Ph.D Biru Zhou
email: biru.zhou@mcgill.ca

Richard Boyer

Stéphane Guay
email: stephane.guay@umontreal.ca

Running head: WORKPLACE VIOLENCE AGAINST BUS DRIVERS 1 Dangers on the Road: A Longitudinal Examination of Passenger Initiated Violence Against Bus Drivers

Keywords: Workplace Aggression, Violence, PTSD, Acute Stress Disorder, Social Support, Work Confidence

This study examined the impact of workplace violence against 109 bus drivers over a one-year span. Workplace violence is related to both psychological and work related consequences. Our findings showed that bus drivers experienced a wide range of violence at work and the psychological consequences were devastating: half of the participants met the diagnostic criteria for acute stress disorder within the first month following the index event.

Majority of them experienced at least moderate levels of PTSD problems over the one year span. About 9.3% of participants showed a delayed onset of PTSD six months after. Furthermore, counter-supportive behaviours and re-exposure to violence played important roles in the maintenance of PTSD symptoms over time. Even though PTSD symptoms per se did not relate to bus driver's confidence in coping with aggressive passengers, the immediate posttraumatic reaction -symptoms of acute stress disorder -showed a significant long term negative effect on bus drivers' confidence in dealing with aggressive passengers 12 months after. This study provided empirical evidence of the changing nature of PTSD symptoms over time among bus drivers.

Dangers on the

to make assaults against on duty public transport operators (e.g., drivers for bus, minibus, trolley and taxi) an aggravating circumstance in sentencing (The Department of Justice Canada, 2015). While the justice system has taken clear actions against violence towards public transit operators, research on violence against bus drivers are also considered as imperative in academia.

Public transit operators' safety and well-being are closely interlocked with millions of passengers' and other road users' safety and well-being on a daily basis. However, international research had painted a terrifying picture of what public transit operators had to face at work everyday. Every year, there are about 2000 assaults against transit employees reported in Canada (Canadian Urban Transit Assocation, 2014). The majority of these assaults (75%) occur on transit vehicles, whereas 15% occur in transit stations. These assaults include littering, causing damage to the vehicle, throwing objects to the operator, verbal threats, physical assaults with or without weapons, and sexual assaults. This malevolence against bus drivers is far from unique in Canada. [START_REF] Lincoln | Moving violations: A study of incivility and violence against urban bus drivers in Australia[END_REF] used a multi-method study to show that Australian bus drivers were suffering from very similar types of violence on the job, and the majority of bus drivers had experienced some form of passenger abuse (i.e., physical violence/antisocial behaviours/incivility) in the past 12 months. [START_REF] Couto | Burnout, workplace violence and social support among drivers and conductors in the road passenger transport sector in Maputo City, Mozambique[END_REF] also reported that more than three-quarters of the public transit drivers had experienced workplace violence in their life time and more than two-thirds had been a victim of workplace violence in the past 12 months. Strikingly similar phenomena were reported also in UK (Smith & Cornish, 2006), Brazil [START_REF] Sampaio | Work ability and stress in a bus transportation company in Belo Horizonte[END_REF], Lithuania [START_REF] Obelenis | Working conditions and health of the employees of public bus and trolleybus transport in Lithuania[END_REF], and Norway [START_REF] Glaso | Bus drivers' exposure to bullying at work: An occupation-specific approach[END_REF]. These past research on workplace violence against public transit operators all point to the conclusion that driving a bus is a very stressful and high risk occupation [START_REF] Kompier | Stress prevention in bus drivers: Evaluation of 13 natural experiments[END_REF]Tse, Flin, & Mearns, 2006).

Bus drivers often work alone, involve in cash handling procedures, work in tight schedules and irregular hours, have direct contact with a large public in an overcrowded space, and often encounter passengers who are intoxicated [START_REF] Essenberg | Violence and stress at work in the transport sector[END_REF]Tse et al., 2006).

All of the above job characteristics of bus drivers are related to high risks of harassments and violence. Previous research consistently show that workplace violence in the service sector is detrimental to workers' physical and mental health (e.g., [START_REF] Dupré | Harm to those who serve: Effects of direct and vicarious customer-initiated workplace aggression[END_REF][START_REF] Guay | Verbal violence in the workplace according to victims' sex -systematic review of the literature[END_REF][START_REF] Hershcovis | Towards a relational model of workplace aggression[END_REF][START_REF] Leblanc | Understanding many faces of workplace violence[END_REF]Vedantham et al., 2001). The "possibility of assault" was reported by UK bus drivers as their top psychosocial stressor at work [START_REF] Duffy | Stress and the bus driver in the uk transport industry[END_REF]. More recent findings continued to support that the risk of being assaulted on the job was still a major concern among public transport workers [START_REF] Budd | Violence at work: New findings from the 2000 British Crime Survey[END_REF]. The stressful work environment combined with the challenging job demands could make any negative interactions with passengers easily escalate to a much more serious situation (e.g., G. W. Evans & Johansson, 1998).

Even though some researchers suggest that psychological distress in the form of post-traumatic stress disorder (PTSD) may be rare for bus drivers (Tse et al., 2006), the prevalence rate for developing PTSD among bus drivers exposed to traumatic events at work (about 10.7%; [START_REF] Boyer | Prévalence du trouble de stress post-traumatique chez des chauffeurs d'autobus [Prevalence of post-traumatic stress disorder in bus drivers[END_REF] was higher than the general population in the U.S.

(about 6.8%; [START_REF] Kessler | Lifetime prevalence and age-of-onset distributions of DSM-IV disorders in the National Comorbidity Survey Replication[END_REF] and comparable with the life time prevalence rate in Canada (9.2%;Van Ameringen, Mancini, Patterson, & Boyle, 2008). Moreover, PTSD is not only detrimental to bus drivers' psychological well-being and physical health (Tse et al., 2006;Vedantham et al., 2001), it can also interfere with victims' normal emotional functioning on a day to day basis [START_REF] Kunst | Exploring the impact of dysfunctional posttraumatic survival responses on crime revictimization[END_REF], which may influence bus drivers' work performance. Therefore, the examination of PTSD among bus drivers is of major interests among the general public and public transport organizations.

PTSD and Social Support

PTSD is a chronic mental illness that may develop among victims (or witnesses) exposed to traumatic event(s). It is characterized by victims (or witnesses) of trauma experiencing intrusive thoughts (e.g., nightmares or difficulties concentrating), avoidance behaviours (e.g., avoiding activities that might remind them of the trauma), negative alterations in cognition and mood (e.g., persistent and distorted blame of self or others, and persistent negative emotional state), as well as alterations in arousal and reactivity (e.g., reckless or self-destructive behaviours, hypervigilance and exaggerated startle responses; American Psychiatric Association, 2013). Empirical evidence showed that social support could have buffering effects on PTSD [START_REF] Adams | Predictors of PTSD and delayed PTSD after disaster[END_REF][START_REF] Andrews | Gender, social support, and PTSD in victims of violent crime[END_REF][START_REF] Avery | Impact of combat and social support on PTSD and alcohol consumption in OEF/OIF veterans[END_REF][START_REF] Guay | Exploring the links between posttraumatic stress disorder and social support: Processes and potential research avenues[END_REF][START_REF] Guay | How do social interactions with a significant other affect PTSD symptoms? An empirical investigation with a clinical sample[END_REF][START_REF] Laffaye | Relationships among PTSD symptoms, social support, and support source in veterans with chronic PTSD[END_REF][START_REF] Pietrzak | Psychosocial buffers of traumatic stress, depressive symptoms, and psychosocial difficulties in veterans of operations enduring freedom and Iraqi freedom: The role of resilience, unit support, and postdeployment social support[END_REF][START_REF] Schnurr | Risk factors for the development versus maintenance of posttraumatic stress disorder[END_REF]. Specifically, lack of social support after the traumatic event is one of the strongest social factors that predicts PTSD. Individuals with lower levels of social support after a traumatic event reported higher levels of PTSD symptoms [START_REF] Brewin | Meta-analysis of risk factors for posttraumatic stress disorder in trauma-exposed adults[END_REF][START_REF] Ozer | Predictors of posttraumatic stress disorder and symptoms in adults: A meta-analysis[END_REF]. [START_REF] Joseph | Understanding post-traumatic stress: A psychosocial perspective on PTSD and treatment[END_REF] in their research suggested that social support could affect PTSD symptoms directly and indirectly. In the direct route, social interactions can affect individuals' psyche on their emotions, cognitions and symptoms. Significant other's point of the view on the traumatic event can influence victim's own interpretations of the event (positively or negatively) and consequently impact symptoms of PTSD [START_REF] Joseph | Understanding post-traumatic stress: A psychosocial perspective on PTSD and treatment[END_REF]. This is consistent with the Interdependence Theory in the field of social relations that members within a close relationship often influence each others' emotions, cognitions and behaviours [START_REF] Rusbult | Why we need interdependence theory[END_REF]Wickham & Knee, 2012). Evidence from previous research supported this direct route, suggesting that perceived social support was negatively associated with PTSD severity (e.g., [START_REF] Adams | Predictors of PTSD and delayed PTSD after disaster[END_REF][START_REF] Avery | Impact of combat and social support on PTSD and alcohol consumption in OEF/OIF veterans[END_REF][START_REF] Pietrzak | Psychosocial buffers of traumatic stress, depressive symptoms, and psychosocial difficulties in veterans of operations enduring freedom and Iraqi freedom: The role of resilience, unit support, and postdeployment social support[END_REF]. Furthermore, according to [START_REF] Joseph | Understanding post-traumatic stress: A psychosocial perspective on PTSD and treatment[END_REF], social support could also influence PTSD indirectly by changing individuals' ways of coping after the trauma. Although social support usually refers to the positive aspects of social interactions, close relationships are also comprised of negative interactions that are counter-supportive. For instance, people can provide poor advice [START_REF] Shumaker | Gender differences in social support and physical health[END_REF] or simply engage in negative behaviours, such as criticism, complains, refusal to help, or making hurtful and insensitive remarks of the situation [START_REF] Crevier | Overt social support behaviors: Associations with ptsd, concurrent depressive symptoms and gender[END_REF][START_REF] Cutrona | Social support communication in the context of marriage: An analysis of couples' supportive interactions[END_REF]. These unhelpful and counter-supportive behaviours from significant others could indirectly contribute to the development and maintenance of PTSD symptoms. Victims encountering negative interactions with their significant others might engage in more avoidance behaviours (e.g., avoid thinking about the event or avoid disclosing feelings to others). Eventually, these negative interactions would lead to slower recovery or even greater psychological distress [START_REF] Guay | Exploring the links between posttraumatic stress disorder and social support: Processes and potential research avenues[END_REF][START_REF] Joseph | Understanding post-traumatic stress: A psychosocial perspective on PTSD and treatment[END_REF]. In short, helpful, uncritical and sensible support has positive impact on the path to recovery, whereas counter-supportive and critical responses are detrimental to trauma survivors.

