Bacem Wali
email: bacem.wali@irisa.fr

Bernard Gibaud
email: bernard.gibaud@irisa.fr

Extending OWL-S for the Composition of Web Services Generated With a Legacy Application Wrapper

Keywords: Ontology, Semantic Web, Web Services Composition I

Despite numerous efforts by various developers, web service composition is still a difficult problem to tackle. Lot of progressive research has been made on the development of suitable standards. These researches help to alleviate and overcome some of the web services composition issues. However, the legacy application wrappers generate nonstandard WSDL which hinder the progress. Indeed, in addition to their lack of semantics, WSDLs have sometimes different shapes because they are adapted to circumvent some technical implementation aspect. In this paper, we propose a method for the semi automatic composition of web services in the context of the NeuroLOG project. In this project the reuse of processing tools relies on a legacy application wrapper called jGASW. The paper describes the extensions to OWL-S in order to introduce and enable the composition of web services generated using the jGASW wrapper and also to implement consistency checks regarding these services.

INTRODUCTION

Web services are a new revolution of software systems. They are considered as self-contained, self-describing, module applications that can be published, located, and invoked through the Web [START_REF] Rao | A Survey of Automated Web Service Service Composition Methods[END_REF] [START_REF] Tsalgatidou | An Overview of Standards and Related Technology in Web Services[END_REF]. They are designed to be manipulated remotely from a network and they have the capability to invoke each other mutually, which raises the issue of their interoperability. Companies implement web services according to their application domain and display them through the web. Consequently, the number of heterogeneous web services is increasing, whose interoperability is severely hampered by this pervasive heterogeneity, inherent to independently developed services. For example: The use of new messaging protocols involves changing WSDL formats according to the domain specific applications and implementation needs. Therefore, when we deviate from standard cases to specific ones, composition of web service becomes a challenging problem that was addressed by many researchers and engineers in the recent years [START_REF] Curbera | The next step in Web services[END_REF].

Different initiatives have been proposed to facilitate the reuse of web services, leading to new languages, protocols and frameworks. For example, UDDI [4] (Universal Description, Discovery and integration), SOA (Service Oriented architecture) [START_REF] Papazoglou | Service oriented architectures: approaches, technologies and research issues[END_REF], BPEL4WS (Business Process Execution Language for Web Service) [START_REF] Andrews | Business Process Execution Language for Web Services Version 1.1[END_REF], SOAP [7] (Simple Object Access Protocol) and WSDL (Web Services Description Language) [START_REF] Chinnici | Web Services Description Language (WSDL) Version 2.0[END_REF] are standards for service discovery, description, and messaging protocols [START_REF] Rao | A Survey of Automated Web Service Service Composition Methods[END_REF] [START_REF] Sirin | Semi automatic composition of web services using semantic descriptions[END_REF]. Those specifications provide a means to syntactically describe a web service. However, they do not deal with semantic web service description and semantic web service composition. Semantic web service is a concept that brings semantics to the aforementioned standards. By adding semantics we can make web services machine understandable and use-apparent form [START_REF] Mcilraith | Semantic Web Services[END_REF]. By adding semantic markup to a web service we can make two aspects of its functionality explicit. First, semantic annotations can define what the service actually does, and second, they can describe its behavioral aspect (i.e., how the service works, the chaining that can be performed according to the sent and received messages).

Several languages have emerged, to add semantic description features to the web services standards. For example, DAML-S [11] (Darpa Agent Markup Language for services) is a revision of DAML+OIL [12] and based on OWL [13] (Ontology Web Language) WSMO [14] (Web Service Modeling Ontology) and OWL-S (Ontology Web Language for Services) [START_REF] Martin | Bringing Semantics to Web Services: The OWL-S Approach Springer[END_REF]. OWL-S is an ontology represented in OWL which aims at applying reasoning capabilities to the functionality, behavior, and execution of web services. OWL-S defines a model with three layers: a service profile describing the service's basic functionalities (function and characteristics, etc), a service model describing how the service works including data and control constructs flow, and a service grounding, describing how to access the service, by grounding its functional elements (input, outputs and operations) in a way consistent with the WSDL's concept of binding.

We worked in the context of the NeuroLOG [16] project which aims to share medical resources (brain images and image processing tools) [START_REF] Michel | Gridwide neuro-imaging data federation in the context of the NeuroLOG project[END_REF]. Image processing tools are wrapped as web services using a software package called jGASW [START_REF] Balderrama | jGASW: A Service-Oriented Framework Supporting High Throughput Computing and Non-functional Concerns[END_REF]. JGASW is a framework for wrapping legacy scientific applications as web services enabling their execution in a SOA environment.

To allow the sharing of neuro-imaging resources, the OntoNeuroLOG [START_REF] Temal | Towards an Ontology for Sharing Medical Imagesand Regions of Interest in Neuro-imaging[END_REF] ontology was designed. It provides common semantics for information sharing throughout the NeuroLOG system and allows the sharing of neuro-imaging resources provided by collaborating actors in the field of neuro-imaging research. The term resources cover both neuroimaging data (such as images) as well as image processing tools (registration, de-noising, and segmentation).

Through the semantic we tend to share the functionalities of images and the functionalities of image processing tools to enable more expressivity from a functional viewpoint.

The goal of our work is to facilitate the sharing, reuse, and invocation for the user of image processing tools wrapped as web services with jGASW and deployed in the site server within the NeuroLOG framework. To this end, we chose to add semantics to jGASW services, to facilitate workflow composition and automate some consistency controls regarding their usage. Our assumption is that this may increase the usability of such tools by people that were not involved in their development. This by addressing some technical aspects that hinder the composition process with OWL-S API and implementing some consistency controls. Such consistency control tends to: (i) ensure interoperability and composition by checking the compatibility between outputs and inputs of web services; (ii) check of the compatibility between the inputs provided by the users and the semantic inputs definition of the service; (iii) check the consistency between the functionality of the image processing tool, like registration or de-noising, and their declared inputs/outputs, with respect to the formal definition of such conceptual actions, modeled in the OntoNeuroLOG ontology.

We used the OWL-S ontology to benefit from its web service description model and its large expressivity in terms of parameters description and behavioral aspect of flows. However, we had to deal with some technical issues regarding jGASW WSDLs which let us extending OWL-S to enable its use in our NeuroLOG framework.

In this paper, we advance the state of the art by (1) specifying an extension of the OWL-S specification to make it adapted to our jGASW framework context without changing the basic structure of the WSDLs, (2) adding some reasoning capabilities to perform consistency checks regarding the usage of our annotated web services. The following of the paper is organized as follows: Section II provides more details about the difficulties related to the WSDLs of service generated by the jGASW tool, together with OWL-S semantic descriptions of services. Section III presents the solutions that we found based on a specific extension of OWL-S that addresses the problem and some associated reasoning mechanisms that validate the services capabilities, consistent with our domain ontology OntoNeuroLOG. Section IV details how the implementation was done to extend OWL-S and solve the problem and describes some technical issues that we tackle. Section V discusses our contribution and situates it in the wider context of semantic workflows, and finally, Section VI opens other perspectives for future work.

