Quantum critical phenomena in the Schrodinger formulation: mapping to classical lattices

Transcription

1 6 January 001 Chemial Physis Letters 333 (001) 451±458 Quantum ritial phenomena in the Shrodinger formulation: mapping to lassial latties Riardo A. Sauerwein 1, Sabre Kais * Department of Chemistry, Purdue University, West Lafayette, IN 47907, USA Reeived 9 Otober 000; in nal form 8 November 000 Abstrat We present a path integral formalism to obtain the statistial mehanis transfer matrix of the system whih leads to a straightforward mapping between the quantum problem and an e etive lassial lattie system. Thus, the quantum ritiality of atomi and moleular systems an be studied by the standard statistial mehanis methods. We illustrate this general tehnique by presenting detailed alulations for the sreened Coulomb potential. Ó 001 Elsevier Siene B.V. All rights reserved. 1. Introdution Phase transitions and ritial phenomena ontinue to be a subjet of great interest in many elds [1]. A wide variety of physial systems exhibit phase transitions and ritial phenomena, suh as liquid±gas, ferromagneti±paramagneti, uid-super uid, ondutor±superondutor and the list ontinues to expand []. Over the last few deades, a large body of researh has been done on this subjet, mainly using lassial statistial mehanis. However, quantum phase transitions have attrated muh interest in reent years. These transitions are zero temperature transitions tuned by parameters in the Hamiltonian [3,4]. Examples from ondensed matter physis inlude the magneti transitions of uprates, superondutor± * Corresponding author. address: kais@power1.hem.purdue.edu (S. Kais). 1 On leave from Universidade Federal de Santa Maria, Santa Maria, RS, Brazil. insulator transitions in alloys, metal±insulator transitions and the Quantum±Hall transitions [4,5]. In the eld of atomi and moleular physis, the analogy between symmetry breaking of eletroni struture on gurations and quantum phase transitions has been established at the large dimensional limit [6]. The mapping between symmetry breaking and mean- eld theory of phase transitions was shown by allowing the nulear harge Z, the parameter whih tunes the phase transition, to play a role analogous to temperature in lassial statistial mehanis [7]. The study of quantum phase transitions and ritial phenomena ontinues to be of inreasing interest in the eld of atomi and moleular physis. This is motivated by the reent theoretial and experimental searhes for the smallest stable multiply harged anions [8,9], experimental and theoretial work on the stability of atoms and moleules in external eletri and magneti elds [10,11], design and ontrol eletroni properties of materials using arti ial atoms [1], the study of seletively breaking /01/$ - see front matter Ó 001 Elsevier Siene B.V. All rights reserved. PII: S ( 0 0 )

2 45 R.A. Sauerwein, S. Kais / Chemial Physis Letters 333 (001) 451±458 hemial bonds in polyatomi moleules [10] and phase transitions of nite lusters [13]. To treat quantum phase transitions and ritial phenomena, it seems that Feynman's path integral is the natural hoie. In this approah, one an show that the quantum partition funtion of the system in d dimensions looks like a lassial partition funtion of a system in d 1 dimensions where the extra dimension is the time [3]. Upon doing so, and allowing the spae and time variables to have disrete values, we turn the quantum problem into an e etive lassial lattie problem. Quantum phase transitions our at zero temperature, T ˆ 0. At this limit the free energy beomes the ground state energy and the various thermal averages beome the ground state expetation values. With the partition funtion, we an use standard statistial mehanis methods, suh as the nite size saling, to get all the quantities of interest for investigating quantum phase transitions. In the following setions, we will show how to arry out the mapping between the quantum problem and an e etive lassial spae-time lattie, give an example to illustrate how alulate the di erent ritial parameters, and then give disussion and onlusions.. Mapping quantum problems to lattie systems In the path integral approah, the probability amplitude K q 0 ; t 0 ; q; t of a partile initially loalized at position q and time t to be found at position q 0 at time t 0 is given by K q 0 ; t 0 ; q; t ˆ Z q t 0 ˆq 0 q t ˆq Dqe i=h S q t Š ; 1 where the symbol R q t 0 ˆq 0 Dq stands for integration q t ˆq over all trajetories onneting the spae±time point q; t to q 0 ; t 0,andS q t Š is the lassial ation for a given trajetory q ˆ q t [14,15]. The analytial ontinuation of the probability amplitude to imaginary time t ˆ is of losed trajetories, q t ˆq t 0, is formally equivalent to the quantum partition funtion Z b, with the inverse temperature b ˆ i t 0 t =h [16]. In pratial alulations it is ommon to use the path integral lattie de nition. In this disrete time approah, the quantum partition funtion reads Z b ˆ lim Ds! 0 exp k XNs 1 m Z Ns= Y N s pdsh `ˆ0 " Ds XN s 1 h `ˆ0 `ˆ0 V q` 1 V q` dq` m q` 1 q` Ds!# ; where k is the strength of the potential V q ; Ds ˆ b=n s is the regular grid spaing between N s points along the imaginary time axis indexed by ` ˆ 0; 1;...; N s. The losed path is made by a periodi boundary ondition in the time diretion suh that q Ns ˆ q 0. Sine we are onsidering a quantum phase transition for a system at its ground state energy, we must onsider only the ase of b!1. Now we introdue the disretization also in the spae diretion. Thus, the totally disrete model an be used as a new sheme to study quantum partiles in the presene of a potential with a point singularity. This kind of potential annot be desribed by the path integral formalism with a nite Ds within the so-alled primitive approximation [17]. Besides the disretization, the present sheme depends upon the thermodynami and ontinuum limits whih we will disuss later. This way the position in time slie ` is given by q i` ˆ q 0 i`dl with i` ˆ 1; ;...; N q ; 3 where DL is the regular grid spaing of the position axis whih has a total of N q points and q 0 is a onstant used to adjust the origin of the oordinate system. Moreover, the size of the spae is limited by L ˆ N q DL. Now we onentrate on the properties of the two-dimensional lattie, the spae±time lattie. The partition funtion, Eq. (), shows that there is oupling only in the time diretion and only between nearest neighbor time slies. This allows us to use the statistial mehanis tehnique of writing the partition funtion Z of the nite system as the trae of a matrix T to the power N s Z DL; Ds Nq;N s ˆ Tr T Ns : 4

