Hanna Lesme

Clémence Alleaume

Said Bouhallab

Marie-Hélène Famelart

Cécile Marzin

Lizeth Lopez-Torres

Carole Prost

Cécile Rannou
email: cecile.rannou@oniris-nantes.fr

Saïd Bouhallab

Aroma-retention capacities of functional whey protein aggregates: Study of a strawberry aroma in solutions and in fat-free yogurts

Keywords:

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Aroma release has a strong influence on the organoleptic qualities, and hence on the acceptability of food by consumers. Among the most common ingredients, it is well known that proteins interact with aroma compounds. Several studies focused on dairy proteins and evidenced a retention of aroma compounds with the increase in protein concentration [START_REF] Guichard | Flavour retention and release from protein solutions[END_REF][START_REF] Kopjar | Retention of aroma compounds: An interlaboratory study on the effect of the composition of food matrices on thermodynamic parameters in comparison with water[END_REF][START_REF] Landy | Retention of aroma compounds by proteins in aqueous solution[END_REF]. The main whey protein, β-lactoglobulin, was particularly studied. The retention of most aroma compounds was increased by increasing the β-lactoglobulin content in a solution, with a stronger effect for compounds with a larger carbon chain length [START_REF] Van Ruth | Influence of b-lactoglobulin, pH and presence of other aroma compounds on the air/liquid partition coefficients of 20 aroma compounds varying in functional group and chain length[END_REF]. The type of interaction and the strength of binding highly depend on the intrinsic properties of the aroma compounds and of the proteins, as well as on external conditions (pH, temperature, ionic strength…) [START_REF] Kühn | Interactions of Milk Proteins and Volatile Flavor Compounds: Implications in the Development of Protein Foods[END_REF].

Aroma perception is governed by the distribution of aroma compounds between gas and matrix phases, which can be influenced by the formulation of the product. The release of aroma compounds from a matrix can be characterized by the partition coefficient between the matrix and the headspace [START_REF] Kühn | Interactions of Milk Proteins and Volatile Flavor Compounds: Implications in the Development of Protein Foods[END_REF]. This coefficient represents the ratio between the concentration of volatile compounds in the matrix and in the gas phase. Gas-matrix partition coefficients can be calculated using static headspace methods, which are based on measurements performed at equilibrium between matrix and gas. The Phase Ratio Variation (PRV) method is widely used for the determination of partition coefficient values by static headspace gas chromatography [START_REF] Ettre | Determination of gas-liquid partition coefficients by automatic equilibrium headspace-gas chromatography utilizing the phase ratio variation method[END_REF]. It is an indirect method that does not require the use of external calibration, nor the exact knowledge of the concentration of the volatile compound in the solution, and it can be applied to mixes of several volatile molecules [START_REF] Atlan | Processing gas chromatographic data and confidence interval calculation for partition coefficients determined by the phase ratio variation method[END_REF]. Complementary results can be obtained by using other methodologies, such as gas chromatography-ion mobility spectrometry, which monitor the release of aroma compounds during food consumption [START_REF] Pu | Characterization of the key odorants contributing to retronasal olfaction during bread consumption[END_REF].

The studies on aroma retention abilities of milk proteins are of great importance, because milk proteins are used in a wide range of food products, including dairy products. This is especially true with the increasing demand of "light" and natural products. Indeed, dairy ingredients are considered as promising natural ingredients to be used as fat replacers and to achieve desirable texture properties [START_REF] Ipsen | Microparticulated whey proteins for improving dairy product texture[END_REF].

Whey protein ingredients are of interest, and research has been done to further improve their functional properties. In this context, whey protein aggregates (WPA) with specific and novel texture properties were produced in controlled conditions (temperature, pH, ionic force). These functional WPA have been shown to be promising natural ingredients to obtain desirable texture in fat-free yogurts [START_REF] Gélébart | Protein aggregates modulate the texture of emulsified and acidified acid milk gels[END_REF][START_REF] Lesme | Controlled whey protein aggregates to modulate the texture of fat-free set-type yoghurts[END_REF].

Besides texture modifications, the use of WPA in fat-free products is likely to change the aroma profile of dairy products, even at very low concentration [START_REF] Kühn | Binding of 2-Nonanone and Milk Proteins in Aqueous Model Systems[END_REF]. Indeed, whey proteins were found to bind aroma compounds more than caseins [START_REF] Fabre | Comparison of Different Methods: Static and Dynamic Headspace and Solid-Phase Microextraction for the Measurement of Interactions between Milk Proteins and Flavor Compounds with an Application to Emulsions[END_REF]. However, the aroma-retention ability of functional WPA has never been studied. Further insights are thus needed to be able to develop low-fat dairy products with desirable organoleptic qualities.

Therefore, the first objective of this work is to investigate the aroma-retention capacities of WPA in comparison to native WPI in order to assess if they are likely to have an impact on aroma release in fat-free dairy products. We hypothesized that the change of structure of WPA impacted their aroma binding abilities in addition to their already known new textural functionalities [START_REF] Lesme | Controlled whey protein aggregates to modulate the texture of fat-free set-type yoghurts[END_REF].

Moreover, up to now, most systems that have been studied regarding aroma retention capacities of proteins, involved one protein and one aroma compound in model aqueous solutions [START_REF] Jouenne | Effect of pH on Retention of Aroma Compounds by β-Lactoglobulin[END_REF][START_REF] Kühn | Binding of 2-Nonanone and Milk Proteins in Aqueous Model Systems[END_REF]. To go further and complement the results obtained in model systems, our study was performed on a mix of twenty aroma compounds, constituting a model strawberry aroma, instead of only one aroma compound as traditionally done. Using a complex aroma is thus closer to a real situation when aromatizing yogurts and allow to consider the competition phenomenon between aroma compounds to link with protein. To determine the binding capacities of WPA, partition coefficients were thus measured in aqueous protein suspensions flavored with the model strawberry aroma.

As the functional WPA are meant to be used in fat-free yogurts, the second aim of this study focused on their impact on aroma release in fat-free strawberry yogurts. Fat free yogurts were chosen as more complex food matrices to get in-depth knowledge on the impact of the retention-capacity of proteins in real food matrices. By this way, this study attempted to study the aroma retention capacities of proteins in model systems but also in more complex conditions to fill the gap between model matrix and real products.

Materials and methods

Raw materials

Low-heat spray dried skimmed milk powder (34.00% proteins, < 1.50% fat, 8.50% ash) and whey protein isolate (WPI) (86.51% (w/w) proteins, 1.98% caseins, 0.40% fat, 1.92% ash) were kindly supplied by local dairy companies (confidential origin). The composition of the powders is given according to manufacturer's information. Food grade sodium chloride (NaCl, ≥ 99%) and sodium hydroxide (1 M) were bought at Sigma Aldrich (Saint-Louis, USA). To produce fat-free yogurts, YFL-812 (Chr Hansen, France) was used as starter culture because of its low ability to produce exopolysaccharides. Milli Q (Merck Millipore, Burlington, USA) water was used to produce WPI solutions and functional WPA. All ingredients were food-grade.

Production of whey protein aggregates

Two types of whey protein aggregates, MFA (Monodisperse Fractal Aggregate) and PFA (Polydisperse Fractal Aggregate) were produced by heat-treatment of a solution of WPI (80 °C for 2 h) at pH 7 according to [START_REF] Lesme | Controlled whey protein aggregates to modulate the texture of fat-free set-type yoghurts[END_REF]. The two different type of aggregates were obtained by varying the ionic strength of a WPI solution (50.00 g.L -1 ). A low ionic strength (15.00 mM) led to MFA (one single population with a diameter of 200 nm), whereas a higher salt concentration (45.00 mM) led to PFA (two distinct populations with a diameter of 200 and 1000 nm).

Experimental design

In the following text, the term "yogurt" will be used to mean "fatfree set-type strawberry yogurt". The experimental design used in the study is depicted in Fig. 1. Yogurt samples or protein suspensions in water were either enriched with native whey protein isolate (WPI), MFA or PFA. WPI, MFA and PFA were incorporated in the yogurts at three different concentration levels: 0.50%, 1.50%, 2.50% (w/w). The aroma retention abilities of protein solutions were investigated using a single protein concentration of 1.00% (w/w).

Strawberry aroma

The strawberry aroma contained 20 odorous compounds mixed with propylene glycol (Mane R&D, France) (Table 1). Among the 20 compounds, 7 compounds were selected with special attention. These compounds belong to different chemical classes with different physicochemical properties and were commonly used in aromas formulated for dairy products. Consequently, they were of importance for the study and their concentrations in the strawberry aroma were boosted to ease their detection in headspace. These concentrations were chosen thanks to preliminary studies to determine the limit of quantification in the yogurt matrix employing HS-GC. The other aroma compounds were selected between the main key food odorants of strawberry and were important to obtain a balanced strawberry aroma [START_REF] Du | Evaluation of Volatiles from Two Subtropical Strawberry Cultivars Using GC-Olfactometry, GC-MS Odor Activity Values, and Sensory Analysis[END_REF]. or Polydisperse Functional Aggregates (PFA). The two types of aggregates were obtained in controlled conditions from a WPI solution. Three different concentrations (0.5%, 1.5%, 2.5%) were used to modify the yogurt texture. Aqueous solutions were enriched with WPI, MFA and PFA in a single concentration (1.0%).

Preparation of the protein solutions

Protein solutions were made using WPI, MFA or PFA at a concentration of 1.00% (w/w) in Milli Q water (Fig. 1). The protein solutions had a neutral pH. Strawberry aroma was added to the diluted protein solutions to reach a final aroma concentration of 0.05% (w/w) as recommended by the aroma manufacturer. The concentration of the volatile compounds in the final suspensions ranged from 0.1 to 30 mg.kg -1 (Table 1). The solutions were stored 24 h at 4 °C before headspace analysis.

Preparation of the strawberry flavored yogurts

Skimmed milk was reconstituted to 100 g.kg -1 milk solids using low-heat spray-dried skimmed milk powder. The reconstituted milk was stored overnight at 4 °C to allow hydration of the powders. After being heated at 90 °C for 5 min, the milk was cooled to the fermentation temperature (43 °C) and inoculated with the yogurt starter culture. At this point, 5.00% sugar (w/w) was added to the milk. Yogurts were also flavored with 0.05% (w/w) strawberry aroma. The concentration of the odorous compounds ranged from 0.1 mg.kg -1 to 30 mg.kg -1 of yogurt (Table 1).

