Transport Process in Multi-Junctions of Quantum Systems

We consider the junction of multiple one-dimensional systems and study how conserved currents transport at the junction. To characterize the transport process, we introduce reflection/transmission coefficients by applying boundary conformal field theory. We compute the reflection/transmission coefficients for some examples to derive the closed formulas. The formulas demonstrate spin-flip transport, where the spin polarization is flipped at the junction.


Introduction
Recent development of nanotechnology allows us to build an electric circuit in nanoscale, which involves quantum mechanical nature of electrons. To control such a nanoscale circuit, we need to investigate fundamental properties of quantum wire junctions. Among the theoretical studies of the quantum wire junction, Ref. [1] pointed out that the number of connected wires interestingly affects the fixed point of the renormalization group flow. In this sense, the quantum wire junction gains interests not only for engineering applications, but also for fundamental theoretical aspects. For this purpose, there have been a number of works based on conformal field theory (CFT). This is because low-energy behavior of the wider class of one-dimensional systems can be described as Tomonaga-Luttinger liquid (TLL) through the bosonization scheme. See, for example, a textbook on this topic [2].
Most works in this field are based on the TLL description of one-dimensional systems, which is just c = 1 free boson CFT. The c = 1 CFT enables us to describe the U (1) degree of freedom, which corresponds to electric charge. However, recent development of spintronics also demands us to incorporate SU (2) spin degree of freedom into such nanoscale devices.
In this case, it is desirable to implement SU (2) symmetry manifestly in order to investigate the spin-dependent property at the junction. Although the c = 1 CFT can treat the spin 1/2 system, corresponding to the SU (2) k=1 Kac-Moody algebra, c = 1 is necessary for describing generic spins, due to the identification of the Kac-Moody level k with the spin s as s = k/2 [3,4].
In this paper we study transport process at the multi-junction of one-dimensional systems that have Lie algebraic symmetries. We work with an arbitrary multiplicity M by generalizing the previous works for M = 2 [5,6]. See also [7]. This junction plays a similar role to an impurity in the one-dimensional system. In fact, one can map both the junction system and the impurity system into a (1+1)-dimensional system with a boundary by using the folding trick [8,9,10]. In this picture the information about the junction is implemented into the boundary state for the two dimensional system. Using the boundary state, we shall define the transmission/reflection coefficient of conserved currents at the junction. In addition, to compute the coefficients explicitly, we shall construct a boundary state corresponding to the multi-junction of SU (2)-symmetric systems. We shall compute both energy and spin-current reflection/transmission coefficients to investigate the spin-dependent property of the junction. In particular the spin transport shows an interesting behavior, namely, the spin-flipping process.
This paper is organized as follows. In Sec. 2, we formulate the transport process at the multi-junction by generalizing the formulation for the multiplicity M = 2 [5,6]. We point out that the R-matrix, which characterizes the transport at the junction, is not symmetric in general for M > 2, while it is always symmetric when M = 2. In Sec. 3, we apply this formalism to the permutation boundary condition, which is the simplest example to demonstrate the asymmetric R-matrix. In Sec. 4, we study the transport with the cosettype boundary condition. We shall propose the associated boundary state by generalizing that shown in [11]. The explicit computation of reflection/transmission coefficients shows that the current transport more strongly depends on the multiplicity M than the energy transport. We also discuss its application to the boundary entropy in Appendix A. We conclude this paper in Sec. 5 with some discussions.

Multi-junction of Quantum Systems
In this paper, we shall consider the system with M one-dimensional quantum systems connected at a point. Each quantum system is characterized by the following Hamiltonian densities in the field theoretical limit: where i = 1, . . . , M is the label of the quantum systems. J i,A is the current taking values in the Lie algebra A i and the index A runs over A = 1, . . . , dim A i . For the moment, we apply Junction at x i = 0 generic Lie algebras A i rather than su (2). d i AB is the inverse of the Cartan-Killing form and h ∨ i is the dual Coxeter number of the algebra A i . The Fourier modes of the current J i,A satisfy the Kac-Moody algebra A i : where f i is the structure constant of A i and k i is the level of A i .
In addition to these Hamiltonians, we also introduce the junction which connects the quantum systems through a local interaction. The interaction occurs if the "spin" S a at the junction takes value in a subalgebra C of A i . One possible local interaction is described by where x i ≥ 0 is the coordinate of the quantum systems and the junction is at x i = 0, as shown in Fig. 1. The index a runs as a = 1, . . . , dim C.
The critical point of this system is M -sheeted CFT glued along the conformal defect corresponding to the world line of the junction. This configuration leads to CFT 1 ×CFT 2 ×· · ·× CFT M , using the so-called folding trick [8,9,10]. See Fig. 2. In the following subsections, we shall define the reflection/transmission coefficient through the boundary CFT (BCFT). The BCFT is characterized by the boundary states |B which describe the boundary conditions.
For example, the energy conservation law along the boundary leads to the gluing condition for the Virasoro generators On the other hand, the current conservation law needs more consideration. If there is a common subalgebra C ⊂ A i for all i, we have j tot,a n + j tot,a −n |B = 0 (2.5) where j tot,a n = M i=1 j i,a n takes value in C. 1 Furthermore, in general, a subset of A i 's has a larger subgroup C ′ ⊃ C. Supposing that A i=1,...,l contains C ′ , the gluing condition can be written as where j i,α takes value in C ′ /C. On the other hand, if A i has no bigger common subalgebra with A j =i 's, we have Now the properties of the junction are encoded in the boundary state |B .