3 R.A. Sauerwein, S. Kais / Chemial Physis Letters 333 (001) 451± The matrix T is alled the transfer matrix. Its elements are given by m 1= T i`; i` 1 ˆDL phds ( " exp Ds m DL i` 1 i` h Ds #) k V i` 1 V i` ; 5 where V i` ˆ V q` is the potential energy of time slie ` evaluated at the spae point i`. The above transfer matrix an be seen as a transfer matrix of a lassial pseudo-system. The Hamiltonian of the pseudo-system reads H ˆ XNs where `ˆ1 H`;` 1 ˆ m H`;` 1 ; DL Ds i` 1 i` k V i` 1 V i` C 6 7 with C ˆ h lnf m= phds Š 1= DLg=Ds being a onstant independent of the state of the lattie. Then, the partition funtion of the lassial pseudo-system beomes T i`; i` 1 ˆexp Ds=h H`;` 1 Š: 8 If the lassial pseudo-system Hamiltonian were independent of Ds, the lassial pseudo-system would behave as a statistial mehanis lattie system with inverse temperature Ds=h. In order to omplete the mapping between the quantum problem and the lassial pseudo-system, one must address the problems of both the ontinuum and the in nite limits. The ground state properties of the original system are obtained by taking both the ontinuum limit, Ds; DL! 0, and the thermodynami limit, b; L!1. We assume that in priniple this limit an be performed in two steps. First, we take the thermodynami limit N s ; N q!1, and seond, we take the ontinuum limit Ds; DL! 0. From the de nition of the nite transfer matrix, (5), we see that both Ds and DL always appear inside the fator mdl =Dsh. So it is lear that we annot hoose DL and Ds independently, otherwise we ould end up with an unde ned transfer matrix when the ontinuum limit is taken. This problem an be overome by taking mdl Dsh ˆ 1 ) DL mdl ˆ h: Ds 9 The above expression an be seen as a quantization of the ation in the semi-lassial theory. Also, this same dependeny Ds ˆ DL is imposed in order to get the di usion equation when taking the ontinuum limit of the Brownian motion of partiles in a lattie []. The number of points along the imaginary time N s does not appear expliitly in any relevant quantity. The requirement N s!1is neessary in order to have a nite partition funtion, Eq. (4), dominated by the leading eigenvalue of the transfer matrix a 0 DL; Ds; N q. We an see how N s is aneled when evaluating the ground state energy of the system E 0. In the lattie system, E 0 is the free energy per time slie whih is related to the partition funtion and onsequently to a 0 DL; Ds; N q by exp N s DsE 0 ˆZ DL;Ds Nq;Ns a 0 DL; Ds;N q Ns. Sine N s appears in both ends of this equation the energy is given by E 0ˆ ln a 0 DL;Ds;N q =Ds. Thus the disretization number whih atually sets the size of the nite system is N q. Setting N qˆn and onsidering impliitly that N s goes also to in nity when taking the thermodynami limit, the original partition funtion for b!1 is reovered by taking the limit Z b ˆ lim DL! e DsˆDL h i lim Z DL; Ds N!1 N ; 10 where is an arbitrarily small number. The inner limit of the above expression N!1 represents the `thermodynami limit' for our system. In previous studies [7] of quantum phase transitions, it was proposed that the role of the `thermodynami limit' be played by the number of terms in the trunated basis set used to expand the exat wave funtion. The thermodynami limit in the present study seems to be more natural, sine it is given by an extensive parameter of the