During yogurt manufacture, the milk was enriched either with WPI or whey protein aggregates (MFA or PFA) which were added respectively before or after the heat treatment of the milk. The mix was conditioned in glass cups of 40 mL and put in an incubator for fermentation at 43 °C during about 6 h until the pH reached 4.6. Yogurts were stored at 4 °C during 5 days prior to analysis to ensure an equilibration of the aroma release from the matrix.

Chromatographic analysis of aroma compounds

The samples were placed in an agitator/incubator of an automatic headspace sampler (GERSTEL MPS, Linthicum, MD, USA). Preliminary analysis showed that equilibrium was reached after 90 min at 30 °C without stirring (results not shown). After equilibration, a 1 mL sample of headspace was automatically taken off by a 2.5 mL gas-tight syringe pre-heated at 35 °C (GERSTEL MPS, Linthicum, MD, USA). The analysis was done using a gas chromatograph (Agilent 7890A, Wilmington, DE, USA) equipped with a Flame Ionisation Detector (FID) and paired to a mass spectrometer (Agilent 5975C, Wilmington, DE, USA). The inlet temperature was 260 °C and the FID detector temperature was 250 °C. Helium was used as carrier gas in splitless mode at a flow of 2 mL.min -1 . The capillary column was DB-WAX (30 m, 0.32 mm, 0.5 µm, Agilent). The program of temperature used was 40 °C for 5 min, ramp to 230 °C at 12 °C.min -1 and 230 °C for 10 min.

Calculation of partition coefficients

The retention of aroma compounds in protein solutions was compared to equally concentrated aqueous solutions according to the following relationship:

= - * R K K K (%) 100 G W G S G W / / / ( 1
)
where R is the percentage of retention in protein solutions compared to aqueous solutions, K G/S and K G/W are the gas/solution and gas/water partition coefficients of aroma compounds respectively. These partition coefficients were determined by the Phase Ratio Variation (PRV) method. Likewise, the partition coefficients of yogurt samples were named K G/Y . The PRV method, described by [START_REF] Ettre | Determination of gas-liquid partition coefficients by automatic equilibrium headspace-gas chromatography utilizing the phase ratio variation method[END_REF], was used to determine the gas/matrix partition coefficient of the aroma compounds detected by headspace analysis, in water K G/W , in protein solutions K G/ S, and in yogurts K G/Y.

The PRV method consists in studying the aroma release in the headspace for different phase ratios (β) inside the headspace vial, according to:

= β V V G S (2)
with V G the gas volume and V S the sample volume.

The phase ratio is the ratio of the volumes of the headspace and the sample solution.

By plotting the reciprocal of the peak area (1/A) against the phase ratio (β), a linear equation is obtained:

= + A aβ b 1 (3)
The partition coefficient K is defined as the ratio between the slope (a) and the intercept (b) [START_REF] Savary | Study of interactions between aroma compounds and acacia gum using headspace measurements[END_REF].

Increasing volumes (50 µl, 100 µl, 250 µl, 500 µl, 1000 µl) of each matrix (water, protein solutions or yogurts) were poured into Table 1 Flavor compounds of the strawberry aroma, their concentrations in protein solutions and yogurts and their physicochemical characteristics. *("The Good Scents Company -Search for product information,"2019). The compounds in bold are the compounds detected and quantified with the PRV method. headspace vials (22 mL). Each vial represented a gas/liquid phase ratio β of 439, 219, 87, 43, 21 respectively. Therefore, a high gas/matrix partition coefficient indicates that the aroma compound is more released in the gas phase at thermodynamic equilibrium.

Statistical analysis

Measurements were performed in triplicate on different productions for yogurts and for protein suspensions. Statistical analysis was performed using the XLStat software from Addinsoft (version 2018.5). A one-way ANOVA analysis was performed on the gas/water (K G/W ), gas/ protein solutions (K G/S ) and gas/yogurts (K G/Y ) partition coefficients. A one-way ANOVA was also performed on the retention (R) of protein solutions for each aroma compound. A two-way ANOVA with interaction (protein type + protein concentration) was performed on the gas/ yogurts partition coefficients (K G/Y ) to study the effect of the protein type (WPI, MFA, PFA) and the protein concentration (0.50%, 1.50%, 2.50%) on the release of aroma compounds.

When a significant effect was present (p < 0.05), the differences between protein solutions were tested using a Fisher test of Least Significant Difference (LSD) for multiple comparisons of means for each aroma compound.

Results and discussion

Detection and quantification of aroma compounds

Seven aroma compounds out of the 22 aroma compounds of the strawberry aroma were detected and quantified by using the PRV method in both protein solutions and yogurts (Table 2). The detected compounds corresponded to the most volatile ones and to the ones present in high concentrations in the strawberry aroma (Table 1). This is in line with [START_REF] Fabre | Comparison of Different Methods: Static and Dynamic Headspace and Solid-Phase Microextraction for the Measurement of Interactions between Milk Proteins and Flavor Compounds with an Application to Emulsions[END_REF], who quantified only three out of ten aroma compounds by static headspace measurement, and this is comparable to the 8 aroma compounds out of 17 detected by Saint-Eve, Juteau, [START_REF] Saint-Eve | Complex Viscosity Induced by Protein Composition Variation Influences the Aroma Release of Flavored Stirred Yogurt[END_REF]. As already reported, the poor sensitivity for compounds with low volatility constitutes a drawback of the static headspace method [START_REF] Kühn | Interactions of Milk Proteins and Volatile Flavor Compounds: Implications in the Development of Protein Foods[END_REF]. For the detected compounds, correlation coefficients R 2 higher than 0.98 were found for the linear regression between 1/A and the phase ratio β. The detected aroma compounds had different physico-chemical properties. This was especially true regarding hydrophobicity with a very hydrophilic compound (diacetyl, logP of -1.34) and a very hydrophobic one (limonene, logP of 4.38) (Table 1). These differences could imply different binding behaviors between proteins and aroma compounds. Indeed, previous studies already reported the importance of hydrophobic interactions regarding aroma compounds retention by proteins [START_REF] Kühn | Interactions of Milk Proteins and Volatile Flavor Compounds: Implications in the Development of Protein Foods[END_REF].

Comparison of K G/W with the results of the literature

Partition coefficients of aroma compounds were first measured in water. The obtained values of the partition coefficients measured were compared to those already reported by other authors (Table 2). The partition coefficients tended to be slightly higher than the ones of the literature, but there was an overall good similarity. It is interesting to notice that having a combination of aroma compounds in a water solution did not impact their individual partition coefficients in comparison with more simple systems made of one aroma compound. In addition, there is an overall good agreement for the partition coefficients among studies, apart for the ones of limonene which go from 1.04E to 3 to 0.25 depending on the study.

As expected, limonene, which is very hydrophobic with a very low solubility in water (Table 1), was the most released compound (Table 2). The release of limonene from water was much higher than the other compounds, which is in line with the results of [START_REF] Savary | Phase ratio variation method as an efficient way to determine the partition coefficients of various aroma compounds in mixture[END_REF]. Ethyl butyrate and 2-nonanone were the two other aroma compounds detected that had a partition coefficient greater than 1, meaning that they were released from water at thermodynamic equilibrium. On the contrary, diacetyl, cis-3-hexenol, benzaldehyde and methyl cinnamate had partition coefficients lower than 1, indicating that they were less released and more water-soluble at thermodynamic equilibrium.

Aroma retention in MFA and PFA versus WPI aqueous protein solutions

The comparison of K G/S and K G/W values showed that the addition of whey proteins in aqueous solutions decreased the partition coefficients of the aroma compounds. However, the observed changes depended on the aroma compounds considered, and on the type of protein added in the solution (WPI, MFA, PFA). A significant decrease of the partition coefficients with the addition of proteins in the aqueous solution was observed for limonene, cis-3-hexenol, 2-nonanone and methyl cinnamate. The same trend was found for ethyl butyrate and benzaldehyde, without being significant. These results indicated a retention of these aroma compounds by the WPI and WPA, probably due to interactions between aroma compounds and proteins. This result coincided with results obtained by van Ruth and Villeneuve (2002) on the aroma retention properties of pure β-lactoglobulin, or by [START_REF] Viry | A predictive model for flavor partitioning and protein-flavor interactions in fat-free dairy protein solutions[END_REF] on the partitioning of aroma compounds in a more complex system where WPI and sodium caseinate were mixed. Differences in aroma partition coefficients exist between native proteins WPI and functional aggregates MFA and PFA (Table 2). These aromadependent differences were further highlighted throughout the retention (R) of the aroma compounds by the proteins (Fig. 2.). Aside from diacetyl, the retention goes from 8.60% for benzaldehyde with MFA to 81.90% for limonene with MFA. The retention ability of WPI, MFA and PFA were significantly different for three compounds: diacetyl, 2-nonanone and methyl cinnamate. Aroma binding ability of proteins is highly dependent on the conformational state of the protein, and more particularly on the sites available to bind aroma compounds. It is well known that β-lactoglobulin is readily denatured by heat treatment [START_REF] Wijayanti | Thermal denaturation, aggregation, and methods of prevention[END_REF], and the resulting change of conformation is likely to influence its retention abilities. Few studies have been performed on the aroma retention abilities of Table 2 Mean partition coefficients (K G/S × 10 2 ) of aroma compounds in water and in protein solutions (WPI, MFA, PFA) flavored with the strawberry aroma. Different letters in the columns indicate significant differences among solutions (p < 0.05). Air/Water partition coefficients (K G/W × 10 2 ) of flavor compounds at 30 °C are compared with literature values (K G/M in water Lit). denatured whey proteins, and the results are contradictory between studies. Increased affinity for β-ionone and guaiacol was achieved after partial unfolding of β-lactoglobulin (Tavel, Moreau, Bouhallab, Li-Chan, & Guichard, 2010) while a decreased binding ability with the denaturation of the proteins was evidenced by [START_REF] Kühn | Interactions of Milk Proteins and Volatile Flavor Compounds: Implications in the Development of Protein Foods[END_REF]. This contradiction between the results evidences that there is no general trend and that the effect depends on the type of the considered aroma compounds (Table 2).

