aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Entanglement of heterogeneous free fermion chains
Yuchi He
Department of Physics, Carnegie Mellon University (Dated: July 27, 2018)We calculate the ground state entanglement entropy between two heterogeneous parts of a freefermion chain. The two parts could be XX chains with different parameters or an XX half chainconnected with a quantum Ising half chain. It is shown that logarithmic behavior holds if the twoparts are conformally critical. In other cases, area law holds with abundant subleading behaviors.In particular, when XX chain at Lifshitz point is connected with a conformally or Lifshitz criticalpart, entanglement entropy converges algebraically with a fractional subleading index.
I. INTRODUCTION
The importance of bipartite entanglement entropy(EE), as a concept or a tool, has been realized by quan-tum physics community in recent years. For illustration,EE is related to the Bekenstein [1] entropy of black holesand can be used to diagnose quantum criticality [2, 3],topological orders [4], and localization transition [5–7].In a quantum ergodicity assumption called ETH, thesame scaling behavior shared by EE and thermal entropybridges quantum mechanics and statistical mechanics [8–10]. Among these studies, the scaling law of EE overpartition’s size usually plays a central role. It has beenproved [11] that the ground state EE of a 1-D gappedsystem is bounded, which is the foundation of a power-ful numerical method named DMRG [12, 13]. Proofs orarguments of ground state area law for a gaped systemin arbitrary dimension has also been presented [14, 15].Solid results have also been obtained for 1-D conformalcritical systems [16], of which ground state EE divergeslogarithmically with the length of the partition.While the entanglement properties of homogeneous [2–4, 8–11, 14–17] and disordered systems [5–7, 18, 19] arewell understood, the entanglement properties of hetero-geneous systems are not. Heterogeneous systems are ofgreat interest in physics. In thermodynamics, the con-tacting of two different systems is a common context. Inquantum transport experiments, the device is a hetero-geneous structure. The entanglement between heteroge-neous subsystems gives a measure of the quantum fluc-tuations which link them together and make observablesnear boundary different from observables of bulk. As willbe made clear in this paper, the study of entanglementof heterogeneous systems may provide new perspectiveson phase transition and quantum transport.In this study, we are interested in the ground stateentanglement of one-dimensional heterogeneous systems.For example, one may wonder what is the scaling law ofEE when two parts are critical and gapped respectively.Apart from the properties of the two halves, the conclu-sion may also depend on the types of interaction betweenthe two parts. A good starting point to study this topicis to look at a simple model. One choice is the heteroge-neous system consisting of two XX (lattice free fermion)chains with different potential and hopping coefficients. Another choice is connecting a quantum Ising (latticeBdG fermion) chain with an XX chain. XX amd Isingchains can either be conformally critical, the correspond-ing CFTs are different. With the components chosenabove, we study two kinds of heterogeneous structures(Fig.1) in one dimension. One is that two parts are ofthe same size L , the other is that a finite subsystem oflength L is embedded in an infinite environment. For thesystems described above, numerically exact calculationscan be implemented for L large enough.Our main results are the properties of functions S ( L )(dependence of EE on L ) in various situations. Those sit-uations can be classified by criticality of two parts of thechains. XX chain with an external field can be confor-mal critical, Lifshitz critical and gapped. Quantum Isingchain can be conformal critical and topologically/triviallygapped. Our results are as follows.EE scales logarithmically if and only if the two partsare conformally critical. Other situations follow area law.Within the area law, the subleading behaviors of EE areabundant. While subleading decay patterns of EE arealaw are always exponential [20] for homogeneous chains,EE subleading terms of heterogeneous chains demon-strate either exponential or algebraic decay. The lengthscale of an exponential decay is found and argued to bethe screening length. The indices of algebraic decays areeither integer or fractional.Fractional indices are found to be signatures of Lifshitzcriticality. ( In contrast, Lifshitz points of homogeneousfree fermion chain have no non-trivial EE behavior [21].)Remarkably, universal crossover of EE is observed whenthe two parts cross over from both gapped to both Lif-shitz critical. Besides heterogeneous chains with two Lif-shitz critical parts, fractional indices are also observedwhen a Lifshitz critical part meets with a conformal crit-ical part.The outline of this paper is as follows. Section II in-troduces the formulas to compute EE of free fermion (in-cluding BdG fermion) from correlation functions. Sec-tion III mainly discusses EE of heterogeneous XX chains.In the beginning, the correlation functions are derived,and particle-hole symmetry of EE is discussed. Afterthis preparation, EE leading behaviors and EE sublead-ing oscillatory and decay patterns are demonstrated andinterpreted. EE leading and subleading behaviors arechecked to be general for heterogeneous free fermion LH RHL L structure I
SE EL structure II
FIG. 1. Two kinds of structures to be studied. The enviro-ment (E) of structure II is infinitely large. Different symbolsrepresent different parts of the chains. The chain is heteroge-neous if the two parts are heterogeneous. chains. Lastly, section IV investigates EE of XX/Isingheterogeneous chains.
