# Strong-randomness infinite-coupling phase in a random quantum spin chain

###### Abstract

We study the ground-state phase diagram of the Ashkin-Teller random quantum spin chain by means of a generalization of the strong-disorder renormalization group. In addition to the conventional paramagnetic and ferromagnetic (Baxter) phases, we find a partially ordered phase characterized by strong randomness and infinite coupling between the colors. This unusual phase acts, at the same time, as a Griffiths phase for two distinct quantum phase transitions both of which are of infinite-randomness type. We also investigate the quantum multi-critical point that separates the two-phase and three-phase regions; and we discuss generalizations of our results to higher dimensions and other systems.

###### pacs:

75.10.Nr, 75.40.-s, 05.70.Jk## I Introduction

Random quantum many-particle systems are easiest to understand if both interactions and disorder are weak. In these cases, the system often behaves analogously to a clean noninteracting one, with small perturbative corrections. If, on the other hand, interactions or disorder are strong, qualitatively new behavior can arise. For instance, repulsive interactions induce a new phase, the Mott insulator, in systems of lattice bosons or electrons. Moreover, strong randomness leads to an Anderson insulator in which the quantum wave functions are localized.

Particularly strong disorder and correlation effects can be expected in the vicinity of zero-temperature quantum phase transitions where the fluctuations extend over large length and time scales. Examples include infinite-randomness criticality Fisher (1992, 1995), quantum Griffiths singularities Thill and Huse (1995); Rieger and Young (1996) and smeared phase transitions Vojta (2003) (for recent reviews see, e.g., Refs. Vojta (2006, 2010)).

Disordered quantum spin chains are a paradigmatic class of materials to study these phenomena, both in theory and in experiment. Theoretically, they have been attacked by strong-disorder renormalization group (SDRG) methods Ma et al. (1979); Igloi and Monthus (2005) that give asymptotically exact results for a number of one-dimensional systems. The ground state of the antiferromagnetic spin- random quantum Heisenberg chain is an exotic random-singlet state controlled by an infinite-randomness renormalization group fixed point Fisher (1994). Similarly, the ferromagnetic-paramagnetic quantum phase transition of the random transverse-field Ising chain is of unconventional infinite-randomness type and accompanied by power-law quantum Griffiths singularities Fisher (1995). Some of these phenomena have been observed in early experiments on organic crystals Theodorou and Cohen (1976); Tippie and Clark (1981) and more recently in MgTiOBO Rappoport et al. (2007).

In this paper we investigate the random quantum Ashkin-Teller model, a prototypical disordered spin chain (or ladder) that can be understood as two coupled random quantum Ising chains. In addition to quantum spin systems, versions of the Ashkin-Teller model are used to describe layers of atoms absorbed on surfaces Bak et al. (1985), current loops in high- superconductors Aji and Varma (2007); *AjiVarma09, as well as the elastic response of DNA molecules Chang et al. (2008).

We explore the ground state phase diagram of the random quantum Ashkin-Teller chain by a generalization of the SDRG technique. In addition to the conventional paramagnetic and ferromagnetic phases, we identify an unconventional partially ordered phase characterized by finite but strong randomness and infinite coupling between the two constituent Ising chains (see Fig. 1).

It plays the role of a “double-Griffiths” phase for two separate quantum phase transitions both of which are of infinite-randomness type. The two-phase region at weak coupling and the three-phase region at strong coupling are separated by a distinct infinite-randomness multi-critical point.

The remainder of this paper is organized as follows. We introduce the random quantum Ashkin-Teller model in Sec. II. The SDRG method is developed in Sec. III. Section IV is devoted to the resulting ground state phase diagram and the properties of the quantum phase transitions between the different phases. In the concluding section V, we discuss generalizations of our results to higher dimensions as well as connections to other random quantum systems.

