Context

An exciting property of a large class of condensed matter lattice systems is the emerging scale invariance at their quantum critical points where a phase transition occurs. At these points, it is understood that the low-energy physics can be captured by conformal field theories (CFTs). Various difficulties arise when it comes to modelling the critical behavior of these strongly correlated systems. One approach to modelling these critical lattice models is through tensor networks (TNs), [1] a theoretical and numerical toolbox that has been developed in our research group and is used worldwide to efficiently deal with strongly correlated quantum systems. [2]

Recently, many communities in theoretical physics have come to exploit the connection between symmetries and so-called topological defects described by a fusion category. Tensor networks fit naturally in this, since the relevant symmetries can be imposed directly in numerical algorithms, allowing for significant computational speedup. Lattice models exhibiting these symmetries can be explicitly constructed as topological TN states, while the symmetries represented by topological defects can be seen as TN operators. The student will further develop this dictionary between CFTs and critical lattice models from a topological perspective.

An approach rising in popularity is the strange correlator, which provides the critical lattice model with corresponding symmetries from top-down. What is yet to be done is to write down the local observables of the related CFT, known as the primary operators, in terms of these topological objects, and to investigate the connection to the TN operators. This is a bottom-up approach which would provide us with the building blocks of more general lattice models. [3,4] It is precisely this bottom-up approach that the student will develop.

Goal

The goal of this project is to construct a tensor network representation of the primary operators of the CFT corresponding to a class of critical lattice models. Any progress in this direction is highly beneficial for efficient and accurate numerical simulations of lattice models near criticality.

The student would first familiarise themselves with fusion category theory, as this is the natural language for discussing the symmetries and topological defects of the considered models. Parallel to this, the student would become accustomed to the numerical tensor network toolbox [2]. With these tools in hand, the first task will consist of developing a good and general TN ansatz for primary operators. Using this ansatz it is possible to calculate many quantities of interest such as correlation functions for many different critical models.

The project will be carried out in an international research environment. This project is well-suited for students with a keen interest in mathematical physics and numerical simulations. However, a background in category theory or tensor networks is not required.

[1] J. I. Cirac, D. Pérez-García, N. Schuch, F. Verstraete, arXiv:2011.12127

[2] Software page

[3] M. Sinha, et al., arXiv:2310.19703

[4] D. Aasen, et al., arXiv:1601.07185

Context

The field of condensed matter physics involves theoretical models to better understand quantum states of matter and critical phenomena, in particular phase transitions. Many of these models exhibit second-order phase transitions which can be described by a conformal field theory (CFT). Most of these CFTs are (unitary) rational CFTs, and in particular minimal models. These models also commonly fall into so-called universality classes: a renormalisation group (RG) flow reduces a theory from its ultraviolet (UV) to its infrared (IR) limit, and only the universal properties remain. This flow converges to fixed points described by CFTs. The universality classes correspond to the models where the same CFT description holds after this RG flow. Within this context, the CFTs are defined to be real, since the running coupling parameter of the RG flow is a real number.

First-order phase transitions, however, cannot immediately be captured by these CFTs. These can be interpreted as transitions from conformal to non-conformal phases, or thus a loss of conformality. This can occur when an IR fixed point annihilates with a UV fixed point and they both disappear into the complex plane. [1] Approximate conformal invariance remains at the weak first-order transition, leading to a recent proposal that this behavior can be captured by a so-called complex CFT. [2] By analytically continuing the RG parameter to imaginary values, the flows will converge to these complex CFTs. This has been shown to apply to various condensed matter/statistical physics systems, as well as in high-energy physics.

CFTs are particularly rich and useful in two dimensions, since they have an infinite amount of conserved quantities. These symmetry considerations allow us to make immense steps towards the solvability of the field theory. For this reason, the project will focus on the two-dimensional Q-state Potts model, which is a generalisation of the famous Ising model and has interesting ferromagnetic phase transition behavior depending on the value of \(Q\). In particular, for \(Q > 4\), the transitions are first-order and can be captured by complex CFTs. This has been shown analytically from the field theory perspective, [2] and recently lattice realisations of this behavior have been brought to attention. [3,4]

Goal

This project will start off with a literature study into the relevance of complex conformal field theories, and particularly in the context of the Q-state Potts model. The student will first familiarise themselves with real CFT, and how these are related to criticality and renormalisation group. Afterwards, the student is fully equipped to tackle complex CFT. Finally, the student is asked to perform some original work related to [3], which can be approached from various angles, and especially from the numerical point of view. For example, a first step could be to reproduce [3] with tensor networks, a most popular theoretical and numerical toolbox that has been developed in our research group to study strongly correlated quantum systems.

The project can be tailored to the wishes and interests of the student in terms of which research topics they want to delve into.

This project is well-suited for students with a keen interest in mathematical physics and theoretical physics. No prior knowledge on any of the mentioned topics above is required; a large part of this project concerns understanding these connections more deeply.

[1] D. Kaplan, et al., PhysRevD.80.125005

[2] V. Gorbenko, et al., arXiv:1807.11512

[3]Y. Tang , et al., arXiv:2403.00852

[4]J. Jacobson, K. Wiese, arXiv:2402.10732

Context

Topological phases of matter are highly-entangled quantum phases which are characterized by a robust ground state degeneracy and exotic anyonic excitations. Their robustness against local noise makes these topological systems prime candidates for storage and manipulation of quantum information.

