Entanglement in composite free-fermion systems
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Entanglement in composite free-fermion systems
Viktor Eisler , Ming-Chiang Chung and Ingo Peschel MTA-ELTE Theoretical Physics Research Group, E¨otv¨os Lor´and University,P´azm´any P´eter s´et´any 1/a, H-1117 Budapest, Hungary Department of Physics, National Chung Hsing University, Taichung 40227, Taiwan Fachbereich Physik, Freie Universit¨at Berlin,Arnimallee 14, D-14195 Berlin, Germany
We consider fermionic chains where the two halves are either metals with different band-widths or a metal and an insulator. Both are coupled together by a special bond. Westudy the ground-state entanglement entropy between the two pieces, its dependence onthe parameters and its asymptotic form. We also discuss the features of the entanglementHamiltonians in both subsystems and the evolution of the entanglement entropy after joiningthe two parts of the system.
I. INTRODUCTION
The entanglement properties of free-fermion systems have been the topic of many studies andvarious different cases have been investigated, see e.g. [1]. In one dimension, these comprisehomogeneous critical chains where the ground-state entanglement entropy S varies logarithmicallywith the size L of the subsystem, and non-critical ones where it approaches a constant for large L . Single defects, both at interfaces [2–10] and in the interior [11–13] were studied, as well asinhomogeneous systems with random [14–17], aperiodic [18] and exponentially decaying [19, 20]couplings or random site energies [21–23]. Finally, systems in external potentials which produce avarying density and surface regions have been investigated [8, 24–30].In the present work, we look at yet another situation, namely at systems formed from twopieces with different properties. Specifically, we study chains composed either of two critical parts,or a critical and a non-critial one. Such systems have been considered previously in the contextof conformal invariance [31–34]. In our case, they are realized in the form of undimerized ordimerized tight-binding models coupled by a special bond. Physically, this corresponds to eithertwo metals with different bandwidths, or a metal and an insulator, and we will use this terminologyin the following. In both cases one has two types of single-particle eigenfunctions: those essentiallyconfined to one of the subsystems, and other ones extending through the whole system but havingdifferent wavelengths in the two parts. This is the same situation as for a potential step in quantummechanics. The occupied single-particle eigenfunctions determine the entanglement, and we studyit between the two different pieces of the composite system in its ground state.For the metal-metal system at half filling, the asymptotic result is very simple. It turns out thatonly the interface bond matters and one comes back to the defect problem solved previously in [4].Thus S varies logarithmically and the coefficient c eff is determined by the transmission through theinterface. The subleading terms, on the other hand, depend on the asymmetry but one can takethis into account by a rescaling of the length and find links to conformal formulae based on thenature of the extended states. Away from half filling, the entanglement is small as long as only thelocalized states are occupied. It increases, when the extended states come into play, but there arestrong variations with the filling which depend on the size and are also connected with the ratioof the bandwidths, which brings an additional length scale into the problem.For the metal-insulator system at half filling, the entanglement lies systematically between thatof the two pure systems. At large sizes, the insulator with its band gap dominates and limits theincrease of S . For sizes smaller than the correlation length, however, there are no states in the gapand one observes