## Ii Random quantum Ashkin-Teller Model

The Hamiltonian of the one-dimensional random quantum Ashkin-Teller model is given by Ashkin and Teller (1943); Kohmoto et al. (1981); Carlon et al. (2001)

(1) | |||||

where and denote the usual Pauli matrices. The model can be understood as two identical random transverse-field Ising chains [first line of (1)], coupled via their energy densities [second line of (1)]. The index that distinguishes the two chains is often called the color index. The strength of the coupling between the colors can be parameterized by the ratios and . Note that the Hamiltonian (1) is invariant under the duality transformation: , , , and , where and are the dual Pauli operators.

We take the interactions and transverse fields to be
independent random variables. Without loss of generality, we can assume the
and to be positive as possible negative signs can absorbed by local
transformations of the spin variables.
For now, we assume the (bare) coupling strengths to be uniform,
^{1}^{1}1Even if we assume uniform, nonrandom values of and ,
they will acquire randomness under renormalization..
Effects of random will be discussed later in Sec. IV.3.

The behavior of the random quantum Ashkin-Teller chain (1) in the weak-coupling regime, , has been studied in Refs. Carlon et al. (2001); Goswami et al. (2008). In the following, we therefore focus on the strong-coupling case where these results do not apply. For strong coupling, the terms in the second line of (1) dominate. It is thus convenient to introduce the product as a new variable. We define

(2) | |||||

(3) | |||||

(4) |

The mapping of the Pauli matrices and can be easily worked out by exploring their action on a complete set of basis states in the 4-dimensional single-site Hilbert space. This gives

(5) | |||||

(6) | |||||

(7) |

Using these transformations, the Hamiltonian (1) can be rewritten as

(8) | |||||

This form immediately gives an intuitive physical picture of the strong coupling regime close to self duality, , i.e., close to the horizonal line in Fig. 1. Here, the typical values of the fields and interactions are defined as and where denotes the disorder average. The behavior of the product variable is dominated by the four-spin interactions while the behavior of the variable which traces the original spins is dominated by the two-spin transverse fields . Moreover, the coupling terms between and are weak. Thus, we expect the system to be in a phase in which the product variables develop long-range order while the spins remain disordered.

## Iii Strong-disorder renormalization group

To confirm this intuitive picture and to work out the properties of the product phase and its transitions, we now develop a strong-coupling SDRG. The basic idea of any SDRG consists in identifying the largest local energy scale and perturbatively integrating out the corresponding high-energy degree of freedom. As the random quantum Ashkin-Teller model contains four competing local energies rather than the usual two, we need to generalize the RG scheme by also considering the second-largest energy in a local cluster. Details of this calculation are outlined in Appendix A. In the strong-coupling regime, , there are four possible SDRG steps.

(a) If the largest energy in the system is the two-spin field , and the second largest energy in the local cluster of sites and is a four-spin interaction, say , the SDRG step decimates the variable but merges and to a new cluster . The unperturbed Hamiltonian for this SDRG step reads . We now keep only the ground state of and treat all other terms that contain or in second order perturbation theory. The resulting Hamiltonian has the same form as (8) with one fewer site and renormalized energies arranged as shown in Fig. 2.

(9) |

As and are renormalized downward while all remaining and are unchanged, the coupling strengths and increase under renormalization.

(b) The same SDRG step is carried out if the largest energy is the four-spin interaction , and the second-largest energy in the cluster of sites and is a two-spin field, say .

(c) If the largest energy in the system is the two-spin field , and the second largest energy in the local cluster of sites and is the field , both and are decimated. This is equivalent to decimating both original spins and and leads to the recursion relations

(10) |

for the interaction energies that emerge between sites and in the renormalized chain. This implies that the renormalized coupling strength increases under renormalization (as we are interested in the strong-coupling regime ).

(d) Finally, if the largest energy is the four-spin interaction , and the second-largest energy associated with the sites and is the interaction , clusters are formed from and as well as and . This is equivalent to forming clusters of both original spin variables, and as well as and . The resulting recursions for the transverse field and two-spin field acting on these clusters read

(11) |

The renormalized coupling strength increases under renormalization.