K G/M in Water Lit K G/S in Water K G/S in WPI K G/S in MFA K G/S
For 2-nonanone and methyl cinnamate, the retention was significantly higher in the aqueous solutions containing whey protein aggregates compared to WPI solutions while the reverse trend was observed for diacetyl (Fig. 2.). To our knowledge, no studies have been performed on binding abilities of denatured whey proteins on methyl cinnamate or diacetyl, and a single study was performed on denatured β-lactoglobulin (heated at 75 °C for 10 and 20 min) and 2-nonanone (O' Neill & Kinsella, 1988). These authors evidenced a decreased binding affinity of 2-nonanone for denatured β-lactoglobulin. However, the denaturation rate of the β-lactoglobulin in this study might be far less important than in our experiment, and the aggregates formed are likely to have a different shape than MFA and PFA, which were obtained by a more intense heat treatment (80 °C for 2 h). Indeed, the heating time was six times shorter and the concentration of the heated β-lactoglobulin solution was five times smaller than in the current protocol. It can be noted that the aroma compounds more retained by whey protein aggregates than that of WPI (2-nonanone and methyl cinnamate) were the 2 compounds with the higher molecular weight, in contrast to diacetyl which had the lowest one. We assume that because of the unfolding of native WPI and subsequent formation of branched aggregates, previously buried binding sites (i.e. hydrophobic residues) become accessible for interaction with non-polar aroma compounds in MFA and PFA, resulting in a lower release of methyl cinnamate and 2nonanone. The increased number of hydrophobic residues present at the surface of MFA and PFA could in the same way limit the interactions with a highly hydrophilic compound such as diacetyl. This could be the reason why diacetyl is much more released in the presence of MFA and PA.

The importance of the structure of WPA was evidenced by the significant difference existing in the retention ability of PFA and MFA regarding methyl cinnamate. The polydispersity of MFA and PFA was varied by changing the ionic strength of the WPI solution. [START_REF] Nicolai | β-Lactoglobulin and WPI aggregates: Formation, structure and applications[END_REF] reported that increasing the ionic strength led to higher denaturation rates, which might correspond to more available binding sites, and therefore inducing a higher retention of hydrophobic compounds such as methyl cinnamate. It was evidenced in a previous study that PFA and MFA have different structures [START_REF] Lesme | Controlled whey protein aggregates to modulate the texture of fat-free set-type yoghurts[END_REF]. PFA have a more branched and expanded structure while MFA are denser aggregates. Moreover, this change of structure might induce a rise in total area of PFA compared to MFA, which could favor the binding of aroma compounds such as methyl cinnamate. Indeed, [START_REF] Tavel | Interactions between aroma compounds and β-lactoglobulin in the heat-induced molten globule state[END_REF] evidenced that a less tightly packed structure induced an easier access to the binding sites and favored aroma binding.

MFA or PFA did not have a significant impact on the retention of ethyl butyrate, limonene, cis-3-hexenol and benzaldehyde (Table 2.). These molecules were retained by whey proteins, regardless of the state of the whey proteins. Benzaldehyde was the only aroma compound for which the retention ability of MFA and PFA tended to be inferior to the one of WPI, even if the difference was not significant. [START_REF] Hansen | Flavor Interaction with Casein and Whey Protein[END_REF] evidenced that heat-treated β-lactoglobulin (30 min at 70 °C) has a higher binding ability for benzaldehyde. However, the authors added the aroma compound before heat treatment, and the elevated temperature might have favored covalent bindings between the aldehyde compound of benzaldehyde with amino groups of β-lactoglobulin. The particularly high retention of limonene might be explained by a preferential binding in the hydrophobic pocket of native β-lactoglobulin that makes up WPI, and by the multiple hydrophobic sites available on MFA and PFA.

The use of a mixture of aroma compounds in a protein solution can also influence the retention and then the partition coefficient of individual aroma components. van Ruth and Villeneuve ( 2002) explained the increased partition coefficients of some esters and aldehydes in the presence of other aroma compounds by a competition for the binding sites on the proteins. For instance, the authors found that the air/liquid partition coefficient of ethyl butyrate was increased when present with others. In our study, ethyl butyrate was not significantly more retained by WPI compared to MFA and PFA (Table 2) while previous studies evidenced a high affinity between ethyl butyrate and WPI [START_REF] Pelletier | Measurement of Interactions between β-Lactoglobulin and Flavor Compounds (Esters, Acids, and Pyrazines) by Affinity and Exclusion Size Chromatography[END_REF][START_REF] Van Ruth | Influence of b-lactoglobulin, pH and presence of other aroma compounds on the air/liquid partition coefficients of 20 aroma compounds varying in functional group and chain length[END_REF]. We can hypothesize that a form of competition between aroma compounds of the strawberry aroma for the binding sites of the proteins impacted the retention of ethyl butyrate. Moreover, [START_REF] Jouenne | Effect of pH on Retention of Aroma Compounds by β-Lactoglobulin[END_REF] evidenced that this competition for the binding sites was concentration and pH dependent, which is probably related to the accessibility of the binding sites. Heat-denatured whey proteins are of great importance for the dairy industry because heat treatment is an important production step of dairy products. According to the results obtained in the first part of the article, MFA and PFA have different aroma binding abilities than native WPI, which evidenced that a modification of the conformation of whey proteins has an impact on their interactions with aroma compounds. However, the aroma binding abilities of WPA were not necessarily reduced, as documented by several studies [START_REF] Chobpattana | Mechanisms of interaction between vanillin and milk proteins in model systems[END_REF][START_REF] Mcneill | Vanillin Interaction with Milk Protein Isolates in Sweetened Drinks[END_REF][START_REF] O'neill | Effect of Heat Treatment and Modification on Conformation and Flavor Binding by β-Lactoglobulin[END_REF]. These modifications are likely to influence the balance of the overall strawberry aroma. Complex food matrixes involve interactions with other food components (proteins, carbohydrates, lipids) which could further modify the impact of functional whey protein aggregates on aroma release.

Impact of WPA on the release of aroma compounds in fat-free yogurts

Fat-free strawberry yogurts were chosen as a model complex food. The yogurts studied are presented in Fig. 1. The PRV method enabled to detect and quantify the same 7 out of the 22 aroma compounds in the strawberry flavored yogurts than in protein aqueous solutions. Measured partition coefficients over yogurts ranged from 2.5 × 10 -4 for methyl cinnamate to 3.1 × 10 -2 for ethyl butyrate (Table 3). The partition coefficients were of the same order of magnitude for the different yogurts. However, the partition coefficients of some compounds such as limonene and methyl cinnamate are very inferior compared to the ones measured in water. Saint-Eve et al. (2006) evidenced the same behavior for the most hydrophobic aroma compounds and explained it by the presence of fat in their yogurts and by a temperature difference between the measurements performed on solutions and on yogurts. As the yogurts investigated in the study were fat-free and all the measurements were performed at 4 °C, it can be hypothesized that hydrophobic interactions between proteins and aroma compounds were responsible for their lower volatility in yogurts.

A significant product effect was evidenced for four compounds out of the seven detected, namely limonene, cis-3-hexenol, 2-nonanone and methyl cinnamate. Aside from cis-3-hexenol, these compounds were the most hydrophobic molecules that make up the strawberry aroma. Interactions between these compounds and the whey proteins were evidenced in the first part of this study as they were the only compounds significantly more retained in protein solutions compared to water solutions (Table 2). Consequently, physico-chemical interactions evidenced between aroma compounds and proteins appeared to also impact the aroma release in more complex matrixes.

Two factors (protein type and concentration) were varied in the fatfree yogurts and a two-way ANOVA with interaction (type of protein and concentration of protein) was applied on the partition coefficients. The main effects of the ANOVA are depicted in Fig. 3. The concentration of proteins had a significant impact on the release of two aroma compounds out of the seven detected (Fig. 3. (A)). The increase of protein concentration led to a significant increased retention of methyl cinnamate (F-value = 5.75 and p-value = 0.009) and limonene (Fvalue = 4.35 and p-value = 0.025) in the yogurts. The impact of the protein concentration was particularly important on the release of methyl cinnamate, with a retention increased by 40.00% in yogurts enriched with 0.50% compared to yogurts enriched with 2.50% proteins.

The type of protein also had an impact on the release of two aroma compounds (Fig. 3. (B)). Cis-3-hexenol (F-value = 3.14 and pvalue = 0.05) and 2-nonanone (F-value = 3.72 and p-value = 0.05) were significantly more retained by MFA and PFA than native WPI. A similar trend was observed, without being significant, for methyl cinnamate and limonene. However, the magnitude of the differences between native WPI and functional WPA partitioning of cis-3-hexenol and 2-nonanone was rather small, as the partition coefficients were only decreased by 12% and 16% respectively. Moreover, the interaction between the type of protein and the protein concentration was significant for 2-nonanone (F-value = 8.93 and p-value = 0.00). This result implied that the impact of the type of protein depended on the concentration considered and vice versa, indicating that the two factors, should be considered an integrated phenomenon.

In a complex matrix such as yogurt, other parameters in addition to direct protein-aroma interactions may be at stake and influence the release of aroma compounds in a different way than in model aqueous solutions. For instance, the aroma release can be impacted by interactions with other food components such as carbohydrates, fat, or by the existence of a three-dimensional protein network [START_REF] Seuvre | Retention of aroma compounds in food matrices of similar rheological behaviour and different compositions[END_REF]. Moreover, the differences in pH between the protein solutions (neutral pH) and the yogurts (pH 4.6) could induce different binding behaviors of the proteins. Knowledge on the structure of the protein network and on the textural properties of protein-enriched yogurts appear to be useful to explain the differences observed. The impact of MFA and PFA on the texture and the protein network of fat-free yogurts was evidenced in a previous study [START_REF] Lesme | Controlled whey protein aggregates to modulate the texture of fat-free set-type yoghurts[END_REF]. While yogurts enriched in MFA had a firmness close to the one of yogurts enriched with WPI, yogurts enriched with PFA were much softer. However, the impact of MFA and PFA on the protein network structure was limited to a slight increase of porosity in yogurts enriched with PFA. The protein concentration had a greater impact on texture properties, and [START_REF] Gélébart | Protein aggregates modulate the texture of emulsified and acidified acid milk gels[END_REF] evidenced that the increase of protein concentration led to a clear increase of the protein network density. These textural modifications can impact the aroma release from yogurts.