a logarithmic increase of the entropy. The system is also a good case to comparethe entanglement Hamiltonians on both sides. They have the same spectra, but their single-particleeigenfunctions and their explicit forms as hopping models are quite different. Basically, one findsthat the features are similar to those of the pure systems on the corresponding side of the chain.For the eigenfunctions this means a decay from the interface into the interior which is slow in themetal and exponential in the insulator [1, 35].For the metal-metal system we also study the behaviour of the entanglement entropy aftera local quench in which the two pieces in their ground states are put together. Here the twoFermi velocities can be seen directly in the time structure and the result can be interpreted in thewell-known picture of two emitted particles [36].The paper is organized as follows. In section 2 we describe the set-up und give the basic formulae.In section 3 we investigate the metal-metal case by forming and diagonalizing numerically thecorrelation matrix. We show the entanglement entropy and discuss its scaling behaviour and fillingdependence together with some entanglement spectra. In section 4 we consider the metal-insulatorcase at half filling and compare all relevant quantities with those of the pure systems. In section 5we present the time evolution of S after connecting two metallic systems and Section 6 contains asummary. Finally, some analytical results for the metal-metal system are given in an appendix. II. SETTING AND BASIC FORMULAE
We consider open chains of free fermions with nearest-neighbour hopping and 2 L sites. Thehopping is different in the left and right half and also between both parts. The Hamiltonian is H = − L − X n = − L +1 ˆ t n ( c † n c n +1 + c † n +1 c n ) (1)For the metal-metal system ˆ t n = t : n < t : n = 0 t : n > H , one can alwaysachieve t t = 1. We will assume this in the following and write the quantities as t = exp(∆), t = exp( − ∆) and t = exp(∆ ). Moreover, we will always consider positive ∆, i.e. t > t . Insome places we also use the ratio r = t /t .For vanishing coupling ( t = 0), the two parts of the chain have single-particle energies ω α = − t α cos( q α ) , α = 1 , q α = πm/ ( L + 1) , m = 1 , , ...L . The corresponding band structure with bandsbetween ± t α is shown on the left of Fig. 1. In the coupled system, one has extended states for | ω | < t , while outside this region they are confined essentially to the left half-chain. Moreover,one can have two states localized at the interface if t is large enough.For the metal-insulator system we takeˆ t n = n < t : n = 01 + ( − n +1 δ : n > ω = ± q cos ( q ) + δ sin ( q ) (5)where the momenta have to be determined from the boundary condition. Thus there is a gapbetween ± δ . The resulting band structure is shown on the right of Fig. 1. For an open chain,there are also states localized at the boundary with exponentially small ω , if the outermost bond ω t −t t δ−δω −t −11 FIG. 1: Schematic band structure of the uncoupled composite systems. Left: Metal-Metal, Right: Metal-Insulator. is a weak one. The coupled system has extended states for | ω | > δ and others, confined essentiallyto the metal, inside the gap. In this sense, it is still critical.It is simple to set up the eigenvalue problem for the full systems, and some details are given inthe Appendix. However, the matching conditions in the center are difficult to handle in general.Therefore we determine the single-particle eigenvalues ω q and the corresponding (real) eigenfunc-tions Φ q ( n ) numerically. The correlation matrix C mn = h c † m c n i is then C mn = X q ∈ F Φ q ( m )Φ q ( n ) (6)where the sum extends over the states q in the Fermi sea. Restricting C mn to the chosen subsystem(either the left or the right half-chain), its eigenvalues ζ k give the single-particle eigenvalues ε k =ln[(1 − ζ k ) /ζ k ] of the free-fermion entanglement Hamiltonian H in ρ = 1 Z exp( −H ) (7)where ρ is the reduced density matrix [37]. From them, the entanglement entropy S follows as S = X k ln(1 + e − ε k ) + X k ε k e ε k + 1 (8)In addition to the eigenvalues ζ k resp. ε k , we also determine the corresponding eigenfunctions ϕ k and construct H in section 4. III. METAL-METAL SYSTEM