4 454 R.A. Sauerwein, S. Kais / Chemial Physis Letters 333 (001) 451±458 pseudo-system; however, we believe that the two de nitions are essentially the same. Coalson [18] showed the onnetion between Fourier oe ients and the disrete path integration. The number of points N of the disrete path integral is related to the number of plane waves, i.e., the number of terms of a trunated basis set used to represent the states of the quantum system. The phase transition is driven by the ontinuous parameter k. There are two possible phases. These orresponding to the presene or absene of a stable bound ground state energy of the quantum system. In the present formalism, these two phases are identi ed by loalized or deloalized partiles. As in ordinary statistial mehanis lattie systems, the transition point an be identi ed as the divergene or disontinuity point in the orrelation length. The two point orrelation funtion an be de ned as C k ` ˆhi 0 i`i hi 0 i ; 11 where the symbol h i stands for the statistial mehanis ensemble or `thermal' average. Its asymptoti behavior for large systems is given by C k ` e `Ds=n k ; 1 where n k is the orrelation length. The physial meaning of this quantity is the distane along the imaginary time within whih the time slies are orrelated. It an be shown [] that in large systems the orrelation length an be evaluated by 1 n k ln a 1 =a 0 Ds; 13 where a 0 and a 1 a 0 > a 1 are the two leading eigenvalues of the transfer matrix. Any operator that is diagonal in the disrete point basis set, like the potential energy or any other funtion of the partile position say f q,has the ensemble average D E hf i` i ˆ w 0 jf q jw 0 ˆ XN f q i jw 0 i j ; 14 where w 0 is the transfer matrix leading eigenvetor. At the limit b!1, this eigenvetor is the ground state eigenstate of the quantum system in the disrete representation. So the ensemble average of the lattie system in the thermodynami iˆ1 limit beomes the ground state quantum average hw 0 jf q jw 0 i. 3. Numerial alulations To illustrate this general approah, we will arry out the alulations for the ritial parameters of the sreened Coulomb potential, V r ˆ e r =r. This potential is frequently used in quantum alulations and the exat values of the ritial parameters k for di erent states are known from nite size saling alulations [19]. In atomi units, the Hamiltonian an be written as H k ˆ 1 e r r k r : 15 A ritial point, k, is de ned as the value of k for whih the bound state energy beomes absorbed or degenerate with a ontinuum [7]. This system is known to exhibit a ontinuous phase transition for states with zero angular momentum, l ˆ 0, and rst-order phase transition for states with nonzero angular momentum [19]. Hamiltonians with spherial symmetry an be investigated with a radial path integral [0]. Using the above proposed sheme, the alulation is performed by taking N!1with DL (and Ds) xed. Exluding the origin with the radial grid r i ˆ idl, with i ˆ 1; ;...; N, the potential is effetively bounded from below. Numerially we found that no matter how small we take DL, for a su iently large value of N, we get the same limiting value. The transfer matrix for the radial path integral equation is T rad i`; i` 1 ˆM l 3= i`; i` 1 T i`; i` 1 ; 16 where T i`; i` 1 is the same transfer matrix de ned in Eq. (5) but with the potential terms given by the radial disretization rule V i` ˆ kexp i`dlš ˆ i`dl. The fator M l 3= i`; i` 1 is introdued by the disrete radial funtional weight r M l 3= i`; i` 1 ˆ p DL Ds i`i` 1 exp DL Ds i`i` 1 DL I l 3= ; 17 Ds i`i` 1