Although previous research acknowledge that social support is an important buffer of stress among bus drivers (Tse et al., 2006), the existing research on the influences of social support among bus drivers are limited in scope (e.g., [START_REF] French | Worksite environment intervention to prevent obesity among metropolitan transit workers[END_REF][START_REF] Sampaio | Work ability and stress in a bus transportation company in Belo Horizonte[END_REF]. In particular, previous research focused primarily on social support at work provided by co-workers and supervisors [START_REF] Sampaio | Work ability and stress in a bus transportation company in Belo Horizonte[END_REF]Tse et al., 2006). Since bus drivers often work alone and lack opportunities to interact with other co-workers on the job, it is important to examine how social support from non-work settings (e.g., friends and family) could buffer stress, especially for the ones who are experiencing symptoms of PTSD after encountering severe violence at work.

Occupational Consequence of PTSD

PTSD is a serious mental health issue among trauma victims. It can negatively impact work functioning, such as absenteeism, unemployment, and work disability, regardless whether it is full PTSD or partial PTSD [START_REF] Breslau | Partial PTSD versus full PTSD: an empirical examination of associated impairment[END_REF][START_REF] Matthews | Prediction of work functioning following accidental injury: The contribution of PTSD symptom severity and other established risk factors[END_REF][START_REF] Taylor | Factors associated with occupational impairment in people seeking treatment for posttraumatic stress disorder[END_REF][START_REF] Zlotnick | Does "subthreshold" posttraumatic stress disorder have any clinical relevance[END_REF].

Previous research showed that symptoms of PTSD, particularly the symptom of hypervigilance, could interfere with normal emotional functioning [START_REF] Kunst | Exploring the impact of dysfunctional posttraumatic survival responses on crime revictimization[END_REF].

The tendency to look for emerging dangers excessively is often accompanied by misinterpretations and negative appraisals of non-threatening cues in the environment among PTSD patients. This dysfunctional emotional reaction due to PTSD could be very problematic for bus drivers when dealing with difficult passengers. Bus drivers might misinterpret non-threatening cues as dangerous and then react irritably and even aggressively (e.g., exaggerated startle responses; American Psychiatric Association, 2013; [START_REF] Kunst | Posttraumatic anger, recalled peritraumatic emotions, and PTSD in victims of violent crime[END_REF].

Reduced concentration at work triggered by PTSD symptoms could also lead to misinterpretations of non-threatening cues in the environment (Work Safe BC, n.d.). This self-precipitating hypervigilant reaction under the influence of PTSD symptoms may severely compromise bus drivers' occupational functioning (S. [START_REF] Evans | Anger and its association to distress and social/occupational functioning in symptomatic disaster relief workers responding to the September 11, 2001, world trade center disaster[END_REF]Work Safe BC, n.d.). Instead of diffusing a potentially dangerous event, the inappropriate or exaggerated startle reactions stemming from PTSD symptoms could intensify a dangerous event while dealing with difficult passengers. Hence, it is essential to examine bus drivers' self perceived confidence in coping with potentially aggressive passengers as an occupational consequence of PTSD.

The Current Study

The objective of this study is to examine psychological consequences of severe violence against bus drivers and to test the buffering effects of social support on PTSD symptoms longitudinally in a one-year span. These research focuses were addressed using a 4-wave longitudinal design spanning over 12 months to follow bus drivers who had encountered a violent event at work. The use of a longitudinal design was necessary in order to capture the changes and dynamics underlying the psychological and occupational consequences of workplace violence over time.

Prior to developing symptoms of PTSD, victims of traumatic event often experience an anxiety disorder -acute stress disorder (ASD; e.g., [START_REF] Bryant | Acute stress disorder scale: A self-report measure of acute stress disorder[END_REF][START_REF] Harvey | Two-year prospective evaluation of the relationship between acute stress disorder and posttraumatic stress disorder following mild traumatic brain injury[END_REF]. ASD describes a set of very intense posttraumatic reactions within the first month following the event. Previous research found that ASD diagnosis is strongly predictive of later PTSD diagnosis [START_REF] Brewin | Acute stress disorder and posttraumatic stress disorder in victims of violent crime[END_REF][START_REF] Bryant | Acute stress disorder scale: A self-report measure of acute stress disorder[END_REF][START_REF] Bryant | Gender differences in the relationship between acute stress disorder and posttraumatic stress disorder following motor vehicle accidents[END_REF][START_REF] Cahill | Posttraumatic stress disorder and acute stress disorder ii: Considerations for treatment and prevention[END_REF]. Therefore, ASD must be accounted for in order to better understand changes in PTSD over time.

In this study, we examined what specific patterns of social support (supportive versus counter-supportive) were associated with changes in PTSD symptoms over time. It was hypothesized that (1) supportive behaviours were associated with a general decline of PTSD symptoms over time, whereas (2) counter-supportive behaviours were positively associated with PTSD symptoms over time, after controlling for symptoms of acute stress disorder and demographic information, such as gender, age and marital status, etc. In this study, we also investigated bus drivers' confidence in coping with aggressive passengers one year after the violent incident. (3) It was hypothesized that high levels of acute stress disorder symptoms after the violent event were related to lower levels of bus drivers' confidence in coping with aggressive passengers one year later. (4) Similarly, PTSD symptoms one month, six and twelve months after the violent event would negatively relate to bus drivers' confidence in coping with aggressive passengers one year later.

Method

Participants

This study was part of a bigger project investigating better management of workplace violence among employees of a public transit company. The data used in this study consisted of 109 bus drivers (women = 46, men = 63) who had recently filed a work incident report regarding a violent event at work. This reported violent work incident is referred to as the index event in this study. Among the 109 participants in our study, about 60.6% were married or in a common-law relationship. Most of them were working in night shifts, over night shifts or on rotation. Furthermore, due to the index event, some of the participants were on sick leave during our four assessments (35.8% of participants at Time 1, 14.7% at Time 2, 6.5% at Time 3 and 5.8% at Time 4).

Procedure

The union representatives and human resources staffs first approached the participants, who recently filed an incident report about a violent event at work, by providing them with a brief description of our study. Potential participants were asked whether they would be interested in participating in our one year study. Interested participants were then asked to sign an authorization form to leave their contact information for our research coordinators to contact them after. The completed contact authorization forms were forwarded to the research team immediately and potential participants were called in the following days to see if they agreed to participate in the study. A packet of questionnaires along with a consent form were sent to the participants who agreed to participate in the study four times by mail following the index event. Questionnaires and the consent form once completed and signed were sent back to our research center by mail. Time 1 assessment was scheduled within 2 to 4 weeks following the index event, Time 2 assessment was scheduled for 1 month after the index event, Time 3 assessment was scheduled for 6 months after and Time 4 was scheduled for 12 months after the index event.

A compensation of 25 Canadian dollars were sent to the participants who completed the questionnaires at each time of assessment. In addition, the public transit company paid a compensation equivalent to 4 hours of wages to bus drivers who have completed the entire study -all four times of assessments. The list of participants who completed the entire study was sent to the heads of the company at the end of the project so that participants can receive the compensation from their employer. This project was conducted between 2013 -2015. A total of 109 participants completed the Time 1 questionnaire, 109 participants returned to complete the Time 2 questionnaire, 108 returned for the Time 3 questionnaire, and 104 returned for the Time 4 questionnaire. This study was approved by the ethics committee in our research institute, and participants were informed prior to the study that their participation was completely voluntary and they could drop out from the study at any time without any penalty.