It is believed that all non-chiral (2+1)d topological orders can be microscopically realized by a mechanism called string-net condensation. This mechanism was introduced by Levin and Wen in 2004 [1]. Soon after their introduction it was appreciated that these systems are efficiently represented in terms of tensor networks [1,2]. In essence, these networks assign to each physical spin a tensor which is connected via virtual bonds to its neighbors. These bonds capture the underlying entanglement patterns of the state. The Ghent quantum group has been at the forefront of both theoretical and numerical research of tensor networks.

The advantages of studying Levin-Wen models in terms of tensor networks are manifold. They not only allow for an elegant computation and representation of the different anyonic excitations but also allow for a characterization of gapped boundaries and edge modes of Levin-Wen models. Indeed, it is understood that the computational capabilities of non-chiral topological orders can be severely influenced by the presence of boundaries.

Two distinct but equivalent approaches to characterize the gapped boundaries and their corresponding Hamiltonians governing the edge dynamics have been proposed in [4] and [5]. Whereas the first approach can be directly interpreted in terms of tensor network representations [6], the second approach is expected to be generalizable to (3+1)d.

Goals

The first goal of this thesis is to compare the approaches in [4] and [5] by rephrasing the approach of [5] in terms of explicit tensors. In particular, it would be interesting to study the edge modes of Levin-Wen models in this language.

A second more ambitious goal would be to generalize the approach presented in [5] to (3+1)-dimensional Walker-Wang models [7]. The models form a generalization of Levin-Wen models to (3+1)d and a systematic study of their gapped boundaries is currently lacking.

This project is suited for a student with a strong interest in theoretical physics and abstract mathematical techniques applied to quantum lattice models.

[1] J. I. Cirac, D. Pérez-García, N. Schuch, F. Verstraete, arXiv:2011.12127

[2] Buerschaper et al., arXiv:0809.2393

[3] Levin, Wen, arXiv:0404617

[4] Kitaev, Kong, arXiv:1104.5047

[5] Hu, et al., arXiv:1706.03329

[6] Lootens, et al., arXiv:2008.11187

[7] Walker, Wang, arXiv:1104.2632

Context

A central problem in quantum sciences and technology - whether it is in the context of quantum computing, quantum sensing or quantum experiments (quantum simulation) - is the preparation of a desired target quantum state using the limited available resources (time, experimental control). The new research field of quantum control merges concepts of classical control theory with the intricacies of quantum physics to find optimal preparation protocols. While quantum systems cover essentially all dynamics in the universe, this thesis will focus on developing quantum control strategies inspired by a particular application: ultra-cold quantum many body states (see e.g. [1-2]) that we are creating in our Bose-Einstein condensate (BEC) lab.

In summary, a BEC consists of >10.000 bosonic atoms (Rb87 in our case) that are cooled to temperatures T<1 microKelvin. At these temperatures a new phase of matter emerges, the so called Bose-Einstein condensate, with all the atoms condensed into one collective quantum state. The unique feature of cold atom experiments such as ours is the high level of control: carefully tuned electromagnets and lasers provide many ‘knobs’ for manipulating the BEC and thereby realizing new experiments that can explore different aspects of the quantum realm. We have launched our experimental effort in 2022 and are now ready to start the first original experiments, in which quantum control will play a crucial role.

[1] Optimal control of complex atomic quantum systems, S. van Frank et al, arXiv:1511.02247

[2] Shortcuts to adiabaticity: concepts, methods, and applications, D. Guéry-Odelin et al, arXiv:1904.08448

Goal

The goal of the thesis is to study, develp and apply control theory in the quantum context, in particular in the context of BEC physics. A specific quantum control task that we have in mind involves the coherent splitting of the BEC in two separate clouds (A and B), with a particular relative complex phase between the clouds. This relative phase can be interpreted as a piece of quantum data that is hidden in the correlation between A and B (see [3]), which can only be read out by looking at the interference patterns that emerge in time of flight images. The aim then is to design the optimal driving protocols that allow for maximal control on this relative phase. On the one hand this will entail simulations of the famous Gross-Pitaevskii equation (with existing numerical packages and/or within a to be determined approximation), that can describe the evolution of the BEC state under different driving conditions. On the other hand, the plan is also to test optimized driving protocols that come out of this numerical work in the actual experiment.

As such this thesis consists of a blend of theory and numerical work in the context of quantum physics and quantum information, in combination with the unique opportunity to collaborate on a cutting-edge cold atom experiment. Depending on the interest and skill of the student the topic can be oriented more towards theory, more towards numerics or more towards the actual experiment.

