Magnetically ordered phase near transition to Bose-glass phase

Magnetically ordered phase near transition to Bose-glass phase

A. V. Syromyatnikov asyromyatnikov@yandex.ru    A. V. Sizanov alexey.sizanov@gmail.com National Research Center ”Kurchatov Institute” B.P. Konstantinov Petersburg Nuclear Physics Institute, Gatchina 188300, Russia Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034 Russia
August 23, 2019
Abstract

We discuss magnetically ordered (“superfluid”) phase near quantum transition to Bose-glass phase in a simple modeling system, Heisenberg antiferromagnet in spatial dimension in external magnetic field with disorder in exchange coupling constants. Our analytical consideration is based on hydrodynamic description of long-wavelength excitations. Results obtained are valid in the entire critical region near the quantum critical point (QCP) allowing to describe a possible crossover from one critical behavior to another. We demonstrate that the system behaves in full agreement with predictions by Fisher et al. (Phys. Rev. B 40, 546 (1989)) in close vicinity of QCP. We find as an extension to that analysis that the anomalous dimension and , where and are critical exponents of the order parameter and the correlation length, respectively. The density of states per spin of low-energy localized excitations are found to be independent of (“superuniversal”). We show that many recent experimental and numerical results obtained in various 3D systems can be described by our formulas using percolation critical exponents. Then, it is a possibility that a percolation critical regime arises in the ordered phase in some 3D systems not very close to QCP.

pacs:
64.70.Tg, 72.15.Rn, 74.40.Kb

I Introduction

The problem of interacting bosons in disordered potential (the so-called dirty-boson problem) has been an active field in recent years beginning with the seminal paper by Fisher et al. Fisher et al. (1989) The recent upsurge of interest to dirty bosons is stimulated by their novel experimental realizations in ultracold atoms in optical lattices with controlled disorder (see, e.g., Ref. Sanchez-Palencia and Lewenstein (2014)) and in doped quantum magnets (see Ref. Zheludev and Roscilde (2013) for review).

The most pronounced effect of disorder is the localization of single-particle low-energy states which cannot host a Bose-Einstein condensate at finite interaction between bosons. As a result, a novel non-superfluid glassy phase arises, the so-called Bose-glass (BG) phase, which intervenes between the superfluid (SF) and Mott insulating (MI) ones and which destroys MI phase at strong enough disorder. Fisher et al. (1989) Due to finite density of states of localized gapless quasiparticles, BG phase (as SF one) has a finite compressibility. In contrast, MI phase is incompressible. The transition from MI to BG phases is found to be of the Griffiths type. Fisher et al. (1989); Griffiths (1969); Gurarie et al. (2009) Properties of the transition from SF to BG phases are in hot debate now. The situation is greatly complicated by the fact that many conventional approaches are not appropriate for dirty bosons. In particular, a perturbative renormalization group treatment breaks down because the upper critical dimension is infinitely large. Fisher et al. (1989) As a consequence, a rigorously motivated scaling form is absent of the free energy so that the simplest scaling ansatz proposed in Ref. Fisher et al. (1989) is called in question recently Yu et al. (2012a); Weichman and Mukhopadhyay (2007) in view of some apparent contradictions to numerical and experimental data.

It is shown in Ref. Fisher et al. (1989) that the finiteness of the compressibility at the quantum critical point (QCP) between SF and BG phases implies that the dynamical critical exponent is equal to the spatial dimension of the system . Numerical results obtained in the largest 2D Prokof’ev and Svistunov (2004) and 3D Yao et al. (2014) systems do support this conclusion. Besides, an alternative derivation of this relation is proposed in Ref. Yu et al. (2012a) which relies only on the assumption that the system is not above the upper critical dimension. Then, the equality seems to be well established at QCP.