Measures

Social Support. The perceived social support was measured using the Questionnaire on Social Support Behaviours in Anxious Situations [START_REF] Guay | How do social interactions with a significant other affect PTSD symptoms? An empirical investigation with a clinical sample[END_REF]. This scale is a 30-item self-report questionnaire assessing perceived frequencies of supportive social interactions (9 items) and counter-supportive social interactions (22 items) with one identified significant other on a 5-point Likert-type scale ranging from 1 = "Never" to 5 = "Always". Examples of the supportive social interactions include when the significant other "asks me how I feel", or "gives me advice". Examples of counter-supportive interactions include when the significant other "criticizes the way I react", or "makes me feel guilty". The one significant other identified by participants in our study included spouse, family members or friends. The average Cronbach's α was .89 across four time points for the supportive social interactions scale, and the average Cronbach's α was .90 across four time points for the counter-supportive social interactions scale.

Psychological consequences of violence. The Acute Stress Disorder Scale (ASDS; [START_REF] Bryant | Acute stress disorder scale: A self-report measure of acute stress disorder[END_REF] was used for Time 1 assessment to measure the immediate psychological consequences of the index event at work. This scale was designed to identify acute posttraumatic stress reactions preceding the potential development of PTSD. This scale consists of 19 items that describe symptoms of acute stress disorder. Each symptom was presented on a 5-point scale ranging from 1 = "Not at all" to 5 = "Very much". An example of the scale includes "Do you feel very upset when you are reminded of the trauma?" Cronbach's α = .87 was obtained for this scale in this study.

The Time 2 to Time 4 assessments on psychological consequences were administered using the PTSD Checklist (PCL; [START_REF] Bliese | Validating the primary care posttraumatic stress disorder screen and the posttraumatic stress disorder checklist with soldiers returning from combat[END_REF]Weathers, Litz, Herman, Huska, & Keane, 1993). PCL1 is a 17-item self assessment of symptoms of PTSD in the fourth edition of the Diagnostic and Statistical Manual of Mental Disorder (DSM-4; American Psychiatric Association, 1994). This scale can be used for monitoring symptom changes over time, screening for PTSD and making provisional PTSD diagnosis. Each symptom was presented on a 5-point scale ranging from 1 = "Not at all" to 5 = "Extremely". An example of the items includes "In the past month, how much were you bothered by repeated, disturbing, and unwanted memories of the stressful experience?" The average Cronbach's α was .94 across three time points for this scale in this study.

Confidence in Coping with Aggressive Passengers. Bus drivers' confidence in coping with aggressive passengers was measured by the adapted version of "Confidence in

Coping with Patient Aggression Instrument" (CCPAI; [START_REF] Thackrey | Clinician confidence in coping with patient aggression: Assessment and Work Safe BC[END_REF], which was originally designed to assess clinicians' self-perceived ability, preparation, and comfort in safety and efficiency for intervening psychologically and physically with aggressive patients. This scale has been widely used in the healthcare sector (e.g., [START_REF] Martin | Clinician perceptions of personal safety and confidence to manage inpatient aggression in a forensic psychiatric setting[END_REF]Verhaeghe et al., 2014). This instrument consists of 10 items measuring different constructs associated with service provider's confidence on a 11-point linear scale. A Cronbach's α of .93 at Time 4 assessment (i.e., one year after the index event) was obtained for this scale in the present study. Example items are as follows:

How comfortable are you working with an aggressive service user? (1 = very uncomfortable to 11 = very comfortable) How safe do you feel around an aggressive service user? (1 = very unsafe to 11 = very safe) Demographic information. Demographic information, such as age, gender (0 = male, 1 = female) and marital status (0 = not married and not common-law, 1 = married or common-law) were assessed at Time 1 only. According to previous research on workplace violence against bus drivers and other occupations, working on day shifts (0 = other shifts, 1 = day shifts) could be a protective factor against workplace violence (e.g., [START_REF] Gilbert | The nature of occupational violence against taxicab drivers: Violence against taxicab drivers[END_REF][START_REF] Harrell | Workplace violence, 1993-2009[END_REF][START_REF] Guay | Verbal violence in the workplace according to victims' sex -systematic review of the literature[END_REF][START_REF] Mayhew | Violent assaults on taxi drivers: Incidence patterns and risk factors[END_REF]Tse et al., 2006). Work shifts were measured at all four assessments. Moreover, studies on trauma and PTSD suggest that prior exposure to trauma and repeated exposure to violence after a trauma are positively related to higher levels of PTSD (e.g., [START_REF] Adams | Predictors of PTSD and delayed PTSD after disaster[END_REF][START_REF] Brunet | Lifetime exposure to traumatic events among a sample of city bus drivers[END_REF][START_REF] Brunet | The effects of initial trauma exposure on the symptomatic response to a subsequent trauma[END_REF][START_REF] Schnurr | Risk factors for the development versus maintenance of posttraumatic stress disorder[END_REF]. Therefore, the number of violent events encountered by bus drivers at work (being a victim or a witness) prior to the index event was measured at Time 1. Repeated exposure to violent events (0 = No, 1 = Yes) after the index event was measured at Time 2, 3 and 4 assessments. All demographic variables were included as control variables in our analyses.

Results

In this section, we will first report results on descriptive statistics pertaining the nature of the violence encountered by bus drivers at work, and percentages for ASD and PTSD diagnosis in our sample. The effects of social supportive and counter-supportive behaviours on changes of PTSD symptoms over time will be analyzed using linear mixed effect models.

Total scores of supportive social interactions scale, counter-supportive interactions scale, ASDS, PCL and CCPAI were used for all analyses. All continuous predictors were grand mean centered before entering into the longitudinal analysis, unless otherwise specified.

Descriptive Statistics

Participants in our study had reported a wide range of violence they experienced at work, including physical (20.2%; e.g., being spat on or punched, threatened at gun point, sexual harassment, etc.), verbal (35.8%; e.g., intimidation, insults, death threats, etc.), both physical and verbal (24.8%; e.g., being spilled hot coffee on the lap with insults), verbal aggression with physical aggression towards objects (13.8%; e.g., damaging the bus with insults), physical aggression towards objects (1.8%; e.g., gun shots fired on the bus but not towards the bus driver), witnessing physical or verbal aggression on the bus (2.8%; e.g., witnessed a severe physical assaults against another passenger), and theft (0.9%; personal belongings being stolen). Participants also indicated they had experienced or witnessed on average 5.48 incidents of violence (ranging from 2 to 22 incidents) at work prior to the index event. Previous exposure to violent events at work appears to be very common among bus drivers in our study.

Within the first month following the index event, more than half of our participants (58.7%) met the diagnostic criteria for acute stress disorder [START_REF] Bryant | Acute stress disorder scale: A self-report measure of acute stress disorder[END_REF]. About 17.4% of participants met the full diagnostic criteria for PTSD 2 (Weathers et al., 1993), with an additional 22.9% of participants reported significant problems in PTSD symptoms which can be described as experiencing "partial" PTSD (PCL scores higher than 32; [START_REF] Dickstein | Diagnostic utility of the posttraumatic stress disorder (PTSD) checklist for identifying full and partial PTSD in active-duty military[END_REF] one month after the index event. Please see Table 1 for percentages of participants meeting different diagnostic criteria across the four time points. 1] here.

Insert [Table

Gender variations were examined for different diagnostic criteria. Results showed that men were more likely to experience moderate problems of PTSD (i.e., moderate level of symptom endorsement in at least one diagnostic cluster measured in PCL) than women one month after the index event, χ2 (1) = 4.22, p = 0.04. No other significant gender differences were observed in our analyses (see Table 1). It has been well documented that women appear to suffer from PTSD more frequently and more intensively than men [START_REF] Brewin | Meta-analysis of risk factors for posttraumatic stress disorder in trauma-exposed adults[END_REF][START_REF] Gradus | Epidemiology of PTSD[END_REF], even after controlling for frequency of exposure and types of violence (e.g., [START_REF] Stein | Gender differences in susceptibility to posttraumatic stress disorder[END_REF]Tolin & Foa, 2006). However, due to the lack of gender differences on PTSD symptoms observed in this study and lack of gender differences on psychological distress among bus drivers in previous research (e.g., [START_REF] Rydstedt | A longitudinal study of workload, health and well-being among male and female urban bus drivers[END_REF], the following analyses would not focus on gender differences in PTSD but would enter gender as a control variable along with other demographic variables. Table 2 presents correlations across all variables. 2] here.

Insert [Table

Furthermore, there were 10 participants (9.2%) showed a delayed onset of full PTSD six months after the index event, and only three participants had full PTSD persisted through Time 2 to Time 4 assessments. The severity of PTSD symptoms appeared to vary within the first 12 months following the index event. Therefore, a longitudinal approach is most appropriate for a thorough understanding of the effects of social support on changes in PTSD symptoms over time.

Social Support and PTSD

Due to the longitudinal design of the study, linear mixed effect models using SAS version 9.3 were employed in the following analyses. Four models in the longitudinal analysis of PTSD symptoms over a one-year span were conducted: Model A was the intercept only model to access whether individuals' PTSD symptoms measured by PCL changed over time;

Model B was a conditional model with supportive behaviours (or counter-supportive behaviours) and time as predictors controlling for levels of acute stress disorder measured by ASDS; Model C was the interaction model where the interaction of supportive behaviours (or counter-supportive behaviours) and Time was entered in the model controlling for the levels of acute stress disorder; Model D was the final model with ASDS and all demographic information (i.e., age, gender, marital status, work shifts, prior and repeated exposure) entered as control variables.

According to our preliminary analysis on collinearity, supportive and counter-supportive behaviours were highly correlated (ranged between -.40 to -.51; see Table 2), and the variance inflation factors for supportive and counter-supportive behaviours ranged between 1.93 to 4.54, which indicated a moderate to serious concern for collinearity.

