# LPTMS PhD Proposal: Entanglement and tensor representation of quantum states

#### Responsable:Olivier GIRAUD 0169153175

**Résumé :**

Quantum information processing holds large promises for increased computational power, communication possibilities, and metrology. A major part of the theoretical research in the field has focused on identifying, quantifying, and understanding the quantum resources that enable such enhancements. Most prominent is the research on entanglement, which was identified as key resource early on: bipartite entanglement is by now well understood, and research efforts focus on multipartite entanglement, where the situation is much richer - and much more complicated.

Understanding or classifying multipartite entanglement for generic quantum states is of course a daunting task. For states symmetric under exchange of particles, one may hope to make progress due to the drastically reduced Hilbert-space dimension. In the case of symmetric states, we have developed a theory of “quantumness” and “classicality”, based on coherent states, which can be reformulated in terms of multipartite entanglement [1,2]. However, the general characterization of quantumness and classicality remains an open question.

The starting point for this internship will be a recently developed tensorial picture of symmetric states, that generalizes the Bloch sphere picture of a two-level system [3]. Many applications open up. Remarkably, this picture allows to map questions about entanglement onto mathematically well-studied problems. As a primary goal, we will exploit this existing mathematical knowledge to obtain more powerful means of identifying entangled symmetric states. The internship may be followed by a thesis. Depending on the taste and background of the student, more mathematically oriented topics (representation theory, generalized coherent states) or more physical aspects (metrology, quantum channels) could be considered.

[1] O. Giraud, P. Braun and D. Braun, *Classicality of spin states*, Phys. Rev. A **78**, 042112 (2008).

[2] O. Giraud, P. Braun, and D. Braun, *Quantifying Quantumness and the Quest for Queens of Quantum*, New J. Phys. **12**, 063005 (2010).

[3] O. Giraud *et al.*, *Tensor Representation of Spin States*, Phys. Rev. Lett. **114**, 80401 (2015).

[4] D. Baguette, F. Damanet, O. Giraud, and J. Martin, *Anticoherence of spin states with point group symmetries*, Phys. Rev. A **92**, 052333 (2015).

# LPTMS PhD Proposal: Mechanical response of branched actin networks

### Responsable: Martin Lenz 01 69 15 32 62

The architecture of living cells is largely determined by a microscopic networks of semiflexible

filaments : the actin cytoskeleton. In addition to ensuring the cell’s mechanical integrity, its growth

enables cellular motion and force exertion. These crucial roles are played by so-called branched actin

networks, which are random fractal assemblies of filaments and branching points.

Despite its importance within the cell, the rigidity of these networks is not understood from a

theoretical standpoint. Indeed, taking into account the sole rigidity of the filaments and attachment

points, we would predict a vanishing elastic modulus, in contradiction with experiments. We will examine

the origin of these networks’ rigidity, considering in particular the effects of the entanglement of the

network with itself, which generates nonlocal interactions between the points of the elastic network.

Given the difficulty of treating such interactions exactly, we will resort to mean-field approaches whose

validity will be assessed numerically.

From an experimental perspective, our collaborators Olivia du Roure and Julien Heuvingh

(ESPCI) operate a setup allowing the first clean characterization of the branched network. We will work

with them to relate our models to the characteristics of a network grown under force (similar to in vivo

conditions), its nonlinear elasticity etc.

Informal inquiries welcome.

# LPTMS PhD Proposal: Corrélations et intrication dans les systèmes “analogues”

#### Responsable:Nicolas PAVLOFF 0169157334

**Résumé :**

Plusieurs systèmes modèles permettent d’étudier -- théoriquement et expérimentalement -- des phénomènes, tels le rayonnement de Hawking ou l'effet Casimir dynamique, que l'on a peu d'espoir d'observer dans le contexte ou ils ont été initialement proposés.

On propose durant cette thèse d’étudier de tels systèmes "analogues", en particulier dans le domaine des condensats de Bose-Einstein et de l'optique non-linéaire. On analysera en particulier plusieurs observables susceptibles de fournir des critères de non séparabilité quantique.

# LPTMS PhD Proposal: Conformal bootstrap and stochastic geometry

#### Responsable: Raoul Santachiara 06 07 84 92 22

The random processes that generate conformally invariant fractal sets are of paramount importance in science. Using conformal bootstrap techniques, this projects aims in finding new conformal field theory which capture the geometry of these extended sets.

