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Zero temperature quantum renormalization group
ZERO TEMPERATURE QUANTUM RENORMALIZATION GROUP* H. W. J. BLOTE, J. C. B O N N E R Univ. of Rhode Island, Kingston, RI 02881, USA
and J. N. F I E L D S Hughes Research Corp., Malibu, CA 90265, USA
A zero-temperature quantum renormalization group which works rather well for simple 1-D quantum models is applied to more complex quantum models. These include the alternating Heisenberg antiferromagnetic chain and the spin anisotropic IsingHeisenberg chain. Results indicate qualitative success.
1. Method This method uses real space renormalization group (RG) transformations reminiscent of those used for classical systems [1]. The lattice is subdivided into blocks of N s sites such that the eigenvalues and eigenvectors of each block may be calculated exactly. The basis of each block (2 Ns levels) is truncated to some n u m b e r N L of levels, and the coupling between adjacent blocks is written within the truncated basis. Since this is a T = 0 approach, the N L levels include the ground state and a dominant set of first excited states. Care is needed to preserve the symmetry character of the N L levels at each step of the iterative process. After each R G step ~ is written in the same form, which leads to recursion relations which express the parameters of the renormalized 0C as a function of those of the original unrenormalized one. Choosing the two lowest states of an odd Ns block, for example, maps the block onto a single spin $1 on the new lattice. An R G transformation for the antiferromagnetic Heisenberg chain using N L = 4 and even N s effectively maps the block into an S=$ dimer (spin-pair). For Ns, N L small, an analytic calculation is possible. A set of recursion relations which define an R G transformation for the ground state of the Hamiltonian can be written down explicitly. Fixed points of the recursion relations and corresponding T = 0 critical singularities are obtained. Some illustrative calculational details have already been published [2, 3]. In principle, the accuracy of the results should be improved by systematically increasing both N~ and N L. An entirely computer based approach is now required, *Work supported in part (at URI) by NSF grant DMR77-24136 and Bunting Institute Fellowship, Radcliffe College.
however, because of the size of the matrix operators. I Models considered are the alternating S - 2 Heisenberg antiferromagnetic linear chain (AH): N/2 ~)C = 2 J ~ { SEi_ l °SEi -.~ aS2i.S2i +l ) i=l
(1)
and the alternating XY counterpart (AXY). The alternation parameter a varies between 1 (uniform limit) and 0 (dimer limit). We have also considered the uniform antiferromagnetic Ising-Heisenberg linear chain ( U I H ) with S = ~I : N x 3C = 2J'5~, ( S i z S i +• , + _y~,~ S xi S i+l-t"
SYS/Y+I)}
i=l
(2) where the anisotropy parameter y varies between I (isotropic limit) and 0 (Ising limit).
2. Results R G calculations have been performed for N L = 2 and N - - 3 , 5, 7, 9 for both the A H and AXY. Calculations have also been performed for N s = 2, 4, 6, 8 and N L ) 4. For the U I H we have examined the cases Ns = 3, N L = 2 and N~ = N L - 4. The accuracy and reliability of this R G approach have been variously assessed. The A X Y and U I H models are exactly solvable, unlike the A H where comparison is made with a Luttinger model calculation [4] and numerical extrapolations on finite chains [5]. We focus our assessment on three areas: (a) location and nature of the fixed points; (b) values
Journal of Magnetism and Magnetic Materials 15-18 (1980) 405-406 ©North Holland
405
H. lee". J. Blote et a l . / T = 0 quantum renormalization group
