Chong Wang

Cyrille Chazallon

Sandrine Braymand

Pierre Hornych

Thermogravimetric analysis (TGA) for characterization of self-cementation of recycled concrete aggregates in pavement

Keywords: Recycled concrete aggregate, Self-cementing properties, Thermogravimetric analysis, Microstructure, Road base

A key question concerning the reuse of recycled concrete aggregates (RCA) in unbound pavement base and subbase layers is the assessment of self-cementing properties, which can improve significantly their performance. In this paper, the self-cementing properties and their influence on the long-term performance, microstructure and mechanical behavior, were studied by subjecting two different RCA (NRCA and ORCA) specimens, after different curing times (1 and 360 days), to thermogravimetric analysis (TGA), scanning electron microscopic analysis (SEM) and monotonic triaxial tests (σ3=20, 40 and 70 kPa). The chemical properties, such as pH value, sulfate content and calcite content, were also investigated. The results show that the unhydrated cement content, responsible for the self-cementing properties of RCA, is low (6.0-7.6% in fine NRCA). However, after curing for 360 days, around half of the unhydrated cement (36% to 45%) was hydrated. The hydration products not only form bonds and fill the macro-pores (width less than 3 μm) between particles, but also consume and decrease the free water content of RCA specimens, turning the open type of soil structure to a rigid tight skeleton structure, increasing both the strength and stiffness of RCA specimens.

INTRODUCTION

Economic and human activities produce a lot of construction and demolition wastes (CDW) each year, occupying a large land space and imposing huge pressures on environment when depositing CDW in landfills. In Europe, over one third of the total waste produced is CDW (over 800 million tons in 2018) (Eurostat, 2018). In terms of CDW, concrete wastes are the main components and represent up to 75% of total CDW by weight [START_REF] Nwakaire | Utilisation of recycled concrete aggregates for sustainable highway pavement applications; a review[END_REF]. To overcome these problems, recycling and reuse of concrete wastes in pavement base and subbase layers has become a topic of global concerns in recent years [START_REF] Lu | Resilient and permanent deformation behaviors of construction and demolition wastes in unbound pavement base and subbase applications[END_REF][START_REF] Ramachandran | Permanent deformation behaviour of pavement base and subbase containing recycle concrete aggregate, coarse and fine crumb rubber[END_REF][START_REF] Sobhan | Durability of a pavement foundation made from recycled aggregate concrete subjected to cyclic wet-dry exposure and fatigue loading[END_REF][START_REF] Yaghoubi | Impact of compaction method on mechanical characteristics of unbound granular recycled materials[END_REF].

It is well known that concrete structures contain partially unhydrated cement even after long term service [START_REF] Lin | Hydration kinetics modeling of Portland cement considering the effects of curing temperature and applied pressure[END_REF][START_REF] Mills | Factors influencing cessation of hydration in water cured cement pastes[END_REF]. However, the crushing process to produce recycled concrete aggregates (RCA) can expose these unhydrated cement particles [START_REF] Amin | Residual Cementing Property in Recycled Fines and Coarse Aggregates: Occurrence and Quantification[END_REF]. Subsequently, when RCA are used in unbound base and subbase layers, instead of a cementitious matrix, this unhydrated cement could hydrate with moisture and create bonds between particles, increasing the strength and stiffness of the pavement layers, which is believed to be the principal reason of self-cementing properties of RCA [START_REF] Bestgen | Mechanical and Environmental Suitability of Recycled Concrete Aggregate as a Highway Base Material[END_REF][START_REF] Bordy | Cement substitution by a recycled cement paste fine: Role of the residual anhydrous clinker[END_REF][START_REF] Oksri-Nelfia | Reuse of recycled crushed concrete fines as mineral addition in cementitious materials[END_REF][START_REF] Poon | The cause and influence of self-cementing properties of fine recycled concrete aggregates on the properties of unbound sub-base[END_REF].

Several approaches can be used to assess the self-cementing properties of RCA, such as the semi-adiabatic calorimeter test, thermogravimetric analysis (TGA), X-ray diffraction (XRD) and scanning electron microscope (SEM) observations [START_REF] Bordy | Cement substitution by a recycled cement paste fine: Role of the residual anhydrous clinker[END_REF][START_REF] Kalinowska-Wichrowska | The Properties of Composites with Recycled Cement Mortar Used as a Supplementary Cementitious Material[END_REF][START_REF] Oksri-Nelfia | Reuse of recycled crushed concrete fines as mineral addition in cementitious materials[END_REF]; J. J. [START_REF] Wang | Recycled cement[END_REF]. Among these methods, TGA is favored by some researchers because it allows quantitative analysis [START_REF] Bordy | Cement substitution by a recycled cement paste fine: Role of the residual anhydrous clinker[END_REF][START_REF] Oksri-Nelfia | Reuse of recycled crushed concrete fines as mineral addition in cementitious materials[END_REF]. [START_REF] Oksri-Nelfia | Reuse of recycled crushed concrete fines as mineral addition in cementitious materials[END_REF] proposed a model to quantify the unhydrated cement content in RCA, prepared with Portland cement, based on the portlandite (Ca(OH)2) content obtained by TGA. The result revealed that the unhydrated cement content was very low in the studied RCA (2.3%). However, the Ca(OH)2 content is easily influenced by the type of cement, carbonation and the use of supplementary cementing materials (Papadakis, 1999a[START_REF] Papadakis | Experimental investigation and theoretical modeling of silica fume activity in concrete[END_REF], which limits the application of this model. Some researchers have studied the influence of self-cementing properties on the long-term performance of RCA [START_REF] Amin | Residual Cementing Property in Recycled Fines and Coarse Aggregates: Occurrence and Quantification[END_REF][START_REF] Kim | Evaluation of Recementation Reactivity of Recycled Concrete Aggregate Fines[END_REF][START_REF] Poon | The cause and influence of self-cementing properties of fine recycled concrete aggregates on the properties of unbound sub-base[END_REF]. [START_REF] Arm | Self-cementing properties of crushed demolished concrete in unbound layers: results from triaxial tests and field tests[END_REF] found that the increase of strength and stiffness in unbound base and subbase layers caused by selfcementing properties is highest in the first few months. [START_REF] Poon | The cause and influence of self-cementing properties of fine recycled concrete aggregates on the properties of unbound sub-base[END_REF] pointed out that the unhydrated cement content in fine aggregates (0-0.6 mm) is the principal cause of selfcementing properties. [START_REF] Jitsangiam | Recycled Concrete Aggregates in Roadways: Laboratory Examination of Self-Cementing Characteristics[END_REF] observed that the unbound RCA specimens evolved into bound specimens after 360 days of curing, as indicated by the microstructure analyses (SEM). To date, however, the mechanisms, and effect on the long-term performance, of self-cementing has not been thoroughly studied.

As a result, this study aims to assess these self-cementing properties and their influence on the improvement of long-term performance (microstructure and mechanical behavior). The chemical properties of two different RCA (NRCA and ORCA), such as pH value, sulfate content and calcite content, were firstly investigated. An improved model based on TGA was then proposed to quantify the unhydrated cement content of RCA. Then, the hydration of unhydrated cement and the influence of self-cementing properties on the microstructure and mechanical behavior of the two RCA (NRCA and ORCA), after 1 day and 360 days of curing, were studied by TGA, SEM and monotonic triaxial tests (σ3=20, 40 and 70 kPa). Furthermore, the mechanisms leading to long-term performance improvement of RCA were also analyzed.

MATERIALS AND METHODS

Materials

The two RCA materials used in this study were sourced from different demolished buildings with unknown compositions, instead of casting and curing in the laboratory, using known component, which is more realistic. The two studied RCA were crushed by impact crushers in two recycling plants in France with a 20 mm maximum particle size:

-One RCA was crushed in 2020 at the beginning of the study (New RCA) at a recycling plant in Strasbourg, France, and is designated by NRCA.

-The other RCA was crushed in 2012 at a recycling plant in Gonesse, France, and stored outdoor in plastic bags to prevent further humidity and carbonation, and is designated by ORCA (for Old RCA). The long-term storage was expected to largely reduce the self-cementing properties of ORCA [START_REF] Kim | Evaluation of Recementation Reactivity of Recycled Concrete Aggregate Fines[END_REF], so the two materials were expected to present different self-cementing properties.

Figure 1 and Figure 2 show the particle size distributions and modified Proctor curves of NRCA and ORCA. More physical properties of the studied materials can be found in C. [START_REF] Wang | Permanent and resilient deformation behaviour of recycled concrete aggregates from different sources, in pavement base and subbase[END_REF].

Test methods

The pH value, water soluble sulfate content and calcite content of RCA were measured to characterize the chemical composition and carbonation of RCA. To avoid the influence of original natural aggregates in RCA on the results, only fine aggregates (0-2 mm), previously dried in the oven at 60 ℃ (in order to avoid damaging RCA) to a constant mass, were used, since the small particles contain less original natural aggregates [START_REF] Juan | Study on the influence of attached mortar content on the properties of recycled concrete aggregate[END_REF] and govern the self-cementing properties of RCA [START_REF] Poon | The cause and influence of self-cementing properties of fine recycled concrete aggregates on the properties of unbound sub-base[END_REF].

For pH value measurement, 30 g of RCA fine aggregates and 75 g of distilled water were mixed and stirred for 2 minutes (BS 1377(BS -3, 2018)). Then, the pH value of each suspension, stored for 24 h, was measured by a pH meter.

Water soluble sulfate content of RCA was measured in accordance with the European standard NF EN 1744-1 (2014). 1 kg of previously heated (60 ℃) distilled water was added to 25 g of fine aggregates, and the mixture was stirred for 15 minutes, while maintaining the temperature of 60 ℃. Then, the solution was extracted using a filter and the water soluble sulfate content was measured by a photometer.

Calcite content was measured by the calcimeter method (NF ISO 10693, 1995) to assess the degree of carbonation of the two RCA, before use, especially for the ORCA after long-term storage. 0.5-2 g of powder from ground fine aggregates were first added to 10 ml of chlorhydric acid and the volume of released carbon dioxide was measured. Then, the calcite content was calculated according to the measured volume of carbon dioxide.

Thermogravimetric analysis (TGA) was conducted to quantify the phases content of RCA, which were used to quantify unhydrated cement content in this study. In addition, to study the hydration of unhydrated cement during the curing stage, fine aggregates, extracted from the samples cured during 360 days, after monotonic triaxial tests, were also subjected to TGA. 50-100 mg of air-dried specimen, without curing and cured for 360 days, were heated at a constant heating rate of 10 ℃/min from 25 ℃ to 1000 ℃ under N2 atmosphere.

Scanning electron microscopic analysis (SEM) was conducted to inspect the microstructures of RCA without curing and cured for 360 days. SEM samples were collected from the specimens after monotonic triaxial tests and coated with gold prior to SEM analysis.

The monotonic triaxial tests were performed under different confining pressures (20, 40 and 70 kPa), and after 1 day and 360 days of curing. The tests were performed at a shear rate of 0.3 mm/min until failure. All the samples were compacted with a vibrating hammer, to the maximum dry density at the specific water content (optimum moisture content minus 1%), corresponding to 11.9% and 10.5% for NRCA and ORCA, respectively, as shown in Figure 2. Then, the specimens were wrapped in plastic foils and stored indoors at a temperature of 18± 5 ℃, during 1 or 360 days.

