Victoria ; Fuste

Victòria Fusté

Marta Del Pino

Assumpta Pérez

Angel Garcia

Aureli Torne

Jaume Pahisa

M.D Jaume Ordi
email: jordi@clinic.ub.es

Detection, p16 INK4A Overexpression and Clinicopathological Correlations"

Keywords: Vagina, Squamous cell carcinoma, p53, p16, Human papilloma virus Vagina -Squamous cell carcinoma-p53-p16-human papilloma virus-HPV HPV, Human papillomavirus, SCCVa, squamous cell carcinoma of the vagina, VaIN, vaginal intraepithelial neoplasia, CIN cervical intraepithelial neoplasia, VIN vulvar intraepithelial neoplasia

Aim: To determine the role of HPV in the pathogenesis of primary squamous cell carcinoma of the vagina (SCCVa), and to evaluate its clinico-pathological significance.

Methods: All cases of SCCVa diagnosed over a 15 year period from two hospitals in Barcelona, Spain (n=32) were retrieved. Patients with a history of carcinoma of the cervix diagnosed less than 5 years before were excluded.

HPV was detected and typed by PCR using the SPF10 primers.

Immunohistochemistry for p16 and p53 was performed.

Results: HPV was detected in 25 cases (78.1%). HPV16 was the most prevalent type. Patients with HPV-positive tumours were frequently associated with a history of carcinoma or intraepithelial neoplasia of the cervix or vulva diagnosed more than 5 years before (56% vs. 0%; p=0.01). HPV-positive tumours were more frequently of non-keratinizing, basaloid or warty type than HPV-negative neoplasms (84% vs. 14.3%; p<0.001), and almost constantly showed diffuse positive staining for p16 INK4a (96%, vs. 14.3%; p<0.001). The sensitivity and specificity of p16 to identify HPV-positive tumours were 96% and 85.7% respectively.

Conclusions:

A high number of SCCVs are related to HPV infection and may be identified by immunostaining for p16. HPV-positive tumours tend to affect women with history of cervical neoplasm.

Introduction

Primary carcinoma of the vagina is a rare disease accounting for about 1-2% of all gynaecologic malignances. [1][2][3][4] Invasive squamous cell carcinoma (SCCVa) is the most frequent tumour representing 65-95% of all cases. 3,[5][6][7][START_REF] Pingley | Primary carcinoma of the vagina: Tata Memorial Hospital experience[END_REF][START_REF] Ptackova | Primary carcinoma of the vagina[END_REF][START_REF] Stock | A 30-year experience in the management of primary carcinoma of the vagina: analysis of prognostic factors and treatment modalities[END_REF] Despite all therapeutic advances, the overall 5-year survival rate is still poor. 1,3,[START_REF] Beller | Carcinoma of the vagina. FIGO 6th Annual Report on the Results of Treatment in Gynecological Cancer[END_REF] Human papillomavirus (HPV) has been clearly implicated in the pathogenesis of malignant tumours of the female lower genital tract. HPV has been detected in almost 100% of cervical carcinomas. [START_REF] Bosch | The causal relation between human papillomavirus and cervical cancer[END_REF][START_REF] Ferreira | HPV DNA detection and genotyping in 21 cases of primary invasive squamous cell carcinoma of the vagina[END_REF][START_REF] Munoz | Epidemiologic classification of human papillomavirus types associated with cervical cancer[END_REF][START_REF] Walboomers | Human papillomavirus is a necessary cause of invasive cervical cancer worldwide[END_REF] In vulvar carcinoma HPV has been identified in 20-35% of all cases, being detected in 75-100% of basaloid and warty carcinomas and in less than 20% of keratinizing squamous cell carcinomas. [16][17][18][19][20] Because of the low frequency of SCCVa, few series have been published focusing on this tumour, and consequently there is scant information on the implication of HPV infection with discrepant results in the published series. There is also limited information on whether there are pathological characteristics associated to HPV-positive tumours and the clinical consequences of HPV infection. Moreover, it has not been determined whether p16 INK4A immunostaining, which in several locations such as the vulva or the head and neck region 18,21 has been shown to be a reliable marker for HPVassociated tumours can also be a useful marker of HPV infection in vaginal neoplasms.

Finally, it has been shown that about 30% of patients with SCCVa have a history of prior dysplasia or carcinoma of the uterine cervix. 3,5,18,[21][22][23][START_REF] Murad | The pathologic behavior of primary vaginal carcinoma and its relationship to cervical cancer[END_REF] A 5 year free period is generally considered the distinction between a recurrent cervical carcinoma and a new primary vaginal carcinoma, as 95% of the recurrences of cervical carcinoma occur within five years. [START_REF] Hellman | Primary carcinoma of the vagina: factors influencing the age at diagnosis. The Radiumhemmet series 1956-96[END_REF][START_REF] Tabata | Treatment failure in vaginal cancer[END_REF] Nevertheless, there is a general concern about this threshold and several authors have expanded that period to 10 years 6 or have even excluded all patients with any previous history of cervical pathology. [START_REF] Ferreira | HPV DNA detection and genotyping in 21 cases of primary invasive squamous cell carcinoma of the vagina[END_REF] Thus, it is not clear whether patients with previous cervical lesions diagnosed more than 5 years before should or should not be considered as primary SCCVa.

Our study was aimed to evaluate the role of HPV in the pathogenesis of SCCVa with its possible clinico-pathological implications, to analyze the usefulness of p16 INK4A as a marker for HPV-associated SCCVa, determining whether it may improve the results of pure histological classification or not, and to evaluate the clinical impact on previous history of cervical or vulvar intraepithelial or invasive neoplasm.

Materials and methods

CASE SELECTION

All cases of invasive squamous cell carcinoma of the vagina diagnosed in a period of 15 years (1994-2008) were retrieved from the surgical pathology files of two tertiary institutions in Barcelona, Spain, the Hospital Clinic and the Hospital Vall d'Hebron. Thus, 32 cases were included in the study.

All tissues had been routinely fixed in 4% buffered neutral formalin and embedded in paraffin. In all cases, both haematoxylin and eosin (H&E) glass slides and paraffin blocks were available for review. In 17 patients (53.1%) a colpectomy or a wide resection was performed. In 15 patients (46.9%) only a small biopsy was available. The clinical charts were reviewed in all cases. The medical records were abstracted to obtain information on patient's age, clinical

HPV DETECTION AND TYPING

A broad-spectrum HPV DNA amplification was performed using the short PCR fragment (SPF10) primer set (Innogenetics Diagnostica, Spain) as previously described. 18 The SPF10 primers amplify a 65-bp fragment from the L1 region of the HPV genome. PCRs were performed in 50 µL reaction following standard conditions. For HPV amplification, a 9 minute denaturation step at 94ºC was followed by 40 cycles of amplification using 1.5 I.U. DNA polymerase (Amplitaq Gold DNA polymerase, Applied Biosystems) with a thermocycler (PTC-200, Peltier Thermal Cycler, MJ Research, Waltham, MA). Each cycle included a denaturation step at 94ºC for 30 seconds, a primer annealing step at 52ºC for 45 seconds, and a chain elongation step at 72ºC for 45 seconds. The final elongation step was prolonged by 5 minutes. The PCR products were subjected to electrophoresis in a 3% agarose gel and the 65-bp product was visualized with ethidium bromide staining. Additional confirmation of the presence of amplified HPV-specific sequences was performed using HPV DNA enzyme immunoassay, a microliter plate-based hybridization assay. All samples were then incubated with the INNO-LIPA HPV research assay (15). Twenty-five individual HPV genotypes (6, 11, 16, 18, 31, 33, 34, 35, 39, 40, 42, 43, 45, 51, 52, 53, 54, 56, 58, 59, 66, 68, 70 and 74) can be identified simultaneously in a single assay. In this assay 10 µL of denatured HPV PCR product was hybridized to the genotype-specific probes immobilized as parallel lines on a nitrocellulose strip. After the washing step, the products of hybridization were detected by a colour reaction with alkaline-phosphatase-streptavidin conjugate and substrate