5 R.A. Sauerwein, S. Kais / Chemial Physis Letters 333 (001) 451± where I l 3= is the modi ed Bessel funtion. In the ontext of the pseudo-system, the pre-fator M` 3= i`; i` 1 plays the role of a entrifugal potential whih leads to another oupling between onseutive time slies. In order to study the properties of the system we x DL and take Ds ˆ DL. Then we numerially alulate the two leading eigenvalues with their respetive eigenvetors for di erent values of k and di erent matrix sizes N. The orrelation length n N k is then readily evaluated using Eq. (15). An `order parameter' an be de ned as the average radial oordinate saled by the system size R=L ˆhi`i=N. Klaus and Simon [3] showed that short range potentials in three dimensions, suh as the Yukawa potential, do not have normalizable ground state eigenfuntions at transition points. Thus, for the ase l ˆ 0, we an foresee that R=L ˆ R =L 6ˆ 0 at the transition point R is R at the transition point); although in the thermodynami limit, this magnitude identi es the two di erent symmetries. A loalized state has R=L < R =L, and a nonloalized state has R=L > R =L. In the ase l ˆ 1, the entrifugal term makes the Yukawa potential e etively long range. Thus the ground state wave funtion of the loalized states are normalizable even at the transition point. Hene, the distintion between the two symmetries beomes learer. In the thermodynami limit, R=L ˆ 0 identi es loalized states while R=L 6ˆ 0 identi es nonloalized states. The e et of the grid spaing and the system size, in the loalization of the pseudo-ritial point, is summarized in Fig. 1. Eah urve orresponds to a di erent value of DL Ds ˆ DL. Notie that there is indeed a `thermodynami limit' sine as onverges towards a de nite value. This value an depend on the grid spaing. In priniple, we ould think that the smaller the grid spaing the loser its thermodynami extrapolation will be to the atual transition point k. However, our results show that the N!1the pseudo-ritial points k N extrapolated value of k N for 0:01 6 DL 6 0:06 leads to the same aurate value k 0:840(1) whih is in perfet agreement with previous alulation using the trunated basis set sheme [19]. In Fig. we show the orrelation length n and the order parameter R=L as a funtion of k for Fig. 1. The pseudo-ritial points, k N vs 1=N for the ground state of the Yukawa potential for di erent grid spaing DL. The exat value of k is also shown by a dot.. The orresponding order parameter R=L urves show that this transition is ontinuous or, of the seond-order type. For angular momentum l ˆ 1, the orrelation length n and the the order parameter R=L show a di erent transition: a rst-order phase transition. There is a lear peak in the orrelation length n N k urves whih sales with L. Comparing the ase l ˆ 0 and l ˆ 1 we see that the behavior of the orrelation length near the transition point is qualitatively di erent. This di erene also appears in the order parameter. We see in Fig. that the R=L urves tend to a step-like funtion as the system size grows. In order to omplete the omparison between the two di erent types of transitions, we plot in Fig. 3 the energy E, the rst derivative de=dk, the orrelation length n, and the order parameter R=L for l ˆ 0 and l ˆ 1 with the system size N ˆ 000 and DL ˆ 0:05. All of these quantities are plotted as a funtion of the saled parameter xed values of DL ˆ 0:05 and Ds ˆ DL. Eah urve in these gures orresponds to alulations with di erent values of N. For l ˆ 0, there is a lear peak in the orrelation length n N k whih sales with L. The position of the peak for eah lattie size determines the pseudo-ritial transition point k N

6 456 R.A. Sauerwein, S. Kais / Chemial Physis Letters 333 (001) 451±458 Fig.. The saled orrelation length n=l and the order parameter R=L vs k for the Yukawa potential with l ˆ 0 and l ˆ 1 for inreasing grid points N ˆ 00 /, 400 (), 600 (}), 800 (M), 1000 () and 000 (), using DL ˆ 0:05. k k N =k N where k N is the pseudo-ritial value for eah nite system. The nature of the phase transitions is now very lear: ontinuous for l ˆ 0 and rst-order for l ˆ 1. The study of phase transitions and ritial phenomena show that physial systems as di erent as uids and anisotropi ferromagnets an be grouped into the same universality lass. Remarkably few universality lasses are de ned by the values of a set of ritial exponents whih are given by the asymptoti behavior of some quantities near the ritial point []. As we have mapped our original problem into a lattie system we are tempted to ask if this pseudo-system has suh a universal behavior. The ritial exponents assoiated with the orrelation length m and the spei heat a are, respetively, de ned as n jk k j m and E 0 jk k j a : 18 Notie that in our system, k plays the role of the temperature. Thus, the seond derivative of the ground state energy of the quantum system (or the free energy of the lattie model) with respet to k is analogous to the spei heat in thermodynami models. Although the alulations in the present work are performed in a nite system, we an use the saling invariane property of systems near ritiality to extrapolate our data to the thermodynami limit. This extrapolation tehnique is based on the theory of nite size saling [1]. We used the algorithm of Bulirsh and Stoer to obtain the extrapolated values of all the magnitudes [7]. The results for the ritial parameters are given in Table Disussion and onlusions We have shown with the use of Feynman's path integral approah that by allowing the spae and time variables to have disrete values, the quantum