Methods to determine the unhydrated cement content by TGA

Non-evaporable water content (Cwater) and portlandite (Ca(OH)2) content, corresponding to the hydration of cement, are usually considered as suitable indicators of cement hydration [START_REF] Taylor | Cement chemistry[END_REF]. In comparison to Ca(OH)2 which is easily influenced by the type of cement, carbonation and the supplementary cementing materials as mentioned above (Papadakis, 1999a[START_REF] Papadakis | Experimental investigation and theoretical modeling of silica fume activity in concrete[END_REF], non-evaporable water is much less sensitive to these factors [START_REF] Chidiac | Cement degree of hydration in mortar and concrete[END_REF][START_REF] Pane | Investigation of blended cement hydration by isothermal calorimetry and thermal analysis[END_REF], which increases the accuracy and reliability of the proposed model. Thus, Cwater was used to calculate the unhydrated cement content of RCA in this study, instead of portlandite (Ca(OH)2) content used by [START_REF] Oksri-Nelfia | Reuse of recycled crushed concrete fines as mineral addition in cementitious materials[END_REF]. Note that the non-evaporable water, retained after freeze-drying, has often been wrongly identified with chemically bound water, present in interlayer spaces, which determination is complicated [START_REF] Taylor | Cement chemistry[END_REF]. Figure 3 and Figure 4 show the derivative thermogravimetry (DTG) and thermogravimetry (TG) curves of NRCA and ORCA before curing. According to the endothermic peaks on the DTG curve (Figure 3), the decomposition of hydration products of NRCA can be divided into four major phases [START_REF] Monteagudo | The degree of hydration assessment of blended cement pastes by differential thermal and thermogravimetric analysis. Morphological evolution of the solid phases[END_REF][START_REF] Ramachandran | Permanent deformation behaviour of pavement base and subbase containing recycle concrete aggregate, coarse and fine crumb rubber[END_REF]. The first phase between 0 and 105 ℃ represents the evaporation of free water. The second phase, between 105 and 430 ℃, and third phase, between 430 and 490 ℃ (T1), correspond to the decomposition of hydration products, such as calcium silicate hydrates (C-S-H), ettringite, gypsum and calcium hydroxide, which are used to calculate non-evaporable water Cwater. The last phase, with the DTG peak between 490 to 850 ℃ (T2), caused by the decomposition of calcite, was also used to determine the calcite content (CCaCO3) and compared with the calcimeter method. It should be mentioned that different materials may have a slightly different decomposition temperature instead of a fixed temperature range. To improve the accuracy of quantification, the corresponding decomposition temperatures (T1 and T2) are chosen carefully according to the slope of the DTG curves, as summarized in Table 1. Then, Cwater and CCaCO3 can be determined from the TG curves (Figure 4) by:

1 105 = water C T C C M M   (1) 3 1 2 100.09 () 44.01 CaCO T C T C C M M   (2)
where the ratio 100.09/44.01 corresponds to the molar mass ratio of CaCO3/CO2, while MT ℃ is the sample weight (%) obtained from the TG curve at temperature T ℃. Cwater and CCaCO3 are the non-evaporable water content and calcite content.

After determining Cwater from the TG curve, the cement content (Ccement) of RCA can be calculated according to the hydration process of cement. For that, it is assumed that a complete hydration of Portland cement, without any supplementary cementing materials (SCMs), such as fly ash, silica fume, etc., will release around 0.23-0.25 gram of non-evaporable water (taken as 0.24 in this study), per gram of cement [START_REF] Bhatty | HYDRATION VERSUS STRENGTH IN A PORTLAND-CEMENT DEVELOPED FROM DOMESTIC MINERAL WASTES -A COMPARATIVE-STUDY[END_REF][START_REF] Mounanga | Predicting Ca(OH)2 content and chemical shrinkage of hydrating cement pastes using analytical approach[END_REF][START_REF] Pane | Investigation of blended cement hydration by isothermal calorimetry and thermal analysis[END_REF]. Then, the relationship between Cwater and cement content (Ccement) of RCA made with Portland cement can be expressed as: =0.24

water cement CC   
(3) where Cwater and Ccement are the non-evaporable water content and Portland cement content in RCA, respectively. α is the degree of cement hydration. For concrete, after a long service life, the ultimate degree of cement hydration is mainly influenced by the initial water cement ratio (w/c), and can be calculated by the following equation developed by [START_REF] Mills | Factors influencing cessation of hydration in water cured cement pastes[END_REF] which is frequently used to model cement hydration [START_REF] Lin | Hydration kinetics modeling of Portland cement considering the effects of curing temperature and applied pressure[END_REF][START_REF] Oksri-Nelfia | Reuse of recycled crushed concrete fines as mineral addition in cementitious materials[END_REF]:

1.031* 0.194 w c w c    (4)
As mentioned before, 0.23-0.25 gram of non-evaporable water is released per gram of fully hydrated Portland cement (CEM I type). Whereas for blended cement, the use of SCMs makes it difficult to estimate such values due to the unknown stoichiometry and type of SCMs [START_REF] Deboucha | Combination effect of limestone filler and slag on hydration reactions in ternary cements[END_REF]. To implement this methodology for blended cements, the SCMs are considered by their equivalent in Portland cement, using the k-value concept [START_REF] Deboucha | Combination effect of limestone filler and slag on hydration reactions in ternary cements[END_REF]NF EN 206-1, 2004;[START_REF] Papadakis | Supplementary cementing materials in concrete -Part II: A fundamental estimation of the efficiency factor[END_REF]. With this approach, the unknown released non-evaporable water for blended cement can be considered as the same as the Portland cement (0.23-0.25 gram). In fact, this concept makes it possible to simply quantify the contributions of SCMs on the hydration of blended cements. (5) Then, Equation (3) can be substituted by: =0.24

water binder CC    (6) Finally, the unhydrated binder content (Cunhydrated) of NRCA can be determined as follows:

=(1 ) unhydrated binder CC    (7) It should be mentioned that the carbonation of concrete can decrease the calculated Cbinder and Cunhydrated values, by decreasing the non-evaporable water content. However, the influence of carbonation was not considered in this study as the high pH value of NRCA (13.07) suggested that carbonation of NRCA is negligible, which will be illustrated hereafter.

Silica fume

Fly ash CEM I 32.5 ≥ CEM I 42.5 k value 0.25 0.9 2 0.2 0.4

RESULTS AND DISCUSSION

Chemical compositions

Table 3 shows the pH values, water soluble sulfate contents and calcite contents of NRCA and ORCA. Although the pH value cannot be related to the self-cementing properties of RCA directly, many scholars still use it to evaluate the self-cementing properties as it can be measured easily and quickly [START_REF] Bestgen | Mechanical and Environmental Suitability of Recycled Concrete Aggregate as a Highway Base Material[END_REF][START_REF] Paige-Green | Preliminary evaluation of the reuse of cementitious materials[END_REF][START_REF] Poon | The cause and influence of self-cementing properties of fine recycled concrete aggregates on the properties of unbound sub-base[END_REF]. In fact, a high pH value implies more soluble alkaline hydration products, suggesting that the RCA were not completely carbonated and may contain some unhydrated cement. The results show that the pH value of NRCA (13.07) is much higher than that of ORCA (10.53). Paige-Green (2010) observed that RCA with a pH value higher than 11 exhibits self-cementing properties, indicating that NRCA should present self-cementing properties while those of ORCA should be negligible. This is consistent with the observations of [START_REF] Kim | Evaluation of Recementation Reactivity of Recycled Concrete Aggregate Fines[END_REF] who concluded that the self-cementing properties of RCA fines, stored in open stockpiles at high temperature and humidity, are negligible. Besides, [START_REF] Chang | The experimental investigation of concrete carbonation depth[END_REF] indicated that the concrete is not carbonated when the pH value (solid/water=1:10) is higher than 11.5, suggesting that carbonation of NRCA is negligible while ORCA is highly carbonated due to the long-term storage.

Sulfate attack is a frequent deterioration mechanism of hydraulically bound materials, which leads to an expansion in solid volume by forming gypsum or ettringite [START_REF] Hewlett | Lea's chemistry of cement and concrete[END_REF][START_REF] Kampala | An investigation of sulfate effects on compaction characteristics and strength development of cementtreated sulfate bearing clay subgrade[END_REF]. According to the European standard NF EN 13285 (2018) and the French national and regional technical guides (CETE de l 'Est, 2009;UNICEM IDF, 2003), the water soluble sulfate content of RCA, used in pavement base and subbase layers, should be less than 0.7%. Test results indicate that the water soluble sulfate content of both NRCA and ORCA are under this limit, although the water soluble sulfate content of NRCA (0.23%) is higher than that of ORCA (0.13%).

Table 3 also shows that the calcite content of NRCA (14.7%), measured by the calcimeter method, is only half that of ORCA (29.97%). This can be attributed to the long-term storage of ORCA and a large specific surface area after crushing, increasing the degree of carbonation while decreasing the pH value of ORCA, although ORCA was stored in plastic bags to prevent humidity and carbonation. However, part of the CaCO3 could also come from the original natural aggregates (NA), although an effort was made to eliminate the influence of original NA by limiting the maximum particle size (2mm).

Unhydrated binder content

Since ORCA were stored for a long period before use (without curing), the partial unhydrated cement has already hydrated, resulting in a much higher α, which cannot be calculated by Equation (4). Besides, the high degree of carbonation of ORCA, as mentioned above, also decreases the accuracy of the proposed method. Thus, only Ccement of NRCA could be calculated in this study. To calculate the unhydrated binder content of NRCA, Erreur ! Référence non valide pour un signet. summaries the weight loss (%) of the studied materials before use (without curing) under different temperature ranges, as well as Cwater and CCaCO3 calculated on the basis of TG curves (see Figure 4). It can be observed that Cwater of NRCA is much higher than that of ORCA, caused by the high degree of carbonation of ORCA after long-term storage. Besides, CCaCO3 calculated by TGA (Table 4) were also compared with the values obtained with the calcimeter method (Table 3). It indicates that the two calculation method lead almost to the same results, showing the reliability of the TGA results. Subsequently, the equivalent binder content (Cbinder) and unhydrated binder content (Cunhydrated) of NRCA can be calculated based on Cwater. Figure 5 presents the Cbinder and Cunhydrated of NRCA, for varying initial water-cement ratios w/c, because the initial w/c and original composition of the concrete structure are not known. Generally, the typical w/c ratio for conventional concrete ranges from 0.4 to 0.5 [START_REF] Shi | A review on ultra high performance concrete: Part I. Raw materials and mixture design[END_REF]. The typical equivalent binder content is between 15.5% and 21.3% at this specified w/c ratio (0.4-0.5), according to hundreds of concrete mixture compositions collected from the literature [START_REF] Ahmad | Prediction of Compressive Strength of Fly Ash Based Concrete Using Individual and Ensemble Algorithm[END_REF]). It appears that within the typical w/c ratio (0.4-0.5), the Cbinder of NRCA (23.3%-24.9%), calculated using Cwater, is slightly higher than the typical equivalent binder content (15.5%-21.3%). This could be due to the fact that the tested fine aggregates (0-2 mm) have a higher mortar content than the global concrete [START_REF] Juan | Study on the influence of attached mortar content on the properties of recycled concrete aggregate[END_REF], resulting in a higher Cbinder of NRCA than for a standard concrete. Thus, within the typical w/c ratio (0.4-0.5), the potential unhydrated binder content of NRCA (0-2 mm) can be estimated between 6.0% and 7.6%.

Hydration of unhydrated cement

To study the hydration of unhydrated cement during the curing phase, the fine aggregates of NRCA and ORCA (0-2mm), extracted from the cured samples (360 days) after monotonic triaxial tests, were also subjected to TGA. The comparison of the TG-DTG curves of NRCA and ORCA without curing and after curing for 360 days are presented in Figure 6. A significant increase of mass loss (TG/%) of NRCA after curing for 360 days can be observed, which is believed to be related to the hydration of unhydrated cement, increasing the non-evaporable water content. On the contrary, for ORCA, this increase of mass loss is negligible, in agreement with the negligible self-cementing properties of ORCA, indicated by the low pH value. Table 4 also summarizes the mass loss for different temperature ranges and the corresponding components (non-evaporable water and CaCO3) of NRCA and ORCA after curing for 360 days. Afterwards, the variation of non-evaporable water and CaCO3 contents of NRCA and ORCA before and after curing for 360 days can be obtained, as presented in Table 5. The amount of initially unhydrated binder, that was hydrated during the 360 days curing phase, named hydrated binder content, can be calculated by Equation ( 6) according to the variation of nonevaporable water content ( equal to 1), also summarized in Table 5. It appears that a slight increase of non-evaporable water (0.65%) can be observed for NRCA after 360 days of curing, indicating that some initially unhydrated binder (2.7%) was hydrated during the curing phase, leading to self-cementing properties. While for ORCA, no obvious variation of non-evaporable water is observed. These behaviors are in good agreement with the observed self-cementing properties of NRCA and ORCA.

