<M> = 1 <M> = 0. h/j J /J 5/6 2/3 1/3 1/ h/j

Transcription

1 Phys. Rev. Lett. 79, (997) cond-mat/9779, La Plata-Th 97/, SISSA 88/97/EP Magnetization Curves of Antiferromagnetic Heisenberg Spin- Ladders D.C. Cabra, A. Honecker ;y, P. Pujol Departamento de Fsica, Universidad Nacional de la Plata, C.C. 7, (9) La Plata, Argentina. International School for Advanced Studies, Via Beirut -, Trieste, Italy. y Work done under support of the EC TMR Programme Integrability, non-perturbative eects and symmetry in Quantum Field Theories, grant FMRX-CT9-. (July 9, 997, revised October, 997) Magnetization processes of spin- Heisenberg ladders are studied using strong-coupling expansions, numerical diagonalization of nite systems and a bosonization approach. We nd that the magnetization exhibits plateaux as a function of the applied eld at certain rational fractions of the saturation value. Our main focus are ladders with legs where plateaux with magnetization one third of the saturation value are shown to exist. PACS numbers: 7..Jm, 7..Ej Recently there has been considerable interest in coupled Heisenberg antiferromagnetic (HAF) chains, socalled `spin ladders', where one of the fascinating discoveries is that the appearance of a gap depends on the number of chains being even or odd (for recent reviews see e.g. []). In this letter we study ladder systems at zero temperature in a strong uniform magnetic eld. This issue has so far only been addressed for two coupled chains with a magnetization experiment on Cu (C H N ) Cl [] and theoretically using numerical diagonalization [], series expansions [] and a bosonization approach []. These studies found a plateau at zero magnetization whose width is given by the spin gap in the otherwise smooth magnetization curve. In this letter we extend the theoretical approaches to three and more coupled chains using strong coupling expansions, numerical diagonalization and a bosonization approach. We nd that in general the magnetization curves exhibit plateaux also at certain non-zero quantized values of the magnetization. To be precise, we concentrate on the zero-temperature behaviour of the following HAF spin ladder with N legs (kept xed) and length L (taken to innity): H (N) = J X i h X i;x x= ~S i;x ~ Si+;x + J NX i= x= ~S i;x ~ Si;x+ S z i;x ; () where the S ~ i;x are spin- operators and h is a dimensionless magnetic eld. We assume periodic boundary condi- tions along the chains but investigate both open (OBC) and periodic boundary conditions (PBC) along the rungs. The magnetization hmi is given by the expectation value of the operator M = LN P i;x Sz i;x. Our main result is that hmi as a function of h has plateaux at the quantized values N ( hmi) Z: () This condition also appears in the Lieb-Schultz-Mattis theorem [] and its generalizations [7,8]. There, it is related to gapless non-magnetic excitations, but plateaux in magnetization curves appear if there is a gap to magnetic excitations. However, there are other arguments which lead to () as the quantization condition for hmi at a plateau. A particularly simple one is given by the limit of strong coupling along the rungs J J (which has also proven useful in other respects [9]). At J = one has to deal only with Heisenberg chains of length N and the only possible values for the magnetization are precisely the solutions of (): hmi f ; + =N; : : : ; =N; g. For N = and J J this consideration predicts a plateau at hmi =. The boundary of this plateau is related to the spin gap simply by h c =. The strongcoupling series for this gap reads [] = J J + J J + J J J 8J + O(J ) ; () where we have extended the result of [] to fourth order (this further order is crucial to obtain a zero of the gap for J ). For N = and small magnetic elds one has a degeneracy which makes already rst-order perturbation theory non-trivial. For OBC and jhm ij =, the low-lying spectrum is then given by a spin- chain in a magnetic eld whose magnetization curve is well-studied (see [] and references therein). On the other hand, for PBC and jhm ij =, the rst-order low-energy eective Hamiltonian for () at N = turns out to be (see also []): H (III;p) e: = J h x= x= + + x x+ + x + x+ ~Sx ~ Sx+ S z x ; ()

