Decimation approach in quantum systems

Nuclear PhysicsB200[FS4] (1982) 45-60 © North-Holland Publishing Company

DECIMATION APPROACH

IN QUANTUM,. SYSTEMS

C. CASTELLANI Istituto di Fisica, Universtd dell'Aquila, Aquila, Italy, Istituto di Fisica "G. Marconi", Universit~ di Roma, Roma, Italy, and Gruppo Nazionale di Struttura della Materia del Consiglio Nazionale delle Ricerche, Sezione dell'Aquila, Italy

C. DI CASTRO Istituto di Fisica "G. Marconi", Universitd di Roma, Roma, Italy and Gruppo Nazionale di Struttura della Materia del Consiglio Nazionale delle Ricerche, Sezione di Roma, Italy

J. RANNINGER Groupe des Transitions arePhases, Centre National de la Recherche Scient~fique, B.P. 166, 38042 Grenoble Cedex, France

Received 2 June 1981 The possibilityof applyingthe decimation method to quantum systemsis criticallyexamined in two systems: the anisotropic Heisenberg hamiltonian and spinless interacting ferrnions. We show that, in general, the decimationtransformationdoes not conserve the ground state of the systemand therefore badly describes the T = 0 fixedpoint. This in turn, within the Migdal-Kadanoffapproach, introduces several spurious effects, whose origin and consequences are analyzed in detail for different dimensions.

1. Introduction The critical properties of quantum systems which cannot be rephrased in terms of equivalent classical systems are still an open problem. Real space-renormalization methods [1] have extensively been applied to such systems. In spite of many attempts which seemed to be very reasonable [2] no conclusive answers have so far been reached within the renormalization approach, especially as far as the x-y model in two dimensions is concerned. The situation is worse for fermions systems to which the usual definition of block variables for linear group transformations or the standard majority rules for nonlinear transformations cannot be generalized in any immediately obvious way. Recently a new technique [3] (originally introduced for problems of field theory on a lattice) of retaining only few excited states for a finite block of the system has been introduced, but this method cannot be easily extended to finite temperature. 45

46

c. Castellani et al. / Decimation approach

Decimation methods [4] have recently been used [5] for quantum spin ½systems in order to obtain recurrence relations for the parameters of the hamiltonian. The main goal was to obtain at least a correct qualitative picture by this fairly simple and straightforward technique. Decimation procedures, avoiding all those complications inherent to more complex real space transformations, also appear at first sight to be suitable for treating Fermi systems. In this paper, we shall critically examine the Migdal-Kadanoff decimation [4] within the approximation proposed by Suzuki and Takano [5] for quantum systems. Spurious behaviour related to the zero-temperature fixed point of the x-y model and to the lower critical dimensionality (dimensionality at which the phase transition occurs at zero temperature and below which it disappears) of the Heisenberg model were already discussed by Barma et al. [6]. We shall extend and complete their analysis as far as the quantum spin ½ system is concerned and critically examine, parallel to that, a spinless interacting fermion system. The major shortcoming in the quantum case appears to be that the decimation transformation does not preserve, in general, the ground state of the system. Therefore, it badly reproduces the physical content of the model near the lower critical dimensionality, precisely where the Migdal-Kadanoff approach finds its most correct application for the classical case.

2. Decimation for quantum systems Let our system be described by a hamiltonian H in terms of operators defined on the sites i of a lattice Z. The coupling constants in H are to be understood as multiplied b y / 3 = 1 / k B T . We restrict ourselves to nearest neighbour interactions only. We construct the decimated hamiltonian on a suitably chosen sublattice Z' with lattice spacing ha, where a is the lattice spacing of the original lattice Z. This is done by carrying out the partial trace over all the degrees of freedom on all the lattice points except those belonging to Z': e - ~ = Tri~z, e -H

(2.1)

The transformation (2.1) does not imply any rescaling of the operator variables in order to compensate for the change of length scale of the lattice. Therefore, at the fixed points of the transformation the space-dependent correlation functions must be constant. Thus, these fixed points can only describe a trivial behaviour associated to either zero or infinite temperature. In general, however, the decimation cannot be carried out exactly either due to the proliferation of couplings, which most likely occurs when the space dimensionality is larger than one, or for the quantum systems even at d = 1 due to non-commutativity among the various terms of the hamiltonian. As far as the first problem is concerned,

