and m + 1, J ( m ) will be between sites 2 N − m and 2 N − m +1 in the chain. Let us write these operatorsexplicitly as J ( m ) = i (cid:16) c † m c m +1 − c † m +1 c m (cid:17) ,J ( m ) = i (cid:16) c † N +1 − m c N − m − c † N − m c N − m +1 (cid:17) . Using the fermionic Green’s function Eq. (14) and Wick’stheorem, we can explicitly calculate the current-currentcorrelation function h J ( m ) J ( m ) i . The real-space formof Wick’s theorem can be generically written in the form h c † m c † n c i c j i = h c † m c j ih c † n c i i − h c † m c i ih c † n c j i . (15)To calculate h J ( m ) J ( m ) i , we write it as a sum of four quartic terms and reduce each term to a sum ofproducts of single-electron Green’s functions. Notice that C ( a, b ) = C ( a − b ), a function of the distance a − b alone.After some algebra, we can then write the current-currentcorrelation function as h J ( m ) J ( m ) i = 2 C (2 N − m ) (16) − C (2 N − m + 1) C (2 N − m − . By plugging the explicit form of C ( a − b ) from Eq. (14)into Eq. (16), we then obtain the following exact expres-sion for the current-current correlation function: h J ( m ) J ( m ) i = 12 N ( sin [ π ( N − m )]sin (cid:2) πN ( N − m ) (cid:3) − sin (cid:2) π ( N − m ) + π (cid:3) sin (cid:2) π ( N − m ) − π (cid:3) sin (cid:2) πN ( N − m ) + π N (cid:3) sin (cid:2) πn ( N − m ) − π N (cid:3) ) . Consider the above correlation function away from thetwo endpoints N − m ≫ m ≫
1. We canthen approximate the denominator of the second termas sin (cid:2) πN ( N − m ) (cid:3) and write h J ( m ) J ( m ) i = 12 h N sin (cid:16) π mN (cid:17)i − . (17)As we will show later, the expression derived above fora simple noninteracting model has a universal form thatsurvives electron-electron interactions.Before proceeding, let us consider a limit of the aboveexpression. If we take the limit of N → ∞ in Eq. (17)above, we are sending off the right junction to infinityand effectively describing a semi-infinite system. In thiscase, for a distance m (lattice spacing set to unity), weobtain the following correlation function: h J ( m ) J ( m ) i = 12 π m . (18)In the next step, we derive a similar expression for asemi-infinite system but with an impurity consisting of asite with hopping amplitude t . It is illuminating to firstgive a short derivation of Eq. (18) formulated directly inthe limit of the semi-infinite system. In this limit, we cantreat the momenta in the continuum and work directlywith the scattering wave functions, which in the specialcase of t = 1 are just plane waves e ikx due to the absenceof back-scattering. We have right-moving (left-moving)plane waves for 0 < k (0 > k ) and at half-filling, the filledmomentum states have − π < k < π . This leads to thefollowing fermion Green’s function: C ( a, b ) = Z π − π dk π e ika e − ikb = sin π ( a − b )2 π ( a − b ) , (19) which is in fact the limit of N → ∞ of Eq. (14). Insertingthe above expression into h J ( m ) J ( m ) i = 2 C ( a − b ) − C ( a − b + 1) C ( a − b − , where m = ( b − a ) / t = 1. Let us nowconsider a semi-infinite system shown in Fig. 5 with animpurity of arbitrary hopping amplitude t . Now, instead FIG. 5: (Color online) The single-particle scattering statesfor a single-impurity infinitely long system. of the simple plane waves e ikx , we can use the followingsingle-particle scattering states. Let us write the right-moving and left-moving scattering states separately forclarity. We have φ Rk ( x ) = (cid:26) e ikx + r k e − ikx , x t k e ikx , x > φ Lk ( x ) = (cid:26) e − ikx + ˜ r k e ikx , x t k e − ikx , x > . In terms of the above scattering states, we can writethe fermionic Green’s function for a < b > C ( a, b ) = Z π dk π (cid:2)(cid:0) e − ika + r ∗ k e ika (cid:1) t k e ikb +˜ t ∗ k e ika (cid:0) e − ikb + ˜ r k e ikb (cid:1)(cid:3) . (20)Note that in the simple case t = 1, where we have noback-scattering and r k = ˜ r k = 0, Eq. (20) simply re-duces to Eq. (19) above. By using the scattering matrixEq. (10) or more directly from plugging in the scatteringstates into the equations ǫ k φ k (1) = − tφ k (0) − φ k (2) , (21) ǫ k φ k (0) = − tφ k (1) − φ k ( − , (22) we obtain r k = t − e − ik − t and t k = t e − ik − e − ik − t . We also have˜ t k = t k and ˜ r k = e − ik r k . By inserting the above trans-mission and reflection coefficients into the expression forthe Green’s function Eq. (20) and after some algebra, wecan write C ( a, b ) = 4 t Z π dk π sin k sin [ k ( a − b + 1)] t − sin [ k ( a − b − − t cos 2 k + t ≡ C ( a − b ) . (23)This leads to the following expression for the current-current correlation function: h J ( m ) J ( m ) i = 2 C ( − m ) − C ( − m − C ( − m + 1)(24)with C ( x ) defined in Eq. (23). We claim that the asymp-totic behavior of the quantity h J ( m ) J ( m ) i is generi-cally given by ∼ m as in the special case Eq. (18). Tocheck this claim we plot m h J ( m ) J ( m ) i as a functionof m for several values of the hopping amplitude t bystraightforward numerical evaluation of the integral inEq. (23). The results are shown in Fig. 6. FIG. 6: (Color online) The quantity m h J ( m ) J ( m ) i calcu-lated from Eq. (24) by a numerical integration of the integralin Eq. (23). For large m , this quantity saturates to a constantvalue that only depends on the hopping amplitude t . This in-dicates that the asymptotic form of h J ( m ) J ( m ) i is given by ∼ m . Since for each t , m h J ( m ) J ( m ) i saturates to a con-stant, a natural question is how this constant depends onthe value of t . A quite remarkable fact is that the sat-uration value is exactly proportional to the conductanceEq. (11) of this simple junction, which we calculated inthe beginning of this section from the Landauer’s formal-ism. In other words, we have the following asymptotic behavior: h J ( m ) J ( m ) i ≃ π t (1 + t ) m , m ≫ . (25)At this level, this observation appears rather myste-rious. It is not very transparent from the expressionfor h J ( m ) J ( m ) i , which is a rather complicated dou-ble integral, why this ground state correlation functionhas such a simple scaling form. By comparing Eq. (25)above with Eq. (17), we claim that for a finite systemshown in Fig. 4, the asymptotic behavior of the correla-tion function is given by h J ( m ) J ( m ) i ≃ t (1 + t ) h N sin (cid:16) π mN (cid:17)i − , m ≫ . (26)We do not attempt to prove the above expression here.The equation is indeed a very special case of the genericrelationship Eq. (2) that we prove in section V in fullgenerality, i.e., in the presence of electron-electron inter-actions for a rather arbitrary junction with an arbitrarynumber of wires and without assuming symmetries suchas time-reversal.The expression above, however, motivates the mainidea of this paper, i.e., the fact that conductance canbe extracted from ground state expectation values in aclosed system, in a very elementary example. It is alsoan example where the relationship can be derived explic-itly on the lattice instead of resorting to the continuumformalism, which helps clarify the application of the con-tinuum CFT results to a lattice calculation. IV. REVIEW OF BOSONIZATION ANDBOUNDARY CONFORMAL FIELD THEORY
The main results of this paper are derived within theframework of boundary conformal field theory. In thissection, we review the required steps for formulating thegeneric system we would like to study, namely a junctionof M quantum wires modeled as a tight-binding Hamil-tonian with spinless electrons, in the language of the con-formal field theory. The material in this section can beskipped by readers familiar with bosonization and CFT. A. Bosonization of quantum wires
The first step in this formulation is the bosonizationprocedure. Bosonization is a powerful nonlocal transfor-mation that, in one space dimension, allows us to describethe low-energy limit of strongly-correlated fermionic sys-tems as a noninteracting theory of bosons (see, for exam-ple, Ref. 49 for a review). Let us start by considering oneinfinitely long quantum wire with the following Hamilto-nian: H = X i (cid:20) − c † i c i +1 − c † i +1 c i + V ( n i −
12 )( n i +1 −
12 ) (cid:21) , (27)where n i = c † i c i . The first two terms describe electronhopping and the last term is the density-density inter-action. At half-filling, this model exhibits a charge den-sity wave phase transition for large repulsive interactions V >