# LPTMS Postdoc: Theoretical Biophysics

We welcome applications from postdoctoral candidates interested in theoretical descriptions of the cytoskeleton and other problems at the **interface between Soft Matter/Statistical Physics and Biology**. Possible projects include collaborations with Margaret Gardel (U. of Chicago), Michael Murrell (Yale U.) and Chase Broedersz (LMU Munich) to predict the out-of-equilibrium structure and contraction mechanisms of the actin cytoskeleton, and with Aurélien Roux (U. of Geneva) on protein-membrane interactions. More details at www.lptms.u-psud.fr/membres/mlenz/research

The postdoc will join a new, dynamic group spearheading research at the Soft Matter/Biology interface within a world-class Statistical Mechanics lab. The position presents ample opportunities for strong interactions with local and international collaborators. **Autonomous interactions with experimentalists** and the **development of creative independent projects** are encouraged. Depending on project and the candidate's expertise and preferences, the work might range from purely analytical to largely numerical. Teaching and outreach opportunities will also be provided.

The postdoc will be employed by CNRS, France's largest and most recognized research institution. Funding is available for at least **two years of employment**. The work is to be conducted at LPTMS, a joint laboratory of CNRS and Université Paris-Sud with a markedly international atmosphere. Located in Orsay, it is **25 minutes away from central Paris via a frequent, direct commuter train**.

The gross salary for the position ranges between 2600 €/month and 3600 €/month depending on experience. Benefits include free full healthcare coverage for the postdoc and his or her dependents, generous vacations, 16-weeks fully-paid maternity leaves, free schooling from age 3 and subsidized child care for younger children. CNRS additionally subsidizes vacations, sports and cultural activities for its employees.

The position will begin at a **flexible date**, preferably in the Winter/Spring of 2017. The successful candidate will hold a Ph.D. by the start date and have strong background and research achievements in Theoretical Soft Matter, Biological and/or Statistical Physics. Applicants coming from Mechanics and Computational Physics will also be considered. Applications will comprise the names of three references, an application letter, a CV and a publications list including preprints. Informal inquiries welcome.

**Contact:**

Martin Lenz

martin.lenz@u-psud.fr

# LPTMS PhD Proposal: Etude des quasiparticules dans l’effet Hall quantique fractionnaire a plusieurs composantes

### Responsable: Thierry Jolicoeur 01 69 15 70 29

L'effet Hall quantique fractionnaire (EHQF) bien connu de la physique des electrons bidimensionnels sous champ magnetique tres fort se manifeste par la formation de liquides d'electrons presentant un gap pour toute forme d'excitations au contraire d'une mer de Fermi ordinaire. De nombreux etats de ce genre ont ete decouverts et caracterises ces vingt dernieres annees. Dans les experiences il est frequent que les electrons ne soient pas totalement polarises de spin. Les fonctions d'onde decrivant ces etats sont alors par la force des choses multicomposantes. Alors que dans les gaz d'electrons de GaAs/GaAlAs seul entre en jeu le spin, d'autres materiaux mettent en jeu encore plus de porteurs comme par exemple AlAs ou on a des degres de liberte de vallee. L'exemple qui est au coeur de ce projet est le graphene monocouche dans lequel on a une degenerescence d'un facteur quatre qui est legerement levee. Des progres experimentaux dans la suspension de monocouches rendent l'etude de l'EHQF tres interessante. De nombreux problemes nouveaux surgissent qui peuvent etre etudies par diagonalisation exacte de systemes de petite taille finie. Dans le cas de l'EHQF, un facteur favorable est la degenerescence de Landau: si l'aire du systeme est bornee alors l'espace des etats du niveau de Landau le plus bas est de dimension finie sans troncation ad-hoc. Dans le cas du graphene des experiences en cours commencent a reveler la physique des etats EQHF pour les remplissages « tiers », 2/3, 4/3, 5/3. Ces etats ne sont pas de simples copies des etats deja connus dans les gaz d'electrons bidimensionnels. Il s'agit d'etudier la nature de ces etats et de caracteriser leurs excitations qui possedent des nombres quantiques nonconventionnels comme la charge electrique fractionnaire. Notamment certains de ces etats ont des excitations dites de herissons de spins, les skyrmions qui gouvernent les proprietes de transport electriques. Il faudra utiliser toute une palette d'outils theoriques pour les comprendre.