The denser protein network of yogurts enriched with high concentrations of proteins might explain the higher retention of hydrophobic compounds such as limonene and methyl cinnamate while less hydrophobic compounds such as diacetyl and benzaldehyde tend to be more released when the concentration of protein was increased. However, the modifications of the structure of the protein network between yogurts enriched with the same concentration of either WPI, MFA and PFA was subtle and might not be enough to induce important variations of aroma release. It is interesting to notice that there is a tendency for limonene, 2-nonanone, methyl cinnamate and cis-3-hexenol to be more retained by WPA whereas diacetyl and benzaldehyde follow the opposite trend. It can be hypothesized that thank to their branched structure, MFA and PFA can both interact with the protein network to strengthen it and with aroma compounds. At the opposite, WPI denatured by the heat treatment applied to the yogurt milk base, are preferentially bound to the protein network, which might explain the lower aroma retention.

Important texture modifications and subtle variations of aroma release were also evidenced in the few studies performed to investigate the impact of the protein content of yogurts on the aroma release [START_REF] Chua | Altering the casein to whey protein ratio to enhance structural characteristics and release of major yoghurt volatile aroma compounds of non-fat stirred yoghurts[END_REF][START_REF] Saint-Eve | Influence of proteins on the perception of flavored stirred yogurts[END_REF]; Anne Saint-Eve, Juteau, [START_REF] Saint-Eve | Complex Viscosity Induced by Protein Composition Variation Influences the Aroma Release of Flavored Stirred Yogurt[END_REF]. [START_REF] Chua | Altering the casein to whey protein ratio to enhance structural characteristics and release of major yoghurt volatile aroma compounds of non-fat stirred yoghurts[END_REF] investigated the effect of the casein to whey protein ratio on the release of key volatile compounds of yogurt aroma, i.e. acetaldehyde, butyric acid and diacetyl. They evidenced that addition of WPI significantly decreased the release of diacetyl and butyric acid, compared to yogurts enriched with casein, showing that WPI can hold volatile aroma compounds in yogurts containing 5.00% total protein content. However, the variations were rather small and the authors concluded that WPI was unlikely to lead to an unbalanced aroma in plain yogurts. Saint-Eve et al. (2006a, 2006b) performed two studies varying the casein to whey protein content of stirred strawberry flavored yogurts and evidenced that the modification of the protein content led to small differences in aroma partitioning (inferior to 30%) and to important texture modifications. They showed that yogurts enriched with casein had a heterogeneous structure with large pores and showed that most of aroma compounds detected had a greater affinity for casein than for whey proteins. The authors hypothesized that the coarse network structure of the yogurts enriched with caseinate could limit the diffusion of aroma compounds in static conditions. Despite the small variations in aroma release, yogurts enriched with caseins were perceived as being the less intense in aroma (Saint-Eve et al., 2006). However, the authors wondered if the difference of aroma perception was due to direct physico-chemical interactions between aroma compounds and proteins or to sensory interactions between texture and aroma perception.

Conclusions

PRV method gave absolute results regarding retention ability of functional whey protein aggregates compared to native WPI for a limited number of aroma compounds present in the strawberry aroma studied. The calculation of partition coefficients confirmed that WPA had different aroma-binding abilities compared to native WPI. The retention ability of MFA and PFA was dependent on the aroma compound considered, but contrary to previous studies performed on denatured whey proteins, a significant higher retention by MFA and PFA was evidenced for some hydrophobic compounds. Moreover, performing the study on a combination of aroma compounds forming a strawberry aroma was relevant to supplement the existing data as a form of competition for the binding sites can influence the release of some compounds.

The impact of functional WPA in a real food such as yogurt was more difficult and complex to assess as the proteins and the aroma compounds can interact with other ingredients. However, the shift from aqueous solutions to real food products is essential to complement the knowledge gained on model systems and gives important information for the formulation of fat-free dairy products.

Obtaining a complete aroma profile of yogurts enriched with functional WPA would be interesting to confirm the results evidenced with the PRV method on a greater number of aroma compounds. It would also allow to investigate the impact of aroma retention by proteins on strawberry aroma perception by correlating the instrumental data with sensory profiles of yogurts.

Fig. 2 .

 2 Fig. 2. Retention (%) of the seven aroma compounds detected by headspace measurements in aqueous solutions containing the strawberry flavor and the different proteins studied: WPI, MFA, PFA. The letters indicate a significant effect of the type of protein added in the aqueous solution (p < 0.05)

Fig. 3 .

 3 Fig. 3. (A) Impact of the concentration of protein on the gas-matrix partition coefficients (KG/M) for the seven aroma compounds detected by headspace measurements in fat-free strawberryflavored yogurts studied enriched with different types of proteins in different concentrations: 0.5%. 1.5%. 2.5% (B) Impact of the type of protein on the gas-matrix partition coefficients (KG/M) for the seven aroma compounds detected by headspace measurements in fatfree strawberryflavored yogurts studied enriched with the different types of proteins: WPI. MFA. PFA. Different letters indicate a significant protein effect (p < 0.005)

  n°CASConcentration in solutions and in yogurts (ppm) M (g/mol) Vapor pressure (mm/Hg @ 25 °C) logP a Flavor note

	Furanone strawberry	3658-77-3 27.0	128.13	0.03	-0.08 Caramelic, strawberry
	Vanillin	121-33-5	6.0	152.15	0.002	1.21	Sweet vanilla
	Pyrazine tetramethyl	290-37-9	30.0	80.10	19.74	-0.26 Pungent, roasted hazelnut
	Diacetyl	431-03-8	25.0	96.09	56.80	-1.34 Butter
	Methyl cinnamate	103-26-4	25.0	162.18	0.011	2.62	Balsamic strawberry
	Ethyl acetate	141-78-6	15.0	88.11	111.72	0.73	Ethereal, fruity, sweet
	Maltol	118-71-8	12.5	126.11	0.000326	0.09	Sweet, caramelic
	Cis-3-hexenol	928-96-1	10.0	100.16	1.04	1.04	Fresh, geen, grassy
	Ethyl butyrate	105-54-4	10.0	116.60	12.80	1.80	Fruity, pineapple
	Benzaldehyde	100-52-7	5.0	106.20	1.27	1.48	Almond
	2-Nonanone	821-55-6	3.0	142.24	0.55	3.14	Fresh, sweet, green
	Limonene	5989-27-5 3.0	136.24	1.54	4.38	Lemon
	γ-decalactone	706-14-9	1.5	179.25	0.005	2.72	Fruity, peach
	γ-undecalactone	104-67-6	1.5	184.28	0.003	2.96	Fruity, peach, creamy
	Acid butyric	107-92-6	1.0	88.11	1.65	0.79	Acetic, cheesy, dairy
	Ethyl isovalerate	108-64-5	0.1	130.19	7.85	2.16	Fruity, sweet, apple
	Linalool	78-70-6	1.0	154.25	0.02	2.97	Citrus, floral, sweet
	Ethyl-2-methyl-2-butyrate 7452-79-1 1.0	130.19	7.85	2.16	Fruity, fresh, berry
	Hexenyl-cis-3-acetate	3681-71-8 1.0	142.20	1.22	2.42	Green, fruity, apple
	Trans-2-hexenal	6728-28-3 0.7	98.15	4.62	1.79	Fresh, green, leafy

a Value of the logarithm of the partition coefficient between n-octanol and water

Table 3

 3 Mean partition coefficients (K G/Y × 10 2 ) of aroma compounds in fat-free strawberry yogurts enriched with different whey proteins (WPI, MFA, PFA) at different concentrations (0.5%, 1.5%, 2.5%). Different letters in the columns indicate significant differences among yogurts (p < 0.005).

		Aroma compounds (K G/M × 10 2 )
		Diacetyl	Ethyl butyrate	Limonene	Cis-3-hexenol	2-nonanone	Benzaldehyde	Methyl cinnamate
	WPI05	0.15	2.81	1.08 ab	0.068 ab	0.60 bc	0.17	0.059 ab
	WPI15	0.19	2.83	1.06 abc	0.074 ab	0.93 a	0.20	0.049 a
	WPI25	0.17	2.66	0.94 bcd	0.072 ab	0.49 c	0.14	0.045 c
	M_FA05	0.15	2.93	1.20 a	0.064 a	0.59 bc	0.15	0.025 ab
	M_FA15	0.18	3.08	0.73 d	0.066 ab	0.46 c	0.15	0.038 bc
	M_FA25	0.17	2.45	0.86 bcd	0.049 c	0.59 bc	0.26	0.041 abc
	P_FA05	0.17	2.54	0.92 bcd	0.069 abc	0.49 c	0.14	0.024 ab
	P_FA15	0.15	2.32	1.02 abc	0.060 bc	0.54 bc	0.16	0.048 bc
	P_FA25	0.18	2.84	0.85 cd	0.059 ab	0.69 b	0.23	0.033 c

Acknowledgements

The authors are grateful to regional councils of Brittany (France, grant n°13008651) and Pays de la Loire (France, grant n°2014-07081) for the financial support, and INRAE for scientific coordination (J. Leonil) through the interregional project PROFIL, supported by Bba industrial association and carried out by the association 'Pôle Agronomique Ouest".


P L Krapivsky
email: pkrapivsky@gmail.com

J M Luck

K Mallick
email: kirone.mallick@cea.fr

Quantum centipedes: collective dynamics of interacting quantum walkers

We consider the quantum centipede made of N fermionic quantum walkers on the one-dimensional lattice interacting by means of the simplest of all hard-bound constraints: the distance between two consecutive fermions is either one or two lattice spacings. This composite quantum walker spreads ballistically, just as the simple quantum walk. However, because of the interactions between the internal degrees of freedom, the distribution of its center-of-mass velocity displays numerous ballistic fronts in the long-time limit, corresponding to singularities in the empirical velocity distribution. The spectrum of the centipede and the corresponding group velocities are analyzed by direct means for the first few values of N . Some analytical results are obtained for arbitrary N by exploiting an exact mapping of the problem onto a free-fermion system. We thus derive the maximal velocity describing the ballistic spreading of the two extremal fronts of the centipede wavefunction, including its non-trivial value in the large-N limit.

