Random Transverse Field Ising Model in dimension d=2,3 : Infinite Disorder scaling via a non-linear transfer approach

Random Transverse Field Ising Model in dimension :
Infinite Disorder scaling via a non-linear transfer approach

Cécile Monthus and Thomas Garel Institut de Physique Théorique, CNRS and CEA Saclay 91191 Gif-sur-Yvette cedex, France
Abstract

The ’Cavity-Mean-Field’ approximation developed for the Random Transverse Field Ising Model on the Cayley tree [L. Ioffe and M. Mézard, PRL 105, 037001 (2010)] has been found to reproduce the known exact result for the surface magnetization in [O. Dimitrova and M. Mézard, J. Stat. Mech. (2011) P01020]. In the present paper, we propose to extend these ideas in finite dimensions via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization of the transfer equations correspond to the transfer matrix for a Directed Polymer in a random medium of transverse dimension , in agreement with the leading order perturbative scaling analysis [C. Monthus and T. Garel, arxiv:1110.3145]. We present numerical results of the non-linear transfer approach in dimensions and . In both cases, we find that the critical point is governed by Infinite Disorder scaling. In particular exactly at criticality, the one-point surface magnetization scales as , where coincides with the droplet exponent of the corresponding Directed Polymer model, with and . The distribution of the positive random variable of order presents a power-law singularity near the origin with so that all moments of the surface magnetization are governed by the same power-law decay with independently of the order .

I Introduction

The quantum Ising model

(1)

where the nearest-neighbor couplings and the transverse-fields are independent random variables drawn with two distributions and is the basic model to study quantum phase transitions at zero-temperature in the presence of frozen disorder. In dimension , exact results for a large number of observables have been obtained by Daniel Fisher [1] via the asymptotically exact Strong Disorder renormalization procedure (for a review, see [2]). In particular, the transition is governed by an Infinite Disorder fixed point and presents unconventional scaling laws with respect to the pure case. In dimension , the Strong Disorder renormalization procedure can still be defined. It cannot be solved analytically, because the topology of the lattice changes upon renormalization, but it has been studied numerically with the conclusion that the transition is also governed by an Infinite Disorder fixed point in dimensions [4, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13]. These numerical renormalization results are in agreement with the results of independent quantum Monte-Carlo in [14, 15].

Even if it is clear that the most natural method to study Infinite Disorder fixed points is the Strong Disorder renormalization approach, it seems useful to determine whether other approaches are able to describe Infinite Disorder scaling. In this paper, we introduce a simple non-linear transfer approximation for the surface magnetization in finite dimension , which is inspired from the ’Cavity-Mean-Field’ approximation developed in Refs [16, 17, 18], and we study numerically the critical properties of this approximation in dimensions and ,

The paper is organized as follows. In Section II, we recall briefly the ’Cavity-Mean-Field’ approximation developed in Refs [16, 17, 18] and introduce the non-linear transfer approach for finite dimensions . Our numerical results in dimension and are presented in sections III and IV respectively. Our conclusions are summarized in section V.

Ii Non-linear transfer approach for the surface magnetization

ii.1 ’Cavity-Mean-Field’ approximation on the Cayley tree [16, 17, 18]

For the random quantum Ising model model defined on a tree of coordinence , the following ’Cavity-Mean-Field’ approximation has been developed [16, 17, 18] : an ancestor is submitted to the effective single spin Hamiltonian

(2)

where is its own random transverse field, and where represents the longitudinal field created by the children (related to by the ferromagnetic couplings ) within a ’Mean-Field approximation’ ( the operator is replaced by its expectation value )

(3)

The effective Hamiltonian of Eq. 2 is only a two-level system that can be solved immediately : the magnetization of the ground state reads

(4)

Using Eq. 3, one obtains the following non-linear recurrence for the magnetizations (see Eq. 4 of [16], Eq. 7 of [17], Eq. 17 of [18] in the limit of zero temperature )

(5)

We refer to Refs [16, 17, 18] for more details on this ’Cavity-Mean-Field’ approximation and on its properties. As a final remark, let us stress that the ’Cavity-Mean-Field’ is not exact for the pure model on the Cayley tree (see Fig. 3 of Ref [18]), but has been argued to become quantitatively correct in the limit of high connectivity [16, 17, 18].