II. BACKGROUND

To address the issue of web service composition using OWL-S and jGASW we need to describe both more closely. So, we first describe WSDL files generated automatically by jGASW, then we study the automatic generation of semantic descriptions with OWL-S and we analyze the mismatch between the two and we study how to address it.

First, we explain how the jGASW [11] framework works: an application GUI allows the user to upload the image processing tools (shell program) and add inputs, outputs arguments and libraries. According to an XML schema and values set by the user, an XML description is generated jGASW descriptor. The generation of the web service consists in transforming the jGASW descriptor into a web service interface by generating the WSDL together with an XSD schema (XML Schema Definition). The XSD schema details the inputs and the outputs of every WSDL operation. In fact, all services have WSDL files with the same content, and identical operations but always have only one input and one output and one output as an exception. In contrast, inputs and outputs are described differently in the XSD schema according to each service. For example, in the Figure 1, column 1 illustrates the description of a WSDL operation named local that is composed of tns:local as input, tns:localResponse as output, and tns:SOAPException as fault message (generated if the execution of the service failed). Column 2 details input tns:local by defining it as a complexType containing two xs:element (i) (simpleinput1 and simpleinput2: input files) and details output tns:localResponse by defining it as a complexType typed as another complexType (ii) (jigsawOutputTest2111: generated automatically). This complexType contains four xs:element representing the different output files (stdout, stderr, simpleoutput1, simpleoutput2).

Column 3 shows SOAP envelopes (call/request). At execution time, jGASW prepares the SOAP envelope, invokes the service and gets back the result according to the sequences described in the complex type jigsawOutputTest2111, but the name of the envelope is ns1:localResult. Thus, the jigaswOutputTest211 is not considered here.

JGASW wraps executables into web services, and produces at least two standard files (std.out: for standard output for any shell output, std.err: error message generated (if execution fails), and other files resulting of the service execution such as image files (e.g. ex2output1.nii, ex2output2.nii).

Column 4 in the figure 1 shows the OWL-S description generated automatically from the WSDL description provided by the jGASW service (using the WSDL2OWLS [20] converter). Green arrows show the grounding of each individual input element whereas the red arrow shows one unique grounding which is localResut. In fact, here, we lost the information that this output contains four files rather one only. Thus, composition of jGASW services is not possible. So, the principal issue concerns the outputs definition, which is not understandable due to their complex schema. This figure shows the main problem faced when we try to automatically generate the semantic description of a jGASW service; Columns 3 and 4 show how inputs are grounded individually whereas outputs are grounded as a single box "locaResult". Therefore, OWL-S Process and Profile of this service contain one single output according to the obtained grounding. Why doesn't OWL-S understand different outputs? Because they belong to a complex type "jigsawOutputTest2111" showed on column 2. Such output types generated automatically by the jGASW software can be even more complex than that. Actually, they can contain multiple nestings of complexType.

Motivation: First, we have shown that WSDL files are only partially understood by OWL-S API. Second, we need control construct to design and execute our image processing workflows. Third, OWL-S is a well defined language for web services composition that offers many functionalities that are not handled by other languages. Moreover it is submitted in the W3C , something important in our context of collaborative research in neuro-imaging. This choice of OWL-S is also consistent with our use of the OWL-Lite OntoNeuroLOG ontology as domain ontology. Indeed, OWL-S is an OWL-DL ontology and it is more expedient to use ontologies that are closer to each other in term of semantic capability and reasoning (i) both are based in OWL and this facilitate the use the same kind of reasoner (ii) if we use WSMO for example we should translate OntoNeuroLOG in WSML. Finally, OWL-S provides the suitable expressivity for representing web service semantics and fits nicely our application's requirements; in fact we cannot modify the structure of WSDL files, nor the XML description of inputs/outputs, since they are intrinsic to the jGASW middleware (otherwise invocation would not work properly).

Approach: First we extend the OWL-S Profile to be adapted to OntoNeuroLOG ontology. Second, we extend OWL-S Process to enable the description of jGASW services and finally, we add a software layer that implements reasoning services ensuring various consistency checks based on knowledge imbedded in the OntoNeuroLOG ontology.

III. METHOD

The OntoNeuroLOG ontology describes the different kinds of brain images in reference to the Dataset taxonomy and the functionality of services in reference to the Data processing taxonomy, each of these classes having its specific characteristics defined using DL axioms.

A.

Extending OWL-S OWL-S is a particular OWL ontology. It allows the semiautomatic composition of Web services. It is composed of three layers. The Service Profile allows the description, publication, and discovery of services. It is used by providers to publish their services and by users to specify their needs. The Service Model is used to compose services. It allows modeling services as processes. Three types of processes exist: atomic processes (AtomicProcess), simple (SimpleProcess) and composite (CompositeProcess). AtomicProcess represents the finest level of action that the service may perform. Composite Process are decomposable into other processes thus their concatenation can be specified using a set of control structures such as Sequence, Split, If-Then-Else, etc. A SimpleProcess is used to provide a view of an atomic process or a simplified representation of a composite process.

The Service Grounding describes how to access the service and provides the mapping between semantic inputs, outputs, message formats, and physical addresses. The purpose of this mapping is to enable the translation of semantic inputs generated by a service consumer into the appropriate WSDL messages for transmission to the service provider, and the translation of service output messages back into appropriate semantic descriptions (i.e., OWL descriptions) for interpretation by the service consumer.

1)

Extending the OWL-S Process model Our extension aims at decomposing the output grounded as a single box (localResult) with OWL-S (described in Figure 1) into its different elements (e.g., stderr, stdout, simpleoutput1, simpleoutput2). To overcome this problem, some classes and data/object properties are added to the OWL-S process model:

-a NlogParameter class to denote parameters that are embedded in a parameter of such a composite nature (e.g., stderr, stdout, simpleoutput1, simpleoutput2); it is defined as: <owl:Class rdf:about="#NlogParameter"> <rdfs:subClassOf rdf:resource="#Parameter"/> <owl:disjointWith rdf:resource="#Output"/> </owl:Class> -a nlogExpandsTo object property, associating a parameter of a composite nature to its essential elements according to the XSD schema of the service; <owl:ObjectProperty rdf:about="#nlogExpandsTo"> <rdfs:domain rdf:resource="#Output"/> <rdfs:range rdf:resource="#NlogParameter"/> </owl:ObjectProperty> -a hasID data property, denoting the markup within the string result after the service invocation <owl:DatatypeProperty rdf:about="#hasID"> <rdfs:domain rdf:resource="#NlogParameter"/> <rdfs:range rdf:resource="&xsd;anyURI"/> </owl:DatatypeProperty> -a hasLabel data property, denoting a non-functional property providing an informal description of the parameter <owl:DatatypeProperty rdf:about="#hasLabel"> <rdfs:domain rdf:resource="#NlogParameter"/> <rdfs:range rdf:resource="&xsd;anyURI"/> </owl:DatatypeProperty> To detect the value of "simpleoutput1" argument first we select its ID using the hasID property and second we parse the string result (SOAP envelop) using the retrieved ID corresponding to the markup of "simpleoutput1". The data type of the NlogParameter is given by the data property Process:parameterType. Extending the OWL-S Profile model As described before, the OWL-S Profile gives information about the capabilities and the behavior of the service. We enrich it by adding a reference to the equivalent data processing class using refers-to, an object property that belongs to OntoNeuroLOG.