II. ENTANGLEMENT OF FREE FERMIONS
Bipartite entanglement entropy (EE) is defined as thevon Neumann entropy of the reduced density matrix ρ : S = − Tr( ρ log ρ ) (1)Reduced density matrix ρ is obtained by tracing the den-sity matrix of the pure state over the degrees of freedomoutside a given region. For one dimensional systems,there are two typical geometries (Fig.1). The structure Iis that the chain is finite with length 2 L and the regionwith length L is one of its halves. The structure II is thata finite subsystem of length L is embedded in an infiniteenvironment. We are interested in how EE depends onthe length L of the region ( S ( L )).For lattice free fermion systems with charge conser-vation, EE can be calculated from equal time two-pointGreen function of either part [22, 23]: C ( r i , r j ) = h c † r i c r j i and S = X i ( − λ i log( λ i ) − (1 − λ i ) log(1 − λ i )) (2)where λ i are the eigenvalues of C .For BdG fermion, the expectation values of pairedcreation (annihilation) operators can be non-zero. Inthis situation, the two-point Green functions F ( r i , r j ) = h c † r i c † r j i are also needed. Let ν i be the eigenvalues of thematrix (2 C + 2 F − C − F −
1) and the EE is givenby Eq. (2) with λ i = √ ν i .Using formulas above, the ground state EE scaling lawsof homogeneous free fermion chains have been studied [2].The key result is that EE diverges with L if and only ifthe system is conformal critical with leading term scales logarithmically with L . The logarithmic behavior canalso be derived from conformal field theory and general-ized to generic 1-D local Hamiltonians [16]. For struc-ture II, conformal field theory predicts that the criticaldegree of freedom contributes c log L to the EE lead-ing term, where c is the central charge of the CFT’s Vi-rasoro algebra. (For a given Hamiltonian on structureI, the EE leading term is always half of that on struc-ture II.) Applying the conclusion to a homogeneous freefermion chain with charge conservation, the correspond-ing EE leading term on structure II is m log L , where m is the number of conformal critical degrees of freedomand equals the number of pairs of Fermi surfaces (points).Near each pair of Fermi surfaces, there is an effective freeboson CFT with central charge c = 1. m can be changedby tuning chemical potential and the transition pointsare called Lifshitz critical points. For a Lifshitz pointseparating m = n and m = n − m = n − log L on struc-ture II. The critical phase is described by free Majoranafermion (Ising) CFT, where c = 1 / III. ENTANGLEMENT OF OFHETEROGENEOUS XX CHAINA. Hamiltonian, single-particle eigenstates andequal time correlation matrix
In section III, we mainly deal with nearest-neighborhopping free fermion chain: H = X i ( − t i c † i c i +1 + h.c. + h i c † i c i ) (3)The chain is heterogeneous in the sense that h and t are homogeneous in two parts respectively but differ fromone part to the other. At their mutual boundary, the twoparts are linked by hopping with parameters t interface .In what follows, we keep t to be the same everywherefor structure II while leaving it possibly different amongleft part, right part and interface of structure I. We solvethis Hamiltonian for both structures (Fig.1) and studythe ground state entanglement entropy between the twoparts.The full chain can be characterized by the criticality ofits parts. A part is denoted as “CFT” if there are Fermisurfaces. If the Fermi energy is right at the band bottomor top, the part is denoted as “Lifshitz”. Otherwise, it isdenoted as “gapped”.For the heterogeneous system, the single-particle statesare piecewise functions of the two parts. In each part,they are linear combinations of the following basis: e − βr , (1) (2)(3) (4) FIG. 2. Possible configurations. Two parts of each figureare band structures of the two parts of the system respec-tively. The two band structures may either describe LH, RHfor structure I or S, E for structure II. One may also changethe sign of t to flip the bands. Fixing the Fermi energies ofthose configurations determines the state