Previous research also showed distinct and independent impact of supportive versus counter-supportive behaviours on PTSD (e.g., [START_REF] Abbey | Effects of different sources of social support and social conflict on emotional well-being[END_REF][START_REF] Andrews | Gender, social support, and PTSD in victims of violent crime[END_REF][START_REF] Borja | Positive and negative adjustment and social support of sexual assault survivors[END_REF]. As a result, supportive and counter-supportive behaviours were entered in separate models for the analyses3 . All longitudinal analyses conducted in this study followed the procedures recommended by [START_REF] Singer | Applied longitudinal data analysis: Modeling change and event occurrence[END_REF]. 3] here.

Insert [Table

Supportive Behaviours on PTSD. Table 3 presents the mixed effect estimates for all the models analyzed in this study. The intercept variance (σ 2 0 = 31.96, p = .003) and the residual variance (σ 2 ε = 135.31, p < .001) in Model A indicated that there were significant between-person and within-person variations in PTSD. ICC calculated based on estimates obtained in Model A showed that there were 19.1% of between-person variations in the model. Thus, it is advantageous to use linear mixed effect models to examine the changes in PTSD in order to capture both the within-and between-person variations over time [START_REF] Singer | Applied longitudinal data analysis: Modeling change and event occurrence[END_REF].

Results from Model B showed that there were significant differences among individual bus drivers' initial levels of PTSD at four weeks after the index event and their levels of PTSD changed significantly over time. There was a general decline of PTSD levels over time, suggested by the negative coefficients of Time in all the models. However, supportive behaviours provided by bus driver's significant other did not relate to the changes in PTSD, as shown in Models B and C. The interaction between supportive behaviours and Time was not significant in Model D either. In other words, the changes in PTSD over time did not depend on the levels of supportive behaviours. Among the control variables, acute stress disorder and repeated exposure to violence significantly predicted higher levels of PTSD.

Counter-Supportive Behaviours on PTSD. Different results were obtained in the analyses regarding the influences of counter-supportive behaviours. Counter-supportive behaviours were positively related to PTSD symptoms (Models B, C and D), and the interaction between counter-supportive behaviours and Time was significant in Model D.

The significant interaction suggested that the weekly rate of change in PTSD symptoms over time differed depending on the levels of counter-supportive behaviours individual bus drivers reported (see Figure 1). More specifically, bus drivers who reported high levels of counter-supportive behaviours from their confidant at 4 weeks after the index event, rated higher on PCL than those reported low levels of counter-supportive behaviours. Furthermore, over the 12 months period of our study, the impact of counter-supportive behaviours on PTSD levels, though destructive, faded away much faster for the ones who received high levels of counter-supportive behaviours from their significant others than the ones who received low levels of counter-supportive behaviours. Similar patterns of control variables emerged in the analyses of counter-supportive behaviours. Acute stress disorder and repeated exposure to violence also significantly predicted higher levels of PTSD symptoms in Model D.

Insert [Figure 1] here.

Negative Influences of PTSD on Driver's Confidence

In order to test the long term effect of posttraumatic reactions to workplace violence among bus drivers, we entered the levels of acute stress disorder, and levels of PTSD at each time point to predict bus drivers' confidence in dealing with aggressive passengers 12 months after the index event. A multiple regression using SPSS version 22 was used for this analysis.

Gender was entered in the regression model as a control variable 4 . Results showed that even though levels of PTSD symptoms over time did not directly impact bus drivers' confidence in coping with aggressive passengers, bus drivers' immediate acute stress reaction within the first month following the index event significantly predicted lower levels of confidence in bus drivers 12 months after (see Table 4). Furthermore, the significant gender differences in the regression model indicated that female bus drivers were less confident in dealing with aggressive passengers than male bus drivers. These results indicated that workplace violence not only had immediate psychological effects on bus drivers but also had long term negative influences at work functioning. 4] here.

Insert [Table

Discussion

Workplace violence has detrimental effects on workers' mental health and well-being, and its devastating consequences on workers' lives can no longer be overlooked (e.g., [START_REF] Barling | Predicting Workplace Aggression and Violence[END_REF][START_REF] Banyard | The impact of interpersonal violence in adulthood on women's job satisfaction and productivity: The mediating roles of mental and physical health[END_REF][START_REF] Couto | Burnout, workplace violence and social support among drivers and conductors in the road passenger transport sector in Maputo City, Mozambique[END_REF][START_REF] Gates | Violence against nurses and its impact on stress and productivity[END_REF][START_REF] Gilbert | The nature of occupational violence against taxicab drivers: Violence against taxicab drivers[END_REF]Health and Safty Executive, 2015;[START_REF] Lanctôt | The aftermath of workplace violence among healthcare workers: A systematic literature review of the consequences[END_REF][START_REF] Lincoln | Moving violations: A study of incivility and violence against urban bus drivers in Australia[END_REF]Tse et al., 2006). This study used a longitudinal approach to examine the psychological and work consequences of workplace violence among a group of bus drivers. Our results showed that bus drivers experienced a variety of terrifying workplace violence ranging from frightful verbal abuse from passengers to death threats at gun point.

Similar to previous research (e.g., [START_REF] Brunet | Lifetime exposure to traumatic events among a sample of city bus drivers[END_REF][START_REF] Glaso | Bus drivers' exposure to bullying at work: An occupation-specific approach[END_REF][START_REF] Obelenis | Working conditions and health of the employees of public bus and trolleybus transport in Lithuania[END_REF]Tse et al., 2006), workplace violence is a common phenomenon experienced by bus drivers in our sample.

Exposure to workplace violence can severely jeopardize bus drivers' mental health.

According to our analysis (Table 1), more than half of the bus drivers were experiencing acute stress disorder, and the majority of them (84.3%) went on to experience at least moderate levels of PTSD symptoms one month after. More strikingly, almost half of the bus we presented the most parsimonious model with gender as a control variable in this analysis.

drivers continued to suffer at least moderate levels of problems in PTSD 12 months after.

The maintenance of PTSD symptoms among bus drivers may be due to their continued exposure to a working environment (i.e., on the bus) where the original traumatic event took place [START_REF] Lerias | Vicarious traumatization: symptoms and predictors[END_REF]. The familiar workplace might serve as a vivid reminder of the violent event for the victims and prompt them to constantly relive the violence at work, which then contributes to the maintenance of PTSD over time.

Moreover, repeated exposure to violence after the the original index event could be related not only to the maintenance of PTSD symptoms, but also to the development of the delayed onset of PTSD. In our sample, there were 10 drivers who met the full PTSD diagnosis 6 months after the index event. A post hoc examination of these 10 bus drivers showed that six of them had been exposed to violent events again after the original index event. Since our participants all had prior exposure to violence at work, they might have a recrudescence of PTSD symptoms triggered by the index event or by the re-exposure to violence after the index event [START_REF] Adams | Predictors of PTSD and delayed PTSD after disaster[END_REF][START_REF] Yehuda | Post-traumatic stress disorder[END_REF]. The high possibility of encountering violence at work could place bus drivers in extreme risks of developing and maintaining PTSD symptoms in this occupation.

Social support is often considered a protective factor for PTSD after a traumatic event [START_REF] Adams | Predictors of PTSD and delayed PTSD after disaster[END_REF][START_REF] Andrews | Gender, social support, and PTSD in victims of violent crime[END_REF][START_REF] Avery | Impact of combat and social support on PTSD and alcohol consumption in OEF/OIF veterans[END_REF][START_REF] Brewin | Meta-analysis of risk factors for posttraumatic stress disorder in trauma-exposed adults[END_REF][START_REF] Crevier | Overt social support behaviors: Associations with ptsd, concurrent depressive symptoms and gender[END_REF][START_REF] Laffaye | Relationships among PTSD symptoms, social support, and support source in veterans with chronic PTSD[END_REF]. We contended that while supportive behaviours could help alleviating symptoms of PTSD, counter-supportive behaviours could have adverse effects on PTSD symptoms [START_REF] Clapp | Understanding the relationship between PTSD and social support: The role of negative network orientation[END_REF][START_REF] Guay | Exploring the links between posttraumatic stress disorder and social support: Processes and potential research avenues[END_REF][START_REF] Guay | How do social interactions with a significant other affect PTSD symptoms? An empirical investigation with a clinical sample[END_REF][START_REF] Lakey | Negative social interactions: Assessment and relations to social support, cognition, and psychological distress[END_REF]. According to our analyses, supportive behaviours were not related to changes in PTSD symptoms over time, nor was it related to levels of PTSD symptoms at each assessment. It is possible that the presence of supportive behaviours per se does not directly contribute to the decline of PTSD symptoms, but the complete lack of support or complete social isolation might have more detrimental effects on PTSD (e.g., [START_REF] Adams | Predictors of PTSD and delayed PTSD after disaster[END_REF][START_REF] Avery | Impact of combat and social support on PTSD and alcohol consumption in OEF/OIF veterans[END_REF][START_REF] Brewin | Meta-analysis of risk factors for posttraumatic stress disorder in trauma-exposed adults[END_REF].

