A one-dimensional Hubbard model with nearest and (negative) next-nearest
neighbour hopping is studied variationally. This allows to exclude saturated
ferromagnetism for $U < U_c$. The variational boundary $U_c (n)$ has a minimum
at a ``critical density'' $n_c$ and diverges for $n \rightarrow 1$.Comment: 5 pages, LateX and 1 postscript figure. To appear in Physica 

D. Baeriswyl

Edwards

Lieb

M. Dzierzawa

Mattis

Mielke

Müller-Hartmann

Nagaoka

P. Fazekas

P. Pieri

Pieri

Preuss

S. Daul

Shastry

Tasaki

Ueda

von der Linden

Wurth

English

arXiv

Crossref

A one-dimensional Hubbard model with nearest and (negative) next-nearest neighbour hopping is studied variationally. This allows to exclude saturated ferromagnetism for U  1

S. DAUL

D. BAERISWYL

M. DZIERZAWA

P. FAZEKAS

PIERI, Pierbiagio

Archivio istituzionale della ricerca - Università di Camerino

ELSEVIER Physica B 230-232 (1997) 1021 1023 Variational study of ferromagnetism in the tl-t2 Hubbard chain S. Daul  a'*, P. Pieri b, M. Dzierzawa a, D. Baeriswyl a, P. Fazekas c alnstitut de Physique Th~orique, Universitk, P~rolles, CH-1700 Fribourg, Switzerland bDipartimento di Fisica, INFM and INFN, Universith di Bologna, Via Irnerio 46, 1-40126, Bologna, Italy CResearch Institute for Solid State Physics, Budapest 114, POB 49, H-1525 Hungary Abstract A one-dimensional Hubbard model with nearest and (negative) next-nearest neighbour hopping is studied variation- ally. This allows to exclude saturated ferromagnetism for U < U~. The variational boundary U~(n) has a minimum at a "critical density" n~ and diverges for n ---, 1. Keywords: Ferromagnetism; Hubbard model; Variational wave function The old question of the existence of ferromag- netism in the single-band Hubbard model has found a renewed interest, when rigorous proofs for a large-spin ground state were discovered for cer- tain special cases [1-4]. While it is in general very hard to show that the ground state is ferromag- netic, it is often rather easy to exclude saturated ferromagnetism, because the lowest-energy fully polarized N-electron state IF)  is trivially construc- ted by occupying each of the N lowest single-par- ticle levels by a spin-up electron. To exclude saturated ferromagnetism one has then simply to find another state of lower energy. This approach has been applied to the Nagaoka problem [5] in terms of variational N-electron wave functions with a single overturned spin [6-8]. We have recently used such an approach to show that for non-patho- logical band structures and dimensions d > 2 the ground state is neither fully nor partially polarized in the low-density limit [9]. On the other hand, a low-density route has been proposed for d = 1 by *Corresponding author. Mfiller-Hartmann [10]. It is this case we want to discuss in the present contribution. We consider the t l -  t2 Hubbard chain described by the Hamil- tonian L H = - y~ ( t , d ~ c , + ~ o  + t2c*~c~+2~ + h.c.) l= l,a-'[,l  L + g ~ nl~nl+. (1) l=1 For  t 2 = 0 we recover the simple Hubbard model with only nearest-neighbour hopping, for which ferromagnetism has been discarded a long time ago by Lieb and Mattis [11]. For  t 2 ~ 0 particles can pass each other and do not preserve a given order. The Lieb-Mattis theorem is then no longer applic- able and indeed, for U = ov and tz < 0 ferromag- netism appears both for one hole in a half-filled band [12] and in the limit tz ~ 0  for arbitrary densities (n < 1) [13]. These results raise the ques- tion about the range of parameters where the ground state is ferromagnetic. Before addressing this problem we turn to the easier question for 0921-4526/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S092  1 - 4 5 2 6 ( 9 6 ) 0 0 7 9 4 - 6  1022 S. Daul et al. / Physica B 230-232 (1997) 1021-1023 which parameter values the fully polarized state is unstable. The fully polarized state IF> is certainly unstable if the energy is reduced by flipping a single spin. We use two different trial states. The first ansatz [6], defined by L I ~ >  = I-[ (1 - qnt,nJctgoCk~tlF > (2) l=1  removes an up spin from the highest occupied level and puts a down spin to the minimum of the single- particle spectrum. It is important to note that this minimum moves away from k = 0 for large nega- tive values of t2, t2 < -  I tll/4, namely to ko = + arccos(- t l /4 t2) .  