The R-matrices
To define the reflection/transmission coefficients, we first introduce the R-matrix for the energy by generalizing that for M = 2, . with an automorphism Ω preserving the energy-momentum tensor. See, for example, [12].
Due to the gluing condition for the total current (2.4), the R-matrix satisfies the following (2.10) These conditions give 2M − 1 constraints for the matrix elements. As a consequence of these constraints, the R-matrix has M 2 − 2M + 1 = (M − 1) 2 degrees of freedom in total. Notice that this reproduces the result of M = 2, which yields only one degree of freedom [5,6]. Let us introduce another basis to express the (M − 1) 2 degrees of freedom in the R-matrix, where α, β = 1, . . . , M − 1 and (2.12) It turns out that this W α −2 |0 forms an orthonormal basis, Then, to express the R-matrix in terms of ω αβ T , we introduce the inverse transformation of (2.12), Using the coefficients A, we get where we have used It is worth to emphasize that the R-matrix is not symmetric in general, while it is always symmetric for M = 2. This is one of the outcomes of the fact that the currents can transmit through more than one channel.
In the same way, we define the R-matrix for the currents j i taking a value in the alge- If we restrict to the common subalgebra C, the symmetry guarantees that the R-matrix is in the product form as R ij,ab Without loss of generality, we can assume that no pair of A i has bigger common subalgebra C ′ ⊃ C. If such a C ′ exists, we can focus on the subsector of the R-matrix associated with C ′ /C and do the same procedure as below. With this setup the matrix elements including the index of On the other hand, the R-matrix for C satisfies the constraints given by replacing a pair (R ij We can now utilize the same argument to obtain using the orthonormal basis

Reflection and Transmission Coefficients
Now we shall define the reflection and transmission coefficients using the R-matrices defined above. As in Refs. [5,6], it is natural to relate the diagonal and off-diagonal elements of the R-matrix to the reflection and transmission coefficients respectively. For the simplest case M = 2 [5,6], the transmission rate is defined by the "average" of the off-diagonal elements since R 12 = R 21 , which is derived from the conservation law. However, for M > 2, the conservation law cannot give such a strong constraint, and thus the average is not suitable to characterize the transport process. Therefore, we set T ij the transmission coefficient from system i to j with i = j, and we shall treat T ij and T ji as independent variables. Physically it is plausible to demand where R i is the reflection coefficient for the i-th system. The constraints (2.10) and (2.21) lead us to define Here R ij J is the R-matrix restricted to the subalgebra C, assuming that no pair of A i 's have a bigger common subalgebra. For A i /C, it is plausible to set R i J = 1 and T ij J = 0 due to the gluing condition (2.8).
This definition does not reduce to the previous ones for M = 2. However, since the R-matrix is symmetric for M = 2, we have where T avr T is the transmission coefficient defined in [5,6]. The similar relation holds for the current. As stated above, the new definition can be naturally extended to M > 2.
In the following two Sections, we shall compute the reflection/transmission coefficients for two examples.

Example I: Permutation Boundary Condition
We first consider the case where the boundary condition is given by with J M +1 = J 1 as shown in Fig. 3. Here C is a subalgebra of all the A i . This gluing condition is consistent when all k i have the same value k. As stated in the previous section, R i J = 1 for A i /C. For C, we can straightforwardly compute R ij J , and non-vanishing components are It is interesting that the R-matrix is not symmetric for M > 2. The transport coefficients for C are On the other hand, the R-matrix for the energy transport R ij T is more non-trivial, where c is the central charge corresponding to the algebra C with level k. Thus the transport coefficients are Again it is easy to check (2.27). Because c i > c, we have T ij T < 1. The physical interpretation of T J and T T is clear: currents for C completely transmits with the contribution c/c i among the total energy, while the rest currents are completely reflected giving 1 − c/c i contribution among the total energy.