Introduction

Quantum walks [START_REF] Aharonov | Quantum random walks[END_REF], the quantum analogues of classical random walks, play a prominent role in quantum information theory [START_REF] Farhi | Quantum computation and decision trees[END_REF]. It has been shown in [START_REF] Childs | Universal computation by quantum walk[END_REF] that any quantum algorithm can be restated in terms of quantum walks. These universal objects are the source of fascinating problems mixing wave dynamics, discrete geometry and probability theory. Quantum dynamics often results in a counter-intuitive phenomenology: quantities like hitting times or survival probabilities of a walker are genuinely different in quantum and classical set-ups; in particular, a quantum search can be far more efficient than a classical algorithm (see e.g. [START_REF] Kempe | Quantum random walks -an introductory overview[END_REF][START_REF] Ambainis | Quantum walks and their algorithmic applications[END_REF][START_REF] Venegas-Andraca | Quantum walks: a comprehensive review Quantum[END_REF] for reviews).

Single-particle quantum walks have been realized in the laboratory, using nuclear magnetic resonance [START_REF] Ryan | Experimental implementation of a discrete-time quantum random walk on an NMR quantum-information processor[END_REF], trapped ions or atoms [START_REF] Schmitz | Quantum walk of a trapped ion in phase space[END_REF][START_REF] Zähringer | Realization of a quantum walk with one and two trapped ions[END_REF][START_REF] Karski | Quantum walk in position space with single optically trapped atoms[END_REF] and photons [START_REF] Schreiber | Photons walking the line: A quantum walk with adjustable coin operations[END_REF]. The behavior of a single quantum walker can be explained by a wave description [START_REF] Knight | Quantum walk on the line as an interference phenomenon[END_REF] and reproduced in an experiment with classical waves [START_REF] Perets | Realization of quantum walks with negligible decoherence in waveguide lattices[END_REF].

Non-classical effects become essential if one considers multiple quantum walkers. Quantum walks of correlated photons have been implemented experimentally by various groups [START_REF] Hillery | Quantum walks through a waveguide maze[END_REF][START_REF] Peruzzo | Quantum walk of correlated photons[END_REF][START_REF] Lahini | Quantum correlations in two-particle Anderson localization[END_REF][START_REF] Sansoni | Two-particle bosonic-fermionic quantum walk via integrated photonics[END_REF]. In such systems, quantum interferences and interactions lead to entanglement and correlations that can not be accounted for by a classical picture, triggering thus much interest among theorists [START_REF] Omar | Quantum walk on a line with two entangled particles[END_REF][START_REF] Gamble | Two-particle quantum walks applied to the graph isomorphism problem[END_REF][START_REF] Štefaňák | Directional correlations in quantum walks with two particles New[END_REF][START_REF] Chandrashekar | Quantum walk on distinguishable non-interacting manyparticles and indistinguishable two-particle Quantum[END_REF][START_REF] Qin | Statistics-dependent quantum co-walking of two particles in one-dimensional lattices with nearest-neighbor interaction[END_REF].

A single quantum walker displays a ballistic rather than a diffusive motion. It spreads over a range of space that grows linearly with time. Surprisingly, the wavefunction displays sharp maxima near the boundaries of that allowed range, whereas it is negligibly small beyond this range. These maxima can be interpreted as ballistic fronts [START_REF] Farhi | Quantum computation and decision trees[END_REF][START_REF] De Toro Arias | Anomalous dynamical scaling and bifractality in the onedimensional Anderson model[END_REF]. In a recent work [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF], we have investigated the dynamics of bosonic and fermionic bound states of two interacting continuous-time quantum walkers in one dimension. The emphasis was on the ballistic spreading of the centerof-mass coordinate. We have demonstrated the existence of multiple internal ballistic fronts, corresponding to singularities of the velocity distribution, besides the two usual extremal ones. This feature is robust and generic, regardless of the statistics and of the precise form of the interaction potential between the two particles.

The aim of the present work is to investigate the center-of-mass dynamics, and especially the ballistic fronts, displayed by a composite object made of N fermionic quantum walkers on a one-dimensional lattice, constrained to remain within a fixed distance ℓ from their neighbors. This problem can be viewed as quantummechanical version of the diffusive dynamics of the N -legged molecular spiders that were considered in [START_REF] Antal | Molecular spiders in one dimension[END_REF], hence the name quantum centipede. In this work we focus our attention onto the simplest of all centipedes, corresponding to ℓ = 2. In this special situation, some analytical results can be derived by exploiting an exact mapping of the problem onto a free-fermion system.

The outline of this paper is as follows. In section 2 we review known results on one-dimensional continuous-time quantum walks, both for a single walker and for a pair of interacting walkers. The fermionic quantum centipede studied in this work is defined in section 3. In section 4 we map the problem onto an integrable XX Heisenberg spin chain, which can be reduced to a free-fermion system and diagonalized by means of a Jordan-Wigner transformation. Explicit results on the spectrum of the quantum centipede are presented in section 5 for the first few values of the fermion number (N = 2 to 5). In section 6 we obtain the maximal spreading velocities V (N ) for arbitrary N , as well as their limit V (∞) . Section 7 contains a discussion of our findings. A derivation of the characteristic equations (4.14), (4.15) is given in Appendix A.

A summary of earlier results on one and two quantum walkers

We consider continuous-time quantum walks on the discrete one-dimensional lattice. There is no need for an internal degree of freedom (quantum coin), as would be required for discrete-time dynamics. We recall some elementary results for the single quantum walk [START_REF] Farhi | Quantum computation and decision trees[END_REF][START_REF] Kempe | Quantum random walks -an introductory overview[END_REF][START_REF] Ambainis | Quantum walks and their algorithmic applications[END_REF][START_REF] Venegas-Andraca | Quantum walks: a comprehensive review Quantum[END_REF][START_REF] De Toro Arias | Anomalous dynamical scaling and bifractality in the onedimensional Anderson model[END_REF], and then discuss the case of two co-walking particles, with an emphasis on the ballistic fronts (see [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF] and the references therein).

The simple quantum walker

The simple continuous-time quantum walk is modeled by a tight-binding Hamiltonian, in which the walker hops from a site to a neighboring site. We denote by ψ n (t) = n|ψ(t) the wavefunction of the particle at site n at time t, and use dimensionless units. The dynamics of the walker is given by

i dψ n (t) dt = ψ n+1 (t) + ψ n-1 (t). (2.1)
Suppose that the particle is launched from the origin at time t = 0: ψ n (0) = δ n0 . The wavefunction at time t is then given by a Bessel function:

ψ n (t) = i -n J n (2t). (2.
2) Asymptotic properties of Bessel functions allow us to analyze the spreading of the quantum walk in the long-time limit [START_REF] De Toro Arias | Anomalous dynamical scaling and bifractality in the onedimensional Anderson model[END_REF]. The asymptotic probability distribution of the effective velocity v = n/t has a compact support and converges to an 'arc-sine law':

f (v) = 1 π √ 4 -v 2 (|v| < 2). (2.3)
We emphasize that, in contrast to the classical case, the convergence is in the weak sense: the probability distribution |ψ n (t)| 2 displays high-frequency oscillations which must be averaged out to derive the function f (v) [START_REF] Baraviera | Hydrodynamic limit of quantum random walks in From Particle Systems to Partial Differential Equations II[END_REF][START_REF] Grimmett | Weak limits of quantum random walks[END_REF][START_REF] Gottlieb | Convergence of continuous-time quantum walks on the line[END_REF][START_REF] Konno | Limit theorem for continuous-time quantum walk on the line[END_REF][START_REF] Strauch | Connecting the discrete-and the continuous-time quantum walk[END_REF]. At late times, the quantum particle is therefore almost surely located in the allowed region (|n| < 2t). A more precise analysis shows that the probabilities |ψ n (t)| 2 display sharp ballistic fronts near the endpoints of the allowed region (n = ±2t), with a height scaling as t -2/3 and a width scaling as t 1/3 . The above generic behavior remains unchanged as long as the initial state is localized in a finite region: the quantum walker spreads ballistically in the allowed region limited by ballistic fronts near n = ±2t, with a forbidden region beyond them. It is however possible to engineer exceptional initial states, for which either one or even both fronts are eliminated by quantum interferences [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF], but these features are non-generic.

The picture changes qualitatively if the particle is allowed to hop to the nextnearest neighboring sites with a transition amplitude g:

i dψ n (t) dt = ψ n+1 (t) + ψ n-1 (t) + g (ψ n+2 (t) + ψ n-2 (t)) . (2.4)
Allowing hopping to second and further neighbors is known to have far reaching consequences in a variety of situations [START_REF] Mülken | Universal behavior of quantum walks with long-range steps[END_REF][START_REF] Mülken | Continuous-time quantum walks: Models for coherent transport on complex networks[END_REF][START_REF] Xu | Continuous-time quantum walks on one-dimensional regular networks[END_REF][START_REF] Xu | Coherent exciton transport and trapping on long-range interacting cycles[END_REF]. For instance, in the case of graphene [START_REF] Geim | The rise of graphene[END_REF][START_REF] Castro-Neto A H | The electronic properties of graphene[END_REF], hopping to second neighbors breaks the chiral symmetry between both sublattices. In the present case, when g > 1/4, the probability distribution of the velocity v = n/t becomes singular at four values. The quantum walker thus exhibits four fronts: two external fronts (as above) at v = ±V + , and also two internal fronts at v = ±V - [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF]. If longer range hopping is allowed and if the corresponding hopping amplitudes exceed critical values, more internal fronts might appear for generic initial conditions.