2 where the S ~ x are su() operators acting in the spin-space and x act on another two-dimensional space which comes from a degeneracy due to the permutational symmetry of the chains. In particular, the usual spin-rotation symmetry is enlarged by an U() coming from the XYtype interaction in the space of the Fourier transforms along the rungs (rst factor in ()). The Hamiltonian () encodes the magnetization curve in the strong-coupling limit of the three-leg ladder with PBC for jhmij =. At least for OBC, the situation is a little more favourable for the hmi = = plateau that one expects for N =. It turns out that one can use non-degenerate perturbation theory to compute the energy cost to ip a spin around this plateau and thus its lower and upper boundaries: h c = J J h c = J J + J 9J 9J J + O(J ) ; () J + J 888J + O(J ) : () Finally, it is straightforward to exactly compute the upper critical eld h uc for the transition to a fully magnetized state for PBC, even L and arbitrary N; (J + J h uc = ) cos N N J + J N even, N odd. (7) Actually, the result (7) at N =, h uc = =J + J also applies to OBC. We now proceed with considerations that follow closely classical work on single Heisenberg spin chains [] and the D triangular HAF []. We have numerically calculated the lowest eigenvalues as a function of the magnetization, wave vectors and coupling constants on nite systems with up to a total of sites. / / <M> / / / 7 FIG.. Magnetization curve for N = at J =J = with OBC. The thin full lines are for L = 8, the long dashed lines for L = and the short dashed lines for L =. The thick full line indicates the expected form in the thermodynamic limit. The two diamonds denote the series () and () for the boundaries of the plateau. Fig. illustrates our main results with a magnetization curve for N = at J =J = with OBC. The thin lines denote curves at nite system size and clearly exhibit a plateau with hm i = =. The strong-coupling approximations () and () for the boundaries of this plateau are in good agreement with the nite-size data. Finite-size eects are also small for the midpoints of the steps in the magnetization curves []. From this one obtains an extrapolation to innite L which is shown by the full line in Fig.. A more compact representation is given by `magnetic phase diagrams' which show the projection of the conventional magnetization curves as in Fig. onto the axis of the magnetic eld. We illustrate this in Fig. with the case of a two-leg ladder which has already been studied in []. On a nite lattice the possible values of hmi are quantized; the values of the magnetic eld h=j where a transition from one such value occurs to the neighbouring one are shown by lines as a function of J =J. Regions without such lines denote plateaux in the magnetization curve while regions where they are close to each other correspond to smooth transitions in the thermodynamic limit. <M> = <M> = -. FIG.. Magnetic phase diagram of the ladder with N = legs. The thin full lines are for L =, the long dashed lines for L = and the short dashed lines for L = 8. The thick full line shows the gap (). The magnetization curves of [,] with J =J = are located somewhat beyond the right border of Fig.. Our gure contains additional information about the dependence on J =J. Note that the numerically obtained values to full magnetization do indeed match with the version of (7) for the ladder: h uc = J + J. For the boundary of the plateau at hmi = one observes that nite-size eects are not substantial for J =J =, and in this region one observes also good agreement with the approximation () for the transition value of the magnetic eld h c =. At weak coupling, eld theoretic arguments [] predict the gap to open linearly, J. This is compatible with Fig., though due to the large nite-size eects in this region much longer chains are needed to conrm