<owl:ObjectProperty rdf:about="&iec;refers-to"> <rdfs:domain rdf:resource="#Profile"/> <rdfs:range rdf:resource="&data-processing-owl-lite;data-processing"/> </owl:ObjectProperty>

3)

Web services composition As mentioned in the background section we cannot modify the WSDL otherwise invocation would no longer work, in consequence we could not extend the Grounding subontology. The extension of the Process Model is enough to allow jGASW services composition. Once OWL-S Outputs have been related to corresponding NlogParameters according to the XSD Schema derived from jGASW processing, we were able to compose jGASW services. To this end, we introduced another object property, links, that binds any OWL-S Parameter to another (also suitable to NlogParameter since they are a subClassOf Parameter):

<owl:ObjectProperty rdf:about="#links"> <rdfs:domain rdf:resource="#Parameter"/> <rdfs:range rdf:resource="#Parameter"/> </owl:ObjectProperty> In this illustrative example, we compose two jGASW services: the first one service 1) has one input and one output. The output is composed of three outputs according to its XSD Schema, and the second (service 2) has 2 inputs and one output. The output is composed of four outputs according to its XSD schema. The profile of the service embedding the whole workflow has two inputs linked respectively to jGASW service 1 and jGASW service 2 and six outputs coming from both jGASW services. One internal parameter only is transmitted from service 1 to service 2.

B. Some reasoning mechanisms 1)

Compatibility check between dataset processing and the OWL-S Profile This service allows users to ensure that the definition of the profile is compatible with the data processing class selected by the user at annotation time. The algorithm is the following: first we create a temporary class tmp_Profile_data-processing class relatively to the current operation, and then we translate relations between profile, inputs, and outputs into axioms and we add profile them to the tmp_Profile_data-processing class. Then, for every relation hasInput/hasOutput we count the number of inputs grouped by dataset class to determine the cardinality of the corresponding axiom; for example: Process:hasInput i1 Process:parameterType Mr-dataset and Process:hasInput i2 Process:parameterType Mr-dataset would lead to a cardinality of 2 concerning Mr-dataset (Mr-dataset denotes a magnetic resonance image dataset). The third step consists in selecting the appropriate object property for the construction of the axiom (e.g. Process:hasInput substituted by has-fordata-at and Process:hasOutput substituted by has-for-resultat. The result of the two first steps is: (Process:hasInput i1 Process:parameterType Mr-dataset and Process:hasInput i2 Process:parameterType Mr-dataset)  (has-for-data-at exactly 2 Mr-dataset). The third step consists of adding those axioms to the tmp_Profile_data-processing class.

The last step is to add the new tmp_Profile_dataprocessing class with axioms added above as subclass of the class referred by the Profile "MyProfile" and selected by the user, in our example (tmp_Profile_data-processing subclassOf Registration), and then, classify and check consistency. If the ontology is consistent then the annotation is considered valid. Semantically, the functionality of the tool is agreed, i.e., the has-for-data-at/has-for-result-at object properties are consistent with respective inputs/outputs specified in the corresponding data processing class in the OntoNeuroLOG ontology. Figure 4 show an illustrative example of the algorithm.

2) Compatibility check between outputs and inputs in a workflow

This service is applied when a user builds a new workflow. The processing aims at ensuring for every link between NlogParameter and Input that corresponding types are compatible. So we distinguish three cases: • Identical data types: the output and the input have exactly the same type. Compatibility is validated and composition is accepted.

• Link to a more specific data type: the output is more general than the input of the next service, so noncompatibility.

• Link to a more general data type: the output is more specific than the input of the next service. The first service will always return results that are semantically compatible with the next service input. Compatibility is validated and composition is accepted.

N.B. workflow is valid if Parameters have the same Type or source is subsumed by target according to the dataset ontology.

3) Compatibility check between values and inputs at invocation time

This service is called when a web service is invoked. It checks whether the actual instances selected by the user (e.g. a Dataset) and assigned to the values actually meet the constraints specified in the semantic annotations of the service. In practice, the semantic service checks whether the class (or the type) of this instance is subsumed by the class type of the input.

IV. IMPLEMENTATION

The semantic annotation of jGASW services is generated automatically using the WSDL2OWLS API. Enrichment of semantic annotation is done using the OWL-S 1.2 specification and the OWLS API 3.0. The semantic annotation of workflow services is generated using the OWL-S 1.2 specification and the OWLS API 3.0.

The consistency check between the profile and the data processing class is implemented using the OWL API, the OWL-S API 3.0 and the HermiT Reasoner.

The web services invocations use the OWL-S API but results and composition issues use the semantic search engine CORESE [START_REF] Corby | Querying the Semantic Web with Corese Search Engine[END_REF] together with the OWL-S API. CORESE is used to select the functional properties (extensions, linked parameters, identifiers …) of OWL-S outputs by querying the triple store containing the semantic annotations of the services. We add here an illustrative example of a workflow composed of two services (Figure 5). First, we prepare the SOAP envelope to invoke the first jGASW service: green markup shows the WSDL operation input (tns:local) and blue markup indicates the concrete input that the service will use. The next section shows the output of the service after invocation: green markup shows the output of the WSDL operation (tns:localResult). It wraps three blue markups that show three files generated by the service. The stderr and stdout are workflow outputs whereas simpleoutput should be transmitted to the second jGASW service. With CORESE we query the triple store to retrieve the nlogParameters to which the output (localResult) is extended. Then, for every nlogParameter retrieved, we find the ID (the markup to extract it from the localResult), its link to another parameter (parameter passing), and its data type. The parameter that we extract is simpleoutput. It should be transmitted to ex002_input2 second input of the second jGASW service. Query: aims at identifying the different outputs that tns:localResult (corresponding semantically to ex001_output1) is expanded to: The output ex001_output1 is expanded to three nlogParameters as seen in the Figure 5 (ex001_stdout, ex001_stderr, ex001_simpleoutput) corresponding respectively to (stdout, stderr, simpleoutput) in the query results (?id fields). Those ID are the markups used in the localResult. The query results show that both ex001_stdout, ex001_stderr are linked to workflow outputs (WF_stdout1, WF_std_err1) as showed in the Figure 5. The query results show that the parameter ex001_simpleoutput is linked to the parameter ex002_input2. Thus it should be passed to the second jGASW service. To this end, the value of ex001_simpleoutput is extracted using the jGASW engine by giving the ID already selected by the query. The result is:http://localhost:80/-~bwali/Test1_1321350928548-9787/testoutput.nii. A new SOAP envelope containing two inputs (as Figure 5 shows) is prepared to invoke the second jGASW service. The simpleinput1 is the file selected by the user for the workflow execution (corresponding semantic id is WF_input2). This parameter is passed to ex002_input2. The simpleinput2 gets the file extracted from localResult. i.e. result of the execution of first jGASW service invoked as described above. The other parameters (stderr, stdout, simpleoutput1, and simpleoutput2 of jGASW service2) are transmitted to the workflow outputs.

