The Ising quantum chain with a peculiar twisted boundary condition is
considered. This boundary condition, first introduced in the framework of the
spin-1/2 XXZ Heisenberg quantum chain, is related to the duality
transformation, which becomes a symmetry of the model at the critical point.
Thus, at the critical point, the Ising quantum chain with the duality-twisted
boundary is translationally invariant, similar as in the case of the usual
periodic or antiperiodic boundary conditions. The complete energy spectrum of
the Ising quantum chain is calculated analytically for finite systems, and the
conformal properties of the scaling limit are investigated. This provides an
explicit example of a conformal twisted boundary condition and a corresponding
generalised twisted partition function.Comment: LaTeX, 7 pages, using IOP style

Alcaraz F C

Coquereaux R

Grimm U

Lieb E H

Petkova V B

Schütz G

Uwe Grimm

English

arXiv

Crossref

Spectrum of a duality-twisted Ising quantum chain

The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalized twisted partition function

Grimm, Uwe

Open Research Online 

Open Research OnlineThe Open University’s repository of research publicationsand other research outputsSpectrum of a duality-twisted Ising quantum chainJournal ItemHow to cite:Grimm, Uwe (2002). Spectrum of a duality-twisted Ising quantum chain. Journal of Physics A: Mathematicaland General, 35(3) L25-L30.For guidance on citations see FAQs.c© [not recorded]Version: [not recorded]Link(s) to article on publisher’s website:http://dx.doi.org/doi:10.1088/0305-4470/35/3/101Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyrightowners. For more information on Open Research Online’s data policy on reuse of materials please consult the policiespage.oro.open.ac.ukSpectrum of a duality-twisted Ising quantum chainUwe GrimmApplied Mathematics Department, Faculty of Mathematics and Computing, TheOpen University, Walton Hall, Milton Keynes MK7 6AA, UKAbstract. The Ising quantum chain with a peculiar twisted boundary condition isconsidered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, whichbecomes a symmetry of the model at the critical point. Thus, at the critical point, theIsing quantum chain with the duality-twisted boundary is translationally invariant,similar as in the case of the usual periodic or antiperiodic boundary conditions. Thecomplete energy spectrum of the Ising quantum chain is calculated analytically forfinite systems, and the conformal properties of the scaling limit are investigated.This provides an explicit example of a conformal twisted boundary condition anda corresponding generalised twisted partition function.PACS numbers: 05.50.+q, 75.10.Jm, 71.10.Fd, 11.25.HfRecently, there has been a lot of interest in generalised twisted partition functionsof conformal field theory [1, 2, 3, 4] and the corresponding conformal twisted boundaryconditions in solvable lattice models of statistical mechanics [5]. Examples of suchboundary conditions leading to the appearance of “exotic” spinor operators in thecorresponding conformal field theory had previously been considered for quantum spinchains, in particular for the Ising and 3-state Potts quantum chains [6, 7]. In these cases,the boundary conditions were derived from a mapping [8, 9, 10, 11, 12, 6] between thesemodels with N sites and the spin-1/2 XXZ Heisenberg model with twisted boundaryconditions, which for an even number of sites 2N yields the “usual” twisted boundaryconditions [8, 10], such as antiperiodic boundary conditions for the Ising quantum chain,and results in rather exotic boundary terms when considered for an odd number ofsites 2N − 1 [6]. The corresponding boundary conditions still allow for translationalsymmetry, but only at the critical point. The corresponding symmetry for the Ising andPotts models was identified as duality, which maps the Ising Hamiltonian of the orderedphase onto that of the disordered phase and vice versa. It becomes a proper symmetryat the critical point, and the corresponding boundary terms and generalised translationand parity operators were analysed in [6, 7] by means of an algebraic approach [13].The conformal partition function corresponding to this duality twisted boundarycondition was derived in [6] from that of the twisted XXZ Heisenberg spin chain [14].However, the spectrum of