C. Castellani et al. / Decimation approach

47

the Kadanoff interpretation of the Migdal recurrence relation [4] for a d-dimensional classical system makes it clear how by bond-moving a d - d i m e n s i o n a l problem is essentially reduced to a 1-dimensional decimation (which can be carried out exactly for the Ising system) thus avoiding the coupling proliferation. As we have recalled, the exact decimation is expected to describe the trivial fixed points at T = 0 and T = 0o correctly. If the model does not show any phase transition, then the first fixed point is unstable and the second one is stable. Iteration of the transformation correctly drives the system towards the T = oo fixed point. If the model does show a phase transition, also the T = 0 fixed point must become attractive, since it controls the ordered phase. The exact decimation cannot produce a non-trivial fixed point and the attractive nature of the two trivial fixed points can only be reconciled by enlarging at each iteration the set of couplings of the hamiltonian. The Migdal-Kadanoff approach instead, by reducing a d-dimensional proble,n to a one-dimensional decimation, limits the coupling space to the original one. Thereby it generates in a system with a phase transition at least one unstable approximate critical fixed point such as to match the flow lines towards the two trivial stable fixed points. The Migdal-Kadanoff recurrence relations act, therefore, as interpolation formulae between the two asymptotic fixed points and give rise to a reasonable critical behaviour for the system only if the approximate critical fixed point is not too far from the asymptotic regions which is the case when the space dimensionality is close to the lower critical dimensionality de. For a quantum system, the non-commutativity of the various terms of the hamiltonian renders it impossible to derive the decimated hamiltonian exactly, even for a one-dimensional lattice. According to Suzuki and Takano [5] it should b e a reasonable approximation in this case to derive the decimated hamiltonian for a finite lattice. We divide the one-dimensional chain into blocks of (A + 1) sites and neglect the non-commutativity among the hamiltonians Hb of the various blocks, Within one block, denoting the initial and final sites by Ai + 1 and a (i + 1) + 1, respectively, the effective hamiltonian H~[Ai + 1, A (i + 1) + 1] acting between these two sites is given by exp [ - H ~ [ai + 1, a (i + 1) + 1] + c] = Y~ (vle-Uqv), {v}

(2.2)

where the sum is carried out over all states It,) of the A - 1 intermediate sites and c is a constant. Provided we know the exact eigenstates Ira) and the exact eigenvalues E,, of the hamiltonian Hb for the block of (A + 1) sites, eq. (2.2) can be written in the form

exp(-Hx[M+l,a(i+l)+l]+c)=

Y~ (vlrn)(rnlv)e -*e~. {,,},{,n}

(2.3)

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C. Castellani et al. / Decimation approach

When diagonalized in the two-site space, the right-hand side of eq. (2.3) has the form of a sum of projection operators o~

where ]a) is a complete set of orthonormalized states for the two-site problem. This enables us to re-write (2.2) in the form exp(-Hx[Ai+l,A(i+l)+l]+c)=exp[~(lnP~)la)lal].

(2.4)

The final approximation consists in defining an hamiltonian for the entire infinite sublattice by merely summing the individual terms appearing in (2.4): HA = X Hx[Ai + 1, A(i+ 1)+ 1].

(2.5)

i

3. Quantum spin systems 3.1. THE A=2 CASE The finite lattice approximation described in sect. 2 was first applied [5] to the case of A = 2 for the anisotropic spin . ~1 Heisenberg hamiltonian H=-K

X S~S , i• - 2 ! D X ( S ,+S i- + S T S 7) . (i,j)

(3.1)

(i,j)

S~ denotes the z-component of the spin operator on site i and S[, S [ the raising and lowering operators respectively. The case K = D corresponds to the isotropic Heisenberg system. Let us consider here the results of the A = 2 decimation in terms of the approach summarized by formulae (2.2) to (2.5) in order to demonstrate for this simple case •the shortcomings of this method in the x-y limit. We shall point out how they are intrinsically related to the decimation procedure rather than to the Migdal-Kadanoff bond-moving mechanism or to the approximation of neglecting the non-commutativity among the hamiltonians of the various blocks into which the infinite chain is divided. For A = 2 we need to consider a finite chain of three sites in which we decimate the central site. The eigenstates and eigenvalues of the three-site problem are listed in table 1 where ,~ 1 D2 2 2 E:~ = -2- -½D2+e 2 , ~ ± = 1 ~w . - . 2 ,-~e±2 .. (3.2) Let us now perform the operations (2.2) to (2.5) with the following set of eigenstates