Table 5 also reveals that the CaCO3 content of both NRCA and ORCA increases after curing for 360 days, but that this increase is much higher for NRCA, which could be attributed to the carbonation. Note that the undried NRCA, after monotonic triaxial test, was not subjected to TGA immediately, but that it was stored in a plastic bag without good seal for another month. Under humid environment, especially for RCA which specific surface area is large, the CO2 present in air dissolves quickly and the carbonation rate is maximum [START_REF] Kurdowski | Cement and concrete chemistry[END_REF]. Thus, the carbonation could have occured during the long-term curing or during the storage phase after the monotonic triaxial test. It should be pointed out that the carbonation during the curing stage can increase the strength and stiffness of RCA specimens [START_REF] Terzis | A decade of progress and turning points in the understanding of bio-improved soils: A review[END_REF][START_REF] Zadeh | CO2-treatment of recycled concrete aggregates to improve mechanical and environmental properties for unbound applications[END_REF], which could also lead to the observed self-cementing properties of RCA. However, this hypothesis needs to be confirmed by further studies.

In comparison with the total unhydrated binder content of NRCA (0-2 mm) without curing (6-7.6%), between 36% to 45% of the total unhydrated binder (2.7%) were hydrated after curing for 360 days. In other words, it can be estimated that about half of the unhydrated binder could hydrate during the first year after the construction of the pavement, and that the rest will continue to hydrate in the following years, which is in good agreement with [START_REF] Arm | Self-cementing properties of crushed demolished concrete in unbound layers: results from triaxial tests and field tests[END_REF]. This phenomenon was also observed in several field tests [START_REF] Korkiala-Tanttu | Long-term behavior of crushed concrete in road structure[END_REF][START_REF] Lancieri | C&D waste for road construction: long time performance of roads constructed using recycled aggregate for unbound pavement layers[END_REF][START_REF] Wang | Re-cementation of Recycled Concrete Aggregate in Pavement Base or Sub-base[END_REF], which indicated that the strength and stiffness of RCA base and subbase layers increases faster in the first few years, and then continue to increase at a lower rate even up to 10 years after construction. Actually, the carbonation of NRCA observed above underestimates the hydration of unhydrated cement (2.7%), by decreasing the amount of non-evaporable water.

Microstructural characteristics

The microstructures of NRCA after 1 and 360 days of curing are presented in Figure 7. Owing to the fact that the self-cementing properties of ORCA are negligible, and that no new hydration products were formed, only the microstructure of NRCA was studied. As shown in Figure 7 (a), NRCA cured for 1 day exhibits a relatively open type of microstructure, with numerous fine particles assembled between coarse aggregates, leaving many different void sizes.

The SEM image of NRCA cured for 360 days (Figure 7 (b)) shows the influence of selfcementing properties on the microstructures of RCA specimens. It appears that the microstructure of NRCA after 360 days of curing is much denser, with a visible decrease of the number and size of small voids (width less than 3 μm). For the larger voids (width larger than 3 μm), the influence of the self-cementing properties on the microstructures appears limited. A compacted soil presents two different types of pores: micro-pores (within aggregates) and macro-pores (between aggregates), with a delimiting diameter around 0.65 μm for micro-and macro-pores [START_REF] Su | Soil-water retention behaviour of fine/coarse soil mixture with varying coarse grain contents and fine soil dry densities[END_REF]. The hydration products of unhydrated cement not only act as bonds between particles, but also fill the voids between particles (macro-pores), as seen in Figure 7 (b), confirmed by the observation made in this study. However, for large pores (large than 3 μm), the low amount of hydration production limits the bonding and filling effects. As a result, the small pores (width less than 3 μm) were filled and the open type of structure (curing 1 day) gradually turns to rigid tight skeleton structure (curing 360 days), increasing the strength and stiffness of RCA and turning unbound pavement layers to the bound layers.

Among the hydration products, it is possible to observe on figure 7 that numerous needleshaped hydration products, ettringite, between few and ten μm long, are formed after 360 days of curing, associated with a decrease of the amount of fine particles. This ettringite (Ca6Al2(OH)12•(SO4)3•26H2O) absorbs a large amount of water and has a large volume, filling the voids between particles. Furthermore, it can be observed that most of these ettringite needles are small (less than 10 μm long), and that only a few large ettringite needles are visible (Figure 7 (b)). In general, there are two types of ettringite: large ettringite (10-100 μm long) and small ettringite (typically 1-2 μm long) [START_REF] Mehta | MECHANISM OF SULFATE ATTACK ON PORTLAND-CEMENT CONCRETE -ANOTHER LOOK[END_REF]). One explanation is that these small needleshaped ettringite particles are formed by the reactions between sulfate and cement hydration products [START_REF] Hewlett | Lea's chemistry of cement and concrete[END_REF][START_REF] Kampala | An investigation of sulfate effects on compaction characteristics and strength development of cementtreated sulfate bearing clay subgrade[END_REF], such as hydrated calcium aluminates. The low water soluble sulfate content of NRCA (0.23%) limits the growth and the size of the ettringite. In addition, the hydration of unhydrated cement can also lead to formation of ettringite.

Despite the water soluble sulfate content of NRCA (0.23%), which is much lower than the limit of the European standard (0.7%), as mentioned above, many small ettringite needles were still formed. It should be mentioned that once the voids are fully filled by the hydration products of unhydrated cement and ettringite, the continuous formation of ettringite can lead to swelling and cracking of the structure, known as sulfate attack of concrete [START_REF] Mehta | MECHANISM OF SULFATE ATTACK ON PORTLAND-CEMENT CONCRETE -ANOTHER LOOK[END_REF], decreasing the strength and stiffness of RCA base and subbase layers. Although this phenomenon has not been observed yet in this study, the sulfate content of RCA should be strictly limited when RCA are used in unbound or bound pavement layers.

Mechanism of strength and stiffness development

Figure 8 presents the results of the monotonic triaxial tests performed on the two recycled concretes, in terms of maximum shear strength, and secant modulus E50. The test were performed with 3 different confining pressures (20, 40 and 70 kPa), and after 1 day and 360 days of curing. For NRCA, the shear strength and stiffness increase significantly (approximately by a factor of two) after 360 days of curing, confirming its strong selfcementing properties. On the contrary, the strength and stiffness of ORCA remain almost constant, despite the sulfate content (0.13%) and carbonation (Table 5) observed in ORCA. It indicates that the hydration of unhydrated cement is the predominant cause of self-cementing properties.

These results indicate that the self-cementing mechanism is complex. On the one hand, the unhydrated cement, especially in fine aggregates, hydrates with moisture, as indicated by TGA (Table 5). The hydration products not only form bonds between particles, but also fill the small voids (width less than 3 μm), modifying the unbound, open structure into a stiffer, dense skeleton structure (Figure 7). In addition, the consumption of moisture due to the hydration process decreases the free water content (Table 5), which can also improve the mechanical properties of RCA specimens [START_REF] Gaillard | Thermo-hydromechanical behaviour of cold reclaimed asphalt aggregates without binder addition[END_REF][START_REF] Jing | Unsaturated mechanical behaviour of a granular material[END_REF][START_REF] Su | Investigation of the effect of water content on the mechanical behavior of track-bed materials under various coarse grain contents[END_REF]. Furthermore, the sulfate present in the RCA, which leads to the formation of ettringite and the carbonation of RCA during the curing phase, can also contribute to the self-cementing properties of RCA. However, the effects of sulfate and carbonation on self-cementing appear limited, compared with the hydration of unhydrated cement, which needs to be confirmed by further studies.

CONCLUSIONS

This study investigated the self-cementing properties of RCA, used as unbound granular materials, and their influence on the improvement of long-term performance, in terms of microstructure and mechanical behavior. The chemical properties of two different RCA (NRCA and ORCA), including the pH value, sulfate content and calcite content, were firstly studied. Then, a model based on the non-evaporable water was proposed to quantify the unhydrated cement content. To study the hydration process of unhydrated cement, and its effect on the microstructure and on the mechanical behaviour, fresh RCA specimen (cured for 1 day), and specimen cured for 360 days were tested, using different methods : TGA, SEM and monotonic triaxial tests. Based on these results, the self-cementing mechanisms were discussed . The following conclusions can be drawn from this research:

1. The first material, NRCA, was recently crushed, and presented self-cementing properties. This was in agreement with its high pH value (13.07). 2. The second material, ORCA, crushed in 2012 and submitted to long term storage, presented only negligible cementation. It was found to be highly carbonated, resulting in a low pH value (10.53) and high calcite content (29.97%). 3. Although the water soluble sulfate contents of both NRCA (0.23%) and ORCA (0.13%) were lower than the permitted level (0.7%), sulfate attack products (ettringite) were observed in NRCA. 4. The non-evaporable water content is a good indicator to quantify unhydrated cement content. The potential unhydrated cement content of NRCA fine aggregates (0-2 mm) was estimated between 5.9% and 7.6%. 5. After 360 days of curing, the non-evaporable water content of NRCA increased, while for ORCA, it remained almost constant. For NRCA, almost half of the unhydrated cement (36% to 45%) was hydrated after curing for 360 days, confirming that the selfcementing effect is particularly important during the first year. 6. SEM observations have shown that the hydration products form bonds and fill the small voids between particles (width less than 3 μm), while for big voids (width larger than 3 μm), the filling effect is limited. The initial open granular structure becomes more rigid and more compact after curing 360 days.

7. The she ar strength and stiffness of NRCA were multiplied by a factor of about 2 after 360 days of curing for all confining pressures. Conversely, the strength and stiffness of ORCA remained almost constant. 8. The hydration of unhydrated cement is responsible for the self-cementing properties of RCA. In addition, sulfate attack products (ettringite) and the decrease of free water also improve the mechanical behavior of RCA.

Figure 1 .

 1 Figure 1. Particle size distributions of NRCA and ORCA.Figure 2. Modified Proctor curves of NRCA and ORCA.

Figure 2 .

 2 Figure 1. Particle size distributions of NRCA and ORCA.Figure 2. Modified Proctor curves of NRCA and ORCA.

Figure 3 .

 3 Figure 3. DTG curves of NRCA and ORCA without curing.Figure 4. TG curves of NRCA and ORCA without curing.

  Figure 3. DTG curves of NRCA and ORCA without curing.Figure 4. TG curves of NRCA and ORCA without curing.

Figure 5 .

 5 Figure 5. Binder content and unhydrated binder content of NRCA calculated by Cwater and CCH for varying w/c.

Figure 6 .

 6 Figure 6. Comparison of TG-DTG curves of RCA without curing and after curing for 360 days: (a) NRCA; (b) ORCA.

  (a) 1 day (b) 360 days Figure 7. SEM images of NRCA after 1day and 360 days of curing: (a) 1 day; (b) 360 days.

Figure 8 .

 8 Figure 8. Development of strength and stiffness of RCA for different confining pressures (20/40/70 kPa), after 1 day and 360 days of curing : (a) Shear strength; (b) Stiffness.

  Table 2 summarizes the k values for different types of SCMs. The equivalent binder content (Cbinder) can be expressed as: *

	binder C	cement C 	k C	SCMs

Table 1 .

 1 Decomposition temperatures of different phases for NRCA and ORCA.

	Temperature/℃	NRCA	NRCA (curing	ORCA	ORCA (curing
		(no curing)	360 days)	(no curing)	360 days)
	T1	490	490	470	470
	T2	820	830	840	860

Table 2 .

 2 k values for different types of supplementary cementing materials.

	Additive Limestone	Blast
	filler	furnace slag

Table 3 .

 3 Chemical compositions of NRCA and ORCA.

	Properties	NRCA	ORCA	Limitation
	pH value	13.07	10.53	No
	Water soluble sulfate content/%	0.23	0.13	0.7
	Calcite content/%	14.7	29.97	No

Table 4 .

 4 Selected components of NRCA and ORCA without curing and after 360 days of curing.

	Materials Curing time	Weight loss/%	Non-	CaCO3 content
		/day	105-T1 ℃ (Cwater)	T1-T2 ℃ (CaCO3)	evaporable water Cwater /%	CCaCO3 /%
	NRCA	0	4.14	6.61	4.14	15.04
		360	4.79		4.79	21.52
	ORCA	0	2.10	13.78	2.10	31.34
		360	2.02	14.28	2.02	32.47

Table 5 .