Example II: Coset-type Boundary Condition
Now we shall consider the M -junction of SU (2) spin chains. To be more specific, we consider the cascade of breaking to the diagonal subgroup SU (2) κ M : We claim that the corresponding boundary state is given by the following generalization of [11]: is a product of (M − 1) Ishibashi states of each coset SU (2) κ i × SU (2) k i+1 /SU (2) κ i+1 and the Ishibashi state of SU (2) κ M . In Appendix B, we shall show that this boundary state satisfies the Cardy condition. The other constraints on the boundary states, i.e. the sewing relations, are assumed.
The parameters (ρ, r) run over 2ρ i = 0, 1, · · · , k i , 2r i = 0, 1, · · · , κ i+1 , and (µ, m) runs over the same region as (ρ, r) satisfying the additional constraints: Not all the states labeled by (ρ, r) are independent. This is because the boundary state is invariant under The two elements of that identification group can be expressed as

Energy Transport
Recall that the R-matrix is defined by the overlap between 0|L i 2 L j 2 and the boundary state |B , which is now written in terms of coset states. The descendant state L i −2 |0 shall be expanded by SU (2) κ M -singlet states with conformal weight h =h = 2, where |Σ A form a complete set of such singlet states. Here we shall use the explicit form with K α,a n = k α+1 α β=1 j β,a n − (κ α + 4)j α+1,a n .
Since the boundary state only has diagonal supports, we have with the normalized state |σ A = |Σ A / Σ A |Σ A and This expression (4.9) shows R ij T = R ji T ; the boundary state (4.2) gives the symmetric Rmatrix for energy. In the next subsection, we shall see that the spin-current R-matrix is also symmetric.
From now we shall compute L C , Σ A |Σ A and σ A | ⊗ σ A |B / 0|B one by one. The straightforward computation gives L C in the form of where U (x) is a unit step function: U (x ≥ 0) = 1, U (x < 0) = 0. We can also compute their norms as where c i and c κ i are the center charges of SU (2) k i and SU (2) κ i respectively.
For the boundary state (4.2), we have intriguing that the result is independent of ρ i , which was also for M = 2 [5,6]. As in the case of the Kondo problem, the parameter r i could be interpreted as the effective spin at the junction [13]. This result implies the energy transport is basically characterized by this residual effective spin of the junction. As we will see below, this property is also observed for the current transport.

Current Transport
Let us now compute the reflection and transmission coefficients for Kac-Moody current with the boundary state (4.2). As shown below, the computation for the current is actually simpler than that for the energy. To compute the ω αβ J (2.23), it turns out to be helpful to write K α,+ −1 |0 in terms of the coset states. K α,+ −1 gives the descendants of SU (2) κα and SU (2) k α+1 , and makes the spin 1 state of SU (2) κ α+1 . Thus we have (4.14) The relevant states to compute R J are the conformal vacuum |0 and the states with the conformal weight h =h = 1 and with spin 1 under SU (2) κ M . From (4.2), these contributions are given by Do not confuse this coefficient W j with the singlet state |W α which appeared in Sec. 4.1.
This leads to (4.17) and the R-matrix for the current This expression shows that the spin-current R-matrix is symmetric and is independent of ρ's as the energy R-matrix. Finally, the reflection and transmission coefficients for the currents It is straightforward to show the conservation law:

Results
From now on, we shall show results for some multiplicity M and parameters r i . First of all, let us check that our formula reproduces the previous results for the simplest case M = 2.
In this case, the Virasoro singlet states are given by |Σ A = {|V , |W 1 , |Y 1 }, and thus we obtain By substituting (4.11)-(4.13), we reproduce the result of Quella, Runnel and Watts [5] up to the conventions; r 1 in this paper is ρ in [5]. In a similar way, for the current transport, the formulas (4.18) and (4.19) reproduce our previous result in [6]. Note that, as addressed in Sec. 2.3, the definitions of reflection/transmission coefficients are different, while R J and R T are the same as the former definitions.
The numerical computation with the new definition involves Table 1 for M = 2. The matrix, which we call the reflection/transmission matrix, in the table is defined by .  Table 2.
In contrast to the current transport, it is expected that the energy transport is always (semi-)positive, because it is not possible to provide a reasonable interpretation for the negative energy transport. Up to now, we do not know how to prove this for generic parameters.
Even for M = 2, the complete proof is missing in spite of an attempt given in [5].

