PHYSICAL REVIEW B VOLUME 60, NUMBER 13 1 OCTOBER 1999-I Bulk and surface spin-flop transitions in an antiferromagnetic XYZ chain J. Karadamoglou* and N. Papanicolaou Department of Physics, University of Crete, and Research Center of Crete, Heraklion, Greece Received 29 April 1999 Spin-flop transitions are studied within a spin-12 antiferromagnetic XYZ chain immersed in a uniform bias magnetic field. For a special choice of anisotropy the bulk spin-flop transition occurs at a critical field Hb characterized by the onset of ``accidental'' degeneracy in the energy spectrum that may indicate a hidden symmetry. On an open chain the bulk transition is preceded by a surface spin-flop transition induced by a surface magnon that turns soft at a new critical field Hs Hb , while an antiferromagnetic domain wall is realized in the ground state as the bias field approaches Hb . These T 0 phase transitions could be observed through electron-spin resonance on magnetic chains doped with nonmagnetic ions. S0163-1829 99 01137-6 I. INTRODUCTION J1 J2 J3 1.2 Surface effects that could be observed in an antiferromag- and thus describes an antiferromagnetic chain in a field net were theoretically predicted some time ago.1­3 However, pointing along the easy axis. To set the stage for the study of the subject was revived only recently with the experimental the quantum model we have carried out a classical observation of a surface spin-flop transition in an Fe/Cr calculation21 which revealed the existence of a surface spin- multilayer.4­6 The precise nature of the transition proved to flop transition analogous to that encountered in multilayers. be more intricate than originally envisaged but has been A peculiarity of the classical ground state is its independence clarified in a number of recent theoretical investigations.7­17 on the specific value of J1 , in the range J1 J2 , a property The advantage of a multilayer is that it can be modeled by a that is not sustained in the quantum model. But the recent classical spin chain characterized by an isotropic exchange work of Kyriakidis and Loss22 on an unrelated subject interaction and a single-ion quadratic anisotropy. Bloch oscillations suggested to us that the most classical In view of this development we return to the problem of antiferromagnet is obtained by the special choice of the ex- crystalline antiferromagnets which are more appropriately change constants described by quantum spin systems. Specifically, we intend J to study spin-flop transitions for a spin-1 1 J2 1, J3 1, 1.3 2 XYZ model whose Hamiltonian which is consistent with inequalities 1.2 . If u and d denote spin-up and spin-down states, the two fully polarized NeŽel 1 states W J x x y y z z z 1Sl Sl 1 J2Sl Sl 1 J3Sl Sl 1 H Sl N l 1 l 1 A d,u,d,u,... , NB u,d,u,d,... 1.4 1.1 are then the two degenerate exact ground states when the applied field is sufficiently weak. is defined on an open chain with sites. Various quasi-one- Some special features of model 1.3 become apparent by dimensional magnets that are thought to be approximated by performing the familiar canonical transformation special cases of Eq. 1.1 have already been identified. Fur- x x y y z z thermore a statistically significant sample of free boundaries Sl Tl , Sl 1 lTl , Sl 1 lTl , 1.5 may be experimentally accessible18 through doping with where the ``pseudospin'' variables T nonmagnetic ions which creates within the crystal open mag- l again satisfy the stan- dard spin commutation relations and map the Hamiltonian to netic chains of varying length. Therefore, a theoretical study of surface effects is now more meaningful than ever. 1 1 In fact, the XYZ model has been the subject of an im- W Tx x y y z z l Tl 1 Tl Tl 1 TlTl 1 4 mense amount of theoretical work based on the Bethe l 1 ansatz19 or any of its variants. In particular, recent activity has concentrated on surface effects in the presense of bound- H 1 lTzl , 1.6 ary fields or other integrable impurities.20 However, the l 1 model is not known to be completely integrable in the pre- which describes an easy-axis ``ferromagnet'' in a staggered sense of the uniform bulk field H in Eq. 1.1 , except in magnetic field. In Eq. 1.6 we have included a trivial addi- special limits such as the U 1 -symmetric XXZ model (J1 tive constant to provide a convenient normalization for the J2) in a field directed along the symmetry axis. Hence, the energy eigenvalues. present work will proceed by a more direct analytical ap- It is clear from Eq. 1.6 that the model 1.3 is also en- proach supported by numerical diagonalization. dowed with a U 1 symmetry because the total azimuthal Hamiltonian 1.1 is here restricted by the inequalities pseudospin 0163-1829/99/60 13 /9477 12 /$15.00 PRB 60 9477 ©1999 The American Physical Society 9478 J. KARADAMOGLOU AND N. PAPANICOLAOU PRB 60 where we have simplified the notation by asserting that l is Tz z l 1 lSl 1.7 the state where l 1/2 but all other 's are equal to 1/2. l 1 l 1 An explicit form of the eigenvalue equations is then given by obviously commutes with the Hamiltonian and thus the Hil- 1 bert space breaks up into 1 sectors characterized by the /2 H E C1 good quantum number /2 with 0,1, . . . , . Al- 2 C2 , though this observation will greatly simplify calculations, the quantum number is not related to a simple physical observ- 1 1 lH E C able. Instead we shall be interested in the total azimuthal l 2 Cl 1 Cl 1 , magnetization l 2,3, . . . , 1, 2.2 M Sz z l 1 lTl , 1.8 1 l 1 l 1 /2 H E C 2 C 1, which does not commute with the Hamiltonian and its ex- and are valid on an open chain with sites. Here and pected values cannot be predicted by simple quantization. throughout the main text we assume that the chain is even One of our objectives in the following is to determine the ( 2N). We further introduce the sublattice variables ground-state expectation value M M( ,H) as a function of anisotropy and applied field H. An C2n 1 , Bn C2n , n 1,2, . . . ,N 2.3 Significant analytical progress can be made for the spe- cific choice 1.3 which will serve as a prototype for further in terms of which the linear system 2.2 reads consideration of the full range of models defined by inequali- ties 1.2 . In Sec. II we provide a complete calculation of 1 /2 H E A one-magnon excitations with or without open boundaries. 