The SDRG steps (a) to (d) are now iterated. As a result, the maximum local energy in the system gradually decreases from its initial (bare) value .

## Iv Phase diagram and phase transitions

### iv.1 Double Griffiths phase

Based on the SDRG recursions (9) to (11), the phase diagram of the random quantum
Ashkin-Teller model shown in Fig. 1 is easily understood. Let us start by recalling that in the weak-coupling
regime, , the local coupling strengths and decrease without limit
under renormalization Carlon et al. (2001); Goswami et al. (2008). This
implies that the two Ising chains that make up the Ashkin-Teller model decouple in the low-energy limit.
Our system thus behaves analogously to the random transverse-field
Ising chain Fisher (1992, 1995): A paramagnetic phase at large transverse fields and a
ferromagnetic phase at large interactions are directly connected by an infinite-randomness critical
point at (transition 1 in Fig. 1)
^{2}^{2}2The position of this phase boundary is fixed by the self-duality of the Hamiltonian..

To understand the strong-coupling regime , let us first focus on the self-duality line . If the bare is just slightly above 1, most of the recursions will initially be site and bond decimations [types (c) and (d)]. In these steps, the local coupling strengths and rapidly increase. When they become larger than the widths of the and distributions, the character of the SDRG changes. Now, most steps are “mixed steps” of types (a) and (b). As a result, the product variable forms larger and larger clusters while the spin variable is decimated.

The system is thus in a “double Griffiths phase:” The -part of the Hamiltonian behaves analogously to an ordered Griffiths phase while the -part behaves as in a disordered Griffiths phase. This double Griffiths phase has a nonzero product order parameter or polarization while the spin variable (and thus and ) remains disordered, . Note that this behavior is valid not just on the self-duality line, , but also in its vicinity because the RG flow of each of the variables and is dominated by a single term in the Hamiltonian and does not rely on the balance between interactions and transverse fields. Thus, we have indeed discovered a bulk phase rather than a special line in the phase diagram. Moreover, as the and flow to infinity, the analysis is asymptotically exact.

To find the extensions of the partially ordered double Griffiths phase we need to locate its transitions to the conventional paramagnetic and ferromagnetic phases. Looking at the first sum in the Hamiltonian (8), it is clear that the long-range order of the product variable will be destroyed if we raise until the transverse fields compete with the four-spin interactions . This leads to a competition between the SDRG steps (a) and (c). From comparing the recursion in (9) with the recursion in (10), we conclude that the phase boundary between the double Griffiths phase and the paramagnetic phase (transition 2 in Fig. 1) is located at or equivalently in the limit of large . Moreover, the transition is governed by an infinite-randomness fixed point in the random transverse-field Ising universality class. The phase boundary to the ferromagnetic phase (transition 3 in Fig. 1) can be found analogously. For large , it is located at or equivalently in agreement with the self-duality of the Hamiltonian.

The thermodynamics of the double Griffiths phase is highly unusual. It can be found in the usual way, i.e., by including conjugate fields into the SDRG. Each of the two order parameters, the magnetization and the polarization , displays power-law quantum Griffiths singularities controlled by different Griffiths dynamical exponents and , respectively, that vary non-universally with and . The exponent diverges at the transition to the ferromagnetic phase while diverges at the transition to the paramagnetic phase. Duality imposes the relation . Thermal quantities such as the entropy and the specific heat pick up contributions from both order parameters. Their Griffiths dynamical exponent thus displays an interesting non-monotonous dependence on , as sketched in Fig. 3.