Two co-walking quantum particles

The continuous-time quantum walk problem can be generalized by considering several interacting quantum walkers. In a recent work [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF], we have investigated the quantum walk performed by two identical particles interacting either through hard-bound constraints or by a smooth confining potential. The statistics of the particles (bosonic or fermionic) turned out to play an important role in the analysis of the ballistic spreading of the bound states thus obtained. We briefly summarize the results of [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF] for the quantum walk of two onedimensional fermions interacting by the hard-bound constraint that their distance is at most ℓ lattice spacings. We denote by n 1 = n + m and n 2 = n the positions of the particles, so that n cm = n + m/2 is the center-of-mass coordinate, whereas m = n 1n 2 is the relative coordinate. The hard-bound constraint imposes that |m| ≤ ℓ. This fermionic system is described by the wavefunction

ψ n,m (t) = (n 1 , n 2 )|ψ(t) = (n + m, n)|ψ(t) , (2.5)
which is odd with respect to m. Because fermionic particles can not cross one another in one dimension, m can be restricted to the range m = 1, . . . , ℓ. The dynamics is then given by

i dψ n,m (t) dt = ψ n,m-1 (t) + ψ n+1,m-1 (t) + ψ n-1,m+1 (t) + ψ n,m+1 (t), (2.6)
with Dirichlet boundary conditions:

ψ n,0 (t) = ψ n,ℓ+1 (t) = 0.
The exact solution of this two-body problem displays the following features. The wavefunction again spreads ballistically in the center-of-mass coordinate. For late times, the components ψ n,m (t) of the wavefunction have appreciable values for a range of n that grows ballistically and symmetrically with respect to the origin. The probability distribution |ψ n,m (t)| 2 of the bound state in its center-of-mass coordinate generically exhibits sharp ballistic fronts for n ≈ V k t, where the front velocities read

V k = 2 cos kπ ℓ + 1 (k = 1, . . . , ℓ). (2.7)
The spreading dynamics is therefore characterized by two extremal fronts, and ℓ -2 internal ones for ℓ ≥ 3. The range of the allowed zone is |n| < V t, the maximal spreading velocity being

V = 2 cos π ℓ + 1 .
(2.8)

In the limit where the extent of the bound state diverges (ℓ → ∞), the above result approaches the free value V = 2, with a 1/ℓ 2 correction. This picture remains qualitatively unchanged if the hard-bound constraint is replaced by a smooth confining potential. The bosonic and fermionic spectra are infinite sequences of dispersive energy levels, each of which giving rise to a ballistic front. These spectra have been studied in detail in the case where the confining potential is homogeneous, i.e., of the form W m = g|m| α . In particular, the maximal spreading velocity of two-fermion bound states departs from its free value V = 2 according to

V ≈ 2 -C (F) g 2/(α+3)
(2.9) at weak coupling (g ≪ 1), where the constant C (F) has been determined [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF].

The fermionic quantum centipede

We now introduce the system we study in this work. It is the quantum centipede made of N interacting fermionic quantum walkers on the one-dimensional lattice.

The interaction is modeled by the simplest of all hard-bound constraints: the distance between two consecutive fermions is either one or two lattice spacings. Besides the number N of fermions, the model is entirely parameter-free. One-dimensional fermions cannot cross each other, and so the discrete positions of the particles along the chain can be assumed to be ordered as x 1 < x 2 < . . . < x N . We label the state of the quantum centipede by the following variables:

• n = x 1 denotes the position of the leftmost fermion along the chain,

• the internal state of the centipede is described by a string ε = (ε 1 , . . . , ε N -1 ) of N -1 binary variables, with ε j = x j+1x j -1 = 0 or 1 for j = 1, . . . , N -1.

We have thus ε j = 0 if fermions j and j + 1 are adjacent, while ε j = 1 if they are separated by a single empty site.

Figure 1 shows a configuration of 6 fermions and the corresponding string ε. The center-of-mass coordinate of the centipede reads

x cm = n + N -1 2 + 1 N N -1 j=1 (N -j)ε j . (3.1)
We have x cm = n + (N -1)/2 for the most compact internal state (ε j = 0 for all j), whereas x cm = n + N -1 for the most extended one (ε j = 1 for all j). The 2 N -1 amplitudes of N -body wavefunction

ψ ε n (t) = x 1 , x 2 , . . . , x N |ψ(t) (3.2)
satisfy coupled continuous-time dynamical equations, which are analogous to (2.6). We shall not need to write down these equations explicitly, except for N = 2 (see (5.1)) and N = 3 (see (5.8)). The system is spatially homogeneous, i.e., invariant under discrete translations. It is therefore convenient to perform a Fourier transform with respect to n and define

ψ ε (q, t) = n e -iqn ψ ε n (t), (3.3)
where the center-of-mass momentum q can be restricted to the first Brillouin zone (|q| ≤ π).

The dynamics of the amplitudes (3.3) is governed by an effective dispersive (i.e., q-dependent) Hamiltonian H, represented by a Hermitian matrix of size 2 N -1 × 2 N -1 (see e.g. (5.2), (5.9), (5.14)). The energy spectrum of the centipede therefore has 2 N -1 branches, i.e., the eigenvalues ω a (q) of H, with a = 1, . . . , 2 N -1 . The corresponding branches of the group velocity read v a (q) = dω a (q) dq .

(3.4)

We shall be mostly interested in the maximal velocity

V (N ) = max a,q |v a (q)|, (3.5)
describing the ballistic spreading of the two extremal fronts of the wavefunction in the center-of-mass coordinate, as well as in the internal ballistic fronts, characterized by all the other stationary values of the group velocity, such that

dv a (q) dq = d 2 ω a (q) dq 2 = 0. (3.6)
In order to proceed, we start by noticing that the action of the quantum Hamiltonian H can be described in purely classical terms. The string ε is interpreted as a classical configuration of particles and holes on an open finite lattice of size N -1. If ε j = 1, site j is occupied by a particle; if ε j = 0, site j is empty. The quantum dynamics generated by H corresponds to the following evolution rules: Bulk:

10 ⇀ ↽ 01 with rate 1. Site 1:

1 → 0 with rate e -iq ,

0 → 1 with rate e iq . Site N -1: 1 → 0 with rate 1, 0 → 1 with rate 1. (3.7)
These rules are reminiscent of the symmetric simple exclusion process (SEP) with open boundaries (see e.g. [START_REF] Krapivsky | A Kinetic View of Statistical Physics[END_REF]). An equivalence with the SEP had already been put forward in the classical situation of the molecular spiders and centipedes investigated in [START_REF] Antal | Molecular spiders in one dimension[END_REF]. There are however several notable differences between quantum-mechanical systems such as the present one and classical stochastic systems such as the SEP: (i) The quantum Hamiltonian H acts on amplitudes, and not on probabilities.

(ii) The transition amplitudes or 'rates' are not necessarily positive real numbers.

(iii) The system is not equivalent to a classical stochastic process, even for q = 0. The Hamiltonian H and the Markov operator for the SEP have the same non-diagonal elements, but the Markov operator contains diagonal loss terms, in order to ensure probability conservation, whereas the Hamiltonian H does not have diagonal entries.

Mapping onto a free-fermion system

The energy spectrum of the quantum centipede can be determined, at least formally, by means of an exact mapping onto an integrable spin chain and finally onto a freefermion system.

The Hamiltonian H that implements the quantum dynamics (3.7) can be written, using Pauli matrices, as

H = e -iq S + 1 + e iq S - 1 + N -2 j=1 S - j S + j+1 + S + j S - j+1 + S + N -1 + S - N -1 .(4.1)
We thus obtain the Hamiltonian of an XX spin chain with non-diagonal boundary terms [START_REF] Nepomechie | Bethe Ansatz solution of the open XX spin chain with non-diagonal boundary terms[END_REF][START_REF] Šamaj | Introduction to the Statistical Physics of Integrable Many-body Systems[END_REF]. By convention, ε j = 0 (site j is empty) corresponds to ↑ j (spin j is up), whereas ε j = 1 (site j is occupied) corresponds to ↓ j (spin j is down). In the local basis {0 j , 1 j } ≡ {| ↑ j , | ↓ j }, the Pauli matrices are given by

S x j = 0 1 1 0 , S y j = 0 -i i 0 , S z j = 1 0 0 -1 (4.2)
and the raising and lowering operators S ± j are defined as

S + j = 1 2 (S x j + iS y j ) = 0 1 0 0 , S - j = 1 2 (S x j -iS y j ) = 0 0 1 0 . (4.
3)

The Hamiltonian H can be diagonalized by means of a Jordan-Wigner transformation [START_REF] Lieb | Two soluble models of an antiferromagnetic chain[END_REF] mapping it onto a free-fermion system. Because of the boundary terms, H is not fully bilinear. This can be rectified [START_REF] Hinrichsen | Solution of a one-dimensional diffusion-reaction model with spatial asymmetry[END_REF][START_REF] Bilstein | The XX-model with boundaries: Part I. Diagonalisation of the finite chain[END_REF] by adding two auxiliary sites, one at each end of the chain, labeled 0 and N . We thus define a new Hamiltonian H long on a chain of N + 1 sites as

H long = e -iq S x 0 S + 1 + e iq S x 0 S - 1 + N -2 j=1 S - j S + j+1 + S + j S - j+1 + S + N -1 S x N + S - N -1 S x N . (4.4)
The boundary operators S x 0 and S x N commute with H long . Hence the eigenstates of H long belong to four distinct sectors, corresponding to the eigenvalues (±1, ±1) of the operators S x 0 and S x N . The restriction of H long to the sector (+1, +1) coincides with the Hamiltonian H of (4.1).

The Hamiltonian H long can be diagonalized using a fermionization procedure, as explained in [START_REF] Bilstein | The XX-model with boundaries: Part I. Diagonalisation of the finite chain[END_REF]. The operators defined as

τ x,y j = j-1 i=0 S z i S x,y i (j = 0, . . . , N ) (4.5)
satisfy the relations

{τ µ j , τ ν k } = 2δ jk δ µν (µ, ν = x, y). (4.6)
The above anti-commutation relations define a Clifford algebra. If we rewrite H long in terms of these operators, we obtain -H long = i cos q τ y 0 τ x 1 + i sin q τ y 0 τ y

1 + i 2 N -2 j=1 τ y j τ x j+1 -τ x j τ y j+1 + iτ y N -1 τ x N . (4.7)
The last step consists in expressing H long as a free-fermion Hamiltonian:

H long = N k=0 Λ k (2a † k a k -1). (4.8)
To do so, we must find a set of annihilation and creation operators a k and a † k of fermionic quasiparticles, satisfying the canonical anti-commutation relations

{a k , a † l } = δ kl , {a k , a l } = {a † k , a † l } = 0 (k = 0, . . . , N ). (4.9)
These quasiparticles are not to be confused with the original fermionic quantum walkers which constitute the centipede. The number operators N k = a † k a k have eigenvalues 0 and 1. It follows from (4.8) that the eigenvalues of H long are given by

ω long = N k=0 (±Λ k ), (4.10)
where the sign ± in front of Λ k depends on whether the k-th quasiparticle is present (N k = 1) or absent (N k = 0).