Another consequence of the scaling ansatz proposed in Ref. Fisher et al. (1989), , is debated now, where and are critical exponents of the critical temperature and the correlation length (here is a control parameter and is its value at QCP). There is a large amount of experimental and numerical data obtained in various 3D systems which are in sharp contradiction with this relation being quite consistent with each other (see Refs. Zheludev and Roscilde (2013); Yu et al. (2012b, a); Hüvonen et al. (2012); Wulf et al. (2013); Yamada et al. (2011); Yu et al. (2010, ) and references therein). To resolve this discrepancy, a numerical investigation has been carried out recently in all-time large 3D systems Yao et al. (2014) which results are fully consistent with Ref. Fisher et al. (1989). It is claimed in Ref. Yao et al. (2014) that previous studies were performed away from quantum critical region (in particular, because of too small system sizes considered in previous numerical works). This statement is disputed in Ref. Yu et al. , where, in particular, the consistency is stressed between data obtained experimentally and numerically in various systems that testifies a universal behavior differing from that predicted in Ref. Fisher et al. (1989). As a possible source of discrepancy authors of Ref. Yu et al. point out that the results of Ref. Yao et al. (2014) characterize the neighborhood of one special point of the phase diagram in the plane of disorder strength and chemical potential whereas previous considerations which contradict Ref. Yao et al. (2014) relate to another part of the phase diagram. Our results below could support a compromise idea: the existence of a crossover in a part of the phase diagram from the critical behavior proposed in Ref. Fisher et al. (1989) to another one which is described by percolation critical exponents (or by critical exponents which are close to percolation ones).

We discuss QCP between magnetically ordered (“SF”) and BG phases in a simple modeling system from the considered universality class, Heisenberg antiferromagnet (HAF) on hypercubic -dimensional lattice whose Hamiltonian has the form

(1)

where denote nearest-neighbor sites, is the value of the external magnetic field, and disorder in exchange coupling constants is implied. As it is well known, model (1) can be mapped onto an extended Bose-Hubbard model that is frequently used for discussion of dirty bosons (see, e.g., Ref. Zheludev and Roscilde (2013)). We develop a hydrodynamic description of long-wavelength excitations in the ordered phase whose spectrum is determined by static observable quantities: the static susceptibility and the helicity modulus, which is characterized by critical exponent . We describe model (1) near QCP using general properties of a system in a critical regime and not involving any scaling ansatz. We demonstrate that model (1) behaves in full agreement with predictions by Fisher et al. (Ref. Fisher et al. (1989)) in the close vicinity of QCP. In particular, we recover many relations between critical exponents derived in Ref. Fisher et al. (1989) from the scaling ansatz (including ). As an extension to that analysis, we find that the anomalous dimension and the critical exponent of the order parameter (the latter is in agreement with numerical results Yao et al. (2014)). The density of states (DOS) is also derived. DOS per spin of low-energy localized excitations are found to be “superuniversal” (i.e., independent of ). An alternative derivation is suggested of the equality at QCP based on common scaling arguments. We demonstrate that many recent experimental and numerical data obtained in various 3D systems can be described by our formulas using percolation critical exponents. This finding implies that a percolation critical regime can arise in the ordered phase not very close to QCP at which is characterized by critical exponents and .

The rest of the present paper is organized as follows. We present a qualitative consideration of the problem and summarize our basic assumptions in Sec. II.1. We remind briefly in Sec. II.2 the approach applied before to diluted HAFs near the percolation threshold and adopt it for consideration of our system in Sec. II.3. Formulas derived in Sec. II.3 which are valid in the whole critical region are applied in Sec. III for discussion of the close vicinity of QCP. The possible percolation scenario in some 3D systems is discussed in Sec. IV. Sec. V contains a summary and our conclusion. One appendix is included with some details of the analysis.