I~) (~ = 1, 2, 3, 4),

11) = Itt),

12) = 15$),

13) =

lt$ + St),

14) = ~ J t $ - St),

C. Castellani et al. / Decimation approach

49

TABLE 1 The eight eigenstates and eigenvalues of a three-site spin-system

State

Energy

Ittt),l~+$) a ~ l t t $++t t)+~,~lt +t) x~l++t +t$+)+~,~l$ t +) ~ltt~-~ t t), ~ l ~ t - t ~ )

]~ = ~ r ± ,~, / E ~ V ~ : 0

denoting the four possible spin configurations of a two-site system. We thus obtain exp

(-Hz[2i + 1, 2(i + 1)+ 1]+c) = Z P,~la)(a[

= {e-K/2 + (/~ 2+e-~+ + ~ 2_ e - , )}(i1)(ll + [2)(2[) + 12(A2+e-e+ + A2_ e-~-)}13)(31 + 214)(41.

(3.3)

Writing (3.3) in the form (2.4) yields the recursion relations

PIP_.__~2

/ ( = In P3P4'

/~ = In P3 M ;

P4

(3.4)

that is / ( = In [(e m2 + F_)2/41"+],

/9 = In F + ,

F± = e-m4 (cosh ¼~/K2 + 8D 2 4- (K/x/g 2+ 8D 2) sinh ¼x/K2 + 8D2),

(3.4')

which were previously obtained by Suzuki and Takano [5]. The new effective coupling constants for a d-dimensional system are obtained within the MigdalKadanoff bond-moving procedure in terms of the ones for one-dimensional systems (3.4) and are given by /(a = 2a-x/~,

/ga = 2a-1/5.

(3.5)

In three dimensions this system of equations has three non-trivial fixed points I*, H* and X* associated to the three universality classes: Ising, Heisenberg and x-y, inherent in the symmetry of the hamiltonian (3.1). Apart from quantitative discrepancies, one would therefore infer from these results that this decimation approach leads to a correct qualitative picture of the topological properties of the flow diagram. However, the flow lines in the entire ordered x-y phase are attracted by the zero-temperature Heisenberg fixed point. This remains true down dimension d = 2, where the non-trivial unstable Heisenberg fixed point H* disappears while the non-trivial unstable x-y fixed point X* persists contrary to expectations of a line of fixed points.

50

C. Castellani et al. / Decimation approach

We have pointed out above that the Migdal-Kadanoff bond-moving approach corresponds to an approximate interpolation scheme between two trivial fixed points. Therefore, it is, in our opinion, indispensable that decimation describes those two fixed points correctly and in particular the zero-temperature one which is the relevant one for describing the d -* dc limit. In the limit T -* 0 only the ground state of the three-site system contributes to the new effective hamiltonian defined by relation (2.3) due to the weighting factor e -~'. For the Ising model (D = 0) for which the ground state for the three-site system is simply the direct product of single-site states, decimation preserves the ground state. For the isotropic Heisenberg system (K = D) the ground state for the three-site block is represented by the four degenerate states

II'tt>,

1~$~>,

4}(It t $>+ I$t t>+ II'~t>),

~(l~t>+lt ~£>+l~t $>). (3.6)

When the decimation is carried out on the states (3.6) we obtain three projection operators acting on the states [1''~),

15~),

x/~[l'~+ $1'),

(3.7)

which are just the three degenerate ground states for the two-site isotropic Heisenberg hamiltonian. For the x-y model (K < D ) the ground state of the three centre model problem is composed of two degenerate states:

x_~(I? t ~ + ~ 1'?)) + ~-I? ~ t),

(3.8)

x - ~ ( l ~ t +t~>)+~-I~t~) • Performing the decimation on these two states we are once more left with projection operators acting on the three sites (3.7), whereas for a two-site hamiltonian with x-y character only the state ~[1' $ + $1') should be involved in the limit T ~ 0. Thus after the decimation the new effective hamiltonian will have the threefold degenerate ground state (3.7) and hence shows an Heisenberg symmetry. It is instructive to consider the limit T ~ 0 explicitly, in which case the right-hand side of expression (3.3) becomes