2. We then see that the ground state spontaneouslybreaks lattice translation symmetry and we get the twodegenerate ground states shown in Fig. 7 for V → ∞ . FIG. 7: The degenerate ground states of the HamiltonianEq. (27) for V → ∞ . Filled circles represent occupied sites. Also at very large attractive interactions, the electronswill clump together and form clusters of neighboring oc-cupied sites. This results in phase separation which, athalf-filling, happens at
V < − − < V <
2, whichincludes the noninteracting fermionic system ( V = 0),the system is in a gapless critical phase known as theLuttinger liquid. This is the regime where bosonizationapplies. Let us first bosonize the noninteracting system.The noninteracting fermions have an ǫ k = − k )dispersion and a ground state consisting of a filled Fermisea up to the Fermi level k F . We can linearize the disper-sion around the Fermi level and only consider the right-moving (left-moving) excitations close to k F ( − k F ).Let us define the following right-moving and left-moving fields: ψ R ( p ) = c ( k F + p ) , ψ L ( p ) = c ( − k F + p ) . (28)We can then write the hopping part of the Hamiltonianas H = v F Z Λ − Λ dp π h pψ † R ( p ) ψ R ( p ) − pψ † L ( p ) ψ L ( p ) i . (29) Upon Fourier transforming, the above equation in realspace reads as H = v F Z dx (cid:20) ψ † R ( x ) 1 i ∂∂x ψ R ( x ) − ψ † L ( x ) 1 i ∂∂x ψ L ( x ) (cid:21) , (30)where ψ ( x ) = e ik F x ψ R ( x ) + e − ik F x ψ L ( x ) . (31)The key step to bosonization is writing the Hamilto-nian (both hopping and interaction part) in terms of thefollowing chiral current operators: j R,L ( x ) = ψ † R,L ( x ) ψ R,L ( x ) (32)instead of the fermionic creation and annihilation opera-tors. One can easily verify using the Hamiltonian H inEq. (30) that these currents satisfy the following commu-tation relations:[ H , j R,L ] = ± iv F ∂∂x j R,L . (33)The interaction term, which is quartic in creation andannihilation operators, will trivially become quadraticin terms of the currents. The nontrivial part of thebosonization procedure is to show that the hopping partof the Hamiltonian, which is quadratic in ψ , will remainquadratic in terms of j .This important result can be shown using the com-mutation relation of the current operators. By Fouriertransforming ψ R as ψ R ( x ) = √ N P k e ikx ψ R ( k ), we canwrite the Fourier transform of the chiral current j R inEq. (32) as j R ( q ) = 1 √ N X k ψ † R ( k − q ) ψ R ( k ) , (34)which leads to the following commutation relation :[ j R ( q ) , j R ( q ′ )] = q π δ q, − q ′ . (35)A similar equation with q π → − q π can be derived for j L .Using the commutation relation Eq. (35), one can showthat the bosonized Hamiltonian H = πv F X q [ j R ( q ) j R ( − q ) + j L ( q ) j L ( − q )] (36)obeys the commutation relations in Eq. (33). We thenargue that Hamiltonian (36) is the bosonized form of thenoninteracting fermionic Hamiltonian (30).By using the commutation relations between j R,L and ψ R,L , one can write the fermionic operators in termsof the currents by introducing bosonic fields φ R,L . Theresult is ψ R,L = 1 √ π e ∓ iφ R,L , j
R,L = 12 π ∂∂x φ
R,L . (37)The interacting Hamiltonian density will now have di-agonal terms of the form j L ( x ), j R ( x ) coming from thehopping and off-diagonal terms of the form j L ( x ) j R ( x )from the electron-electron interaction (back-scattering)and is quadratic in terms of the currents. By perfrominga linear transformation (cid:18) J R J L (cid:19) = (cid:18) cosh β sinh β sinh β cosh β (cid:19) (cid:18) j R j L (cid:19) (38)that preserves the commutation relations, we obtain H ( x ) ∝ (cid:2) J R ( x ) + J L ( x ) (cid:3) . (39)It is convenient to define new bosonic fields ϕ and θ that are linear combinations of φ R,L in Eq. (37) such that ψ R,L ( x ) = 1 √ π e i ( ϕ ( x ) ± θ ( x )) / √ . (40)In terms of these fields and by introducing the renormal-ized charge carrier velocity v and the dimensionless Lut-tinger parameter g , the low energy effective Hamiltoniandensity for a wire can be written as H ( x ) = v π (cid:20) g ( ∂ x ϕ ) + 1 g ( ∂ x θ ) (cid:21) . (41)The bosonized Hamiltonian Eq. (41) is the generic de-scription of 1D metallic quantum wires that we work within this paper. Notice that π ∂ x ϕ is the momentum con-jugate to θ and we have the following commutation rela-tions: [ ϕ ( x ) , θ ( x ′ )] = iπ sgn( x ′ − x ) . (42)The above review covers the main ingredients of thebosonization scheme that we need for setting up the prob-lem at hand in this paper. B. Boundary conformal field theory
Here, we briefly review the basics of CFT and BCFTthat we need in the remainder of this paper. This reviewis largely based on Ref. 51. An important property ofcritical theories, i.e., theories where certain fields knownas the scaling fields have critical correlation functions, isscale invariance. This simply means that the correlationfunctions of scaling fields O i satisfy h O ( b r ) O ( b r ) . . . O n ( b r n ) i = b − P i x i h O ( r ) O ( r ) . . . O n ( r n ) i , (43)where the exponent x i is known as the scaling dimensionof the operator O i and b is an arbitrary scaling factor.A powerful leap from scaling symmetry to conformalsymmetry is by allowing the scaling factor b to varysmoothly, i.e., considering more general transformation r → r ′ than r → br with b ( r ) = | ∂r ′ ∂r | . If we have a local theory and the transformation r → r ′ locally resemblesa scaling transformation (modulo a local rotation) thenwe expect h O ( r ′ ) O ( r ′ ) . . . O n ( r ′ n ) i = Y i b ( r i ) − x i h O ( r ) O ( r ) . . . O n ( r n ) i . (44)In three dimensions, the transformations that locally re-semble scaling are limited but in two dimensions (2D)(or 1+1D quantum systems) any analytic mapping [ z → w ( z ) on the complex plane z ] is conformal (preserves an-gles) and requiring conformal symmetry leads to highlynontrivial results. An important quantity in a CFT is thestress-energy tensor. Consider the action S of a 1+1Dquantum system that is defined on the complex plane z .The stress-energy tensor T µν is defined in terms of thevariations of this action through δS = − π Z T µν ∂ ν α µ d r, (45)where δS is the variation of S due to the infinitesimaltransformation r µ → r µ + α µ ( r ). Another importantingredient of CFT is the operator product expansion(OPE), which describes the nature of the singularitiesin h O i ( r i ) O j ( r j ) . . . i as r i → r j as h O i ( r i ) O j ( r j ) . . . i = X k C ijk ( r i − r j ) (cid:28) O k ( r i + r j . . . (cid:29) . (46)In any CFT, the primary fields are fields for which themost singular term in the OPE of T ( z ) O j ( z j , ¯ z j ) is order( z − z j ) − where T ( z ) ≡ T zz .Let us consider primary fields with the following scal-ing transformation on the complex plane z : O j ( λz, ¯ λ ¯ z ) = λ − ∆ j ¯ λ − ¯∆ j O j ( z, ¯ z ) , where ∆ j and ¯∆ j are called the complex scaling dimen-sions. Notice that the above equation is only a shorthandto describe the scaling behavior of correlation functionsinvolving O . An important result in CFT is that undera generic conformal transformation z → w ( z ), the corre-lation functions of primary fields transform as h O ( z , ¯ z ) O ( z , ¯ z ) . . . i = Y i (cid:12)(cid:12)(cid:12)(cid:12) dw i dz i (cid:12)(cid:12)(cid:12)(cid:12) ∆ i (cid:12)(cid:12)(cid:12)(cid:12) d ¯ w i d ¯ z i (cid:12)(cid:12)(cid:12)(cid:12) ¯∆ i h O ( w , ¯ w ) . . . i . (47)This is in fact the same as Eq. (44) which we intuitivelywrote down in the beginning of this section. We shallemphasize that the transformation (47) only holds forthe primary fields of the theory.So far our discussion has been limited to CFTs onthe entire complex plane z . A CFT on a domain withboundaries is known as boundary conformal field the-ory (BCFT) where in addition to the bulk properties, we0need to specify appropriate boundary conditions (BCs).An important task in BCFT is classifying the conformallyinvariant boundary conditions for a given bulk CFT. The BCFT techniques have proved powerful in study-ing various quantum impurity problems such as theKondo model and Luttinger liquids with impurities.