3 this linear behaviour conclusively []. 8 7 <M> = <M> = / <M> = -. FIG.. Magnetic phase diagram of N = chains with OBC. The thin full lines are for L = 8, the long dashed lines for L = and the short dashed lines for L =. The two thick full lines denote () and (), respectively. Figs. and show the magnetic phase diagrams for N = with OBC and PBC, respectively (Fig. is a section of Fig. at J =J = ). Both gures clearly exhibit a plateau with hmi = = at least in the region J J. The strong-coupling series () and () for the boundaries of this plateau are also shown in Fig. and in the aforementioned region J J, where nite-size eects are small, one observes good agreement between the expansions and the numerical results. 8 7 <M> = <M> = / <M> = -. FIG.. Same as Fig., but with PBC. Figs. and dier at least in their details. For example, one observes that at the upper boundary of the hmi = = plateau in Fig. it becomes favourable to ip two spins at once rather than one for J J which may be interpreted as one signal of the frustration introduced into the system. At strong coupling the excitations above the /- plateau in Fig. are described by () with all spins aligned along the eld (Sx z = ). This is an XY-chain and therefore massless, providing us with an example of a plateau in the magnetization curve with gapless nonmagnetic excitations above it. The strong-coupling expansions as well as the numerical results obtained so far clearly show the existence of plateaux for suciently strong coupling. To see if the plateau persists in the weak-coupling region J J we use abelian bosonization (see e.g. [,7]). The computation to be presented below is similar to the ones performed recently in [8,8], so we refer the interested reader to these two references for more details. The starting point for this weak-coupling expansion is the observation that at J = one has N decoupled spin- HAFs in a magnetic eld whose low-energy properties are described by a c = Gaussian conformal eld theory [9]. This theory is characterized by the radius of compactication R(hM i; ) which depends on the magnetization as well as the XXZ-anisotropy which we have here introduced for convenience. At zero magnetic eld one has [7]: R (; ) = cos. In general, this radius can be computed using the Bethe-Ansatz solution [9] for the Heisenberg chain. For a qualitative understanding it may be helpful to notice that the radius of compactication R decreases if either becomes smaller or if the magnetization hm i increases. Now the low-energy properties of the Hamiltonian () at small coupling are described by the following Tomonaga Hamiltonian with interaction terms: H (N) = Z + + X i dx " X i NX i= n i (x) + R (hmi; ) (@ x i (x)) o (@ x i (x)) (@ x i+ (x)) : cos k F x + p ( i + i+ ) : (8) + : cos p(i i+ ) + : cos p( ~ i ~ i+ ) : : # with i ~ x i, and i J =J. It is convenient to rescale the elds by the radius of compactication R in (8), i.e. i! p R i. For N = chains with PBC we now change variables from the elds,, to p ( + + ), p ( ), p ( + ). Following [] one can show that the perturbation terms with coecients or in (8) give rise to a mass for the latter two elds. Let us now consider the remaining eld diag := P p N N i= i. Radiative corrections to (8) generate the interaction term : cos( diag =R) : []. For N = and at hmi = this operator is irrelevant (in the region of close to ), which conrms that three chains weakly coupled in a periodic manner are massless. ;

4 Now we address the question of the appearance of plateaux for hmi =. Since this requires a gap for the (magnetic) excitations, we expect such a plateau to occur if the remaining eld diag acquires a mass. In fact, the additional interaction term : cos( diag =R) : (9) survives in the continuum limit on a plateau []. The operator (9) can appear only if () holds, as can easily been inferred from translational invariance of the original Hamiltonian () and the fact that k F = ( hmi) and a one-site translation of the lattice Hamiltonian () translates into the internal symmetry transformation diag! diag + NRk F. For N =, () requires that hmi = =. If one now estimates the radius of compactication R( ; ) following e.g. [8], one nds that at J = the operator (9) is slightly irrelevant for =. The dimension of this operator decreases with J implying that the =-plateau in Fig. extends down into the region of small J. It should be noted that the appearance of the plateau for a given small J crucially depends on the value of, explaining why the numerical evidence in Fig. is not conclusive in this region. For simplicity we have concentrated on PBC. The case of OBC is qualitatively similar