the duality-twisted Ising quantum chain itself has never beencalculated explicitly. In fact, using the standard free-fermion approach [15], it is possibleto obtain the complete spectrum even for finite chains. This is the purpose of thisSpectrum of a duality-twisted Ising quantum chain 2Letter. The calculation proceeds along the same path as for Ising quantum chains with“generalised defects” [16], as the boundary terms are actually rather similar. However,the defects considered in [16], albeit featuring rather general couplings of the last andthe first spin in the chain, did not include the duality-twisted boundary considered here.For comparison, we treat the usual periodic and antiperiodic boundary conditions in thesame way.Consider an Ising quantum spin chain consisting of N sites, so the Hilbert space isHN = (C2)⊗N ∼= C2N. We introduce local spin operators, acting at site j, byσwj =(j−1⊗k=1I)⊗ σw ⊗(N⊗k=j+1I), (1)where w stands for x, y and z, and I denotes the unit operator. In the canonical basis,the spin operators σw are represented by the Pauli matricesσx =(0 11 0), σy =(0 −ii 0), σz =(1 00 −1), (2)and I is the 2× 2 unit matrix.The critical Ising quantum chain with periodic (+) and antiperiodic (−) boundaryconditions is defined by the HamiltoniansH(±) = −12(N−1∑j=1σxj σxj+1 +N∑j=1σzj ± σxNσx1). (3)The Ising quantum chain is related to the transfer matrix of the classical zero-fieldsquare-lattice Ising model at the critical temperature through an anisotropic limit. Thenormalisation factor is chosen such that the fermion velocity in the scaling limit is equalto unity. The duality-twisted Ising quantum chain is given by [6, 7]H(d±) = −12(N−1∑j=1σxj σxj+1 +N−1∑j=1σzj ± σyNσx1). (4)Note that, besides the coupling between σyN and σx1 , the Hamiltonians (4) and (3) alsodiffer in the diagonal terms, as there is no operator σzN in (4). This is also the reasonwhy such boundary terms were not considered as a “generalised defect” in [16]. Becausethe generalised parity transformation for the duality-twisted Hamiltonian maps the twosigns at the boundary onto each other [7], and since the two Hamiltonians are related bycomplex conjugation, they have the same spectrum, and the momenta of the eigenstatesdiffer by a sign.The Hamiltonians H (±) and H(d±) commute with the operatorQ = (σz)⊗N =N∏j=1σzj , (5)so they have a global C2 symmetry. We denote the corresponding projectors byP± =12(IN ±Q) , (6)Spectrum of a duality-twisted Ising quantum chain 3where IN = I⊗N is the identity operator on HN . In addition, they commute withproperly defined translation operators, which take the boundary condition into account,see [13, 6, 7] for details.In order to obtain local boundary conditions in the fermionic language, we play theusual trick and pass to “mixed-sector” Hamiltonians [17] defined asH˜(±) = H(±)P+ + H(∓)P−, (7)H˜(d±) = H(d±)P+ + H(d∓)P−. (8)Performing a Jordan-Wigner transformation [15]ck =(k−1∏j=1σzj)σ−k , c†k =(k−1∏j=1σzj)σ+k , (9)we rewrite H˜(±) and H˜(d±) as bilinear expressions in the fermionic creation andannihilation operators c†k and ck,H˜ =N2IN +N∑j,k=1[Dj,kc†jck +12(Ej,kc†jc†k + E∗k,jcjck)], (10)where the N ×N matrices D and E depend on the boundary conditions2D±j,k = − 2δj,k + δj+1,k(1− δj,N) + δj,k+1(1− δk,N)∓ δj,1δk,N ∓ δj,Nδk,1, (11)2E±j,k = δj+1,k(1− δj,N)− δj,k+1(1− δk,N)± δj,1δk,N ∓ δj,Nδk,1, (12)2Dd±j,k = − 2δj,k(1− δj,N) + δj+1,k(1− δj,N) + δj,k+1(1− δk,N)∓ i δj,1δk,N ± i δj,Nδk,1, (13)2Ed±j,k = δj+1,k(1− δj,N)− δj,k+1(1− δk,N)∓ i δj,1δk,N ± i δj,Nδk,1, (14)and ∗ denotes complex conjugation. The quadratic form (10) can be diagonalised toH˜ =N−1∑k=0Λkη†kηk + e IN (15)by means of a Bogoliubov-Valatin transformation. Here, η†k and ηk are new fermioniccreation and annihilation operators, e is a constant, and the squared fermion energiesΛ2k are given by the eigenvalues of the 2N × 2N matrixA =(D E−E∗ −D∗)2=(D2 − EE∗ (D −D∗)E−(D −D∗)E∗ (D∗)2 − EE∗), (16)see [16] for details. The matrix A has the property CAC = A∗, where C is the 2N ×2Nmatrix with elementsCj,k = δj+N,k + δj,k+N . (17)An eigenvector Φ of A corresponds to a proper Bogoliubov-Valatin transformation if itsatisfies Cφ = φ∗. As was shown in [16], the spectrum of A is doubly degenerate, andone can always choose the eigenvectors such that they obey the conjugation condition.Spectrum of a duality-twisted Ising quantum chain 