Ii Antiferromagnets with disorder near QCP

ii.1 Qualitative consideration and basic assumptions

In the absence of disorder, a transition takes place in model (1) at some field value from magnetically ordered (“SF”) phase with Goldstone excitations to fully saturated (magnetically disordered “MI”) phase with gapped spectrum (the gap value is given by ). Mean transverse spin component plays the role of the order parameter in this transition. This quantum phase transition belongs to the universality class of Bose-Einstein condensation characterized by mean-field critical exponents , , and . Sachdev (2001)

We assume below that values of a small part of randomly chosen exchange constants are increased. As a result, one has to apply a field greater than to saturate spins around defect bonds. It is natural to expect also that local values of the saturation field are greater in regions with more dense distribution of defects. Then, the following qualitative picture arises. At , all spins have finite mean transverse components. As the field exceeds , less dense regions having smaller local critical fields become magnetically disordered (i.e., saturate) and the magnetically ordered part of the system acquires the form of an infinite network. Some regions (which are surrounded by areas with lower local critical fields) leave the infinite network before approaches their local critical field values. These regions, being isolated from each other and from the infinite network, do not contribute to the net order parameter of the sample which is determined solely by that of the infinite network. The number of sites decreases in the infinite network upon the field increasing and it disappears (falls to clusters of finite volume) at a critical field . This is the transition point from the magnetically ordered (“SF”) phase to the disordered BG one. It resembles qualitatively the conventional percolation transition that was pointed out before in many papers (see, e.g., Ref. Yu et al. (2010)).

It is worth to remind that it is the self-similar (fractal) geometry of a lattice at the percolation threshold that makes the percolation transition to be universal. Then, geometric properties of a system are characterized by a unique characteristic length scale (the correlation length) which diverges at the percolation threshold. Many critical indexes of systems near percolation threshold are related to (universal) geometric characteristics of the random fractal (such as the fractal dimension, for example). We assume below that the universality of the transition from SF to BG phases has the same origin: finite clusters form a random fractal at QCP with some characteristics which differ in general from those of the percolation fractal. Geometric properties of regions with unsaturated magnetization are characterized close to QCP by the correlation length . We assume also (as in percolation theory) that one can find finite clusters of all characteristic linear sizes smaller than near QCP while the probability to find larger clusters is exponentially small.

Under such assumptions, it is natural to consider model (1) near QCP using methods which were successfully applied before for discussion of diluted HAFs near percolation threshold at zero magnetic field (see Refs. Nakayama et al. (1994); Harris and Kirkpatrick (1977); Shender (1978, 1976) and references therein). We remind first briefly the main aspects of that analysis and point out then similarities and distinctions between diluted HAF and our system.

Previous considerations Nakayama et al. (1994); Harris and Kirkpatrick (1977); Shender (1978, 1976) of diluted HAFs rely heavily on the assumption that low-energy elementary excitations in the infinite network near percolation threshold are weakly damped gapless spin waves (hydrodynamic excitations). The existence of the hydrodynamic excitations is closely related to commutativity of the Hamiltonian with the total spin operator. To the best of our knowledge, value has not been established above which the hydrodynamic description of excitations is correct in diluted HAFs. It is well known that it is valid in diluted 3D HAFs up to the percolation threshold. Nakayama et al. (1994) On the other hand, it is found in Ref. Chernyshov et al. (2002) that a small concentration of vacancies in 2D HAFs leads to a localization of spin waves with wavelengths greater than (see also Ref. Harris and Kirkpatrick (1977)).

We rely below on the assumption that well-defined long-wavelength quasiparticles exist at . Bearing in mind results of Ref. Chernyshov et al. (2002) and a good agreement of formulas obtained below to numerical findings at , we indicate as a rough estimation of the range of our results validity. Indeed, further consideration is required for a more accurate determination of the value above which our discussion is valid.

ii.2 Diluted HAF near percolation threshold

The hydrodynamic excitations can be described phenomenologically using the following expression for the system energy in the continuum limit accounting for fluctuations in transverse components of sublattices magnetizations and : Harris and Kirkpatrick (1977)

(2)

where x,y at zero magnetic field, z axis is directed along sublattices magnetizations, we consider 3D HAF for definiteness, and , , and are phenomenological constants which have to be found from an analysis of the corresponding microscopic theory.