[3] Quantum nonlocality in the presence of superselection rules and data hiding protocols, F. Verstraete, J.I. Cirac, arXiv:0302039

Context

The physics of interacting many-body systems has been of great interest to researchers for decades, as they host a variety of wildly exotic phenomena emerging from the rich interplay of quantum effects. However, a large class of these systems remain theoretically and numerically intractable due to an exponential growth of the system description. The concept of quantum simulation is a particularly elegant formulation designed to tackle this, originally proposed by Feynman [1]. It is based on the idea of engineering the Hamiltonian of an experimentally controllable quantum system, such that its dynamics can be used as a proxy to understand the physics of seemingly unrelated models that are encoded within it. Such a construction is, in fact, so flexible that it not only provides insight into condensed matter systems, but also into systems of relevance to cosmology and high-energy physics. In recent years, the spectacular advances in experimental techniques facilitating a high degree of control and manipulation of atoms has led to the emergence of cold‐atom experiments as a versatile platform for realizing such quantum simulators.

One such experiment is hosted within our own Bose-Einstein condensate (BEC) lab, wherein, thousands of Rb87 atoms are cooled down to the order of micro-Kelvins such that a new phase of matter emerges (i.e., a BEC) that is fundamentally quantum in nature. Such a phase arises from the sudden condensation of all the atoms into a collective state, resulting in a macroscopic quantum object that can be manipulated with laser spots and magnetic fields. This experimental effort was originally initiated in 2022 and we are now ready to start with the first original experiments. A particularly powerful feature of the setup is its flexible projection system that allows us to ‘paint’ arbitrary potentials onto the BEC, paving the way to encode Hamiltonians into the system. One possible route to achieve this is by applying a periodically driven external potential on the system, such that the time-averaged dynamics are governed by the encoded Hamiltonian, an approach that is broadly known as Floquet engineering [2].

While there exist several techniques for engineering new Hamiltonians, a particularly interesting one involves the manufacturing of synthetic dimensions [3]. The basic idea is to introduce a coupling between internal degrees of the system and re‐interpret their behavior as dynamics along an effective spatial dimension. This allows us to then simulate the physics of lattice models using our continuous BEC system. A simple way to realize such a construction is by leveraging the formalism of periodically driven systems as mentioned above, and early attempts at this have been corroborated by experiments as well [4]. As a result, this line of research shows considerable promise in realizing exotic quantum dynamics and is an exciting avenue to pursue, motivated by the access to our very own cold-atom experiment.

[1] Simulating Physics with Computers, Richard P. Feynman, doi:10.1007/BF02650179

[2] Periodically-driven quantum systems: Effective Hamiltonians and engineered gauge fields, N. Goldman, J. Dalibard, arXiv:1404.4373

[3] Hannah M. Price, Tomoki Ozawa, Nathan Goldman, Synthetic dimensions for cold atoms from shaking a harmonic trap, arXiv:1605.09310

[4] Bloch oscillations along a synthetic dimension of atomic trap states, C. Oliver, et. al., arXiv:2112.1064

Goal

Initial attempts at manufacturing synthetic dimensions involve rather simple approaches such as the periodic driving of a linear-gradient potential on a non-interacting BEC system [4]. The effects of including interactions as well as the capabilities of more carefully constructed driving schemes remains unexplored.

The goal of this thesis is to understand and develop one such novel periodic driving scheme to engineer a non-trivial lattice Hamiltonian, with a focus on realizing it in our BEC setup. This will involve experimenting with various driving schemes and identifying relevant observables and their experimental signatures that can be detected in our setup. To begin with, the student will familiarize themselves with the formalism of Floquet dynamics of periodically driven systems. This will enable them to formulate theoretically-motivated driving potentials and study their dynamics through numerical simulations of the Gross-Pitaevskii equation using existing software libraries. An ambitious student may also attempt to augment their schemes to mitigate certain unwanted effects, such as the micro-motion [2] resulting from Floquet driving.

As such this thesis consists of a blend of theory and numerical work in the context of quantum physics, in combination with the unique opportunity to collaborate on a cutting-edge cold atom experiment. Depending on the interest and skill of the student the topic can be oriented more towards theory, more towards numerics or more towards the actual experiment.

Context

“Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.” The new blooming field of quantum simulation [1] is turning these memorable words of Feynman, uttered more than 40 years ago, into reality. It is based on the idea of engineering the Hamiltonian of an experimentally controllable quantum system, such that its dynamics can be used as a proxy to understand the physics of seemingly unrelated models that are encoded within it. Such a construction is, in fact, so flexible that it not only provides insight into condensed matter systems, but also into systems of relevance to cosmology and high-energy physics.

In recent years, the spectacular advances in experimental techniques facilitating a high degree of control and manipulation of atoms has led to the emergence of cold‐atom experiments as a versatile platform for realizing such quantum simulators. One such experiment is hosted within our own Bose-Einstein condensate (BEC) lab, wherein, thousands of Rb87 atoms are cooled down to the order of micro-Kelvins such that a new phase of matter emerges (i.e., a BEC) that is fundamentally quantum in nature. Such a phase arises from the sudden condensation of all the atoms into a collective state, resulting in a macroscopic quantum object that can be manipulated with laser spots and magnetic fields. This experimental effort was originally initiated in 2022 and we are now ready to start with the first original experiments.

The crucial part of the set-up that gives us the finest control both in space and in time is the laser projection system that allows us to shape the light-field that interacts with the atoms. This system is based on Acousto Optical Deflectors (AOD), see for example Ref [2] where they consider attractive light fields.