+ a3 e -E1/kBr [~/~11'$ - $1')(1' ~ - $ l'lx/~],

(3.9)

with al = tt 2, a2 = h2--, 33 = 2 andEo = e_, E1 = 0. In (3.9) the factor 1 / k a T multiplying the coupling constants has been made explicit. We notice that the weights of the first three operators derive their contributions from the original x-y ground state with energy E0 of the three-site system, while the last operator is weighted by a contribution arising from an excited state with energy E~ of the initial system. Rewriting the left-hand side of (3.3) in the more explicit form

51

c. Castellani et al. / Decimation approach

introducing the new coupling constants/~,/~ and the constant energy shift c, we find exp {c + 1/~[I t t)(t tl + I~$)($ $1] - (¼/~ - 1/~)x/~lt $ + $ t)(t ~ + $ t I,/~ -

(}/~ + ½15)~ltt - $ tXt l - $ tl~.

(3.10)

-Equating the various different operators appearing in (3.10) and (3.9) term by term yields the following set of equations: c + kK = -Eo/ks

T + In a 1,

c -¼I~+½1) = -Eo/ksT+ln

a2,

(3.11)

c - l ffi - ½1~)= - E 1 / kB T + ln a3 .

In the limit T--*0 we find/~ = / ) = ( - E o + E 1 ) / k B T ~ o o and hence the x-y hamiltonian with which we started is mapped by decimation into the zero-temperature Heisenberg fixed point. The Migdal-Kadanoff procedure generalizes this behaviour derived for d = 1 to any dimension. This result, however, is specific for the choice A = 2 only. For a different A a different situation can arise. The ground state of the original system gives a contribution to all four terms of the two-site effective hamiltonian appearing in (3.9). In this case there will no longer be a separation in energy between a ground state and an excited state for the decimated hamiltonian. The decimation shifts the system from zero temperature to finite temperature. As we shall see, for A = 3 this is indeed the case. Completing our analysis of the A = 2 case, we now show why as the dimension is lowered below d = 2, a spurious stable fixed point Y* of the x-y type emerges from the zero-temperature Heisenberg one and approaches the fixed point X* [6]. Y* and X* eventually merge into a marginal fixed point at d = 1.95 before they disappear altogether. This value of d has been assumed to coincide with the approximate lower critical dimensionality [5]. We believe that one cannot derive any conclusion from the behaviour of the fixed point Y* since its appearance is strictly linked to the previously stated spurious result that the low-temperature phase of the x-y model is attracted by the zero-temperature Heisenberg fixed point. In fact, at d = 2 the Heisenberg unstable critical fixed point H* disappears, as it should. This means that the corresponding zero-temperature fixed point becomes unstable. Since the x-y unstable fixed point X* is still present, another spurious stable fixed Y* point must emerge at finite temperature within the approximated interpolation formula and reconcile the situation according to the schematic picture of fig. 1. 3.2. THE a =3 CASE Barma et al. [6] have already shown that the lower critical dimensionality for the Heisenberg model is not the same for A = 3 and ,~ = 2. We shall present a detailed

52

C. Castellani et aL / Decimation approach

d-2 T=°° fixed point

H* T=O Heisenbe~9 fixed point

×.

1.95
Y=O Heisenberg fixed point

Fig. 1. Schematicpicture of the variation with dimensionalityof the stabilityof variousfixedpoints for an anisotropic Heisenberg system. analysis of the decimation results for A = 3, for the general anisotropic Heisenberg case (3.1). For this purpose we determine the complete set of eigenstates for the four-site problem and then decimate with respect to the entire set of states for the two intermediate sites, obtaining the recursion relations (3.4) and (3.5) with the following expressions for the P~ :

Pz(K,D)=P2(K,D)=lesr/4 +en/4 [cosh !x/!, 2 4,K K-D

. •

_ D ) 2+ D 2

1/1

2x/¼(K - D ) 2 + D 2 smh ~ ~(K + e - ° / 4 [cosh

D)2 + D 2 ]

