January 2012 2 1 0 Entanglement Entropy of Quantum Wire Junctions 2 n a J Pasquale Calabresea,b, Mihail Mintchevb,a and Ettore Vicaria,b 0 2 ] a Dipartimento di Fisica dell’Universit`a di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy h c e b Istituto Nazionale di Fisica Nucleare, Largo Pontecorvo 3, 56127 Pisa, Italy m - t a t s . t a m - d n Abstract o c We consider a fermion gas on a star graph modeling a quantum wire junction and [ derive the entanglement entropy of one edge with respect to the rest of the junction. 2 The gas is free in the bulk of the graph, the interaction being localized in its vertex v and described by a non-trivial scattering matrix. We discuss all point-like interactions, 3 1 which lead to unitary time evolution of the system. We show that for a finite number of 7 particles N, the R´enyi entanglement entropies of one edge grow as lnN with a calculable 5 . prefactor,whichdependsnotonlyonthecentralcharge,butalsoonthetotaltransmission 0 probability from the considered edge to the rest of the graph. This result is extended to 1 1 the case with an harmonic potential in the bulk. 1 : v i X r a IFUP-TH 21/2011 1 Introduction Quantum field theory on graphs attracted recently much attention mainly in relation with the study [1]-[6]of the transport propertiesof quantumwire networks. Different frameworks[7]-[20] have been developed to investigate the phase diagram and the conductance of these structures. Despite of the fact that the universal properties in the bulk are described by the well known Luttinger liquid theory, the different boundary conditions at the junctions lead to exotic phase diagrams[10,11,12,15,16,17,18,19]whosedegreeofuniversalityisnotcompletelyunderstood and is still under investigation. The results, concerning the charge transport, confirm that the conductance properties of the quantum wire networks are strongly affected by the boundary conditions as well. In the present paper we analyze another physical quantity - the entanglement entropy of one edge of the junction with respect to all the others edges. Lots of studies on the entanglement properties of many-body systems in the last decade have unveiled new (universal) features of these systems and somehow put their global understanding on a deeper level (see e.g. the reviews [21]). In particular, von Neumann and R´enyi entanglement entropies of the reduced density matrix ρ of a subsystem A turned out to be particularly useful for 1D systems. R´enyi A entanglement entropies are defined as 1 S(α) = lnTrρα . (1.1) 1−α A For α → 1 this definition gives the most commonly used von Neumann entropy S(1) = −Trρ lnρ , while for α → ∞ is the logarithm of the largest eigenvalue of ρ also known A A A as single copy entanglement [22]. Furthermore, the knowledge of the S(α) for different α char- acterizes the full spectrum of non-zero eigenvalues of ρ [23]. A One of the most remarkable results is the universal behavior displayed by the entanglement entropy at 1D conformal quantum critical points, determined by the central charge [24] of the underlying conformal field theory (CFT) [25, 26, 27, 28]. For a partition of an infinite 1D system into a finite piece A of length (cid:96) and the remainder, the R´enyi entanglement entropies for (cid:96) much larger than the short-distance cutoff a are (cid:18) (cid:19) c 1 (cid:96) S(α) = 1+ ln +c , (1.2) α 6 α a where c is the central charge and c a non-universal constant. α Given the importance of this result (and also many others not mentioned here) for homo- geneous systems, it is natural to wonder whether in the case of junctions the entanglement entropies can share some light on the universality and on the relevance of the parameters defin- ing the junction. Previous studies in the subject [29, 30, 31, 32, 33, 34, 35, 36, 37] have been limited to the case of only two edges (i.e. an infinite line with a defect) and most often per- formedforlatticemodels. Theseresultsprovideastrongevidencethatthelogarithmicbehavior in Eq. (1.2) remains valid even in the presence of defects, but the prefactor does not depend 1 only on the central charge of the bulk CFT when the defect is a marginal perturbation (in renormalization group sense) as it is known to happen for free fermions [1]. In order to tackle the problem of entanglement in a junction with an arbitrary number of wires, we use the recently developed systematic framework [38, 39] for calculating the bipartite entanglement entropy of spatial subsystems of one-dimensional quantum systems in continuous space. We only consider a free fermion gas in bulk in which the junction introduces a marginal perturbation. The junction boundary conditions define a specific scattering matrix S, encoding all possible point-like interactions in the vertex which give rise to unitary time evolution. Focussing on the scale invariant case, we show that for a finite number of particles N and for edges of equal length L, the R´enyi entanglement entropies of any of the edges grow as lnN. Oppositely to the case in the absence of the point-like interaction (i.e. Eq. (1.2)), the prefactor of this logarithm does not depend only on the central charge, but also on the total transmission probability (1−|Sii|2) from the considered edge i to the rest of the graph. Some of the results presented here have been anticipated in the short communication [38]. We show also that the presence of an external harmonic potential in the bulk (acting identically on all edges) does not alter this result. The paper is organized as follows. In the next section we describe the basic features of the model and the scattering matrices generated by the point-like interactions at the junction. We discuss in detail the scale invariant case and derive the two-point correlation function. The entanglement entropy, associated with this system, is analytically computed in section 3. In section 4 we extend our considerations, adding a harmonic potential in the bulk. Section 5 is devoted to the conclusions and the discussion of some further developments in the subject. 2 Schro¨dinger junction 2.1 The general setting In this section we consider a gas of N spinless fermions on a quantum wire junction. We consider only the ground-state of such system and we refer to it as the ground-state of N particles, having in mind that the particles are the original fermions and not the excitations above the ground-state (that are usually referred as particles in field theory literature). A simple model, describing the junction, is represented by a star graph Γ with M edges of finite length L, as shown in Fig. 1. Each point P in the bulk of Γ is parametrized by (x,i), where 0 ≤ x ≤ L is the distance of P from the vertex V of the graph and i labels the edge. We assume that in the bulk (x (cid:54)= 0 and x (cid:54)= L) the gas is free and is described by the Schro¨dinger field ψ (t,x), which satisfies i (cid:18) (cid:19) 1 i∂ + ∂2 ψ (t,x) = 0 (2.1) t 2m x i