is a mean staggered magnetization per unit volume which is proportional to the number of sites in the infinite network. Harris and Kirkpatrick (1977) scales near the percolation threshold as

(3)

is a measure of the energy needed to create a spatial variation of the staggered magnetization (i.e., is the helicity modulus). It is related to the conductivity of the system which is obtained from the diluted HAF by replacing bond between neighboring sites and by a resistor with conductance (in diluted HAF, is taken to be , where for an occupied site and for a vacancy): Harris and Kirkpatrick (1977)

(4)

To demonstrate this, periodic boundary conditions are applied along x and z directions and open boundaries are assumed in the y direction. Let us consider a spin fluctuation having the form

(5)

where is the antiferromagnetic vector, is equal to and on sites from different sublattices, , is the unit vector directed along z axis, and is a unit vector from xy plane. Fixing on one open boundary and on another boundary, we obtain for the equilibrium conditions at site lying inside the system

(6)

Eqs. (6) together with the boundary conditions are equivalent to Kirchoff’s law of the network of conductances connecting electrodes of potentials and . Harris and Kirkpatrick (1977)

Applying a small uniform transverse field and minimizing the energy , one finds that , where is a transverse susceptibility. It is demonstrated in Ref. Harris and Kirkpatrick (1977) that so that

(7)

Landau-Lifshitz equations 111The effective fields acting on which appear in Landau-Lifshitz equations are determined as , where is the system volume. for Fourier components of and give the spectrum of the doubly degenerate (due to fluctuations along equivalent x and y directions) Goldstone mode Harris and Kirkpatrick (1977)

(8)

These propagating excitations exist on the length scale greater than the correlation length . On smaller length scale, excitations (the so-called “fractons”) are localized. Nakayama et al. (1994)

ii.3 Model (1) near QCP

As the infinite network in our system is surrounded by areas having saturated magnetization and a gapped spectrum (the gap in such areas depends linearly on ), the system resembles the diluted HAF in many respects. In particular, we can analyze the low-energy dynamics as it is done above which is governed by low-energy excitations in the infinite network and large finite clusters. The difference arises from the fact that HAF sublattices are not collinear and they are directed by an angle to the magnetic field (see Fig. 1). is an extra phenomenological parameter in the theory. As soon as the infinite network breaks up into finite clusters which magnetizations are not saturated, one concludes that is finite at .

Figure 1: (Color online). Sublattices magnetizations (per unit volume) in external magnetic field .

It can be easily shown that the energy of fluctuations in the transverse components of the sublattices magnetizations is given by Eq. (2) as in diluted HAF (the angle arises only in Landau-Lifshitz equations). However fluctuations within xz plane (within which the field and sublattices magnetizations lie) and those along y direction have different properties. Analysis of the Landau-Lifshitz equations shows that the former have gapped spectrum whereas the latter are characterized by the Goldstone mode with dispersion (8), where should be multiplied by . We imply below that y in Eq. (2) in the Goldstone mode discussion.

Phenomenological constants and have the same meaning as in diluted HAFs and we try them in the form (3) and (4), respectively. Due to the finiteness of at , is the critical exponent of the order parameter. The equilibrium condition for fluctuations of the type (5) (where is directed along y axis, is the angle between -th magnetic moment and xz plane, and ) reads as (cf. Eq. (6))

(9)

where is the angle between -th magnetic moment and the field and is a site from the infinite network. The last term is negligible in Eq. (9) and the analog of the Kirchoff’s law is recovered provided that . The latter inequality is expected to hold at least not very close to the infinite network edges. Then, it seems to us likely that the helicity modulus is related to the conductivity of the corresponding system however further consideration may be required to proof it rigorously. We use below the relation between and the conductivity only in the discussion of the possible percolation regime.

is related in our system to the susceptibility to a small field directed along y axis. The important difference from diluted HAFs is that remains finite at . Indeed, the origin of divergence in diluted HAFs is uncompensated net spins arising on large length scales which lead to large magnetization of the whole system in the external field (see Refs. Harris and Kirkpatrick (1977); Nakayama et al. (1994) for more details). In contrast, there are no such objects in our system and a small additional field along y axis leads only to a small turn of the external field in the y direction. This rotation produces just a small magnetization along y direction proportional to the magnetization of the whole sample. As a result, one arrives at Eq. (7) for , where

(10)

This circumstance leads to some important differences in dynamical properties of our system in comparison with diluted HAFs.