The quasiparticle operators a k and a † k are obtained from the Jordan-Wigner operators τ x,y j by a Bogoliubov transformation of the form

a k = 1 2 N j=0
x k;j τ x j + y k;j τ y j ,

a † k = 1 2 N j=0
x k;j τ x j + y k;j τ y j ,

where the bar denotes complex conjugation. The complex coefficients (x k;j , y k;j ) are found by requiring that a k and a † k satisfy the canonical fermionic anti-commutation relations (4.9) and that H long takes the diagonal form (4.8). These constraints are implemented by writing the commutation relations between H long and a k ,

a † k : [H long , a k ] = -2Λ k a k , [H long , a † k ] = 2Λ k a † k . (4.12)
Details are given in Appendix A. The quasiparticle eigenvalues are given by

Λ k = cos p k , (4.13)
where the discrete values p k of the internal momentum p satisfy the characteristic equation (A.13), which can be further simplified by dealing separately with even and odd values of N . We obtain after some algebra N even: sin((N + 1)p) -3 sin((N -1)p) = ±4 sin p sin q, (

N odd: sin((N + 1)p) -3 sin((N -1)p) = ±4 sin p cos q. (4.15)

Explicit results for the first few values of N

In this section we present explicit results for the first few values of the fermion number (N = 2 to 5).

• N = 2
This is a special case of the more general two-body problem considered in [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF], where the maximal distance between the two quantum walkers is an arbitrary integer ℓ.

With the notation (3.2), the amplitudes ψ 0 n and ψ 1 n obey the equations

i dψ 0 n (t) dt = ψ 1 n-1 (t) + ψ 1 n (t), i dψ 1 n (t) dt = ψ 0 n (t) + ψ 0 n+1 (t), (5.1)
which can be viewed as a special case of (2.6). The corresponding Hamiltonian reads

H = 0 1 + e iq 1 + e -iq 0 . (5.2)
We thus readily obtain

ω 1,2 = ±2 cos q 2 .
(5.

3)

The associated group velocities read

v 1,2 = ∓ sin q 2 .
(5.4)

In particular, the maximal velocities ±V (2) , with V (2) = 1, (5.5) are reached for q = ±π. There is no other stationary value of the group velocity, and consequently no internal front besides the extremal ones, in agreement with the findings of [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF] for ℓ = 2, recalled in section 2.2. Figure 2 shows plots of the energy spectrum (left) and of the group velocities (right) against q/π in the first Brillouin zone. (5.6) Figure 3 shows the quasiparticle spectrum against q/π. The correspondence (4.10) relies on the following identities:

ω 1 = -ω 2 =    Λ 2 -Λ 1 (-π ≤ q ≤ -π/2), Λ 1 + Λ 2 (-π/2 ≤ q ≤ π/2), Λ 1 -Λ 2 (π/2 ≤ q ≤ π).
(5.7)

• N = 3
The wavefunction amplitudes obey the equations i dψ The associated characteristic equation is ω(ω 3 -5ω -4 cos q) = 0.

(5.10)

Figure 4 shows plots of the energy spectrum (left) and of the group velocities (right) against q/π. The maximal velocities ±V (3) , with

V (3) = 4 5 , (5.11)
are respectively reached for q = ±π/2. The group velocity also exhibits a flat (i.e., non-dispersive) band, as well as four non-trivial stationary points obeying (3.6). Differentiating twice the characteristic equation (5.10), we obtain after some algebra the stationary velocities ±V (3,1) , with These results show that the wavefunction of the 3-fermion centipede generically exhibit five ballistic peaks in the center-of-mass coordinate: two extremal ones at n ≈ ±V (3) t, two internal ones at n ≈ ±V (3,1) t, and possibly a central one at the origin, corresponding to the flat band. These predictions are illustrated in figure 5, showing plots of the probability profiles |ψ 00 n (t)| 2 (left) and |ψ 01 n (t)| 2 (right) at time t = 200 against n (n serves as a proxy for the center-of-mass coordinate x cm ) for the N = 3 centipede launched at t = 0 at sites 0, 1 and 2, i.e., with a single non-zero amplitude ψ 00 0 (0) = 1. The first profile exhibits a central peak, whereas the second one does not. (3) t and ±V (3,1) t.

V (3,1) = √ 5
Equation (4.15) yields a cubic equation for the quasiparticle eigenvalues: 4Λ 3 -5Λ ± 2 cos q = 0.

(5.13)

A comparison with (5.10) demonstrates that the correspondence (4.10) goes as follows for N = 3: the energies ω are twice as large as (some of) the quasiparticle eigenvalues Λ. Figure 6 shows the quasiparticle spectrum against q/π. • N = 4

The Hamiltonian reads

H =            0 1 0 0 e iq 0 0 0 1 0 1 0 0 e iq 0 0 0 1 0 1 1 0 e iq 0 0 0 1 0 0 1 0 e iq e -iq 0 1 0 0 1 0 0 0 e -iq 0 1 1 0 1 0 0 0 e -iq 0 0 1 0 1 0 0 0 e -iq 0 0 1 0            . (5.14)
The associated characteristic equation is

ω 8 -12ω 6 + 4(8 -3 cos q)ω 4 -24(1 -cos q)ω 2 + 4(1 -cos q) 2 = 0.(5.15)
Figure 7 shows plots of the energy spectrum (left) and of the group velocities (right) against q/π. The maximal velocities ±V (4) , with

V (4) = 1 √ 2 , (5.16)
are reached for q = 0. Figure 8 shows the quasiparticle spectrum against q/π. This is the first case where the correspondence (4.10) exhibits its generic nature, in the sense that it involves non-trivial linear combinations. Left: energy spectrum of the N = 4 centipede against q/π. Right: associated group velocities.

• N = 5

We shall not write down the 16 × 16 Hamiltonian matrix H explicitly. The associated characteristic equation is ω(ω 5 -7ω 3 + 9ω -4 cos q)(A(ω) -B(ω) cos q -16 cos 2 q) = 0, (5.17)

with Figure 9 shows plots of the energy spectrum (left) and of the group velocities (right) against q/π. The maximal velocities ±V (5) , with

A(ω) = ω 2 (ω 2 -1)(ω 2 -13)(ω 4 -7ω 2 + 9), B(ω) = 4ω(11ω 4 -28ω 2 + 13). (5.18)
V (5) = 26 + 14 √ 13 117 = 0.653 655 . . . , (5.19)
are respectively reached for q = ∓π/2. Figure 10 shows the quasiparticle spectrum against q/π. The above results illustrate the general feature that the complexity of the energy spectrum of the centipede grows very fast as the fermion number N is increased. In particular the number of stationary values of the velocity satisfying (3.6), which are responsible for the occurrence of internal ballistic fronts, grows very rapidly with N . The quasiparticle spectrum however remains regular and simple, as the number of quasiparticle eigenvalues only grows linearly with N . Quasiparticle spectrum of the N = 5 centipede against q/π.

Maximal spreading velocity for arbitrary N

The aim of this section is to obtain an exact expression of the maximal velocity V (N ) of the centipede for an arbitrary fermion number N . This quantity, defined in (3.5), characterizes the ballistic spreading of the two extremal fronts of the wavefunction in the center-of-mass coordinate. We shall also derive the exact value of the limit V (∞) . The explicit results given in section 5 for the first few values of N suggest the following pattern. The maximal velocity V (N ) is reached for the values of the center-of-mass momentum q such that the right-hand side of the characteristic equations (4.14), (4.15) vanishes, i.e., q = 0 or ±π when N is even, whereas q = ±π/2 when N is odd. In these situations, all the quasiparticle eigenvalues Λ k are twofold degenerate. The simultaneous linear lifting of these degeneracies yields, by means of (4.10), the largest possible value of the group velocity.

Let us consider for definiteness the case where N is even, and set sin((N + 1)p) -3 sin((N -1)p) sin p = P N (Λ), (

with Λ = cos p (see (4.13), (A.9)). The function P N (Λ) thus defined is a polynomial with degree N , which can be expressed as a linear combination of two Chebyshev polynomials of the second kind [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF]:

P N (Λ) = U N (Λ) -3U N -2 (Λ). (6.2)
The polynomials P N obey the recursion

P N +1 (Λ) = 2ΛP N (Λ) -P N -1 (Λ). (6.3)
We have P 0 (Λ) = 4, P 1 (Λ) = 2Λ, P 2 (Λ) = 4(Λ 2 -1), P 3 (Λ) = 2Λ(4Λ 2 -5), and so on. The characteristic equation (4.14) can thus be recast as P N (Λ) = ±4 sin q. (6.4)

For q = 0 or q = π, the right-hand side vanishes. The doubly degenerate quasiparticle eigenvalues therefore coincide with the N roots Λ k of the polynomial P N . The lifting of these twofold degeneracies in the vicinity of q = 0 or q = π is described by the slopes

dΛ k dq = ± 4 P ′ N (Λ k ) . (6.5)
Using (4.10), taking care about avoiding multiple counting, we obtain the expression

V (N ) = N k=1 4 |P ′ N (Λ k )| (6.6)
for the maximal spreading velocity. The case where N is odd can be dealt with in a similar way and yields the same expression. The general formula (6.6) allows us to recover (5.5), (5.11), (5.16), (5.19), i.e., and to predict that V (6) = 0.620 924 . . . , V (7) = 0.600 722 . . . (6.8) are the largest roots of the polynomial equations 229V 4 -78V 2 -8V + 1 = 0 and 79 937V 3 -49 192V 2 -3 664V + 2 624 = 0, with respective degrees 4 and 3. More generally, the maximal velocity V (N ) is an algebraic number whose degree d N grows exponentially fast with N . Let us skip details and give the following result:

V (2) = 1, V (3) = 4 5 = 0.8, V (4
N = 2m even: d 2m = 2 m-1 , N = 2m + 1 odd: d 2m+1 = m Int(m/2) , (6.9)
where Int(.) denotes the integer part.

As the fermion number N increases, the velocities V (N ) converge to a finite limit V (∞) , which can be obtained as follows. The last two roots satisfy |Λ| > 1. They correspond to evanescent modes with complex momenta p = iζ and p = π + iζ, such that Λ = ± cosh ζ, with tanh(N ζ) = 2 tanh ζ. In the large-N limit, we have tanh ζ → 1/2, and so ζ → (ln 3)/2 (see (6.14)) and Λ → ±2/ √ 3. By differentiating (6.1), using (6.10) and (6.11), we obtain the following estimate

|P ′ N (Λ k )| ≈ 2N sin 2 p k (cos θ k cos p k + 2 sin θ k sin p k ) 1 + 3 sin 2 p k (6.12)
for large N and real momenta p k . Finally, inserting the above expression into (6.6), and replacing the sum by an integral, we obtain

V (∞) = 4 π π/2 0 sin 2 p dp 1 + 3 sin 2 p = 2 3π 4E( √ 3
2

) -K( √ 3
2 ) = 0.570 349 449 . . ., (6.13) where E and K are the complete elliptic integrals [START_REF] Gradshteyn | Table of Integrals, Series, and Products[END_REF].