1 2 B1 , These results already suggest the occurrence of a bulk spin- flop BSF transition at a critical field H 1 b , which is preceded H E A by a surface spin-flop SSF transition at a critical field H n s 2 Bn 1 Bn , n 2,3, . . . ,N, Hb in the presence of open boundaries. The two types of transition are analyzed in Secs. III and IV. Antiferromagnetic 1 domain walls arise naturally in the description of the SSF H E Bn 2 An An 1 , n 1,2, . . . ,N 1, transition and are thus also discussed in Sec. IV. In Sec. V 2.4 we summarize some of our main conclusions, in addition to presenting some preliminary results for other quantum spin 1 models in the range 1.2 . For clarity of presentation we /2 H E BN 2 AN . found it useful to restrict attention to an even chain ( 2N) in the main text, while the necessary modifications for It is instructive to consider in parallel a cyclic or periodic the case on an odd chain ( 2N 1) are discussed in the chain with the same number of sites, in order to establish a Appendix. simple reference case which will enable us to better appreci- ate surface effects that may arise on an open chain. The first II. BULK AND SURFACE MAGNONS and fourth equations in Eq. 2.4 are absent on a cyclic chain, and the remaining two equations are valid for all n The eigenstates of the Hamiltonian 1.6 are linear super- 1,2, . . . ,N. It is then a straightforward matter to eliminate positions of states of the form 1 , 2 ,..., where the l's Bn to obtain the equivalent system take the values 1/2 or 1/2 in any combination that pre- serves their sum . The NeŽel states 1.4 are mapped to two 1 An An 1 completely polarized ``ferromagnetic'' states, with Bn 2 H E , /2, which are exact eigenstates, with energy E 0, for any value of the applied field. It is also clear that these are 1 the two degenerate ground states at vanishing field. The first E 2 H2 An 4 2An An 1 An 1 , 2.5 question is then to determine the field region over which the polarized states persist as the ground states of the system. whose solution is An eikn, where k 2 /N with One is thus lead to study excitations, the simplest possi- 0,1, . . . ,N 1 is a sublattice crystal momentum, provided bilities being one-magnon modes with /2 1 or /2 that the energy is given by 1. It is sufficient to consider only the former case, infor- mation about the latter being inferred by extending the field E H2 cos2 k/2 . 2.6 region to both positive and negative values of H. The one- magnon eigenvalue problem reads Viewed as functions of the applied field the energy eigenval- ues are contained within the two shaded regions of Fig. 1 and their distribution becomes increasingly dense in the bulk W E , C limit N . The shaded regions are bounded from above l l , 2.1 l 1 and below by the two curves E H2 1 and display a PRB 60 BULK AND SURFACE SPIN-FLOP TRANSITIONS IN . . . 9479 and should be viewed as a system of two algebraic equations for the unknowns and E. Detailed examination of the roots that satisfy the condition 1 yields two distinct surface modes which we discuss in turn. The most interesting surface mode is given by the root 1 and E E1 with 1 1 2 2 2 4 H 1 2 4 2 2 4 H 1 , 1 E1 H2 4 2 1 1/ 1 , 2.11 and its energy labeled as curve 1 in Fig. 1 lies below the one-magnon continuum. This curve emanates from the con- tinuum at the characteristic field H0 with FIG. 1. Field dependence of the one-magnon spectrum for the specific anisotropy 1.5. 2 1 H0 2 2.12 middle gap extended between the curves E H. The lowest gap closes (E 0) at H Hb where and persists in the field region H H0 . The parameter 1 is equal to unity at H0 but lies in the interval 0 1 1 for Hb 2 1 2.7 H H0 . This surface state exists as a distinct gap mode below the continuum even at vanishing field (H 0) where is a critical field of special importance in the following. The preceding derivation already indicates that the simple NeŽel 1 1/ 2 and E1 ( 2 1)/2 . A second root satisfying the condition 1 is given by states 1.4 can no longer be the ground states for field values outside the interval H 2 and E E2 with b ,Hb . Further analysis of the cy- clic chain given in Sec. III will establish that Hb provides the 1 critical boundary of the BSF transition. 2 2 2 2 4 H 1 2 4 2 2 4 H 1 , We now return to the main theme and examine the possi- bility of one-magnon surface modes on an open chain. Solu- tion of the linear system 2.4 is complicated by the appear- 1 ance of the two distinct equations at the outer layers. H2 4 2 2 1/ 2 H0 H 0, E However, the essential new ingredients can be obtained ana- 2 1 lytically on a semi-infinite chain. The fourth equation in Eq. H2 4 2 2 1/ 2 H H0. 2.4 may then be ignored and the third may again be used to 2.13 eliminate Bn as in Eq. 2.5 . The first two equations are written as The two branches in E2 join smoothly at H0 and their union is labeled as curve 2 in Fig. 1. Therefore, the second 1 root describes a surface mode with energy in the middle gap /2 H E H E A1 4 A1 A2 , of the magnon continuum. The parameter 2 lies in the in- terval 1 2.8 2 0 and thus the middle-gap mode decays 1 away from the boundary in an oscillatory manner. E 2 H2 A To complete the description of Fig. 1 we must now rec- n 4 2An An 1 An 1 , oncile the preceding analytical results on a semi-infinite where n 2,3, . . . , . A surface mode is described by a spe- chain with those obtained by numerical diagonalization of cial solution of the form the linear system 2.2 or 2.4 on a finite open chain. For any given 2N the majority of eigenvalues fall within the 1 1 shaded regions of Fig. 1, but a finite number of eigenvalues A occur outside the continuum for each field H. Also note that n n, Bn 2 H E n 2.9 a dublication of gap modes should be expected on an open supplemented by the requirement 1 which guarantees chain because surface states can now be formed near either that the state decay exponentially away from the boundary. one of the two free ends. Indeed, in addition to confirming