The effect of the impurity, in this approach to quantumimpurity problems, is to select the appropriate confor-mally invariant BC corresponding to the low-energy RGfixed point. At far away from the boundaries, the bulkCFT is expected to adequately describe the system, i.e.,the boundary conditions do not matter. Also very closeto the boundary, there is nonuniversal short-distancephysics from the microscopic degrees of freedom. Ondistances from the boundary that are much larger thanthe microscopic length scales, but still much smaller thanthe domain size, the correlation functions are affected bythe conformally invariant boundary conditions. As pointed out by Cardy, the BCs in a BCFT are en-coded in boundary states. To understand the notion ofa boundary state, it is helpful to consider the partitionfunction of a CFT on a cylinder with boundary condi-tions A and B on two sides of the cylinder as in Fig. 8. One can write down the partition function using transfermatrices running parallel or perpendicular to the bound-ary. The transfer matrix in the direction parallel to the
FIG. 8: Boundary states in the partition function of a CFTon a cylinder. boundary depends on the boundary conditions A and B and can be written as exp ( − H AB ) (the lattice spacingis set to unity). The imaginary time runs parallel to theboundary and H AB is the Hamiltonian with boundaryconditions A and B . We then have the partition function Z = tr (cid:2) e − β H AB (cid:3) . (48)Alternatively one can construct the transfer matrix in thedirection perpendicular to the boundary. The Hamilto-nian H for this transfer matrix is determined by the bulkonly and is independent of the boundary conditions. Towrite the partition function we need to construct states |Ai and |Bi in the space where the transfer matrix actssuch that the partition function Eq. (48) is also equal to Z = hA| e − ℓ H |Bi . (49)The two different representations of the partition func-tion above illustrate the notion of the boundary state. V. A GENERAL RELATIONSHIP BETWEENTHE CONDUCTANCE AND STATICCORRELATION FUNCTIONS
Let us recall the two difficulties with calculating theconductance of a junction of multiple interacting quan-tum wires, which we discussed in the Introduction. First,conductance is a property of an open quantum system.This means that we think of the wires emanating from thejunction as being attached to reservoirs that can serve assources and drains for the electrons passing through themolecular structure at the junction. To study the con-ductance, we either need to faithfully model the reser-voirs in our theoretical description or assume that wehave infinitely long wires. In the latter case, we basicallysend the reservoirs to infinity. An appropriate boundarycondition is assumed at infinity, which does not enter thetheory of the junction explicitly. For a numerical calcula-tion of conductance, it then seems that we would need tomodel large enough systems that approximate the semi-infinite ones.The second difficulty is that, in the standard linear-response framework, the linear conductance is related todynamical rather than static correlation functions. Thiscan be seen explicitly from the Kubo formula for theconductance tensor G αβ = lim ω → + − e ~ ωL Z ∞−∞ dτ e iωτ (50) × Z L dx hT τ J α ( y, τ ) J β ( x, i , where T τ indicates imaginary-time ordering and thequantity hT τ J α ( y, τ ) J β ( x, i is a dynamical current-current correlation function for currents J α and J β onwires α and β , respectively. It seems that, generically,a numerical calculation of this quantity requires time-dependent methods such, as for example, time-dependentDMRG.With a brute force approach, the numerical calcula-tion of conductance is not feasible for junctions of threeor more wires. In this section, we derive a key relation-ship between the universal conductance and certain staticcorrelation functions in a finite system. This importantrelationship is the basis of a method for numerically cal-culating the conductance that we will discuss in Sec. VI. A. CFT for independent wires
Let us begin by formulating the problem in the BCFTframework reviewed in Sec. IV. We have M quantumwires connected to a junction. Let us first consider thesystem of M independent quantum wires in the absenceof the junction. As argued in Sec. IV A, the effectiveLuttinger-liquid Hamiltonian for one wire can be written1as H = v π Z dx (cid:20) g ( ∂ x ϕ ) + 1 g ( ∂ x θ ) (cid:21) . (51)Using the fact that Π θ = π ∂ x ϕ is the conjugate mo-mentum of θ , the Lagrangian density can be written as L = 14 πg (cid:20) v ( ˙ θ ) − v ( ∂ x θ ) (cid:21) . Let us set the charge carrier velocity v to unity for con-venience. We can then write the Euclidean action as S = 14 πg Z dτ dx ∂ µ θ∂ µ θ. (52)Alternatively the action can be written in terms of thedual field ϕ as S = g π R dτ dx ∂ µ ϕ∂ µ ϕ . It is convenientto represent the points on the ( x, τ ) plane with complexcoordinate z = τ + ix . The system described by themassless action Eq. (52) is a CFT on the complex plane z . Physically, covering the whole complex plane implies1D quantum wires extending from −∞ to ∞ at zero tem-perature ( β → ∞ ).Let us begin by calculating the correlation functions ofthe bosonic fields θ from the action Eq. (52). We have hT τ θ ( x , τ ) θ ( x , τ ) i = R D [ θ ] θ ( x , τ ) θ ( x , τ ) e − S R D [ θ ] e − S = K ( x − x , τ − τ ) , (53)where the propagator K ( x − x , τ − τ ) is the inverseof the operator − πg (cid:0) ∂ x + ∂ τ (cid:1) , i.e., − πg (cid:0) ∂ x + ∂ τ (cid:1) K ( x − x ′ , τ − τ ′ ) = δ ( x − x ′ ) δ ( τ − τ ′ )(54)and is given by K ( x − x ′ , τ − τ ′ ) = − g (cid:2) ( x − x ′ ) + ( τ − τ ′ ) (cid:3) + const . (55)Up to an unimportant additive constant, we can thenwrite the correlation function of the bosonic fields θ as hT τ θ ( z , ¯ z ) θ ( z , ¯ z ) i = − g z − z ) + ln (¯ z − ¯ z ) ] . (56)By differentiating the above Eq. (56) with respect to z and ¯ z , we obtain the following correlation functions: (cid:28) T τ ∂θ∂z ( z , ¯ z ) ∂θ∂z ( z , ¯ z ) (cid:29) = − g z − z ) , (57) (cid:28) T τ ∂θ∂ ¯ z ( z , ¯ z ) ∂θ∂ ¯ z ( z , ¯ z ) (cid:29) = − g z − ¯ z ) . (58)Also we can similarly show that hT τ ∂θ∂z ( z , ¯ z ) ∂θ∂ ¯ z ( z , ¯ z ) i vanishes. The chiral currents below, where we have put in a wireindex α , are then primary fields for this CFT: J αL ( z ) = i √ π ∂ θ α ( z, ¯ z ) , J αR (¯ z ) = − i √ π ¯ ∂ θ α ( z, ¯ z ) , (59)with the notation ∂ ≡ ∂ z = 12 ( ∂ τ − i∂ x ) , ¯ ∂ ≡ ∂ ¯ z = 12 ( ∂ τ + i∂ x ) . In the absence of a boundary, the chiral currents indifferent wires are uncorrelated and the only correlationsare between chiral currents of the same chirality in thesame wire. The only nonvanishing correlation functionsof the chiral currents are then hT τ J αL ( z ) J βL ( z ) i = g π δ αβ ( z − z ) , hT τ J αR (¯ z ) J βR (¯ z ) i = g π δ αβ (¯ z − ¯ z ) . (60)Note that the fact that there is no correlation betweenthe left-movers and the right-movers does not mean thatthere is no back-scattering in the system. Indeed, weactually have back-scattering in the presence of electron-electron interactions. The chiral currents J L,R are thechiral eigenmodes of the interacting theory which, as ex-plained in Eq. (38), are in fact linear combinations of thebare (noninteracting) chiral currents j L,R . Interactionsinduce back-scattering for these bare currents j L,R .The form of the correlation functions above is highlyconstrained by conformal symmetry and is basically de-termined by the scaling dimension of the primary opera-tors. A very important observation is that space and timeare tied together in the above correlation functions anda measurement of static correlation functions uniquelydetermined the dynamical ones. Let us note that thesechiral currents are related to the physical current J andthe density fluctuation ρ through J α = v ( J αR − J αL ) , ρ α = J αR + J αL . The velocity of the charge carriers v coincides with theFermi velocity only in the absence of electron-electroninteractions. For simplicity, we shall set v to unity in theremainder of this section. B. BCFT for the semi-infinite system
Let us now return to the problem of the quantum junc-tion. The system of M independent wires is described bya CFT of M bosonic fields living on the full complex z plane. The junction of the M quantum wires at the RGfixed point is described by BCFT on the upper-half com-plex plane. The real axis x = 0 is the boundary of thedomain where the CFT lives. We expect to have thesame bulk CFT as the system of M independent wires ifwe are infinitely far away from this boundary.2This is an example of a quantum impurity problem. Animportant hypothesis, which has been repeatedly verifiedin a multitude of quantum impurity problems, such asthe single-channel and multichannel Kondo model, isthat at the RG fixed points, the conformal symmetry inthe bulk terminates smoothly. More precisely, the hy-pothesis states that the BC on the boundary of the do-main in the complex plane is conformally invariant andtherefore the quantum impurity problem at the RG fixedpoint is described by a BCFT.How can the presence of the boundary change the cor-relation functions? In other words, what is the differencebetween the correlation functions in the CFT on the full z plane and the BCFT on the upper half plane? A keyobservation is that the presence of the boundary does notchange the correlation functions between chiral currentsof the same chirality. On physical grounds, it is easy tosee that the left movers coming from infinity toward thejunction have not yet felt the presence of the junctionand, just by causality, can not have different correlationfunctions than in the system of M independent wires. Forthe right movers, on the other hand, we need the confor-mal invariance of the boundary conditions to make sucha statement.If we have a conformally invariant BC on the real axis, h J αR (¯ z ) J αR (¯ z ) i will be the same as on the full complexplane. This is generically the case in the limit when thetwo points z and z are far away from the boundary anddeep in the bulk of the system. With conformally invari-ant BCs, however, these points can be brought close tothe boundary through conformal transformations, suchas z → λz or z → − /z , from the upper half-plane ontoitself, which leaves h J αR (¯ z ) J αR (¯ z ) i invariant. The aboveargument is not a rigorous proof, but may give some intu-ition. Notice that irrelevant boundary operators can givecorrections to h J αR (¯ z ) J αR (¯ z ) i in the semi-infinite system,which decay as a power law with the distance from theboundary with exponents larger than 2. In a finite sys-tem, both h J αL ( z ) J αL ( z ) i and h J αR (¯ z ) J αR (¯ z ) i will havesuch corrections.The presence of a boundary, however, does introducenew correlations between the left and the right movers.From the scattering picture, we can understand this inthe sense that left movers must go through the junctionbefore turning into right movers. The form of the newcorrelation functions is determined by the scaling dimen-sion of the current operators, and as far as these corre-lation functions are concerned, all the information aboutthe boundary condition at x = 0 is encoded in an overallcoefficient A αβ B .In summary, in the BCFT describing the junction atthe RG fixed point, in addition to the correlation func-tions Eq. (60), we have the following nonvanishing corre-lation functions : hT τ J αR (¯ z ) J βL ( z ) i = − g π A αβ B z − z ) (61)and a similar equation for hT τ J αL ( z ) J βR (¯ z ) i . The coefficients A αβ B are determined by the confor-mally invariant boundary condition on the real axis. Infact we can write this coefficient in terms of the boundarystate |Bi as g π A αβ B = h J αR J βL , |Bih |Bi , (62)where | O, i is the highest weight state corresponding toa generic operator O (here O = J αL J βR ) and | i is theground state. The definition of the highest weight stateand a derivation of Eq. (62) is given in appendix B.One important observation is in order. While theHamiltonian is local in terms of the fermionic degreesof freedom, the mapping from fermion to bosons, i.e,the bosonization procedure, is highly nonlocal. Onemust notice, however, that regardless of the number ofwires, there is a fundamental difference between a two-dimensional system and this quasi-1D system. The non-locality only shows up along the boundary itself. Thestatistics of the microscopic degrees of freedom can af-fect what the boundary condition B , and consequentlythe coefficient A αβ B , is but it does not change the factthat the effect of the junction can be reduced to a BC at x = 0. The generic form Eq. (61) holds even though thebosonic fields are nonlocal objects in terms of the originalfermions.So far, we have established that the current-currentcorrelation functions appearing in Eq. (50) have a univer-sal form modulo coefficients A αβ B , which depend on theboundary conditions. We should then be able to expressthe conductance G αβ in terms of these coefficients A αβ B by performing the integrals in the Kubo formula. Con-sider the conductance G αβ between two distinct wires α = β . Using J = J R − J L (velocity set to unity), we canwrite hT τ J α ( z , ¯ z ) J β ( z , ¯ z ) i = g π (cid:20) A αβ B z − z ) + A βα B z − ¯ z ) (cid:21) . Note that the presence of the boundary does notchange the correlation between currents of the same chi-rality, and the chiral correlation functions are obviouslyequal to zero in the half-plane for α = β (because theyare proportional to δ αβ ). Plugging the above equationinto the Kubo formula Eq. (50) (with z = τ + iy and z = ix ) gives two similar terms. Let us first perform theintegral over the imaginary time τ . We can write Z + ∞−∞ dτ e iωτ ( τ − iu ) = − πωH ( u ) e − ωu , u = 0 (63)which can be easily derived by a contour integration.Here H ( u ) is the Heaviside step function H ( u ) = 1(0)for u > u < G αβ = lim ǫ → g e h L Z Lǫ dx h A αβ B H ( x + y ) + A βα B H ( − x − y ) i , which simply yields G αβ = A αβ B g e h , α = β. (64)In the above expression, since both x and y are posi-tive (the wire extends from x = 0 to + ∞ ), the secondterm identically vanishes. Notice that determining theoff-diagonal elements G αβ with α = β is enough to alsodetermine the diagonal elements of the conductance ten-sor G . The elements are not independent and satisfy therelations X β G αβ = 0 , X α G αβ = 0 . (65)The first relation follows because applying the same volt-age to all the wires leads to zero current, and the secondrelation follows from current conservation.Notice that in deriving the above expression (64), weassumed that the interacting Luttinger-liquid wires ex-tend to infinity. In many experimental situations, theinteracting wires are attached to noninteracting Fermiliquid leads. If the interacting region is long enough,the effect of the Fermi-liquid leads can be taken into ac-count by considering a contact resistance as discussedlater in Sec. VIII. Short interacting wires attached toFermi-liquid leads can be considered as part of the junc-tion, while leads constitute quantum wires with g = 1,which extend to infinity. The possible crossover behavioras a function of the length of the interacting region in thepresence of infinite noninteracting leads is an interestingopen question.One of the two difficulties with a numerical calcu-lation of the universal conductance is now effectivelysolved. The conductance is uniquely determined by thecoefficient in front of the dynamical correlation functionEq. (61). But, conformal symmetry determines the uni-versal form of this correlation function. In other words,space and time are tied together by conformal symme-try, and the coefficient A αβ B can be extracted from astatic correlation function alone. More explicitly, con-sider the static correlation function h J αR ( x ) J βL ( x ) i that ismerely a special case of the generic correlation function hT τ J αR ( ¯ z ) J βL ( z ) i in Eq. (61) for z = z = ix and isgiven by h J αR ( x ) J βL ( x ) i = g π A αβ B x ) . (66)The above correlation function is just a ground-state ex-pectation value in a semi-infinite system. If we can nu-merically compute this ground-state expectation value and, as a function of the distance x from the boundary,fit it to the power-law form above, we can extract thecoefficient A αβ B and uniquely determine the conductance.Because the BCFT description applies only at low en-ergies and large distances, the behavior of the system atshort distances (in units of the lattice spacing) is nonuni-versal and depends on the microscopic details rather thanthe continuum limit. Even at distances where the con-tinuum limit is valid, the expression above only givesthe asymptotic behavior of the correlation function. Theconformally invariant boundary condition B describes theRG fixed point, but, generically, there are corrections tothe BCFT prediction Eq. (66) from irrelevant boundaryoperators at finite distances. These corrections of coursedie out faster than x , and at large distances the asymp-totic behavior is given by the leading term Eq. (66). C. Conformal mapping from the semi-infinite to afinite system
Having solved one of the difficulties of calculating theconductance, we now turn to the other difficulty. Namely,the fact that modeling an open, i.e., semi-infinite, sys-tem requires finding correlation functions in very largesystems. Let us go back to the BCFT description ofthe semi-infinite junction for which the conductance iswell defined. We argued that this system is described bya BCFT on the upper half-plane. As explained below,we can use a conformal mapping to go from the upperhalf-plane to a strip of width ℓ , which describes a finitesystem of length ℓ at zero temperature. We will show inwhat follows that the correlation function h J αR ( x ) J βL ( x ) i in this finite system also has a universal form and allowsthe extraction of the key coefficient A αβ B .Let us first explain the mapping, shown in Fig. 9, fromthe upper half-plane z = τ + ix to a strip w = u + iv .The mapping is given by w = ℓπ ln z, z = e πℓ w . (67)In polar coordinates we can write z = re iφ on the half-plane with 0 ≤ r ≤ ∞ and 0 ≤ φ ≤ π which gives u = ℓπ ln r and v = ℓπ φ . The x > z plane is mapped to the boundary v = 0 and the x < v = ℓ .In the finite system of length ℓ , which we obtain fromthe conformal mapping above, the correlation functionsbehave in a universal manner. Let us formally considerthe static correlation function of Eq. (66), which is onlya function of the distance from the boundary x , as afunction of z = τ + ix but with no dependence on τ . Ifwe have a primary operator O (here O = J αR J βL ) in thesemi-infinite plane with h O ( z ) i = A O B (2 x ) − X O , (68)4 x v uτ ℓ FIG. 9: (Color online) The conformal transformation fromthe upper half-plane to a strip of width ℓ . The physical sys-tem described by the half-plane is a semi-infinite junction atzero temperature with the imaginary time running from −∞ to + ∞ and the position along the wires running from 0 to+ ∞ . The physical system corresponding to the strip is a fi-nite system of length ℓ at zero temperature. The structure ofthe junctions at the two end-points of this finite system mustgive rise to the appropriate boundary conditions consistentwith the conformal mapping. we can use the expression Eq. (47) to write the expecta-tion value of the same operator in the strip as h O ( w ) i = | dwdz | − X O h O ( z ) i = A O B (cid:18) ℓπ x | z | (cid:19) − X O . (69)Using | z | = e π ℓ ( w + w ∗ ) = e πℓ u and x = Im e πℓ w = e πℓ u sin (cid:0) πℓ v (cid:1) , we obtain h O ( w ) i = (cid:20) ℓπ sin (cid:16) πℓ v (cid:17)(cid:21) − X O , (70)where v = Im( w ) is physically the distance from the leftboundary ( v = 0) in the finite system (see the right-handside of Fig. 9). Since the zero-temperature finite system isinvariant under translations in u , the static ground-stateexpectation value of the local operator J αR ( x ) J βL ( x ), inthe finite system of length ℓ obtained from the transfor-mation Eq. (67), is expected to behave as h J αR ( x ) J βL ( x ) i GS = g π A αβ B h (cid:16) πℓ x (cid:17) / πℓ i − (71)with x the distance from aboundary.A simple check of the above expression is that it re-produces, as expected, the correct power-law expectationvalue of the semi-infinite system Eq. (66) in the limit ℓ → ∞ . By combining the above expression with theresult of Eq. (64), we obtain G αβ = h J αR ( x ) J βL ( x ) i GS h ℓ sin (cid:16) πℓ x (cid:17)i e h . (72)Notice that as long as the correlation function in theabove expression is calculated for large enough x , i.e.,larger than the microscopic length scales and the re-quired healing length for the contributions of the irrele-vant boundary operators to become negligible, the valueof G αβ should be independent of x . Practically, one canextract G αβ by fitting the calculated h J αR ( x ) J βL ( x ) i GS tothe form of the universal sine function. Notice that the assumption that all wires have the same Luttinger pa-rameter g was not essential for the derivation of Eq. (72).This is because the generic form of Eq. (61) holds evenif the wires α and β have different Luttinger parameters.The overall coefficient of z − z ) however will dependon the two Luttinger parameters as well as the boundarycondition B .The Eq. (72) is one of the key results of this paper. It issignificant both from a fundamental and a practical pointof view. From a fundamental viewpoint, we think aboutconductance when we have a state with currents flowingthrough the system from a source and into a drain. It isremarkable that one can construct a closed finite system,the ground state of which fully encodes the conductance,a quantity defined in open systems. From the practicalpoint of view, the above expression is the basis for amethod of calculating the universal conductance that wediscuss in the next section where we show how to applythis continuum result to a lattice computation and howto implement the boundary conditions. VI. A METHOD FOR CALCULATING THECONDUCTANCE