In contrast, counter-supportive behaviours have significant impact on PTSD according to our analysis. Even though levels of PTSD symptoms decreased over time in our study, individuals who reported high levels of perceived counter-supportive behaviours rated higher on their PTSD symptoms at each assessment than those who reported low levels of counter-supportive behaviours. This adverse impact of counter-supportive behaviours on PTSD faded away over a one-year span. People who received high levels of counter-supportive behaviours from significant others initially would reach the same levels of PTSD symptoms one year after as those who received low levels of counter-supportive behaviours initially. In other words, the presence of counter-supportive behaviours may have contributed to the persistence of PTSD symptoms during the first year after the violent event.

PTSD and workplace violence have great impact on individuals' psychological well-being. Epidemiological studies showed that men were more reluctant to seek help in times of duress. Support from family members were more important to men's than to women's well-being because women were able to draw multiple sources of support (e.g., friends, families, religious affiliations, etc.; [START_REF] Cable | Friends are equally important to men and women, but family matters more for men's well-being[END_REF][START_REF] Galdas | Men and health help-seeking behaviour: literature review[END_REF]. It is crucial for victims of workplace violence, especially for men, to draw on support from multiple sources and engage in direct support seeking [START_REF] Barbee | Effects of gender role expectations on the social support process[END_REF] indicating clearly what kind of support is needed in order to avoid the mismatch between support seeking and support provision [START_REF] Cutrona | Optimally matching support and perceived spousal sensitivity[END_REF].

Violence against bus drivers not only can lead to serious psychological distress, but it can also negatively influence bus drivers' confidence in coping with passenger aggression one year after, especially among female drivers. This study provided evidence for the long term effect of psychological distress on bus drivers' confidence in coping with aggressive passengers. After being exposed to a trauma, it is common for victims to experience a wide range of negative emotions, such as fear and anxiety, especially among female victims (e.g., [START_REF] Bryant | Gender differences in the relationship between acute stress disorder and posttraumatic stress disorder following motor vehicle accidents[END_REF]. These negative emotions could hinder bus drivers' confidence in dealing with aggressive passengers effectively due to their reduced awareness and intensified fear of the surroundings, as well as heightened startle responses (Work Safe BC, n.d.). Even though there was no evidence suggesting gender differences on acute stress disorder or PTSD (except moderate levels of PTSD) in our sample, the significant gender effect on bus drivers' confidence in coping with aggressive passengers one year later may point to an indirect effect of PTSD on work consequences among bus drivers. In other words, gender differences of work consequences (e.g., confidence in coping with aggressive passengers) after a traumatic event might be transmitted through the effect of PTSD symptoms. Given the job characteristics of bus drivers (e.g., working along and highly stressful; G. W. Evans & Johansson, 1998), their ability and confidence to cope with aggressive passengers are not only related to bus drivers' own safety but also other passengers and road users' safety. Therefore, it is essential to conduct more longitudinal studies to provide further comprehensive evidence of long term psychological and work related consequences of workplace violence among bus drivers.

Conclusions, Limitations and Implications

Our findings in this study supported previous research that workplace violence against bus drivers is not only a common phenomenon but also a repeated phenomenon [START_REF] Boyer | Prévalence du trouble de stress post-traumatique chez des chauffeurs d'autobus [Prevalence of post-traumatic stress disorder in bus drivers[END_REF][START_REF] Brunet | Lifetime exposure to traumatic events among a sample of city bus drivers[END_REF][START_REF] Couto | Burnout, workplace violence and social support among drivers and conductors in the road passenger transport sector in Maputo City, Mozambique[END_REF][START_REF] Glaso | Bus drivers' exposure to bullying at work: An occupation-specific approach[END_REF][START_REF] Lincoln | Moving violations: A study of incivility and violence against urban bus drivers in Australia[END_REF]Smith & Cornish, 2006;Tse et al., 2006), and it had short term and long term effects on individuals' mental health. It is important to have some social support after the traumatic event, but having low levels of counter-supportive interactions with significant others is even more potent for the well-being of the victims. A subset of participants (9.2%)

shows a delayed onset of PTSD 6 months after, which provides empirical evidence of the changing nature of PTSD symptoms among bus drivers who are victims of workplace violence. Without the longitudinal design of the study, this interesting finding of delayed onset of PTSD would have been lost in a cross-sectional design.

Due to collinearity issues among supportive versus counter-supportive behaviours, it was not possible to analyze these two aspects of social support simultaneously in the same analytical model in the current study. Moreover, all participants in this study were able to identify a confidant from whom they have been receiving support since the index event. As a result, the measure for social support used in this study was not able to capture someone who had no social support available to them at all (i.e., complete lack of social support).

Furthermore, we did not distinguish sources of support in this study, such as spousal support versus friend support. [START_REF] Laffaye | Relationships among PTSD symptoms, social support, and support source in veterans with chronic PTSD[END_REF] examined different sources of support among a group of male veterans with PTSD and found that veteran peer support was the most important, stress-free and highly valued component of the support resources for veterans, in comparison to family or non-veteran peer support. Our null finding of the positive social support on changes of PTSD symptoms could attest to the importance of considering social support and PTSD symptoms according to the sources of the support provider in future studies.

Gender differences on PTSD were not observed in our sample, which is seemingly contradictory with previous research on gender differences of PTSD (e.g., [START_REF] Andrews | Gender, social support, and PTSD in victims of violent crime[END_REF][START_REF] Bryant | Gender differences in the relationship between acute stress disorder and posttraumatic stress disorder following motor vehicle accidents[END_REF][START_REF] Olff | Gender differences in posttraumatic stress disorder[END_REF]Van Ameringen et al., 2008). However, the lack of gender differences on PTSD have been observed among traditionally male dominant professions, such as military or law enforcement (e.g., [START_REF] Brewin | Meta-analysis of risk factors for posttraumatic stress disorder in trauma-exposed adults[END_REF][START_REF] Lilly | Gender and PTSD: What can we learn from female police officers[END_REF][START_REF] Pole | Effects of gender and ethnicity on duty-related posttraumatic stress symptoms among urban police officers[END_REF]. The organizational culture that is traditionally male oriented might encourage female bus drivers to adopt a more masculine gender role and to minimize their emotional reactions, which then may have contributed to the reduced PTSD symptoms among female drivers. With the implementation and enforcement of equal employment opportunity policy in recent years, there will be more and more female employees entering traditionally male-dominant occupations, and vice versa. In order to thoroughly understand the reasons for gender differences in PTSD related to workplace violence, future studies should consider comparing levels of PTSD symptoms among participants who are working in traditionally gender specific occupations, such as male nurses and female bus drivers. Cross comparisons between male drivers versus male nurses or female drivers versus female nurses will prove fruitful in separating the gender role effects from biological sex effects on PTSD 5 .

Finally, according to our results, passenger initiated workplace violence appears to be very common among bus drivers. It is important for organizations to regularly analyze and survey employees' knowledge regarding workplace violence policies in the organization [START_REF] Dillon | Workplace violence: Impact, causes, and prevention[END_REF] in order to create a safe and open environment for victims to use appropriate services effectively to overcome their psychological and work difficulties after encountering workplace violence. Trainees and students from high risk professions, such as bus drivers, police officers and healthcare workers, should learn about workplace violence and related policies in their work sector before they enter the profession and during orientation training.

5 The participating public transit company has less than 25% female employees (including bus drivers, administrative staff, office workers and technicians, etc.) according to their latest public record, which may be considered a limitation of the current study. However, the gender distribution of bus drivers in this company was not available to the researchers at the time the study was conducted. With the company's effort to implement equal employment opportunity policy, the gender distribution of bus drivers could be changing drastically with more and more female bus drivers being presence in this occupation. Therefore, more research focusing on gender differences among bus drivers will be crucial to reflect a growing population of female bus drivers. T1 32.38(6.62) .17

  Road: A Longitudinal Examination of Passenger Initiated Violence Against Bus Drivers On March 23rd, 2015 the Department of Justice Canada joined by the Canadian Senator Bob Runciman to celebrate Bill S-221, which amends the Criminal Code of Canada

  enhancement. Professional Psychology:Research and Practice, 18 (1), 57-60. doi: 10.1037/0735-7028.18.1.57 The Department of Justice Canada.(2015). Government of Canada celebrates legislation to ensure the safety and security of hardworking public transit operators [Press Release]. Retrieved from http://news.gc.ca/web/article-en.do?nid=953849 Tolin, D. F., & Foa, E. B. (2006). Sex differences in trauma and posttraumatic stress disorder: A quantitative review of 25 years of research. Psychological Bulletin, 132 (6), 959-992. doi: 10.1037/0033-2909.132.6.959 Tse, J. L., Flin, R., & Mearns, K. (2006). Bus driver well-being review: 50 years of research. Transportation Research Part F: Traffic Psychology and Behaviour, 9 (2), 89-114. doi: 10.1016/j.trf.2005.10.002 Van Ameringen, M., Mancini, C., Patterson, B., & Boyle, M. H. (2008). Post-traumatic stress disorder in Canada. CNS Neuroscience & Therapeutics, 14 (3), 171-181. doi: 10.1111/j.1755-5949.2008.00049.x Vedantham, K., Brunet, A., Boyer, R., Weiss, D., Metzler, T., & Marmar, C. (2001). Posttraumatic stress disorder, trauma exposure, and the current health of canadian bus drivers. Canadian Journal of Psychiatry, 46 (2), 149-155. doi: 10.1177/070674370104600206 Verhaeghe, S., Duprez, V., Beeckman, D., Leys, J., Meijel, B. V., & Hecke, A. V. (2014).Mental health nurses' attitudes and perceived self-efficacy toward inpatient aggression:A cross-sectional study of associations with nurse-related characteristics. Perspect Psychiatr Care, 52 (1),12-24. doi: 10.1111/ppc.12097 Weathers, F., Litz, B., Herman, D., Huska, J., & Keane, T. (1993). The PTSD checklist (PCL): Reliability, validity, and diagnostic utility. San Antonio, TX. Wickham, R. E., & Knee, C. R. (2012). Interdependence theory and the actor-partner interdependence model: Where theory and method converge.Personality and Social Psychology Review, 16 (4), 375-393. doi: 10.1177/1088868312447897

.