(5-bromo-4-chloro-3-indolyl phosphate and nitroblue tetrazolium), which resulted To confirm DNA amplification a second PCR was conducted with primers βglobin-PCO3 and β-globin-PCO5. In each series of cases we included a positive control (squamous cell carcinoma of the cervix known to harbour HPV16), as well as a negative control (distilled water).

IMMUNOHISTOCHEMISTRY

Immunohistochemical studies were done with the automated immunohistochemical system Autostainer Link 48® (Dako Co, Carpinteria, CA, USA). p16 INK4a was detected using the CINtec histology kit (clone E6H4, mtm Laboratories, Heidelberg, Germany), following the manufacturer's protocol. In each series a negative control consisting of a normal cervix, as well as a positive control consisting of a CIN3, were included.

The evaluation of the p16 INK4A immunostained slides was performed on 3 different staining patterns: negative, focal, or diffuse staining. Negative staining was defined as non-immunoreactive; focal staining was defined as non continuous staining of isolated cells or small cell clusters. Diffuse staining was defined as a continuous staining of cells of the basal and parabasal layers, independently of the positive or negative staining in the centre of the tumour nests. p16 INK4A expression was nuclear, as well as cytoplasmic. Only cases with diffuse staining were considered as positive for analytical purposes. 23 p53 was detected with the monoclonal antibody DO-7 (Dako). In each series a positive control consisting of an endometrial serous papillary carcinoma for p53 was included. p53 immunostaining was scored as negative when <1% of cells were positive, (+) when less than 5% were positive, (++) when less that 25% of cells were positive and (+++) when more than 25% of positive cells were observed. 18 STATISTICAL ANALYSIS Data was analyzed with the program SPSS (Version 15.0, SPSS Inc, Chicago, IL, USA).Chi-square analysis or Fisher's exact test were used for comparisons between qualitative variables and Student's t test and ANOVA were applied for quantitative variables according to application conditions. One of them was negative for p16 INK4a . In this case a HPV52 was repeatedly identified. In the HPV-negative group only one case showed intense and diffuse positivity for p16 INK4a (score +++). This positive staining coexisted with a strong and staining for p53. Figure 2 illustrates the histological as well as the p16 INK4a staining of two HPV-negative tumours with keratinizing and non-keratinizing.

Results

HPV DETECTION AND CLINICAL CORRELATIONS

Table 3 shows the sensitivity, specificity, and predictive values for the diagnosis of HPV-positive carcinoma of the vagina according to the following diagnostic approaches: diffuse positivity for p16 INK4a , negative staining for p53 and nonkeratinizing, basaloid or warty histology.

TREATMENT AND FOLLOW-UP DATA

Table 4 shows the treatment received by each patient, and the follow-up results. Twelve out of 23 women (47.8%) with HPV-positive tumours and adequate follow-up and none out of 5 (0%) women with HPV-negative tumours were alive without disease (p=0.052). Five out of 25 (24.0%) women with HPVpositive tumours and 2/5 women (40.0%) with HPV-negative tumours died of disease (p=0.587). The median survival was 113.9 months for women with HPV-positive and 19.7 months for women with HPV-negative tumours respectively (p=0.152). In the HPV-positive group of patients no differences were observed in terms of survival between patients with a previous history of cervical carcinoma (0 out of 7 died of disease), high grade premalignant cervical

Discussion

Our results confirm previous evidence showing that, similarly to vulvar invasive squamous cell carcinoma, invasive SCCVa develops following two different ethiopathogenic pathways: one group is related to HPV infection, whereas the second group of tumours arises via independent-HPV pathways.

However, in contrast with the 20-35% of tumours positive for HPV in the vulva 18,[START_REF] Kurman | Basaloid and warty carcinomas of the vulva. Distinctive types of squamous cell carcinoma frequently associated with human papillomaviruses[END_REF][START_REF] Toki | Probable nonpapillomavirus etiology of squamous cell carcinoma of the vulva in older women: a clinicopathologic study using in situ hybridization and polymerase chain reaction[END_REF] an overwhelming number of SCCVa (over 75% of the invasive SCCVa in the present series) are HPV-positive. The high percentage of HPV-positive tumours observed in our series is similar to the 81% observed in a recent series published by Ferreira et al. [START_REF] Ferreira | HPV DNA detection and genotyping in 21 cases of primary invasive squamous cell carcinoma of the vagina[END_REF] but is clearly higher than the percentages reported in older series. Although these variations might in part reflect different prevalence of HPV infection in different areas, they are likely caused mainly by the higher sensitivity of the HPV detection techniques used in these recent reports. In this regard, in our study, like in the report by Ferreira et al, the SPF10 primers have been used to detect HPV. These primers, which amplify a small region of 65 bp, are particularly useful for studies based on formalin-fixed, paraffin embedded tissue, and have been shown to be more sensitive than other PCRs in this type of material. 18 In our study, HPV16 was identified in 72% of HPV-positive SCCVa. This percentage is higher than the percentage of HPV16 reported in the uterine cervix. [START_REF] Munoz | Epidemiologic classification of human papillomavirus types associated with cervical cancer[END_REF] This high percentage is however, similar to that observed in other reports focused on SCCVa. [START_REF] Ferreira | HPV DNA detection and genotyping in 21 cases of primary invasive squamous cell carcinoma of the vagina[END_REF] These high percentages of HPV16 involvement observed in the vagina are as well similar to those observed in the vulva. 18 Moreover, percentages even higher than those (around 90%) have been reported in most studies focusing on head and neck tumours. 21,[START_REF] Gillison | Evidence for a causal association between human papillomavirus and a subset of head and neck cancers[END_REF][START_REF] Herrero | Human papillomavirus and oral cancer: the International Agency for Research on Cancer multicenter study[END_REF][START_REF] El-Mofty | Prevalence of human papillomavirus type 16 DNA in squamous cell carcinoma of the palatine tonsil, and not the oral cavity, in young patients: a distinct clinicopathologic and molecular disease entity[END_REF] Thus, our data indicate that, as in all regions and particularly in sites other than the uterine cervix, HPV16 is by far the most frequently involved type and that currently developed vaccines may prevent the large majority SCCVa associated with HPV infection.