### iv.2 Multicritical point

Finally, we consider the infinite-randomness multicritical point (MCP) located at .
It has two independent unstable directions, the lines and .
On the line that separates the weak-coupling and strong-coupling regimes,
all SDRG steps are site and bond decimations (types (c) or (d)). The recursions
(10) and (11) reduce to the well-known transverse-field
Ising forms
and ^{3}^{3}3The extra factors of
2 in the denominator are irrelevant in the low-energy limit.. The SDRG flow of the
and distributions on the line is thus identical to the
corresponding flow of the random-transverse-field Ising chain.
We emphasize, however, that although the SDRG flow of the and distributions
at is identical to the weak-coupling regime , the fixed-point Hamiltonian differs
because the two Ising chains that make up the Ashkin-Teller model do not decouple.

The flow along the line can be characterized by the following critical singularities: correlation length , magnetization , and correlation time with exponents

(12) |

In contrast, the SDRG flow on the self-duality line for close to the MCP is determined by the evolution of under repeated site and bond decimations (steps (c) and (d)). It can be worked out (see Appendix B) by including as an auxiliary variable in the SDRG flow of the and distributions. We find different critical singularities and with exponents

(13) |

The tunneling exponent remains 1/2. Combining these results to write a scaling form of the polarization gives

(14) |

where is an arbitrary scale factor. The phase transition between the partially ordered and paramagnetic phases corresponds to a singularity of for and . Using (14), we find that the phase boundary behaves as

(15) |

sufficiently close to the multicritical point. The phase boundary to the ferromagnetic phase can be found analogously.

### iv.3 Random coupling strength

So far, we have considered systems in which the (bare) coupling strengths are uniform . In the present section, we discuss what changes for random coupling strengths.

If all and are below the multicritical value of 1, the renormalized values are also smaller than 1 and decrease under renormalization. Thus, the two Ising chains that make up the Ashkin-Teller model decouple in the low-energy limit, just as in the case of uniform bare . Conversely, if all and are above the multicritical value of 1, the renormalized values are also larger than 1 and increase under renormalization. The system thus flows to the strong coupling region, also just as in the case of uniform bare . Consequently, none of our results change in these two cases, except for unimportant modifications of nonuniversal quantities. This also implies that the three bulk phases shown in Fig. 1 are stable against weak randomness in . The same holds for the phase transitions (1), (2) and (3) sufficiently far away from the multicritical point discussed in Sec. IV.2.

In contrast, the multicritical point at itself is unstable against weak disorder in the and . To show this, we analyze how the width of a narrow -distribution around flows under repeated SDRG site and bond decimations. By including as an auxiliary variable in the SDRG and using the methods of Ref. Fisher (1994), we find

(16) |

(Note that we need to consider the flow of a symmetrically distributed auxiliary variable; the exponent is therefore denoted as .) This means that a narrow bare distribution broadens under the SDRG, destabilizing the uniform- multicritical point of Sec. IV.2.

We have not found an analytic solution of the multicritical behavior in the case of random and . Instead, we implement the SDRG numerically. We study systems with up to sites. To place the system on the self-duality line , we employ identical power-law distributions and for the interactions and transverse fields, with being a measure of the disorder. The coupling strengths are drawn from a box distribution between and . The results of a strongly disordered () example system are summarized in Figs. 4 and 5. We fix and tune the multicritical point by varying . The data are averages over 50 different chains of sites each. Fig. 4 shows how the average and standard deviation of the coupling strength evolve under the SDRG.

From the inset, we determine the multicritical point to be located between and 644. Moreover, increases as with with the SDRG length scale. Here, the number in brackets gives the error of the last digit. This error is mostly due to the uncertainty in precisely locating the multicritical point. The statistical error is much smaller. As the tunneling exponent remains at , this implies

(17) |

The value of the exponent
^{4}^{4}4We call this exponent rather than (as was done in Ref. Fisher (1994))
to avoid confusion with the tunneling exponent.
fulfills the constraint derived by Fisher Fisher (1994).
Interestingly, it is not very different from the value that describes
the initial increase of near the uniform- multicritical point.

In Fig. 5, we study how the distance from the multicritical point increases with SDRG length scale in the regime .