Equations 2.8 reduce to the gap modes 1 and 2 predicted on a semi-infinite chain, the numerical diagonalization also yields the two mirror modes 1 1 and 2 shown in Fig. 1. /2 H E H E The energies of the gap modes calculated analytically on 4 1 , a semi-infinite chain with 1.5 agree with the numerical 2.10 results to several significant figures for 10 while the 1 E 2 H2 agreement improves rapidly with increasing or . Hence 4 2 1/ , the numerical simulations described in Secs. III and IV for 9480 J. KARADAMOGLOU AND N. PAPANICOLAOU PRB 60 spin chain in the presence of suitable boundary fields proved to be very instructive for the current work on a cyclic chain in a bulk bias field H. We may also invoke the cluster argument of Bader and Schilling24 on a cyclic chain noting that the Hamiltonian may then be written as a sum of cell Hamiltonians, namely W Wl , l 1 1 W x x y y z z l Tl Tl 1 Tl Tl 1 TlTl 1 4 1 z z 2 H 1 l Tl Tl 1 , 3.1 and thus the ground-state energy E0 satisfies the inequality FIG. 2. The T 0 phase diagram for the quantum model 1.3 . The bulk and surface critical boundaries Hb and Hs are given ana- 1 lytically by Eqs. 2.7 and 2.14 , and the domain-wall boundary 2 E E E0 , 3.2 Hw was obtained numerically as described in Sec. IV. where E chains of modest size are expected to provide a reliable pic- are the ground-state energies of the two-spin Hamiltonians ture for most values of of practical interest. The most important conclusion derived from the one- 1 1 magnon calculation becomes apparent by simple inspection W x x y y z z z z T1T2 T1T2 T1T2 T2 . of Fig. 1. The surface modes in the lower gap are degenerate 4 2 H T1 with the NeŽel states at the critical fields H 3.3 s . The field Hs is computed from the condition E1 0 where E1 is given by The eigenvalues of both W and W are given by Eq. 2.11 . A straightforward calculation shows that 1 1 1 H 2 H2 1 , 0, 0, 2 H2 1 , 3.4 s 4 2 1 9 2 1 2 1 . 2.14 where the first eigenvalue is always positive and the fourth Outside the interval Hs ,Hs the NeŽel states cease to be one may be positive or negative depending on the field the lowest-energy states. Therefore, the BSF transition an- strength H. Therefore, inequality 3.2 is written as ticipated to occur at the critical field Hb of Eq. 2.7 is pre- ceded on an open chain by a SSF transition at Hs Hb . The 1 critical fields H min 0, s and Hb coincide with those found in the 2 H2 1 E0 0, 3.5 classical calculation21 except for an overall factor of 2 which originates in the normalization of the classical spin to unity. and has been supplemented by E0 0 which follows from The curves H Hs( ) and H Hb( ) provide the main the fact that the polarized states are eigenstates of the com- critical boundaries in the T 0 phase diagram provisionally plete Hamiltonian with vanishing energy for any value of the shown in Fig. 2 and further analyzed in Secs. III and IV. applied field. An immediate consequence of Eq. 3.5 is that The surface magnon that drives the SSF transition in the the true ground-state energy vanishes when Hb H Hb , present quantum model is the analog of the Mills-Saslow where Hb is precisely the critical field 2.7 , and thus coin- mode2 which was derived semiclassically in a model with cides with the energy of the polarized states. single-ion anisotropy and played an important role in the The preceding result strengthens the conclusion that the theory of classical multilayers.17 Interestingly, the currently BSF transition occurs at the critical field Hb but a more calculated ratio Hb /Hs approaches the value & in the iso- detailed argument is required to determine the precise nature tropic limit 1 , in agreement with the corresponding of the transition. The work of Sec. II already established that ratio in the limit of vanishing single-ion anisotropy.2 In the the lowest-energy state in the one-magnon sector becomes opposite limit, , Hb /Hs approaches 2 which may also degenerate with the polarized states at Hb . We shall further be deduced from an elementary analysis of the Ising chain. show that the lowest-energy states of all multimagnon sec- tors become degenerate at the same critical field, in analogy III. BULK SPIN-FLOP TRANSITION with a similar result in the model of Ref. 23. When H Hb it is convenient to introduce the parametrization In this section we focus on a cyclic chain with an even number of sites and examine in greater detail the BSF tran- 1 1 1 1 sition at the critical field H q , H q , 3.6 b of Eq. 2.7 suggested by the 2 q 2 q one-magnon calculation. In a curious turn of events, the re- cent work of Alcaraz, Salinas, and Wrezinski23 on an open where 1 and q 2 1 1. PRB 60 BULK AND SURFACE SPIN-FLOP TRANSITIONS IN . . . 9481 To motivate the demonstration we return to the one- A numerical calculation of all eigenstates and eigenval- magnon calculation and specifically consider the lowest- ues, for various 12, confirms that the ground states of all energy state at Hb obtained by setting k 0 and E 0 in Eqs. sectors become degenerate at the critical field. Excited states 2.5 and 2.6 to yield An 1 and Bn 1/q for all n also exhibit some accidental degeneracy that cannot be ac- 1,2, . . . ,N. In the notation of Eq. 2.1 this special one- counted for by the U 1 symmetry. One is thus tempted to magnon state reads conclude that the model acquires a larger symmetry at the critical point; perhaps analogous to the quantum-group Cl q 1/2 1 l, 3.7 Uq SU 2 symmetry of the model with boundary fields studied in Refs. 23, 25. We shall not pause to examine the where we have also included an overall normalization factor possibility of a hidden symmetry in the present model be- q. cause the explicit result 3.12 proves to be sufficient to illu- We next consider the two-magnon eigenvalue problem minate the nature of the BSF transition. Thus the emerging qualitative picture is substantiated W E , C l with an explicit calculation of the ground-state expectation 1 ,l2 l1 ,l2 , 3.8 l1 l2 value of the total magnetization M of Eq. 1.8 which van- ishes in the region H H where l b but acquires finite values for H 1 ,l2 is a state with l 1/2 and all other 's 1 l2 Hb . A