Figure 1

 1 Figure 1 . Counter-Supportive Behaviours.High Counter-Supportive Behaviours, Low Counter-Supportive Behaviours.

Table 1

 1 Participants meeting PTSD Diagnostics at Each Time Assessment

	Time 1 Time 2	Time 3	Time 4

Table 2

 2 Correlations and Descriptives Among All Variables

PTSD Checklist for DSM-5 (PCL-5) was not used in this study because PCL-5 was not published when we first launched our study. In order to have consistent measures throughout our study, we have decided to keep using PCL even after PCL-5 became available.

Full PTSD diagnosis was given when participant had endorsed at least 1 item in cluster B, at least

items in cluster C, at least 2 items in cluster D, and obtained a total PCL score greater than

A response greater than 2 was considered an endorsement towards an item(Weathers et al., 1993).

Only random intercept models were presented here in this study because they were the most parsimonious models in comparison with random slope models. The same sets of analyses were conducted with random slopes entered in Models B, C and D. No significant changes of the results were obtained except the goodness of fit decreased in the random slope models.

During a preliminary analyses, all demographic variables were entered in the regression model. However, all the demographic variables were not significant, and did not improve the model fit, except gender. Therefore,

This project was supported in part by a team grant from the Canadian Institutes of Health Research (TVG 89969). The authors would also like to thank Charles-Edouard Giguere for his insightful comments and suggestions on our statistical analyses in this project.


Rukhsan Ul Haq

Renormalization group study of the interplay between Majorana-Kramers pair and Kondo effect in normal lead-Quantum Dot-Superconductor junctions

In this paper, we study a set-up in which quantum dot is tunnel-coupled to Majorana-Kramers pair(MKP) which are the edge modes of time reversal invariant topological superconductor(TRITOPS), which is an experimental realization time reversal invariant Kitaev chain model.

Our motivation is to study the effect of Majorana-Kramers pair on Kondo fixed point and conversely the effect of Kondo interaction on Majorana-Kramers pair. This way we can capture the signatures of Majorana fermions on Kondo effect in quantum dots which can be probed feasibly in experiemts. Employing the renormalization group study we report the first study of the signatures of Majorana-Kramers pair in the dynamical spin susceptibility of the quantum dot electrons. We not only calculate the flow equations for coupling constants but also study the renormalization of observables and especially calculate dynamical susceptibility. We propose that since these dynamical quantities have signatures of Majorana-Kramers pair, so they provide an unambiguous experimental probe for detection of Majorana fermions in TRITOPS.

INTRODUCTION

Majorana fermions(Majorana zero modes) have recently attracted a lot of attention in condensed matter physics community due to their special topological properties which make them a promising candidate for quantum computing. Majorana fermions make topological qubits which can store the information non-locally and hence are immune to local perturbations. Their topological protection comes from Z 2 parity symmetry. Majorana fermions are Ising anyons and obey non-abelian statistics. Though it was known that Majorana fermions exist in topological superconductors [1][2] but after Kitaev [3] introduced a simple Hamiltonian and it was realized that it could be experimentally realized [4], it gave a boost to the research on Majorana fermions. So the search for Majorana fermions in solid state systems started. Since as of now there is no conclusive evidence for the topological superconductivity in nature, it was realized that we could engineer systems which harbour Majorana fermions. The recent successes in the detection and control of the Majorana fermions [5][6] [7] have created a lot of hope about Majorana fermion based quantum computing [8]. However, the experimental detection of Majorana fermions is based on spectroscopic signatures, and there are physical effects which contribute to the zero-bias spectral peaks including Kondo impurities and disorder. So the search for unambiguous signatures of Majorana fermions is still on. In [11] it was proposed that Kondo effect can be used for the detection of Majorana fermions. Kondo effect is known to have a high level of experimental tunability in quantum dots. So quantum dot systems offer a very feasible and controllable way for the detection of Majorana fermions.There are other reasons also to consider a setup where Majorana fermions are coupled to quantum dots. It was shown in [9] [10] that Majorana fermions tunnel-coupled to conduction leads do not preserve the information in qubit and also parity time of Majorana fermions can not be determined in those experiments. They proposed an experimental setup in which Majorana fermions are coupled to quantum dot which can be coupled to normal lead(s) as well. To explore the signatures of Majorana fermions in Kondo effect and hence the interplay between Majorana fermions and Kondo effect, there have been recent theoretical studies [11] [12][13] [14]. In [11] poor man's scaling was applied to study the stability of Kondo effect. Though they find that Majorana fermions have a drastic effect on Kondo effect, they have got the same scaling equations as for the standard Kondo model. In [12] based on numerical renormalization group(NRG) calculations, it has been found that there is Majorana fermion induced Zeeman field which shifts the spectral function of one species of fermions and also there is the contribution of Majorana fermions to differential conductance. In [14] similar results have been obtained using recursive Greens function method. However, none of these studies has found a new fixed point arising due to the Majorana fermions. In [13] based on perturbative renormalization, slave boson mean field theory and density matrix renormalization group calculations, the authors have come to the conclusion that in the strong Majorana-Dot coupling regime, physics of the system is governed by a new fixed point rather than Kondo fixed point. The signatures of the new fixed point can be found in spin susceptibility, and they find that, quite unlike Kondo fixed point, spin susceptibility is dependent on gate-voltage(particle-hole asymmetry). Though the interplay between Majorana fermions and Kondo effect has been well studied now and essential aspects of the ensuing physics are well-understod, the similar studies of the time reveresal symmetric version of Majorana fermions called Majorana-Kramers pair are still lacking. In [16], based on perturbative renormalization method and slave boson mean field theory, authors have concluded that in the regime of the strong coupling to MKP, Kondo fixed point becomes unstable and low energy physics is governed by Andreev fixed point.

In this paper, we have employed flow equation renormalization method to confirm this new fixed point. Flow equation method has recently been extensively applied to study Kondo models both in equilibrium and in non-equilibrium [START_REF] Kehrein | The flow equation approach to many particle systems[END_REF]. Flow equation method is one of the few renormalization methods which captures analytically,the full cross-over from weak coupling to strong coupling regimes of the Kondo(and Anderson impurity) model. Combined with bosonization, flow equation method gives access to the Tolouse point of the Kondo model where the flow equation can bee solved analytically to get closed form solution. We have calculated the flow equatios for the parameters of the effective Hamiltonian and solved them numerically. Ours is the first study of the calculating the renormalization effects of MKP on dynamical spin susceptibility and hence we propose that these signatures of MKP in spin response should be feasibly probed in experiments on quantum dots.

The rest of this paper is organized in following way. First, we consider the Hamiltonian for the system in which a quantum dot is coupled to a normal lead on one side and is sidecoupled to TRITOPS. In the long chain limit, there is only one Majorana-Kramers pair in the topological superconductor. We calculate the effective Hamiltonian, Majorana-Kondo model, which has both Kondo interaction and terms arising from the coupling to Majorana fermion. To find the stability of Kondo fixed point in the presence of Majorana fermions, we calculate the flow equations for the couplings constants of the Majorana-Kramers-Kondo model. We solve the flow equations numerically. Then we calculate the flow equations of Kondo spin observable and solving the flow equations numerically, we calculate dynamic spin susceptibility, and we find clear signatures of MKP in this key observable quantity. Once again we explore both the cases, with particle-hole symmetry being present and absent.

Finally, we summarize our conclusions and discuss the interplay between Majorana fermions and Kondo effect and hence point to the unambiguous ways for the detection of Majorana-Kramers pair.

HAMILTONIAN FOR NORMAL LEAD-QUANTUM DOT-TRITOPS SYSTEM

We consider a set-up in which a quantum dot is connected to a topological superconductor on one side and to a normal lead on another side. Topological superconductor is an experimental realization of Kitaev chain model and hence has two Majorana fermions on the edges of the chain. We work in the long chain limit and couple only one of the Majorana fermions to the quantum dot. The Hamiltonian for this system is:

H = kσ k c † kσ c kσ + σ d d † σ d σ + U n d↑ n d↓ + kσ t σ (c † kσ d σ + h.c.) + σ iλ σ γ(d σ + d † σ ) (1)
The first four terms constitute the Anderson impurity model for quantum dot connected to a normal lead. The last term represents the coupling of Majorana Kramers pair(MKP) with quantum dot electron, λ being the coupling strength. We have taken general coupling, but later on, we will see that Majorana fermion gets coupled to only one species of electrons.

Due to the time reversal symmetry(TRS), t ↑ = t ↓ = t and λ ↑ = -λ ↓ = λ where t and λ are taken to be real.

