Classical Ising chain in transverse field

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 310 (2007) e477–e479 www.elsevier.com/locate/jmmm

Classical Ising chain in transverse field A. Cuccolia,, A. Taitia, R. Vaiab, P. Verrucchib a

Dipartimento di Fisica, Universita` di Firenze, Via G. Sansone, 1 - I-50019 Sesto Fiorentino (Fi), Italy Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Via Madonna del Piano, 10 - I-50019 Sesto Fiorentino (Fi), Italy

b

Available online 7 November 2006

Abstract The spin 12 Ising chain in transverse field is considered the prototypical system for quantum phase transitions. However, very little is apparently known in literature about its classical counterpart, not to be confused with the standard classical Ising model: while the latter is constructed from classical discrete variables, the model we consider is a chain of classical vectors of modulus 1, interacting via an Isinglike Hamiltonian. When an uniform field is applied perpendicular to the exchange interaction, both the quantum model and its classical counterpart get to be characterized by a critical field separating a ferromagnetically ordered state of minimal energy from a paramagnetic one. The properties of the classical model, and especially the behaviour of the correlation length, are investigated at low temperature around the critical field and compared with those of the quantum model, in order to single out the role played by quantum and classical fluctuations at finite temperature; the possibility to experimentally observe peculiar quantum critical effects in Ising spin chains is discussed. r 2006 Elsevier B.V. All rights reserved. PACS: 75.10.Pq; 75.10.Hk; 75.10.Jm; 05.70.Jk Keywords: Quantum phase transitions; Classical ising chain; Ising model in transverse field

The genuine phase transitions displayed by some physical model systems (e.g., spin-chains) at T ¼ 0 against the variation of external parameters (e.g., magnetic field) are commonly known as quantum phase transitions (QPT), as they are a direct consequence of the sole zero-point quantum fluctuations: the S ¼ 12 quantum Ising chain in transverse field (QIC), being the simplest, exactly solvable [1], system displaying QPT, is considered the prototypical example [2]. However, experiments are necessarily performed at finite temperature, so the properties of a given QPT can be experimentally addressed only after a careful analysis of the finite temperature behaviour of observable quantities possibly affected by the presence of the T ¼ 0 quantum critical point. In order to understand the relative relevance of thermal (classical) and quantum fluctuations, and ascertain the claimed essential role played by the latter, a good knowlCorresponding author. Tel.: +39 0554572045; fax: +39 0554572121.

E-mail address: cuccoli@fi.infn.it (A. Cuccoli). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.10.422

edge of the expected behaviour of the quantum system at finite temperature is not sufficient: also the classical behaviour of the model must be investigated and compared with the quantum one. It is thus somewhat surprising that very little is apparently known in literature [3] about the properties of the classical Ising chain in transverse field (CIC) described by the Hamiltonian: X X H ¼ Sxi S xiþ1  2h S zi , J i i

(1)

are the components of a classical vector where Sx;y;z i y2 z2 (classical spin) sitting on site i with S x2 i þ S i þ S i ¼ 1, i.e. y x z Si ¼ sin yi cos ji , Si ¼ sin yi sin ji and Si ¼ cos yi , with yi 2 ½0; p and ji 2 ½0; 2p ; h is the applied magnetic field in reduced units and J!is the exchange interaction between neighbouring spins S . Even if no fluctuations are present in a classical system at T ¼ 0, the CIC shares with the QIC the presence of a critical value hc ¼ 1 of the applied field h separating an ordered, symmetry broken, minimum-energy configuration

ARTICLE IN PRESS A. Cuccoli et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e477–e479

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with hSxi ia0 for hohc , from a disordered, paramagnetic one, with hS xi i ¼ 0 for h4hc . By using the numerical transfer-matrix (TM) technique, we have investigated different static thermodynamic quantities of the CIC, focusing our attention on their behaviour in the neighbourhood of the critical point h ¼ hc , T ¼ 0. In the following we will mainly concentrate on the correlation length x of the Ising order parameter S x , but the crossover structure between different regimes in the phase-diagram clearly emerges also from other quantities as the specific heat or the longitudinal and transverse susceptibility [4,5]. In Fig. 1 the TM results for the correlation length x are reported as a function of t  kB T=J in the region jh  hc j5t (a) and as a function of ðh  hc Þ for ðh  hc Þbt (b), showing a power-law behaviour with respect to the chosen independent variable; for a given hohc , x diverges exponentially in 1=T as T ! 0, as in the standard Ising model when the T ¼ 0 ordered state is approached.