andstandardequal-timecanonicalanticommutationrelations. Theonlynon-trivialinteractions are localized in the vertex V of Γ and are encoded in boundary conditions at x = 0. These conditions are fixed in turn by imposing that the bulk Hamiltonian −∂2 admits a self-adjoint x extensiononthewholegraph. Insuchawayallpoint-likeinteractions, leadingtoaunitarytime 2 (cid:2)(cid:2)• 2 (cid:14)(cid:2) (cid:0)• (cid:0) 1 (cid:2) (cid:0)(cid:18) . . (cid:2) (cid:0) i . (cid:27) (cid:27)S(k) (cid:24) • L . . @ . (cid:26)(cid:25)@R @@ M • Figure 1: (Color online) A star graph Γ with scattering matrix S(k) in the vertex and all edges of length L. We consider the entanglement entropy of the edge i (red) with respect to all the others. evolution of the system, are covered. The most general boundary conditions, implementing this natural physical requirement, are [41, 42] at the vertex M (cid:88) [λ(I−U)ijψj(t,0)−i(I+U)ij(∂xψj)(t,0)] = 0, (2.2) j=1 where U is an arbitrary M ×M unitary matrix and λ a real parameter with the dimension of mass. To fully specify the problem we also need to impose boundary conditions at the external ends of the edges. The most general ones are (∂ ψ )(t,L) = µ ψ (t,L), (2.3) x i i i where µ are again real parameters with the dimension of mass. Eq. (2.3) is the familiar Robin i (mixed) boundary condition. Notice that Eq. (2.2) extends this condition to the vertex V of the graph. It has been established in [41, 42] that the point-like interaction, induced by (2.2), generates the scattering matrix [λ(I−U)−k(I+U)] S(k) = − . (2.4) [λ(I−U)+k(I+U)] Besides of unitarity S(k)S†(k) = I, (2.5) S(k) satisfies Hermitian analyticity S†(k) = S(−k) (2.6) as well. Notice also that S(λ) = U, S(−λ) = U−1, (2.7) 3 showing that the unitary matrix U, entering the boundary conditions (2.2), is actually the scattering matrix at the scale λ. The main difficulty in solving the Schro¨dinger equation (2.1) on the graph Γ is the mixing between the different edges codified in the boundary conditions (2.2) and (2.3). In order to simplify this problem, we impose that the boundary conditions at the ends of each arm are all the same, in such a way to restore (at the level of the Hamiltonian) permutation symmetry of the edges of the graph. It should be clear from the physical point of view that, being interested in the thermodynamic limit with L,N → ∞, the boundary conditions at L must not affect the final result. Thus we assume from now on that µ = µ = ··· = µ ≡ µ. (2.8) 1 2 M Under this condition, Eqs. (2.2) and (2.3) can be rewritten in equivalent forms without mixing. Indeed, let us introduce the unitary matrix U diagonalizing U, namely π π (cid:0) (cid:1) U UU† = Ud = diag e−2iα1,e−2iα2,...,e−2iαM , − < αi ≤ . (2.9) 2 2 Remarkably enough, U diagonalizes also S(k) for any k: (cid:18) (cid:19) k +iη k +iη k +iη Sd(k) = U†S(k)U = diag 1, 2,..., M , (2.10) k −iη k −iη k −iη 1 2 M where η ≡ λtan(α ). (2.11) i i It is quite natural at this point to introduce the fields M (cid:88) ϕ (t,x) = U ψ (t,x), (2.12) i ij j j=1 which obviously satisfy Eq. (2.1). In terms of ϕ the boundary conditions (2.2,2.3) decouple, i (∂ ϕ )(t,0) = η ϕ (t,0), (2.13) x i i i (∂ ϕ )(t,L) = µϕ (t,L), (2.14) x i i defining a simple spectral problem on the tensor product H = (cid:78)M L2[0,L], which is analyzed i=1 below. It is worth stressing that ϕ (t,x) is a superposition of the values of the original field ψ (t,x) i i at the same distance x from the vertex, but on different edges of the