Like Anderson impurity model, our Hamiltonian has many parameter regimes but to study the interplay between Majorana fermions and Kondo effect; we need to project our Hamiltonian into its Kondo regime. We will use the projection operator method to calculate the effective Hamiltonian in the Kondo regime of our model. Effective Hamiltonian in singly occupied space is given by

H ef f = H 11 + H 10 1 E -H 00 H 01 + H 12 1 E -H 22 H 21 (2)
To get the effective Hamiltonian we need to calculate the projected Hamiltonian terms H 12 and H 10 . Since there are only two tunneling terms so only these will contribute to off-diagonal part of the effective Hamiltonian.

H 12 = σ (tc † kσ + iλ σ γ)d σ n dσ H 21 = H † 12 (3)
H 01 = σ (tc † kσ + iλ σ γ)d σ (1 -n dσ ) H 10 = H † 01 (4)
To calculate the effective Hamiltonian, we need to evaluate following two terms.

H 12 1 E -H 22 H 21 = σ tc † kσ d σ n dσ 1 E -H 22 k σ td † σ c k σ n d σ + kσ tc † kσ d σ n dσ 1 E -H 22 k σ iλ σ γd † σ c k σ n d σ + kσ iλ σ γd σ n dσ 1 E -H 22 k σ it σ d † σ c k σ n d σ + kσ iλ σ γd σ n dσ 1 E -H 22 k σ iλ σ d † σ n d σ (5)
H 10 1 E -H 00 H 01 = kσ k σ td † σ c kσ (1 -n dσ ) 1 E -H 00 tc † k σ d σ (1 -n d σ ) + kσ k σ td † σ c kσ (1 -n dσ ) 1 E -H 00 iλ σ γd σ (1 -n d σ ) + kσ k σ iλ σ γd † σ c kσ (1 -n dσ ) 1 E -H 00 tc † k σ d σ (1 -n d σ ) + kσ k σ iλ σ γd † σ c kσ (1 -n dσ ) 1 E -H 00 iλ σ γd σ (1 -n d σ ) (6)
Using the Abrikosov's fermion representation of spin operators, we can write the effective Hamiltonian as(also calculated in [12][13]):

H ef f = kσ k c † kσ c kσ + kk J(k, k )S.s kk -J 0 (k)iγ ↑ γ ↓ S y + k J 1 (k)i(γ ↑ (c k↑ + c † k↑ ) -γ ↓ (c k↓ + c † k↓ )) + k J 2 (k)i(γ ↑ (c k↑ + c † k↑ ) + γ ↓ (c k↓ + c † k↓ ))S z + k J 3 (k)i(γ ↑ (c † k↓ S + + c k↓ S --γ ↓ (c k↑ S † ) + c † k↑ S -)) (7)
where

J(k, k ) = t 2 ζ 1+ , J 0 = λ 2 ζ 1+ , J 1 = tλζ -, J 2 = J 3 = tλζ 2+ . ζ 1+ = 1 d + U -k + 1 k -d + 1 d + U -k + 1 k -d (8)
ζ 2+ = 1 d + U -k + 1 k -d (9)
ζ -= 1 d + U -m - 1 m -d (10)

Majorana-Kramers-Kondo Model

The effective Hamiltonian that we have obtained has terms arising from the Quantum Dot-Majorana coupling. We call this model Majorana-Kondo model (MKKM) to differentiate it from topological Kondo model. In Majorana-Kramers-Kondo model there are four other terms in addition to the standard Kondo interaction term. 1. Andreev Scattering term. Since there is no direct tunneling between lead and Majorana so this term arises due to the virtual charge fluctuations. 2. In presence of time reversal symmetry Zeeman field like term is not present in this model, however there is a term which couples S y of the dot electrons with S y m which is constututed from Majorana-Kramers pair. This terms makes this model different from the Majorana-Kondo model [15][12] [13]. Then there are two more terms in which impurity spin couples both to Majorana and lead electrons.

Unlike the Majorana-Kondo model, where the particle-hole asymmetry is a relevant perturbation, in MKKM, the corresponding term is present both at particle-hole symmetry and away from it.

FLOW EQUATIONS FOR MAJORANA-KRAMERS-KONDO MODEL

In this section, we will apply the flow equation renormalization group method to Majorana-Kramers-Kondo model. We will calculate the flow equations for the parameters of the model. In the case of Majorana-Kramers-Kondo model, our interest is to explore the interplay between the Kondo effect and Majorana-Kramers pair, so we will find out what happens to Kondo divergence(scale) in the presence of MKP. We will increase the QD-TSC coupling λ and find out whether Kondo effect survives in the strong λ regime. To calculate the flow equation for our model, we first need to calculate the generator. Generator for the flow equations for Majorana-Kramers Kondo model is given below:

η = 1 2 pq ( p -q )(J ↑ (p, q) : c † p↑ c q↑ : -J ↓ (p, q) : c † p↓ c q↓ :)S z + pq ( p -q )J ⊥ (p, q)(: c † p↑ c q↓ : S --: c † q↓ c p↑ : S + ) + p p J 1 (p)i(γ ↑ (c † p↑ -c p↑ ) -γ ↓ (c † p↓ -c p↓ )) + p p J 2 (p)i(γ ↑ (c † p↑ -c p↑ + γ ↓ (c † k↓ -c k↓ )S z + p ( p )J 3 (p)i(γ ↑ (c † p↓ S † ) -c p↓ S -γ ↓ (c † k↑ S --c k↑ S † )) (11)
η = η K + η ⊥ K + η 1 M + η 2 M + η 3 M (12)
In equation 12 flow equation generator has been written as sum of generators corresponding to different terms in the interaction of the Hamiltonian. To calculate the flow equations we need to evaluate the commutators of the generators with the full Hamiltonian. Putting all the commutators together and comparing with the original Hamiltonian we obtain the flow equations for the MKKM.

For spin-up Kondo coupling, the flow equation is

dJ ↑ (p, q) dl = -( p -q ) 2 J ↑ (p, q)+ 1 2 s 2( s ) -( p + q ))J ⊥ (p, s)J ⊥ (q, s)(1 -2n(s) (13)
Similarly for spin down Kondo Coupling, flow equation is

dJ ↓ (p, q) dl = -( p -q ) 2 J ↓ (p, q)+ 1 2 s 2( s ) -( p + q ))J ⊥ (s, p)J ⊥ (s, q)(1 -2n(s) (14)
Flow equation for transverse Kondo coupling is

dJ ⊥ (p, q) dl = -( p -q ) 2 J ⊥ (p, q)+ 1 4 s (1 -2n f (s)) (2 s -( p + q ))J ⊥ (s, q)J ↑ (p, s)) + (2 s -( p + q ))(J ⊥ (p, s))J ↓ (q, s)) (15)
The flow equation for J 0 coupling constant is:

dJ 0 dl = 2 p p J 3 (p)J 1 (p) (16)
Flow equation for Andreev scattering term is

dJ 1 (p) dl = -p J 1 (p) + 1 4 q ( p -q )J ↑ (p, q)J 2 (q) + 1 2 q ( p -q )J 3 (q)J ⊥ (p, q) - 1 2 q ( q )J 3 (q)J ⊥ (p, q)) - 1 4 q q J 2 (q)J ↑ (p, q) (17)
Flow equation for Majorana coupling J 2 is:

dJ 2 (p) dl = -p J 2 (p) + q ( p -q )J ↑ (p, q)J 1 (q) - q q J 1 (q)J ↑ (p, q) - q ( p -q )J 3 (q)J ⊥ (p, q)(1 -2n(q)) + q ( q )J 3 (q)J ⊥ (p, q)(1 -2n(q)) (18)
Flow equation for Majorana coupling J 3 is:

dJ 3 (p) dl = -p J 3 (p) - 1 2 q ( p -q )(1 -2n(q))(J ↓ (p, q)J 3 (q) - q ( q -p )J ⊥ (q, p)J 1 (q) + 1 2 q ( q -p )J ⊥ (q, p)J 2 (q)(1 -2n(q))+ - q q J 1 (q)J ⊥ (p, q) - 1 2 q q J 2 (q)J ⊥ (q, p)(1 -2n(q)) - 1 2 q ( q )J 3 (q)J ↓ (p, q)(1 -2n(q)) (19)

Numerical Solution of Flow Equations

Flow equations need to be solved numerically because they are non-linear coupled differential equations and analytical solution can be obtained only in some special limits like infrared limit where momentum dependence of coupling constants can be dropped off. In case of MKKM, since there is larger number of coupling constants so the system of differential equations becomes larger and even more coupled which increases the computational complexity of the problem. Also there are more energy scales in addition to the Kondo band, the dimension of systems of flow equations is of the order 10 4 . The computational ex-pense however scales as ON 3 . We have used DOPRI5 which is the fifth order Runga-Kutta method for solving ODEs. The method has proved stable for the flow equations except that due to the stiffness of the flow equations we often had to tweak the discretization of the flow parameter l grid so that the lowest energy scale gets resolved. Now we will present the numerical solutions of the flow equations. In Figure 1, J1 and J2, the flow of the J1 and J2 coupling constants has been plotted. As we can see that they grow very rapidly and show the logarithmic divergence. Similar behaviour is shown by J0 coupling constant as can been seen from the left panel of Figure 2. We can clearlu see that these Majorana induced coupling constants grow very fast as compared to the Kondo coupling constant, as shown in Figure 2(right panel). We can conclude that in this regime of the model, there is no Kondo divergence and it is the Majorana couplings which are relavtn for the low energy physics. This signifies that there may be a new fixed point which governes the low energy physics of the model as has been proposed by [16]. It also needs to be noted that though there is no Kondo divergence, but all the curves of the Kondo couplings collapse on the same curve as shown in Figure 1(right panel) and hence there is not strong renormalization of the Kondo coupling constant in this regime of the model. One main difference between flow equations and conventional renormalization methods is that in flow equations observables also undergo unitary flow and hence renormalization.