½~/¼(K+ 0) 2 + D 2

K + D

1

1"

2

2

,, - - - - J . ~sinh~/~(K+D) +D ] 2~/~(K + D ) ~ + D ~ a

+!--KI4±l ^-KI4 2~ T2~

~" 2 v Z~,t v

o ) = e,(K, - o ) = { 2 e o,, [cosh

e-e~} ,

(3.12)

- 0)2 + O2

K-D

sinh ½4¼(K - D)2 + D 2]

q 2x/¼(K - D ) 2 + D 2

~sinh~~] D

+ e -K/" [cosh ½ ~

1

+ e -K/4 ~ t2~ e-'vl, v

J

where tiv and ev are eigenstates and eigenvalues, respectively, of the following 3 x 3 matrix: !K

--2

4o

!2 D

-,~D

-4D

o

•

(3.13)

53

c. Castellani et aL / Decimation approach

1

2

3

4

5

6

7

D

Fig. 2. Anisotropic Heisenberg system. Flow diagram obtained by decimation with ~ = 3. (a) d = 2, accumulation of flow lines is observed along the dashed line. (b) d = 2.1; (c) d = 3, K*(X*)='0.06. For d = 2 only the non-trivial Ising point I* appears. A strong concentration of flow lines with very low rate of flow is observed (fig. 2a) in the region of x-y symmetry, thus foreshadowing the appearance of some fixed points at a slightly higher dimension. In fact, for d > 2.08 two x-y fixed points X* and Y* emerge, X* being unstable, Y* being stable (fig. 2b). On increasing the dimensionality d they tend to m o v e away from each other as illustrated in fig. 2c for d = 3; in the same figure the critical Heisenberg fixed point H* (which appears [6] at d = 2.11) is also indicated. As we have seen, for )t = 2 an x-y system at T = 0 is m a p p e d into the T = 0 Heisenberg fixed point and this gives rise, for 1.95 < d < 2, to the spurious fixed point Y*. For A = 3 we now show that an x-y system at T = 0 is m a p p e d into a finitet e m p e r a t u r e system. As a consequence, for d < 2.08, iteration of the transformation drives the system to the infinite-temperature fixed point. For d > 2.08 instead, a spurious fixed point Y* at finite t e m p e r a t u r e emerges together with the critical X* fixed point and attracts the x-y ordered phase. As for A = 2, any conclusion (such as d c = 2.08) based on the presence of this fixed point would be highly questionable, The mechanism by which the T = 0 x-y system is shifted to finite t e m p e r a t u r e is the

54

C. Castellani et al. / Decimation approach

following. For h = 3 in the limit T--> 0 the right-hand side of expression (3.3) can be cast once again in the form (3.9). Apart from the obvious fact that al, a2 and a3 are now more complicated, the main difference with respect to the h = 2 case is that the weight of all four projection operators derive their contributions from the original x-y ground state. In the limit K = 0 this ground state becomes simply

Therefore one and only one energy-dependent factor e -E°/r appears in the four terms, where E0 is the ground-state energy. This leads to a marked change in the behaviour compared to the h = 2 decimation. Eq. (3.11) for/~,/~ is still valid with E1 replaced by E0. Instead o f / ~ = / 9 = ( - E o + E x ) / T ~ oo, we now find /~ = In (a2/a2a3),

1~ = In (a2/a3).

(3.14)

B o t h / ~ a n d / ~ being finite, they represent a system at finite temperature. So that for h = 3 it is no longer the zero-temperature Heisenberg fixed point which induces a spurious behaviour at finite temperature for the x-y model as for h = 2, but the zero-temperature fixed point itself is shifted to finite values.