junction. Being so delocal- ized, ϕ (t,x) is unphysical and provides only a convenient basis for dealing with the boundary i conditions. Thephysicalobservablesandcorrelationfunctionswillbealwaysexpressedinterms of the physical fields ψ (t,x). i The eigenfunctions of −∂2, obeying (2.13) and (2.14) are x (cid:18) (cid:19) k +iη φ (k,x) = c eikx + ie−ikx , k ≥ 0, (2.15) i i k −iη i 4 where c are some constants to be fixed below and k satisfy i (cid:18) (cid:19)(cid:18) (cid:19) k +iη k −iµ e2ikL = i . (2.16) k −iη k +iµ i In order to determine the spectrum of k explicitly, we simplify the problem further by requiring scale invariance. 2.2 The scale invariant case Scale invariance of the boundary conditions (2.13) and (2.14) implies the values (cid:40) (cid:40) 0, 0 (α = 0), Neumann b.c., i µ = η = (2.17) i ∞, ∞ (α = π/2), Dirichlet b.c. i In other words, the critical points are fixed by the (M + 1) vector (µ,η ,η ,...,η ) whose 1 2 M components take the values (2.17). For an edge i one has the following possibilities: (a) µ = 0 (Neumann condition at x = L): Eq. (2.16) gives (cid:18) (cid:19) k +iη e2ikL = i , (2.18) k −iη i which gives (cid:114) 2 (cid:104) x(cid:105) η = 0 =⇒ φ (n,x) = cos (n−1)π , n = 1,2,... (2.19) i i L L (cid:114) (cid:20)(cid:18) (cid:19) (cid:21) 2 1 x η = ∞ =⇒ φ (n,x) = sin n− π , n = 1,2,...; (2.20) i i L 2 L (b) µ = ∞ (Dirichlet condition at x = L): Eq. (2.16) implies (cid:18) (cid:19) k +iη e2ikL = − i , (2.21) k −iη i which gives (cid:114) (cid:20)(cid:18) (cid:19) (cid:21) 2 1 x η = 0 =⇒ φ (n,x) = cos n− π , n = 1,2,... (2.22) i i L 2 L (cid:114) 2 (cid:16) x(cid:17) η = ∞ =⇒ φ (n,x) = sin nπ , n = 1,2,... (2.23) i i L L Notice that any of the sets (2.18, 2.19, 2.22, 2.23) represent a complete ortho-normal system in L2[0,L]. 5 2.3 Scale invariant scattering matrices Observing that the eigenvalue of S is 1 for ηi = 0 and −1 for ηi = ∞ one concludes that the most general scale-invariant scattering matrix, compatible with a unitary time evolution, is given by S = U SdU†, Sd = diag(±1,±1,···±1), (2.24) where U is a generic M × M unitary matrix. From the group-theoretical point of view, any critical S matrix is a point in the orbit of some Sd under the adjoint action of the unitary group U(M). Obviously, one can enumerate the edges in such a way that the first p eigenvalues of S are +1 and the remaining M −p are −1. The cases p = M and p = 0 correspond to S = I and S = −I and are not interesting for the entanglement. In these two cases in fact, the single wires are decoupled (there is no transmission), which implies a vanishing entanglement. It follows from (2.24) that besides being unitary, S is also Hermitian (in agreement with (2.6) at criticality). Therefore, all diagonal elements Sii are real. Notice however that in general S is not symmetric. If this is the case, time reversal invariance is broken [19]. In the nontrivial case p = 1 the most general 2×2 scale-invariant scattering matrix depends on two parameters and can be written in the form (cid:18) (cid:19) 1 (cid:15)2 −1 2(cid:15)eiθ S((cid:15),θ) = , (cid:15) ∈ R, θ ∈ [0,2π). (2.25) 1+(cid:15)2 2(cid:15)e−iθ 1−(cid:15)2 Time reversal invariance is broken for θ (cid:54)= 0,π. For M = 3 one has two families corresponding to p = 1 and p = 2. In order to avoid cumbersome formulae, we display only two representatives of these families, namely   2(cid:15) 2(cid:15) 1−(cid:15)2 −(cid:15)2 1 1 2 1 2 Sp=1((cid:15)1,(cid:15)2) = 1+(cid:15)2 +(cid:15)2  2(cid:15)2 −(cid:15)21 +(cid:15)22 −1 2(cid:15)1(cid:15)2  , (2.26) 1 2 (cid:15)2 −(cid:15)2 −1 2(cid:15) (cid:15) 2(cid:15) 1 2 1 2 1   (cid:15)2 −(cid:15)2 −1 2(cid:15) (cid:15) 2(cid:15) −1 1 2 1 2 1 Sp=2((cid:15)1,(cid:15)2) = 1+(cid:15)2 +(cid:15)2  2(cid:15)1(cid:15)2 −(cid:15)21 +(cid:15)22 −1 2(cid:15)2  , (2.27) 1 2 2(cid:15) 2(cid:15) 1−(cid:15)2 −(cid:15)2 1 2 1 2 where (cid:15)1,2 ∈ R. 2.4 Two-point correlation function Nowweareinpositiontoconstructthephysicalfieldψ (t,x)andtherelativetwo-pointfunction i needed in the computation of the entanglement entropy. First of all, we write the unphysical field in terms of the eigenfunctions φ (n,x) i ∞ (cid:88) ϕ (t,x) = e−iωi(n)tφ (n,x)a (n), (2.28) i i i n=1 6 where the fermion annihilation and creation operators satisfy standard anti-commutation rela- tions [a (m), a†(n)] = δ δ , [a (m), a (n)] = [a†(m), a†(n)] = 0, (2.29) i j + ij mn i j + i j + and the energies are given by (cid:40) 1 (cid:2)(n−1)π(cid:3)2 , if 1 ≤ i ≤ p, ω (n) = 2m L (2.30) i 1 (cid:2)(2n−1) π (cid:3)2 , if p < i ≤ M , 2m 2L (cid:40) 1 (cid:2)(2n−1) π (cid:3)2 , if 1 ≤ i ≤ p, ω (n) = 2m 2L (2.31) i 1 (cid:2)nπ(cid:3)2 , if p < i ≤ M , 2m L for µ = 0 and µ = ∞ respectively. Notice that different “unphysical” edges may have different dispersion relation, which is not a problem because these edges are totally isolated from each other. By means of (2.12) one gets the physical fields M M ∞ (cid:88) (cid:88)(cid:88) ψ (t,x) = U† ϕ (t,x) = U† e−iωj(n)tφ (n,x)a (n). (2.32) i ij j ij j j j=1 j=1 n=1 One easily verifies that [ψ (t,x), ψ†(t,y)] = δ δ(x−y), (2.33) i j + ij which fixes the normalization of the fields. The equal time two-point correlation function of the physical field ψ (t,x) on a given state i |Ψ(cid:105) is CΨ(x,y) ≡ (cid:104)Ψ|ψ†(t,x)ψ (t,y)|Ψ(cid:105) ij i j M ∞ (cid:88) (cid:88) = U† U ei[ωj(m)−ωi(n)]tφ (n,x)φ (m,y)(cid:104)Ψ|a†(n)a (m)|Ψ(cid:105), (2.34) jk li k l k l k,l=1n,m=1 where the correlator (cid:104)Ψ|a†(n)a (m)|Ψ(cid:105) can be deduced from the action of the algebra generated k l by {a (m),a†(n)} on the state |Ψ(cid:105). In particular, we are interested in the case when |Ψ(cid:105) is the i j ground-state of the system formed by N fermions in the whole junction. It is then useful to rewrite N as N = M N (2.35) whereN representstheaveragenumberofparticlesforeachwire. Theactionoftheannihilation and creation operators on the ground-state is obvious since it is annihilated by all a (m) with l m > N and so (cid:104)Ψ|a†(n)a (m)|Ψ(cid:105) = δ δ θ(N −n). Using this relation, Eq. (2.34), restricted k l kl nm to the same edge (i = j) which is needed actually for computing the entanglement entropy, becomes M N (cid:88)(cid:88) CN(x,y) = |U |2φ (n,x)φ (n,y), (2.36) ii ki k k k=1 n=1 where N can also be interpreted as an ultraviolet cut-off for the series in (2.34). 7 It is convenient for what follows to rewrite (2.36) in more explicit terms. For this purpose we consider any critical point characterized by the integer 1 < p < M, i.e. a scale invariant scattering matrix with p eigenvalues equal to +1. The two sums in (2.36) factorize and one gets p N M N (cid:88) (cid:88) (cid:88) (cid:88) CN(x,y) = |U |2 f (n,x)f (n,y)+ |U |2 f (n,x)f (n,y), (2.37) ii ki + + ki − − k=1 n=1 k=p+1 n=1 where (cid:113) (cid:2) (cid:3)  2 cos (n−1)πx , µ = 0, L L f+(n,x) = (cid:113) (cid:2)(cid:0) (cid:1) (cid:3) (2.38)  2 cos n− 1 πx , µ = ∞, L 2 L (cid:113) (cid:2)(cid:0) (cid:1) (cid:3)  2 sin n− 1 πx , µ = 0, L 2 L