The AODs contain a crystal whose refractive index can be altered via perturbing it with a sound wave. This creates effectively a diffraction grating for a light beam that passes through this crystal. By programming a simple sinusoidal waveform into the crystal, the diffraction angle of the light can be adjusted. Using different frequencies in the sine wave, this diffracts the light beam in different deflection angles. However, recently a new promising approach emerged for programming optimal diffraction gratings into the AODs, based on the holographic framework. This approach comprises of a more advanced description of the complete light field in terms of changes in amplitude and phase, see Ref [3].

[1] Simulating Physics with Computers, Richard P. Feynman, doi:10.1007/BF02650179

[2] Precise shaping of laser light by an acousto-optic deflector, D. Trypogeorgos, et al., arXiv:1307.6734

[3] Artifact-free holographic light shaping through moving acousto-optic holograms, D. Treptow, et al., doi:10.1038/s41598-021-00332-4

Goal

The goal of this thesis is to study and engineer the holographic method unto our AODs with the aim to create specific target potentials, thereby further developing our system towards a full-fledged quantum simulator. As such, the student has the opportunity to develop and apply numerical methods and quantum optics theory on a cutting-edge quantum experiment.

The student will start by familiarizing themselves with the workings of an AOD and with the theory of the holographic method on these components. Studying which algorithms, e.g. the Gerchberg-Saxton algorithm, can be adapted to the usage in our system to optimize the information of the target potential into the encodings of the AODs. Then the student will implement the holographic method on the system and compare with the naïve deflection angle method. This encompasses writing code to program the sound wave into the AODs, and taking and analyzing the resulting data of the projected potential. Depending on the interest of the student, machine learning tools such as neural networks, can be applied to improve the light shaping in an automated fashion.

Context

The renormalisation group (RG) is one of the cornerstones of modern theoretical physics and exists in various implementations across different areas of classical as well as quantum physics. One particularly insightful RG scheme in the context of classical spin models in statistical mechanics is Kadanoff’s spin blocking [1], a particular example of a real-space coarse-graining scheme [2]. While very instructive, this particular scheme does not result in a quantitatively precise description of critical phenomena. More recently, highly accurate real-space coarse-graining schemes have been constructed using the formalism of tensor networks. Hereto, the partition function of a statistical or quantum mechanical system is represented as a network of contracted tensors, where the contractions between different tensor indices encode the summations over the configuration space variables. By decomposing these tensors and contracting them, real-space transformations are obtained that result in coarse grained partition functions in terms of novel statistical variables. After the initial proposal by Levin and Nave [3], different improved and more advanced schemes have been proposed [4,5,6] (one of which is depicted below), and various strategies have been developed to extract the scaling behaviour of the system from these numerical simulations [7,8].

This technique can also be applied to quantum systems with fermionic degrees of freedom, which are then encoded using Grassmann numbers [9]. While this approach has been used in a number of studies, there are various interesting open questions:

• A more recent formalism to deal with fermions in tensor networks is based on super vector spaces [10], but this formalism has not yet been employed in the context of tensor-based RG. It would be interesting to understand whether this formalism provides a greater flexibility in how to implement the sequential coarse graining transformations.

• For relativistic quantum field theories, there are various ways to discretise the action or Hamiltonian, and thus to encode the partition function as a tensor network. This is particularly relevant in relation to the notion of the fermion doubling problem, i.e. the fact that discretising Dirac fermions necessarily gives rise to spurious low-energy degrees of freedom. These can be avoided in the one-dimensional Dirac Hamiltonian when discretising space, and it is an interesting question how this relates to the partition function where the action is isotropically discretised in (1+1) dimensional spacetime.

• Finally, different flavors of (1+1)-dimensional massless Dirac fermions can interact in a particular way that shares several features with quantum chromodynamics. This is known as the Gross-Neveu model. In particular, the Gross-Neveu interaction term is marginally relevant and results in dynamical mass generation. The behaviour of such marginally relevant interactions in tensor-based RG schemes is not well understood, and could be investigated further using the techniques from Ref [7].

Goal

It is the goal of this thesis to address some (or all) of the above questions. We will start with replicating some of the existing results from fermionic tensor-based RG (TRG) studies, thereby translating the Grassmann number formalism to super vector spaces. Code to manipulate tensors in super vector spaces is available (in both Julia and Matlab) and can be used to implement the TRG schemes. Next, we will look at the different strategies to discretise the free Dirac fermion in (1+1) spacetime dimensions and build the tensor network representation of the partition function, and study it using our TRG approach, in order to understand the differences in relation to the fermion doubling problem. Finally, if time permits, we can also study the Gross-Neveu model, thereby the selecting the discretisation strategy that seems most appropriate for the problem, and try to understand how to extract the scaling behaviour (the running coupling constant or beta function).

[1] Kadanoff, L. P. (1966). Scaling laws for Ising models near T c. Physics Physique Fizika, 2(6), 263.

[2] Efrati, E., Wang, Z., Kolan, A., & Kadanoff, L. P. (2014). Real-space renormalization in statistical mechanics. Reviews of Modern Physics, 86(2), 647. arXiv:1301.6323.

[3] Levin, M., & Nave, C. P. (2007). Tensor renormalization group approach to two-dimensional classical lattice models. Physical review letters, 99(12), 120601. arXiv:cond-mat/0611687.