4. Fermions systems 4.1. THE a =2 CASE It is of general interest to construct a suitable renormalization group transformation for interacting fermions. For instance, it has been shown [7] that the Hubbard model cannot be simply reduced by functional integral methods to an effective polynomial lagrangian in terms of classical spin and charge fields in order to apply either the Wilson or the field theoretic renormalization group. In effect this approach encounters serious difficulties when requiring that the original Fermi character of the system should be maintained. On the other hand as we stressed in the introduction, it is not simple to generalize both linear and non-linear ordinary real space renormalization groups to Fermi systems. In fact, neither the idea of block variables nor the majority rules have yet found a direct application to the case of anticommuting variables. Recently a new zero-temperature real space renormalization group [3] has been applied with some success to a large class of quantum system including interacting fermions [8]. However, this method has not yet found an extension to finite temperature where, for instance, the metal-insulator transition for the Hubbard model should be present. In the decimation approach, the non-decimated variables maintain their original properties. It seems, therefore, that decimation could be suitable to avoid all the difficulties mentioned above. We shall, however, see that, as in the x-y model, also in this case there is an intrinsic difficulty in handling by decimation the ground state of

C. Castellani et al. / Decimation approach

55

the system. Repeating the arguments given for the spin case, the recursion formulae obtained by decimation are therefore questionable. This strengthens our conviction that in general the Migdal-Kadanoff approach is not suitable to analyze the topological structure of the flow diagram in the hamiltonian space for quantum systems. For simplicity we shall analyze the case of spinless interacting fermions described by the following hamiltonian:

H=-K

Y. o¢ z l¢ zo j - 2 *l _. r,~ ~ ( c +i c j + c j c+ i ) - h

g.j)

(i.D

z

(4.1)

~Si, i

where S~ ----Ci+c i - - ~1= ni - 1 , and c~ and c~ are the Fermi creation and annihilation operators at site i of the d-dimensional lattice Z. In (4.1) we allow for both attractive (K > 0) and repulsive (K < 0) interactions. The external field h plays the role of the chemical potential. We shall assume h = 0 corresponding to half an electron per site on average. In one dimension the model (4.1) is equivalent (via the Wigner-Jordan transformation) to the anisotropic Heisenberg hamiltonian (3.1), whereas for d # 1 this equivalence does not hold. For d = 1 the T = 0 behaviour of (4.1), or equivalently (3.1), is well established [9], while the T # 0 thermodynamic properties have been recently evaluated by Monte Carlo techniques [10]. To carry out the decimation procedure of sect. 2 with A = 2 for the hamiltonian (4.1) we need the eigenstates and eigenvalues of the three-centre problem. They are listed in table 2 which reflects the previously mentioned correspondence between (4.1) and (3.1) in one dimension, where A~,/z ~ are given in (3.2) and O or ® indicate that the corresponding site is empty or occupied. Labelling the three sites by -1, 0, 1, for instance the state ]® ® ®) is given by c+-lc~c-~ 10) where 10) is the vacuum state for the three-centre problem. Table 2 is equal to table 1 provided the correspondence O'~-~,, ®"-->1' is made. However, we note that while all the spin states are symmetric under exchange of site TABLE 2 The eight eigenstates and eigenvalues of a three-site spinless interacting Fermi system State

I®®®). Iooo)

•~.~1®®o+ o®®)+ ~.1®o®) x . ~ l o o ® + ®oo:, + ~.1o®o)

~1®®o-o®®:,. ~1®oo-oo®;,

Energy

-~K

C. Castellani et al. / Decimation approach

56

indices, this is no longer true for the fermion states, due to the anticommutation rules o f the c-operators. When these anticommutation rules are taken properly into account, the result of the decimation for fermions is different from that for spins and (3.3) has to be replaced by 4

exp ( - H 2 1 2 i + 1 , 2 ( i + 1 ) + 1 ] + c ) =

E

P~la)(a})

= (e "/2 + r-){l® ®><® ®1 + Io o>
© + © ®)(® 0 + ©

(4.2)

© - © ®)(® © - ©

The 1-dimensional recursion formulae are still given by (3.4), but now using the P~ defined by (4.2) we obtain /~ = In

(P~P2/P3P4)= In [(e ~¢/2+ I_)2/(1 + f,+)2],

15 = In

(P3/P4) = In 1 = O.