In the disordered phase where the magnetizations flows towards zero, the non-linear recurrence of Eq. 5 can be linearized to give the following recursion

(6)

which is equivalent to the problem of a Directed Polymer on the Cayley tree [16, 17, 18]. This equivalence can be justified directly at the level of lowest-order perturbation theory (i.e. without invoking the ’Cavity-Mean-Field’ approximation of Eq. 5), and can be in this way extended to the finite-dimensional case [19].

ii.2 ’Cavity-Mean-Field’ approximation in [18]

For , the Cayley tree of coordinence discussed in the previous section becomes a one-dimensional chain, and Eq. 5 becomes the one-dimensional non-linear recurrence [18]

(7)

Assuming one starts with the boundary condition at site , one obtains the following explicit expression for the surface magnetization at the site [18]

(8)

As stressed in [18], this expression exactly coincides with the rigorous expression that can be obtained from a free-fermion representation [20, 21], and from which many critical exponents can be obtained [21, 22, 23].

The reason why the ’Cavity-Mean-Field’ approximation turns out to become exact for the surface magnetization in is not clear to us, and seems rather surprising : usually ’mean-field approximation’ are exact in sufficiently high dimensions or on trees, and are not exact in low dimensions, the ’worst case’ being precisely . Here we have exactly the opposite conclusion : the ’Cavity-Mean-Field’ is not exact for the pure model on the tree (for the disordered case, it is not known), but turns out to be exact in , both for the pure and the disordered case. In the absence of any satisfactory explanation for this unusual situation, we tend to think that the exactness of the ’Cavity-Mean-Field’ in is likely to be a ’coincidence’ specific to this particular case, from which one cannot draw general conclusions for the validity of this approach in higher dimensions or for other quantum disordered models.

ii.3 Non-linear transfer approach in finite dimension

ii.3.1 Description

Figure 1: Notations to define the non-linear transfer approach in : we impose the boundary conditions on the left boundary, and we study the surface magnetizations on the right boundary (see text for more details).

In finite dimensions , the authors of Ref [18] have proposed to use the ’Cavity-Mean-Field’ approximation on a Cayley tree with parameter (to reproduce the connectivity of each spin). In the present paper, we propose instead to extend Eqs 5 towards an appropriate non-linear transfer approach for the surface magnetizations of a finite sample of volume .

For clarity, let us first explain the procedure for the case . As shown on Fig. 1, we consider a lattice containing spins : when is even (), the coordinate takes the integer-values ; when is odd (), the coordinate takes the half-integer-values . The boundary conditions are periodic in with . At , we impose the boundary condition of unity magnetization

(9)

and we are interested in the surface magnetizations at the opposite boundary .

For this situation, we propose to use the ideas of Eqs 5 within the following transfer approach. We assume that we have already found the surface magnetizations on the column , and we add another column of sites at . From the Cavity point of view, the new spin at is submitted to its own random transverse field and to the longitudinal field (see Eq. 3) created by its two neighbors on the column

(10)

so that its surface magnetization reads (Eq. 4)

(11)

Eqs 10 and 11 define a non-linear transfer procedure that can be iterated from the boundary condition on the column of Eq. 9 up to , where we analyze the statistics of the final surface magnetizations . It is clear that the generalization of this procedure to is straightforward : we add another direction with periodic boundary conditions that plays exactly the same role as .

ii.3.2 Linearized transfer matrix within the disordered phase

Within the disordered phase, the surface magnetizations are expected to decay typically exponentially in , so that one may linearize the transfer Eqs 10 and 11 to obtain

(12)