f−(n,x) = (cid:113) (cid:2) (cid:3) (2.39)  2 sin nπx , µ = ∞. L L Because of (2.24), the sums over k in (2.37) give p M (cid:88) 1 (cid:88) 1 |Uki|2 = (1+Sii) , |Uki|2 = (1−Sii) . (2.40) 2 2 k=1 k=p+1 Moreover, using that S is both unitary and Hermitian, one has M (cid:88) S2ii = 1− |Sij|2 ≡ 1−Ti2, (2.41) j=1 j(cid:54)=i where T2 is the total transmission probability from the edge i to the rest of the graph. i In conclusion, the correlator (2.36) can be fully expressed in terms of the transmission probability T2 and the one-particle wave functions as follows i 1 (cid:18) (cid:113) (cid:19)(cid:88)N 1 (cid:18) (cid:113) (cid:19)(cid:88)N CN(x,y) = 1+ 1−T2 f (x,n)f (y,n)+ 1− 1−T2 f (x,n)f (y,n), ii 2 i + + 2 i − − n=1 n=1 (2.42) which is the basic input for deriving the entanglement entropy in the next section. We observe that CN involves only the diagonal elements of S and consequently, does not depend on the be- ii havior of S under transposition. Therefore, contrary to the conductance [19], the entanglement entropy in our case is not sensitive to the breaking of time-reversal invariance. 3 Entanglement entropy In order to compute the bipartite R´enyi entanglement entropies defined as in Eq. (1.1) of a subsystem A in the ground-state our star graph, we use the method recently introduced in 8 Refs. [38, 39]. The starting point to deal with a system made of a finite number of particles in continuous space is the Fredholm determinant DA(λ) = det[λδA(x,y)−CA(x,y)] , (3.43) where C (x,y) is the restriction of the correlation matrix C(x,y) defined in Eq. (2.34) to A, A i.e. C = P CP , where P is the projector on A. The same definition holds for δ (x,y) = A A A A A PAδ(x−y)PA. Following the ideas for the lattice model [43], DA(λ) can be introduced in such a way that it is a polynomial in λ having as zeros the eigenvalues of C . Since we are dealing A only with free fermions in the bulk, the reduced density matrix ρ is Gaussian [44] and so one A can easily derive [43, 38, 39] S(α) ≡ lnTrραA = (cid:73) dλ e (λ)dlnDA(λ), (3.44) α 1−α 2πi dλ where the integration contour encircles the segment [0,1], and 1 e (λ) = ln[λα +(1−λ)α] . (3.45) α 1−α For α → 1, e (λ) = −xlnx−(1−x)ln(1−x) and Eq. (3.44) gives the von Neumann entropy. 1 The Fredholm determinant is turned into a standard one by introducing the reduced overlap matrix A (also considered in Ref. [45]) with elements (cid:90) x2 Anm = dzφn(z)φm(z), n,m = 1,...,D, (3.46) x1 where in general φ (x) represent the eigenfuctions corresponding to the D lowest energy level n which are occupied in the ground-state of the system with D degrees of freedom. The matrix A satisfies TrCk = TrAk and so [39] A (cid:88)∞ TrCk (cid:88)∞ TrAk (cid:88)D lnDA(λ) = − kλkA = − kλk = lndet[λI−A] = ln(λ−am), (3.47) k=1 k=1 m=1 where am are the eigenvalues of A and D is its dimension to be specified later. Inserting (3.47) in the integral (3.44), we obtain (cid:73) D D dλ (cid:88) e (λ) (cid:88) 1 S(α) = α = eα(am) = Trln[Aα +(I−A)α] , (3.48) 2πi λ−a 1−α m m=1 m=1 as a consequence of the residue theorem. In the following we will be interested only in the entanglement entropy of any edge i of the wire with respect to the rest of the junction in the global ground-state of the star graph. As we have seen above in Eq. (2.42), the two-point correlation function for finite number of particles N in the full star graph, can be written in the form (we omit the edge index i hereafter) 2N (cid:88) CN(x,y) = χ(x,n)χ(y,n), (3.49) n=1 9