(4.3)

The main difference from the spin formulae (3.4) is that the first iteration of (4.3) yields/~ = 0. This result does not depend on the finite lattice approximation used in deriving the recursion formulae (4.3) but it is an exact consequence of the choice A = 2. Since the hamiltonian (4.1) is invariant under c ~ c ÷ provided D ~ - D (and h ~ - h ) , it can be shown [11] that when the decimation is performed on a cubic lattice and A is an even number, the decimated h a m i l t o n i a n / t must be invariant under the same operation c ~-~c ÷ and t h e r e f o r e / ~ = 0. In effect a cubic lattice can be divided in two equivalent sublattices, each one formed by the nearest neighbour sites of the other. The change ÷ D ~ - D can always be intended as an irrelevant change of sign of the operators c, c on the sites of that sublattiee whose degrees of freedom are eliminated when the decimation is carried out with A even. Obviously the exact decimation (on the chain) would introduce (for K # 0) further invariant couplings beyond the t e r m / ~ ( n i ½)(nj -½). While reproducing the symmetry properties of the exact/-I, our approximate approach, on the contrary, reduces the number of the allowed parameters in the effective hamiltonian and the first iteration maps the hamiltonian (4.1) in an Ising model with K given in (4.3) for the 1-dimensional case. The Migdal-Kadanoff bond-moving procedure extends this result to a d-dimensional system Via eq. (3.5) and the entire ( D - K ) plane is mapped on the K-axis at the first iteration of the approximated decimation, regardless of dimension. The flow diagram will therefore show only the critical Ising fixed point I* for d > 1, and no x-y or Heisenberg fixed point will be present at any dimension. In fig. 3 the flow diagram for d = 2 is illustrated. No new features appear for K < 0. Similar diagrams are obtained for d > 1 and in particular for d = 3.

C. Castellani et aLI Decimation approach

57

region I

1°

region IT

l/

I

I

I

I

I

I

0

1

2

3

4

5

6 D

Fig. 3. Spinless interacting fermions. Flow diagram obtained by decimation with A = 2 and d = 2.

The critical line attracted by the fixed point I* is formed by the points m a p p e d at the first iteration in I* and it is described by the equation

[e i

21nt

~

3

n)]}

--K*

(4.4)

where K * = 2.4 is the 2-dimensional Ising fixed point value for A = 2. This line tends asymptotically to the bisectrix K = D after having crossed it at a finite distance. In the D-K plane it separates a region of disordered phase which is a uniform mixture of e m p t y and occupied sites with half an electron per site on average, from a coexistence region of two phases, one with m o r e than and the other with less than half an electron per site. A small variation of the chemical potential would destroy the degeneracy between these last two states. The infinite end point of the critical line, i.e. the point K = oo, D = oo, KID = 1, is not a fixed point of the transformation. By the same procedure described in the A = 3 x-y-spin case this T = 0 (KID = 1) point is m a p p e d into a finite-temperature point as well as any other T = 0 point with D > K. In particular we obtain that in the limit T ~ 0 K --~l-:K=-41n2, D

K - - ~ 1+:/~ = oo, D

K --= 1:/~=41n2. D

(4.5)

58

C. Castellani et aL / Decimation approach

All the ordered phase I is first mapped o n the K axis and then attracted by the T = 0 (D = 0, K = oo) Ising fixed point. The non-ordered phase II is attracted by the T = oo (D = K = 0) fixed point. A closer inspection of the recursion formulae shows the existence of a third region for very high values of D and K. This region is mapped at the first iteration on the interval f - K * , - 4 In 2] of the negative K axis and is eventually attracted by the T = 0 Ising fixed point since the recursion formulae are even in K for D = 0. This region is bounded by a line which is mapped first into - ( K * ) and then in K*. The presence of this region is a totally spurious effect driven by the incorrect behaviour of the recursion formulae around the K = D infinite point. The absence of the x-y fixed point at any dimension appears to be a reasonable result since no phase transition is expected at finite temperature for a Fermi system with a pure x-y symmetry i.e. a free Fermi gas. But even this property is not maintained by varying the decimation parameter )t, as we shall see next. 4.2. THE A=3 CASE In the A = 3 case the i t e r a t e d / 9 is no longer constrained to be zero by symmetry. The one-dimensional recurrence relations are also in this case given by eq. (3.4) where the expressions for the weights Pa =/>2 are identical to the ones for the spin case (3.12) and P 3 ( K , D ) = P4(K, - D ) =