In the previous section, we derived a general expres-sion that related the off-diagonal elements G αβ of theconductance tensor to the static correlation functions h J αR ( x ) J βL ( x ) i . The key to this relation is that both theconductance and the static correlation function above areproportional to only one coefficient that depends on theconformally invariant boundary condition at the junc-tion. This relationship can be used as the basis of amethod for calculating the conductance.If we measure, numerically, the static correlation func-tion h J αR ( x ) J βL ( x ) i in a finite system obtained from theconformal transformation Eq. (67), we can then fit thecorrelation function (a function of x ) to the universalform Eq. (71) and obtain the coefficient A αβ B . It is impor-tant to note that the universal expression is the asymp-totic form of the correlation function and the fit shouldbe done in a region far away from the boundary. Weneed to consider large enough x so that the microscopicdetails become unimportant and the corrections due toirrelevant operators become negligible.To carry out this scheme, we need to compute h J αR ( x ) J βL ( x ) i in the ground state. The computation isamenable to the time-independent DMRG method. Inthe quasi-1D system obtained by having all wires run-ning parallel to one another, this correlation function isactually the expectation value of a local operator, whichmakes it a particularly easy quantity to calculate withDMRG. We need to answer two questions, however, be-fore we can proceed.1. Since we cannot model chiral creation and annihi-lation operators on a lattice, how can we model theoperator J αR ( x ) J βL ( x )?52. What exactly is the finite system that we needto put into a computer? In other words, givena tight-binding Hamiltonian for the original semi-infinite system, what is the Hamiltonian of thefinite system for which the correlation function h J αR ( x ) J βL ( x ) i has the universal form we derived inthe continuum?To answer the first question, we make use of the follow-ing continuum relationship between the chiral currentsand the total current and the total charge density: J α = v ( J αR − J αL ) , N α = J αR + J αL . (73)Notice that since with electron-electron interactions thesechiral currents are not the bare left and right-movingcurrents, the velocity v is not the Fermi velocity but therenormalized velocity of the charge carriers that can becalculated from the Bethe ansatz and is given by thefollowing expression at half-filling: v = π p − ( V / arccos ( V / , (74)where we have set the hopping amplitude to unity and V is the interaction strength in the bulk lattice HamiltonianEq. (27). From the Bethe ansatz, we can also obtain theeffective Luttinger parameter in terms of the microscopiclattice Hamiltonian and, at half-filling, we have g = π − V / . (75)Can anything go wrong if we apply the continuum rela-tion (73) to a lattice computation? The relation Eq. (73)is only exact in the continuum. However, we find thatit gives very accurate results on the lattice for correla-tion functions between different wires α = β . We will seethis explicitly in the numerical studies of the followingsections. The simple lattice calculations we performed inSec. III also shed some light on this issue.These off-diagonal correlation functions are those thatwe actually need for the conductance calculation, as inour approach, we first find the off-diagonal elements ofthe conductance tensor, which are simply proportionalto A αβ B . The diagonal ones can be later deduced fromEq. (65). Now, we would like to find the off-diagonalelements of the conductance tensor G αβ , and we useEq. (73) as if it were an operator identity. Using Eq. (73),we can then write, for α = β , h J α ( x ) J β ( x ) i = − v (cid:16) h J αL ( x ) J βR ( x ) i + h J αR ( x ) J βL ( x ) i (cid:17) , h N α ( x ) J β ( x ) i = v (cid:16) h J αL ( x ) J βR ( x ) i − h J αR ( x ) J βL ( x ) i (cid:17) . The correlation function h J αR ( x ) J βL ( x ) i is then simplygiven by h J αR ( x ) J βL ( x ) i = − v α v β h J α ( x ) J β ( x ) i− v β h N α ( x ) J β ( x ) i , (76) where v α ( v β ) is the velocity in wire α ( β ). Throughoutthis paper, we work with wires with the same Luttingerparameter and therefore v α = v β = v but the more gen-eral expression above can be used in cases where the wireshave different Luttinger parameters.In writing the above expressions, we have made use ofthe fact that the chiral correlation functions of the form h J αR ( x ) J βR ( x ) i vanish in this finite system for α = β . Weargued in section V that these correlation functions van-ish in the semi-infinite system (upper half-plane). Sincethe finite system corresponds to the strip obtained fromthe upper half-plane by a conformal transformation, theyalso vanish in the finite system.Note that if we have time-reversal symmetry, as in thecase of the simple two-wire problem studied in section III,the second term in the above equation vanishes. Thisis because (nonchiral) currents J are odd under time-reversal and the densities N are even. If the structureat the junction does not include magnetic fluxes, time-reversal symmetry remains unbroken and we only needto calculate one current-current correlation function.The operators J and N are easily modeled in termsof the lattice creation and annihilation operators. Twoimportant observations are in order. First, since the cur-rent operator is defined for a bond, we have to write aneffective bond density so the current and density are as-sociated with the same position x . This is done by takingthe average of the two site densities. The second obser-vation is that bosonization is done for the charge densityfluctuations and the background charge should be sub-tracted to construct the operator N . With these twoobservations, we can explicitly write J α ( m + 12 ) = i ( c † α,m +1 c α,m − c † α,m c α,m +1 ) ,N α ( m + 12 ) = 12 ( n α,m + n α,m +1 − h n α,m i − h n α,m +1 i ) . We now address the question of implementing theboundary conditions. To measure the expectation valuesabove, we need the ground state of a finite system. Whatwe need to do is write a lattice Hamiltonian that corre-sponds to the continuum system we obtained from a con-formal mapping. As discussed in the Introduction, thestarting point is the Hamiltonian of the junction, whichgenerically can be written as H = H boundary + H bulk . The finite system has two boundaries on the left and onthe right and as shown in Fig. 10, generically, a Hamil-tonian H ′ = H + H ′ bulk + H ℓ . (77)Since the bulk theory is conformally invariant, we canwrite H ′ bulk by using exactly the same terms as in H bulk ,but just a finite number of them. The more subtle issue isto identify the boundary Hamiltonians H and H ℓ that6 FIG. 10: (Color online) Given the bulk and boundary Hamil-tonians of a semi-infinite system, the goal is to obtain the twoboundary Hamiltonians H and H ℓ of a finite system corre-sponding to the conformal transformation to the strip. give the correct conformally invariant boundary condi-tions. There are some details regarding persistent cur-rents and the parity of the number of electrons that wediscuss later, but for now let us give a generic argumentregarding the boundary conditions.The semi-infinite system that corresponds to theupper-half plane has one boundary and a conformallyinvariant boundary condition on this boundary, i.e., thereal axis. Now, a conformally invariant boundary con-dition, by definition, does not change under a conformaltransformation. The strip on the w = u + iv plane, how-ever, has two boundaries at v = 0 and v = ℓ . Let usreview the conformal transformation w = ℓπ ln z . Writ-ing z in polar coordinates as z = re iφ gives v = ℓπ φ ,so the left boundary comes from the positive real axis( φ = 0) and the right boundary from the negative realaxis ( φ = π ). Therefore, both these boundaries are ex-pected to have the same boundary condition as the realaxis in the semi-infinite system.Now, in the semi-infinite system, this unknown confor-mally invariant boundary condition is stabilized by thestructure and interactions of the junction. That is tosay a cap with Hamiltonian H boundary gives rise to theboundary condition B at x = 0. It is then a completelynatural assumption that the same cap should give theboundary condition B at the left boundary of the finitesystem. This simply yields H = H boundary . The boundary Hamiltonian on the other side ( H ℓ ),however, is generically different than H . Let us for con-venience represent the distance from the left boundary by x in the finite system as well as in the semi-infinite one.As argued above, we wish to have the same boundarycondition on both sides. This does not mean, however,that we should place the same cap on both ends of thefinite system. To understand the issue, let us considerthe case of noninteracting electrons, where the boundarycondition can be simply written in the form of a single-particle scattering matrix.Suppose the S matrix of the left boundary ( x = 0) is S . As seen in Fig. 11, the incoming scattering statesat x = 0 are the right-moving fermions ψ R (0) and theout-going states are the left-moving fermions ψ L (0). Theboundary condition can then be written in terms of S as ψ R (0) = S ψ L (0) . (78)As evident from Fig. 11, the left and the right movers FIG. 11: (Color online) A generic finite system with twoboundaries. For noninteracting fermions, the boundary con-ditions can be written in the form of scattering matrices thatrelate the in-coming and out-going states. The correspon-dence of these states with the chiral fermions is illustrated atboth boundaries. switch roles at the two boundaries. Namely, the leftmovers (right movers) are the out-going (in-coming) scat-tering states for the right boundary and the in-coming(out-going) scattering states for the left boundary. If thescattering matrix corresponding to the Hamiltonian ofthe right cap H ℓ is S ℓ , we have by definition ψ L ( ℓ ) = S ℓ ψ R ( ℓ ) . (79)However, we want to have the same boundary condi-tion at the two boundaries. This means that to write theboundary condition of the right boundary at x = ℓ , weshould write exactly the same expression as Eq. (78) butreplace the argument x = 0 with x = ℓ , which yields ψ L ( ℓ ) = S − ψ R ( ℓ ) = S † ψ R ( ℓ ) . (80)Comparing Eqs. (79) and (80) leads to the simple butimportant equation S ℓ = S − = S † . (81)Still working with a noninteracting system with theboundary condition (78) for the semi-infinite system, letus derive the boundary conditions of the finite systemmore carefully. The chiral fermionic creation and anni-hilation operators are in fact primary fields with scalingdimension 12 and transform asΨ L ( w ) = (cid:18) dwdz (cid:19) − Ψ L ( z ) , Ψ R ( ¯ w ) = (cid:18) d ¯ wd ¯ z (cid:19) − Ψ R (¯ z ) . (82)By using the derivatives dwdz = ℓπ z = ℓπ e − πℓ w , d ¯ wd ¯ z = ℓπ z = ℓπ e − πℓ ¯ w , we find that, at the two endpoints of the finite system, d ¯ wd ¯ z = dwdz (cid:12)(cid:12) v =0 = ℓπ e − πℓ u , (83) d ¯ wd ¯ z = dwdz (cid:12)(cid:12) v = ℓ = − ℓπ e − πℓ u . (84)7The minus sign above indicates that there could besubtle issues with the assumption of having the sameboundary condition at both ends, which lead us to writeEq. (80). Of course, notice that the minus-sign differ-ence between the transformed chiral fermions at the twoendpoints of the finite system is not inconsistent withEq. (80) because we also need to take the square rootof dwdz and d ¯ wd ¯ z as seen from Eq. (82). At this level, itis not clear, generically, which branch of the square rootwe should pick, but the two choices lead to the follow-ing two possibilities for the scattering matrix at the rightendpoint: I : S ℓ = S − , II : S ℓ = − S − . The above relations can be thought of as two micro-scopic