sudden jump occurs at the critical field which can equal to 1/2. On a cyclic chain the generic eigenvalue equa- be calculated analytically on an even cyclic chain of any tion is size. The calculation is based on the important observation that the 0 sector prevails above H 2 1 l b , in the sense that it 1 1 l2 H E C l1 ,l2 contains the unique absolute ground state for H Hb . The 1 last statement is corroborated by numerical diagonalization on short chains of varying size. Therefore the magnetization 2 C l1 1,l2 C l1 1,l2 C l1 ,l2 1 jump at the critical field is given by C l1 ,l2 1 , 3.9 z 0 Tl 0 and should be completed with the meeting condition19 M 1 l l 1 0 0 , 3.14 1 where C l,l 1 0 is the state 3.12 applied for 0. The special 2 C l,l C l 1,l 1 , 3.10 structure of this state permits us to write where one formally extends the definition of C(l1 ,l2) to co- inciding arguments (l M q 1 l2) which are absent in Eq. 3.8 . q ln 0 0 3.15 When and H are given by Eq. 3.6 the wave function and the problem reduces to the calculation of the norm C l 1 ,l2 q 1/2 1 l1 1 l2 3.11 0 0 . A reasonable amount of combinatorics leads to satisfies Eq. 3.9 , with E 0, as well as the meeting condi- N N! 2 tion 3.10 . We have thus obtained a special two-magnon 0 0 qN 2 3.16 N ! ! eigenstate which is degenerate with both the one-magnon 0 state 3.7 and the completely polarized states at the critical and field Hb . These elementary results possess a simple generalization 1 N 2 to an arbitrary sector. A set of exact eigenstates with vanish- M N 2 N! qN 2 . 0 0 0 N ! ! ing energy is given by 3.17 It is also possible to provide an integral representation of the 1 l q1/2 l 1 l 1 , 2 ,..., , 3.12 sums to obtain where the sum extends over all configurations M 1 IN 1 ( 2 1 , 1 , 2 ,..., ) which are consistent with a definite azi- 2N 2 IN muthal pseudospin 3.18 1 I N cos Nd , 0 l N,N 1, . . . , N, 3.13 l 1 where is the average magnetization per site, a quantity that where 2N. As a check of consistency one may apply Eq. is more appropriate for the discussion of the thermodynamic 3.12 for N 1 and N 2 to recover the one- and two- limit. magnon wave functions 3.7 and 3.11 . For other values of Analytical calculation of the magnetization for H Hb is our basic result 3.12 can be established by a straightfor- impossible at present because the 0 ground state becomes ward generalization of the two-magnon calculation given increasingly complex away from the critical point. We have above. thus resorted to numerical diagonalization on short chains. 9482 J. KARADAMOGLOU AND N. PAPANICOLAOU PRB 60 netization at the critical field computed on chains with 10 does not differ from its limit by more than 13%, for all 1, and the agreement improves at large . It is also likely that the thermodynamic limit for H Hb could be reached by extrapolation, but this is clearly an issue of no special urgency at this point. The analytical result 3.19 reveals yet another surprise, for it coincides with the magnetization obtained within the classical calculation21 at the onset of the BSF phase which is described by a canted spin configuration. This curious fact could be investigated further by generalizing the special states 3.12 to arbitrary spin s, also in analogy with a similar calculation in the model of Ref. 23. One should then be able to explicitly study the classical large-s limit on any finite chain and eventually explain its coincidence with the quan- tum prediction 3.19 in the thermodynamic limit. To summarize, the T 0 phase diagram is rather simple on a cyclic chain and consists of a single boundary H FIG. 3. Field dependence of the average magnetization per site Hb( ) given by Eq. 2.7 . To the left of this boundary the M/ in the ground state of a chain with 1.5 and 10. ground state is purely antiferromagnetic AF and to the right The dashed line corresponds to a cyclic chain and the solid line to it may be called a bulk spin-flop BSF state which becomes an open chain with the same number of sites. The inset focuses on increasingly ferromagnetic F . No transition to a pure F the first step of the main figure and compares the total magnetiza- state occurs at any finite field H. Since a cyclic chain pre- tion M to the analytical prediction 4.4 depicted by open circles. serves translation invariance, and thus provides a faithful Memory requirements restrict us to the range 12 if we representation of the bulk limit, we have in effect described wish to compute all eigenstates. However, we have been able the BSF transition on an infinite chain. The additional struc- to extend the range to 22 for the calculation of the ture shown to the left of the boundary Hb in the phase dia- ground-state energy via a Lanczos algorithm. The latter gram of Fig. 2 is due to surface effects that are analyzed in proved to be somewhat less reliable as well as time consum- Sec. IV. ing in the calculation of the magnetization. Therefore, the Lanczos algorithm is used on chains with 22 for the IV. SURFACE SPIN-FLOP TRANSITION investigation of those issues that depend exclusively on the ground-state energy. But results for the magnetization are The numerical calculation described in Sec. III was sub- limited to 16, using either complete diagonalization or sequently repeated on an open chain with the same size the Lanczos algorithm, or both. In all cases we provide ex- 10, and the result is depicted by a solid line in Fig. 3. The plicit estimates of finite-size effects which suggest that the magnetization is now seen to exhibit a sudden jump at a new derived overall picture is indeed reliable. critical field Hs 0.609 225 70 which is in excellent agree- For instance, the average magnetization per site ment with Eq. 2.14 applied for 1.5. This result is con- M/ calculated on a cyclic chain with 1.5 and sistent with the theoretical development of Sec. II which pre- 10 is depicted by a dashed line in Fig. 3 and exhibits a dicts that the one-magnon surface mode becomes the ground sudden jump at the critical field H state beyond H b 1.118 033 99 given by s . One may then use the analytical results Eq. 2.7 . The size of the jump was found equal to