[4] Xie, Z. Y., Chen, J., Qin, M. P., Zhu, J. W., Yang, L. P., & Xiang, T. (2012). Coarse-graining renormalization by higher-order singular value decomposition. Physical Review B, 86(4), 045139. arXiv:1201.1144.

[5] Evenbly, G., & Vidal, G. (2015). Tensor network renormalization. Physical review letters, 115(18), 180405. arXiv:1412.0732.

[6] Hauru, M., Delcamp, C., & Mizera, S. (2018). Renormalization of tensor networks using graph-independent local truncations. Physical Review B, 97(4), 045111. arXiv:1709.07460.

[7] Gu, Z. C., & Wen, X. G. (2009). Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Physical Review B, 80(15), 155131. arXiv:0903.1069.

[8] Ueda, A., & Oshikawa, M. (2023). Finite-size and finite bond dimension effects of tensor network renormalization. arXiv preprint arXiv:2302.06632.

[9] Gu, Z. C., Verstraete, F., & Wen, X. G. (2010). Grassmann tensor network states and its renormalization for strongly correlated fermionic and bosonic states. arXiv preprint arXiv:1004.2563.

[10] Bultinck, N., Williamson, D. J., Haegeman, J., & Verstraete, F. (2017). Fermionic matrix product states and one-dimensional topological phases. Physical Review B, 95(7), 075108. arXiv:1610.07849.

Context

When studying quantum matter or quantum field theories, one of the interesting quantities is dynamical correlation functions. Upon Fourier transformation to the momentum and frequency/energy domain, these are known under various names such as spectral functions, Green’s functions, or dynamic structure factors. These quantities are of great interest because they enable to relate analytical or computational predictions with experimental data. Indeed, dynamic structure factors of the magnetic moments in materials are directly accessible via for example inelastic neutron scattering experiments. Similarly, the retarded Green’s function of the stress-energy tensor can be related to the shear viscosity via the Kubo formula, and is of great interest in the study of quantum field theories. Indeed, this parameter serves as an input to a relativistic hydrodynamical description of nonequilibrium processes in quantum field theories, e.g. to model the quark gluon plasma that is created in heavy ion collision experiments.

All of these quantities give direct information about the spectrum or dynamics of the quantum many body system. The study of strongly interacting quantum many body systems is of course notoriously hard, because of the exponential increase of the Hilbert space. Equilibrium properties can sometimes be captured by a quantum-to-classical mapping, in which thermal quantum states are represented as classical probability distributions, which are then probed via Monte Carlo sampling. However, this approach has a number of restriction. Many interesting quantum systems (especially those involving nonzero fermion densities, such as appearing in the quark gluon plasma) give rise to nonpositive “probabilities” under this mapping, which is known as the sign problem. Furthermore, Monte Carlo sampling techniques cannot access real-time phenomena. Instead, they attempt to evaluate dynamic correlation functions in imaginary time, and then to reconstruct the proper quantities at real time via “analytic continuation”, a technique which is numerically unstable and not under good control.

An alternative technique is that of tensor networks, in which quantum wave functions are directly targeted and approximated as a contraction of a network of tensors. The feasibility of this approach is motivated by insights regarding the entanglement structure of low-energy quantum states. For one-dimensional quantum lattice systems, such as quantum spin chains, methods based on the variational class of one-dimensional tensor networks known as matrix product states, have become the de facto default computational approach and yield near-exact results. Tensor network methods are not restricted to equilibrium settings and can capture real-time evolution. While quantum entanglement does grow in nonequilibrium settings, which affects the computational cost, spectral properties at zero temperature can also be captured more directly via tensor network techniques that work directly in momentum and frequency space, at least at zero temperature. However, these techniques have not yet been developed to cope with finite temperature.

Goal

The goal of this thesis is to investigate different strategies to compute spectral functions of interacting quantum quantum lattice systems at finite temperature, using the formalism of matrix product states. As a first part, the student will follow the standard approach. Here, one first constructs the finite temperature quantum state, using the formalism of thermofield double states [1]. On top of these states, one can then compute the real-time dynamical correlation function using algorithms for time evolution, in particular, the time-dependent variational principle [2]. Green’s functions are then obtained by numerically computing the Fourier transforms to frequency and momentum space. Various extensions are possible. For example, the maximal time interval that can be simulated will limit the resolution in frequency space, but can be extended by using complex (as opposed to purely real or purely imaginary) time [3], or by signal processing techniques [1]. As a second part to this thesis, a completely novel approach will be investigated. Here, one will directly linearize the equations that result from applying the time-dependent variational principle, which would then yield direct access to the spectral information in the form of a generalised eigenvalue problem. This technique has been investigated at zero temperature [4], but not at finite temperature, where one expect to see a more interesting spectrum of resonances with a finite lifetime.

An extensive set of MPS algorithms is available in the form of open source software packages within the Quantum Group of promotor Jutho Haegeman [5]. These algorithms can be used to perform the simulations of the first part, and on top of which the new algorithm for the second part can be constructed.