We first consider half-filled systems where the Fermi level is in the middle of the bands. Theresulting entanglement entropies for t = 1 and three values of the parameter ∆ are shown in Fig.2, both for even and for odd L . One sees that the typical increase with the size, known for thehomogeneous case ∆ = 0, persists in the composite systems. The plot against lnL shows thatalso the asymptotic law 1 / L + k is unchanged. However, the value of k becomes smaller andthe finite-size effects, in particular the even-odd alternation, increase dramatically as ∆ becomeslarger. For ∆ = 3 (not shown), the values of S are initially close to zero and to ln 2, respectively.In this case, the ratio of the bandwidths t /t ≃ /
400 is already very small, the states localized onthe left drop rapidly on the right, while the extended states have very small amplitudes on the left(see the Appendix). For even L , this leads to a small entanglement which only increases as morestates come into play at larger L . The value ln 2 for odd L is a consequence of the particle-holesymmetry of the problem which, in this case, forces one of the ε k to vanish. ∆ =0 ∆ =1 ∆ =2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 2.5 3 3.5 4 4.5 5 5.5S ln L ∆ =0 ∆ =1 ∆ =2 FIG. 2: Entanglement entropy S for ∆ = 0 and three values of ∆. Left: as function of L . Right: asfunction of ln L . The upper (lower) curves correspond to odd (even) values of L . The function S ( L, ∆) has an intriguing universal behaviour. If one defines a length L ∗ = L/α and chooses the scale factor α properly, one can achieve S ( L ∗ , ∆) = S ( L, S ( L, ∆) = S ( αL,
0) (9)In the asymptotic region, where one has straight lines in a logarithmic plot, this feature is not sur-prising: A rescaling can always generate the necessary shift to make the curves coincide. However,the phenomenon is not restricted to this region. As shown in Fig. 3 on the left, it also holds forsmall sizes where there is still curvature in the graph. The variation of α with ∆ is given on theright hand side of Fig. 3. One sees that the results determined from different (even) L agree verywell and that α becomes rapidly smaller as ∆ increases. The curve can be described approximatelyby 1 / ch ∆. Note that the relation (9) can only be applied to values of L such that αL is largerthan 1. α L ∆ =0 ∆ =1 ∆ =1.5 ∆ =2log(x)/6+0.45 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 α = L / L * ∆ L=20L=50L=100CFT
FIG. 3: Scaling behaviour of the entanglement entropy. Left: S as a function of the effective length αL forfour values of ∆ (logarithmic scale). Right: Variation of L/L ∗ = α with ∆. The dotted line is the conformalfactor in (10). At this point, an observation from the appendix is helpful. Namely, near the Fermi level thewavefunctions in the right subsystem are the same as for a homogeneous system of total length L (1 + r ). Thus, ignoring the states outside the narrow band, the correlation matrix on the right isthe same as for an unsymmetrically divided homogenous chain and one can invoke the conformalresult for the entropy [38] extended in [39] to include 1 /L corrections. This gives S F C = 16 ln z − ( − ℓ z + const ., z = 4 π L (1 + r ) sin πℓL (1 + r ) , (10)with ℓ = L . The r -dependent factor in z provides a similar rescaling of L as α , and a correspondingshift of the curves. It turns out, however, that this is not enough to make them coincide. Rather,one has a residual difference S − S F C = S I (∆), which one can attribute to the interface, and whichrises smoothly from 0 to about 0 .