This linearized equations can be derived directly within a lowest-order perturbative approach [19] (i.e. without invoking the ’Cavity-Mean-Field’ approximation) and corresponds to the transfer matrix satisfied by the partition function of a Directed Polymer with transverse directions, as discussed in detail in [19]. We refer to [19] for the description of the consequences of this correspondence, and for the analogy with Anderson localization, where the droplet exponent of the Directed Polymer also appears in the localized phase [24, 25, 26]. Here our conclusion is that the non-linear transfer approach describes at least correctly the disordered phase, where it coincides with the lowest-order perturbative approach [19].

ii.3.3 Discussion

Besides its correctness in the disordered phase that we have just discussed, the validity of the non-linear transfer exactly at criticality and in the ordered phase has to be studied for the disordered case in . Since it has been found to be exact in (see section II.2), one could hope that it is not ’too bad’ in (even if it is clear that this approximation is not valid for the pure model) : we believe that it should capture correctly the nature of the transition between ’Infinite-Disorder’ or ’Conventional’ scaling. In the following, we present our numerical results in and and discuss the scaling properties in the two phases and at criticality.

Iii Numerical results in dimension

In this section, we present the numerical results obtained with the following sizes and the corresponding numbers of disordered samples of volume

(13)

For each sample , we collect the values of the surface magnetization at the different points of the surface (see Fig. 1). Average values and histograms are then based on these values.

We have chosen to consider the following log-normal distribution for the random transverse fields

(14)

of parameter and , whereas the ferromagnetic couplings are not random but take a single value that will be the control parameter of the quantum transition.

iii.1 Disordered phase ()

iii.1.1 Exponential decay of the typical surface magnetization

Figure 2: Disordered phase in (a) Exponential decay of the typical surface magnetization : we show the linear decay of as a function of the length (see Eq. 16). (b) Log-log plot of the width of the distribution of the logarithm of the surface magnetization as a function of (here for ) : the slope is of order (see Eq. 20).

In the disordered phase , one expects that the typical surface magnetization defined by

(15)

decays exponentially with

(16)

where represents the typical correlation length that diverges at the transition as a power-law

(17)

On Fig. 2 (a) we show our numerical results : concerning the exponential decay with of Eq. 16 for various values of . We find that the corresponding slope vanishes near the critical value with the exponent

(18)

iii.1.2 Growth of the width of the distribution of the logarithm of the surface magnetization

In the disordered phase , one expects that the width of the distribution of the logarithm of the surface magnetization defined by

(19)

grows as a power-law of

(20)

Our numerical data shown on Fig. 2 (b) correspond to the value

(21)

in agreement with the argument presented in [19] where should coincide with the Directed Polymer droplet exponent [27, 28, 29, 30].

iii.1.3 Distribution of the logarithm of surface magnetization

Figure 3: Disordered phase in (here ) : (a) Evolution with of the probability distribution of the logarithm of surface magnetization : (b) Corresponding fixed distribution of the rescaled variable in log-scale to show the tails, compared to the exact Tracy-Widom GOE distribution (thick line).

We show on Fig 3 (a) our numerical results concerning histograms of the logarithm of the surface magnetization in the disordered phase. Our conclusion is that the surface magnetization follows the scaling

(22)

where the behaviors of the typical value and of the width have been already discussed above in Eqs 16 and 20 respectively. On Fig. 3 (b), we show that the stable distribution of the rescaled variable coincides with the GOE Tracy-Widom distribution, as expected from the correspondence with the Directed Polymer model in the disordered phase [19].

iii.2 Ordered phase

iii.2.1 Behavior of the typical surface magnetization in the ordered phase

Figure 4: Ordered Phase in : (a) Behavior of the asymptotic typical surface magnetization as a function of the ferromagnetic coupling : our numerical data are compatible with an essential singularity (Eq. 24) of exponent (b) the probability distribution (here for ) of the logarithm of surface magnetization remains fixed and attached at the origin (as a consequence of the bound ).