[1] Banuls, M. C., Heller, M. P., Jansen, K., Knaute, J., & Svensson, V. (2020). From spin chains to real-time thermal field theory using tensor networks. Physical Review Research, 2(3), 033301. (https://arxiv.org/abs/1912.08836)

[2] Haegeman, J., Cirac, J. I., Osborne, T. J., Pižorn, I., Verschelde, H., & Verstraete, F. (2011). Time-dependent variational principle for quantum lattices. Physical review letters, 107(7), 070601. (https://arxiv.org/abs/1103.0936)

[3] Grundner, M., Westhoff, P., Kugler, F. B., Parcollet, O., & Schollwöck, U. (2023). Complex Time Evolution in Tensor Networks. arXiv preprint arXiv:2312.11705. (https://arxiv.org/abs/2312.11705)

[4] Hackl, L., Guaita, T., Shi, T., Haegeman, J., Demler, E., & Cirac, I. (2020). Geometry of variational methods: dynamics of closed quantum systems. SciPost Physics, 9(4), 048. (https://arxiv.org/abs/2004.01015)

[5] https://quantumghent.github.io/software/ or https://github.com/quantumghent/

Context

In recent years, cold atom experiments world-wide have rapidly claimed their position at the forefront of the remarkable advances taking place in our understanding of the quantum world. Such endeavours routinely push the boundaries of experimentally accessible regimes and uncover novel quantum phenomena, necessitating the advancement of ab-initio numerical techniques to understand these systems. Despite the popularity and effectiveness of tensor network techniques in simulating quantum systems, they have not seen many significant applications in these kinds of cold atom setups, aside from the setting of optical lattices.

In order to use tensor networks on such continuous systems, it is necessary to construct a network of local Hilbert spaces, effectively encoding the system on a lattice. However, a naive approach of discretizing the spatial dimension to generate a grid (i.e. finite differences) scales poorly with the number of grid points and introduces convergence issues during the variational optimization [1]. A common alternative involves expanding the wave-function with respect to a set of global basis functions (i.e. spectral methods) that are chosen based on the geometry and boundary conditions of the system. While these generally have excellent convergence properties, they introduce highly non-local terms in the Hamiltonian which invalidates the usual entanglement considerations that tensor network states are based upon.

A promising middle-ground seems to exist in the form of Discrete Variable Representations (DVR) [2], closely related to Pseudo-spectral methods. This approach amounts to the construction of a localized basis set through the application of an appropriate transformation on a spectral basis set. This allows the evaluation of local potentials on a discrete grid, while only introducing a slightly non-local kinetic energy term in the Hamiltonian. As a result, the DVR method is expected to maintain locality to a good extent as with grid-based methods, while also possessing similar convergence properties as spectral methods, giving us the best of both worlds. Such an approach has significant potential as it opens up the possibility of studying inhomogenous cold atom systems in the continuum.

[1] Michele Dolfi, Bela Bauer, Matthias Troyer, and Zoran Ristivojevic, Multigrid Algorithms for Tensor Network States, https://arxiv.org/abs/1203.6363

[2] John C. Light, Tucker Carrington Jr., Discrete Variable Representations and their Utilization, http://light-group.uchicago.edu/dvr-rev.pdf

Goal

The overall goal of this thesis is to understand the utility and limitations of Discrete Variable Representations in augmenting tensor network techniques (particularly, matrix product states) to study strongly interacting one-dimensional systems in the continuum. This will be largely facilitated by gaining familiarity with the numerical tensor network toolbox [3] developed by our group.

Over the course of the project, the student will develop modular numerical routines that facilitates experimentation with various DVRs to discretize the continuum, followed by the application of tensor network machinery to study the system. This will involve performing benchmarks with an alternative discretization scheme that has been explored recently [4], based on dividing the continuous space into segments and choosing carefully tuned basis functions within each of them (akin to finite element methods). Based on the results of these attempts, a motivated student may also wish to explore natural extensions to higher dimensions.

This project is well-suited for a student with a strong interest in computational physics and in the development of efficient numerical techniques.

[3] MPSKit.jl, https://quantumghent.github.io/software/

[4] Shovan Dutta, Anton Buyskikh, Andrew J. Daley, Erich J. Mueller, Density-Matrix Renormalization Group for Continuous Quantum Systems, https://arxiv.org/abs/2108.05366

Context

In condensed-matter physics, the everlasting search for strongly-correlated materials that do not form simple metal or insulating phases has recently led to the discovery of twisted bilayer graphene. This material is obtained by taking two sheets of graphene, and placing them on top of each other with a small twisting angle. The twisting of the two honeycomb lattices leads to a so-called moiré superlattice, which at special “magic” angles exhibits a plethora of exciting strongly-correlated phases such as superconductivity or fractional Chern insulators. This groundbreaking discovery has opened up a completely new field of designing and exploring new materials with many new physical phenomena.

One particularly interesting set of materials is the class of transition metal dichalcogenides (TMD) such as the WSe2 or WS2 Moiré superlattices. For these materials, it was shown by theory [1] and experiment [2] that the flat bands around the Fermi level can realize the physics of strongly-correlated Hubbard models on the triangular lattice. The latter model is known to host metal-insulator transitions and putative spin-liquid regimes, but the precise determination of the phase diagram and the right parameters for observing these phases are currently not known.