As it has been shown in the vulva, women with HPV-positive SCCVa were significantly younger than patients with HPV-negative tumours. Nevertheless, although it has been suggested that SCCVa associated to HPV could be similar to cervical cancers in terms of risk factors and prognostic, 3,5,[22][23][START_REF] Murad | The pathologic behavior of primary vaginal carcinoma and its relationship to cervical cancer[END_REF] there are important differences between both neoplasm, particularly in age of presentation. SCCVa and cervical cancer occur in different age groups: while cervical carcinomas affect women younger than 60 years, SCCVa, is mainly a disease of elderly postmenopausal women, as showed in previous studies. 3, 7-9, 22, 32 On our study, only 7 tumours, all of them positive for HPV (21.9% of all SCCVa; 28% of HPV-positive tumours) developed in women under 60 years.

In agreement with previous reports 3,5,[22][23][START_REF] Murad | The pathologic behavior of primary vaginal carcinoma and its relationship to cervical cancer[END_REF] our results show that a high percentage of women with SCCVa have a history of prior dysplasia or invasive carcinoma of the uterine cervix. Following general consensus we have used in this report a 5 year free period between any previous cervical or vulvar neoplasm or relapse and the diagnosis of the SCCVa as the limit to distinguish a recurrent cervical carcinoma and a new primary vaginal carcinoma.

Nevertheless, there is some concern about this threshold and several authors have expanded that period to 10 years 6 or have even excluded all patients with any previous history of cervical pathology. [START_REF] Ferreira | HPV DNA detection and genotyping in 21 cases of primary invasive squamous cell carcinoma of the vagina[END_REF] In our study no significant differences were observed in age, and more importantly in terms of survival, between patients with SCCVa-HPV positive and previous history of cervical or vulvar neoplasia, and those with SCCVa-HPV positive and no previous history of cervical or vulvar disease. Thus, our findings suggest that this 5 year free period is an adequate threshold to distinguish a recurrent cervical carcinoma and a new primary vaginal carcinoma.

Our series confirms previous data showing that most HPV-positive SCCVa are of non-keratinizing, basaloid or warty types, whereas most HPV-negative tumours are commonly of keratinizing type. [START_REF] Ferreira | HPV DNA detection and genotyping in 21 cases of primary invasive squamous cell carcinoma of the vagina[END_REF] However, as observed in the vulva, 18 some overlap exists between the histological types and the association with HPV. Thus, 16% of all HPV-positive SCCVa were of keratinizing type, whereas 14.3% of the HPV-negative SCCVa showed non-keratinizing features.

The present study confirms previous data from vulva, 18,33 penis, 34 head and neck region, 21,[START_REF] Gillison | Evidence for a causal association between human papillomavirus and a subset of head and neck cancers[END_REF][START_REF] El-Mofty | Prevalence of human papillomavirus type 16 DNA in squamous cell carcinoma of the palatine tonsil, and not the oral cavity, in young patients: a distinct clinicopathologic and molecular disease entity[END_REF]35 and anal canal 36 showing that, as observed in the uterine cervix, 37-39 an intense and diffuse expression of p16 INK4A in a SCCVa indicates a high-risk HPV-associated process, this being irrespective of the histological type. An intense and diffuse p16 INK4A expression was observed in 96% HPV-positive SCCVa and only in one (14.3%) non-HPV associated SCCVa. In our series, the sensitivity and specificity of p16 INK4a immunostaining to detect HPV-associated carcinomas (96% and 87.5% respectively) were higher than the sensitivity and specificity of histological classification and those of p53 negative staining. Thus, our results clearly indicate that p16 INK4A immunostaining can be a useful tool to properly classify SCCVa into HPV- Finally no significant differences in terms of clinical stage, local recurrence or survival were observed between HPV-positive and HPV-negative tumours.

However, the results are limited in this respect due to the small number of HPVnegative tumours observed in our series, and should be confirmed in larger studies.

In summary, our study confirms that SCCVa occur in two different groups of patients. HPV-positive SCCVa, which account for over 75% of all SCCVa, arise in younger women, and are frequently associated with previous HPVpositive tumours or premalignant lesions of the uterine cervix, or vulva.

Histologically, HPV-positive SCCVa are frequently of non-keratinizing, basaloid or warty type, and overexpress p16 INK4a . In contrast, HPV-negative tumours, arise in elderly women, are histologically of keratinizing type and do not overexpress p16 INK4a . Our study indicates that p16 INK4a is a reliable marker for HPV-positive SCCVa and improves the results of pure histological classification.

F

F o r P e e r R e v i e w

  , other previous or concurrent neoplasms or premalignant lesions, stage, treatment and clinical outcome. Patients with a previous history of carcinoma of the uterine cervix diagnosed less than 5 years before were excluded from the study.[START_REF] Hellman | Primary carcinoma of the vagina: factors influencing the age at diagnosis. The Radiumhemmet series 1956-96[END_REF][START_REF] Tabata | Treatment failure in vaginal cancer[END_REF] HISTOLOGICAL EVALUATION Representative blocks of all cases were chosen. Review of H&E stained slides of the tumour specimens was performed independently by two pathologists who did not know the results of HPV testing or the tumour staging and clinical outcome. Tumours were classified and graded according to the World Health Organization (WHO) classification. When differences between the two independent evaluations were detected, a new evaluation was conducted and a consensus diagnosis was reached.DNA EXTRACTIONIn all cases a representative block including both the invasive carcinoma as well as the associated vaginal intraepithelial neoplasia (VaIN) lesion was selected. 10 µm thick sections of formalin-fixed, paraffin embedded tissues were cut using the sandwich technique, which implies that the first and last sections are stained with H&E and used to confirm the adequacy of the sample, whereas the central sections are used for DNA extraction. The microtome blade was replaced after each case; a blank paraffin block was cut prior to cutting a new sample. A vacuum cleaner was used to avoid cross-contamination between paraffin blocks. The tissue sections were digested with proteinase K in a volume of 250 µl at 56º C overnight. Proteinase K was heat inactivated at 95º C for 10 minutes. A 1/10 dilution of the sample was used for PCR (10 µl).

  precipitate. The results of the hybridization were assessed visually by comparing to the standard grid.

  associated neoplasms, as well as the results of HPV DNA detection for each individual tumour. Of all tumours, 25 out of 32 (78.1%) were HPV positive, whereas only 7 SCCVa were HPV-negative. All HPVs detected were of high risk types and HPV16 was the type most frequently identified (18 out of 25 [72.0%] HPV-positive cases).Patients with HPV-positive tumours were significantly younger than patients with HPV-negative neoplasms (62.6 ±13.8 vs. 74.0 ± 8.5; p=0.049). All tumours arising in women under 60 years (7 women) were positive for HPV. A previous history of invasive carcinoma (7 patients) or high-grade intraepithelial neoplasia of the uterine cervix (CIN2-3, 6 cases) or vulva (basaloid VIN3, 1 patient) was identified in 56.0% (14/25) of women with HPV-positive tumours and none of HPV-negative tumours (p=0.010). In all the cases the cervical or been diagnosed more than 5 years before the diagnosis of SCCVa and no evidence of relapse of the cervical tumour or premalignant lesion had been detected during a period of at least 5 years. No differences in mean age were observed between women with history of previous cervical or vulvar neoplasm and women with HPV-positive tumours and no previous history of neoplasm (61.7 ± 15.1 vs. 63.8 ± 12.8; p=0.715). Nineteen patients (59.4%) presented because abnormal vaginal bleeding, in 11 women (34.4%) the tumour was detected because of abnormal Pap test and in two a vaginal mass was detected in a routine gynaecological examination. The upper third of the vagina was exclusively involved in 8/25 (32.0%) HPV-positive and 0/7 (0%) HPV-negative tumours (p=0.149).HISTOPATHOLOGICAL AND IMMUNOHISTOCHEMICALCHARACTERISTICSThe histological and immunohistochemical characteristics of HPV-positive and negative tumours are shown in table 2. Overall, 15 SSCVa were nonkeratinizing, 10 were keratinizing, 4 were basaloid and 3 were of warty type. 21 out of 25 HPV-positive tumours (84.0%) were of non-keratinizing, basaloid or warty types, whereas only 4/25 (16.0%) tumours were of keratinizing type. In contrast, 6 out of 7 HPV-negative tumours (85.7%) were of keratinizing type (p<0.001). There was no association between HPV-type and any histological type of tumour.All but one HPV-positive tumours were strongly positive, showing diffuse basal staining (score +++) for p16 INK4a regardless of their histological type. The staining for p16 INK4a was intense and clear both in the nuclei and in the and no background was observed in negative cases or in negative areas of positive cases. Figure1illustrates the histological as well as the p16 INK4a staining of four HPV-positive tumours with non-keratinizing, basaloid and keratinizing features. p53 was positive in 3 out of 25 HPV-positive tumours.