In the case of fermionic systems, fermion operators get transformed under unitary flow. The new fermion operators are then written as operator product expansion of fermion operators.

In the case of Kondo model, the spin operator gets transformed under unitary flow. To check the consistency, there are many checks which one can make using the properties of the unitary flow. Unitary flow keeps the operator algebra(commutation/anti-commutation) invariant. There are other sum rules also which can be used to check the consistency of the operator product expansion. There are physical plausibility arguments as well which can be used to make sense of these operator product expansions. In the case of Kondo model, it is known that as renormalization flow proceeds to strong coupling fixed point, the magnetic moment of Kondo impurity gets quenched. This physical process has an algebraic mapping in flow equation method as shown below. Using the transformed observables, we can calculate correlation and response functions for a given Hamiltonian.

In this section, we will calculate flow equations for the Kondo impurity spin and upon solving the flow equations, obtain the dynamic spin susceptibility for the Majorana-Kramers-Kondo model. This quantity will give us access to the signatures of Majorana-Kramers pair which can be detected easily in an experiment. Hence dynamic spin susceptibility is a very important quantity to understand the interplay between MKP and Kondo effect and hence for the detection of MKP.

We make following ansatz for the spin operator S z :

S z (l) =h z (l)S z + pq γ pq (l)(: c † p↑ c q↓ : S -+ : c † q↓ c p↑ : S + ) + p ζ p (l)i(γ ↑ (c k↑ + c † k↑ ) + γ ↓ (c k↓ + c † k↓ )S z )) + p η p (l)i(γ ↑ (c † k↓ S + + c k↓ S --γ ↓ (c k↑ S † ) + c † k↑ S -)) (20)
where h z (l = 0) = 1 and initial values of all other parameters are zero. For the Kondo model, the ansatz for the spin operator [START_REF] Kehrein | The flow equation approach to many particle systems[END_REF] is motivated by the fact that Kondo effect quenches the magnetic moment. So for isotropic Kondo model, the decomposition of the spin operator has only two terms referring to the Kondo spin and the Kondo interaction.

For the anisotropic case, since there is Zeeman field and hence magnetization also, so the ansatz for spin observable needs to include these terms as well [START_REF] Fritsch | [END_REF]. For the case of Majorana-Kramers-Kondo model, there are Majorana fermion induced couplings due to which Kondo impurity spin couples to Majorana fermion also and hence these interactions need to be included in the ansatz. Andreev scattering term does not couple to Kondo impurity spin and hence has not been included in the ansatz.

To calculate the flow equation of the spin operator , we will use one-loop generator as given in equation 11. .

dS z dl = [η(l), S z (l)] (21)
In the following we will present the flow equations for the various parameters in Equation 20.

Flow equation for h z co-efficient is:

dh z dl = - pq ( p -q )(J ⊥ (p, q)γ pq )(n(p) + n(q) -2n(p)n(q)) (22)
Flow equation for γ co-efficient is:

dγ pq dl = 1 2 r J ↑ (p, r)( r -p )γ rq (l) + J ↓ (r, q)( r -q )γ pr (l) (1 -2n(q)) + 1 2 ( p -q )J ⊥ (p, q)h z (l) (23)
Flow equation for ζ co-efficient is:

dζ p dl = q ( q )J 3 (q)γ pq (1 -2n(q)) (24) - q ( p -q )J ⊥ (p, q)η q (1 -2n(q)) (25)
Flow equation for η co-efficient is:

dη p (l) dl = -( p )J 3 (p)h z (l) - q q J 1 (q)γ qp (l) (26) - 1 2 q q J 2 (q)γ qp (l)(1 -2n(q)) + 1 2 q ( p -q )J ⊥ (p, q)ζ q (l)(1 -2n(q)) (27) - 1 2
q ( pq )J ↓ (p, q)η q (1 -2n(q)) (28)

Numerical solution

In this section we will solve the flow equations for the spin operator and calculate the dynamic spin susceptibility. For the Majorana-Kramers-Kondo model, spin-spin correlation function is given by

C(ω) = π 4 p γ2 p, p+ω n f ( p )(1 -n f ( p + ω)) + n f ( p + ω)(1 -n f ( p )) ( 29
)
This correlation is related to dynamical spin susceptibility via fluctuation-dissipation theorm [START_REF] Kehrein | The flow equation approach to many particle systems[END_REF]. The tilde sign denotes that γ co-efficient is in the limit l → ∞.

χ(ω) = tanh( ω 2T )C(ω) (30)
To compute dynamic susceptibility we need to solve the flow equations for the spin observable numerically. plotted dynamic spin susceptibility on frequency axis in Figure 3 . In left panel, we see the effect of increasing J 2 Majorana coupling. As we increase J 2 , spin susceptibility decreases because of the competition from MKP induced couplings which coupled spin to the parity of the MKP. In presence of time reversal symmetry, there is an additional pairing which couples S y of the dot electron spin to the S y m constituted by Majorana-Kramers pair and due to its transverse nature, it also gives competition to the Kondo effect and hence leads to the decrease in spin susceptibility.

In the right panel of Figure 3 we see the effect of increasing Kondo coupling constant on the spin susceptibility and we can clearly see spin susceptibility gets enhanced as we increase Kondo coupling constant. This should be contrasted with the effect seen on left panel of Figure 3 where we find that spin susceptibility decreases with increasing values of J2.

CONCLUSION

In this paper, we have done a detailed study of the interplay between Majorana-Kramers pair and Kondo effect considering an experimental set-up in which a quantum dot is coupled to a normal lead on one side and tunnel-coupled to a TRITOPS, which itself is an experimental realization of time reversal invariant Kitaev p-wave chain model. Our motivation comes from the recent surge in the research on Majorana fermions( and MKPs) as being the promising candidates for topological quantum computing. Hence searching for umambiguous ways of their detection has assumed utmost importance. Kondo effect in quantum dots offers a lot of tunability and hence very feasible way to detect Majorana fermions. However, to single out the signatures of Majorana fermions in Kondo physics, needs careful and detailed study. There are questions which need to be addressed to have conclusive evidence for the signatures of Majorana fermions. Firstly Kondo effect is associated with the strong coupling fixed point of Anderson impurity model and is very stable to perturbations and hence to show that Majorana fermions make Kondo fixed point unstable and lead to new fixed point, one has to take recourse to methods which can be relied upon in the strong coupling regime of the model. In our work, we have employed flow equation method which has already been applied to Kondo model to study its strong coupling physics both in equilibrium and non-equilibrium. Based on this method we did the renormalization group study of the Majorana-Kramers-Kondo model which is the effective Hamiltonian in the strong coupling regime of the Hamiltonian for NL-QD-TRITOPS system. We calculated the flow equations for the coupling constants of the MKKM, and after solving them numerically, we clearly found how MKP induced terms, cut off Kondo divergence. We find that MKP induced coupling constants grow faster as compared to the Kondo coupling constant and hence they are more relevant for the low energy physics. Our calculations suggest that there may be a new fixed point determined by MKP induced coupling constants. Dynamic spin susceptibility is an experimentally accessible quantity. We find clear signatures of MKP in this quantity and we propose these signatures can be feasibly found in experiments on quantum dots. Our studies have confirmed and consolidated the interplay between MKP and Kondo effect and hence showed that Kondo effect in quantum dots provides an experimentally feasible and unambiguous way for the detection of Majorana-Kramers pair.

  scale. Due to the interplay of various energy scales present in the system flow equations generally become stiff. For the numerical solution of the flow equation of Majorana-Kondo model we have taken conduction bath to have flat density of states. Conduction band has been discretized and typically we have kept 200 energy states and since number of flow equations to be solved scales as O(N 2 ) where N is the number of states kept in conduction

FIG. 1 .

 1 FIG. 1. In left panel, J1 coupling constant is plotted versus flow parameter. In right panel, J2 coupling constant is plotted versus flow parameter. Other parameters are: U= 6.0, d = -3.0, t= 4.0.

  To capture the effects of Majorana-Kramers pair in spin susceptibility we have Dynamical spin susceptibility plots for Majorana-Kramers-Kondo model : In left panel, spin susceptibility has been plotted on frequency axis for different J 2 coupling constants. In right panel, spin susceptibility has been plotted for increasing values of Kondo coupling constant.

	0.15 0.2				J 2 =0.01 J 2 =0.02 J 2 =0.03 J 2 =0.04 J 2 =0.05	0.25	J k =0.6 J k =0.65 J k =0.68 J k =0.70
								0.2	J k =0.73
								χ
	χ	0.1					0.15
								0.1
	0.05
								0.05
		0	0	0.2	0.4	0.6	0.8	1	0 0	0.2	0.4	0.6	0.8	1
					ω						ω
	FIG. 3.

ACKNOWLEDGEMENTS I would like to acknowledge Prof. N. S. Vidhyadhiraja for various discussions related to this work. I would like to acknowledge Department of Science and Technology(DST) for funding and JNCASR Bangalore for the facilities and vibrant research environment.