BCs, which in conjunction with other microscopicdetails such as the number of electrons in the systemgive rise to the correct BC in the continuum. Althoughwe can microscopically implement both of these scatter-ing matrices, we work with the relation I above in thispaper. Our approach is to use the relation I and choosethe number of electrons such that, as discussed below,no persistent currents are generated in the finite systemwith this microscopic BC.The presence of persistent currents can give correc-tions to the CFT predictions. The reason persistent cur-rents need to be avoided is that the best region for fittingthe correlation function h J αR ( x ) J βL ( x ) i is away from theboundaries and close to the center of the finite systemwhere the effect of the irrelevant boundary operators isnegligibly small. In the center of the finite system, thecorrelation function will be O ( 1 ℓ ) and since persistentcurrents go as 1 ℓ , the corrections to h J αR ( x ) J βL ( x ) i , whichare not accounted for in our formulation, become of thesame 1 ℓ order as the value of the h J αR ( x ) J βL ( x ) i correla-tion function.As an example for how to avoid persistent currents con-sider a two-wire system with the boundary condition I,i.e., the system shown in Fig. 4 of Sec. III. We find thatthe persistent currents are avoided by having an evennumber of electrons. This can be seen by noting that forhopping amplitude at the junction t = 1 (a simple loopwith a π flux and no actual impurity), the allowed mo-mentum levels exclude k = 0 and are symmetric aroundit. Therefore, filling in an odd number of electrons leadsto degeneracies.Having gained some intuition regarding the bound-ary conditions in the absence of electron-electron inter-actions, we now tackle the problem of constructing theHamiltonian H ℓ of the right cap from that of the left,i.e., H = H boundary . We do this in two steps describedbelow.First, consider the case where the interactions are onlyin the wires and H boundary is quadratic. In this case weimagine turning off the bulk interactions. We then en-gineer a Hamiltonian H ℓ in a completely noninteracting system, which gives the scattering matrix S ℓ = S † at thedesired filling. We now have a microscopic system withthe same boundary conditions at both endpoints. Turn-ing on the interactions causes the system to flow awayfrom the original noninteracting fixed point. The bound-ary conditions at both endpoints also flow away. How-ever, since they were the same initially, we expect themto flow together and reach a new BC. The two bound-aries are then expected to have the same BC even withinteractions.The next step is to find symmetry transformations thatautomatically generate the desired Hamiltonian H ℓ fromthe given H . In what follows we argue that at half-filling, a combination of time-reversal and particle-holetransformations does the job. Having the boundary con-ditions implemented with such simple transformationsstrongly suggests that these transformation should alsoimplement the correct BCs even when we have interac-tions in the junction itself as well as in the bulk of thewires.Our starting point for this procedure is an explicit ex-pression for the scattering matrix from Sec. III, whichwe repeat here for convenience: S ( k ) = − (Γ + e − ik ) − (Γ + e ik ) , (85)where Γ is an M × M Hermitian matrix defining theboundary Hamiltonian as H boundary = Ψ † Γ Ψ , with the notation defined in Sec. III right above Eq. (7).At half-filling, we have k F = π S F = − Γ + i Γ − i . (86)Note that it is important that the correct boundary con-dition is implemented at the Fermi level. Neither theconductance nor the behavior of the static h J αR ( x ) J βL ( x ) i correlation functions are strongly affected by what hap-pens deep inside the Fermi sea. It is evident from theform of the scattering matrix in Eq. (86) that we need tochange Γ → − Γ to invert the scattering matrix. In otherwords, if we have H = Ψ † Γ Ψ , H ℓ = Ψ † ℓ Γ ℓ Ψ ℓ , (87)and we require S R ( k F ) = S † ( k F ) at the half-filling, wemust have Γ ℓ = − Γ . This is one example where we in fact engineered theHamiltonian H ℓ to implement the boundary condition I.It is very helpful to find symmetry transformations thatautomatically reproduce H ℓ from H . We argue that thetransformation Γ → − Γ for the quadratic Hamiltonian8 H is a combination of time-reversal and particle-holetransformations.Let us call these transformations K and C , respec-tively. Since here we are considering spinless fermions,the time-reversal transformation is simply equal to thecomplex conjugation. Note that generically the time-reversal operator is anti-unitary and can be written asthe product of the complex conjugation and a unitary op-erator. The particle-hole transformation simply switchesthe role of creation and annihilation operators. We canthen write K ( i ) = − i, C ( c ) = c † , C ( c † ) = c. (88)Acting with the particle-hole transformation on the leftboundary Hamiltonian of the form Eq. (87) yields C ( H ) = C (cid:16) Ψ † Γ Ψ (cid:17) = M X α,β C (cid:16) Γ αβ c † α, c β, (cid:17) = − X α,β Γ αβ c † α, c β, + const . Neglecting an unimportant constant, we can write theabove transformation in a more compact form as C ( H L ) = − Ψ † Γ T0 Ψ = − Ψ † Γ ∗ Ψ , where we have used the Hermiticity of Γ in the secondstep. We can then write K ( C ( H )) = − H = H ℓ . (89)So far we have verified that for a simple junction de-scribed by the boundary Hamiltonian Eq. (87), actingwith K and C transformation on the boundary Hamil-tonian inverts the Fermi-level scattering matrix at half-filling. We conjecture here that this is a more genericstatement, and acting with these transformations onthe boundary Hamiltonian gives a microscopic boundaryHamiltonian that correctly implements the same bound-ary conditions at both endpoints. To support this conjec-ture, we test it on a more complex junction constructedby adding one more column of sites to the system. Thiscalculation is presented in Appendix A.The method developed in this paper can also be ex-tended to study systems with spin- electrons. If there isno electron-electron interaction in the bulk of the wires,we can visualize the spin-up and down sectors as twodistinct wires of spinless fermions. This constructioncan be used to engineer the Hamiltonian of the mirror-image junction from the scattering approach. In the pres-ence of interactions in the bulk, the Hamiltonian of eachwire can be written as a sum of a charge-sector Lut-tinger liquid with velocity v c and Luttinger parameter g c and a spin-sector Luttinger liquid with velocity v s and Luttinger parameter g s . The conductance is thenrelated to the the chiral current-current correlation func-tion h J αcR ( x ) J βcL ( x ) i in the charge sector. To measure this correlation function, we need to calculate v c in terms ofthe microscopic parameters such as the strength of theon-site Hubbard interaction U n ↑ n ↓ , where ↑ ( ↓ ) indicatesspin up (down). This is a bulk property and can becalculated numerically or via the Bethe ansatz. The cor-relation function h J αcR ( x ) J βcL ( x ) i can then be computednumerically using the following relations: J α ↑ + J α ↓ = v c ( J αcR − J αcL ) , N α ↑ + N α ↓ = J αcR + J αcL , which are analogous to Eq. (73). These relations lead toa similar equation to Eq. (76).One final comment is in order regarding the efficientcomputation of the required correlation functions withDMRG. When interactions are confined to the junctionitself, i.e., in the absence of interactions in the bulk of thewires, it is in some cases possible to perform a canonicaltransformation, which decouples some of the degrees offreedom in the system as recently shown in Ref. 59. Suchtransformations when possible can reduce the dimensionof the local Hilbert space in DMRG calculations and leadto significant performance improvements. VII. NUMERICAL TESTS AND BENCHMARKSA. Noninteracting Y junction
The first set of numerical tests and benchmarks we per-form is with a noninteracting Y junction. The system weconsider has three noninteracting fermionic chains withhopping amplitude of unity. At the junction, each chainis coupled to the other two chains with hopping ampli-tude t and the loop formed at the junction is threadedwith a magnetic flux Φ. The gauge-invariant flux Φ canbe modeled in many different ways in the tight-bindingHamiltonian such as, for example, the following bound-ary Hamiltonian: H = − Ψ † te i Φ tte − i Φ tt t Ψ . (90)Using Eq. (9) and the Landauer formula (6), we obtainthe following exact expression for the conductance of thissimple junction: G , = 4 t (1 + t ± t sin Φ)1 + 6 t + 9 t + 4 t cos Φ e h . (91)We can now use exact diagonalization of the single-particle Hamiltonian, which can be written as an 3 N × N matrix (with N the number of sites in each leg and 3 N the total number of sites in the system), to obtain the h J α ( x ) J β ( x ) i and h N α ( x ) J β ( x ) i ground-state correla-tion functions. Notice that since the total Hamiltonian isquadratic, Wick’s theorem [Eq. (15)] holds. Let us labelall the 3 N sites in the multi-wire system in some orderas, for example, in Fig. 12, i.e., assign an annihilation9operator c α , α = 1 . . . N to every site. All we need tocalculate are then fermionic Green’s functions of the form C ( α, β ) = h c † α c β i . (92) FIG. 12: A labeling of system sites for exact diagonalization.
Diagonalizing the 3 N × N single-particle Hamiltoniangives single-particle wave functions | Ψ ǫ i = P α φ ǫα | α i where | α i is a basis state of the single-particle Hilbertspace with site α occupied and all other sites empty and ǫ represents the energy of the single-particle level. Thesewave functions are obtained directly from exact diagonal-ization. At a given filling ν , the many-body wave functionis a Slater determinant of 3 νN low-energy single-particlewave functions. The wave functions of the filled levelsare then φ ǫα for ǫ < ǫ F where ǫ F is the Fermi energy. Wecan then write C ( α, β ) = X ǫ<ǫ F φ ∗ ǫα φ ǫβ . (93)We calculate, through exact diagonalization, the cor-relation function h J L ( x ) J R ( x ) i as explained above andplot their logarithms versus ln (cid:2) ℓπ sin (cid:0) x πℓ (cid:1)(cid:3) for differentvalues of the hopping t and for two different fluxes Φ = 0and Φ = π/
2. The results for Φ = 0 are shown in Fig. 13.As expected from the analysis of previous sections, faraway from the junction, these plots are lines of slope − B. Junction of two interacting quantum wires
The next benchmark we perform is for two interactingwires with an impurity. As shown in Ref. 60, this systemexhibits universal behavior. For attractive interactions,any impurity should heal at the RG fixed point, leadingto a conductance of g e h , while for repulsive interactions,any impurity should effectively cut the chain resulting inzero conductance.First, we show benchmarks for the attractive case.Here, of course, we are dealing with a strongly correlatedsystem and exact diagonalization of the single-particleHamiltonian can not be used. Due to the exponen-tially large dimension of the Hilbert space, we cannot t = 1 t = 2 t = 3 t = 4 t = 5 ln π sin( x π ) FIG. 13: (Color online) The inset shows the noninteracting Yjunction used for the numerical test of the method. The thickbonds at the junction have a different hopping amplitude t than the thin bonds in the bulk of the wires, the hoppingamplitude of which is 1. The static correlation function isfor a finite system of length ℓ = 99 (100 sites in each wire)for Φ = 0 and five different values of the hopping amplitude t . The lines drawn through the data points show the CFTprediction with the exact conductance Eq. (91). The strongagreement with the numerical data confirms that our method,i.e., extracting the conductance from fitting the correlators,yields accurate results. -3 -2 -1 0 1 2 3 4 500.20.40.60.81 G , Φ = 0, Exact G , Φ = 0, Numerical G , Φ = π , Exact G , Φ = π , Numerical G , Φ = π , Exact G , Φ = π , Numerical G t FIG. 14: (Color online) The conductance of the noninteract-ing Y junction calculated with our method versus the exactsolution. exactly diagonalize the many-body Hamiltonian either.The DMRG method is an ideal tool for performing thesecalculations, and the results presented in this section areobtained by this method.The results of these calculations are shown in Figs. 15and 16 for the Luttinger parameter g = 2 . g = 6 . g = 6 . V = 2 andwe need a larger healing length. The universal behavioris evident from the fact that the value of the hoppingamplitude at the junction has no effect on the asymptoticform of the correlation functions.Let us now consider the case of repulsive interactions.This is a special case because the theoretical predictionfor the conductance is G = 0 so we do not expect tobe able to fit the correlation function to the CFT form.The behavior of the correlation function, however, shedslight on the nature of the problem. As noted earlier, the0 FIG. 15: (Color online) The correlation functions are plottedfor two values of hopping at the impurity t = 0 . t = 0 . g = 2 .