4In this approach, we effectively doubled the system, which is the reason why eachfermion energy occurs twice. We only need to consider “half” the eigenvalues of A, thusremoving this trivial degeneracy. Furthermore, because we can change the sign of thefermion energies Λk in (15) by a transformation that exchanges creation and annihilationoperators, resulting only in a different value of the constant in (15), we may choose allenergies to be positive, and ordered, so 0 ≤ Λ0 ≤ Λ1 ≤ . . . ≤ Λk−1. In this case, theconstant in (15) is just the corresponding ground-state energy, e = E0.For the periodic and antiperiodic mixed-sector Hamiltonians H˜± of (7), it turnsout that A is in fact block-diagonal, and both blocks are identical, because all matrixelements are real. So in this case we can limit ourselves to one block immediately, anddiagonalise the resulting N ×N matrix. The solution of the eigenvalue equations(D2 − EE∗)(±)Ψ(±)k = (Λ(±)k )2Ψ(±)k , (18)with k = 0, 1, . . . , N − 1, are simply given byΛ(±)k = 2 sin(p(±)k2), (Ψ(±)k )j = exp(i p(±)k j), (19)where the eigenvalues and the unnormalised eigenvectors are parametrised by themomentap(+)k =(2k + 1)piN, p(−)k =2kpiN. (20)For the duality-twisted mixed-sector Hamiltonians H˜(d±) (8), the situation is morecomplicated, because the matrix A(d) := A(d+) = A(d−) does not have block form.Nevertheless, the bulk part of the equations, which does not involve boundary terms,is still the same and can be solved as above. The modified equations correspond tothe rows 1, N − 1, N , N + 1, 2N − 1 and 2N of the matrix A(d). These yieldadditional conditions, so one should look for a solution involving terms exp(±i p(d)k j)with site-dependent coefficients at the boundary. Whereas it was not possible to solvethe equations in closed form for the generalised defects in [16], it is possible in this case.One arrives at the solutionΛ(d)k = 2| sin(p(d)k2)|, p(d)k = ±4kpi2N − 1, (21)and the corresponding unnormalised eigenvectors of A(d) are(Φ(d)k )j = (Φ(d)k )∗j+N = cos(p(d)k2) exp(i p(d)k j)+ i [sin(p(d)k2)− 1] exp(−i p(d)k j), 1 ≤ j ≤ N−1,(Φ(d)k )N = (Φ(d)k )∗2N = 1− sin(p(d)k2). (22)In general, positive and negative momenta in (21) give two linearly independenteigenvectors (22) for the same eigenvalue. However, there is only one vector for k = 0,which corresponds to the zero mode. A second, linearly independent eigenvector is(Φ(d)0 )j = i (δj,N − δj,2N), (23)as can easily be checked.Spectrum of a duality-twisted Ising quantum chain 5The fermion excitation spectra (19) and (21) determine the complete energyspectrum of the mixed-sector Hamiltonians H˜(±) (7) and H˜(d±) (8), respectively, apartfrom the constant in (15), which corresponds to the ground-state energy of the respectiveHamiltonian. This can be calculated by considering the trace of the Hamiltonian. Astr(H˜(±)) = tr(H˜(d±)) = 0, andtr(N−1∑k=0Λkη†kηk + E0 IN)= N(12N−1∑k=0Λk + E0), (24)the ground-state energies are given by−E(+)0 =N−1∑k=0sin((k+ 12)piN) =1sin( pi2N)=2Npi+pi12N+ O(N−3), (25)−E(−)0 =N−1∑k=0sin(kpiN) = cot( pi2N) =2Npi−pi6N+ O(N−3), (26)−E(d)0 =N−1∑k=0sin( kpiN−12) =1 + cos( pi2N−1)2 sin( pi2N−1)=2(N − 12)pi−pi24(N− 12)+ O[(N− 12)−3]=2Npi−1pi−pi24N+ O[N−2] (27)Comparing those with the finite-size corrections of the ground-state energy expectedfrom conformal field theory−E0 = A0N +pic6N+ o(N−1), (28)we find that the ground-state energy per site in the thermodynamic limit is given byA0 = limN→∞(−E+0N) = limN→∞(−E−0N) = limN→∞(−Ed0N) =2pi(29)and the central charge c = 12for the periodic chain. The finite-size corrections forthe other two cases result in “effective central charges” c˜− = −1 and c˜d = −1/4,which correspond to excitations with conformal dimension x− = (c− c˜−)/12 = 1/8, themagnetisation operator, and xd = (c− c˜d)/12 = 1/16, the “exotic” spinor operator withconformal spin 1/16 [6]. Note that, for the latter result, the result (27) shows that it ispreferable to define the scaling limit by using N − 12as the effective system size in (28);for the naive scaling with N , a constant surface term A1 = 1/pi appears in (28), similarto what one finds for free or fixed boundary conditions, or in the case of defect lines[16, 17]. In that sense, the duality-twisted chain acts as a toroidal chain of N − 12sitesrather than N sites, in accordance with the relation to the odd length spin-1/2 