of the critical singularities; and (c) quantitative a c c u r a c y of the g r o u n d state energy per spin E 0, and the energy gap AE between g r o u n d and first excited states. We observe at the outset that the m e t h o d has some qualitative successes. F o r A X Y and A H a stable fixed point (FP) is f o u n d at c~* = 0 (also ct* = oo), and an unstable F P at a* = l, for Ns odd systems. This has the important physical implication that the system with non-zero alternation i.e. ct = 1, will renormalize into a dimer system, characterized by a singlet-triplet energy gap which vanishes only in the uniform limit, a = 1. This agrees with exact results for the A X Y , and gives a powerful a r g u m e n t for the presence of a similar alternation gap in the Heisenberg counterpart [2, 3]. W h e n N~ even systems are considered, however, there is a minor problem. Stable F P ' s are again f o u n d at a * = 0 ( ~ ) , but the unstable F P is no longer exactly at unity. F o r N s = 2, 4: t~* = 0.962; for N s = 6: a * = 1.001; a n d for N s = 8: a * = 1.021. F o r the X Y model the results are rather worse. For N = 2, 4: a* = 0.78; for N~ = 6: a* = 0.82; and for N~ = 8: a* = 0.86. These results apply to N L ---4; however, increasing N L to 8 does not systematically improve a*. F o r larger NL, finite size effects cause levels of different s y m m e t r y character to cross as a varies, and this is a problem for the R G method. F o r the case of N~ odd, there is an important s y m m e t r y relation between the two alternation exchange constants J a n d a J , which fixes an unstable F P at 1. F o r even Ns this alternation s y m m e t r y is inevitably lost. F o r the U I H , F P ' s are f o u n d at 3'* = 0 (Ising, stable), y* = 0o (XY, stable) and 3'* = 1 (Heisenberg, unstable). This scheme is in agreement with exact results [6] which say that the gap vanishes for ), = 1 only. As an example of critical singularities, we consider the behavior of the A H energy gap near 2
a = 1. The C r o s s - F i s h e r theory [4] gives E 0 ~ 8 7, where the dimerization parameter 8 = ( 1 - a ) / (I + a). The numerical extrapolations indicate an exponent less than one, in qualitative agreement. The N s o d d R G calculations give E 0 ~ 8 0.76 a n d the N~ even R G calculations give E 0 ~ 8 °"64. H e n c e these methods agree satisfactorily. F o r the X Y
model, however, E 0 ~ 8 (exact), whereas the N s odd R G method gives 8 0"72, a more serious discrepancy. For both A H (strong evidence) a n d A X Y (rigorously), the energy gap vanishes as a power of 8, whereas we k n o w exactly that the U I H g a p v a n i s h e s e x p o n e n t i a l l y slowly ( A E exp( {-- A / ~/1 - ~, )). The R G m e t h o d applied to the U I H energy gap near ~, = 1 gives the result that the gap vanishes as a power law (with power 1.59, N s odd, N L ---- 2). Regarding global properties, our R G m e t h o d is not always accurate. F o r the g r o u n d state energy neither N s odd nor even gives accurate results. However, extrapolation of the sequence N s = 3, 5, 7, 9 gives m a r k e d i m p r o v e m e n t [3]. The N s odd R G calculations, which give the correct value for the unstable c~*, are p o o r for the energy gap near the dimer limit. This is attributable to the 2-level truncation, which does not utilize the singlet-triplet character of the low-lying levels. The N L -- 4 scheme, which does, is rather accurate near the dimer limit (but ct* ~ l) [3]. For the X Y model, an N s even, 3-level (singlet-doublet) truncation scheme is rather accurate near the dimer limit but, again, c~* ~ 1. F o r the U I H energy gap, a 2-level truncation with N s = 3 fails to p r o d u c e a term linear in ), near the Ising limit. In conclusion, it appears that this R G approach, which describes the simple q u a n t u m model of a I-D transverse Ising model rather successfully [7], has some quantitative limitations when applied to more complex l-D q u a n t u m models. Attempts at improvement will be reported elsewhere.
References [1] Th. Niemeijer and J. M. J. van Leeuwen, Physica 71 (1974) 17. [2] J. N. Fields, Phys. Rev. B 19 (1979) 2637. [3] J. N. Fields, H. W. J. Blote and J. C. Bonner, J. Appl. Phys. 50 (1979) 1807. [4] M. E. Cross and D. S. Fisher, Phys. Rev. B 19 (1979) 402. [5] J. C. Bonnet et al., J. de Phys. 39 (1978) C6-710; J. C. Bonner and H. W. J. Blote, unpublished work. [6] J. D. Johnson and B. M. McCoy, Phys. Rev. A 6 (1972) 1613. [7] R. Jullien et al., Phys. Rev. B 18 (1978) 3568.
and J. N. F I E L D S Hughes Research Corp., Malibu, CA 90265, USA
A zero-temperature quantum renormalization group which works rather well for simple 1-D quantum models is applied to more complex quantum models. These include the alternating Heisenberg antiferromagnetic chain and the spin anisotropic IsingHeisenberg chain. Results indicate qualitative success.