{2e°/4 [cosh ~ / ~ ( K - D ) Z + D 2

+ 2x/41(K _ D l z + D

2

t4!'K-DlZ+D ~

+ e -x/4

2 e -~} , + ,4~_D+ D ~ sinh 2ix/K--~-+~]J + e - ~ / 4 Y. tz~

cosh ~ x / ~ (4.6)

where tlv and e~ are the eigenvectors and the eigenvalues of the matrix (3.13). When the recurrence relations are analyzed at different space dimensions, we have the following results. At d = 2 according to fig. 4 only the Ising type fixed points I*(K*) and I * ( - K * ) appear just as for A = 2. At d = 3 new fixed points are present as shown in fig. 5. On the line K = 0 (free fermion system) there are two spurious fixed points F* and F*. This once again is related to the fact that the decimation does not preserve the ground state. The T = 0 fixed point, being shifted to a finite temperature, must be accompanied by another fixed point to match the flow line. For K ~ 0 there are two other fixed points Z* and Z* for both K* and D* finite. Z* is unstable with respect to any direction in the D - K plane as the critical Heisenberg H* fixed point for the spin case. Z* instead has a direction of stability.

C. Castellani et al. / Decimation approach

59

-3

1

1

I

I

2

3

" I

4

/

J

I

I

5

6

7

e

\

I

9

_ _

Iol

Fig. 4. Spinless interacting fermions. Flow diagram obtained by decimation with A = 3 and d = 2.

1

2

3

4

5

6

7

8

9 IOl

Fig. 5. Spinless interacting fermions. Flow diagram obtained by decimation with ~ = 3 and d = 3, Z * = ( - 9 . 7 3 , 9.85).

60

C. Castellani et al. / Decimation approach

The fixed points F*, F* and Z*, Z* arise at different lower critical dimensionalities. Whereas F* and F* are completely spurious, a physical interpretation may be attached to Z~* and Z2*. Along the line connecting I*(-K*) and Z* one has a second-order phase transition between a charge ordered phase and a disordered phase. For the h = 2 case this critical line extends to infinity. In the present case the appearance of Z* could indicate that the second-order phase transition switches into a first-order one. Within this interpretation the fixed point Z* should correspond to a T = 0 fixed point which has been shifted to finite temperature by the decimation procedure. References [1] Th. Niemeijer and J.M.J. Van Leeuwen, Phys. Rev. Lett. 31 (1973) 1411; Physica 71A (1974) 17 [2] J. Rogiers and R. Dekeyser, Phys. Rev. B13 (1976) 4886; Z. Friedman, Phys. Rev. Lett. 36 (1976) 1326; K. Subbarao, Phys. Rev. Lett. 37 (1976) 1712; A.L. Stella and F. Toigo, Phys. Rev. B17 (1978) 2343; R. Dekeyser, M. Reynaert, A.L. Stella and F. Toigo, Phys. Rev. BI8 (1978) 3486 [3] S. Jafarey, R. Pearson, D.J. Scalapino and B. Stoeckly, unpublished; STD. Drell, M. Weinstein and S. Yankielovicz, Phys. Rev. D14 (1976) 487; R. Jullien, J. Fields and S. Doniach, Phys. Rev. Lett. 38 (1977) 1500 [4] A.A. Migdal, JETP (Soy. Phys.) 42 (1976) 743; L.P. Kadanoff, Ann. of Phys. 100 (1976) 359 [5] M. Suzuki and H. Takano, Phys. Lett. 69A (1979) 426; M. Suzuki and H. Takano, preprint (1980) [6] M. Barma, D. Kumar and R.B. Pandey, J. Phys. C12 (1979) L909 [7] C. Castellani and C. Di Castro, Phys. Lett. 70A (1979) 37 [8] R. Jullien and P. Pfeuty, Phys. Rev. B19 (1979) 4646; P. Pfeuty, R. Jullien and V.A. Penson, to be published; S.T. Chui and J.W. Bray, Phys. Rev. B18 (1978) 2426; C. Dasgupta and P. Pfeuty, to be published [9] C,N. Yang and C.P. Yang, Phys. Rev. 150 (1966) 321,327 [10] H. De Raedt and Ad. Lagendijk, Phys. Rev. Lett. 46 (1981) 77 [11] C. CasteUani, C. Di Castro, D. Feinberg and J. Ranninger, Phys. Rev. Lett. 43 (1979) 1957