  vulvar neoplasia (3 out of 7, 37.5% died of disease), or no previous history (3 out of 11, 27.3% died of disease) (p=0.200).

  or HPV-negative and can be considered as almost equivalent to HPV detection.

a:Figure 1 .

 1 Figure 1. Human papillomavirus-positive squamous cell carcinomas of the

Figure 2 .

 2 Figure 2. Human papillomavirus-negative vulvar squamous cell carcinomas. A, B, C: keratinizing type. A: H&E, B: negative staining for p16 INK4A ; C: negative

F

Figure 1 .Figure 1 .

 11 Figure 1. Human papillomavirus-positive squamous cell carcinomas of the vagina. A: Basaloid; B: warty, C: keratinizing types, H&E staining. Inset in B koilocytotic change. Diffuse immunohistochemical expression of p16INK4A (A',B',C'). 297x209mm (300 x 300 DPI)

Figure 2 .

 2 Figure 2. Human papillomavirus-negative vulvar squamous cell carcinomas. A, B, C: keratinizing type. A: H&E, B: negative staining for p16 INK4A ; C: negative

Table 1

 1 lists the age, location, previous VaIN, previous or concurrent other

Table 1 .

 1 Cases of primary squamous cell carcinoma of the vagina with age, tumour location, other associated neoplasms and HPV type

	Case	Age	Tumour location Previous VaIN Associated neoplasm HPV type
	1		Lower	Unknown	No	16
	2		Upper	Unknown	CIN3	16
	3		Upper	Yes	Carcinoma cervix	16
	4		Upper	Unknown	CIN3	16
	5		Unknown	Yes	Carcinoma cervix, VIN3	16
	6		Lower	Yes	No	16
	7		Upper	No	Carcinoma cervix	16
	8		Lower	Yes	CIN3, Melanoma	16
	9 10 11		Lower o Middle r All	Yes Unknown Yes	VIN3 No No	16 16 16
	12 13 14 15 16		Lower Upper Upper Middle and lower P e e Yes Yes Yes No r Middle and lower No	Carcinoma cervix Carcinoma cervix Carcinoma cervix CIN3 No	16 16 16 16 16
	17 18		All All	No R Unknown	No CIN3	16 16
	19 20 21 22 23 24		All Upper Middle Middle All Upper	No Yes e v i Carcinosarcoma endometrium No No No e Unknown No Unknown Carcinoma cervix w Yes No	35 51 52 52 58 59
	25		All	Yes	CIN2	68
	26		All	No	No	Negative
	27		Middle and lower	No	No	Negative
	28		Middle	Unknown	No	Negative
	29		Upper and middle	Unknown	No	Negative
	30		All	Unknown	No	Negative
	31		Lower	Unknown	No	Negative
	32		Middle and lower	No	No	Negative

Table 2 .

 2 Histological and immunohistochemical characteristics of human papillomavirus-positive and -negative primary squamous cell carcinomas of the

	vagina
		High-risk human papillomavirus
		Positive (n=25)	Negative (n=7)	p value
	Histologic type					0.006
	Keratinizing	4 (16.0 %)	6 (85.7 %)
	Non-keratinizing	14	(56.0%)	1	(14.3%)
	Basaloid	4 (16.0 %)	0	(0 %)
	Warty	3 (12.0 %)	0	(0 %)
	Immunohistochemistry
	p16 INK4A positivity	24 (96.0 %)	1 (14.3 %)	<0.001
	p53 positivity	3 (12.0 %)	4 (64.5 %)	0.025

Table 3 .

 3 Sensitivity, specificity, positive and negative predictive values for the diagnosis of human papillomavirus-associated carcinoma based on diffuse positivity for p16INK4A , negative staining for p53 and basaloid or warty histology.

		p16 INK4A	p53 (-)	Non-keratinizing/
		diffuse		basaloid/warty histology
	Sensitivity	96.0%	88.0%	95.5%
	Specificity	85.7%	57.1%	60.0%
	Positive predictive value	96.0%	88.0%	84.0%
	Negative predictive value	85.7%	57.1%	85.7%
	p16

INK4A diffuse: Continuous staining of cells of the basal and parabasal layers p53 (-): Less than 25% of positive cells (score 0, + or ++)

Table 4 .

 4 Primary squamous cell carcinoma of the vagina with treatment, relapse, and follow-up

	Stage			Months follow-
	Case	Treatment	Relapse	up	Follow-up
	II	Radiotherapy	No	159.5	AWOD
	I	Radiotherapy	Pelvis	33.2	DOD
	Unknown	Radiotherapy + Surgery a	Vagina	15.6	AWOD
	IV	Radiotherapy + Surgery c, d	Pelvic	15.6	DOD
	Unknown	Surgery e	Vagina	148.3	AWD
	II	Radiotherapy	Vagina	9.4	AWOD
	I	Radiotherapy + Surgery a	No	134.2	AWOD
	II	Radiotherapy + Surgery a	Vagina	136.4	AWD
	I	Radiotherapy	Pelvis	24.3	AWOD
	III	Radiotherapy	No	106.0	AWOD
	I	Radiotherapy + Surgery e	Pelvic and abdominal	104.0	AWOD
	II	Surgery a	No	2.2	AWOD
	I	Radiotherapy + Surgery a	No	42.1	AWOD
	I	Radiotherapy + Surgery a	Pelvis	83.7	AWOD
	II	Radiotherapy + Chemotherapy	No	26.1	AWOD
	IV	Chemotherapy	No	40.1	AWD
	III	Palliative	Progression	31.2	DOD
	IV	Surgery a	Pelvis and abdomen	57.4	AWD
	IV	Radiotherapy	Progression	9.0	AWD

Table 1 .