of the systems. Inthe second sub-figure, when the Fermi surface is right at thedashed line, the two systems are both at Lifshitz point. e ikr or ( − r e − βr . The linear combinations and the val-ues of k and β are determined by matching energies andboundary conditions. For example, single-particle statesof structure II are listed in Fig.3. The quantities relevantto behaviors of EE are k and (or) β of the single-particlestates with Fermi energy. Except for trivially fully filledor empty systems, a CFT part is characterized by Fermivector k F and a gapped part is characterized by inverseFermi deay length β F . A Lifshitz part is denoted by ”0”or π . We refer a system on structure I as ( a LH , b RH ) or A LH /B RH . a LH and b RH are k F or β F of left and rightparts. The overline in β F is used to distinguish β F from k F . For example, (0 . , π ) means the right chain hasFermi vector | k F | = π , while the left is gapped with β F = 0 . A LH and B RH can be “CFT”, “Lifshitz”or “gapped”. Similarly, ( a E , b S , a E ) and A E /B S /A E areused to denote systems on structures II. A E and a E de-note environments while b S and B S denote subsystems.Equal time correlation matrices can be con-structed from single-particle eigenstates: C ( r i , r j ) = P E ( m ) S E FIG. 3. Classification of the single-particle states of struc-ture II. Single-particle states exist in the energy range exap-nded by the band of subsystem and the band of envi-ronment (Fig. 2). The unnormalized single-particle wavefunctions of each region are as follows. 1:( P W, e − β | x | ); 2,5:( P W, P W ); 3 :(( − x cosh( βx ) or ( − x sinh( βx ) , P W ); 4:(cosh( βx ) or sinh( βx ) , P W ); 6: ( P W, ( − x e − β | x | ). Thefirst components denote subsystem. PW means plane waves.Due to the potential flipping symmetry of EE and for com-putational convenience, we calculate the situations when theFermi surface are in the region 1, 4, 5. mining the EE’s oscillatory period. B. Entanglement entropy: leading terms andsubleading oscillatory pattern By numerical calculation, we obtain entanglement en-tropy of structure I and structure II for different partitionsize L . We find that EE leading term of structure I is halfof that of structure II if the bulks and the interface(s) arerespectively the same. EE diverges if and only if the twoparts are conformally critical (Fig. 4). The correspondingleading term is c ′ log( L ), where c ′ is determined by thetransmission ratio of the mode at the Fermi point [25].Otherwise, the leading behavior is area (constant) law.The subleading term of logarithmic law is a possibly os-cillatory o(1) term. The subleading term of area law mayalso be modulated by oscillation.The subleading term is oscillatory only if either part ofstructure I or the subsystem of structure II is CFT, due tothe existence of Fermi vectors. In contrast, for an infinitehomogeneous free fermion chain, EE subleading term hasno oscillatory behavior [26]. The period of the oscillationis found to be determined by k F,L , k F,R (structure I) and k F,S (structure II), while k F,E (structure II) is not relatedto the oscillation.The oscillation pattern is given by the wave vector(s)2 k F,j , where j denotes LH, RH or S. The factor 2 comesfrom the fact that EE is invariant under particle-holetransformation | k F | → π −| k F | . The oscillation is strictlyperiodic if and only if πk F,j is (are) rational and the pe-riod is (are) given by T j , which is the denominator ofthe irreducible fraction πk F,j . For CF T /CF T systems,the period is the least common multiple of T LH and T RH (Fig.4). Note that k F,L and k F,R influence the oscillatorypattern symmetrically since one can either pick reduceddensity matrix of the left or right part to compute the bi-partite EE. (Fig.4). Once the oscillation is periodic, thearray S ( L ) divides into several non-oscillatory branches.For ( CF T / ) CF T /CF T systems, the number of branchesmay be smaller than the period. However, such degener-acy is not robust under tuning parameters t i in the twoparts or at the interface (Fig.4).We