PREFIX p1: <http://localhost/
V. DISCUSSION Several semantic languages and frameworks have been proposed based on W3C web service languages to support web service composition. However, web service composition is hampered by the heterogeneity of web services. Our work is an extension of OWL-S at the concrete service level to address the issue of jGASW web services composition.

We relied on OWL-S because it is a well defined Ontology [START_REF] Corby | Querying the Semantic Web with Corese Search Engine[END_REF] based on manifold earlier solutions and it is currently submitted in the W3C. It is also a semantic framework that provides more complete specifications than any other alternative solutions. It is represented in OWL which is a standardized language and exploits its reasoning capability [START_REF] Yang | Towards Dynamic Integration of Collaborative Virtual Enterprise using Semantic Web Services[END_REF]. Thus, it enables us to leverage our domain ontology in reasoning aiming at performing various consistency checks regarding the use of our services. OWL-S is a multi-layered language thus, it is easy to handle. In our contribution, extending the Profile layer and the Process layer leverages this characteristic. OWL-S differs from other specifications by providing conditions, effects, sequences and control constructs. We reused conditions and effects definitions to verify the consistency of service compositions and control construct specifying the behavioral aspect [START_REF] Guo | Mapping to business process model to semantic web service model[END_REF] of composed jGASW services. The OWL-S Service Grounding is conceived to be adapted for grounding any kind of service. Unfortunately, our WSDL files are really specific and cannot be grounded entirely. Getting service output as a unique box and as a string format actually hampers generating the grounding automatically and therefore the semantic description. Nevertheless, OWL-S is still the nearest solution and its adoption and extension allowed overcoming the problem. WSDL-S [24] and SAWSDL [25] define how to add semantic annotations to WSDL specifications. In fact, they let WSDL components refer to semantic concepts via the ModelReference attribute, added to WSDL elements to assign one or more semantic concepts, via the schemaMapping property to map complex types and elements with a semantic model, via Precondition and effect for service discovery, via serviceCategory to help in case of service advertisement. In contrast to OWL-S they externalized domain application and let the reasoning mechanisms free. Grounding should be interpreted manually and service composition is not explicit. They do not deal with context of execution, behavior aspect and therefore, the reasoning aspect is really neglected, so we preferred use a more sophisticated and developed language for reasoning mechanisms.

Web service composition is still a complex task [START_REF] Rao | A Survey of Automated Web Service Service Composition Methods[END_REF][26] [START_REF] Dustdar | A Survey on Web Services Composition[END_REF]. Numerous surveys on web service composition present an overview of methods that deal with web service composition. Based on a large background, Dustdar and Schreiner [START_REF] Dustdar | A Survey on Web Services Composition[END_REF] discussed the need of web service composition and related issues. They outline the importance of the context in web service composition. The context should be formatted in some customized and personalized manner for relevant use by the next service. In our work we had to face the same requirements regarding the composition problem. The enrichment of OWL-S aims to format outputs in order to make them adequate for the next service that will be invoked. Enabling jGASW services composition is the added value of this enrichment and key factor of our work. It enabled us to add algorithms to check consistency Rao and Su [START_REF] Rao | A Survey of Automated Web Service Service Composition Methods[END_REF] investigated automated web service composition and propose an abstract framework for automatic service composition. They discuss abstract process model and business workflow involving the impact of heterogeneity of web services sources. We conclude that web service composition becomes more difficult if ever we deviate from the standard cases to specific cases. For example, automatic selection, matching, and composition work well while using standards. It is against hindered if we are out of standards. In our work, jGASW WSDL files are different from standard WSDL files. They differ by their XML schema, thus, they are heterogeneous compared to standard ones. This shows nevertheless, the dependency of semantics model on thin technical details and with the manner how to access services.

Without OWL-S the composition of jGASW service is not possible. In fact the form of SOAP envelop of the result not allows the chaining of web services. If we would like to compose jGASW services without semantics we should add interoperability within the jGASW engine. The first benefit from extension and use of OWL-S facilitate this task by enable the composition process. The second benefit from OWL-S is the multilayered structure that it has. In fact with the ServiceProfile it enables us to add semantic verification according to the neuro-imaging expectations. This is shown throw the implementation of the validation algorithm.

Casati et al [START_REF] Casati | Adaptive and dynamic service composition in eFlow[END_REF] uses the notion of process template to model composite services and composers need to browse the process library to search for process templates of interest [START_REF] Dustdar | A Survey on Web Services Composition[END_REF]. Rao et al [START_REF] Rao | A Survey of Automated Web Service Service Composition Methods[END_REF] and Dustdar et al [START_REF] Dustdar | A Survey on Web Services Composition[END_REF] distinguish in workflow composition static and dynamic workflow generation, static defines the business tasks and dynamic linking the concrete e-services. Both help for monitoring e-services. OWL-S does not provide explicit support for monitoring and errors handling [START_REF] Vaculín | Monitoring execution of OWL-S web services[END_REF]. OWL-S service profile is just a service categorization and still lacks semantics. In our work we add some semantics to augment workflow monitoring. For example while users compose their workflows our consistency checking algorithm verifies that the service profile and the related data processing are consistent, which ensures that the chaining of the service can make sense from the point of view of processing. Cardoso and ShethIn [START_REF] Cardoso | Semantic e-Workflow Composition[END_REF] try to overcome e-workflow composition problems by making services interoperable. They use a multidimensional approach based on ontology mediation. Medjahed et al [START_REF] Medjahed | Composing Web Services on the Semantic Web[END_REF] address the interoperability issue by using composability rules. Currently, this task must be performed by a human who might use a search engine to find a service, and connect the service manually. However, a couple of verification algorithms were implemented within our application framework using OWL-S markup of services, and the necessary information from the OntoNeuroLOG ontology. At this stage, our work is still basic, the automatic discovery and mediation process are not well handled.