The reason is that ab-initio calculations for these types of materials are notoriously difficult, because they exhibit very strong electron correlations. Indeed, standard ab-initio methods such as density-functional theory (DFT) fail in the presence of strong correlations and more accurate methods become unwieldy very quickly for realistic materials. Recently, the group of Profs. Frank Verstraete and Jutho Haegeman on the one side and the group of Prof. Veronique Van Speybroeck have teamed up for tackling precisely this question. The strategy is to use advanced ab-initio methods such as extensions of DFT, GW-methods and random-phase approximations to derive effective models for the electron bands that exhibit strong correlations, and afterwards use variational entanglement-based methods (known as tensor-network methods) for solving this effective model in the regime of strong electron correlations.

Goal

In this project, we will focus solely on the second step. This means that we will assume we have derived an effective model for the WSe2 or WS2 bilayer systems, which takes the form of a Hubbard model on the triangular lattice. We will therefore solve this model using tensor networks.

In the literature, there is a well-established tensor-network method for simulating these types of model, which boils down to placing the triangular-lattice model on a cylindrical geometry and using matrix product state (MPS) techniques for solving this effective one-dimensional model. Although quite successful [2-3], this method scales exponentially with the cylinder circumference, and it is unclear whether the physics of TMD materials is accurately represented by this method.

Therefore, we will develop a methodology where we tackle the model directly on the infinite triangular lattice and impose all possible lattice symmetries. This can be done using the formalism of projected entangled-pair states (PEPS), where variational wavefunctions can be built without breaking any symmetries. However, the application of PEPS methods in this context requires some methodological advances. In particular, we will need a method for computing observables of triangular-lattice PEPS wavefunctions.

In the course of this proposal, we hope that the student will develop a performant code for simulating PEPS wavefunctions on the triangular lattice. If this first part is successful, the student can proceed to simulate the physics of the triangular-lattice Hubbard model, which is still under heavy debate in the community. Furthermore, the methodology as developed by the student will serve as an integral part of the global strategy for the ab-initio simulation of triangular-lattice materials such as the TMD class of systems.

[1] Wu, F.; Lovorn, T.; Tutuc, E.; MacDonald, A. H., Hubbard Model Physics in Transition Metal Dichalcogenide Moiré Bands. Physical Review Letters 2018, 121 (2), 026402.

[2] Wang, L.; Shih, E.-M.; Ghiotto, A.; Xian, L.; Rhodes, D. A.; Tan, C.; Claassen, M.; Kennes, D. M.; Bai, Y.; Kim, B.; Watanabe, K.; Taniguchi, T.; Zhu, X.; Hone, J.; Rubio, A.; Pasupathy, A. N.; Dean, C. R., Correlated electronic phases in twisted bilayer transition metal dichalcogenides. Nat. Mater. 2020, 19 (8), 861-866.

[3] Wilhelm Kadow, Laurens Vanderstraeten, and Michael Knap, Hole spectral function of a chiral spin liquid in the triangular lattice Hubbard model, Phys. Rev. B 106, 094417 (2022)

Context

The theoretical understanding and numerical simulation of strongly-correlated quantum materials has been one of the long-standing challenges in physics: Whereas the microscopic quantum-mechanical equations for describing e.g. electrons in a material have been known for over a century, solving these equations for a large number of them is prohibitively expensive. As a result, the use of approximate methods such as mean-field theory, Hartree-Fock theory, perturbation theory, etc have flourished and can actually yield very accurate descriptions of the relevant physics of many materials. For example, band theory often describes the electronic structure of many metallic or insulating systems, and Fermi-liquid theory describes the occurrence of quasiparticles in these systems. Another example is density functional theory (DFT), which is an independent particle model for many-electron systems that revolves around the use of exchange-correlation functionals for capturing the electron correlations; in the last fifty years, DFT has worked itself up to the default method for simulating large many-electron systems.

There are, however, still a number of materials that resist such approximate descriptions, and for which still no satisfactory understanding has been acquired. The most outstanding example in the field of condensed-matter physics is the mystery of high-Tc superconductivity: layered materials for which superconductivity persists up to temperatures at which any conventional description of superconductivity (i.e., BCS theory) would predict that a normal conductor has formed. It appears that strong quantum correlations in these systems are responsible for the superconducting properties, which falls outside the scope of traditional approximate methods such as mean field of perturbation theory. The most famous class of high-Tc superconductors are cuprates.

In the last years, a new theoretical and numerical framework has been developed that can account for strong quantum correlations or entanglement in quantum many-body systems: tensor networks. Because of their direct modeling of the quantum entanglement in many-body systems, they are typically applied to microscopic Hamiltonians for simplified toy models, which somehow capture the essential physics of a given materials. In the case of the cuprate superconductors, an effective model is the single-band Hubbard Hamiltonian, for which, indeed, tensor networks can predict the phase diagram with unprecedented precision.

Goal

In order to make progress in understanding real-life high-Tc materials such as the cuprates – and for designing other materials with superconducting properties, with an eye on technological applications – we need to go beyond toy models and come up with ab-initio calculations of these materials. In this thesis proposal, we will make use of a hybrid DFT/tensor-network method for simulating a specific cuprate material La2-xSrxCuO4, which has been verified experimentally and that probably goes beyond the simple Hubbard model.