XXZHeisenberg spin chain [6].Using the appropriate scaling, the linearised low-energy spectrum becomeslimN→∞[N2pi(H˜+ − E+0 IN)] =∞∑r=0[(r + 12)a†rar + (r +12)b†rbr] (30)Spectrum of a duality-twisted Ising quantum chain 6limN→∞[N2pi(H˜− − E+0 IN)] =∞∑r=0[ra†rar + (r + 1)b†rbr] +18(31)for the periodic and antiperiodic mixed-sector Hamiltonians, and simplylimN→∞[N− 122pi(H˜d± −N− 12NE+0 IN)] =∞∑k+0[ra†rar + (r +12)b†rbr] +116(32)for the duality-twisted case, where the fermionic operators ar and br are obtainedfrom the ηk by renumbering. Taking into account the momenta, and bosonising thelow-energy theory in terms of two generators L0 and L¯0 of two commuting Virasoroalgebras with central charge c = 12by means of a Sugawara construction, the firsttwo yield the usual conformal torus partition functions Z+ = (χ0 + χ1/2)(χ¯0 + χ¯1/2)and Z− = 2χ1/16χ¯1/16 for the Ising model, where χ∆ denote the character function ofthe irreducible representations of the c = 12Virasoro algebra with highest weight ∆.Depending on the sign of the boundary term, the last corresponds to the combinations(χ0 + χ1/2)χ¯1/16 or χ1/16(χ¯0 + χ¯1/2) of characters, as derived in [6] from the relation tothe XXZ Heisenberg chain with toroidal boundary conditions [8, 10, 6].Hence, for this particular instance of conformally twisted boundary conditions,the complete spectrum is known even for finite systems, and the scaling limit can beperformed explicitly. It would be interesting to know whether the more general boundaryconditions introduced recently [1, 2, 3, 4, 5] can be interpreted in a similar way, and whatthe corresponding symmetries are that allow the definition of twisted toroidal boundaryconditions.References[1] Petkova V B and Zuber J-B 2001 The many faces of Ocneanu cells Nucl. Phys. B 603 449–96[2] Petkova V B and Zuber J-B 2001 Generalised twisted partition functions Phys. Lett. B 504 157–64[3] Petkova V B and Zuber J-B 2001 Conformal field theories, graphs and quantum algebras Preprinthep-th/0108236[4] Coquereaux R and Schieber G 2001 Twisted partition functions for ADE boundary conformal fieldtheories and Ocneanu algebras of quantum symmetries Preprint hep-th/0107001[5] Chui C H O, Mercat C, Orrick W P and Pearce P A 2001 Integrable lattice realizations of conformaltwisted boundary conditions Phys. Lett. B 517 429–35[6] Grimm U and Schu¨tz G 1993 The spin-1/2 XXZ Heisenberg chain, the quantum algebra Uq[sl(2)],and duality transformations for minimal models J. Stat. Phys. 71 921–64[7] Schu¨tz G 1993 ‘Duality twisted’ boundary conditions in n-state Potts models J. Phys. A: Math.Gen. 26 4555–63[8] Alcaraz F C, Grimm U and Rittenberg V 1989 The XXZ Heisenberg chain, conformal invarianceand the operator content of c < 1 systems Nucl. Phys. B 316 735–68[9] Alcaraz F C, Baake M, Grimm U and Rittenberg V 1989 The modified XXZ Heisenberg chain,conformal invariance and the surface exponents of c < 1 systems, J. Phys. A: Math. Gen. 22L5–11[10] Pasquier V and Saleur H 1990 Common structures between finite systems and conformal field-theories through quantum groups Nucl. Phys. B 330 523–56[11] Grimm U and Rittenberg V 1990 The modified XXZ Heisenberg chain: conformal invariance,Spectrum of a duality-twisted Ising quantum chain 7surface exponents of c < 1 systems, and hidden symmetries of finite chains Int. J. Mod. Phys.B 4 969–78[12] Grimm U and Rittenberg V 1991 Null states of the irreducible representations of the Virasoroalgebra and hidden symmetries of the finite XXZ Heisenberg chain — A story about movingand frozen energy levels Nucl. Phys. B 354 418–40[13] Levy D 1991 Algebraic structure of translation-invariant spin-1/2 XXZ and q-Potts quantum chainsPhys. Rev. Lett. 67 1971–4[14] Alcaraz F C, Baake M, Grimm U and Rittenberg V 1988 Operator content of the XXZ chain J.Phys. A: Math. Gen. 21 L117–20[15] Lieb E H, Schultz T D and Mattis D C 1961 Two soluble models of an antiferromagnetic chainAnn. Phys. 16 407–66[16] Grimm U 1990 The quantum Ising chain with a generalized defect Nucl. Phys. B 340 633–58[17] Baake M, Chaselon P and Schlottmann M 1989 The Ising quantum chain with defects. II. Theso(2n) Kac-Moody spectra Nucl. Phys. B 314 625–45

http://oro.open.ac.uk/1146/1/grimm1146.pdf

Spectrum of a duality-twisted Ising quantum chain

Abstract

Similar works

Full text

Available Versions

Crossref

Open Research Online