It is precisely in this re- gion where according to Miiller-Hartmann ferro- magnetism should appear at low density and infinite U [10]. The variational parameter t/ in Eq. (2) reduces the double occupancy and steadily increases from 0 to 1 as a function of U. The energy change due to the spin flip consists essentially of a gain in band energy and a loss due to the Coulomb repulsion [9]. Thus, one expects the fully polarized state to be unstable for small U. This is confirmed by explicit calculations shown in Fig. 1. As a second trial state we use the wave function introduced by Edwards [14]: 1 ~ N - I  IZ> = ~  t= ,  eiqlcltl a=lI~ c~,(1)10>, (3) where L ctT(l) = Z q~(m-  l)c~ T (4) m = l  creates an up-spin electron in an orbital that is determined variationally. The variational para- meters are the wave vector q and the ( N -  1)L amplitudes q~(l). The ansatz (3) is exact for t2 = 0 [14]. For t2 # 0 we add the Gutzwiller projection operator, which amounts to replace the one-par- ticle orbitals q~(n) by ( 1 -  t/f,.o)~p~(n). Then the wave function (3) includes the ansatz (2) as a special case. The variational energy for orthonormal one- 1 6 -  i i i i 1 2 -  8 4 0 0.0 0.2 0.4 0.6 0.8 n Fig. 1. Variational phase diagram of the tx-t2 Hubbard chain in the n-U plane, for t2 = -  0.8[q[. Full line: ansatz (2) for L = 1000, dots: ansatz (3) for L = 100. particle orbitals reads g(q, {~p~(1)}) = 2q cos(q) det S ") + 2t2 cos(2q) det S(2) + 2tl tr S t~) + 2t2 tr S (2) + U y~ ~o*(O)~op(O) (5) with overlap matrices S~(~ defined as L S(i) ,# = ~ qg*(l)q~#(l + i). (6) l = l  We have used the conjugate gradient method [15] to minimize the energy. The derivatives have been calculated analytically, and after each iteration the orbitals have been orthonormalized using the modified Gram-Schmidt method. As initial or- bitals, which have to be close enough to the final wave functions q~(l), we use the N -  1 lowest eigenfunctions for a single-site impurity. With this second ansatz, which involves much more varia- tional freedom than the first, the instability line is moved to higher values of U, as shown in Fig. 1. It is worthwhile to add that a still more refined varia- tional wave function [8] yields a similar instability line [16]. Our variational calculations delimitate a low U region where the ground state is certainly not fully polarized. For the present model and large enough negative values of t2 there is still a large 5'. Daul et al. / Physica B 230-232 (1997) 1021 1023 1023 region left where saturated ferromagnetism cannot be excluded, as illustrated in Fig. 1 for the particu- lar value t2 = -0.81t11. In order to find out whether these variational lower bounds for the critical value Uc are likely to be representative for the exact boundary, we have carried out exact diag- onalizations for L ~< 20. The numerical results show that for densities around 0.4 and t 2 ~ - -0 .51 t1[  the exact boundary lies typically about 20-50% above the variational bound of the second ansatz, Eq. (3). An interesting question is what happens above the "critical density" nc where the Fermi surface is reduced from four to two points. Both instability lines shown in Fig. 1 have a minimum at this density. In addition, for the Edwards ansatz the optimal wave vector q is found to jump from q = ko (the location of the minimum in the band structure) for n < n~ to q = n for n > n~. In our previous exact diagonalization studies (for an even number of particles and periodic boundary conditions) we tentatively associated n¢ with the phase boundary for ferromagnetism. This would indicate that the low-density route to ferromag- netism is not directly connected to the Nagaoka problem. In the mean time we have extended our numerical studies both by imposing other bound- ary conditions and by increasing the chain length, using the density-matrix renormalization group method. The results obtained so far do not repro- duce the phase boundary at no. Instead, at least for U ~ ~ and not too large It2 l, ferromagnetism seems to exist for all densities n < 1. Further calcu- lations will be needed to establish the full phase diagram. Financial support by the Swiss National Foundation through the grant No. 20-40672.94 is gratefully acknowledged. We also thank E. Miiller-Hartmann, K. Penc, K. Ueda and M. Sigr- ist for useful discussions and specially H. Castella for very helpful correspondence on technical de- tails. References [-1] E. Lieb, Phys. Rev. Lett. 62 (1989) 1201. [2] A. Mielke, Phys. Lett. A 174 (1993) 443. [3] H. Tasaki, Phys. Rev. Lett. 69 (1992) 1608; A. Mielke and H. Tasaki, Commun. Math. Phys. 158 (1993) 341. [4] H. Tasaki, Phys. Rev. Lett. 75 (1995) 4678. [5] Y. Nagaoka, Phys. Rev. 147 (1966) 392. [6] B.S. Shastry, H.R. Krishnamurthy and P.W. Anderson, Phys. Rev. B 41 (1990) 2375. [7] W. yon der Linden and D.M. Edwards, J. Phys.: Condens. Matter 3 (1991) 4917. [8] P. Wurth, G. Uhrig and E. Miiller-Hartmann, Ann. Physik 5 (1996) 148. [9] P. Pieri, S. Daul, D. Baeriswyl, M. Dzierzawa and P. Fazekas, Phys. Rev. B 54 (1996) 9250. [10] E. Miiller-Hartmann, J. Low Temp. Phys. 99 (1995) 349. [11] E. Lieb and D.C. Mattis, Phys. Rev. 125 (1962) 164. [12] D.C. Mattis and R.E. Pefia, Phys. Rev. B 10 (1974) 1006. [13] K. Ueda, T. Nishino and H. Tsunetsugu, Phys. Rev. B 50 (1994) 612. [14] D.M. Edwards, Prog. Theor. Phys. Suppl. 101 (1990) 453. [15] W.H. Preuss, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes (Cambridge University Press, Cambridge, 1995). [16] E. Miiller-Hartmann, private communication. 

Variational study of ferromagnetism in the $t_1$-$t_2$ Hubbard chain

Daul, S.

Pieri, P.

Dzierzawa, Michael

Baeriswyl, D.

Fazekas, P.

OPUS Augsburg

Variational study of ferromagnetism in the t1-t2 Hubbard chain