We find
with
^{5}^{5}5Our exponent is equivalent to the exponent used in Ref. Fisher (1994)
to describe the scaling of the average anisotropy in an XXZ spin chain..
Expressed in terms of , this means

(18) |

Again, the error is mostly due to uncertainties in the location of the multicritical point as well as the fit range.

We have performed analogous calculations for a number of different parameter sets. For the weaker disorder case of and , the multicritical point is located at . In this case, our analysis of 180 chains of sites gives the same value as above, . The exponent is somewhat harder to determine in the weak-disorder case because the available fit range becomes very narrow. We find in agreement with the strong-disorder value. Further calculations for even weaker disorder and shorter chains (between and sites) are less precise but compatible with the values given above.

Once , almost all are on the same side of the multicritical point. The further analysis therefore follows the steps outlined in Appendix B. The resulting multicritical behavior along the self-duality line on the strong-coupling side of the MCP is characterized by the power laws and with exponents

(19) |

The shape of the phase boundary close to the multicritical point can be found as in Sec. IV.2 yielding .

## V Discussion and Conclusions

In summary, we have investigated the ground state phase diagram of the random quantum Ashkin-Teller spin chain. The topology of the phase diagram, shown in Fig. 1, is analogous to that of the clean quantum Ashkin-Teller model (see, e.g., Ref. Igloi and Solyom (1984)). However, the properties of the phases and phase transitions are different. In addition to the usual paramagnetic and ferromagnetic phases, we have identified a partially ordered phase characterized by strong randomness and infinite coupling between the colors. This phase acts as a Griffiths phase for two distinct quantum phase transitions leading to an unconventional non-monotonic variation of the Griffiths dynamical exponent throughout the phase.

We now turn our attention to the phases boundaries between the three phases. The direct transition at weak intercolor coupling between the paramagnetic and ferromagnetic (Baxter) phases (transition (i) in Fig. 1) is in the infinite-randomness universality class of the random transverse-field Ising chain, as was already found in Refs. Carlon et al. (2001); Goswami et al. (2008). In contrast, the corresponding phase boundary in the clean quantum Ashkin-Teller chain shows an unusual line of fixed points with continuously varying exponents Kohmoto et al. (1981); Baxter (1982). The quantum phase transitions separating the partially ordered phase from the paramagnetic and ferromagnetic phases (transitions (ii) and (iii)) are also of infinite-randomness type and in the universality class of the random transverse-field Ising chain, while they are in the (1+1)-dimensional Ising universality class in the clean model.

We have also studied the quantum multicritical point separating the two-phase and three-phase regions. It is in one of two different universality classes (both of infinite-randomness type), depending on whether the intercolor coupling strengths are uniform or random. This differs from the infinite-order multicritical behavior seen in the clean case Kohmoto et al. (1981); Baxter (1982).

Generalizations of the Ashkin-Teller Hamiltonian (1) to colors have recently reattracted considerable attention because they have been used to analyze the fate of first-order quantum phase transitions under the influence of disorder Goswami et al. (2008); Greenblatt et al. (2009); Hrahsheh et al. (2012). Interestingly, for colors, the paramagnetic and ferromagnetic phases meet directly at the self-dual line for all coupling strength . Thus an analog to the partially ordered strong-coupling phase does not exist. For three and four colors, this question is not yet solved to the best of our knowledge.

The random quantum Ashkin-Teller chain (1) with sites can be mapped onto a random XXZ quantum spin chain with sites Alcaraz et al. (1988). Under this mapping, the transverse fields in the Ashkin-Teller model map onto the even bonds of the XXZ chain while the interactions map onto the odd bonds. The coupling strengths and map onto the local anisotropies of the XXZ chain. Importantly, the mapping is nonlocal as it involves (semi-infinite) chains of operators. Thus, although the energy spectra of the Ashkin-Teller model and the XXZ chain are analogous, their order parameters are not directly related. This explains, for example, why the correlation length exponent given in (13) takes the same value as the exponent that describes the effects of weak anisotropy about the Heisenberg fixed point of the XXZ chain Fisher (1994). In contrast, our order parameter exponent does not have a direct counterpart in the XXZ chain.