 1 Shrivastava SK, Sarin R, et al. Primary carcinoma of the vagina: Tata Memorial Hospital experience. Int. J. Radiat. Oncol. Biol. Cases of primary squamous cell carcinoma of the vagina with age, tumour location, other associated neoplasms and HPV type

	o r
	P
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11. Beller U, Benedet JL, Creasman WT, et al. Carcinoma of the vagina. FIGO 6th Annual Report on the Results of Treatment in Gynecological Cancer. Int. J. Gynaecol. Obstet. 2006 Nov;95 Suppl 1:S29-S42. 12. Bosch FX, Lorincz A, Munoz N, Meijer CJ, Shah KV. The causal relation between human papillomavirus and cervical cancer. J. Clin. Pathol. 2002

Table 2 .

 2 Histological and immunohistochemical characteristics of human papillomavirus-positive and -negative primary squamous cell carcinomas of the

	vagina
		High-risk human papillomavirus
		Positive (n=25)	Negative (n=7)	p value
	Histologic type					0.006
	Keratinizing	4 (16.0 %)	6 (85.7 %)
	Non-keratinizing	14	(56.0%)	1	(14.3%)
	Basaloid	4 (16.0 %)	0	(0 %)
	Warty	3 (12.0 %)	0	(0 %)
	Immunohistochemistry
	p16 INK4A positivity	24 (96.0 %)	1 (14.3 %)	<0.001
	p53 positivity	3 (12.0 %)	4 (64.5 %)	0.025

Table 3 .

 3 Sensitivity, specificity, positive and negative predictive values for the diagnosis of human papillomavirus-associated carcinoma based on diffuse positivity for p16INK4A , negative staining for p53 and basaloid or warty histology.

		p16 INK4A	p53 (-)	Non-keratinizing/
		diffuse		basaloid/warty histology
	Sensitivity	96.0%	88.0%	95.5%
	Specificity	85.7%	57.1%	60.0%
	Positive predictive value	96.0%	88.0%	84.0%
	Negative predictive value	85.7%	57.1%	85.7%
	p16

INK4A diffuse: Continuous staining of cells of the basal and parabasal layers p53 (-): Less than 25% of positive cells (score 0, + or ++)

Published on behalf of the British Division of the International Academy of Pathology Histopathology

Acknowledgements

The authors are grateful to Ms. Rosana Millan and Ms. Ingrid Victoria for their technical assistance in the preparation of histological sections and Ms.

Montserrat Tortosa and Ms. Elena Gonzalvo for their help with the immunohistochemical studies. We thank Mr. Ray Stanvius for the English revision of the manuscript.

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Hiroyuki Yamase

Cuprate superconductors in the vicinity of a pomeranchuk instability

Keywords: numbers: 74.25.Ha, 71.18.+y, 74.72.-h, 74.20.Mn

We propose that cuprate superconductors are in the vicinity of a spontaneous d-wave type Fermi surface symmetry breaking, often called a d-wave Pomeranchuk instability. This idea is explored by means of a comprehensive study of magnetic excitations within the slave-boson mean-field theory of the t-J model. We can naturally understand the pronounced xy anisotropy of magnetic excitations in untwinned YBa 2 Cu 3 O y and the sizable change of incommensurability of magnetic excitations at the transition temperature to the low-temperature tetragonal lattice structure in La 2-x Ba x CuO 4 .

In addition, the present theoretical framework allows the understanding of the similarities and differences of magnetic excitations in Y-based and La-based cuprates.

A c c e p t e d m a n u s c r i p t I. INTRODUCTION

Usually the symmetry of the Fermi surface (FS) satisfies the point-group symmetry of the underlying lattice structure. However, recently, symmetry breaking of the FS with the dwave order parameter was discussed in the t-J [1] and Hubbard [2] models on a square lattice:

the FS expands along the k x direction and shrinks along the k y direction, or vice versa.

This d-wave type Fermi surface deformation (dFSD) is often called a d-wave Pomeranchuk instability, referring to Pomeranchuk's stability criteria of isotropic Fermi liquids. [3] But the dFSD occurs even without violating Pomeranchuk's criteria since the transition is usually of first order at low temperature, [4] and also occurs in strongly correlated electron systems such as those described by the t-J model.

The dFSD competes with superconductivity. In the slave-boson mean-field theory of the t-J model [1] superconductivity becomes more dominant and the dFSD instability does not occur, which was recently confirmed by a variational Monte Carlo calculation. [5] However, substantial correlations of the dFSD still remain and lead to a giant response to a small anisotropy, e.g. in a lattice. [1,5] Recently a pronounced anisotropy was reported in magnetic excitation spectra of untwinned YBa 2 Cu 3 O y (YBCO). [6,7] In the superconducting state [6] the peak of the imaginary part of the dynamical magnetic susceptibility χ(q, ω) at various frequencies appear at q = (π ± 2πη x , π) and (π, π ± 2πη y ) with η x = η y and different peak intensity; η x(y)

parameterizes the degree of incommensurability. In the pseudogap phase [7] the incommensurate (IC) peaks are smeared, and the spectral weight forms a broad distribution centered at (π, π), but with a strongly enhanced anisotropy characterized by an elliptic shape. In La-based cuprates, IC signals appear up to a temperature much higher than the superconducting transition temperature T c . While the IC signals retain fourfold symmetry, namely η x = η y = η, the value of η turns out to show a sizable change when the lattice undergoes the structural phase transition from the low-temperature orthorhombic structure (LTO) to the low-temperature tetragonal structure (LTT) in La 2-x Ba x CuO 4 (LBCO) with x = 0.125. [8] In this paper, we show that the above experimental data are naturally understood in terms of dFSD correlations. While sizable dFSD correlations were found in the slave-boson, [1] exact diagonalization, [9] and variational Monte Carlo [5] techniques in the t-J model, and in various renormalization group analyses [2,[10][11][12] in the Hubbard model, we employ the t-J

II. MODEL AND FORMALISM

The t-J model

H = - r, r , σ t (l) τ c † r σ cr σ + r,r J τ S r • S r (1)
is defined in the Fock space with no doubly occupied sites. The operator c † r σ (c r σ ) creates (annihilates) an electron with spin σ on site r, and S r is the spin operator; J τ (> 0) is a superexchange coupling between the nearest neighbor sites and t

† r b r + σ f † rσ f rσ = 1.
This is an exact transformation known as the slave-boson formalism. We then decouple the interaction with the so-called resonating-valence-bond (RVB) mean fields:

χ τ ≡ σ f † r σ f r σ , b † r b r , and ∆ τ ≡ f r ↑ f r ↓ -f r ↓ f r ↑ .
These mean fields are assumed to be real constants independent of site r. We approximate the bosons to condense at the bottom of the band, which leads to b † r b r = δ, where δ is the hole density. The order parameter of the dFSD is defined as φ = (χ yχ x )/2 and d-wave singlet pairing is given by ∆ = |∆ x -∆ y |/2; both orders are generated by the second term in the Hamiltonian (1). After determining the RVB mean fields by minimizing the free energy, we compute the dynamical magnetic susceptibility χ(q, ω) numerically in the renormalized random phase approximation (RPA). [13,14] The material dependence of high-T c cuprates is taken into account mainly by different choices of band parameters. While the parameters t (l) τ and J τ are considered on a square lattice for La-based cuprates, the bilayer coupling is included to perform a more realistic calculation for Y-based cuprates. The details of our formalism are presented in Refs. 15 and 16 for Y-based and La-based cuprates, respectively. We show results for δ = 0.12, which may be complement to those references.