 5 Variation of selected components and hydrated binder content of NRCA and ORCA .

	Materials ΔCwater	ΔCCaCO3	Hydrated binder
		/%	/%	content /%
	NRCA	0.65	6.48	2.70
	ORCA	-0.08	1.13	-0.34

ACKNOWLEDGMENTS

The authors are grateful to several colleagues from Lorraine university for the laboratory tests, especially to Cecile Dilberto who conducted TGA test, Jean Michel Mechnling who measured soluble sulfate content and Sebastien Roux who measured calcite content by calcimeter method. This work is supported by the China Scholarship Council.


J M Luck
email: jean-marc.luck@ipht.fr

Scaling laws for weakly disordered 1D flat bands

We investigate Anderson localization on various 1D structures having flat bands. The main focus is on the scaling laws obeyed by the localization length at weak disorder in the vicinity of flat-band energies. A careful distinction is made between situations where the scaling functions are universal (i.e., depend on the disorder distribution only through its width) and where they keep depending on the full shape of the disorder distribution, even in the weak-disorder scaling regime. Three examples are analyzed in detail. On the stub chain, one central flat band is isolated from two lateral dispersive ones. The localization length remains microscopic at weak disorder and exhibits disorder-specific features. On the pyrochlore ladder, the two flat bands are tangent to a dispersive one. The localization length diverges with exponent 1/2 and a non-universal scaling law, whose dependence on the disorder distribution is predicted analytically. On the diamond chain, a central flat band intersects two symmetric dispersive ones. The localization length exhibits two successive scaling regimes, diverging first with exponent 4/3 and a universal law, and then (i.e., further away from the pristine flat band) with exponent 1 and a non-universal law. Both scaling functions are also derived by analytical means.

Introduction

Tight-binding electronic spectra of crystalline structures having more than one atom per unit cell may exhibit flat bands. A flat band is a dispersionless branch of the spectrum, such that energy E is independent of wavevector q. Equivalently, the tightbinding Hamiltonian has an extensively degenerate set of eigenstates at that value of energy, with a basis of eigenstates that are strictly localized, i.e., supported by finitely many sites. Such an extensive (i.e., macroscopic) degeneracy induces an extremely high sensitivity to various kinds of perturbations, including interactions, disorder, or external fields. A weak disordered potential provides an example of foremost importance of such a perturbation. Any amount of disorder is indeed sufficient, in generic circumstances, to induce enough hybridization to entirely lift the extensive degeneracy. Flat-band systems have actually been shown to exhibit Anderson localization with various kinds of unconventional features [1,2,3,4,5,6,7,8]. More specific predictions have been made in the one-dimensional situation, where eigenstate are exponentially localized, irrespective of the disorder width, at least in generic circumstances [9,[START_REF] Lifshitz | Introduction to the Theory of Disordered Systems[END_REF][START_REF]50 Years of Anderson Localization[END_REF]. The key question therefore resides in the scaling behavior of the localization length ξ as a function of the disorder width w in the weak-disorder regime. For a usual dispersive band, ξ is known to diverge with exponent 2 inside the band, and with exponent 2/3 in the vicinity of band edges. More exotic exponents have been recently reported for various weakly disordered 1D structures having flat bands, including 0 for the stub chain, 1/2 for the cross-stitch and pyrochlore ladders, and 4/3 for the diamond chain and Lieb ladder [START_REF] Leykam | [END_REF]13,14,15,16].

The goal of this paper is to corroborate the above predictions on weakly disordered 1D flat-band systems and, more importantly, to complement them with quantitative results on the corresponding scaling laws (i.e., full scaling functions) in the weak-disorder regime. The main focus will be on whether these laws are universal (i.e., depend on the distribution of disorder only through its width w) or depend on the full shape of the disorder distribution. We shall study three examples in detail. In all three cases, the analysis will be reduced to investigating the Lyapunov exponent of products of random 2 × 2 transfer matrices in the weak-disorder regime. The key object of the approach is the invariant distribution of the Riccati variables (section 2). The stub chain (section 3) has an isolated flat band and scaling exponent 0, meaning that the localization length remains microscopic at weak disorder. Its non-universal features are only accessible by perturbative and numerical means. On the pyrochlore ladder (section 4), the two flat bands are tangent to a dispersive one. The localization length has scaling exponent 1/2. The associated non-universal law is predicted analytically. On the diamond chain (section 5), a central flat band intersects two symmetric dispersive ones. This is the richest of all examples. The localization length exhibits two successive scaling regimes, diverging first with exponent 4/3 and a universal law, and then (i.e., further away from the pristine flat band) with exponent 1 and a nonuniversal law. Both scaling functions are predicted by analytical means. Section 6 summarizes our findings. Two appendices contain more technical material. Cauchy tails of probability distributions are investigated in Appendix A, whereas Appendix B is devoted to the effective disorder on the pyrochlore ladder and the diamond chain.

A reminder on localization on the chain

This section is a reminder on localization properties of the tight-binding model on the chain with a weak diagonal disorder. This gives us the opportunity of introducing concepts and tools which will be at work in the sequel.

The tight-binding equation on the chain reads

Eψ n = ψ n+1 + ψ n-1 + v n ψ n . (2.1)
The random potentials v n are independent from each other and drawn from some symmetric (i.e., even) distribution ρ(v). On a clean chain (i.e., without disorder), the spectrum consists of a single band with dispersion relation

E = 2 cos q (|q| ≤ π).
(2.2)

Transfer-matrix approach and Lyapunov exponent

The transfer-matrix approach consists in recasting (2.1) as

ψ n+1 ψ n = T n ψ n ψ n-1 , (2.3)
where the transfer matrix

T n = E -v n -1 1 0 (2.4)
obeys det T n = 1. We have thus

ψ n+1 ψ n = T n . . . T 1 ψ 1 ψ 0 .
(2.5)

The exponential growth (or decay) of a generic wavefunction is therefore dictated by the exponential growth of a large product of random matrices. In particular, the Lyapunov exponent of the matrix product,

γ = lim n→∞ 1 n ln |T n . . . T 1 | , (2.6)
identifies with the inverse localization length [17,18,19,20,[START_REF] Luck | Systèmes désordonnés unidimensionnels[END_REF][START_REF] Comtet | [END_REF].

In the following, we recall the scaling properties of the Lyapunov exponent, both inside the band and near the band edges. The cases of regular disorder with a finite variance and of Cauchy disorder will be dealt with successively.

Regular disorder

Regular disorder corresponds to the case where the disorder variance v 2 = w 2 is finite (i.e., convergent). The quantity w is referred to as the disorder width. The scaling properties of the Lyapunov exponent at weak disorder are universal, in the sense that the disorder distribution only enters through w.

Inside the band, the Lyapunov exponent scales as

γ ≈ w 2 8 sin 2 q ≈ w 2 2(4 -E 2 ) . (2.7)
This result, known as the Thouless formula [START_REF] Thouless | Percolation and localization Ill-Condensed Matter[END_REF], breaks down near band edges (E → ±2), indicating that eigenstates are more strongly localized there than inside the band. The Lyapunov exponent obeys the scaling law [START_REF] Frisch | [END_REF]25,26,27,28] γ ≈ (w 2 /2)

1/3 F |E| -2 (w 2 /2) 2/3 , (2.8)
where the universal scaling function

F (x) = Ai(x)Ai ′ (x) + Bi(x)Bi ′ (x) Ai(x) 2 + Bi(x) 2 (2.9)
involves the Airy functions Ai and Bi.

Cauchy disorder (the Lloyd model)

The Cauchy distribution

ρ(v) = W π(v 2 + W 2 ) (2.10)
is a very special instance of a disorder distribution with an infinite variance. First, the tight-binding model with Cauchy disorder, known as the Lloyd model [29], is solvable on any regular lattice in any dimension, in the sense that the average one-particle Green's function can be evaluated exactly. Second, an effective Cauchy disorder emerges in several instances of 1D structures possessing flat bands [START_REF] Leykam | [END_REF]13,14,15,16], including the pyrochlore ladder and the diamond chain, investigated in sections 4 and 5.

Let us give a full derivation of the Lyapunov exponent for Cauchy disorder on the chain, following the approach introduced by Dyson [30] and Schmidt [31]. The key quantities are the Riccati variables, defined as the ratios

R n = ψ n+1 ψ n . (2.11)
They obey the recursion

R n = E -v n - 1 R n-1 , (2.12)
so that R n depends on R 0 and on the random potentials v 1 , . . . , v n . Its probability density f n (R n ) obeys the integral recursion formula

f n (R n ) = ∞ -∞ ρ(v n )dv n ∞ -∞ f n-1 (R n-1 )dR n-1 × δ R n -E + v n + 1 R n-1 = ∞ -∞ ρ(v n )dv n (E -v n -R n ) 2 f n-1 1 E -v n -R n . (2.13)
When n becomes infinitely large, f n (R n ) converges to a limiting distribution f (R), referred to as the invariant distribution associated with the random recursion (2.12). The Lyapunov exponent reads

γ = Re ln R = ln |R| , (2.14)
where the average is taken w.r.t. the invariant distribution.

For Cauchy disorder, (2.13) can be worked out explicitly. Assuming R n-1 has a Cauchy distribution of the form

f (R n-1 ) = b n-1 π((R n-1 -a n-1 ) 2 + b 2 n-1 ) , (2.15)
the recursion (2.13) implies that R n is also Cauchy distributed, with parameters

a n = E - a n-1 a 2 n-1 + b 2 n-1 , b n = W + b n-1 a 2 n-1 + b 2 n-1
.

(2.16)

The above recursion assumes a simpler form in terms of the complex variables

z n = a n + ib n , E = E + iW, (2.17)
namely

z n = E - 1 z n-1 .
(2.18)

The complex variables z n converge to the stable fixed point of this transformation,

z ⋆ = a ⋆ + ib ⋆ = E + √ E 2 -4 2 . (2.19)
The invariant distribution f (R) is therefore a Cauchy distribution, whose parameters a ⋆ and b ⋆ are the real and imaginary parts of z ⋆ . Equation (2.14) yields

γ = ln(a 2 ⋆ + b 2 ⋆ ) 2 = ln |z ⋆ |. (2.20)
More explicit predictions can be made at weak-disorder (W ≪ 1). Inside the band, the Lyapunov exponent scales as γ ≈ W 2 sin q .

(2.21)

The above estimate again breaks down near the band edges. Consider the upper band edge for definiteness. Introducing the scaling variable

x = E -2 W , (2.22)
we have

z ⋆ ≈ 1 + (x + i)W . (2.23)
The Lyapunov exponent therefore obeys the scaling law

γ ≈ W 1/2 F (L) (x), (2.24)
whose scaling function reads

F (L) (x) = Re √ x + i = 1 √ 2 x + x 2 + 1 1/2 . (2.25)
Anderson localization on the chain thus exhibits different scaling laws for a weak regular disorder and a weak Cauchy disorder. For regular disorder, the universal formulas (2.7) within the band and (2.8) near band edges have respective exponents 2 and 2/3. For Cauchy disorder, the formulas (2.21) within the band and (2.24) near band edges have respective exponents 1 and 1/2. It is also worth noting that the universal scaling function F (x) entering (2.8) in the regular case has a much richer structure than F (L) (x) entering (2.24) in the Cauchy case.

The stub chain

The stub chain (see figure 1, left) is the simplest of all 1D structures exhibiting a flat band [13,14,16]. We equip it with a tight-binding model with diagonal disorder. All hopping amplitudes between neighboring atoms equal unity. All disordered potentials v A n , v B n , v C n are drawn from the same symmetric distribution ρ(v) with variance v 2 = w 2 . The tight-binding equations at energy E read The dispersion relation of a clean stub chain is obtained by expressing that E is an eigenvalue of the matrix

(E -v A n )ψ A n = ψ B n-1 + ψ B n + ψ C n , (E -v B n )ψ B n = ψ A n + ψ A n+1 , (E -v C n )ψ C n = ψ A n . ( 3
M (q) =   0 1 + e -iq 1 1 + e iq 0 0 1 0 0   (3.2)
encoding the right-hand sides of (3.1). The spectrum (see figure 1, right) consists of one flat band and two dispersive bands:

E = 0, E = ± 3 + 2 cos q. (3.3)
The flat band is isolated, i.e., separated from the dispersive ones by a finite gap ∆E = 1. It consists of the extensively degenerate states

ψ B n = -σ n , ψ C n = σ n-1 + σ n , (3.4)
where σ n are arbitrary amplitudes. The most localized states, corresponding e.g. to σ n = δ n0 , live on three sites belonging to two consecutive cells, such as e.g. B 0 , C 0 and C 1 .