13 as ∆ increses from 0 to 3. It has its origin in the neglectedexponentially decaying states below the narrow band.This can be seen clearly, if one considers a partition of the system with ℓ < L sites on the rightand calculates the entanglement between this subsystem and the remainder. As demonstrated inFig. 4, the expression (10) then fits the data very well as soon as one moves away from the interfaceby a few lattice sites. One can also consider a division located in the left part of the system wherethe extended wave functions are the same as in a homogeneous system with total length L (1 + 1 /r ),but in this case the conformal formula does not work well because the additional states are moreimportant. These results also show that the symmetrical division is actually a somewhat specialcase.We now turn to the effect of t . If one varies the central bond in a homogeneous chain (∆ = 0), ∆ =0 ∆ =1 ∆ =2 94 96 98 100 -0.04 0 0.04 0.08 0.12 S - S F C PSfrag replacements ℓ FIG. 4: Entanglement entropy between the outermost ℓ sites of the right subsystem and the remainder for L = 100 and three values of ∆. The inset shows the deviation from the formula (10). the asymptotic behaviour of S is S = c eff L + k (11)where c eff depends on t and is given by an explicit, though lengthy formula in which only thetransmission coefficient at the Fermi energy T ≡ s enters [4]. But the calculation in the Appendixshows that T is independent of ∆ and given by T = 1ch ∆ (12)Therefore one expects the result of the homogeneous problem also in the composite systems. Thisis, in fact, the case and demonstrated in Fig. 5 on the left, where results obtained by fitting thedata between L = 100 and L = 400 to (11) plus a 1 /L term are collected. For ∆ = 2, the averagebetween even and odd sizes was taken, while for ∆ = 1 this was unnecessary. This verifies theFermi-edge character of c eff also in the present case.The constant k , on the other hand, depends on ∆, as was found above already for the specialcase t = 1. It is shown in Fig. 5 on the right. The values for odd L are always larger than thosefor even L with the maximal difference of ln 2 appearing for t →
0. This is connected with thehalf-integer particle number in each subsystem for odd L which leads to a kind of singlet stateeven in the decoupling limit. The minimum in k for even L is also present for ∆ = 0 and was alsofound for a segment in an infinite chain [2]. eff t ∆ =0 ∆ =1 ∆ =2 -0.4-0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1k t ∆ =1 odd ∆ =1 even ∆ =2 odd ∆ =2 even FIG. 5: Effective central charge c eff and constant k in (11) as functions of t for different values of ∆. Theline ∆ = 0 in the left figure is the theoretical result, see [4]. Finally, we consider the dependence of S on the filling ν = N/ L of the system, where N is thenumber of particles. It is shown in Fig. 6 for a fixed length L = 200 and four values of ∆. Thesmooth curve (red) is the result for the homogeneous system and given, up to the small even-oddoscillations [39–42] and a constant, by 1 / q F ), where q F = πν [43]. Turning on ∆, regionsof very small entanglement develop for small and large filling, where only states localized on theleft are occupied or remain empty. At the same time, the oscillations of S in the central regionbecome slower and slower. ν∆ =0 ∆ =0.1 ∆ =1 ∆ =2 FIG. 6: Entanglement entropy S as a function of the filling ν = N/ L for L = 200 and different ∆. The origin of these oscillations, at the level of the eigenvalues ε k , can be seen in Fig. 7. There,the low-lying ε k are shown for all fillings between 0 . .
8. For the case ∆ = 2, shown on theright, the variation of the ε k with ν is rather slow and the curves cross zero only at two fillings. Atthese points, the crossing eigenvalue gives the maximal contribution to S , namely ln 2, and theycoincide with the locations of the maxima in Fig. 6. For ∆ = 1, shown on the left, the variation ismuch faster and one has altogether 16 crossings. This is again the number of maxima of S in thecenter of Fig. 6. -10-5 0 5 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ε k ν -10-5 0 5 10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 ε k ν FIG. 7: Variation of the low-lying single-particle eigenvalues ε k with the filling ν = N/ L for L = 200. Left:∆ = 1. Right: ∆ = 2. The colours are the same as in Fig. 6. These structures can again be understood from the relation to a smaller, unsymmetricallydivided homogeneous system. In such a case, if L ′ is the size of the smaller subsystem, one has L ′ crossings of the eigenvalues and the effect on S is given by a factor sin ( q F (2 L ′ + 1)) / sin q F replacing the alternating sign of the 1 /z term in (10). This leads to slow oscillations, although lesspronounced than the observed ones. However, L ′ = rL gives L ′ = 3 . L ′ = 27 for∆ = 1 and thus not the correct numbers (2 and 16), even if one takes the nearest integer. Butone can argue that the length rL only refers to the states in the center of the band, while near theedges sin k appears in (26) which is smaller than k by a factor of 2 /π . Working with ˜ L ′ = 2 rL/π gives values 2 . .