A c c e p t e d m a n u s c r i p t III. RESULTS FOR UNTWINNED Y-BASED CUPRATES

The crystal structure of untwinned YBCO is orthorhombic in the carrier doping region where superconductivity is realized at low T , yielding a small xy anisotropy to the electronic band structure. We introduce 5% anisotropy to hopping integrals between the x and y direction, and 10% anisotropy between J x and J y , twice as large, as imposed by the superexchange mechanism. Since CuO chains run along the y direction, we assume the band parameters are enhanced more along the y direction.

The order parameter of the dFSD φ is plotted in Fig. 1 Momentum space maps of Imχ(q, ω) for a sequence of temperatures are shown in Fig. 2 for ω = 0.30J and δ = 0.12 in the odd channel; a similar result is obtained in the even channel. [15] The strong intensity region forms a deformed diamond shape at low T [Fig. 2(a)],

and the spectral weight inside the diamond gradually increases with T [Fig. 2(b)]. The spectral weight around q = (π, π) becomes dominant at temperature slightly below T RVB = 0.124J and an enhanced anisotropy appears with an elliptic-shape distribution as shown in Fig. 2(c). When T increases further, however, the anisotropy is reduced [Fig. 2(d)]. This strong T dependence is characteristic of the effect of dFSD correlations.

IV. RESULTS FOR LA-BASED CUPRATES

Three different crystal structures are realized in La-based high-T c cuprates, depending on temperature and carrier doping: high-temperature tetragonal structure (HTT), LTO, and LTT. In the former two structures, the lattice does not produce a xy anisotropy and thus not couple to the underlying dFSD tendency. We expect an electronlike FS for δ = 0.12 as

A c c e p t e d m a n u s c r i p t

shown in Fig. 3(a). The LTT, however, yields a small xy anisotropy, the direction of which alternates along the z axis. Such a small anisotropy is then strongly enhanced by dFSD correlations as we have seen in Fig. 1(a). Through a coupling to the LTT, therefore, we expect a strongly deformed FS as shown by solid lines in Fig. 3(b); the FS is deformed in the opposite direction (gray lines) in neighboring CuO 2 planes and thus the superimposed FS recovers fourfold symmetry. A weak interlayer coupling then leads to two FSs, an inner electronlike FS and an outer holelike FS as shown in Fig. 3(c). Note that dynamical fluctuations of the dFSD are expected in the presence of the soft phonon mode toward the LTT phase transition even in the LTO phase. [17] For simplicity we do not consider the interlayer coupling, although it is important in the sense that it can yield a one-dimensional-like incommensurate peak along the direction q = 1 √ 2 (q, q). [18] We compute Imχ(q, ω) for the FSs shown in Figs. 3(a) and (b). We find that similar magnetic excitations are obtained for both FSs. [16] The spectral weight distribution has fourfold symmetry around q = (π, π). At low T , the strong intensity region forms a diamond shape as shown in Fig. 2(a), and pronounced IC signals appear at (π ± 2πη, π) and (π, π ± 2πη). With increasing T , the spectral weight around q = (π, π) increases and the IC signals become less clear, but they are still discernible even at high temperature (∼ 0.15J), [16] different from the result in Fig. 2(d). Since dFSD correlations produce a distinct change of the FS shape from Fig. 3(a) to (b) when the lattice undergoes a phase transition to the LTT, such a FS change in general leads to a sizable change of η at the LTT transition. This features the underlying dFSD correlations in La-based cuprates.

V. DISCUSSION AND CONCLUSION

dFSD correlations lead to a strong anisotropy of magnetic excitations at relatively high T [Fig. 2(c)], which accounts for the recent observation in the pseudogap phase in untwinned YB 2 Cu 3 O 6.6 . [7] In La-based cuprates, dFSD correlations are expected to lead to a change of the FS (Fig. 3) through the LTT phase transition, which is in general accompanied by a sizable change of incommensurability of magnetic excitations; the FS shown in Fig. 3(b) tends to favor a larger η. Such a change was recently observed in LBCO with x = 0.125, [8] although the authors [8] interpreted the data differently as due to a charge stripe formation.

Magnetic excitations in high-T c cuprates, especially in La-based cuprates, are frequently Observations of weak charge order signals [20] do not necessarily mean that charge stripes are crucial to magnetic excitations. The effect of charge order on magnetic excitations can be higher order corrections beyond the present renormalized RPA.

A well-known distinction of magnetic excitations between La-based and Y-based cuprates is that low-energy IC signals are realized up to a temperature much higher than T c in the former, [21,22] while they are realized only below T c or possibly below pseudogap temperature T * in the latter. [23] This difference comes from a FS difference in the present theory.

For the FSs shown in Figs. 3(a) and (b), there are no particle-hole scattering processes with q = (π, π) for low energy, yielding a robust incommensurate structure even in the normal state, while the FS shown in Fig. 1(b) allows such scattering processes, smearing incommensurate signals in the normal state. The FS geometry is essential for the material dependence of magnetic excitations, the same insight as the early proposal, [24,25] although we have extended the early studies in the t-J model [26][27][28] by introducing an idea of the dFSD. More detailed comparisons of magnetic excitations between Y-based and La-based cuprates are presented in Ref. 16.

We have shown that many salient features of magnetic excitations in high-T c cuprates are well-captured in terms of particle-hole excitations. In particular, dFSD correlations are essential for the pronounced anisotropy observed in untwinned YBCO [6,7] and the sizable change of incommensurability at the onset temperature of the LTT phase transition in LBCO. [8] Cuprate superconductors are expected to be in the vicinity of the dFSD instability, FIG. 2: q maps of Imχ(q, ω) for a sequence of T in 0.6π ≤ q x , q y ≤ 1.4π for ω = 0.30J and δ = 0.12 in the odd channel; here T RVB = 0.124J.

A c c e p t e d m a n u s c r i p t

  model in the slave-boson mean-field theory, which has the advantage of dealing with both d-wave singlet pairing and dFSD correlations on an equal footing as well as performing a systematic calculation of magnetic excitations.

τ

  is a hopping amplitude between the lth nearest neighbors (l ≤ 3); τ = rr.We introduce the slave particles, f rσ and b r , as crσ = b † r f rσ , where f rσ (b r ) is a fermion (boson) operator that carries spin σ (charge e), and S r = 1 2 f † rα σ αβ f rβ with the Pauli matrices σ = (σ x , σ y , σ z ). The slave bosons and fermions are linked by the local constraint b

  (a) together with the d-wave singlet pairing ∆. φ increases with decreasing T and exhibits a cusp at the onset of ∆, which is denoted as T RVB ; T RVB is interpreted as pseudogap crossover temperature T * in the underdoped regime and as T c in the overdoped regime of high-T c cuprates. Below T RVB , φ is suppressed because of competition with singlet pairing but still enhanced compared with the value of φ at high T . The FSs at low T are shown in Fig. 1(b). The inner (outer) FS is formed by the antibonding (bonding) band due to the bilayer coupling. The inner FS can easily open in the presence of an anisotropy while the outer FS is still holelike.