obtained on a semi-infinite chain to actually predict the mag- 0.249 732 68, also in perfect agreement with the analytical netization, at least for some nontrivial field region above Hs . result 3.18 applied for 1.5 and N /2 5. Away from The ground-state total magnetization again vanishes for the critical point the magnetization increases smoothly to H Hs but is equal to achieve the saturated ferromagnetic value 1/2 in the limit H . z Although H 1 Tl 1 b is independent of chain size, the magnetiza- M Ml , Ml 1 l tion does depend on 2N for H H b . A good estimate of l 1 1 1 4.1 its size dependence is obtained by taking advantage of the analytical result 3.18 . In the limit N the integral is for H Hs where 1 is now the state of the surface mode dominated by the maximum value of the integrand achieved in the lower gap of Fig. 1. Therefore, the norm of this state is at 0 to yield IN ( 1)N and given by 1 lim M 1 N 2N 2 1. 3.19 1 1 An 2 Bn 2 n 1 Applied for 1.5 the above expression leads to 2 0.223 606 80 which is overestimated by the value at 1 2H 2E 1 10 quoted earlier by about 12%. More generally, the mag- 1 2 1 2 , 4.2 1 PRB 60 BULK AND SURFACE SPIN-FLOP TRANSITIONS IN . . . 9483 where An and Bn are taken from Eq. 2.9 applied for 1 and E E1 given by Eq. 2.11 . Accordingly the local magnetization Ml is given by A 1 1 B M n 2 n 2 2n 1 1 1 2 , M2n 2 1 1 4.3 for odd and even sites, respectively, and the total magnetiza- tion by 1 2H 2E M 1 2 1 2H 2E1 2 . 4.4 It should be clear that the preceding results are valid also for H Hs where the surface mode is not the ground state. In particular, 2 1 M H 0 2 1 4.5 is the total magnetization of the one-magnon surface state at vanishing field. At the critical field Hs , where E1 0, Eq. 4.4 yields FIG. 4. Snapshots of the local magnetization Ml on a chain with 1 2H 1.5 and 10, for a characteristic set of field values described M H H s 2 s 1 2H in the text. s 2 , 4.6 which agrees with the jump observed at Hs in Fig. 3 to eight field becomes apparent in the H 1.12 entry, where the field significant figures; as expected on the basis of our discussion was chosen to be slightly greater than the bulk critical value of the size dependence of the surface modes in Sec. II. Hb 1.118 of Eq. 2.7 . It should be noted here that the Such an excellent agreement of the analytical prediction anticipated BSF transition is replaced on an open chain by a 4.4 with the finite-size results of Fig. 3 persists over a non- rapid but rounded crossover which becomes increasingly trivial field region Hs H Hw , as demonstrated in the inset sharp with increasing chain size. The inflation of the domain which focuses on the first step of the main figure. Clearly a wall is more rampant at higher field values and the local new transition takes place at Hw and the one-magnon surface magnetization approaches a nearly uniform ferromagnetic mode ceases to be the ground state for H Hw . One would configuration within the bulk, with some nonuniformity per- think that the SSF transition proceeds beyond Hs by a cas- sisting near the edges of the open chain; as is completely cade of level crossings induced by multimagnon surface apparent in the last, H 2, entry of Fig. 4. However, com- modes. But the results of Fig. 3 clearly indicate that there plete ferromagnetic order is achieved only when H . exists only one additional crossing at the critical field Hw The picture was completed with a detailed examination of 1.06 Hb for 1.5. the pertinent level crossings. Thus we calculated the ground- In order to appreciate the precise nature of the transition state energies of all sectors N,N 1, . . . , N as func- at Hw we have examined the evolution of the local magne- tions of the applied field H at the given anisotropy 1.5. tization Ml with increasing bias field. Results for 1.5 and The first transition occurs at the critical field Hs where the 10 are shown in Fig. 4 for a selected set of field values, NeŽel state NA , with N, is crossed by the one-magnon using a more or less obvious notation. We begin with the surface mode, with N 1. For higher fields, multiple first NeŽel state NA of Eq. 1.4 whose local magnetization level crossing take place among the lowest-energy states of is depicted in the H 0 entry of Fig. 4. The NeŽel state per- the multimagnon sectors N 2,N 3, . . . , and are likely sists as the absolute ground state until the field crosses the to play an important role in the low-temperature dynamics. critical value Hs 0.609 of Eq. 2.14 . Just above Hs a sur- But most of these crossings are irrelevant for the determina- face magnon is realized in the ground state, as shown in the tion of the absolute ground state because the one-magnon H 0.61 entry which may be reproduced to great precision surface mode is eventually overtaken only by the ground using the analytical prediction 4.3 obtained on a semi- state of the 0 sector which is an antiferromagnetic do- infinite chain. With further increase of the applied field the main wall located at the center of the open chain. surface magnon slowly approaches a boundary Ising domain The last remark prompted us to examine the local magne- wall of the type uududu . . . . However, a sudden change tization in the lowest-energy states of all sectors, even occurs at the critical field Hw 1.06, as demonstrated in the though most of these states do not become the absolute H 1.07 entry where a bulk domain wall appears at the cen- ground state for any field value. The result for 2N 10 is ter of the chain. This state is a slightly depleted version of an presented in Table I using a symbolic notation that is strictly ideal Ising domain wall of the type . . . uduudu . . . , thanks appropriate only in the extreme Ising limit . But the in part to the finite value of and to the applied field. The essence of the derived picture at finite and 0 H Hb is tendency for inflation of the domain wall with increasing well represented in Table I. Thus the ground state in each 9484 J. KARADAMOGLOU AND N. PAPANICOLAOU PRB 60 TABLE I. Symbolic illustration of the ground state in each sec- tor on an open chain with 10. Pseudospin Spin M 5 uuuuuuuuuu dududududu 