2, respectively, which are quite close to the numerical findings. One alsoarrives at ˜ L ′ by counting the number of levels which the left subsystem contributes to the totalnumber of band states. To accommodate them, one needs just this number of additional sites.One should mention that one can see basically the same structures also in the energy ω astiny deviations from the otherwise smooth level spacing of the extended states. In this case, theinterpretation is that ˜ L ′ also gives the number of periods of the slowly varying tangent in (26) as k sweeps through the band, and thus the number of possible peculiarities in the allowed momenta.0 IV. METAL-INSULATOR SYSTEM
In the following, we always consider half-filled systems. To avoid boundary states in the dimer-ized subsystem, we work with even L and choose strong bonds 1 + δ at the borders. The entan-glement entropies for two relatively small dimerizations and t = 1 are shown in Fig. 8, togetherwith those for the pure systems. One sees that the mixed system always has an intermediateentanglement, which is very plausible. Replacing one half of the metal by an insulator reduces themetallic result, while replacing one half of the insulator by a metal increases the insulator result.For δ = 0 . δ =0.001 M-MM-II-I 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0 50 100 150 200 250 300 350 400S L δ =0.01 M-MM-II-I FIG. 8: Entanglement entropies for metal-metal, metal-insulator and insulator-insulator systems as a func-tion of L for two values of the dimerization parameter δ . while for δ = 0 .
01 a saturation is clearly visible for the lower ones. This can be interpreted interms of the correlation length ξ = 1 /δ in the insulator. It is ξ = 1000 in the first case so that thenon-criticality does not fully show up. In the second case, where ξ = 100, it does and the crossovertakes place roughly around this value.In the region L ≪ ξ , one can fit the curves for the composite system to a form a ln L + b and findsvalues of a quite close to 1 /
6. This can also be done for t < c eff which liesomewhat below the metallic one shown in Fig. 5, but approach it as δ goes to zero. In the oppositelimit, L ≫ ξ , the entropy saturates and for t = 1 one finds the asymptotic law S ≃ / ξ +const . as for the pure insulator but with a larger constant. To obtain the coefficient 1 / ξ , one has to include subleading corrections of the form ( d ln ξ + e ) /ξ which oneknows in the pure case from the connection to a transverse Ising chain sketched below. Again, theanalysis can be extended to t <
1, and in this case one finds c eff to an accuracy of 3-4 digits.On the level of the wavefunctions entering the correlation matrix, the (pseudo-) critical be-1haviour can be understood in the following way. For L < ξ one has | ω | ∼ /L > δ and there areno states in the metal with energy in the band gap of the insulator and none in the insulator tooclose to the band edge. Thus one only has extended wavefunctions which are similar to those inthe pure metal. This can be seen analytically or from the numerics. -30-20-10 0 10 20 30 20 22 24 26 28 30 32 ε k k I-IM-IM-M FIG. 9: Eigenvalues ε k for metal-metal, metal-insulator and insulator-insulator systems with L = 50 and δ = 0 . It is also instructive to look at the ε k spectra. In Fig. 9 an example is shown for the case δ = 0 . L = 50. The lowest curve with the slight bend is the well- known result for the metal,while the highest one is for the insulator and strictly linear as for the transverse Ising chain or theXY model [1]. In fact, one can find a relation between the ε k for the dimerized hopping modeland the transverse Ising chain using the method in [44]. The latter model then has field h = 1 andcoupling λ = (1 − δ ) / (1 + δ ) and the ε k are ε k = (2 k + 1) ε, k = 0 , ± , ± , ..., ε = π I ( λ ′ ) /I ( λ ) (13)where I ( λ ) denotes the complete elliptic integral of the first kind and λ ′ = √ − λ , see also [45].The MI result lies in between, with a small break after the first eigenvalue, which becomes moreapparent for larger values of δ , when the curve becomes steeper and more linear.For the spectrum, it does not matter in which of the two subsystems one considers the correla-tion matrix. Formally, this follows from the property C = C for the full system [46]. However, theeigenvectors of the two reduced matrices, and therefore the corresponding entanglement Hamilto-nians H , are quite different. This is illustrated in Fig. 10 for the lowest | ε k | and δ = 0 .