In the presence of a weak disorder, the flat band is broadened symmetrically over an energy range of order w. If the disorder distribution is supported by a finite interval, i.e., ρ(v) is non-zero only for |v| < V , the disordered flat band has well-defined edges at E = ±V . The density of states vanishes exponentially fast in the vicinity of these band edges, which correspond to internal Lifshitz tails [32,33]. For V < 1/2, the disordered flat band remains isolated from the dispersive ones. For the uniform distribution on [-V, V ], with w 2 = V 2 /3, the band edges read E = ±w √ 3. The weakly disordered flat band of the stub lattice has been investigated in several recent works [13,14,16]. The subsequent analysis corroborates earlier findings and allows us to make more precise statements on some aspects.

Whenever energy E and all disordered potentials are simultaneously small, the tight-binding equations (3.1) simplify to

ψ A n+1 ≈ -ψ A n + η n φ B n , φ B n+1 ≈ -φ B n -χ n ψ A n+1 , (3.5)
where the amplitudes ψ C n have been eliminated, the rescaling φ B n = Eψ B n has been done, and

η n = 1 - v B n E , χ n = 1 - v C n E -1 (3.6)
are reduced disorder variables. Furthermore, the rescaled equations (3.5) can be recast as a product of 2 × 2 independent random matrices, namely

ψ A n+1 φ B n+1 ≈ T n ψ A n φ B n , (3.7)
with

T n = - 1 -η n -χ n 1 + η n χ n . (3.8)
We have again det T n = 1. The associated Riccati variables obey the recursion

R n+1 = R n -η n 1 + χ n (η n -R n ) . (3.9)
The associated Lyapunov exponent

γ = ln R n -η n R n (3.10)
gives the inverse localization length, in units of the size of a unit cell. The above formalism involves E only through the dimensionless quantities η n and χ n . As a consequence, the Lyapunov exponent depends on energy only through the dimensionless ratio [14,16] x = E w .

(3.11)

In the absence of disorder (η n = χ n = 1), the recursion (3.9) has two fixed points, R + (unstable) and R -(stable), with

R ± = (1 ± √ 5)/2. The associated Lyapunov exponent is γ ∞ = ln(1 -1/R -), i.e., γ ∞ = ln 3 + √ 5 2 = 0.962423. (3.12)
This is the value of the inverse localization length for w ≪ |E| ≪ 1, i.e., |x| ≫ 1. This result can be recovered by more elementary means, along the lines of [16].

The pristine chain has extensively degenerate states given by (3.4). An infinitesimal disorder lifts this extensive degeneracy by allowing the flat-band states to hybridize with the dispersive ones. Setting E = 0 in the expression (3.3) of the dispersive bands yields a complex momentum q ⋆ such that cos q ⋆ = -3/2, i.e., e iq⋆ = -(3 ± √ 5)/2. The latter momentum describes the falloff of flat-band states in the limit of an infinitesimal disorder. The identification | Im q ⋆ | = γ ∞ allows one to recover (3.12).

A systematic weak-disorder expansion around (3.12) can be derived by considering the variables

Z n = R n -R - R n -R + , (3.13)
which represent the deviation of the Riccati variables from the stable fixed point, and are therefore small at weak disorder. The Z n obey the recursion

Z n+1 = N n+1 D n+1 , (3.14)
with

N n+1 = √ 5 -5 + (1 - √ 5)v B n + 4v C n + (1 + √ 5)v B n v C n (Z n -1) + √ 5 -5 + (5 + √ 5)v C n , D n+1 = -(4 + 2 √ 5)v B n -(1 + √ 5)v C n + (1 + √ 5)v B n v C n (Z n -1) -4 √ 5 -10 + (5 + √ 5)v C n , (3.15)
whereas (3.10) reads

γ = ln 1 + ( √ 5 + 1) (1 -v B n )(1 -Z n ) 2 + (3 + √ 5)Z n . (3.16)
Expanding the recursion (3.14) in powers of the disordered potentials allows one to derive a weak-disorder expansion of the moments of the invariant distribution of Z.

We thus obtain

Z = - 5 + √ 5 50 v 2 E 2 + 29 √ 5 -75 250 v 4 E 4 + 27 -12 √ 5 125 v 2 2 E 4 + • • •(3.17)
and analogous expressions for Z p for p = 2, 3 and 4. Putting everything together, we obtain a fourth-order expansion of the Lyapunov exponent. Once recast in terms of the reduced quantities x (see (3.11)) and κ = v 4 /w 4 , the kurtosis of the disorder distribution, the latter expansion appears as a large-x expansion:

γ = γ ∞ + √ 5 -1 5 x 2 + (6 √ 5 -8)κ + 5 -3 √ 5 25 x 4 + • • • (3.18)
Higher-order dimensionless ratios such as κ p = v 2p /w 2p for p = 3, 4, . . . enter higherorder terms of the above expansion. This demonstrates that the Lyapunov exponent depends on all moments of the disorder distribution, i.e., on its full shape, and not only on the disorder width w.

Besides the above expansion, no analytical result is available in general. In order to proceed, we must have recourse to a numerical study. A very efficient approach consists in building up the invariant measure by iterating the rescaled recursion (3.9) along a very long trajectory (e.g. of length 10 8 ). Figure 2 shows the Lyapunov exponent thus obtained. Throughout this work, we compare results for three disorder distributions: uniform, Gaussian and exponential (see Appendix B.3 for details).

The Lyapunov exponent (inverse localization length) is an even smooth function of the reduced energy x (see (3.11)), which depends on the shape of the disorder distribution. It exhibits a non-monotonic dependence on the reduced energy, with a non-trivial maximum γ max , put forward in Reference [16] as an anomalous minimum localization length. For large x, it approaches the limit γ ∞ from above, according to the expansion (3.18), which also predicts the observed ordering of the curves at large x. The coefficient of the kurtosis κ in (3.18) is positive, and the curves are indeed ordered as the kurtosis of the distributions, which are recalled in table 1 below.

For the uniform distribution, the disordered flat band exhibits a band edge at x = √ 3, shown as a vertical segment. The Lyapunov exponent reaches its maximum within the band but not far from the band edge. It seems to have only a very mild singularity there, in agreement with the Lifshitz behavior of the density of states recalled above. The Lyapunov exponent often reaches its minimum at the band center (γ min = γ(0)), in agreement with the intuitive picture that the states closest to the band center are the most hybridized, so that their localization length is the largest. This is however not true for the exponential distribution, where the Lyapunov exponent exhibits a non-trivial minimum γ min slightly below γ(0). All the characteristic values mentioned above depend on the shape of the disorder distribution (see table 1). The numbers for uniform and Gaussian disorders agree with those to be read off from the figures of [16]. Finally, for the distributions we have considered, all these quantities are monotonic functions of the kurtosis κ.

Distribution

The pyrochlore ladder

The pyrochlore ladder (see figure 3, left) is a well-known example of a 1D structure exhibiting flat bands [13,14]. We equip it with a tight-binding model with diagonal disorder. All disordered potentials are again drawn from the same distribution. The tight-binding equations at energy E read The dispersion relation of a clean ladder is obtained by expressing that E is an eigenvalue of the matrix

(E -v A n )ψ A n = ψ C n-1 + ψ D n-1 + ψ B n + ψ C n , (E -v B n )ψ B n = ψ C n-1 + ψ D n-1 + ψ A n + ψ D n , (E -v C n )ψ C n = ψ A n + ψ D n + ψ A n+1 + ψ B n+1 , (E -v D n )ψ D n = ψ B n + ψ C n + ψ A n+1 + ψ B n+1 . ( 4
M (q) =    0 1 1 + e -iq e -iq 1 0 e -iq 1 + e -iq 1 + e iq e iq 0 1 e iq 1 + e iq 1 0    . (4.2)
The spectrum (see figure 3, right) consists of two flat bands and two dispersive ones:

E = 0, E = -2, E = 1 ± 5 + 4 cos q. (4.3)
The lower dispersive band touches both flat bands. The latter consist of the extensively degenerate states

E = 0 : ψ A n = ψ C n = σ n , ψ B n = ψ D n = -σ n , E = -2 : ψ A n = ψ D n = τ n , ψ B n = ψ C n = -τ n , (4.4)
where σ n and τ n are arbitrary amplitudes. The most localized states, corresponding e.g. to σ n = δ n0 or τ n = δ n0 , live on the four sites of a single unit cell. The weakly disordered flat bands of the pyrochlore ladder have been investigated recently [13,14]. Here, at variance with the case of the stub chain, a dispersive band touches both flat bands, so that the localization length is expected to be large when disorder is weak. From a quantitative viewpoint, a scaling of the localization length in w -1/2 is predicted in References [13,14], based on the occurrence of an effective Cauchy disorder. The subsequent analysis corroborates this finding, provides an exact mapping onto the Lloyd model at weak disorder and yields the analytical prediction (4.23) for the corresponding non-universal scaling function F (pyro) (x).

Let us focus our attention onto the flat band at E = 0 and introduce the combinations

s n = ψ A n + ψ B n + ψ C n + ψ D n , t n = ψ A n + ψ B n -ψ C n -ψ D n , u n = ψ A n -ψ B n + ψ C n -ψ D n , v n = ψ A n -ψ B n -ψ C n + ψ D n . (4.5)
In the absence of disorder, the tight-binding equations (4.1) imply that s n and t n obey the recursion

s n+1 = t n -s n , t n+1 = -t n , (4.6)
while v n vanishes and u n is left arbitrary. The latter property is expected, as u n = 4σ n describes the compactly localized states on the pristine ladder (see (4.4)).

The idea of the subsequent analysis is to follow how the recursion (4.6), involving two variables only, is deformed when energy E and all random potentials are simultaneously small. A careful treatment of the tight-binding equations (4.1) to first order in the variables

η A n = E -v A n , η B n = E -v B n , η C n = E -v C n , η D n = E -v D n (4.7) yields s n+1 t n+1 ≈ M n s n t n , (4.8)
with

M n = -1 -α n+1 1 + α n+1 -β n+1 γ n + α n+1 -1 -δ n + β n+1 -α n+1 ≈ -1 -α n+1 1 + α n+1 -β n+1 α n+1 -1 + β n+1 -α n+1 1 -γ n δ n -γ n 1 + δ n (4.9)
and

α n = 1 ∆ n (2η A n η B n + η A n η D n + η B n η C n ), β n = 1 ∆ n (2η A n η B n + η A n η C n + η B n η D n ), γ n = 1 ∆ n (2η C n η D n + η A n η D n + η B n η C n ), δ n = 1 ∆ n (2η C n η D n + η A n η C n + η B n η D n ), (4.10)
with

∆ n = 2(η A n + η B n + η C n + η D n ). (4.11)
We have

α n + δ n = β n + γ n .
In the weak-disorder regime, the Lyapunov exponent of the above matrix product can be determined by means of an exact mapping onto the Lloyd model (see section 2.3). The mapping proceeds in two steps. First, the transfer matrices M n are not independent from each other, as they involve the disordered potentials in two neighboring cells. This can be remedied by swapping both matrices entering the second line of (4.9), i.e., by replacing M n by

M n = - 1 -γ n δ n -γ n 1 + δ n -1 -α n 1 + α n -β n α n -1 + β n -α n ≈ 1 + a n -1 + c n b n 1 + d n , (4.12)
to leading order, with

a n = α n -γ n , b n = -α n -γ n , c n = 2δ n , d n = 2γ n . (4.13)
We have a n + b n + d n = 0, and so det M n = 1, to leading order. A minus sign has been introduced for convenience in the definition of M n . Second, a change of basis defined by the matrix

P = 1 0 -1 1 (4.14)
amounts to replacing M n by

T n = P -1 M n P ≈ 2 + a n -c n -1 + c n 1 -c n -2d n c n + d n . (4.15)
At weak disorder, the above transfer matrix is close to that of the Anderson model on the chain, given by (2.4), near the upper band edge (E ≈ 2). The Riccati variables associated with (4.15) obey the recursion

R n = (2 + a n -c n )R n-1 -1 + c n (1 -c n -2d n )R n-1 + c n + d n . (4.16)
Let us anticipate for a while that the disorder entering (4.15) has Cauchy tails. An appropriate identification of the random recursions (2.12) and (4.16) yields a mapping of the present problem onto the Lloyd model (see section 2.3). Let us denote quantities pertaining to the latter model with a superscript (L). Near the upper band edge (E (L) ≈ 2) and at weak disorder, the invariant distribution

f (L) (R) is a narrow distribution concentrated around R = 1. Equation (2.23) indeed shows that R = a (L) ⋆
is close to unity, whereas the width b L) . It is therefore sufficient to compare the formulas (2.12) and (4.16) in the vicinity of R = 1. Both equations obey ∂R n /∂R n-1 ≈ 1 at weak disorder. Equating the values of R n for R n-1 = 1 yields the identification

(L) ⋆ scales as √ W (
E (L) -2 -v (L) n ≈ a n + d n = α n + γ n = η (1) n η (2) n η (1) n + η (2) n , (4.17)
to leading order, with

η (1) n = η A n + η C n = 2E -v A n -v C n , η (2) n = η B n + η D n = 2E -v B n -v D n . (4.18)
The effective disorder defined in the rightmost side of (4.17) is a homogeneous but non-linear combination of the four disordered potentials of the nth unit cell.