Indeed, this process requires further development to overcome the heterogeneity of semantic web services using mediation. Especially WSMO, that uses the mechanism of mediation between semantic services coming from different heterogeneous frameworks. In our work the semi-automatic composition does not need mediation, however it needs a semantic validation through a reasoning aspect implementing verification of the consistency of the flows.

Gannod et al. [START_REF] Gannod | An Interactive Approach for Specifying OWL-S Groundings[END_REF] the authors present a generic approach to ground services with OWL-S. Users can ground automatically or manually the service to its description. Although it is a generic approach this kind of grounding does not meet our needs. In fact, it considers that every end point (WSDL or others) defines the outputs individually. However in our case the outputs are embedded in the unique box and are not explicit for the WSDL API and are understandable only by our jGASW Engine. Semantically, this editor considers that service grounding and service model are two distinct layers. In our work, we no longer keep those two layers separate, which is a limitation of our solution. In fact the process:hasID data property that was added is the unique way to access to the WSDL elements (input/output) as explained in the implementation section . We are required to do that because the way jGASW gets back the result obliges us to have a link to the parameter in the process specification. Otherwise, if we try to extend service grounding, invocation would no longer work.

Web services differ in form, technique or design point of view. Application wrappers provide outputs and inputs in different forms due to the functional requirements of the application domains. In this case, semantic solutions are not enough. The extension that we proposed is adequate for every kind of service so we augment the flexibility of web service development. Even the service has different technical details, the proposed idea, when reused in another context, is still valid and address both technical and semantic problems.

VI. CONCLUSION AND FUTURE WORK

In this paper, we introduced a method to extend the OWL-S specification to cope with jGASW web services description. We succeeded to address the problem of semantic web services composition and to add some semantic validation and verification mechanisms. This solution addresses several issues concerning the web services composition in the neuro-imaging domain, but it is not sufficiently tested by NeuroLOG users. For us to assess its added value from an end user point of view, moreover, automatic composition still needed.

The next step of this work tends to, ensure and validate this work by adding serious test through the neuro-imaging framework and developing an algorithm for automatic selection and composition of jGASW web services and OWL-S workflows. We are trying to add reasoning capability over the description of data processing and the validation with profile algorithm.

Figure 1 :

 1 Figure 1: Automatic grounding of a jGASW service using the WSDL2OWLS API

Figure 2

 2 Figure 2 provides an illustrative example of the use of the previous extensions: we present one extension of the output of jGASW service 2.

Figure 2 :

 2 Figure 2: Enrich the output in its essential parameters

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Figure 3 :

 3 Figure 3 : How to link NlogParameters with OWL-S parameters and workflow parameters in case of workflow composition

Figure 4 :

 4 Figure 4: Transformation of Profile to data processing class (1) Represents the description of Registration data processing, (2) represents the semantic description of the registration tool according to enriched OWL-S that should do registration if invoked and (3) shows the transformation of the profile into data processing.

Figure 5 :

 5 Figure 5: semantic annotation of workflow using the OWL-S Process layer and the extension described in this work

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	http:/

ACKNOWLEDGMENT

The NeuroLOG project was funded by the French National Agency for Research (ANR-06-TLOG-024). The authors gratefully acknowledge all contributors, including our clinical colleagues providing the image data exploited in the NeuroLOG test-bed.


Ettore ; Vitali

Pierantonio Arrighetti

Maurizio Rossi

Davide Emilio Galli
email: davide.galli@unimi.it

Long Range Correlations in Quantum Solids

Keywords: quantum solids, long range correlations, supersolidity Quantum solids; long range correlations, supersolidity

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

Introduction

Quantum solids provide a very interesting research field both from a theoretical and from an experimental point of view; they are characterized by a large zero point motion, letting the atoms explore a sizable region in the unit cell of the crystal lattice. This allows quantum statistics to enter the game, giving rise to a scenario suitable for investigating quantum coherence effects in spatially localized systems. 4 He at very low temperature, a quite unique pure quantum system, as the external pressure exceeds about 25 atm becomes a typical example of a quantum solid and a microscopic understanding of the correlations governing its behavior deserves naturally strong interest. Recently Non Classical Rotational Inertia (NCRI) effects have been detected in bulk and confined crystalline 4 He, a signal that coherence phenomena take place in the system. The experimental findings about the solid phase of 4 He, in connection with the controversial topic of a possible supersolid state of matter (i.e. superfluidity taking place in the crystalline phase) can be found in many excellent reviews [2][3][4][5][6].

From the theoretical side, the strong correlations among Helium atoms make Quantum Monte Carlo (QMC) methods the most powerful technique to study the behavior of such a system. Since 4 He atoms obey Bose statistics, QMC are able to provide exact estimations of equilibrium properties, relying on imaginary time Schrödinger equation. As far as coherence phenomena are concerned, it has been possible to show that in a crystalline commensurate sample of 4 He Bose Einstein Condensation (BEC) does not take place, a result that had been initially obtained at finite temperature [7,8] and, afterwords, has been confirmed also at T = 0 K [5,9], preventing the possibility of a phase transition in the perfect crystal taking place at T ≤ 0.2 K.

The formalism of exact Path Integral QMC methods, both a finite and at zero temperature, is made of multidimensional integrals involving propagators and this prevents the possibility of keeping under control the physical correlations underlying the observed properties. Nevertheless, among the several interesting features of QMC, during last decades such methods have provided the possibility to build up and test variational approximations for the Ground State wave function, comparing their predictions with exact ones. This opens the possibility to put on a firm ground the microscopic comprehension of the behavior of the system: an accurate variational model explicitly contains the physical interatomic correlations underlying the rich phenomenology of 4 He systems.

The simplest way to build up a variational model of solid Helium is to correct a traditional Jastrow correlation term φ J (R):

φ J (R) = 1≤i<j≤N exp (-u (| r i -r j |)) , R ≡ ( r 1 , . . . , r N ) (1)
via an harmonic particle-site interaction introducing a priori a crystal lattice R i , obtaining the Jastrow-Nosanow wave function [10][11][12]:

ψ JN (R) = φ J (R) N i=1 exp -α| r i -R i | 2 (2)
This model has the disadvantage that it breaks Bose symmetry, preventing automatically the possibility of describing coherence phenomena. One possible way to recover the Bose symmetry is provided by the Shadow Wave Function (SWF) [13],

where the Jastrow term is improved via an harmonic coupling of Helium atoms positions, the real variables, with additional shadow variables S = ( s 1 , . . . , s N ), which are subsequently integrated over:

ψ sh (R) = φ J,r (R) dS N i=1 exp -c| r i -s i | 2 φ J,s (S) (3)
The label in φ J,r,s means that the pseudopotential u(r) among real variables is different from the one among shadows. The SWF is Bose symmetric and translationally invariant, features shared by the exact Ground State wave function [14], and describes the solid phase as a result of a spontaneous translational symmetry breaking mechanism: when the density is above a certain value the many-body correlations present in the SWF become so strong that the solid phase is stable and its energy is below the energy of the metastable liquid phase. SWF still provide the most accurate variational description of liquid and solid 4 He with an accurate prediction of freezing and melting densities [15]: the energy per particle obtained with a fully optimized SWF turns out to be no more than 3% above the one computed with "exact" QMC methods and with an almost constant gap in a wide density range, leading to a very accurate equation of state.