but more subtle in the details and will be discussed in []. In this letter we have shown that spin ladders exhibit plateaux in their magnetization curves when subjected to strong magnetic elds. We have mainly concentrated on the plateau with hmi = = in three coupled chains, but also other rational values can be obtained by varying the number of chains N. Quantum uctuations (i.e. the choice of the spin S = ) do play an important r^ole { the plateaux disappear in general when one inserts classical spins into (). Nevertheless, by analogy to the two-dimensional triangular antiferromagnet [] it seems likely that such plateaux would appear in rst-order spinwave theory. Also with Ising spins the behaviour of () is dierent: At zero temperature the magnetization changes only discontinuously between plateaux values (i.e. no smooth transitions occur), and a gap opens both for odd and even numbers of chains. It would be highly interesting to check experimentally whether the plateaux can indeed be observed, in particular since nowadays materials with a given number of legs can be engineered. The rst non-trivial check would be to look for the hmi = = plateau in a three-leg ladder. Here, the material Sr Cu O [] comes to mind, which is however not very well suited for these purposes since the necessary order of magnetic elds is not accessible today due to its large coupling constants. However, there are at least two-leg ladder materials such as the conventional (VO) P O 7 [] or Cu (C H N ) Cl [] with much weaker coupling constants. A three-leg analogue of such materials could provide a testing ground for our predictions, in particular if such a material can be found with J J where we would expect a clearly visible plateau in the magnetization curve at suciently low temperatures (cf. Fig. ). A.H. thanks Profs. K.D. and U. Schotte for drawing his attention to magnetization curves. We are grateful to them, M. Kaulke, G. Mussardo, A.A. Nersesyan, V. Rittenberg and F.A. Schaposnik for useful discussions and comments on the manuscript as well as to G. Chaboussant and L.P. Levy for explanations concerning the experimental possibilities. D.C.C. thanks CONICET and Fundacion Antorchas for nancial support. We are indebted to the Max-Planck-Institut fur Mathematik, Bonn-Beuel for allocation of CPU time on their computers. [] E. Dagotto, T.M. Rice, Science 7, 8 (99); T.M. Rice, Z. Phys. B, (997). [] G. Chaboussant, P.A. Crowell, L.P. Levy, O. Piovesana, A. Madouri, D. Mailly, Phys. Rev. B, (997). [] C.A. Hayward, D. Poilblanc, L.P. Levy, Phys. Rev. B, R9 (99). [] Z. Weihong, R.R.P. Singh, J. Oitmaa, Phys. Rev. B, 8 (997). [] R. Chitra, T. Giamarchi, Phys. Rev. B, 8 (997). [] E. Lieb, T. Schultz, D. Mattis, Ann. Phys., 7 (9). [7] I. Aeck, Phys. Rev. B7, 8 (988). [8] M. Oshikawa, M. Yamanaka, I. Aeck, Phys. Rev. Lett. 78, 98 (997). [9] E. Dagotto, J. Riera, D.J. Scalapino, Phys. Rev. B, 7 (99); T. Barnes, E. Dagotto, J. Riera, E.S. Swanson, Phys. Rev. B7, 9 (99). [] M. Reigrotzki, H. Tsunetsugu, T.M. Rice, J. Phys.: Condensed Matter, 9 (99). [] J.C. Bonner, M.E. Fisher, Phys Rev. A, (9); J.B. Parkinson, J.C. Bonner, Phys. Rev. B, 7 (98). [] K. Totsuka, M. Suzuki, J. Phys. A: Math. Gen. 9, 9 (99); H.J. Schulz, preprint cond-mat/97. [] H. Nishimori, S. Miyashita, J. Phys. Soc. Jpn., 8 (98). [] K. Totsuka, M. Suzuki, J. Phys.: Condensed Matter 7, 79 (99); D.G. Shelton, A.A. Nersesyan, A.M. Tsvelik, Phys. Rev. B, 8 (99). [] M. Greven, R.J. Birgeneau, U.-J. Wiese, Phys. Rev. Lett. 77, 8 (99). [] H.J. Schulz, Phys. Rev. B, 7 (98). [7] I. Aeck, in Fields, Strings and Critical Phenomena, Les Houches, Session XLIX, edited by E. Brezin and J. Zinn- Justin (North-Holland, Amsterdam, 988). [8] K. Totsuka, Phys. Lett. A8, (997). [9] F. Woynarovich, H.-P. Eckle, T.T. Truong, J. Phys. A: Math. Gen., 7 (989). [] The appearance of the factor here is intimately linked to the fact that we are considering an odd number of chains.

5 [] D.C. Cabra, A. Honecker, P. Pujol, in preparation. [] A.V. Chubukov, D.I. Golosov, J. Phys.: Condensed Matter, 9 (99). [] M. Azuma, Z. Hiroi, M. Takano, K. Ishida, Y. Kitaoka, Phys. Rev. Lett. 7, (99). [] D.C. Johnston, J.W. Johnson, D.P. Goshorn, A.J. Jacobson, Phys. Rev. B, 9 (987).