20 11 9 7

ξx

5 3

1

0.4 0.005

0.01

0.1

1

t

(a)

1.6 1.5 1.4

ξx

1.3 1.2 1.1

1 0.4 (b)

0.5

0.6 2(h-hc)

0.7

0.8

Fig. 2. Low-T phase diagram of the classical Ising model in transverse field. Shaded regions refer to the three lines of Eq. (2).

Also in the CIC three different regimes can thus be singled out, where the numerical TM data for the correlation length can be well fitted by the following functional forms: 8 xt1=3 for jh  hc j5t; > > > < xðh  hc Þ1=2 for ðh  hc Þbt;   (2) > ðhc  hÞ > >  hÞbt; x exp a for ðh c : t where a is a constant. The qualitative behaviour of x in the CIC closely resembles that obtained in the corresponding quantum model: we only observe different values of the exponents describing the power law behaviour of x as a function of t ( 13 in CIC, 1 in QIC) and h  hc ( 12 in CIC, 1 in QIC) in the critical and paramagnetic region, respectively. The low-T (T5J) phase-diagram of the CIC emerging from our results is finally reported in Fig. 2: its structure is very similar to that observed in the QIC, with crossover lines at t  jh  hc j separating a central critical regime (jh  hc j5t) from long range ordering (hc  hbt) and paramagnetic (h  hc bt) ones. In conclusion, the finite-T phase-diagrams of the classical and quantum Ising models in transverse field are essentially the same, with only quantitative differences; in particular, also in the CIC a central critical regime, controlled by the T ¼ 0 critical point, is present where the correlation length x is almost independent of h and has a power-law behaviour as a function of T: what discriminates between the classical and quantum critical regime (QCR) is thus just the value of the exponent, and this may represent a challenge for getting clear experimental evidences of QCR and the underlying QPT properties [6–9].

0.9

Fig. 1. Correlation length x of spin fluctuations along the ordering direction x in lattice constant units. In (a) results for jh  hc j5t are reported for 0:98php1:02, with h increasing by 0:005 moving downwards from the upper line; in (b) results for ðh  hc Þbt are shown for t ¼ 0:005; 0:01; 0:015; 0:02, respectively, starting from the upper line.

References [1] S. Katsura, Phys. Rev. 127 (1962) 1508; P. Pfeuty, Ann. Phys. 57 (1970) 79. [2] S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, UK, 1999.

ARTICLE IN PRESS A. Cuccoli et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e477–e479 [3] C.J. Thompson, J. Math. Phys. 9 (1968) 241; T. Horiguchi, J. Phys. Soc. Japan 59 (1990) 3142; K. Minami, J. Phys. A 29 (1996) 6395; K. Minami, J. Phys. Soc. Japan 67 (1998) 2255. [4] A. Taiti, Thesis, Universita` di Firenze, 2006. [5] A. Cuccoli, A. Taiti, R. Vaia, P. Verrucchi, in preparation.

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[6] D. Bitko, T.F. Rosenbaum, G. Aeppli, Phys. Rev. Lett. 77 (1996) 940. [7] H.M. Ronnow, D.F. McMorrow, A. Harrison, Phys. Rev. Lett. 82 (1999) 3152. [8] P. Carretta, et al., Phys. Rev. Lett. 84 (2000) 366. [9] H.M. Ronnow, et al., Science 308 (2005) 389.