Although SWF has been able to provide an extremely accurate description of many properties of liquid and solid Helium systems [16][17][18][19][20][21][22][23][24], in the solid phase it predicts [25,26] Bose Einstein Condensation (BEC) which, for a perfect crystal at T = 0 K, is not confirmed by exact calculations [5,9]. This suggests that some important physical correlations are not present at the SWF level of the theory. As suggested also in Ref. [8], we argue that such correlations are long-range, consistently with the behavior of the exact One Body Density Matrix (OBDM) ρ 1 ( r-r ′ ), which remains non-zero for distances much larger than the lattice spacing, before decaying exponentially as | r -r ′ | → ∞.

In a admirable paper about liquid 4 He [1], Reatto and Chester performed an hydrodynamic analysis of large scale dynamics of Helium, relying on a "quantization" of classical sound field, and provided a mathematical model describing zero-point motion of density fluctuations: compressional waves (longitudinal phonons). Such a zero-point dynamics corresponds to collective motions of the atoms in the Bose fluid, resulting in a long-range two-body interparticle correlation of the form:

u lr (r) ∝ 1 r 2 + a 2 (4)
Such a correlation has been shown to improve the variational description, in particular correcting the small wave vector limit of the static structure factor (see Fig. 1), but still is not able to account for the suppression of the condensate fraction in the crystalline phase [1,27]. The aim of this work is to improve the variational model for the wave function in the solid phase, where also transverse phonons exist and the exact ground state has to contain their zero-point motion, resulting in additional long-range correlations. Our idea has been to mimic the procedure in [1] in the solid phase, relying on elasticity theory. Starting from continuum mechanics, we have defined a quantum mechanical operator which adds the zero-point motion of transverse modes to a given wave function. Under a "small currents" approximation we have obtained a possible form for the resulting long-range interatomic correlations and inserted them into a SWF-like model. From variational calculations we realized that our correction makes the condensate fraction decrease with respect to SWF results, but it is still non vanishing, making interesting in the future to study different ways of exploiting the "quantum elasticity" approach we have here introduced.

This paper is structured as follows: in the next section, after a review of Reatto-Chester approach, we will present the methodology of "quantization" of transverse dynamics in an elastic continuum, providing a new Shadow Wave function containing zero point motion of transverse phonons; in Sec. III, we will present variational Monte Carlo calculations relying on the new wave function and discuss possible future perspectives. As a first step, we find useful to briefly sketch the derivation of the result (4), which will provide a guide for our analysis. The idea is to start from an "hydrodynamic Hamiltonian" in the local density variable ρ( r, t) of the fluid, at lowest order in deviations from global thermal equilibrium:

E({ v( r, t)}, {ρ( r, t)}) = 1 2 mρ d r| v( r, t)| 2 + 1 2K T ρ 2 d r (ρ( r, t) -ρ) 2 (5)
In ( 5) v( r, t) is the velocity field of the continuum and the prefactor of the potential energy derives from standard equilibrium thermodynamic relations between

∂ 2 E
∂ρ 2 and the isothermal compressibility K T . The linearized form of the continuity equation in wave vector space:

∂ρ q (t) ∂t = iρ q • v q (t) ( 6
)
where it is transparent that we are dealing with longitudinal dynamics, together with ( 5), makes straightforward to obtain an Hamiltonian describing a gas of harmonic oscillators for the degrees of freedom {ρ q }, Fourier components of the local density. Interpreting (5) as a quantum Hamiltonian, {ρ q } playing the role of position operators (one for each wave vector), we get immediately a ground state wave function of the form:

Ψ 0 {ρ q } ∝ e -ms 2N P q |ρ q | 2 | q| (7)
s being the longitudinal sound velocity. The link with many body physics arises naturally from the definition of the density fluctuation operator:

ρ q = N i=1 e -i q• ri (8)
which maps (7) into a simple multiplication operator in the Hilbert space of the Helium sample: the strategy is to apply such an operator to a given model of the ground state wave function of the Helium system in order to "correct" it by "adding" the zero-point motion of longitudinal phonons. Performing the sum over the wave vectors in the continuum limit via the introduction of a suitable ultraviolet cutoff we get a two-body Jastrow wave function with a pseudopotential precisely of the form (4). Such a term has been proved to improve the variational SWF description, the most important novelty being the accuracy of the low wave vector limit of the static structure factor of the system (see Fig. 1); despite that, exact analytical results indicate that BEC survives in the model [1], so one has to look for other motivations to explain the vanishing BEC fraction resulting from exact calculations.

Elasticity and transverse modes

Here we try to devise an analogous strategy for transverse modes in the crystalline phase, starting from classical elasticity theory (for a general review many excellent textbooks exist, for example Ref. [START_REF] Marder | Condensed Matter Physics[END_REF]). Let's start considering an elastic medium described in terms of a displacement field u( r, t), a dynamical variable physically representing deformations with respect to a given equilibrium configuration; in the language of continuum mechanics, deformations give rise to a stress field in the sample generating internal forces described by the stress tensor T α,β ( r, t). The equations of motion of the medium, in the infinitesimal deformations regime, may be written as:

ρ ∂ 2 u α ( r, t) ∂t 2 = 3 β=1 ∂T β,α ( r, t) ∂r β + b α ( r, t) (9)
where b( r, t) represents external fields, volume forces acting on the system. In the linear approximation, under the assumption of homogeneity and isotropy, 1 symmetry arguments lead to the following constitutive equation:

T α,β ( r, t) = λδ α,β ∇ • u( r, t) + µ ∂u α ( r, t) ∂r β + ∂u β ( r, t) ∂r α (10)
λ and µ being phenomenological constants, the Lamé coefficients. Putting together ( 9) and ( 10) we arrive to the famous Navier equation for elastic media:

ρ ∂ 2 u α ( r, t) ∂t 2 = (λ + µ) ∂ ∂r β ∇ • u( r, t) + µ 3 β=1 ∂ 2 u α ( r, t) ∂r 2 β + b α ( r, t) (11)
Focusing on the dynamics of transverse displacement fields, defined by:

∇ • u( r, t) = 0 , (12)
if all the external forces vanish, we get the simple equation:

ρ ∂ 2 u α ( r, t) ∂t 2 = µ 3 β=1 ∂ 2 u α ( r, t) ∂r 2 β ( 13
)
which, if we are allowed to consider, at first order, the medium as incompressible ρ = cost, is simply a D'Alembert equation describing waves propagating at velocity v T = µ ρ . The transverse contribution to the elastic energy of the continuum 1 To be precise, the crystalline structure hcp of Helium samples at T = 0 and P ≃ 25 bars is not isotropic; nevertheless we are making a "coarse-grained" large scale description, and, at a preliminary level, we make an isotropy hypothesis, but a generalization could be straightforward. medium may be written as follows:

E({ j q }, { u q }) = 1 V q j q • j -q 2ρ + 1 2 µ| q| 2 u q • u -q , (14)
where

j( r, t) = ρ ∂ u( r, t) ∂t = 1 V q e i q• r j q (t) (15)
is the local current density and u q and j q are the Fourier component of the displacement field and of the local current respectively, with

q • j q (t) = 0 = q • u q (t) (16)
to fulfill the transversality condition.