7 R.A. Sauerwein, S. Kais / Chemial Physis Letters 333 (001) 451± Fig. 3. Comparison of the bound state energy E, the rst derivative de=dk, the saled orrelation length n=bn, and the order parameter R=L as a funtion of the saled k for the Yukawa potential at seond-order phase transition, l ˆ 0 and rst-order transition, l ˆ 1. Table 1 Critial parameters for the Yukawa potential for l ˆ 0 and l ˆ 1 Quantity l ˆ 0 l ˆ 1 Transition-order Seond-order First-order k (1) () a 0.00 (1) 0.99 () m 1.00 (1) (5) problem an be mapped to an e etive lassial lattie problem. The lattie an also be used to remove an existing point singularity of the original quantum problem without ompromising its ritial behavior. Thus, the quantum ritiality of the original system an be studied by the powerful statistial mehanis methods developed for latties. In partiular, the Monte Carlo tehniques an be diretly applied. Hene, opening the possibility of studying larger atomi and moleular systems. With this analogy we have obtained very aurate ritial parameters for the sreened Coulomb potential for both rst-order and ontinuous phase transitions. The eld of quantum ritial phenomena in atomi and moleular physis is still in its infany and there are many open questions about the interpretations of the results inluding whether or not these quantum phase transitions really do exist. The possibility of exploring these phenomena experimentally in the eld of arti ial atoms [4] o ers an exiting hallenge for future researh. This approah is general and might provide a powerful way in determining ritial parameters for the stability of atomi and moleular systems in external elds, for seletively breaking hemial bonds and for design and ontrol eletroni properties of materials using arti ial atoms.

8 458 R.A. Sauerwein, S. Kais / Chemial Physis Letters 333 (001) 451±458 Aknowledgements We would like to aknowledge the nanial support of the National Siene Foundation and the O e of Naval Researh. Referenes [1] B. Benderson, More Things in Heaven and Earth, A Celebration of Physis at the Millennium, Springer, APS, 1999, p [] J.J. Binney, N.J. Dowrik, A.J. Fisher, M.E.J. Newman, The Theory of Critial Phenomena: An Introdution to the Renormalization Group, Oxford University Press, Oxford, 199. [3] S.L. Sondhi, S.M. Girvin, J.P. Carini, D. Shahar, Rev. Mod. Phys. 69 (1997) 315. [4] S. Sahdev, Quantum Phase Transions, Cambridge University Press, New York, [5] H.L. Lee, J.P. Carini, D.V. Baxter, W. Henderson, G. Gruner, Siene 87 (000) 633. [6] P. Serra, S. Kais, Phys. Rev. Lett. 77 (1996) 466. [7] S. Kais, P. Serra, Int. Rev. Phys. Chem. 19 (000) 97. [8] M.K. Sheller, R.N. Compton, L.S. Cederbaum, Siene 70 (1995) [9] X. Wang, L. Wang, Phys. Rev. Lett. 83 (1999) 340. [10] R. Rost, J. Nygard, A. Pasinski, A. Delon, Phys. Rev. Lett. 78 (1997) [11] Y.P. Kravhenko, M.A. Lieberman, Phys. Rev. A 56 (1997) R510. [1] F. Remale, R.D. Levine, Pro. Natl. Aad. Si. USA 97 (000) 553. [13] R.S. Berry, Nature (London) 393 (1998) 1. [14] H. Kleinert, Path Integrals, World Sienti, New Jersey, [15] N. Makri, Ann. Rev. Phys. Chem. 50 (1999) 167. [16] M.L. Bella, Quantum and Statistial Field Theory, Oxford University Press, New York, [17] J.M. Thijssen, Computational Physis, Cambridge University Press, Cambridge, [18] R.D. Coalson, J. Chem. Phys. 85 (1986) 96. [19] P. Serra, J.P. Neirotti, S. Kais, Phys. Rev. A 57 (1998) R1481. [0] C. Groshe, F. Steiner, Handbook of Feynman Path Integrals, Springer, Berlin, [1] V. Privman (Ed.), Finite Size Saling and Numerial Simulations of Statistial Systems, World Sienti, Singapore, [] B.D. Hugues, Random Walks and Random Environments, vol. 1, Random Walks, Clarendon, [3] M. Klaus, B. Simon, Ann. Phys. 130 (1980) 51. [4] L. Jaak, P. Wawrylak, A. Wojs, Quantum Dots, Springer, New York, 1998.