DOS of the infinite network can be found as it was done in Refs. Harris and Kirkpatrick (1977); Shender (1978); Nakayama et al. (1994) for diluted HAFs, the result being

(11)
(12)

One can further simplify Eq. (12) using the geometrical relation , foo (a) where is the fractal dimension (i.e., the space dimension on the length scale smaller than ). The second line in Eq. (11) is simply DOS of propagating excitations (8). The quantity is given by Eq. (8) at . The first line in Eq. (11) is found by taking into account that properties of excitations (energy and DOS normalized to one spin) on the length scale smaller than (corresponding to ) do not depend on the proximity to QCP (i.e., on ). Shender (1978); Halperin and Hohenberg (1969) Then, DOS of localized excitations can be represented as that should match DOS of the hydrodynamic mode at . As a result one arrives at Eq. (12) for .

The dynamical critical exponent is determined by the scaling of (see Eq. (11)) that gives

(13)

Because in our system, Eqs. (11)–(13) differ from their counterparts in diluted HAFs.

To estimate the Néel temperature, we follow Ref. Shender (1978) and introduce first an auxiliary quantity

(14)

where is the energy (2) of the spin density wave characterized by momentum . To find it at , we express via using Landau-Lifshitz equations, substitute the result to Eq. (2) and obtain (the derived expression differs from that in diluted HAF Shender (1978) by a factor of 2 due to different number of Goldstone modes). As soon as , one easily finds from this result at . As soon as does not depend on at , we try in the form in this regime, where is found by matching at . One obtains as a result

(15)
(16)

The reduction of sublattices magnetization due to thermal fluctuations at small temperature at a given is estimated as Shender (1978) , where we use Eq. (14) and denotes thermal average. Replacing the summation over momenta by integration over energy, one obtains Shender (1978)

(17)

We find from Eqs. (11), (15), and (17) that at . At larger temperatures, , the region of gives the main contribution to the integral in Eq. (17) provided that

(18)

We assume below that this inequality is fulfilled while it is not generally the case. For instance, Eq. (18) holds in the percolation theory at but it does not hold at (i.e., for the mean-field critical percolation exponents). Stauffer and Aharony (1994) As will be seen below, Eq. (18) is fulfilled in the close vicinity of QCP at . As a result, one finds from Eq. (17) . As soon as this expression is valid also by the order of magnitude at , when , we obtain the following estimation for the Néel temperature:

(19)

The corresponding expression is also readily written out when Eq. (18) does not hold.

The order parameter static correlation function having the form at QCP (see Ref. Fisher et al. (1989)) should behave as the fractal correlation function giving the probability that a site a distance apart from the given site belongs to the same cluster. foo (a) Using also the geometrical relation one derives

(20)

Eq. (20) is obtained in Ref. Fisher et al. (1989) using scaling relations and implying that is the order parameter exponent.

One needs also DOS of finite clusters to describe the system behavior in the ordered phase near QCP. According to our assumptions, one can find finite clusters of all characteristic sizes smaller than near QCP while the probability to find larger clusters is exponentially small. As a result, at . Then, one try in the form at because the number of sites in finite clusters does not vanish at . To find , one notices that DOS per spin is of the order of at in the infinite network (see Eq. (11)). DOS per spin should have the same form in the largest clusters which characteristic linear size is of the order of . Thus, one arrives at the estimation , where is the total volume of the largest clusters. Because the infinite network breaks predominantly into finite clusters with linear size of the order of , is of the order of the infinite network volume . One obtains as a result

(21)
(22)

where Eq. (13) is also taken into account.