0 4 duuuuuuuuu uududududu 1 3 dduuuuuuuu uddudududu 0 2 ddduuuuuuu uduudududu 1 1 dddduuuuuu ududdududu 0 0 ddddduuuuu ududuududu 1 1 dddddduuuu udududdudu 0 2 ddddddduuu udududuudu 1 3 dddddddduu ududududdu 0 4 dddddddddu ududududuu 1 5 dddddddddd ududududud 0 sector ranges between the two pure NeŽel states of Eq. 1.4 which correspond to the two extreme values of the azimuthal pseudospin N. For intermediate values of N , with 1,2, . . . ,2N 1, a domain wall is formed at a dis- tance equal to lattice units from the left boundary of the chain. Therefore, the good quantum number provides a rigorous definition of the relative location of a domain wall on an open chain, even at finite where the wall expands and thus departs from its ideal Ising shape. Domain walls with opposite pseudospin are displayed symmetrically FIG. 6. Snapshots of the local magnetization Ml on a chain with about the center and carry the same energy. The least-energy 1.5 and 12, for a characteristic set of field values described state for H H in the text. w is a 0 domain wall located at the center of the chain. results of the local magnetization M To rule out the possibility of an accident that might have l given in Fig. 6, in con- junction with an obvious extension of Table I to 12. occurred for the specific size 10 used so far, we repeated At this point it is useful to address the last column of the calculation for 12 and the results for the total mag- Table I which quotes the possible values of the total magne- netization are shown in Fig. 5. Surprisingly, two instead of tization of domain walls in the extreme Ising limit, M 0 or one additional level crossings may now be discerned: the 1. One may say that the two values correspond to domain first at Hw 1.06, which is virtually identical to the value walls of dd or uu type, respectively. Although these values obtained earlier for 10, and the second at Hw 1.09 are significantly modified at finite , they nevertheless sug- which indicates the existence of yet another critical field. gest that a level crossing is more likely to be induced by a uu This interesting twist in the general picture is clarified by the state whose negative Zeeman energy in a positive field H is greater in absolute value. The 0 domain wall at the center of the 10 chain is indeed a uu state. However, when Table I is extended to 2N 12, where N 6 is even, the 0 domain wall becomes a dd state, whereas uu domain walls that are closest to the center are those with 1 and are likely to be energetically favorable. Simple comparison of the H 1.07 entries in Figs. 4 and 6 reveals that the same domain wall of the uu type appears in both cases, but the wall in the second case is displaced by one lattice unit from the center of the chain. Since a bulk domain wall is rather narrow for the specific anisotropy 1.5 used so far, its energy is relatively insensitive to the precise location about the center even for short chains. This explains why the transition observed for 12 occurs at virtually the same critical field Hw 1.06 found earlier for 10, even though it now corresponds to a level crossing of the one-magnon mode by the ground state of the 1 sector. However, the 0 ground state, which originates in a dd FIG. 5. Same as Fig. 3 for a chain with 12. The inset now domain wall at lower fields, ultimately becomes sufficiently focuses on a narrow region near the critical field Hw to reveal the frustrated to overtake the 1 sector at a new critical field existence of a secondary critical field Hw . Hw 1.09, as demonstrated by the H 1.10 entry of Fig. 6. PRB 60 BULK AND SURFACE SPIN-FLOP TRANSITIONS IN . . . 9485 Hs ,Hw ,Hw , . . . which stabilize to size-independent values. The main SSF transition is still given by the crossing of the N NeŽel state by the N 1 surface magnon or bound- ary domain wall at the critical field Hs of Eq. 2.14 . But the next transition at Hw now corresponds to a crossing of the N 1 surface magnon by the N 3 ground state which is a uu-type domain wall located three lattice units away from the left end of the chain. Subsequent transitions at a sequence of critical fields Hw ,Hw ,... correspond to a se- quense of hoppings of the domain wall in steps of two lattice units until it arrives at the center of the chain. Once the domain wall reaches the center, its future evolution is similar to the one described earlier for 1.5. A completely satisfactory description is not possible on the short chains used in the numerical calculations, because the size of the relevant domain walls increases to lattice di- mensions in the limit 1 . However, the observed pattern is sufficiently clear to provide unambiguous numerical evi- dence for the new critical boundary H Hw( ) which com- pletes the phase diagram of Fig. 2. Although this basic phase diagram does not reflect the fine structure in Hw H Hb alluded to in the preceding paragraphs, it certainly contains all those elements that are likely to be important in practical applications. Thus the ground state is purely NeŽel and the correspond- ing phase is labeled as antiferromagnetic AF for H Hs . The region Hs H Hw is characterized by a ground state which is a surface mode and is called a surface spin-flop SSF phase. The domain-wall DW phase extends in the region Hw H Hb where a bulk domain wall is realized in the ground state. Finally the region H Hb corresponds to the bulk spin-flop BSF phase, studied in Sec. III, which becomes increasingly ferromagnetic F . This explains the composite designation BSF, F in Fig. 2. The union of the AF, SSF, and DW phases would become an extended AF FIG. 7. The results of Figs. 3 and 5 now iterated on longer phase in the absence of free boundaries. chains with 14 and 16 to establish the alternating pattern de- The description of the phase diagram is completed with a scribed in the text. comment on the thermodynamic limit. Although the critical boundaries reach size-independent values, for practically all The absolute ground state remains in the 0 sector for H 1, bulk quantities such as the average magnetization per Hw and is again rendered increasingly ferromagnetic in the site M/ become