0. The strong agreementwith the CFT prediction of G = g e h supports the correctnessof our method.FIG. 16: (Color online) The correlation functions are plottedfor two values of hopping at the impurity t = 0 . t = 0 . g = 6 .
0. Here a larger healinglength of ℓ = 90 is needed to observe the universal behaviorand verify the CFT prediction. CFT prediction for the universal form of the h J R J L i is anasymptotic form, and we expect subleading correctionscoming from irrelevant boundary operators.In the repulsive case, since the conductance is zero, h J R J L i will decay faster than (cid:2) sin( x πℓ ) (cid:3) − . As seen inFig. 17 for g = 0 . <
1, where the coefficient in front ofthe leading term is zero, we only observe these subleadingcorrections. As a function of ln (cid:2) sin( x πℓ ) (cid:3) , ln h J R J L i de-cays with a larger negative slope than − k F oscillatory behavior.The studies of this section verify that the method de-veloped in this paper correctly reproduces established re-sults in the presence or absence of electron-electron in-teractions. In the next section we apply the method to FIG. 17: (Color online) The correlation functions are plottedfor hopping at the impurity t = 0 . g = 0 .
65. Since the universal conductance vanishes, weonly observe the nonuniversal subleading corrections. an open problem for three interacting wires.
VIII. UNIVERSAL CONDUCTANCE IN ANINTERACTING Y JUNCTION OF QUANTUMWIRES
In this section, we employ the method developed in thispaper to study a Y junction of interacting quantum wireswith spinless electrons, each connected to a distinct siteon a loop at the junction as seen in Fig. 3. The tunnelingamplitude between the wires is t and the flux throughthe loop is φ . This system was analytically studied inRefs. 45 and 46 where several nontrivial fixed points wereidentified. A similar analysis for a Y junction with spinfulelectrons is given in Ref. 61.In this work, we focus on the Luttinger parameterrange 1 < g <
3. The RG flow diagram for this range,which was conjectured in Ref. 46, is shown in the right-hand side of Fig. 18. We have two types of fixed points:the chiral fixed points χ + and χ − in the presence of atime-reversal-symmetry-breaking flux and the M fixedpoint for unbroken time-reversal symmetry. A. Chiral fixed point
For the chiral fixed point, the time-reversal symmetryis broken by a magnetic flux and therefore G = G .It was predicted in Ref. 46 that these conductances ac-tually have a different sign and depend on the Luttingerparameter as follows: G = − g (3 + g ) ( g + 1) e h , (94) G = 2 g (3 + g ) ( g − e h . (95)1 FIG. 18: The conjectured RG flow diagram for the Y junctionof Fig. 3, given in Ref. 46, for the Luttinger parameter in therange 1 < g < This conductance is universal, i.e., it is independent ofthe hopping amplitude t at the junction for nonvanish-ing t . It is worthwhile to emphasize that the emer-gence of universal conductances strictly relies on thepresence of interactions, i.e., by taking the noninteract-ing limit g →
1, we shall not expect to recover correctresults for noninteracting cases. In the RG language,the noninteracting point is a marginal fixed point andeach value of hopping amplitude t gives a unique conduc-tance as in Eq. (91). Because we are studying junctionswith Z symmetry, the conductances obey the relations, G = G = G and G = G = G . In the time-reversal symmetric case (the M fixed point), the conduc-tances in addition follow G ij = G ji . To verify these pre-dictions, we numerically calculated the correlation func-tion, h J R ( x ) J L ( x ) i , for two values of t and two values of g in a system size of ℓ = 40. As shown in Fig. 19, thenumerical results strongly agree with the prediction for G . The consistency serves both as the first numericalverification of the prediction Eq. (94) as well as a highlynontrivial benchmark for the correctness of our method.It is quite remarkable that we can verify the G con-ductance in a system as small as ℓ = 40. Note that due tothe asymptotic form of the relation between the univer-sal conductance and the computed correlation functions,the results are reliable once there is convergence in thesystem size. Also, one needs to check for the conver-gence of the DMRG method by increasing the number ofstates used in the calculations to make sure the DMRGresults are in fact quasi-exact. We show some of theseconvergence tests in Fig. 20.In the last part of this section, we present the numeri-cal verification of the G conductance. It turns out thatfor G , a longer healing length is needed and we areable to see the agreement with the theoretical predictionin a system of ℓ = 60. It is remarkable that the correla- FIG. 19: (Color online) A plot of the ln h J R ( x ) J L ( x ) i for g = 1 . g = 2 . g = 1 . t = 0 .
7. As seen, the correlation functionscollapse for ℓ = 40 and ℓ = 60. In the right panel, we checkthe DMRG convergence as the truncation error is reduced byincreasing m . The data are for g = 2 . t = 0 . ℓ = 60. tion function here actually changes sign and the universallong-distance behavior, corresponding to a positive con-ductance, has opposite sign to the nonuniversal values ofthe correlation function near the junction.Due to the sign change of the correlation function, herewe plot the logarithm of minus the correlation function ina region far away from the junction (where the argumentof the logarithm is positive). The results are shown inFig. 21 and, once again, there is good agreement betweenthe data and the theoretical prediction. B. M fixed point
Let us now turn to the conductance of the time-reversalsymmetric M fixed point for which theoretical predictionsdo not exist. The benchmarks in the previous section andthe study of the chiral fixed point for three interactingwires increases our confidence in the correctness of the2
FIG. 21: (Color online) The universal G conductance for g = 2 . t = 1 . t = 0 .
7. Notice the sign of the argument of thelogarithmic function labeling the vertical axis. Here, G ispositive and the correlation function h J R ( x ) J L ( x ) i actuallychanges sign at a certain distance from the boundary. method developed here. We are now ready to apply themethod to a thus-far unsolved quantum impurity prob-lem.We perform the calculations of the M fixed-point con-ductance for three values of the Luttinger parameter g and two values of the hopping amplitude t . The indepen-dence from the hopping amplitude can be seen at ℓ = 60.The results are shown in Figs. 22, 23, and 24, respectivelyfor g = 1 . , . , . FIG. 22: (Color online) The universal G = G conductancefor g = 1 . t = 1 . t = 0 . To obtain the conductance, we fit the data for a slightlylarger system of ℓ = 70. The three correlation functionsare plotted in Fig. 25 in addition to the best fit in theasymptotic region for each value of g . We obtain the FIG. 23: (Color online) The universal G = G conductancefor g = 2 . t = 1 . t = 0 . G = G conductancefor g = 2 . t = 1 . t = 0 . following conductances for the M fixed point: G ( g = 1 .
5) = G ( g = 1 . ≈ − . e h ,G ( g = 2 .
0) = G ( g = 2 . ≈ − . e h ,G ( g = 2 .
5) = G ( g = 2 . ≈ − . e h . Is there a pattern to these values of the conductance?It is well known that when a Luttinger liquid with at-tractive interactions ( g >
1) is coupled to Fermi-liquidleads, the ge /h conductance renormalizes to e /h . One way to think about this phenomenon is to assumethat we have an effective contact resistance at the inter-face between each interacting wire and the noninteractingleads. If this contact resistance is 1 /g c , we must have1 /g + 2 /g c = 1 ⇒ g c = 2 / (1 − g − ) . FIG. 25: (Color online) Fitting the correlation function to theCFT prediction to extract the conductance. The data in thisfigure are for t = 0 . ℓ = 70, with three different valuesof g . For the multi-wire conductance tensor, we can considerthe following transformation, which, if the effective con-tact resistance is the correct mechanism, renormalizesthe conductance in the presence of noninteracting leads.Here, we think of the transformation¯ G = ( G/g c ) − G as a mathematical mapping of the conductance tensor,which can be inverted as G = ¯ G (cid:0) − ¯ G/g c (cid:1) − . It turns out that for all the other fixed points of the Yjunction as well as in the two-wire case, ¯ G is independentof the Luttinger parameter. We conjecture here thatthis is also true for the M fixed point. We can checkexplicitly that for the three values of g studied above,¯ G ≈ γ − γ − γ − γ γ − γ − γ − γ γ e h with γ ≈ .
42. This conjecture leads to a general formfor the conductance of the M fixed point as a function ofthe Luttinger parameter and one dimensionless number γ ≈ .
42 as G ( g ) = G ( g ) = − gγ g + 3 γ − gγ e h . As stated above, this conjecture works to a good ap-proximation for the three values of g studied above nu-merically. The functional form of the dependence of theconductance of the M fixed point on the Luttinger param-eter is plotted in Fig. 26. This simple plot, which hadremained elusive since the prediction of the existence ofthe M fixed point in , is the main result of this section FIG. 26: The dependence of the off-diagonal conductance ofthe M fixed point on the Luttinger parameter g . and an important achievement for the method developedin this paper.Notice that the measured value of γ differs from 4 / ≈ .