Our study has focused on one space dimension. Let us briefly comment on random quantum Ashkin-Teller models in higher dimensions. The crucial step in our understanding of the strong-coupling regime was the transformation defined in eqs. (2–7) from the original spins to the product variable. This transformation is purely local and can be performed in the same way in any space dimension. We therefore believe that the basic features of the phase diagram in higher dimensions will be similar to the one dimensional case. In particular, for small , we expect a direct transition between the ferromagnetic and paramagnetic phases while a partially ordered product phase is expected to intervene between them for large . Obtaining quantitative results in higher dimensions will be significantly more complicated than in one dimension. First, the Hamiltonian is not self-dual in , thus the phase diagram is not symmetric under the exchange of transverse fields and interactions. Second, the SDRG can only be implemented numerically in because the decimation steps change the topology of the lattice. This work remains as a task for the future.

## Acknowledgements

This work was supported by the NSF under Grant Nos. DMR-1205803 and PHYS-1066293, by Simons Foundation, by FAPESP under Grant No. 2013/09850-7, and by CNPq under Grant Nos. 590093/2011-8 and 305261/2012-6. R.N. acknowledges the hospitality of the Physics Department of Missouri S&T where this works was initiated. J.H. and T.V. acknowledge the hospitality of the Aspen Center for Physics.

## Appendix A SDRG recursion relations

A single step of the SDRG consists in identifying the largest local energy scale in the Hamiltonian and perturbatively integrating out the corresponding high-energy excitations. This is done using the projection technique described, e.g., in Ref. Auerbach (1998). The Hilbert space is divided into a low-energy subspace and a high-energy subspace. Any wave function can be decomposed as with in the low-energy subspace and in the high-energy subspace. This allows us to write the Schroedinger equation in matrix form

(20) |

with . Here, and project on the low-energy and high-energy subspaces, respectively. Eliminating from these two coupled equations gives . Thus, the effective Hamiltonian in the low-energy Hilbert space is

(21) |

The second term can now be expanded in inverse powers of the large local energy scale.

The quantum Ashkin-Teller Hamiltonian has four competing local energy scales, viz., , and rather than two. We therefore generalize the usual SDRG scheme by considering the largest and second-largest energies in a local cluster to define the SDRG step. In the strong-coupling regime, , the largest local energy is always either a four-spin interaction or a two-spin field. In total, there are four possible steps.

(a) The largest local energy is a two-spin field . The second largest energy in the three-site cluster of sites , , is a four-spin interaction, either or . Let us assume that it is for definiteness. In this case, the low-energy Hilbert space is spanned by states for which or . and contain all terms in the Hamiltonian that do not flip the spins ; their leading terms are . All terms that flip at least one of the variables are contained in and . Specifically,

(22) | |||||

takes the same form but with and exchanged. We now insert and into (21) and approximate the denominator by or depending on which of the spins is flipped. The resulting effective Hamiltonian has the same form (8) as the initial one, but with one fewer site. The arrangement of the renormalized energies and between the remaining sites is shown in Fig. 2 and their values are given in (9).

(b) Exactly the same SDRG step is carried out if the largest local energy is the four-spin interaction , and the second-largest energy in the two-site cluster of sites and is a two-spin field, either or .

Steps (a) and (b) are the dominant SDRG steps for . More precisely, most steps are of types (a) or (b) if is larger than the width of the and distributions (on a logarithmic scale). In the opposite case, strong disorder and not too large , most SDRG steps are site and bond decimations of types (c) and (d).

(c) The largest energy in the system is the two-spin field , and the second largest energy in the local cluster of sites and is the field . In this case, the low-energy Hilbert space is spanned by all states having . and contain all terms in the Hamiltonian that do not flip and , with the leading terms being . All terms that flip and/or are part of and . Specifically,

(23) | |||||

and takes the same form but with and exchanged. After inserting this into (21) and approximating the denominator by or depending on which spins are flipped, site is eliminated (i.e., both and are decimated). The effective interaction energies between the neighboring sites and are given in (10).