In (14) the local current plays the role of the linear momentum in the harmonic oscillators, of "mass" ρ and "elastic constant"µ| q| 2 . Now, in classical mechanics, minimum energy would naturally correspond to vanishing of the currents, but, from a quantum point of view, we may expect a zero-point motion governed by a "wave function" either in the "displacement field variable" or in the "current variable". It is evident now that our result is completely analogous to the hydrodynamic one discussed above, the only difference being that now the basic dynamical degree of freedom is the displacement field. While u( r, t) has no natural counterpart in a many body microscopic theory, the local current has a well known expression, so that we are induced to use the "momentum representation". In the basic harmonic oscillator with Hamiltonian:

p2 2m + 1 2 kx 2 (17)
the ground state in momentum representation is given by:

c 0 (p) ∝ e -p 2 2 √ km (18)
so that we are induced to write the ground state of (14) in "current representation" in the following way:

Ψ 0 { j q } ∝ exp   - q j q • j -q 2 √ ρµ| q|   (19)
How can we use now (19) to "correct" a given variational model for a collection of N bosons in the crystalline phase?

In order to clarify the problem we face, let's consider for a while the simple one-dimensional Hamiltonian (17), and suppose we have constructed two different "variational models", pretending not to know the exact ground state:

F o r P e e r R e v i e w O n l y φ 1 (x) ∝ 1 φ 2 (x) ∝ δ(x) (20)
Of course both φ 1 (x) and φ 2 (x) do not describe the correct zero-point motion, the first one because it is "too delocalized", lacking confinement, while the latter because it is "frozen", and has to be somehow delocalized. How can we correct?

The idea is the following: the classical variables x and p display simple oscillatory dynamics and then we are induced to "quantize" their zero point motion via expressions of the form e -ax 2 and e -bp 2 , precisely as it has been done for the local density in the description of longitudinal motion, but what does it mean? In the quantum mechanics formalism, we define two operators of the form:

Â1 ∝ e -ax 2 Â2 ∝ e -bp 2 (21)
acting on (20). It is easy to check that what we obtain is:

Â1 φ 1 (x) ∝ e -ax 2 Â1 φ 2 (x) ∝ φ 2 (x) Â2 φ 1 (x) ∝ φ 1 (x) Â2 φ 2 (x) ∝ e -cx 2 (22)
We observe that Â1 provides the desired result when it acts on φ 1 (x), but it doesn't have any effect on φ 2 (x); on the other hand the role of Â2 is completely the opposite. So, in some sense, if we use the operator Â2 we have to act on a function which is "too localized" so that it may suitably "add" zero-point motion. Now, keeping these observations in mind, let's come back to solid Helium: in a many-body first principles quantum model, we may define the local current density operator of the system, whose Fourier components are defined by:

ĵ q = 1 2m N i=1 -i ∂ ∂ r i e -i q• ri + e -i q• ri -i ∂ ∂ r i (23)
The transverse components are given by:

ĵT q = ĵ q - q | q| • ĵ q q | q| (24)
and from (23) it is possible to construct the operator:

Ĵ def = exp   -γ q 1 | q| ĵT q • ĵT -q   γ ≥ 0 ( 25
)
which plays the same role as Â2 defined above. The similarity to the hydrodynamic approach is transparent. We have learned that we may hope that Ĵ could provide the desired result only if we apply it to a model wave function which is somehow "too localized". So, as a preliminary step, a natural idea appears to be that of trying to act on an element of the Hilbert space of the system of the Hartree form:

where { R k } are sites of a crystal lattice. The technical difficulty of applying ( 25) is formidable, and we decide to work at first order in the exponent, relying on the following expression:

ĵT q • ĵT -q f = - 2 m 2 N i,l=1   e -i q•( ri-rl) 3 α,β=1 δ α,β - q α q β | q| 2 ∂ 2 f ∂r i,α ∂r l,β   ( 27
)
which holds for any element f ( r 1 , ..., r N ) such that the r.h.s. of ( 27) does make sense. The derivation of ( 27), though a bit lengthy, is a straightforward application of the definitions ( 23) and (24). Performing the second derivative appearing in (27) when f is φ defined in (26), and remembering the definition of the cross product it is quite simple to show that:

q ĵT q • ĵT -q | q| φ ∝   - N i,l=1 q e -i q•( ri-rl) | q| q × ( r i -R i ) • q × ( r l -R l )   φ (28)
q being the versor in the direction of q.

Our "small currents" approximation strategy works as follows:

Ĵ φ ≃  Î -γ q ĵT q • ĵT -q | q|   φ = (1 -(...)) φ ≃ e -(...) φ ( 29
)
so that we obtain the following function:

Ĵ φ ( r 1 , ..., r N ) ∝ e - P N k=1 α| rk-Rk| 2 e γ P N i,l=1 P q e -i q•( r i -r l ) | q| (q×( ri-Ri))•(q×( rl-Rl))
(30) It is evident that the last term in the exponential introduces a long-range correlation depending on transverse oscillations of atoms around the lattice positions. The expression (30) suggests the following picture: as in the Gaussian model (26) the atoms are still localized around the lattice sites but transverse vibrations of different atoms are no more independent; the new pseudopotential favor a cooperative motion, and this is what we expected from the approach we have followed. ( 30) is our main result: the Hartree wave function, φ, has been improved with a new microscopic term which describes the coupling between transverse vibrations; We argue that in an accurate model of a quantum solid, long range correlation of the form appearing in (30) should be present. The main difficulty is that we should be able to recover the Bose symmetry. In the following we will present a first attempt to pursue this aim relying on the observation that ( 26) is formally equivalent to the kernel which, in the SWF picture, localizes the variables to the shadow ones; we are thus induced to substitute it with (30) "inventing" a model of the form:

S(q)

Figure 1. Static structure factor S(q) in the liquid phase at the equilibrium density with (filled circles)

and without (open circles) the contribution of the zero point motion of longitudinal phonons; error bars are lower than the symbol size. Experimental data [START_REF] Hallock | [END_REF] (crosses) at low wave vectors are also shown.