444 by about 4 . γ . The DMRG truncation erroris very small in this case and the calculated correlationfunctions are quasi-exact. The error from fitting the datais around 2% and there are errors from finite-size correc-tions that with our system sizes are of the order a fewpercent. It seems plausible that the renormalized conduc-tance of the M fixed point, when attached to noninteract-ing leads, is equal to G = − e /h as indicated by theresults of Ref. 38 which were obtained by an approximatefunctional renormalization group scheme. This stronglysuggests a simple understanding of the M fixed point con-ductance because e /h is the largest conductance | G | we can have, with time reversal symmetry, for three non-interacting quantum wires connected by a unitary scat-tering matrix. The g -dependence of the M fixed pointconductance is such that when attached to noninteract-ing leads the system acts as an effective scattering matrixwith the largest possible conductance G . IX. CONCLUSIONS AND OUTLOOK
In summary, we have developed a method in this pa-per that allows the computation of the conductance ten-sors of rather arbitrary quantum junctions with standardnumerical techniques such as time-independent DMRG.We presented specific studies of an interacting Y junctionand resolved an outstanding open question regarding theconductance of the M fixed point.Our generic method has only one practical limitation.Namely, the size ℓ of this system must be large enoughfor the universal scaling behavior to emerge. Two factorsaffect this minimum length. First, an intrinsic healinglength required to observe the continuum long-distanceLuttinger-liquid behavior. For large attractive interac-tions (large g ), this intrinsic healing length is large. Sec-ond, the scaling dimension of the irrelevant boundaryoperators. If these operators are only slightly irrelevant,4we need to move away a larger distance from the bound-ary to observe the fixed-point behavior. The D p fixedpoint of the Y junction, for example, which occurs for3 < g <
9, presents these difficulties. The scaling dimen-sion of the leading irrelevant boundary operator increaseswith g , but making g larger brings the Luttinger liquidclose to the phase separation point and increases the in-trinsic healing length. We have attempted to verify thetheoretical prediction for the D p fixed point but have notbeen able to reach the required length with our compu-tational resources. As demonstrated in this paper, how-ever, this practical limitation is by no means a genericproblem.The method developed in this paper opens up the pos-sibility of applying the powerful technology of DMRGand matrix product states, which in the last decades havebeen successfully applied to quasi-1D systems such as lad-ders, to transport calculations in the presence of electron-electron interactions. For decades, quantum transportwas formulated for noninteracting electrons, and interac-tion effects were treated by mean-field and approximatemethods. The results of this paper make it possible toutilize the quasi-exact numerical methods such as time-independent DMRG to study the transport properties ofstrongly correlated systems.Many interesting questions can now be answered withstandard computations. A long-standing open question isthe conductance of a chain of interacting spinful fermionsat the nontrivial Kane and Fisher fixed point. Themethod also makes it possible to study fixed points ofmore than three quantum wires. For two wires of spin-less fermions, there are only two interacting fixed points,while the Y junction presents a much richer flow dia-gram with multiple fixed points. What are the possibleuniversal conductances with four, five wires, etc.? Arethere hierarchies or other mathematical structures?Another direction of interest is to identify the uni-versality class that particular junctions, which are ex-perimentally accessible, fall into. Transport measure-ments are done on a multitude of organic moleculessuch as fullerenes, for example. Effective tight-bindingmodels can be derived for such molecular structures viafirst-principles calculations. Our method combined withDMRG computations is then helpful for extracting theuniversal behavior of such molecular junctions.In summary, the technology developed in this paperprovides a systematic way to study the effects of strongelectron-electron interactions in the transport propertiesof quantum impurity problems and molecular electronicdevices. In this framework, and with the help of efficientDMRG computations, strong correlations can be studiedin a quasi-exact manner without resorting to mean-fieldor perturbative treatments.
Acknowledgments
We are grateful to A. Polkovnikov and A. Sand-
FIG. 27: A junction with two columns of fermionic sites atthe boundary and a general quadratic boundary Hamiltonianas in Eq. (A1). vik for helpful comments and discussions. This workwas supported in part by the DOE Grant No. DE-FG02-06ER46316 (CC, CH, and AR), the Dutch Sci-ence Foundation NWO/FOM (CH), NSF DMR-0955707(AF), KAKENHI No. 50262043 (MO), NSERC (IA) andCIfAR (IA).
Appendix A: Symmetry transformation for atwo-column noninteracting junction
In this appendix, we test the H R = K ( C ( H L )) con-jecture of Eq. (89) for a two-column noninteracting junc-tion. The boundary Hamiltonian for the junction, whichis shown in Fig. 27, is as follows: H L = Ψ † Γ Ψ + Ψ †− Γ − Ψ − + Ψ †− T Ψ + Ψ † T † Ψ − , (A1)where the two Γ matrices are Hermitian and T is anarbitrary M × M matrix. Let us first calculate the S matrix. We have ǫψ − = Γ − ψ − + T ψ , (A2) ǫψ = Γ ψ + T † ψ − − ψ . (A3)We work at half-filling where ǫ = 0. From Eq. (A2), wethen obtain ψ − = − (Γ − ) − T ψ . Plugging this into Eq. (A3) and replacing the wave func-tions ψ with scattering plane waves givesΓ ( A out + A in ) − T † (Γ − ) − T ( A out + A in ) − i ( A out − A in ) = 0from which we get the following scattering matrix: S = h i − Γ + T † (Γ − ) − T i − h i + Γ − T † (Γ − ) − T i . We have seen already that the transformation KC takes Γ to − Γ . It similarly changes Γ − to − Γ − .5In order for this transformation to work, we need tohave T † (Γ − ) − T → − T † (Γ − ) − T under KC . Wewill show this by arguing that the matrix T goes to − T .Let us begin by applying the particle-hole transformationto the T term Ψ †− T Ψ + Ψ † T † Ψ − . We have C (Ψ †− T Ψ + Ψ † T † Ψ − )= − X j,j ′ (cid:16) T jj ′ c − j c † j ′ + T † jj ′ c j c †− j ′ (cid:17) = − Ψ †− T ∗ Ψ − Ψ † T T Ψ − and, therefore, K (cid:16) C (Ψ †− T Ψ + Ψ † T † Ψ − ) (cid:17) = Ψ †− ( − T )Ψ + Ψ † ( − T † )Ψ − . So, we find that indeed T † (Γ − ) − T → − T † (Γ − ) − T . Appendix B: Relationship between A αβ B and theboundary state |Bi In this appendix, we review the notion of the highest-weight states and derive Eq. (62), which explicitly re-lates the coefficient A αβ B to the boundary state and thehighest-weight states of some primary operators in theCFT. The derivation in this appendix is merely a reviewof the generic result obtained in Ref. 56 and is presentedfor completeness.In field theory, it is customary to quantize the theoryfor constant times. The Hamiltonian of the system is thegenerator of time translation. In CFT, however, becauseof scale invariance, it is convenient to quantize the the-ory on a fixed circle in the complex z = τ + ix plane.This is called radial quantization and the generator ofscale transformations is called the dilatation operator D ,which can be written as the integral of the radial compo-nent of the stress tensor.The operator-state correspondence in CFT is formu-lated in the radial quantization framework. There is avacuum state | i in radial quantization, and acting bythe operator corresponding to a scaling field on the vac-uum gives a state | φ i = ˆ φ (0 , | i . It can be shown that these states are eigenstates of thedilatation operator D. The state constructed in thismanner for a primary field is called the highest-weightstate of that field.Let us now turn to the derivation of Eq. (62) follow-ing Ref. . We would like to show that if h O ( x ) i = A O B (2 x ) − X O for a scaling operator O living on the upper-half plane, we have A O B = h O |Bih |Bi , where |Bi is the boundary state on the real axis.The key to this derivation is calculating the expec-tation value of O on a semi-infinite cylinder of radius R seen in Fig. 28 in two different ways. First, we usea conformal mapping from the upper half-plane to thesemi-infinite cylinder. Consider ˜ w = − iz iz . It is easy toshow that for z on the real axis, ˜ w ˜ w ∗ = 1 and the upperhalf-plane is mapped to the outside of a unit disk. It isthen easy to see that the conformal mapping w ( z ) = R π ln (cid:18) − iz iz (cid:19) takes the upper-half plane to the semi-infinite cylinder.By applying this transformation to the expectation valueof O on the semi-infinite plane, we obtain h O ( y ) i = A O B (cid:18) πR (cid:19) X O exp( − πX O y/R )[1 − exp( − πy/R )] X O . (B1) FIG. 28: The expectation value of O is calculated on thesemi-infinite cylinder of radius R in two different ways. Now a second way to calculate this quantity is directlyon the semi-cylinder and by using the boundary state.If the transfer matrix in the y direction (parallel to thecylinder axis) is exp( −H ) for a Hamiltonian H , we canwrite h O ( y ) i = lim T →∞ h | e − ( T − y ) H O ( y ) e − y H |Bih | e − T H |Bi , where | i is the ground state of H .We now insert the resolution of the identity between O ( y ) and e − y H in the above expression. Since O is aprimary operator only states that are not annihilated by O enter the sum. The highest-weight state O | i = | O i has the smallest eigenvalue and, in the limit of large y , will be the leading term. We then obtain h O ( y ) i = h | O | O ih O |Bih |Bi e − ε O y , (B2)where ε O = 2 πX O /R . We now argue that h | O | O i be-haves as (2 π/R ) X O . This can be seen from calculating h | O ( y ) O ( y ) | i on a cylinder. By mapping the full com-plex plane to a cylinder of radius R , we obtain h | O ( y ) O ( y ) | i = (cid:20) Rπ sinh πR ( y − y ) (cid:21) − X O . In the limit of ( y − y ) → ∞ , the above correlationfunction reduces to h | O ( y ) O ( y ) | i ∼ (cid:18) πR (cid:19) X O e − πR X O ( y − y ) . Q ( y ) and O ( y ) and taking the limitof ( y − y ) → ∞ , which yields h | O ( y ) O ( y ) | i ∼ |h | O | O i| e − ε O ( y − y ) . We then find by comparison that h | O | O i ∼ (2 π/R ) X O . (B3) Comparing Eq. (B1) (in the limit of large R ) and Eq. (B2)and making use of the above expression Eq. 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