relatively insignificant in the limit limit H . for H H The foregoing analysis suggests that a transition into a b . For example, in the SSF phase, the total magnetization M is given analytically by Eq. 4.4 and is of domain-wall state is always present on an open chain at a order unity. Therefore, the average moment decreases lin- critical field Hw , and is merely decorated by a secondary early with 1/ , a fact that is progressively apparent in Figs. transition at a slightly higher field Hw when N in 2N is 3, 5, and 7 which depict results in the range 10 16. even. This alternating pattern is confirmed by the calculated Nevertheless surface effects are always present on open even total moment for 14 and 16 shown in Fig. 7, and by chains of any size and could be observed in a magnetic me- further analysis of level crossings using the Lanczos algo- terial that is sufficiently doped to produce a statistically sig- rithm on chains with 22. The main new critical field Hw nificant number of such chains. quickly stabilizes to the size-independent value Hw 1.0625, for 1.5, which is distinct from the bulk critical field H V. CONCLUSION b 1.118 and thus clearly suggests the appearance of a definite domain-wall DW phase in Hw H Hb . The main advantage of the bulk and surface spin-flop The picture just derived for the specific anisotropy transitions studied in this paper is that they are induced by a 1.5 is more or less sustained for a wide range of anisotro- uniform bias field which can be easily applied and tuned to pies in the region 1.25. However, this simple picture any desired value. This situation should be contrasted with becomes more involved as the anisotropy approaches the iso- the case of boundary fields23 that are generally difficult to tropic limit 1 . Already at 1.125 a cascade of level implement, especially in doped materials where open mag- crossings are induced by the least-energy states of the sectors netic chains are produced within the crystal in a random N 1,N 3,N 5, . . . at a sequence of critical fields manner. It is thus conceivable that the current theoretical 9486 J. KARADAMOGLOU AND N. PAPANICOLAOU PRB 60 FIG. 8. Field dependence of the average magnetization per site M/ in the ground state of a model with J FIG. 9. Field dependence of the average magnetization per site 1 0, J2 1, J3 1.5, and 14. The dashed line corresponds to a cyclic chain M/ in the ground state of the XXZ model J1 1 J2, J3 and the solid line to an open chain with the same number of sites. 1.5 with 14. The dashed line corresponds to a cyclic chain and the solid line to an open chain with the same number of sites. work will eventually find an experimental realization analo- Note the transition to a pure F phase above the critical field Hf gous to that obtained in classical Fe/Cr multilayers.4­6 2.5. Suppose that a quasi-one-dimensional magnetic material is found22 with exchange constants that are approximately chain,26 and the problem of spin-flop transitions amounts to given by Eq. 1.3 after suitable normalization. Doping such studying the density of level crossings induced by the linear a material with nonmagnetic ions18 would produce open Zeeman shift. In fact, the T 0 phase diagram on an infinite magnetic chains of varying size. Since the critical boundaries chain was studied by Johnson and McCoy27 and consists of of Fig. 2 are practically independent of chain size, it would an AF phase for H Hb , a BSF phase for Hb H Hf , and be possible to tune the applied field to the various regions of a pure F phase for H Hf . If we set cosh the critical the phase diagram and thus probe the predicted magnetic field Hb is given by28 phases. Electron-spin resonance at low temperature seems to be an appropriate experimental tool, in analogy with reso- 1 n nance experiments already performed6 and theoretically Hb sinh discussed6,17 for classical Fe/Cr multilayers. A corresponding n cosh n study in the present quantum model would require explicit calculation of the relevant dynamic susceptibilities, an issue 1 sinh to which we hope to return in the future. n , 5.1 The prospects for experimental realization would be sig- cosh 2n 1 2 2 nificantly enhanced if the present theoretical work could be extended to the full range of models defined by inequalities which differs significantly from Eq. 2.7 especially at weak 1.2 . For example, an interesting special case is the aniso- anisotropies 1 or 0 where the field 5.1 van- tropic XY model J ishes exponentially. Furthermore, a transition to a pure F 1 0, J2 1, J3 1 , or YZ model in current notation, in the presence of an in-plane field applied state now takes place above the critical field along the easy axis. Analytical solution of this model does not seem possible at nonvanishing field, and theoretical Hf 1. 5.2 analysis is further complicated by the lack of a U 1 symme- try. A preliminary numerical calculation of the ground-state Comparison of the results for the total magnetization total magnetization on a chain with 14 is shown in Fig. computed numerically on an open and a cyclic chain with 8. While a trace of both a surface (Hs) and a bulk (Hb) 14, shown in Fig. 9, again suggests a SSF transition at a critical field is again present, the spin-flop transition obvi- new critical field Hs Hb . However, both Hb and Hs are ously proceeds by multiple level crossings which are difficult now size dependent and hence the results of Fig. 9 are not to study in detail by numerical simulations on short chains. A sufficient to establish the existence of a SSF transition. Ex- notable feature of Fig. 8 is that ferromagnetic order at high trapolation of the relevant magnon gaps calculated on chains fields is now more robust. with 22 indicates a ratio Hb /Hs that remains remarkably The picture could again simplify in the XXZ model J1 close to its Ising value 2 for a wide range of strong anisotro- 1 J2 , J3 1 where a U 1 symmetry is restored. pies in the region 2. Nevertheless extrapolation becomes The effect of a uniform field pointing along the symmetry problematic at weak anisotropies and thus a definite predic- axis is simply a linear Zeeman shift of the zero-field energy tion near the isotropic limit is difficult to obtain numerically. eigenvalues. The latter may, in principle, be obtained by the It should be mentioned that a considerable amount of work Bethe ansatz19 known to apply to both a cyclic and an open has been devoted to the study of the XXZ model in the pres- PRB 60 BULK AND SURFACE SPIN-FLOP TRANSITIONS IN . . . 