ψ tr (R) ∝ φ r (R) dSe - P N k=1 α| rk-sk| 2 e γ P N i,l=1 P q e -i q•( s i -s l ) | q| (q×( ri-si))•(q×( rl-sl)) φ s (S) (31)
in which, of course, all the parameters and the pseudopotentials have to be variationally optimized. To take into account the fact that short range correlation in the SWF model have been already optimized, we have introduced also a cut-off in the q summation in the new term, limiting it to introduce only long range correlations, similarly to what was done in Ref. [1].

Results for a first variational implementation

Now that we have built up a theoretical ground to face the topic of long range zero-point transverse modes, the next step is the characterization of the properties of such a wave function. Before considering the solid phase we will present some results in the liquid phase obtained using only the long range term arising from longitudinal phonons. In this case the SWF turns out to be

ψ tr (R) ∝ φ r (R) dSe - P N k=1 v(| rk-sk|) e γl P N i,l=1 P q e -β| q| | q| e -i q•( s i -s l ) φ s (S) . (32)
In (32) the cut-off in the q summation has been explicitly written, and this introduces the variational parameter β. In the liquid phase, we have considered N = 256 particles in a cubic box at the equilibrium density ρ = 0.0218 Å-3 , and as functional form we employed the one fully-optimized in Ref. [15]. In the optimization of the variational parameters connected with the long range term arising from longitudinal phonons we found β = 4 Å and γ l = 0.012 Å-1 . The long-range term provides only a 1% gain in the total energy, as found also previously in Ref. [15], and, as expected, it reproduces the correct linear behavior at small wave vectors of the static structure factor; this is shown in Fig. 1.

We have studied the role of the long-range corrections in the solid phase by considering N = 180 and N = 360 particles in a simulation box chosen to fit a regular hcp crystal at ρ = 0.0294 Å-3 . In the ground state variational SWF we have considered long-range correlations related to zero-point motion of both longitudinal and transverse phonons. The optimization of the variational parameters turned out to be quite difficult: by following the steepest descendant direction of the energy as function of the variational parameters we arrived to a set of parameters corresponding to an unstable crystalline phase. This suggested us to follow a different procedure: we have fixed the standard variational parameters in the SWF for the solid phase to those optimized in Ref. [30]; the parameter related to the long-range term of the longitudinal phonons has been simply rescaled with respect to the value in the liquid phase according to the change in the sound velocity: from s ≃ 248m/s to s ≃ 471m/s which gives γ l = 0.0228 Å-1 ; note, in fact, that the multiplicative constant in the exponent of ( 7) is proportional to the sound velocity s. We have then optimized the parameter γ, controlling the intensity of the transverse phonon term. No sharp minimum was found, but for values of γ in the range [0, 0.016] Å-2 the energy shows a shallow valley with an almost constant value; we decided to fix γ = 0.016 Å-2 . The crystal described by the new SWF is slightly stiffer than that described by a SWF without the correlations related to the zero-point motion of phonons; this can be inferred from the increased height of the Bragg's peaks in the static structure factor S(q), plotted in Fig. 2. We have computed also the one-body density matrix ρ 1 along the first neighbor direction in order to explore the effects of the long-range corrections on the off-diagonal properties. It is noteworthy that, by including the zero point motion of phonons (longitudinal and transverse), the long distance tail of ρ 1 is found depressed with respect to the SWF without long-range correlations. The condensate fraction reduces from n 0 = (9.8 ± 1.5) × 10 -6 to n 0 = (4.2 ± 0.6) × 10 -6 ; the contribution of the transverse phonons is about 2.5 times larger than the longitudinal one, and this was measured by computing ρ 1 by considering separately the two long-range terms. Our results for ρ 1 are reported in Fig. 3. Despite the quantitative change in the condensate fraction, no qualitative changes in ρ 1 (r) were found: we were not able to capture the exponential decay of the exact OBDM [5,7,8]. We argue that the weakness of our first variational attempt to include correlations related the zero point motion of transverse phonons is connected with the use of shadow variables to define equilibrium positions of the lattice in (31) is correct only on average. Recently a very interesting Bose symmetric Jastrow-Nosanow wave function, not cumbersome from the numerical point of view, has been introduced [31], of the form:

ψ BJN (R) = φ J (R) N I=1 N i=1 g | r i -R I | , (33)
where g can be a Gaussian or another one-body localization factor. A symmetric Bose model that could be able to better exploit the physics contained in (30) could be built inserting the long-range correlations in (33), where the equilibrium positions of the lattice appear explicitly without breaking the Bose symmetry. We plan to investigate such a possibility in the future.

Conclusions

Quantum simulation methods have been of the greatest importance in the study of quantum liquids and solids. Notwithstanding the presence of exact simulation methods for the calculation of ground state properties, the variational method, the first historically introduced, continues to attract special interest. It is in fact the only method able to make evident the microscopic structure of the wave function: the role of different correlations on observable properties of the system. Our main result is that, in the solid phase, we are able to deduce a functional form for additional long range correlations related to the zero-point motion of transverse phonons. This has been obtained via an analysis of classical elasticity theory which provides an elastic energy offering the possibility of "quantization" in suitable variables, mimicking the approach of Reatto and Chester in the liquid phase [1]. This opens many possibilities of defining variational models which contain contribution from correlations among transverse vibrations of the lattice; we propose a first attempt, consisting in a modified Shadow Wave Function. The crystal described by this new variational model of solid 4 He turns out to be slightly stiffer with higher Bragg's peaks in the static structure factor and with a lower tail in the one-body density matrix, but no qualitative change is found for the long-range behavior of As discussed in the text, we plan to further investigate new variational models with long range correlations coming from the zero-point motion of longitudinal and transverse phonons.

Acknowledgments

In entering the PhD School in Physics my Supervisor, L. Reatto, was particularly determined in suggesting, as research topic, the study of quantum liquids and solids via quantum Monte Carlo techniques. I (DEG) have to thank his determination: his suggestion led me into a fascinating adventure of scientific research, which lasts to this day. We would also like to thank L. Reatto for useful discussions.

1 .

 1 Reatto-Chester theory

  Vitali, P. Arrighetti, M. Rossi and D.E. Galli φ( r 1 , ..., r N ) ∝ e - P N k=1 α| rk-Rk| 2

Figure 2 .

 2 Figure 2. Static structure factor in the solid phase at ρ = 0.0294 Å-3 along different crystalline axis with (filled symbols) and without (open symbols) the contribution of the zero point motion of longitudinal and transverse phonons; error bars are lower than the symbol size

Figure 3 .

 3 Figure 3. One-body density matrix ρ 1 (r) computed along the first neighbor direction in an hcp 4 he crystal at ρ = 0.0294 Å-3 with (filled symbols) and without (open symbols) the contribution of the zero point motion of longitudinal and transverse phonons.