The crucial idea in this proposal is to use DFT for obtaining an effective band model for this material, such that we can isolate the electronic degrees of freedom that exhibit strong correlations. Indeed, it is expected that most electron bands are, in fact, only weakly-correlated, and that only a few bands contribute to the phenomenon of superconductivity. The DFT solution allows to project out all these weakly-correlated bands, after which we can find real-space localized Wannier orbitals that describe the relevant electrons. We can compute hopping amplitudes between the orbitals and interaction terms. In the next stage, we use tensor networks to treat this effective strongly-correlated model head-on, using tensor networks. We will not only study the ground-state properties of this materials, but also the dynamics of the charge carriers (which are supposed to determine the mysterious Cooper pairs and the superconductivity), which can be readily measured in angle-resolved photo-emission spectroscopy.

We expect the student to collaborate with researchers of the Center for Molecular Modeling and the Quantum Group; the former group has a large expertise in DFT calculations and Wannier-localization, whereas the latter has developed the main tensor-network algorithms that will be used in this work.

[1] M. Imada, Charge Order and Superconductivity as Competing Brothers in Cuprate High-Tc Superconductors. J. Phys. Soc. Jpn. 2021, 90 (11), 111009.

[2] J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021)

[3] Boris Ponsioen, Sangwoo S. Chung, and Philippe Corboz, Period 4 stripe in the extended two-dimensional Hubbard model, Phys. Rev. B 100, 195141 (2021)

Context

New quantum technologies like quantum key distribution, quantum computation and quantum sensing are nowadays a hot topic. The main ingredient for many of these new technologies is quantum entanglement. By exploiting the sometimes counter-intuitive properties of quantum mechanics, one can do many things that is impossible for a classical computer or system. For example when two distant parties have a source of maximally entangled pairs one can create a secure communication line, which is highly attractive for our current global digital world.

The main difficulty to practically implement these quantum technologies is the creation of entanglement over long distances. Due to decoherence and attenuation one cannot easily create an entangled qubit pair and send it over to the different parties. To overcome this obstacle one can use a quantum repeater. Instead of trying to create the entanglement between two parties in one go, one can first try to establish entanglement between the parties and a trusted node. This trusted node or quantum repeater is situated at the midpoint between the two parties. After the quantum repeater is entangled with both parties it performs an entanglement swapping protocol to establish the long range entanglement between the two parties. The principle of a quantum repeater looks straightforward, but it is notoriously hard to implement due to imperfect measurements and noise.

The framework of tensor networks finds its origin in the field of quantum information and provides the ideal language to study entanglement. Furthermore it allows for a plethora of numerical tools to perform simulations of quantum mechanical systems. One particular instance of tensor networks are matrix product states (MPS). These MPS form a variational class of quantum states for one dimensional systems. MPS are very succesful in capturing the groundstates of many different spin chains. One way to construct such an MPS is by considering different ‘virtual’ pairs of maximally entangled particles. Then we can perform local projections to project these pairs to a ‘physical’ particle level. This construction is called the projected entangled pairs construction. In the case of 1D systems this construction is very reminiscent of a quantum channel made out of quantum repeaters. This is the central starting point of the thesis subject.

Goal

The overall goal of the thesis is to study and explore quantum repeaters by using tensor networks. The first part of the thesis consists of a literature study on quantum repeaters and quantum entanglement distillation. Secondly the student tries to form a tensor network description of a quantum repeater. For this goal the construction of a MPS from projected entangled pairs will be a good starting point. The student can investigate, by using this tensor network description, which quantum repeater protocols are more optimal. Lastly the student can also investigate the effect of noise and imperfect measurements on the protocols. Depending on the interest of the student one can also look at how to perform entanglement distillation with MPS. Here the student can investigate what measurements are optimal to extract maximally entangled pairs from multiple copies of a MPS.

[1] J. I. Cirac, D. Pérez-García, N. Schuch, F. Verstraete, arXiv:2011.12127

[3] K. Azuma, et al., arXiv:1309.7207

[3] K. Azuma, et al., arXiv:2212.10820

Context

The master thesis investigates the implementation of rigorous renormalization group (RRG) algorithm and its application to disordered models. Tensor Network (TN) methods [1] have been pivotal in understanding the behavior of various quantum many-body systems, especially in 1D. However it remains a challenge for traditional TN approaches to simulate models with strong disorder. Recently an algorithm called rigorous renormalization group (RRG) [2,3] has been proposed for the analysis of disordered systems. This algorithm differs fundamentally from traditional TN algorithms. RRG does not utilize variational tensor network methods, instead it operates by constructing approximations of the global groundstate in a tree-like manner. This tree-like approach is able to tackle challenging problems like critical systems and disoredered systems, even outperforming traditional algorithms.

Goal

The overall goal of the thesis is to study and explore disordered quantum systems by RRG. The first part of the thesis consists of a literature study on RRG and disordered models. The student will try to make his own implementation of RRG by using the tensor network toolbox developed by our group. Secondly the student will need to compare the RRG method to standard TN approaches to identify the benefits of this approach. Lastly the student can explore a plethora of disordered models by using RRG, to get insight in the phase diagram and behaviour of these models.

[1] J. I. Cirac, D. Pérez-García, N. Schuch, F. Verstraete, arXiv:2011.12127

[3] Roberts, et al., arXiv:1703.01994

[3] Roberts, et al., arXiv:2107.12937