A c c e p t e d m a n u s c r i p t

  discussed in terms of charge stripes or electronic nematic order envisaged as a partial stripe order.[19] Although the state with a d-wave deformed FS has the same symmetry as the nematic order, dFSD correlations originate from forward scattering processes of quasiparticles[1,2] and do not require charge stripe correlations. An interesting open question is whether the dFSD state can have an instability toward a charge ordered state such as stripes. Even if it were the case, our calculation shows that many salient features of magnetic excitations observed in La-based cuprates are already well-captured without stripes.

FIG. 1 :

 1 FIG. 1: The mean-field solution in the presence of an anisotropy for δ = 0.12. (a) T dependence of φ and ∆. (b) Fermi surface at low T .

A c c e p t e d m a n u s c r i p t FIG. 3 :

 t3 FIG. 3: (a) Fermi surface expected in the HTT and LTO phases for δ = 0.12. (b) and (c) Fermi surface expected in the LTT phase; a weak interlayer coupling is considered in (c) but not in (b).

Acknowledgments

The author is grateful to W. Metzner for collaboration on a related work and to R. Zeyher for a critical reading of the manuscript.


J P Hague

P E Kornilovitch

J H Samson

A S Alexandrov

Singlet and triplet bipolarons on the triangular lattice

Keywords: Bipolarons, electron-phonon interactions, unconventional superconductivity

We study the Coulomb-Fröhlich model on a triangular lattice, looking in particular at states with angular momentum. We examine a simplified model of crab bipolarons with angular momentum by projecting onto the low energy subspace of the Coulomb-Fröhlich model with large phonon frequency. Such a projection is consistent with large long-range electron-phonon coupling and large repulsive Hubbard U . Significant differences are found between the band structure of singlet and triplet states: The triplet state (which has a flat band) is found to be significantly heavier than the singlet state (which has mass similar to the polaron). We test whether the heavier triplet states persist to lower electron-phonon coupling using continuous time quantum Monte Carlo (QMC) simulation. The triplet state is both heavier and larger, demonstrating that the heavier mass is due to quantum interference effects on the motion. We also find that retardation effects reduce the differences between singlet and triplet states, since they reintroduce second order terms in the hopping into the inverse effective mass.

Introduction

The possible role of local pairs in the cuprates and the origin of the pairing mechanism are subject to intense debate in the superconductivity community. Many experiments now demonstrate that there is a strong electron-phonon (e-ph) interaction in the cuprates. In particular, isotope effects show that lattice vibrations do play a role in determining the specific electronic structure of high-temperature superconductors [1]. However, consensus has not yet been reached about the effects of lattice vibrations. Theorists proposing a view of intrinsic repulsionmediated superconductivity claim that the isotope effects are simply due to change of the polaron mass, which are the underlying quasi-particles for * Tel: +44-1509-228208. Fax: +44-1509-223986 Email address: J.P.Hague@lboro.ac.uk (J.P.Hague). 1 We acknowledge support from EPSRC grants EP/C518365/1 and EP/D07777X/1 their theories, but that otherwise the e-ph interaction does not lead to the pairing required for superconductivity [2]. Here, we explore a scenario in which the e-ph interaction is so strong that pairing is into local bipolarons, which form a Bose-Einstein Condensate with no electronic resistance [3].

The possibility of bipolaronic superconductors in real materials has been critisised on the grounds that, in the presence of strong Coulomb repulsion, the e-ph interaction would have to be huge to form bound pairs. Thus, individual polarons forming the bipolaron would be extremely heavy [4]. For strong e-ph coupling on rectangular lattices with only nearest-neighbour hopping, it can be demonstrated that bipolaron mass is proportional to polaron mass squared [5], indicating that bipolaronic superconductivity on such lattices could only be a low temperature phenomenon. However, a more recent scenario involves the possibility that e-ph interactions are longer range than the Coulomb For strong e-ph coupling, the energy space is such that bipolarons must be composed of dimers between nearest neighbour sites. There are six possible dimer configurations on the triangular lattice, categorised according to the relative positions of the first and second inserted particles. Configurations A and D etc. can be related through and exchange and translation. The most general 2 particle wavefunction is a linear combination of these six states, symmetrised according to translational (Bloch) and rotational (angular momentum) invarience.

repulsion [6], leading to inter-site pairs which can move freely on lattices formed from triangular plaquettes (crab bipolarons). Here, we consider the difference between singlet and triplet bipolarons on the triangular lattice. Given recent results concerning the possibility of d-wave superconductivity mediated via phonons [7], we consider this a first step towards understanding unconventional pairing on a quantitative footing using numerically exact QMC computations.

We study the Coulomb-Fröhlich model for longrange e-ph interactions in quasi-2D materials,

H = -t nn σ c † n σ c nσ + nn σ V (n, n )c † nσ c nσ c † n σc n σ + m P 2 m 2M + m ξ 2 m M ω 2 2 - nmσ f m (n)c † nσ c nσ ξ m .
ξ m is the ion displacement, sites are numbered by n / m for electrons / ions respectively and c annihilate electrons. The phonons are independent oscillators with frequency ω and mass M . Pm = -i ∂/∂ξ m is the ion momentum operator. Since electrons are mobile in the plane, in-plane Coulomb repulsion V (nn ) is heavily screened. We consider only infinite on site repulsion U . The Fröhlich interaction is specified via the force function,

f m (n) = κ (m -n) 2 + 1 -3/2
, where κ is a constant [8]. The dimensionless e-ph coupling λ is defined as λ = m f 2 m (0)/2M ω 2 zt (z is the coordination number). Long-range interactions make polarons light, and less dependent on lattice type [8][9][10][11].

Unconventional pairing and the crab bipolaron

To obtain insight into the crab bipolaron, we consider pairs in the limit where on-site Coulomb repulsion is infinite, inter-site attraction is large and phonon frequency is large (anti-adiabatic limit ω > ztλ). Thus the lowest energy states span a subspace of near-neighbour pairs. The most general wavefunction in this basis is,

|ψ = 1 NR ri e ik.ri R l=1 e ilm2π R R n=1 a n A (n+l) † ri |0 (1)
that is that Bloch's theorem and the equivalent for rotated angular momentum states have been applied to a linear combination of all neighbour pairs. R is the order of rotational symmetry, m is the angular momentum. A There are a number of symmetries that can be used to simplify this expression: A

(N +R/2) † ri = -A (N ) † ri-τN , A (N +R) † ri = A (N ) † ri
, making the substitution ān = a n + (-1) m a n+R/2 and noting that e 2πim(l+R/2)/R = (-1) m e ilm2π R

, the wavefunction can be simplified to:

|ψ = 1 NR ri e ik.ri R/2 l=1 e ilm2π R × R/2 n=1 ān-l (A (n) † ri + (-1) m A (n+R/2) † ri )|0 (2)
i.e. the problem is reformulated as R/2 singlet and R/2 triplet bands. The effect of the hopping term on pairs can be written as a rotation, showing that hopping also has some of the properties of an angular momentum operator (see figure 2). However, this makes the heavier mass more surprising, since larger (i.e. less well bound) bipolarons would be expected to be lighter. The apparent convergence of the singlet and triplet properties at large λ is due to retardation effects.