1. Method This method uses real space renormalization group (RG) transformations reminiscent of those used for classical systems [1]. The lattice is subdivided into blocks of N s sites such that the eigenvalues and eigenvectors of each block may be calculated exactly. The basis of each block (2 Ns levels) is truncated to some n u m b e r N L of levels, and the coupling between adjacent blocks is written within the truncated basis. Since this is a T = 0 approach, the N L levels include the ground state and a dominant set of first excited states. Care is needed to preserve the symmetry character of the N L levels at each step of the iterative process. After each R G step ~ is written in the same form, which leads to recursion relations which express the parameters of the renormalized 0C as a function of those of the original unrenormalized one. Choosing the two lowest states of an odd Ns block, for example, maps the block onto a single spin $1 on the new lattice. An R G transformation for the antiferromagnetic Heisenberg chain using N L = 4 and even N s effectively maps the block into an S=$ dimer (spin-pair). For Ns, N L small, an analytic calculation is possible. A set of recursion relations which define an R G transformation for the ground state of the Hamiltonian can be written down explicitly. Fixed points of the recursion relations and corresponding T = 0 critical singularities are obtained. Some illustrative calculational details have already been published [2, 3]. In principle, the accuracy of the results should be improved by systematically increasing both N~ and N L. An entirely computer based approach is now required, *Work supported in part (at URI) by NSF grant DMR77-24136 and Bunting Institute Fellowship, Radcliffe College.
however, because of the size of the matrix operators. I Models considered are the alternating S - 2 Heisenberg antiferromagnetic linear chain (AH): N/2 ~)C = 2 J ~ { SEi_ l °SEi -.~ aS2i.S2i +l ) i=l
(1)
and the alternating XY counterpart (AXY). The alternation parameter a varies between 1 (uniform limit) and 0 (dimer limit). We have also considered the uniform antiferromagnetic Ising-Heisenberg linear chain ( U I H ) with S = ~I : N x 3C = 2J'5~, ( S i z S i +• , + _y~,~ S xi S i+l-t"
SYS/Y+I)}
i=l
(2) where the anisotropy parameter y varies between I (isotropic limit) and 0 (Ising limit).
2. Results R G calculations have been performed for N L = 2 and N - - 3 , 5, 7, 9 for both the A H and AXY. Calculations have also been performed for N s = 2, 4, 6, 8 and N L ) 4. For the U I H we have examined the cases Ns = 3, N L = 2 and N~ = N L - 4. The accuracy and reliability of this R G approach have been variously assessed. The A X Y and U I H models are exactly solvable, unlike the A H where comparison is made with a Luttinger model calculation [4] and numerical extrapolations on finite chains [5]. We focus our assessment on three areas: (a) location and nature of the fixed points; (b) values
Journal of Magnetism and Magnetic Materials 15-18 (1980) 405-406 ©North Holland
405
H. lee". J. Blote et a l . / T = 0 quantum renormalization group
of the critical singularities; and (c) quantitative a c c u r a c y of the g r o u n d state energy per spin E 0, and the energy gap AE between g r o u n d and first excited states. We observe at the outset that the m e t h o d has some qualitative successes. F o r A X Y and A H a stable fixed point (FP) is f o u n d at c~* = 0 (also ct* = oo), and an unstable F P at a* = l, for Ns odd systems. This has the important physical implication that the system with non-zero alternation i.e. ct = 1, will renormalize into a dimer system, characterized by a singlet-triplet energy gap which vanishes only in the uniform limit, a = 1. This agrees with exact results for the A X Y , and gives a powerful a