note that choosing oscillation to be strictly periodicis convenient for studying EE subleading terms’ decaypattern. C. Entanglement entropy: subleading decaypattern of area law With the exception of ( CF T / ) CF T /CF T systems, thenumerical results indicate that S ( L ) converges to a singlelimit. The patterns of convergence are observed to beeither exponential or algebraic decays.For ( gapped/ ) CF T /gapped systems, we find the con-vergence of each S ( L ) branch is algebraic (Fig.5): S = S + dL , (4)where d is different for different branches. Tuning thehopping parameters or gap can change the sign d of somebranch (Fig.5).For CF T /gapped/CF T systems, S ( L ) is not oscilla-tory and S ( L ) shows an exponential convergence (Fig.6): S = S − d ′ e − αL , (5)where d ′ is a constant. Here, α ≈ β F,S = 2 arccosh(1 + g t S ), where g is the gap and t S is the hopping amplitude.The inverse decay length 2 β F,S can be considered as an-alytical continuation of oscillation wave vector 2 k F,S inthe previous subsection.For ( gapped/ ) gapped/gapped systems, S ( L ) also con-verges exponentially (Fig.9). Similarly to the previoussituation, α is estimated as 2 β F,S = 2 arccosh(1 + g t S )for structure II. For structure I, α is approximately2 min( β F,LH , β F,RH ) = 2 arccosh(1 + g t LH ,t RH ) ).Remarkably, we find that systems with Lifshitz part(s)have algebraic EE subleading behavior with fractionalindices: S ( L ) = S − d ′′ L γ (6)Systems with Lifshitz part(s) are classified and labeledas :[a] CF T /Lif shitz/CF T [b] ( Lif shitz/ ) CF T /Lif shitz [c] ( Lif shitz/ ) Lif shitz/Lif shitz . (Fig. 2. (2))The S ( L ) of [a], [c] has a single branch while S ( L ) of[b] has multiple branches. We extract γ by fitting dS ( L ) dL of each branch in a log-log plot. The fitting results arevalidated by the linearity of − /L γ vs. S plots. In such plots, all the branches of [b] cross at one point when − /L γ is extrapolated to zero.The fitted γ for [a], [b] and [c] are close to rationalnumber , and respectively. Errors of the fittingare estimated and listed in Table I. We demonstrate the − /L γ vs. S plots in Figs. 5, 6 and 8. The fitting re-sults of [a] and [c] on structure II is shown in Fig.7. For[b], like d in Eq. 4, d ′′ of some branch can also changesign by tuning hopping parameters. When d and d ′′ isclose to 0, the branch appears flat (Fig. 5), and gives anindetermined value for γ .To have some understanding of the fractional sub-leading behavior, it is helpful to look at crossover fromgapped to Lifshitz phase. The above [a], [b] and [c] sys-tems can be approached by decreasing the gap of [a ′ ],[b ′ ]and [c ′ ] respectively, where[a ′ ] CF T /gapped/CF T [b ′ ] ( gapped/ ) CF T /gapped [c ′ ] ( gapped/ ) gapped/gapped During this procedure, the numbers of electrons (orholes) in the gapped part and the particle number fluctu-ations increase. The EE is a measure of fluctuation [27]and is expected to increase with more fluctuation. Asthe gap of one part vanishes, the part is Lifshitz criticaland those almost extended bound states β ≈ /β F is the length scale of the interface, wherethe gapped part has some electrons or holes.) Intuitively,such almost extended states make EE more sensitive to L in the large L limit. Hence, the convergence is ex-pected to be slower and the subleading behavior (Eq. 4and Eq. 5) should change (Fig.5, Fig.6). For the crossoverfrom [a ′ ] to [a] and from [b ′ ] to [b], the change is fromalgebraic decay with index 1 to algebraic decay with anindex smaller than 1. The fitting results are γ = < γ = < ′ ]to [c], the change is from exponential decay to algebraicdecay with γ = .The above crossovers feature the divergence of lengthscale. Universal crossover is expected to be observed. Fora given ( gapped/ ) gapped/gapped systems with a small β F , the subleading term of S ( L ) seems to be algebraicas ( Lif shitz/ ) Lif shitz/Lif shitz systems for small L ,but really is exponential for large L . S ( L ) is a univer-sal function after being rescaled by the length scale 1 /α of each gapped/gapped/gapped