# LPTMS PhD Proposal: Separation of Variables and Correlation Functions of Quantum Integrable Systems

### Responsable: Véronique Terras 01 69 15 78 83

Quantum integrable systems are physical models, essentially in 1+1 dimensions (quantum 1D lattice models, quantum 1D field theories, 2D exactly solvable models of statistical physics...), for which it is possible to calculate exactly some quantities of physical interest such as the spectrum of commuting quantum integrals of motion, or correlation functions. More precisely, this terminology is commonly used for models associated to an R-matrix satisfying the Yang-Baxter relation, enabling one to construct an algebra (the Yang-Baxter algebra) containing an abelian sub-algebra generated by the so-called ''transfer matrix'', which is a generating functional of the conserved quantities (including the Hamiltonian) of the system. The Yang-Baxter structure is then used to construct the eigenstates and compute the eigenvalues of the Hamiltonian, for instance by means of the algebraic Bethe ansatz (ABA) [1] or by the quantum separation of variables (SOV) [2] methods. The range of applications of the study of such integrable systems covers various domains of physics (from condensed matter and statistical physics to field and string theories) and of mathematics (quantum groups, knot theories, combinatorics...).

In recent years, important progresses have been made concerning the computation of form factors, correlation functions and structure factors, in particular for simple models solvable by ABA such as the XXZ Heisenberg spin-1/2 chain or the Lieb-Liniger model. The ABA approach has notably led to exact representations for the form factors in finite volume [3] which could be used both for the numerical of for the analytical (determination of the large-distance asymptotic behavior of the correlation functions at the thermodynamic limit [4]) study of the correlation functions and of the structure factors. These important new results led to new applications, in condensed matter physics (the numerical analysis permitted to establish connections with experimental measurements for spin chain materials) as well as in high energy physics (with notable applications in the framework of the AdS/CFT correspondence).

It happens that several integrable models, very interesting from the point of view of their applications, are not directly solvable by Bethe ansatz, whereas they are solvable by SOV, a method which therefore appears to be more general. Moreover, the SOV approach presents several interesting features with respect to ABA, one of them being the fact that it provides by construction a complete construction of the spectrum and eigenstates; this is in contrast to the ABA approach where the proof of completeness is in general a non trivial task. However, despite its promising features, the SOV approach has not yet been fully developed for the effective computation of form factors and correlation functions.

In this context, the purpose of this PhD is to understand how one can develop the SOV method towards the exact computation of form factors and correlation functions. The idea is to start with the study of the simple XXZ Heisenberg spin ½ chain, for which the results from ABA should in principle be recovered in the thermodynamic limit, but which already exhibits some interesting features which makes the application of SOV not completely trivial and probably very instructive. Then, one could try to apply the SOV approach to several models of various physical interest, not or hardly solvable by ABA. For instance one could considers spin chains with general integrable boundary conditions (interesting for the consideration of out-ofequilibrium problems), or higher rank spin chains (with possible applications in the context of the AdS/CFT correspondence).

**Keywords:** Quantum integrable models, Yang-Baxter algebra, Separation of variables, form factors,

correlation functions

**References:**

[1] L.D. Faddeev, How Algebraic Bethe Ansatz works for integrable model, arXiv:hep-th/9605187.

[2] E.K. Sklyanin, Quantum Inverse Scattering Method. Selected Topics, arXiv:hep-th/9211111.

[3] N. Kitanine, J.M. Maillet and V. Terras, Nucl. Phys. B 554 (1999) 647, arXiv:math-ph/9807020.

[4] see for instance:

N. Kitanine, K. Kozlowski, J.M. Maillet, N. A. Slavnov and V. Terras, J. Stat. Mech. (2011) P12010,

arXiv:1110.0803.

N. Kitanine, K. Kozlowski, J.M. Maillet, N. A. Slavnov and V. Terras, J. Stat. Mech. (2012) P09001,

arXiv:1206.2630.

N. Kitanine, K. K. Kozlowski, J. M. Maillet and V.Terras, J. Stat. Mech. (2014) P05011, arXiv:1312.5089.