r g u m e n t for the presence of a similar alternation gap in the Heisenberg counterpart [2, 3]. W h e n N~ even systems are considered, however, there is a minor problem. Stable F P ' s are again f o u n d at a * = 0 ( ~ ) , but the unstable F P is no longer exactly at unity. F o r N s = 2, 4: t~* = 0.962; for N s = 6: a * = 1.001; a n d for N s = 8: a * = 1.021. F o r the X Y model the results are rather worse. For N = 2, 4: a* = 0.78; for N~ = 6: a* = 0.82; and for N~ = 8: a* = 0.86. These results apply to N L ---4; however, increasing N L to 8 does not systematically improve a*. F o r larger NL, finite size effects cause levels of different s y m m e t r y character to cross as a varies, and this is a problem for the R G method. F o r the case of N~ odd, there is an important s y m m e t r y relation between the two alternation exchange constants J a n d a J , which fixes an unstable F P at 1. F o r even Ns this alternation s y m m e t r y is inevitably lost. F o r the U I H , F P ' s are f o u n d at 3'* = 0 (Ising, stable), y* = 0o (XY, stable) and 3'* = 1 (Heisenberg, unstable). This scheme is in agreement with exact results [6] which say that the gap vanishes for ), = 1 only. As an example of critical singularities, we consider the behavior of the A H energy gap near 2
a = 1. The C r o s s - F i s h e r theory [4] gives E 0 ~ 8 7, where the dimerization parameter 8 = ( 1 - a ) / (I + a). The numerical extrapolations indicate an exponent less than one, in qualitative agreement. The N s o d d R G calculations give E 0 ~ 8 0.76 a n d the N~ even R G calculations give E 0 ~ 8 °"64. H e n c e these methods agree satisfactorily. F o r the X Y
model, however, E 0 ~ 8 (exact), whereas the N s odd R G method gives 8 0"72, a more serious discrepancy. For both A H (strong evidence) a n d A X Y (rigorously), the energy gap vanishes as a power of 8, whereas we k n o w exactly that the U I H g a p v a n i s h e s e x p o n e n t i a l l y slowly ( A E exp( {-- A / ~/1 - ~, )). The R G m e t h o d applied to the U I H energy gap near ~, = 1 gives the result that the gap vanishes as a power law (with power 1.59, N s odd, N L ---- 2). Regarding global properties, our R G m e t h o d is not always accurate. F o r the g r o u n d state energy neither N s odd nor even gives accurate results. However, extrapolation of the sequence N s = 3, 5, 7, 9 gives m a r k e d i m p r o v e m e n t [3]. The N s odd R G calculations, which give the correct value for the unstable c~*, are p o o r for the energy gap near the dimer limit. This is attributable to the 2-level truncation, which does not utilize the singlet-triplet character of the low-lying levels. The N L -- 4 scheme, which does, is rather accurate near the dimer limit (but ct* ~ l) [3]. For the X Y model, an N s even, 3-level (singlet-doublet) truncation scheme is rather accurate near the dimer limit but, again, c~* ~ 1. F o r the U I H energy gap, a 2-level truncation with N s = 3 fails to p r o d u c e a term linear in ), near the Ising limit. In conclusion, it appears that this R G approach, which describes the simple q u a n t u m model of a I-D transverse Ising model rather successfully [7], has some quantitative limitations when applied to more complex l-D q u a n t u m models. Attempts at improvement will be reported elsewhere.
References [1] Th. Niemeijer and J. M. J. van Leeuwen, Physica 71 (1974) 17. [2] J. N. Fields, Phys. Rev. B 19 (1979) 2637. [3] J. N. Fields, H. W. J. Blote and J. C. Bonner, J. Appl. Phys. 50 (1979) 1807. [4] M. E. Cross and D. S. Fisher, Phys. Rev. B 19 (1979) 402. [5] J. C. Bonnet et al., J. de Phys. 39 (1978) C6-710; J. C. Bonner and H. W. J. Blote, unpublished work. [6] J. D. Johnson and B. M. McCoy, Phys. Rev. A 6 (1972) 1613. [7] R. Jullien et al., Phys. Rev. B 18 (1978) 3568.