In the ordered phase, the typical surface magnetization remains finite in the limit where the number of generations diverges

(23)

and one expects an essential singularity behavior

(24)

Our data shown on Fig. 4 (a) can be fitted with the value

(25)

that can be related to other exponents via finite-size scaling (see below around Eq. 34)

iii.2.2 Distribution of the logarithm of the surface magnetization

In the ordered phase, the probability distribution of the logarithm of surface magnetization remains fixed as varies, and terminates discontinuously at the origin, as a consequence of the bound corresponding to (see Fig. 4 (b))

iii.3 Critical point

iii.3.1 Behavior of the typical surface magnetization and of the width

Figure 5: Critical point (here ) : (a) Log-log plot of the logarithm of the typical surface magnetization and of the width : both slopes are of order (see Eqs 26 and 27) (b) Finite-size scaling of the typical surface magnetization according to Eq. 29 with , and .

Exactly at criticality, one expects that the typical surface magnetization follows an activated behavior of exponent (compare with Eq. 16 in the disordered phase)

(26)

and that the width defined in Eq. 19 is also governed by the same exponent

(27)

Our numerical data at shown on Fig. 5 (a) are compatible with these behaviors with the value

(28)

i.e. coincides with the fluctuation exponent measured in the disordered phase (see Eq. 21)

This last property implies that the finite-size scaling for the typical surface magnetization involves some correlation length exponent different from

(29)

The matching with the behavior of Eq. 22 in the disordered phase implies that

(30)

and that reads

(31)

This relation can be understood within a rare events analysis for the averaged correlation in the disordered phase [19]. The values and yield

(32)

The matching of the finite-size scaling form of Eq. 29 with the essential singularity of Eq. 24 in the ordered phase implies that

(33)

with

(34)

The values and yield

(35)

in agreement with the estimate of Eq. 25.

As shown on Fig 5 (b), our numerical data collapse well with the finite-size scaling form of Eq. 29 with .

iii.3.2 Distribution of the logarithm of surface magnetization

Figure 6: Critical point in (here ) : Evolution with of the probability distribution of the logarithm of surface magnetization. Inset : Corresponding fixed distribution of the rescaled variable .

At criticality, the rescaled variable

(36)

remains a positive random variable of order as . Our numerical measure of its probability distribution shown on Fig. 6 is compatible with a power-law singularity near the origin

(37)

with an exponent of order that we do not measure precisely. Note that this is different from the case where is finite (). We have not been able to find a physical argument to predict the value of in . This exponent will directly influence the scaling of the moments of the surface magnetization, as we now discuss.

iii.3.3 Moments of the surface magnetization

In contrast to the activated behavior of the typical surface magnetization of Eq. 26, the moments of the surface magnetization are expected to follow a power-law, as a consequence of the following rare events analysis : the surface magnetization of Eq. 36 will be of order if the random variable happens to be smaller than . Taking into account the behavior of Eq. 37, this will happen with probability

(38)

and all moments will be governed by this power-law

(39)

independently of the order .

Our numerical data for the three first moments and various sizes are compatible with Eq. 39 with an exponent

(40)

The relation of Eq. 39 then corresponds to

(41)

In the ordered phase, our numerical data are compatible with the power-law

(42)

with

(43)

Iv Numerical results in dimension

In this section, we present the numerical results obtained with the following sizes and the corresponding numbers of disordered samples of volume

(44)

For each sample , we collect the values of the surface magnetization at the different points of the surface. Average values and histograms are then based on these values. We consider again the disorder distribution of Eq. 14 and take as the control parameter of the transition.

iv.1 Disordered phase

Figure 7: Disordered phase in (here ) : (a) Evolution with of the probability distribution of the logarithm of surface magnetization : (b) Corresponding fixed distribution of the rescaled variable in log-scale to show the tails.