Iii Close vicinity of QCP

In the dirty-boson problem considered in Ref. Fisher et al. (1989), the equality

(23)

comes from the finiteness of the compressibility at QCP. Arguments against that derivation are presented in Ref. Weichman and Mukhopadhyay (2007). In Ref. Yu et al. (2012a), Eq. (23) is derived in a different way from the scaling of the particle-hole gap. Our consideration above suggests another way to obtain this equality. The counterpart of the compressibility in our system is the longitudinal susceptibility . It is clear from a physical point of view that is a constant because a large amount of arbitrary large clusters with unsaturated magnetization remains in the system. But we cannot relate to any quantity consider above using just its definition because we discuss only energy change by fluctuations in the direction transverse to the field. However, one can obtain Eq. (23) by assuming that the singular part of the free energy scales inversely with the space-time correlation volume, i.e., . Weichman and Mukhopadhyay (2007); Yu et al. (2012a); Fisher et al. (1989) This assumption has not been questioned yet in close vicinity of QCP. One has as a result for the helicity modulus Fisher et al. (1989) so that (see Eq. (4)). foo (b) Substituting the latter equality to Eq. (13) we come to Eq. (23).

One obtains from Eqs. (13), (19), and (23)

(24)
(25)

It is seen from Eqs. (21)–(23) that DOS of finite clusters is a constant at . In particular, for at QCP. As a consequence, one derives for the specific heat at

(26)

One recognizes in Eqs. (24) and (25) expressions for critical exponents of the helicity modulus and the critical temperature obtained using the scaling ansatz in Ref. Fisher et al. (1989). The relation is also found in Ref. Fisher et al. (1989) using scaling arguments. Then, we come to the same results from another direction. We are able also to derive some new expressions. It is argued in Ref. Fisher et al. (1989) that one-particle DOS scales as at QCP so that our finding signifies that 222Remember that DOS in disordered HAFs corresponds to DOS of uncondensed particles in a system of dirty bosons. However, the condensate disappears at QCP so that this comparison of DOS is correct.

(27)

as in disordered Fermi systems (see Ref. Fisher et al. (1989) for a more detailed discussion of the similarity between disordered Fermi and Bose systems). Notice that inequality is proposed in Ref. Fisher et al. (1989) that is valid, e.g., at in which case . Giamarchi and Schulz (1988) Then, we establish that coincides with the upper bound of this interval at . One obtains from Eqs. (20) and (27)

(28)

To the best of our knowledge, Eqs. (27) and (28) have not been obtained before. Notice that Eqs. (25) and (28) are consistent with values of , , and found in Ref. Yao et al. (2014) for . Table 1 summarizes all expressions for critical exponents derived in this section and demonstrates the consistency of the theory at with numerical results of Ref. Yao et al. (2014).

numerical values at
theory numeric
3 3
3.5(2)
1.5(2)
Table 1: Expressions for critical exponents obtained in the present consideration at in close vicinity of QCP (expressions for , , and coincide with those obtained in Ref. Fisher et al. (1989)). Predictions of the theory for are compared with numerical results of Ref. Yao et al. (2014). Values in the column ”theory” are found by substituting (observed in Ref. Yao et al. (2014)) to the expressions.

It should be pointed out that Eqs. (23), (24), and (28) simplify greatly some general relations obtained in Sec. II.3. In particular, one finds

(29)

which appear to be independent of . Thus, similar to fractons in diluted HAFs at (see Ref. Nakayama et al. (1994)), DOS per spin of fractons both in the infinite network and in finite clusters shows in our system superuniversal properties at .