It is typically small at weak disorder. It may however become large in the rare instances where its denominator nearly vanishes. As a consequence, it has a broad probability distribution exhibiting symmetric Cauchy tails (see Appendix A). In the present context, this specific feature of disorder has been put forward for the first time in Reference [START_REF] Leykam | [END_REF]. The above effective disorder is therefore characterized by two parameters, its Cauchy strength and its mean value, namely

G 2 (E) = δ η (1) n + η (2) n η (1) n η (2) n , H 2 (E) = η (1) n η (2) n η (1) n + η (
G 2 (E) = wg 2 (x), H 2 (E) = wh 2 (x), x = E w . (4.20)
Working out explicitly the identification (4.17) yields

W (L) ≈ πwg 2 (x), x (L) ≈ h 2 (x) πg 2 (x) . (4.21)
We finally obtain our prediction for the scaling behavior of the Lyapunov exponent of the weakly disordered flat band near E = 0 by inserting the estimates (4.21) into (2.24). This reads

γ ≈ w 1/2 F (pyro) (x), (4.22)
where the scaling function reads

F (pyro) (x) = 1 √ 2 h 2 (x) + h 2 (x) 2 + π 2 g 2 (x) 2 1/2 . ( 4.23)
This result simplifies right at E = 0. We have h 2 (0) = 0 by symmetry, and so

F (pyro) (0) = πg 2 (0) 2 1/2 . ( 4.24)
The scaling function F (pyro) (x) is non-universal, as it depends on the full shape of the disorder distribution through the functions g 2 (x) and h 2 (x), investigated in Appendix B.2. For the uniform, Gaussian and exponential distributions, explicit expressions of g 2 (x) are given in Appendix B.3, and the values of g 2 (0) in table B1. The function h 2 (x) is expressed as an integral that can be carried out in closed form only for Gaussian disorder.

The scaling function F (pyro) (x) is plotted in figure 4 for the uniform, Gaussian and exponential disorder distributions. The right panel, showing an enlargement of the central part of the left one, demonstrates that the analytical prediction (4.22) is in excellent agreement with numerical results for the Lyapunov exponent (symbols), based on products of 10 8 matrices for each type disorder with w = 0.001.

The most salient feature of these plots is that they are very asymmetric. It is indeed expected that the Lyapunov exponent (inverse localization length) is much smaller for x negative, i.e., on the side of the dispersive band, than for x positive, i.e., on the side of the gap. From a quantitative viewpoint, deep in the gap (x ≫ 1, i.e., E ≫ w), g 2 (x) falls off rapidly to zero, whereas h 2 (x) ≈ x (see (B.19)). We thus obtain the universal law irrespective of the disorder distribution. This collapse is clearly seen in the right part of the left panel. The formula (4.25) can be recast as γ ≈ √ E. This estimate, which does not involve disorder at all, is in agreement with the quadratic form of the lower dispersive band near E = 0, i.e., E ≈ -(π -q) 2 (see (4.3)), and so γ = | Im q | ≈ √ E. For the uniform distribution, g 2 (x) vanishes identically for |x| > √ 3 (see (B.24)). As a consequence, F (pyro) (x) vanishes for x < -√ 3, implying that γ vanishes more rapidly than √ w as w tends to zero while E < -w √ 3. The Lyapunov exponent is indeed expected to exhibit the usual scaling in w 2 at any fixed energy inside the dispersive band. In the central part, enlarged in the right panel, the scaling function F (pyro) (x) exhibits a rather strong dependence on the disorder distribution. It is non-monotonic for the uniform and Gaussian distributions.

F (pyro) (x) ≈ √ x, (4.25)

The diamond chain

The diamond chain (see figure 5, left) is another 1D structure known to exhibit a flat band [START_REF] Leykam | [END_REF]14]. We equip it with a tight-binding model with diagonal disorder. All disordered potentials are again drawn from the same distribution. The tight-binding equations at energy E read

(E -v A n )ψ A n = ψ C n + ψ C n+1 , (E -v B n )ψ B n = ψ C n + ψ C n+1 , (E -v C n )ψ C n = ψ A n-1 + ψ B n-1 + ψ A n + ψ B n . (5.1)
The dispersion relation of a clean chain is obtained by expressing that E is an eigenvalue of the matrix The spectrum (see figure 5, right) consists of one flat band and two dispersive ones:

M (q) =   0 0 1 + e iq 0 0 1 + e iq 1 + e -iq 1 + e -iq 0   . ( 5
E = 0, E = ± 2 √ 2 cos q 2 . ( 5.3)
Both dispersive bands intersect the flat one at the edges of the Brillouin zone (q = ±π).

The flat band consist of the extensively degenerate states

ψ A n = -ψ B n = σ n , (5.4)
where σ n are arbitrary amplitudes. The most localized states, corresponding to e.g. σ n = δ n0 , live on two sites. The weakly disordered flat band of the diamond chain has been investigated recently [START_REF] Leykam | [END_REF]14]. Here, too, as dispersive bands intersect the flat band, the localization length is expected to be large when the disorder is weak. From a quantitative viewpoint, a scaling of the localization length in w -4/3 was predicted in References [START_REF] Leykam | [END_REF]14]. The subsequent analysis refines this finding. We shall indeed demonstrate that the Lyapunov exponent obeys two different scaling laws in two successive regimes as energy goes away from the pristine flat band, scaling first as w 4/3 in Regime I (E ∼ w 5/3 ) and then as w in Regime II (E ∼ w).

The scaling analysis of the model goes as follows. Introducing the combinations

s n = ψ A n + ψ B n , t n = ψ A n -ψ B n , (5.5)
one can eliminate t n from the tight-binding equations (5.1), whereas the remaining equations can be recast -without any approximation -as

s n ψ C n+1 = T n s n-1 ψ C n , (5.6)
where

T n = - 1 -η C n χ n 1 -χ n η C n , (5.7)
with

η A n = E -v A n , η B n = E -v B n , η C n = E -v C n (5.8)
and

χ n = η A n η B n η A n + η B n .
(5.9)

The above non-linear combination of disorder variables again has a broad probability distribution with symmetric Cauchy tails (see Appendix A). It is therefore characterized by its Cauchy strength and its mean value:

G 1 (E) = δ 1 χ n = δ η A n + η B n η A n η B n , H 1 (E) = χ n = η A n η B n η A n + η B n . ( 5
G 1 (E) = wg 1 (x), H 1 (E) = wh 1 (x), x = E w .
(5.11)

The functions G 1 (E) and g 1 (x) are even, whereas H 1 (E) and h 1 (x) are odd. The transfer matrix (5.7) obeys det T n = 1. The associated Riccati variables obey the recursion

R n = χ n + R n-1 1 -η C n R n-1 , (5.12)
where the Cauchy distributed random quantity χ n enters additively. Equivalently, their reciprocals S n = 1/R n obey

S n = S n-1 -η C n 1 + χ n (S n-1 -η C n ) . (5.13)
The associated Lyapunov exponent reads

γ = ln |1 -η C n R n-1 | . (5.14)
It gives the inverse localization length, in units of the size of a unit cell.

It is worth beginning with a heuristic scaling analysis. Consider first the situation where E = 0 exactly. One can get an idea of the dynamics generated by the recursion (5.12) at weak disorder by comparing two extreme initial conditions. If the Riccati variables are initialized to R 0 = 0, (5.12) yields R n ≈ R n-1 + χ n , so that R n is the sum of n Cauchy variables, hence R n ∼ nw. Setting R 0 = ∞, i.e., S 0 = 0, (5.13) yields S n ≈ S n-1 + η C n , with η C n = E = 0, and so S n is the sum of n random variables with zero mean and finite variance, hence S n ∼ n 1/2 w. Equating both estimates yields R ⋆ ∼ w -1/3 . This is the predicted scaling of the typical value of the Riccati variable. Furthermore, the number of steps needed to relax to this typical scale from either initial condition scales as n ⋆ ∼ w -4/3 . This provides an estimate of the correlation length. We thus obtain γ ∼ w 4/3 . At non-zero energy E, the above scaling laws hold as long as the cumulative effect of η C n does not take the lead. This condition reads En ⋆ R ⋆ ∼ 1, i.e., E ∼ w 5/3 , or x ∼ w 2/3 . We shall refer to this range as Regime I. Consider now Regime II, defined as E ∼ w, or x ∼ 1. The dynamics with R 0 = 0 is unchanged, whereas the initial condition S 0 = 0 yields S n ∼ nE. Equating both estimates yields R ⋆ ∼ 1 and γ ∼ w.

The quantitative analysis of the weak-disorder regime exposed below corroborates the existence of two successive scaling regimes. The random recursion (5.12) can be expanded as

R n -R n-1 ≈ χ n + η C n R 2 n-1 + (η C n ) 2 R 3 n-1 , (5.15)
where each term is small, and higher-order terms are negligible. The distribution of χ n has been described above (see (5.10), (5.11)). The distribution of η C n is a narrow regular one, with η

C n = E = wx and (η C n ) 2 = E 2 + w 2 = w 2 (1 + x 2
). As a consequence, the invariant distribution f (R) of the Riccati variable obeys

f (R) ≈ -wh 1 (x) d dR f (R) + wg 1 (x) ∞ -∞ f (R -χ) dχ χ 2 + (πwg 1 (x)) 2 -wx d dR (R 2 f (R)) + w 2 2 d 2 dR 2 (R 4 f (R)) -w 2 (1 + x 2 ) d dR (R 3 f (R)).
(5.16)

The three lines correspond to the three contributions in the r.h.s. of (5.15), in the same order. In the third line, the term involving a second derivative has been skipped as it is negligible.

In order to solve the integro-differential equation (5.16), it is advantageous to introduce the Hilbert transform

L(p) = ∞ -∞ f (R) dR p + iR , (5.17)
where p is a complex variable obeying Re p > 0. The use of Hilbert transforms in the analysis of one-dimensional disordered systems dates back to Dyson's pioneering work [30]. It has been instrumental in a range of other investigations [28,34,35,36,37].

In the present situation, the Hilbert transform of the second term in the first line on the r.h.s. of (5.16) is simply

L(p + πwg 1 (x)) ≈ L(p) + πwg 1 (x)L ′ (p)
, where accents denote derivatives w.r.t. p. Integrations by parts readily yield the Hilbert transforms of all other terms. We are left with

0 ≈ w(ih 1 (x) + πg 1 (x))L ′ (p) -iwx(p 2 L ′ (p) + 2pL(p) -1) + w 2 2 (-p 4 L ′′ (p) -8p 3 L ′ (p) -12p 2 L(p) + 6p) + w 2 (1 + x 2 )(p 3 L ′ (p) + 3p 2 L(p) -2p), (5.18)
where terms are written in the same order as in (5.16). The above equation will be solved successively in both regimes evidenced by the heuristic analysis.