Our data follow the scaling of Eq. 22, with the following properties :

(i) the scaling of the typical surface magnetization is given by Eq 16, and the typical correlation length exponent of Eq. 17, seems again very close to unity

(45)

(ii) the width of Eq. 19 grows as the power-law of Eq. 20 with the exponent

(46)

that coincides with the numerical values of the droplet exponent of the Directed Polymer model with transverse dimensions [31, 32, 33, 34, 35, 36, 37], in agreement with the argument presented in [19].

(iii) As grows, the evolution of the probability distribution is shown on Fig. 7 (a). The corresponding fixed distribution of the rescaled random variable is shown on Fig 7 (b).

iv.2 Critical point

Figure 8: Typical surface magnetization in the critical region in (a) Log-log plot of the logarithm of the typical surface magnetization and of the width at criticality : both slopes are of order (see Eqs 26 and 27 ) (b) Finite-size scaling of the typical surface magnetization in according to Eq. 29 with .

At criticality , we find that the exponent of Eqs 26 and 27 coincides with the value of of Eq. 46 concerning the disordered phase (see Fig. 8 (a))

(47)

We show on Fig. 8 (b) the finite-size scaling analysis of Eq. 29 for the logarithm of the typical surface magnetization with the averaged correlation length exponent .

Figure 9: Critical point in (here ) : Evolution with of the probability distribution of the logarithm of surface magnetization. Inset : Corresponding fixed distribution of the rescaled variable .

We show on Fig. 9 our numerical data for the probability distribution of the surface magnetization at criticality : the fixed point distribution of the rescaled variable of Eq. 36 displays a power-law singularity near the origin (Eq 37), that will determine the scaling of all moments of the surface magnetization according to Eq. 39. Our numerical data for the moments are compatible with Eq. 39 with an exponent

(48)

that would correspond to

(49)

and to the exponent (Eq. 42)

(50)

V Conclusion

Since the ’Cavity-Mean-Field’ approximation developed for the Random Transverse Field Ising Model on the Cayley tree [16, 17, 18] has been found to reproduce the known exact result for the surface magnetization in [18], we have proposed to extend these ideas in finite dimensions via a non-linear transfer approach for the surface magnetization. In the disordered phase, the linearization (Eq 12) of the transfer equations correspond to the transfer matrix for a Directed Polymer in a random medium of transverse dimension , in agreement with the leading order perturbative scaling analysis [19].

We have presented numerical results of this non-linear transfer approach in dimensions and , where large system sizes can be easily studied. In both cases, we have found that the critical point is governed by Infinite Disorder scaling. In particular exactly at criticality, the one-point surface magnetization scales as , where coincides with the droplet exponent of the corresponding Directed Polymer model, with and . The distribution of the positive random variable of order presents a power-law singularity near the origin with so that all moments of the surface magnetization are governed by the same power-law decay with independently of the order . Our conclusion is thus that this non-linear transfer approach is able to lead to Infinite Disorder scaling, that had been found previously via Monte-Carlo in [14, 15] and via Strong Disorder RG in [4, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Exactly at criticality, the presence of activated scaling means that the linearization of Eq. 12 is typically still valid also at criticality (and not only in the disordered phase), so that the identity can be understood. The rare cases where this linearization is not valid at criticality is when the positive random variable happens to be smaller than . Our conclusion is thus the following :

(i) in the disordered phase and for ’typical’ situations exactly at criticality, the linearization of Eq. 12 is valid and coincides with the leading order perturbative scaling analysis [19] : it is thus likely to give exact values for critical exponents, in particular for the exponent of activated scaling.

(ii) in the ordered phase and for ’rare’ situations at criticality, the non-linear terms of the transfer approach plays an important role. Since they come from an uncontrolled approximation, the critical exponents like and that are determined by these non-linear contributions could be different from the exact ones. To judge the accuracy of this approximation, it would be very helpful to compare with other approaches like Quantum Monte-Carlo and Strong Disorder RG (but up to now, these other approaches have not studied surface properties).

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