Iv Possible crossover to percolation scenario near QCP at

It should be noted that the results obtained in Ref. Yao et al. (2014) which are consistent with formulas discussed in the previous section are obtained in 3D models, where the disorder strength is a driving parameter, the average chemical potential is fixed, and the transition takes place at quite large . On the other hand there is a large amount of results obtained experimentally and numerically in another part of the phase diagram in the plane (at smaller ) by varying at fixed (this part of the phase diagram is also examined in Ref. Yao et al. (2014) on larger systems, main findings being consistent with those from previous works). Yu et al. These results are in sharp contradiction with Ref. Fisher et al. (1989). We show now that they could be described by formulas obtained in Sec. II.3 using percolation critical exponents. This points to a possibility of a crossover in the critical region in a part of the phase diagram from the percolation regime to that predicted in Ref. Fisher et al. (1989) and discussed in the previous section.

Notice first that values of the critical exponent of the correlation length obtained in all previous numerical works Yu et al. (2012b, a, 2010); Hitchcock and Sørensen (2006); Yao et al. (2014) are close to the percolation one . On the other hand, the value of obtained in Ref. Yao et al. (2014) very close to QCP is equal to within the error to . Then, the necessary (but not sufficient) condition seems to be fulfilled for the possibility of the crossover scenario in the critical region.

critical expression via percolation numerical values
exponent critical exponents percolation experiment numerics
0.876 not available (Refs. Hitchcock and Sørensen (2006); Yu et al. (2012a, 2010))
0.418 (Refs. Hüvonen et al. (2012); Wulf et al. (2013)) 0.6(1) (Ref. Yao et al. (2014))
(Refs. Yu et al. (2012b); Hüvonen et al. (2012); Yamada et al. (2011)) (Refs. Yu et al. (2012b, a, 2010, ); Yao et al. (2014))
not available (Refs. Hitchcock and Sørensen (2006); Yu et al. (2012a));

Numerical data from Ref. Yu et al. (2012a) are consistent also with these values (see Appendix A).

Table 2: Comparison of critical exponents obtained before experimentally and numerically in various 3D systems with corresponding values in the percolation regime discussed in Sec. IV (percolation exponents have index ). Values of percolation critical exponents are taken from Refs. Wang et al. (2013); Xu et al. (2013); Bergman and Stroud (1992).

Critical exponents calculated from Eqs. (13) and using percolation critical exponents (assuming that is the critical exponent of the conductivity) are compared in Table 2 with results of previous experimental and numerical works. It is seen that all experimental and many numerical findings are consistent with the percolation scenario. However some remarks are in order.

First, some previous numerical calculations Yu et al. (2012a, 2010) give values of which are substantially greater than . However consideration of the corresponding model in Ref. Yao et al. (2014) gives the value of not very close to QCP which is much closer to . Notice that system sizes in Ref. Yao et al. (2014) are at least an order of magnitude larger than in previous numerical works. Then, there is a tendency for decreasing upon the system size increasing. That is why we indicate the value of for in Table 2.

Second, values of were reported before for 3D systems Yu et al. (2012a, 2010); Hitchcock and Sørensen (2006) which are larger than the proposed percolation value . However we demonstrate in Appendix A that numerical data taken from Ref. Yu et al. (2012a) are not very sensitive to a variation in in quite a broad interval (this interval is for some data). It should be stressed also that equality (23) holds only in the close vicinity of QCP so that it can be violated in the percolation regime. Certainly, we are unable to reexamine all relevant data from all previous numerical papers. We choose Ref. Yu et al. (2012a) for this purpose recognizing that it is one of the most detailed and reliable works. Further numerical consideration may be necessary to find value not very close to QCP.

As it is shown above, the common assumption that the singular part of the free energy scales inversely with the space-time correlation volume leads to Eq. (23) that would be violated in the percolation regime. To be consistent with Eq. (13), would scale as

(30)

not very close to QCP. In accordance with Eq. (13), Eq. (30) signifies that . foo (b) Notice that Eq. (30) works also close to QCP, where . The finite-size scaling would change accordingly for the helicity modulus:

(31)

DOS per spin would not be superuniversal in the percolation regime: and given by Eqs. (12) and (22) do depend on .