system (Fig. 10). Theperfect collapse of different curves implies 11 / S S S S S FIG. 4. S ( L ) of ( CF T / ) CF T /CF T systems. Two small figures of the first row correspond to ( π , π ), ( π , π ) with uniform t . Their periods are both 6, but the numbers of branches differ. The first small figure of the second row shows that tuning t RH /t LH away from 1 ( t RH /t LH = 1 . π , π ). The second shows the case ( π , π , π ) with uniform t . Its inset is a detailed look at the lower branch, which shows the period is indeed 3. The large figure is log L vs. S fitting ofthe ( π , π ) figure. S S S S S S −0.05 0.00 −0.05 0.00 0.00.0 FIG. 5. ( gapped/ ) CF T /gapped and ( Lifshitz/ ) CF T /Lifshitz systems on structure I and structure II. The EE of the structureII (blue curves) is divided by 2 to be compared with structure I. The first row shows transition from ( gapped/ ) CF T /gapped to( Lifshitz/ ) CF T /Lifshitz . t is uniform. The systems from left to right are ((2 . , ) π , . . , ) π , . π, ) π , π ).When we are tuning β F , a transition of monotonicity of the middle branch happens. As the gap vanishes ( β F → t LH changes the monotonicity of one branch of ( π , 0) andmakes it well fitted by index . t LH of the three figures are 1, 1 . 22 and 1 . 5. The interface t does not influence the shape ofbranches. Insets without solid lines are − L − vs. S while those with solid lines are − L − / vs. S fitting. The solid lines arethe linear regression of results from structure I. The lines in each inset cross at 0. IV. ENTANGLEMENT OF HETEROGENEOUSQUANTUM ISING/XX CHAIN In this section, we calculate EE for one kind of hetero-geneous quantum Ising/XX chain which can be mappedto BdG fermion. We study this model on structure I.The Ising part and the XX part of Hamiltonian are re-spectively: H QIsing = X i ( − σ x,i σ x,i +1 + gσ z,i ) H XX = X i ( − t ( σ x,i σ x,i +1 + σ y,i σ y,i +1 ) + hσ z,i ) (7) The coupling between two parts is an XX coupling withcoefficient t interface . It is found that that the amplitudeof t interface and switching to Ising type coupling do notinfluence the results qualitatively.The results (Fig. 11) are quite similar to the last sec-tion. The oscillatory period is determined by the Fermivector of the XX part. The EE diverges (logarithmi-cally) if and only if both parts are conformally critical.Subleading terms are found to be either exponential andalgebraic decay. Algebraic decay with a fractional indexis again a signature of Lifshitz criticality. We only ob-serve integer index 1 in the last section, but for Ising/XXchain, we observe both 1 and 2. It is found that whether S S FIG. 6. CF T /gapped/CF T and CF T /Lifshitz/CF T sys-tems of structure II. The first figure represents a gapped sub-system ( π , . , π ). Inset is L vs. log( S − S ) fitting. Slopeis fitted to be 0 . . π , , π .). Inset is − L − / vs. S fitting. π4 πk F, E γ FIG. 7. Subleading indices fitting of CF T /Lifshitz/CF T and Lifshitz/Lifshitz/Lifshitz systems. All the sys-tems can be labeled as ( k F,E , , k F,E ). Two ends are the Lifshitz/Lifshitz limit. The left end is trivially direct prod-uct states. The EE curves of the right end is showed in Fig8. S FIG. 8. S ( L ) of ( Lifshitz/ ) Lifshitz/Lifshitz systems. TheEE of the structure I is multiplied by 2 to be compared withstructure II. The blue curve (slightly upper) is for structureII. The insets are − L − / vs. S fitting. S structure Istructure II S FIG. 9. S ( L ) of gapped/gapped/gapped systems. The gapof the figures are all set to be 0 . 05. The first figure de-scribes structure I and structure II with uniform t = 1. EE ofstructure II is divided by 2 to compare with structure I. Thesecond figure is for structure I, tuning t RH = 2. Insets are L vs. log( S − S ) fitting. The fitting slopes are 0 . . . 331 in figure2, comparing to our estimation 0 . . 