Discussion
In this paper we have discussed the transport process at the multi-junction with respect to both of energy and current flows. We have defined the transport coefficients with arbitrary multiplicity M by modifying and generalizing the previous one for M = 2. We have applied this formalism to some examples. The permutation boundary condition gives a simple, but important example such that the transport process becomes asymmetric between the channels, which cannot occur in the situation with M = 2. We have also considered the coset-type boundary condition, in order to study the spin-dependent transport at the junction. Proposing the corresponding boundary state, we have seen the multiplicity-dependence of the transport coefficients. By increasing the multiplicity, we have obtained more examples with the negative current transport, while the energy transport coefficients are always positive. This behavior suggests the spin-flip at the junction, which is more specific to high-multiplicity. In addition to the boundary states used in this paper, there are a number of solutions to the boundary condition. For the corresponding boundary states, it is interesting to study the transport coefficients. For example, Fredenhagen and Quella [14] proposed a new type of boundary states, which is a generalization of the permutation boundary state, used in Sec. 3. However, the form of the boundary states is not well known. One way to find them is to solve string field theory with SU (2) k 1 × SU (2) k 2 symmetry. As a first step, the authors are solving string field theory with the single SU (2) k with collaborators.

A Boundary entropy
In addition to the transport process discussed in the main part of this paper, another interesting application of the boundary state is the boundary entropy, which is also called the g-factor [15]. The boundary entropy is obtained by the inner product of the boundary state and the conformal vacuum, with a proper subtraction of the "bulk" contribution, This bulk contribution S 0 = ln 0|B 0 corresponds to the situation in the absence of the interaction between the junction and the bulk. Therefore |B 0 is given by where |0 's are Cardy's boundary states for SU (2) km for m = 1, . . . , M .
As pointed out in the previous work [6], |0 ⊗M is obtained by setting all the parameters to be zero in the boundary state (4.2), Since the overlap between the boundary state and the conformal vacuum is given by we obtain the boundary entropy as follows, Here W bdry implies the ground state degeneracy for the boundary, which is referred to the g-factor. This expression can be directly applied to the spin-chain junction under the identification of the Kac-Moody level k i with the spin representation s i as s i = k i /2 [3,4].
It is interesting to check the formula (A.5) by studying the discrete lattice models with the Bethe ansatz method.

B Cardy Condition
Boundary states should satisfy consistency relations: the Cardy condition and the sewing relations. In this Appendix, we shall show that (4.2) satisfies the Cardy condition. While we have considered M products of SU (2) in Sec. 4, we shall treat M products of a generic group G in this Appendix. Now ρ i and r i are the weight of G k i and G κ i respectively. In the same way, the region of (µ, m) is specified by All G (see eq. (12) of Ref. [11]) with G the root lattice of G. This reduces to (4.4) when G = SU (2). Notice that the projection operator, which appears in [11], is trivial in this case. Now let us compute the partition function on the cylinder, where χ = χ( q). Using the modular S-matrix, we get where χ = χ(q) and we have used the modular transformation: and χ [ν,n,p] = χ ν 1 ,ν 2 ,n 1 χ p 1 ,ν 3 ,n 2 · · · χ p M −2 ,ν M ,n M −1 χ p M −1 . G id is the identification group of G × G/G and |G id | is its dimension. Rep G is obtained by taking the quotient with respect to G id in each coset. For example, (ν 1 , ν 2 , n 1 ) ∼ (J 11 ν 1 , J 12 ν 2 , J 13 n 1 ) (B.5) where (J 11 , J 12 , J 13 ) ∈ G id . Now we can use the identity This holds if and only if (µ, m) ∈ Rep G otherwise the right hand side vanishes. By substituting this we get Z (µ,r),(τ,t) = J,d µ i ,m i ∈Rep(G) [ν,n,p]∈Rep G e 2πi(Q J 11 (µ 1 )+Q J 12 (µ 2 )−Q J 13 (m 1 )) Obviously, the coefficients of characters are semi-positive integers. In addition, the number of identity operator (ν = n = p = 0) is (B.9) Here we have used the following properties of the fusion matrix.
The second factor can be transformed as (B.11) Finally, we get (B.12) The dots include the contribution from the other states. In the meantime, the |(ρ, r) is invariant under (4.5). Thus we conclude that the unique identity operator appears if and only if boundary states are equivalent up to G id . This completes the proof of the Cardy condition.