1. We haveplotted ϕ k ( n ) for + ε k on the left and for − ε k on the right, since these two are directly related.Namely, if one partitions C into four blocks according to the location of the sites and ϕ is an2eigenvector of C for part 1 with eigenvalue ζ , then ϕ = C ϕ (14)is an eigenvector of C for part 2 with eigenvalue 1 − ζ .One sees the typical features of the pure systems on the two sides, namely a slow power-lawdecay in the metal (left) and a fast one within a distance ξ = 10 in the insulator (right). The purecases are shown for comparison, and one notes that the differences in the composite system arerelatively small, lying more in the amplitudes than in the general behaviour. -0.4-0.2 0 0.2 0.4 0.6-50 -40 -30 -20 -10 0 ϕ ( n ) n δ =0.1M-MM-I -0.4-0.2 0 0.2 0.4 0.6 0 10 20 30 40 50 ϕ ( n ) n δ =0.1M-II-I FIG. 10: Eigenvectors of the MI correlation matrix for the smallest | ε k | in the metallic (left) and theinsulating (right) subsystem for L = 50 together with the results for the homogeneous cases (MM and II). With the eigenvectors, one can explicitly determine the entanglement Hamiltonians H . Theyhave again the form of hopping models with basically only nearest-neighbour hopping as in H , butthe amplitudes ˜ t n increase from the center towards the boundaries.In the pure metal, this increase is monotonous and the ˜ t n vary as n (2 L − n ) [1], while in theinsulator one has an additional alternation which is coupled to the bond alternation in H . Due tothe form of the eigenvectors, the result in the composite system is again close to that of the puresystem on the corresponding side. This is shown in Fig. 11 for a system with a total of 2 L = 16sites, which is the largest size attainable before numerical errors in the large ε k set in.3 n nM-MM-I 0 5 10 15 20 1 2 3 4 5 6 7-t~ n n M-II-I FIG. 11: Nearest-neighbour hopping in the entanglement Hamiltonians for the left and right subsystem asa function of the position together with the results for the pure systems for L = 8 and δ = 0 . V. EVOLUTION AFTER A QUENCH
In this section, we return to the metal-metal systems and study how the entanglement evolvesafter one joins the two initially disconnected parts. This type of local quench has been consideredrepeatedly in the past [11, 47–59]. For the homogeneous case, one finds oscillations of S ( t ) whichone could call “entanglement bursts”, see e.g. [51]. In our case, the situation is more complex,because of the interface and the two Fermi velocities.The two pieces are coupled at time t = 0 by switching on t and the evolution of C ( t ) iscalculated via the Heisenberg operators c n ( t ) , c † n ( t ). The resulting S ( t ) is shown in Fig. 12 for twodifferent values of ∆ and of t . The system on the left with ∆ = 0 . τ τ τ + τ ∆ = 0 ∆ = -1 0 0.5 1 1.5 2 0 100 200 300 400 500 600 700 800S t τ τ τ + τ ∆ = 0 ∆ = -1 FIG. 12: Evolution of the entanglement entropy S after connecting two metallic systems with L = 100.Left: ∆ = 0 .
5. Right: ∆ = 1. The times indicated are the theoretical values, see the text. = 0 (upper curves) one has the following general features: An initial burst ending witha decrease at a time τ followed by a plateau, another burst at time τ which is a kind of mirrorimage of the first ending at time τ + τ and then a rough repetition. For ∆ = 1, the plateau islonger and shows six additional structures, whereas for ∆ = 0 . t = 0, travelling in opposite directionsand spreading the entanglement. In the present case, they have the velocities v = t and v = t and the corresponding space-time diagram is given in Fig. 13. −L τ τ τ + t L τ FIG. 13: Trajectories of the particles in the Calabrese-Cardy picture.