9487 ence of boundary fields,29,30 but the more direct questions raised here in the presence of a uniform bulk field do not seem to have been addressed. To conclude, we note that interesting variations of the main picture may occur for various J1 J2 J3 . In this respect, we recall that the classical21 ground state is indepen- dent of J1 and thus combines features of the entire class of quantum models in the above range. Therefore, although the earlier classical calculations for single-ion14 and exchange21 anisotropy provided extremely valuable motivation for the present work, their detailed results cannot be readily applied to the study of the respective quantum models. ACKNOWLEDGMENT We are grateful to Xenophon Zotos for assistance in the development of the Lanczos algorithm. APPENDIX: THE ODD CHAIN FIG. 10. Field dependence of the average magnetization per site M/ in the ground state of our standard model 1.3 with In a material that is randomly doped with nonmagnetic 1.5 on an open chain with an odd number of sites 15. ions, approximately half of the produced open magnetic chains are composed of an odd number of sites ( 2N picture that is fairly similar to that of Fig. 1, with the follow- 1). It is thus important to also examine the ground state of ing notable difference. The gap mode 1 is now missing from odd chains in the presence of a bias field H. The two NeŽel the spectrum, while mode 1 is duplicated. As a result there states will be no SSF transition at the critical field Hs . Instead an odd open chain will proceed directly to a BSF transition NA d,u,d,u,...,d,u,d , which occurs by a cascade of successive level crossings in A1 N the vicinity of the critical field Hb . This picture is similar but B u,d,u,d,...,u,d,u not identical to the BSF transition on an even cyclic chain, are again mapped by Eq. 1.5 to two completely polarized studied in Sec. III, where all crossings take place at precisely ``ferromagnetic'' states which are exact eigenstates of the the same critical field Hb . Putting it differently, the antici- Hamiltonian 1.6 for any value of the applied field. How- pated hidden symmetry of an even cyclic chain at the critical ever, degeneracy is now lifted by the bias field because the point is broken on an open chain. states A1 carry nonvanishing total magnetization M The lack of a SSF transition on an odd chain becomes 1/2 and the corresponding energy eigenvalues are given by apparent with an explicit calculation of the total magnetiza- H/2. Therefore, when H is taken to be positive, NB is the tion for 1.5 and 15 shown in Fig. 10. At low field unique ground state with M 1/2. Similarly, when H is nega- values the total magnetization is given by M 1/2, or tive, the unique ground state is NA with M 1/2. For 1/2 , and coincides with that of the pure NeŽel state NB . definiteness, we assume that the bias field is positive, the The BSF transition near the critical field Hb is also apparent case of H being completely analogous. in Fig. 10, whereas the chain is set on a more or less smooth Our task is then to determine the critical field above course toward ferromagnetic order for H Hb . Therefore, a which a spin-flop transition may take place. Examination of clear distinction between even and odd chains is present, in the one-magnon spectrum around the state NB leads to a analogy with the situation in classical Fe/Cr multilayers.4,10 *Electronic address: gkara@physics.uch.gr 9 R. L. Stamps, R. E. Camley, F. C. Nošrtemann, and D. R. Tilley, Electronic address: papanico@physics.uch.gr Phys. Rev. B 48, 15 740 1993 . 1 D. L. Mills, Phys. Rev. Lett. 20, 18 1968 . 10 R. W. Wang and D. L. Mills, Phys. Rev. B 50, 3931 1994 . 2 D. L. Mills and W. Saslow, Phys. Rev. 171, 488 1968 . 11 L. Trallori, P. Politi, A. Rettori, M. G. Pini, and J. Villain, Phys. 3 F. Keffer and H. Chow, Phys. Rev. Lett. 31, 1061 1973 . Rev. Lett. 72, 1925 1994 . 4 R. W. Wang, D. L. Mills, E. E. Fullerton, J. E. Mattson, and S. D. 12 L. Trallori, P. Politi, A. Rettori, M. G. Pini, and J. Villain, J. Bader, Phys. Rev. Lett. 72, 920 1994 . Phys.: Condens. Matter 7, L451 1995 . 5 R. W. Wang, D. L. Mills, E. E. Fullerton, S. Kumar, and M. 13 L. Trallori, Phys. Rev. B 57, 5923 1998 . Grimsditch, Phys. Rev. B 53, 2627 1996 . 14 C. Micheletti, R. B. Griffiths, and J. Yeomans, J. Phys. A 30, 6 S. Rakhmanova, D. L. Mills, and E. E. Fullerton, Phys. Rev. B L233 1997 . 57, 476 1998 . 15 C. Micheletti, R. B. Griffiths, and J. Yeomans, Phys. Rev. B 59, 7 F. C. Nošrtemann, R. L. Stamps, A. C. Carrico, and R. E. Camley, 6239 1999 . Phys. Rev. B 46, 10 847 1992 . 16 N. Papanicolaou, J. Phys.: Condens. Matter 10, L131 1998 . 8 F. C. Nošrtemann, R. L. Stamps, and R. E. Camley, Phys. Rev. B 17 N. Papanicolaou, J. Phys.: Condens. Matter 11, 59 1999 . 47, 11 910 1993 . 18 H. Asakawa, M. Matsuda, K. Minami, H. Yamazaki, and K. Kat- 9488 J. KARADAMOGLOU AND N. PAPANICOLAOU PRB 60 sumata, Phys. Rev. B 57, 8285 1998 . 25 V. Pasquier and H. Saleur, Nucl. Phys. B 330, 523 1990 . 19 M. Gaudin, La Fonction d'Onde de Bethe Masson, Paris, 1983 . 26 F. C. Alcaraz, M. N. Barber, M. T. Batchelor, R. J. Baxter, and G. 20 Zhan-Ning Hu, Phys. Lett. A 250, 337 1998 . R. W. Quispel, J. Phys. A 20, 6397 1987 . 21 J. Karadamoglou and N. Papanicolaou, J. Phys. A 32, 3275 27 J. D. Johnson and B. M. McCoy, Phys. Rev. A 6, 1613 1999 . 1972 . 22 J. Kyriakidis and D. Loss, Phys. Rev. B 58, 5568 1998 . 28 J. D. Johnson and J. C. Bonner, Phys. Rev. B 22, 251 1980 . 23 F. C. Alcaraz, S. R. Salinas, and W. F. Wrezinski, Phys. Rev. 29 M. Jimbo, R. Kedem, T. Kojima, H. Konno, and T. Miwa, Nucl. Lett. 75, 930 1995 . Phys. B 441, 437 1995 . 24 H. P. Bader and R. Schilling, Phys. Rev. B 19, 3556 1979 . 30 A. Kapustin and S. Skorik, J. Phys. A 29, 1629 1996 .