Regime I (E ∼ w 5/3 ). This first regime can be parametrized by the scaling variable

y = E w 5/3 = x w 2/3 . (5.19)
The reduced variable x is small, and so g 1 (x) can be approximated by g 1 (0), while h 1 (0) vanishes by symmetry. Setting all terms in the r.h.s. of (5.18) contribute on the same footing, confirming the relevance of the scaling E ∼ w 5/3 . We are left with a dimensionless second-order differential equation for M (s), which can be integrated once, yielding

λ = w 1/3 (2πg 1 (0)) 1/3 , Y = 2 2/3 (πg 1 (0)) 1/3 y, ( 5
s 4 M ′ (s) + (2s 3 + iY s 2 -1)M (s) = s 2 + iY s -φ(Y ).
(5.22)

The complex function φ(Y ), which entered (5.22) as an integration constant, is of central importance. It will indeed determine the universal scaling function F I (Y ) involved in the result (5.31) for the Lyapunov exponent. It can be determined as follows (see [37] and [28]). Using the complex variable u = 1/s instead of s, and setting M (u) = M (s), the general solution to (5.22) reads

M (u) = u 2 e iY u-u 3 /3 K(u), (5.23)
with

K ′ (u) = - 1 u 2 + iY u -φ(Y ) e -iY u+u 3 /3 . (5.24)
Consider the behavior of M (u) for u = ρ e ±iπ/3 , with ρ real and positive, so that u 3 = -ρ 3 . In the ρ → +∞ limit in either direction, the growth of the exponential factor in the expression of M (u) must be compensated by an exponential falloff of K(u). We thus obtain the condition K(e iπ/3 ∞) = K(e -iπ/3 ∞) = 0, hence

C K ′ (u)du = 0, (5.25)
where the contour C joins e -iπ/3 ∞ to e iπ/3 ∞. An integration by parts yields

C (φ(Y ) -u)e -iY u+u 3 /3 du = 0. (5.26)
Recognizing the integral representations of the Airy function and of its derivative, we obtain

φ(Y ) = - Ai ′ (iY ) Ai(iY ) . (5.27)
The scaling behavior of the Lyapunov exponent can be investigated by averaging the expression (5.14) first over the invariant distribution of the Riccati variable R n-1 , then over the disorder η C n . The first average goes as follows. The definition (5.17) implies

Λ(p) = ln 1 + iR p = ∞ p 1 q -L(q) dq. (5.28)
Only the first two terms of the large-p expansion of Λ(p) will be needed. The differential equation (5.24) yields the expansion

K(u) = 1 u + K 0 (Y ) + φ(Y ) - Y 2 2 u + • • • (u → 0), (5.29)
where the explicit value of the complex integration constant K 0 (Y ) will not be needed.

The above expansion translates into

Λ(p) = - iY + K 0 (Y ) λp + Y 2 -iY K 0 (Y ) -φ(Y ) 2λ 2 p 2 + • • • (5.30)
Performing the substitution p → i/η C n in the above expansion, taking the real part and averaging over η C n , we obtain the following prediction for the scaling behavior of the Lyapunov exponent all over Regime I:

γ ≈ (πg 1 (0)) 2/3 2 1/3 F I (Y ) w 4/3 , (5.31)
where

F I (Y ) = Re φ(Y ), (5.32)
with Y defined in (5.20) and φ(Y ) given by (5.27). The result (5.31) depends on energy through the universal scaling function F I (Y ). It depends on the disorder distribution through g 1 (0), which enters both (5.31) explicitly and the definition (5.20) of the scaling variable Y . As predicted by the heuristic reasoning, the Lyapunov exponent scales as w 4/3 , whereas the typical Riccati variable scales as 1/λ, i.e., as w -1/3 .

The scaling function F I (Y ) is even. Its behavior at small Y reads

F I (Y ) = f 0 + f 2 Y 2 + • • • , (5.33)
with

f 0 = - Ai ′ (0) Ai(0) = 3 1/3 Γ(2/3) Γ(1/3) = 0.729011, f 2 = 1 2 + Ai ′ (0) Ai(0) 3 = 1 2 -3 Γ(2/3) Γ(1/3) 3 = 0.112561. (5.34)
Its behavior at large Y ,

F I (Y ) ≈ |Y | 2 1/2 , (5.35)
yields the estimate

γ ≈ πg 1 (0)|E|w 2 1/2 , (5.36)
in agreement with the small-x behavior in Regime II (see (5.45)). The scaling function F I (Y ) entering the prediction (5.31) is plotted in figure 6. A good agreement is observed with numerical results for the Lyapunov exponent (symbols), based on products of 10 9 matrices for uniform, Gaussian and exponential disorders with w = 0.0025. The corresponding values of g 1 (0) are given in table B1.

Regime II (E ∼ w). This second regime is parametrized by the scaling variable

x = E w . (5.37)
The heuristic reasoning predicts that typical Riccati variables are of order unity. As a consequence, no rescaling of the variable p is involved in the analysis of (5.18). Keeping only terms proportional to w, the latter equation boils down to

(p 2 -µ(x) 2 )L ′ (p) + 2pL(p) = 1, (5.38)
where we have introduced the complex function

µ(x) = α(x) -iβ(x) = h 1 (x) -iπg 1 (x) x 1/2 , (5.39)
hence We recall that g 1 (x) is even and h 1 (x) is odd, and so α(x) is even and β(x) is odd. The solution to (5.38) is

α(x) = 1 |x| √ 2 |x| h 1 (x) 2 + π 2 g 1 (x) 2 + xh 1 (x) 1/2 , β(x) = 1 x √ 2 |x| h 1 (x) 2 + π 2 g 1 (x) 2 -xh 1 (x)
L(p) = 1 p + µ(x) . (5.41)
There can indeed be no pole at p = µ(x), whose real part α(x) is positive. The corresponding invariant distribution of the Riccati variable is Cauchy:

f (R) = α(x) π ((R + β(x)) 2 + α(x) 2 )
.

(5.42)

The scaling behavior of the Lyapunov exponent is obtained by linearizing the expression (5.14). We thus obtain γ ≈ -η C R , with η C = E = xw and R = -β(x), by symmetry, and so γ ≈ w F II (x), (5.43) in qualitative agreement with the heuristic reasoning. The corresponding scaling function,

F II (x) = xβ(x) = 1 √ 2 |x| h 1 (x) 2 + π 2 g 1 (x) 2 -xh 1 (x) 1/2 , (5.44)
is non-universal, as it depends on the full shape of the disorder distribution, through the functions g 1 (x) and h 1 (x), investigated in Appendix B.2. The function F II (x) is even, as should be. For small values of x, we have

F II (x) ≈ πg 1 (0)|x| 2 1/2 , (5.45)
so that a smooth crossover to Regime I is observed (see (5.36)).

For the uniform, Gaussian and exponential distributions, explicit expressions of g 1 (x) are given in Appendix B.3, whereas h 1 (x) is expressed as an integral that can be carried out in closed form only for Gaussian disorder. The resulting scaling functions F II (x) are plotted in figure 7. The prediction (5.43) is found in excellent agreement with numerical results for the Lyapunov exponent (symbols), based on products of 10 8 matrices for each type disorder, with w = 0.0025. For the uniform distribution, g 1 (x) vanishes identically for |x| > √ 3 (see (B.23)). As a consequence, F II (x) also vanishes there, implying that γ falls off more rapidly than linearly in w in that range. Gathering our findings on both scaling regimes at weak disorder, we predict that the Lyapunov exponent is a highly non-monotonic function of energy. Considering only positive energies for definiteness, γ exhibits a minimum at the center of the dispersive band (E = 0), scaling as w 4/3 (Regime I), a maximum for E ∼ w, scaling as w (Regime II), a second minimum near the middle of the dispersive band, scaling as w 2 , before it reaches values of order unity if E exceeds the outer band edge (E > 2 √ 2) and then keeps growing indefinitely. The highly non-monotonic profile sketched above is a weak-disorder feature, which only takes place below some non-universal threshold. The minimum at the band center only occurs below w c ≈ 0.32 for the uniform distribution, and below w c ≈ 0.90 for the Gaussian distribution, with the conventions used in this work. This explains why it is not visible on the data shown in Reference [START_REF] Leykam | [END_REF]. For the exponential distribution, the maximum for E ∼ w disappears at w c ≈ 1.69, and so the Lyapunov exponent increases monotonically with energy whenever w is above w c .

Discussion

We have investigated the scaling behavior of the localization length in the weakdisorder regime of various 1D structures having flat bands. The foremost feature is the power-law divergence ξ ∼ w -ν of the localization length at weak disorder. Earlier studies [START_REF] Leykam | [END_REF]13,14,15,16] have shown that the two well-known values of the exponent ν pertaining to the Anderson problem on the chain, ν = 2/3, 2, (

are supplemented by a few more more exotic values ν = 0, 1/2, 1, 4/3, (

pertaining to 1D flat-band systems. To our knowledge, whether the above list is complete or whether it can be derived by systematic reasoning are still open questions.

The main focus of the present work has been on the full scaling functions describing the dependence of the Lyapunov exponent (inverse localization length) γ on energy E and disorder width w throughout weakly disordered flat bands. A careful distinction has been made between situations where the scaling functions are universal (i.e., depend on the disorder distribution only through its width) and where they depend on the shape of the disorder distribution. Three examples (stub chain, pyrochlore ladder, diamond chain), exhausting the full list (6.2) of exotic exponents known to date, have been analyzed in detail. In each case, the analysis of the localization length at weak disorder has been reduced to investigating the Lyapunov exponent of products of random 2 × 2 transfer matrices. The latter reduction, which is asymptotically exact to leading order in the weak-disorder regime, can be thought of as a thorough implementation of the detangling procedure put forward in Reference [13].

Our main specific findings are as follows. On the stub chain (section 3), where one central flat band is isolated from two lateral dispersive ones, the localization length remains microscopic at weak disorder. Its behavior is only accessible by perturbative and numerical means. Its non-universal dependence on the ratio x = E/w exhibits several features of interest (see figure 2 and table 1). In particular, the Lyapunov exponent takes a constant value γ ∞ (see (3.12)) for |x| ≫ 1, i.e., w ≪ |E| ≪ 1. The latter constant is dictated by the band structure of the pristine system. It is formally infinite in situations where all bands are flat. An interesting example is provided by the diamond ladder subjected to a magnetic field, in the case where each unit cell is pierced by half a flux quantum [38,39]. The spectrum of that system consists of three flat bands at energies E = 0, ±2. There, omitting details, we predict a logarithmic divergence of the Lyapunov exponent of the form γ ≈ ln |x| + K for |x| ≫ 1, where K is an explicit non-universal constant, depending on the full shape of the disorder distribution.

In the other geometries considered here, the flat bands are not isolated from the dispersive ones, and so the localization length diverges in the weak-disorder regime. On the pyrochlore ladder (section 4), the two flat bands are tangent to a dispersive one. The localization length has scaling exponent 1/2 and obeys the non-universal scaling law (4.22). The dependence of the scaling function F (pyro) (x) on the disorder distribution is given by (4.23) in terms of the functions g 2 (x) and h 2 (x), investigated in Appendix B.2 and characteristic of the Cauchy effective disorder. On the diamond chain (section 5), a central flat band intersects two symmetric dispersive ones. The situation at weak disorder is richer than the previous cases. The localization length now exhibits two successive scaling regimes. In Regime I (E w 5/3 ), it diverges with exponent 4/3, according to (5.31), and a universal scaling function F I (Y ) given by (5.32). In Regime II (E ∼ w), it diverges with exponent 1, according to (5.43), and a non-universal scaling function F II (x) (see (5.44)) depending on the full shape of the disorder distribution, through the functions g 1 (x) and h 1 (x) investigated in Appendix B.1. Here, too, the lack of universality in this second regime originates in the occurrence of a broad probability distribution of the effective disorder, exhibiting symmetric Cauchy tails.

The Cauchy strength G, introduced above, characterizes atypical large values of y, whereas its typical small values are characterized by its symmetric mean value, i.e., H = y = lim Λ→∞ Λ -Λ y f y (y) dy.