After time τ = 2 L/v , the left particle returns to the center, and if it simply moved into theright subsystem, the entanglement would continue to drop. This happens in a somewhat relatedsituation where the velocities are the same, but the size of the right subsystem is larger [51]. Here,however, there is a probability for reflection at the interface, such that the entanglement rises againand another small “burst” follows. For ∆ = 1, where v = 7 . v , the left particle can make sevenround trips before the right particle retuns to the center at time τ = 2 L/v = τ /r . These are veryclearly visible in the Figure. There are, however, some remarks to make. According to (12) there isno reflection right at the Fermi level for ∆ = 0. The effect can therefore only come from somewhatslower particles away from it. In fact, the numerical values for τ and τ are somewhat larger thanthe theoretical ones, which supports this argument. Also, one would expect the reflection effect tobecome weaker with each cycle, which is only barely the case. On the other hand, for ∆ = − R = 0 .
58 at the Fermi level and the effect should be stronger, which is also not the case.Nevertheless, the picture gives a good overall description.One could presume that at least the first burst can be described by the conformal result [51] S h ( vt ) = c (cid:12)(cid:12)(cid:12)(cid:12) Lπ sin πvt L (cid:12)(cid:12)(cid:12)(cid:12) + const . (15)valid for a homogeneous system with particle velocity v . Thus we tried the Ansatz S ( t ) = [ S h ( v t ) + S h ( v t )] / . (16)which, on the scale of Fig. 12, fits the data for ∆ = 0 very well. However, a closer look at the finaldecrease shows that it comes too early since, as mentioned, the value of τ is too small. Finally,we note that the first-burst results for a homogeneous but unequally divided system with the sameround-trip times τ = 2 L and τ = 2 L lie above ours, and the difference increases with ∆. Thusthe two problems are not trivially connected. VI. SUMMARY
We have considered the entanglement in fermionic chains composed of two different halves. Insolid-state terminology, they were metals and insulators, while in statistical-physics terms theycorresponded to critical and non-critical systems. The metal-metal system at half filling showedthe same logarithmic behaviour of S as a homogeneous chain with a defect. In the subleadingterms, the asymmetry showed up, but the entropy had a remarkable scaling property. Its fillingdependence, finally, showed oscillatory behaviour again linked to the asymmetry. Both featurescould be understood from the nature of the extended eigenfunctions which are the same as inhomogeneous systems with different lengths.The difference in the Fermi velocities showed up directly in the quench experiment where twohalf-filled metals were joined and the entanglement was monitored. In that case, the value of r determined, how many cycles one sees in S ( t ), before a certain return to the initial situationtakes place. This pattern would occur in unsymmetrically divided homogeneous systems only withadditional defects.The metal-insulator system was somewhat simpler. Its entanglement properties were seen tolie always between the two pure systems and as a function of the size, one has a crossover froman initial critical behaviour with a logarithmic increase of the entanglement to a saturation typicalfor a non-critical system with finite correlation length. It also provided an instructive example6for entanglement Hamiltonians which are quite different for the two subsystems although theyhave the same spectra, as they must. One should note that the dimerized hopping model used forthe insulator, and investigated already in [45], plays a central role in the theory of polyacetylene[60, 61]. In that sense, we studied a particular polymer system.We considered the geometry with open ends, because then one has only a single interface.However, one could equally well look at rings. For composite transverse Ising models this was donefirst in [31, 32] but without particular interface bonds and with a view on the spectra. Obtainingthe entanglement entropy is more complicated, and we could only give an analytical result for theasymptotic behaviour of the metal-metal system.While we focussed on the entanglement entropy, it is known that the particle-number fluctua-tions in the subsystems behave similarly [62]. Thus, in a homogeneous metal, the prefactor of theln L term in the fluctuations is given by T / π with the transmission coefficient T [8] and variesqualitatively like c eff . The same result is found in the composite systems, and this could offer away to access the entanglement experimentally. Acknowledgments
MCC and VE acknowledge the hospitality of Freie Universit¨at Berlin, where part of this workwas done. IP thanks National Tsing Hua University, Hsinchu, Taiwan for an invitation. The workof VE was supported by OTKA Grant No. NK100296 and MCC acknowledges NSC support underthe contract No. 102-2112-M-005-001-MY3.