Figure 11 illustrates the above results. The left panel shows a plot of V (N ) against the fermion number N . The limit V (∞) (see (6.13)) is shown as a blue line. The right panel shows a logarithmic plot of the difference V (N ) -V (∞) against N . The data points are observed to become extremely close to the straight line with slope -ζ, where ζ = ln 3 2 = 0.549 306 . . . (6.14) is the inverse penetration length of the evanescent modes. This clearly demonstrates that the velocities converge to their limit as V (N ) -V (∞) ∼ e -N ζ , i.e., exponentially fast in N . Left: plot of the extremal velocity V (N) against the fermion number N . Blue horizontal line: limit V (∞) (see (6.13)). Right: logarithmic plot of difference V (N) -V (∞) against N . The blue straight line has slope -ζ.

Discussion

We have investigated a quantum centipede made of N fermionic quantum walkers on the one-dimensional lattice interacting by the hard constraint that the distance between two successive fermions is either one or two lattice spacings. Besides the number N of fermions, the model is entirely parameter-free. As in our previous work [START_REF] Krapivsky | Interacting quantum walkers: Two-body bosonic and fermionic bound states[END_REF], the main emphasis has been put on the ballistic spreading of the wavefunction of the centipede in its center-of-mass coordinate. For a generic initial state located in the vicinity of the origin, the distribution profile of the velocity v = n/t of the center of mass generically exhibits two extremal ballistic fronts at ±V (N ) , as well as internal ballistic fronts, whose number grows rapidly with the number N of fermions.

The energy spectrum of the centipede and the corresponding velocity dispersion curve have been analyzed by direct means for the first few values of N , whereas some analytical results have been derived for arbitrary N by exploiting a mapping of the problem onto a free-fermion system. We have thus obtained the expression (6.6) of the maximal spreading velocity V (N ) , and the non-trivial result (6.13) for the limit V (∞) .

It is interesting to put the present findings in perspective with the results of [START_REF] Antal | Molecular spiders in one dimension[END_REF] on the diffusive dynamics of N -legged molecules dubbed polypeds and spiders [START_REF] Pei | Behavior of polycatalytic assemblies in a substrate-displaying matrix[END_REF]. The classical analogue of the present situation is that of symmetric molecular centipedes, whose diffusion coefficient is given (for all N ≥ 2) by

D (N ) = 1 4(N -1) . (7.1)
There is a stark contrast between the fall-off of the diffusion coefficient D (N ) for large N in the classical case and the convergence of the spreading velocity to a finite limit V (∞) in the quantum case. This is yet another manifestation of the qualitatively different dynamical behavior of classical and quantum walkers. Some of the properties of the quantum centipede depend on the parity of the fermion number N . The symmetries of the energy spectrum ensure the existence of a flat (i.e., non-dispersive) band when N is odd. As a consequence, the wavefunction of the centipede may exhibit a central peak near the origin for odd N . (This is illustrated in figure 5 for N = 3.) The occurrence of a central peak has been underlined in other types of quantum walks. For a single discrete-time walker equipped with a three-dimensional quantum coin, a localization phenomenon has been put forward, in the sense that a finite fraction of the probability stays forever in the vicinity of the particle's starting point [START_REF] Inui | One-dimensional three-state quantum walk[END_REF][START_REF] Štefaňák | Limit distributions of three-state quantum walks: The role of coin eigenstates[END_REF]. From a different perspective, parity effects are also known to affect transport properties of some quasi-one-dimensional systems. Disordered strips made of N coupled channels with purely off-diagonal disorder are known to exhibit conventional Anderson localization for even N , albeit unconventional localization properties for odd N , with a subexponential scaling of the conductance at the band center [START_REF] Brouwer | Delocalization in coupled onedimensional chains[END_REF][START_REF] Mudry | Random magnetic flux problem in a quantum wire[END_REF][START_REF] Mudry | Crossover from the chiral to the standard universality classes in the conductance of a quantum wire with random hopping only[END_REF][START_REF] Brouwer | Nonuniversality in quantum wires with off-diagonal disorder: a geometric point of view[END_REF].

In this work we have demonstrated that the simplest fermionic quantum centipede, with maximal separation ℓ = 2 between neighboring particles, is tractable by analytical means. It would be interesting to investigate fermionic or bosonic quantum centipedes with larger maximal separations as well. Classical centipedes with ℓ ≥ 3 however lead to extremely complicated results, so that the general case seems intractable [START_REF] Antal | Molecular spiders in one dimension[END_REF]. Another variant that has been studied in the classical case is a centipede whose total length never exceeds some given length L [START_REF] Antal | Molecular spiders in one dimension[END_REF]. Its quantum analogue also appears to be interesting. Finally, it might also be worth considering bound states of N quantum walkers, either fermionic or bosonic, on higher-dimensional lattices with various kinds of hard-bound constraints.

Appendix A. Derivation of the characteristic equations (4.14), (4.15) In this appendix we provide a detailed characterization of the quasiparticle operators defined in (4.11) and a derivation of the characteristic equations (4.14), (4.15).

Our starting point is the quadratic identity [τ µ j τ ν j+1 , τ λ k ] = 2 δ j+1,k δ νλ τ µ jδ jk δ µλ τ ν j+1 , (A.1)

where µ, ν, λ = x, y, that follows from the Clifford algebra (4.6). We can then write explicitly the commutation relation [H long , a k ] = -2Λ k a k (see (4.12)) as follows i cos q (x 1 τ y 0y 0 τ x 1 ) + i sin q (-y 0 τ y 1 + y 1 τ y 0 )

+ i 2 N -2 j=1
x j+1 τ y j + x j τ y j+1y j τ x j+1y j+1 τ x j + i x N τ y N -1y N -1 τ x N = Λ N j=0

x j τ x j + y j τ y j .

(A.

2)

The characteristic equation thus obtained is a polynomial equation with degree N + 1 in the variable cos 2p = 2Λ 2 -1. The value p = 0 is however not allowed. The eigenvectors constructed as above indeed vanish identically for p = 0. We are thus left with N pairs of opposite quasiparticle eigenvalues ±Λ k . This spectrum is to be completed by Λ = 0 with multiplicity two. For e ip = ±i the system (A.12) indeed always admits the solution A = B = C = D, irrespective of q. This elementary non-dispersive solution had been discarded in the algebra leading to (A.13).

Inserting the above quasiparticle spectrum into (4.10) yields the 2 N +1 (possibly degenerate) eigenvalues ω long of H long . A quarter of them, corresponding to the sector (+1, +1) of the boundary operators S x 0 and S x N , coincide with the 2 N -1 (again possibly degenerate) eigenvalues ω of H.

Finally, the characteristic equation (A.13) can be further simplified to (4.14) and (4.15), by dealing separately with even and odd values of N .

Figure 1 .

 1 Figure 1. A configuration of 6 fermions obeying the hard-bound constraint and the corresponding string ε of 5 binary variables.

Figure 2 . 1 , 2 = 1 ±

 2121 Figure 2. Left: energy spectrum of the N = 2 centipede against q/π. Right: associated group velocities.

5 ΛFigure 3 .

 53 Figure 3. Quasiparticle spectrum of the N = 2 centipede against q/π.

-17 1 Figure 4 .

 14 Figure 4. Left: energy spectrum of the N = 3 centipede against q/π. Right: associated group velocities.

2 Figure 5 .

 25 Figure 5. Two probability profiles at time t = 200 for the N = 3 centipede launched in its most compact state near the origin (ψ 00 0 (0) = 1). Left: |ψ 00 n (t)| 2 exhibits a central peak (not to scale). Right: |ψ 01 n (t)| 2 does not. Vertical blue lines: nominal positions of the ballistic fronts at ±V(3) t and ±V(3,1) t.

5 ΛFigure 6 .

 56 Figure 6. Quasiparticle spectrum of the N = 3 centipede against q/π.

Figure 7 .

 7 Figure 7. Left: energy spectrum of the N = 4 centipede against q/π. Right: associated group velocities.

5 ΛFigure 8 .

 58 Figure 8. Quasiparticle spectrum of the N = 4 centipede against q/π.

Figure 9 .

 9 Figure 9. Left: energy spectrum of the N = 5 centipede against q/π. Right: associated group velocities.

5 ΛFigure 10 .

 510 Figure 10. Quasiparticle spectrum of the N = 5 centipede against q/π.

  For N ≥ 3, N -2 roots Λ k of the polynomial P N obey |Λ k | < 1. They correspond to real momenta p k , such that N p k = kπ + θ k (k = 1, . . . , N -2), (6.10) with |θ k | ≤ π/2 and tan θ k = 2 tan p k . (6.11)

Figure 11 .

 11 Figure 11. Left: plot of the extremal velocity V (N) against the fermion number N . Blue horizontal line: limit V (∞) (see (6.13)). Right: logarithmic plot of difference V (N) -V (∞) against N . The blue straight line has slope -ζ.

(The subscript k = 0, . . . , N has been omitted for ease of reading.) After identifying the coefficients of each τ x,y j , we obtain the following system of linear equations for the coefficients x j and y j : For j = 0:

x 0 = 0, -iΛy 0 = x 1 cos q + y 1 sin q.

(A.3)

For j = 1:

For j = 2, . . . , N -2:

The above equations can be interpreted as an eigenvalue problem for the vector (x 0 , y 0 , x 1 , y 1 , . . . , x N , y N ) of length 2(N + 1). The quasiparticle eigenvalues Λ are obtained as follows. The bulk equations (A.5) yield the recursion

The corresponding characteristic equation is bi-quadratic: 4Λ 2 = r -2 + 2 + r 2 = (r -1 + r) 2 . The four values r = ±e ±ip lead to the dispersion relation Λ = cos p. (A.9) Using (A.5) and (A.9), the corresponding eigenvectors read

x j = Ae ijp + Be -ijp + (-1) j (Ce ijp + De -ijp ), -iy j = Ae ijp + Be -ijp -(-1) j (Ce ijp + De -ijp ).

(A.10)

The values of p are yet to be determined by the boundary conditions. First, using (A.3) and (A.7), we substitute y 0 and x N in (A.4) and (A.6) to obtain

Then, imposing that the generic forms (A.