(d) The largest local energy is a four-spin interaction , and the second largest energy in the cluster consisting of sites and is the interaction . The low-energy Hilbert space is spanned by states having or or or . After projection into the low-energy Hilbert space, the two sites and can thus be represented by a single site with variables and . and contain all terms in the Hamiltonian that do not flip or . The leading terms are . In contrast, and consist of the terms that flip and/or . This gives

(24) | |||||

Inserting this into the effective Hamiltonian (21) as before yields the transverse field and two-spin field acting on the cluster variables and . Their values are given in (11).

## Appendix B Multicritical point

The multicritical point separating the two-phase and three-phase regions is located at . In this appendix, we sketch the derivation of the SDRG flow on the self-duality line for but close to the multicritical point.

Let us begin with a qualitative discussion. For and strong disorder, initially almost all SDRG steps are site decimations (c) or bond decimations (d), thus the RG flow is identical to that of the random transverse-field Ising chain. Under these steps, increases rapidly. When the typical reaches the width of the and distributions, the character of the SDRG flow changes. Now, most steps are “mixed” decimations of types (a) and (b). Under these steps, the magnetization rapidly drops to zero while the polarization (product order parameter) stops decreasing and reaches a nonzero asymptotic value. Thus, the RG scale at which reaches the width of the and distributions determines the correlation length and the polarization.

For a quantitative analysis of this SDRG flow, we start from the recursion relations for the coupling strengths and defined by (10) and (11). Written in terms of logarithms, they read

(25) | |||||

(26) |

We follow the -flow from to where and are the inverse widths of the and distributions. Two regimes need to be distinguished, and .

For , we expand in , and equations (25) and (26) simplify to

(27) | |||||

(28) |

The recursions can be understood as special cases of the general recursion with
^{6}^{6}6 comprises both site and bond , arranged in an alternating fashion..
The flow of variables governed by such recursions close to the infinite-randomness fixed point (of the random transverse-field
Ising chain) was studied in detail by
Fisher Fisher (1994). He found that the typical scales like
with decreasing SDRG energy scale . The exponent is given by .
(In contrast to (16), we need to use the “asymmetric” version of Fisher’s results because all our .)
Thus, in the first regime (), the typical scales as

(29) |

In the second regime, , we can approximate the recursions (25) and (26) for by

(30) | |||||

(31) |

These recursions are of the same type as (27) and (28), but with . Thus, in the second regime, scales as

(32) |

To test the predictions (29) and (32), we implemented the strong-disorder renormalization group numerically. Figure 6 shows as a function of for systems located on the self-duality line . We employed identical power-law distributions and for the interactions () and transverse fields (), with being a measure of the disorder. The coupling strength is uniform and close to the multicritical value .

The figure shows that the data in the range lie on straight lines, i.e., they follow (32) as predicted. For the data curve downward suggesting a smaller exponent. In fact, the data in the range can be very well fitted with functions of the form , in agreement with (29).

Let us now combine the two regimes. We consider a (bare) system close to the multicritical point, , with strong initial disorder, i.e., the widths of the bare distributions of and are large, . Under repeated site and bond decimations (SDRG steps c and d), while the typical scales as once . Setting gives the crossover SDRG scale

(33) |

The correlation length is given by the length scale corresponding to ,

(34) |

The correlation length exponent thus takes the value as given in (13). The product order parameter (polarization) can be found by noting that -clusters are not decimated anymore once . is thus given by its value at .

(35) |

where and are the number and moment of clusters surviving at SDRG scale . Using yields the order parameter exponent given in (13).

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- (26) Our exponent is equivalent to the exponent used in Ref. Fisher (1994) to describe the scaling of the average anisotropy in an XXZ spin chain.
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- (33) comprises both site and bond , arranged in an alternating fashion.