Appendix: Some formulae for the metal-metal system
Let ¯ φ ( n ) and φ ( n ) denote the wave function at site n on the left and right, respectively. The bulksolutions on the two sides are waves with momenta q and q and energy ω = − t cos q = − t cos q .At the interface, the equations are t ¯ φ (0) = t φ (0) , t ¯ φ (1) = t φ (1) (17)For a wave coming in from the left and being partially reflected and transmitted, the amplitudesare ¯ φ ( n ) = A exp ( iq n ) + B exp ( − iq n ) φ ( n ) = A exp ( iq n ) (18)7Inserting this into the relations (17), one finds for the reflection coefficient R = (cid:12)(cid:12)(cid:12)(cid:12) B A (cid:12)(cid:12)(cid:12)(cid:12) = ch 2 ν − cos ( q − q )ch 2 ν − cos ( q + q ) (19)where the quantity ν is defined as exp (2 ν ) = t /t t . In our parametrization, where t t = 1, onehas ν = ∆ . The transmission coefficient follows from T = 1 − R and can be written T = sin q sin q sh ν + sin (( q + q ) /
2) (20)If the left and right subsystems are identical, q = q = q , and one finds the result for a bond defectin an otherwise homogeneous chain. The same holds in a general system if one is in the middle ofthe bands. Then ω = 0 , q = q = π/ T = 1ch ν (21)Due to the factor sin q in (20), the transmission vanishes at the edges of the narrow band.The extended eigenfunctions of H have the form¯ φ ( n ) = A sin q ( n + L ) φ ( n ) = A sin q ( n − L −
1) (22)and the conditions (17) lead to sin q ( L + 1) sin q ( L + 1)sin q L sin q L = t t t (23)which, together with t cos q = t cos q , determines the allowed momenta. Setting q α = π/ k α and assuming L even, it takes the formcos k ( L + 1) cos k ( L + 1)sin k L sin k L = t t t (24)The relation between the k α readssin k = r sin k , r = t /t (25)and for small r simplifies to k = r sin k . Inserting this into (24) and setting L + 1 ≃ L in thenumerator then gives cot ( k L ) = t t t tan ( rL sin k ) (26)which contains only the momentum k . The solutions can be discussed graphically, but one seesdirectly that besides L a second length L ′ = rL appears. If it is large enough, the tangent8completes several cycles as k varies, and this can lead to additional features in the momenta or inthe amplitude ratio which is, again with L + 1 ≃ L , A A = − t t cos( rL sin k )sin k L (27)If k is small, k = rk for all r < t /t t = 1 can be written in the two formstan ( k rL ) tan ( k L ) = 1 , tan ( k L ) tan ( k L/r ) = 1 (28)Both equations are the same as for a homogeneous chain built up from two pieces with differentlengths and can therefore be solved explicitly. For k , the homogeneous system has length L + L ′ ,while for k it has length L + L ′′ , where L ′′ = L/r . Thus the small momenta are equidistant, i.e. k = π L + L ′ ) (2 n + 1) , n = 0 , ± , ± , ... (29)Using (28) in (27), one finds ( A /A ) = t = 1 /r , i.e. the amplitudes are larger on the right thanon the left. Including normalization to leading order, this gives prefactors A = 2 / ( L + L ′′ ) and A = 2 / ( L + L ′ ) which correspond exactly to these chain lengths.9 References [1] Peschel I and Eisler V 2009
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