(A.5)

In the above example, we have

H = 1 x = - ∞ -∞ f x (x) dx x = lim ε→0 -ε -∞ + ∞ ε f x (x) dx x , (A.6)
where the bar denotes the Cauchy principal value, i.e., explicitly,

H = lim ε→0 -ε -a + b ε dx (a + b)x = 1 a + b ln b a . (A.7)
This expression vanishes in the symmetric situation where a = b, as should be.

In spite of their interpretation, the Cauchy strength G and the mean value H of y have the same dimension. As a consequence, they are of the same order of magnitude in the weak-disorder regime where a and b are simultaneously large.

Appendix B. Effective disorder on the diamond chain and on the pyrochlore ladder

This appendix is devoted to an investigation of the effective disorder on the diamond chain and the pyrochlore ladder. We consider a symmetric continuous distribution ρ(v) of the random potentials, such that all moments are finite. The disorder width w is defined by setting v 2 = w 2 . The kurtosis is defined by

κ = v 4 v 2 2 = v 4 w 4 . (B.1)
The Fourier transform of the distribution reads

ρ(q) = e iqv = ∞ -∞ e iqv ρ(v)dv. (B.2) Appendix B.

The effective disorder on the diamond chain

The effective disorder on the diamond chain is characterized by its Cauchy strength and its mean value (see (5.10)):

G 1 (E) = δ η A + η B η A η B , H 1 (E) = η A η B η A + η B , (B.3) with η A = E -v A , η B = E -v B . (B.4)
The Cauchy strength can be evaluated as

G 1 (E) = ∞ -∞ ρ(E -η A )dη A ∞ -∞ ρ(E -η B )dη B δ η A + η B η A η B = ∞ -∞ ρ(E -η)ρ(E + η)η 2 dη. (B.5)
The second line is obtained by integrating over η B , denoting η A as η. This can be recast in Fourier space as

G 1 (E) = - 1 2π ∞ -∞ ρ(q)e iqE d 2 dq 2 ρ(q)e iqE dq. (B.6)
The latter expression is more amenable to an explicit evaluation for specific disorder distributions.

The mean value can be evaluated as

H 1 (E) = (E -v A )(E -v B ) 2E -v A -v B = E 2 - 1 4 (v A -v B ) 2 2E -v A -v B . (B.7)
At this stage it is advantageous to add an infinitesimal imaginary part to E. We have

Im H 1 (E + i0) = π 4 (v A -v B ) 2 δ(2E -v A -v B ) = π 4 ∞ -∞ ρ(v A )dv A ∞ -∞ ρ(v B )dv B (v A -v B ) 2 × δ(2E -v A -v B ). (B.8) Setting v A = E -η, v B = E + η, we get Im H 1 (E + i0) = π ∞ -∞ ρ(E -η)ρ(E + η)η 2 dη = πG 1 (E). (B.9)
The real part H 1 (E) ≡ Re H 1 (E + i0) can be reconstructed from the imaginary part by an integral equation somehow similar to the Kramers-Kronig dispersion relations:

H 1 (E) = E 2 + - ∞ -∞ G 1 (E ′ )dE ′ E ′ -E , (B.10)
where the bar again denotes the Cauchy principal value.

The scaling dependence of G 1 (E) and H 1 (E) on w reads

G 1 (E) = wg 1 (x), H 1 (E) = wh 1 (x), x = E w . (B.11)
The functions G 1 (E) and g 1 (x) are even, whereas H 1 (E) and h 1 (x) are odd, and (B.10) translates to

h 1 (x) = x 2 + - ∞ -∞ g 1 (x ′ )dx ′ x ′ -x . (B.12)
Appendix B.2. The effective disorder on the pyrochlore ladder

The effective disorder on the pyrochlore ladder is characterized by its Cauchy strength and its mean value (see (4.19)):

G 2 (E) = δ η (1) + η (2)
η (1) η (2) , H 2 (E) = η (1) η (2) η (1) + η (2) , (B.13) with whose Fourier transform is ρ 2 (q) = ρ(q) 2 . Applying these changes to the results (B.5), (B.6) and (B.10), we readily obtain

η (1) = 2E -v A -v C , η (
G 2 (E) = ∞ -∞ ρ 2 (2E -η)ρ 2 (2E + η)η 2 dη = - 1 2π ∞ -∞
ρ(q) 2 e 2iqE d 2 dq 2 ρ(q) 2 e 2iqE dq (B. 16)

and

H 2 (E) = E + - ∞ -∞ G 2 (E ′ )dE ′ E ′ -E . (B.17)
The scaling dependence of G 2 (E) and H 2 (E) on w reads The functions G 2 (E) and g 2 (x) are even, whereas H 2 (E) and h 2 (x) are odd, and (B.17) translates to Inserting the latter expression into (B.6) and (B.16), we obtain the following expressions for g 1 (x) and g 2 (x) for x ≥ 0 (these functions are even):

h 2 (x) = x + - ∞ -∞ g 2 (x ′ )dx ′ x ′ -x . (B.
g 1 (x) =    1 18 ( √ 3 -x) 3 (0 < x < √ 3), 0 (x > √ 3), (B.23) g 2 (x) =            g ⋆ (x) 0 < x < √ 3/
)
where erf is the error function. We have v 2 = 2V 2 and v 4 = 24V 4 , and so w = V √ 2, κ = 6. (B.32) Its Fourier transform reads ρ(q) = 1 1 + q 2 V 2 .

(B.33)

Inserting the latter expression into (B.6) and (B.16), we obtain the following expressions for g 1 (x) and g 2 (x) for x ≥ 0 (these functions are even): .

g 1 (x) = √ 2
The functions h 1 (x) and h 2 (x) have not been evaluated in closed form.

To close, we give in table B1 the values of the Cauchy strength of the effective disorder at the center of the flat band at E = 0 for the two flat-band structures and the three disorder distributions studied in this appendix.

Figure 1 .

 1 Figure 1. Left: the stub chain has 3 atoms per unit cell. Right: its spectrum (see (3.3)) has one central flat band (red) isolated from two symmetric dispersive ones (blue).

Figure 2 .

 2 Figure 2. The Lyapunov exponent γ against reduced energy x = E/w for three disorder distributions: uniform, Gaussian and exponential. Vertical segment: edge of flat band for the uniform distribution (x = √ 3).

Figure 3 .

 3 Figure 3. Left: the pyrochlore ladder has 4 atoms per unit cell. Right: its spectrum (see (4.3)) has two flat bands (red) and two dispersive ones (blue).

Figure 4 .

 4 Figure 4. Left: the non-universal scaling function F (pyro) (x) (see (4.23)) entering the prediction (4.22) for the Lyapunov exponent of the pyrochlore ladder near E = 0, against reduced energy x = E/w, for three disorder distributions: uniform, Gaussian and exponential. Right: enlargement of the central part of the left plot. Symbols: Numerical results for w = 0.001.

Figure 5 .

 5 Figure 5. Left: the diamond chain has 3 atoms per unit cell. Right: its spectrum (see(5.3)) has one flat band (red) and two dispersive ones (blue).

. 10 )

 10 These quantities are investigated in Appendix B.1. Their scaling dependence on w reads (see (B.11))

Figure 6 .

 6 Figure 6. The universal scaling function F I (Y ) entering the prediction (5.31) for the Lyapunov exponent of the diamond chain in Regime I, against the rescaled energy variable Y . Symbols: Numerical results for three disorder distributions: uniform, Gaussian and exponential, with w = 0.0025.

Figure 7 .

 7 Figure 7. The non-universal scaling functions F II (x) entering the prediction (5.43) for the Lyapunov exponent of the diamond chain in Regime II, against the rescaled energy variable x, for three disorder distributions: uniform, Gaussian and exponential. Symbols: numerical results for w = 0.0025.

2 )

 2 = 2E -v B -v D . (B.14)The expressions (B.13) and (B.14) are similar to (B.3) and (B.4), up to two differences: energy is doubled (E → 2E), and each random potential is replaced by a sum of two independent potentials (vA → v A + v C , v B → v B + v D ). The latter operation amounts to replacing the disorder distribution ρ by ρ 2 = ρ * ρ, i.e.,

G 2 (

 2 E) = wg 2 (x), H 2 (E) = wh 2 (x), x = E w . (B.18)

19 )Appendix B. 3 .

 193 Explicit formulas for specific disorder distributionsIn the body of this work we use the following results for three disorder distributions: uniform, Gaussian and exponential.The uniform distribution. It readsρ(v) = 1 2V (-V < v < V ). (B.20)We have v 2 = V 2 /3 and v 4 = V 2 /5, and so

  The exponential distribution. The symmetric exponential distribution, also referred to as the Laplace distribution, readsρ(v) = e -|v|/V 2V . (B.31)

Table 1 .

 1 Characteristic values of the Lyapunov exponent (inverse localization length) of the weakly disordered flat band of the stub lattice for three disorder distributions, whose kurtosis κ is recalled (see Appendix B.3 for details). The minimum γ min is given when it is non-trivial, i.e., different from γ(0).

		κ	γ max	γ(0)	γ min
	Uniform	1.8 1.219 0.591	-
	Gaussian	3	1.057 0.668	-
	Exponential	6	1.006 0.788 0.771

  The vanishing of g 1 (x) and g 2 (x) for |x| > √ 3, i.e., |E| > V , is expected. Indeed in this range none of the η variables entering (B.4) or (B.14) can vanish. The functions h 1 (x) and h 2 (x) have not been evaluated in closed form.We have v 2 = w 2 and v 4 = 3w4 , and so κ = 3. (B.27) Its Fourier transform reads ρ(q) = e -q 2 w 2 /2 . (B.28) Inserting this expression into (B.6) and (B.16), we obtain the simple results

	with	g ⋆ (x) =	2 15 √ 3	-	4	√ 3 9	x 2 +	8 9	x 3 -	8 27 √ 3	x 4 +	8 45	x 5 .	(B.25)
	The Gaussian distribution. It reads
		ρ(v) =	e -v 2 /(2w 2 ) w √ 2π	.						(B.26)
		g 1 (x) =	e -x 2 4 √ π	,				g 2 (x) =	e -2x 2 √ 8π	.	(B.29)
	The expressions (B.12) and (B.19) yield h 1 (x) = x 2 + i √ π e -x 2 erf(ix), 4 h 2 (x) = x + i √ 2π 4 e -2x 2 erf(ix √	2),	(B.30
															2 ,
				8 135 0	( √ 3 -x) 5		(x > √ 3/2 < x < 3), √	√ 3 ,	(B.24)

Table B1 .

 B1 Values of the Cauchy strength of the effective disorder at the center of the flat band at E = 0 for the two flat-band structures studied in this appendix, namely g 1 (0) on the diamond chain (see section 5 and Appendix B.1) and g 2 (0) on the pyrochlore ladder (see section 4 and Appendix B.2), and for three disorder distributions, whose kurtosis κ is recalled.

	Distribution		κ		g 1 (0)		g 2 (0)
	Uniform	9 5	= 1.8	√ 6 3	= 0.288675	2 15 √ 3	= 0.230940
	Gaussian Exponential		3 6	1 √ π √ 4 2 16	= 0.141047 = 0.088388	1 √ 8π 7 √ 2 64	= 0.199471 = 0.154679

Acknowledgments

It is a pleasure to thank Alexei Andreanov, Sergej Flach and Julien Vidal for useful correspondence.

Appendix A. Cauchy tails

Cauchy tails in probability distributions play a central part in the analysis of the pyrochlore ladder (section 4) and the diamond chain (section 5).

Let us introduce this notion on a simple and explicit example. Consider a random variable x with a uniform distribution on an interval containing 0, i.e.,

Its reciprocal y = 1/x has the probability distribution

These distributions are sketched in figure A1. The occurrence of symmetric Cauchy tails falling off as 1/y 2 as y → ±∞ is a generic feature of the distribution of a variable y equal to the reciprocal of a variable x which may vanish. The strength G of these Cauchy tails, or the Cauchy strength of the variable y for short, is defined as

In the above example, this reads

in agreement with (A.2). Consider for definiteness the regime where a and b are simultaneously large. There, x is typically large, and so y is typically small. There are however rare circumstances where x is close to zero, and so y is much larger than its typical value.