446 and 0 . S ξ=4.949ξ=6.982ξ=15.563ξ=21.964 l o g ( ˜ S ) − l o g ( ξ ) ξ=4.949ξ=6.982ξ=15.563ξ=21.964 FIG. 10. Universal crossover of gapped/gapped sys-tems (structure I). For L < ξ , the system behaves as Lifshitz/Lifshitz system. The second figure shows the scal-ing behavior: ˜ S = ξ / f ( ˜ L ), where ˜ S = S ( ∞ ) − S , ˜ L = L/ξ . the gapped Ising phase is topological or not is irrelevantto the behavior of EE. V. SUMMARY AND DISCUSSION In summary, we have investigated the EE scaling law ofheterogeneous free fermion chains. The EE behaviors ofvarious situations are summarized in the Table I. We findthat Logarithmic law applies for ( CF T / ) CF T /CF T sys-tems while area law applies for other situations. Remark- S S S S S FIG. 11. S ( L ) of XX/Ising systems. From left to right, the figures of first row are CF T XX /CF T Ising , CF T XX /gapped Ising and Lifshitz XX /CF T Ising while the figures of second row are gapped XX /CF T Ising and gapped XX /gapped Ising . By the sameorder, the insets are are log L vs. S , − L − vs. S , − L − / vs. S , − L − vs. S and L vs. log( S − S ) fitting. ably, we also find both exponential and algebraic area lawsubleading behaviors. The algebraic indices are found tobe non-integer when one of the subsystems is Lifshitz crit-ical. Algebraic behavior is consequence of divergence oflength scale; to see this, universal crossover near Lifshitzcriticality is illustrated. We noted that a special case ofour structure I has been studied [28]. Their focus on theeffective central charges of logarithmic behavior togetherwith our results of subleading EE behaviors complemen-tarily present the features of heterogeneous free fermionsystems.The heterogeneous systems entanglement problem canbe considered as a further generalization of the entan-glement problem of homogeneous systems with localizedimpurity [29–34]. Here, we solve the heterogeneous prob-lem for free fermion systems. One open question is toget the fully analytical result by finding or conjecturingspectral function in Eq. 2 [26, 35]. We conjecture thatthose fractional exponents might be found to be exactfractions. It is also interesting to consider the problemfor interacting systems. In particular, some interactionmight change the fixed point. For example, the algebraicdecay is ”gapped out” to become exponential decay andthe junction becomes open boundary in RG sense. Thereis an even more interesting possibility, that interactiondrives the interface to another ”algebraic” fixed point.In this situation, it’s interesting to seek a field theoret-ical derivation formalism. Conformal field theory andconformal perturbation theory have been used to explainEE behaviors of homogeneous systems [16] and homo-geneous systems with localized impurity [30]. However,because Lifshitz criticality rather than conformal critical-ity plays the main role in the current problem, formalismbeyond CFT is possibly needed.The heterogeneous systems can also be considered asintermediate between homogeneous systems and disor-dered systems. Thus, the study of EE of heterogeneoussystems might be able to shine light on the problem of (a)XX chain, structure I LH RH EE behaviorCFT CFT logarithmicCFT Lifshitz γ = ± . γ = 1 ± . γ = ± . (b)XX chain, structure II S E EE behaviorCFT CFT logarithmicCFT Lifshitz γ = ± . γ = ± . γ = gapped CFT γ = 1 ± . (c)XX/quantum Ising chain XX QI EE behaviorCFT CFT logarithmicCFT gapped γ = 1 ± . γ = ± . γ = 2 ± . ± σ . σ are estimated from system ensembles: σ = pP i ( γ i − γ ) /N . Each ensemble includes more than10 points. Those points are away from crossover region and(or) accidental flat region. The indices of (a) and (c) are fit-ted from S ( L ) between L = 300 and L = 500. The indices of(b) are fitted from S ( L ) between L = 200 and L = 400. many-body localization. One might use EE and othermutual information to decide if two parts of heteroge-neous systems are well connected. Likewise, in reso-nant cluster picture of many-body localization [36], theremight be ways to use entanglement structure to defineif two subsystems belong to the same cluster. The en-tanglement and quantum fluctuation involving dynamicexponent = 1 is also interesting. It might be studied insystems such as multilayer graphene [37]. 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