Htb A (N ) † ri = -t A (N +1) † ri + A (N -1) † ri (3) -A (N +1+R/2) † ri+τN -A (N -1+R/2) † ri+τN
If the basis didn't just contain nearest neighbour pairs, then Htb would introduce m dependent functions into the result. However for only nearest neighbour pairs, eq. 2 can be rewritten as,

|ψ = 1 NR ri e ik.ri R/2 l=1 e ilm2π R R/2 n=1 ān-l Ā(n) † ri |0 (4)
where,

Ā(n) † ri = A (n) † ri + (-1) m A (n+R/2) † ri
and ān = a n + (-1) m a n+R/2 . Thus, there is no dependence on angular momentum. s, d etc. states are degenerate singlets, p, f etc. are degenerate triplets. Initially, the degeneracy of the angular momentum states seems surprising. However, since only the near-neighbour subspace is considered, there are only 6N degrees of freedom. As an increasing number of possible pairing arrangements is considered (for example if U were not infinite), the total number of degrees of freedom would increase, and the degeneracy would be lifted.

As we have previously discussed, singlet and triplet states can be mapped onto a dimer lattice, with hopping -t for singlets and t for triplets (the difference in hopping sign can be demonstrated by considering only two particles on a singlet trianglular molecule, symmetrised according to eq. 2) [12]. Thus one can determine the secular equations:

E N |ψ = N | H|ψ ( 5
)
where |N = Ā(N) † |0 . The solution to those equations is given by the equation,

det|H k -E| = ∓ t ±E/ t 1 + γ * k 1 + β k 1 + γ k ±E/ t 1 + α * k 1 + β * k 1 + α k ±E/ t = 0 (6) with α k = exp -iak x /2 + i √ 3ak y /2) , β k = exp -ik x a/2 -i √ 3k
y a/2 and γ k = exp (ik x a). There are 6 bands resulting from the diagonalisation of the Hamiltonian. Three singlet (corresponding to even m) and three triplet (odd m states),

E 1 (k) = V min ± t(-1 -3 + (k)/ t) ( 7 ) E 2 (k) = V min ± t(-1 + 3 + (k)/ t) ( 8 ) E 3 (k) = V min ± 2 t (9)
where (k) = -2 t(cos k x a+cos k x a/2 -√ 3k y a/2 + cos k x a/2 + √ 3k y a/2 ) is the polaron band structure in the antiadiabatic limit. An unusual result is that the singlet bipolaron mass is proportional to the polaron mass, m * * s = 6m * while the triplet bipolaron mass is infinite. In the rest of this article, we compute singlet and triplet states computed using a continuous time QMC algorithm to determine which of these attributes survive into the intermediate coupling and phonon frequency regime [13,12].

Quantum Monte-Carlo results

QMC simulations of the crab bipolaron state have been carried out for singlet states [13,12]. We have also simulated triplet bipolarons on the chain [14].

A c c e p t e d m a n u s c r i p t

It is of interest to determine if the limiting analytic results discussed in the previous section can be realised in numerical simulations. We have previously determined that there are significant differences between the masses of bipolarons on staggered and rectangular ladders [12,5]. We have also found that masses of polarons and bipolarons on lattices with triangular components have similar magnitude [13].

We compute numerical results from the continuous time QMC algorithm for a bipolaron on a triangular lattice. The singlet and triplet properties are computed by symmetrising the waveunctions. So the triplet bipolaron picks up a sign change on exchange, whereas the singlet always has positive sign. This affects the estimators, which depend on the sign as,

R bp = s 1 β β 0 (r 1 (τ ) -r 2 (τ )) 2 dτ ( 10
)
m 0 m * * i = lim β→∞ 1 β s∆r 2 i ( 11
)
For configurations where the end points of the paths are together, s = 0. ∆r is defined as in ref. [9]. Figure 3 shows the effective mass and size of singlet and triplet states on a triangular lattice with infinite local Coulomb repulsion. Results are computed for ω/t = 4. Panel (a) shows the inverse masses of singlet and triplet bipolarons, and panel (b) their radii. It can be seen that the masses of the triplet bipolarons are generally larger than their singlet counterparts. The radii of triplet bipolarons are also larger, as should be expected due to the node in the wavefunction. However, this makes the heavier mass more surprising, since larger bipolarons would be expected to be lighter since they are less well bound. The larger mass is thus due to quantum interference effects in the motion. This can be best understood by imagining the tight binding model constructed for hopping of bipolarons. The bipolaron overlap integrals giving the effective hopping are highly sensitive to the form of the binding wavefunction, and it is clear that the node in the triplet wavefunction will make those integrals (i.e. the effective hopping) much smaller leading to smaller effective mass. For larger couplings, ztλ > ω, the antiadiabatic approximation used in the previous section does not hold. Thus the largest differences between triplet and singlet states are found around λ = 1.

The apparent convergence of singlet and triplet states at large coupling is initially surprising. However, analysis of the sign can shed light on this ef-fect. For weak coupling, any bipolarons are weakly bound, so singlet-triplet splitting must be very small, since it can't be larger than the singlet binding energy. Since ∆E st = -(ln s )/β, the average sign must be approximately 1. At large coupling, paths must be nearly straight at very strong coupling due to retardation effects. Since there is infinite Coulomb repulsion, at least two kink insertions are required on each path for an exchange, exchanges become increasingly rare, and the sign must return to 1. So the total sign dips (and thus the difference in mass is largest) when ztλ ∼ ω.

Conclusion

We have computed properties of bipolarons with angular momentum on triangular lattices. The mass of the singlet bipolaron is of the order of the polaron mass for a wide range of couplings. However, we find that triplet bipolarons have much larger mass for coupling ztλ ∼ ω. A surprising result here is that retardation effects at very large λ reduce the difference between singlet and triplet properties when the local Coulomb repulsion is infinite. Understanding of the full range of bipolaron behaviours, especially on unconventional lattices is clearly important, and will form the subject of future publications.

Fig. 1 .

 1 Fig. 1. (Colour online) Schematic of the small bipolaron on a triangular lattice with infinite Hubbard repulsion and long range attraction.For strong e-ph coupling, the energy space is such that bipolarons must be composed of dimers between nearest neighbour sites. There are six possible dimer configurations on the triangular lattice, categorised according to the relative positions of the first and second inserted particles. Configurations A and D etc. can be related through and exchange and translation. The most general 2 particle wavefunction is a linear combination of these six states, symmetrised according to translational (Bloch) and rotational (angular momentum) invarience.

Fig. 2 .

 2 Fig.2. Schematic of the hopping term as a set of rotations, showing that the hopping operator is also related to the angular momentum operator.

(

  N ) † ri = c † ri c † ri+τN and τ N are the nearneighbour vectors.

Fig. 3 .

 3 Fig. 3. Singlet and triplet states on a triangular lattice with infinite Coulomb repulsion. β = 3.5. Paths are constrained to be within 200 lattice spacings of each other. Measurements were taken every 10 MC steps, with error bars computed by blocking groups of 1000 measurements. Results are computed for ω = 4. Panel (a) shows the inverse masses of singlet and triplet bipolarons, and panel (b) their radii.The masses of the triplet bipolarons are generally larger than their singlet counterparts. The radii of triplet bipolarons are also larger, as should be expected due to the node in the wavefunction. However, this makes the heavier mass more surprising, since larger (i.e. less well bound) bipolarons would be expected to be lighter. The apparent convergence of the singlet and triplet properties at large λ is due to retardation effects.