V Summary and conclusion

In summary, we discuss the magnetically ordered phase of HAF (1) with bond disorder near quantum phase transition to the disordered BG phase. Our consideration is based on the assumption that long-wavelength quasiparticles are well-defined at . Besides, we assume that the transition bear a qualitative resemblance to the percolation one. We derive formulas for the Néel temperature (19), the dynamical critical exponent (13), DOS of infinite network (11) and DOS of finite clusters (21) which are valid in the entire critical region. In the close vicinity of QCP, where Eq. (23) holds, we recover by these formulas some results obtained before using physical arguments and the scaling ansatz suggested in Ref. Fisher et al. (1989) (see Eqs. (20), (24)–(26)). As an extension of the previous analysis, we derive Eqs. (27) and (28) (the latter is in quantitative agreement with Ref. Yao et al. (2014)). DOS per spin of localized excitations (fractons) are found to be superuniversal (i.e., independent of ) both in the infinite network and in finite clusters (see Eq. (29) and first lines in Eqs. (11) and (21)). An alternative way is proposed to derive the equality close to QCP. We show that all experimental data and many numerical ones obtained before in various 3D systems would be relevant to the percolation critical scenario which could arise not very close to QCP before the onset of the regime predicted in Ref. Fisher et al. (1989). However further numerical and experimental considerations are necessary to support this idea. The results obtained are expected to be valid starting from a value which lies between 2 and 3 and determination of which is a subject of future work.

Acknowledgements.
This work is supported by Russian Science Foundation (grant No. 14-22-00281).

Appendix A Dynamical critical exponent in previous numerical consideration

In this appendix, we revisit numerical data obtained in Ref. Yu et al. (2012a) by quantum Monte Carlo (QMC) simulations. The system under consideration is widely used model of the DTN compound (NiCl 4SC(NH)). DTN is an antiferromagnet whose Hamiltonian has the form (1) in which a large (compared to the exchange coupling constants) single-ion easy-plane anisotropy is added having the form , where . At low temperature, pure DTN has two QCPs at fields and . A magnetically ordered (“SF”) phase with canted antiferromagnetic order arises at whereas “MI” (paramagnetic) phases are stable outside this interval. Disorder leads to BG phases near both critical fields. Both transitions from BG to “SF” phases are discussed in Ref. Yu et al. (2012a), where two types of quenched disorder are considered: 1) random substitution of some Cl atoms by Br ones (the so-called Br-DTN) that changes exchange constants and values around the defect; 2) substitution of magnetic Ni ions by non-magnetic ones (i.e., dilution of the magnetic subsystem). We revisit QMC data only for the case of dilution because the finite-size scaling in Br-DTN is somewhat unsatisfactory as it is explained in Ref. Yu et al. (2012a).

The dynamical critical exponent is extracted in Ref. Yu et al. (2012a) from finite-size scaling for the spin stiffness (the helicity modulus) . Corresponding plots presented in Fig. 8 of Ref. Yu et al. (2012a) for the site-diluted DTN with dilution are shown in left panels of Figs. 2 and 3. In right panels of those figures, we present the same data drawn using smaller values. We also shift a little values of and in right panels so that they remain consistent with those found in Ref. Yu et al. (2012a) in corresponding analysis of data for the correlation length and the order parameter. It is seen that collapses of data obtained for different system sizes in right panels are at least no worse than those in left panels. This signifies that smaller values are also consistent with numerical data.

Figure 2: (Color online). Finite-size scaling plots for the spin stiffness in the site-diluted DTN at . Numerical data are taken from Fig. 8 of Ref. Yu et al. (2012a). Left panel is drawn with parameters from original paper Yu et al. (2012a) (, , and ). We change values of and in the right panel in comparison with the left one.
Figure 3: (Color online). Same as in Fig. 2 but for .

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