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One-dimensional models are a theoretical playground for the investigation of strongly interacting many-body dynamics that have in recent years become experimentally realizable. While mean-field methods and other standard approaches are not applicable, the models can often be investigated with techniques of integrability and generalized hydrodynamics. Fluctuations around hydrodynamic values display remarkable robustness and have recently been observed to be anomalous, indicating additional processes besides normal diffusion.

We propose a theory of diffusive-scale dynamics in integrable and closely related models. The theory is similar to nonlinear fluctuating hydrodynamics but with important structural differences, including linear degeneracy that ensures its stability. Diffusive-scale dynamics of integrable models stems from fluctuations of ballistically propagating modes, a mechanism termed convective diffusion, which can coexist with normal diffusion. A family of cellular automata with ballistically propagating charged particles and stochastic scattering provides an exactly solvable toy model where the hydrodynamic predictions can be matched to microscopic results, revealing how anomalous fluctuations are generated by convective diffusion.

We explain the structural similarity between the hydrodynamics of the cellular automata and the quantum XXZ spin chain in the easy-axis regime where a similar phenomenology of anomalous fluctuations has been observed in equilibrium spin current fluctuations. We also discuss the Heisenberg point where the spin structure factor matches that of the Kardar-Parisi-Zhang universality class while spin fluctuations are anomalous but distinct from those of the KPZ class.

Website: courses.ipht.fr
Livestream: youtube.com/IPhT-TV (free access)
Videoconference (non-IPhT attendees): to receive the links join the course newsletter, see courses.ipht.fr/subscriptions

Website: courses.ipht.fr
Livestream: youtube.com/IPhT-TV (free access)
Videoconference (non-IPhT attendees): to receive the links join the course newsletter, see courses.ipht.fr/subscriptions

Website: courses.ipht.fr
Livestream: youtube.com/IPhT-TV (free access)
Videoconference (non-IPhT attendees): to receive the links join the course newsletter, see courses.ipht.fr/subscriptions

Abstract: 

These lectures will provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS).

• Basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation.
• Main MPS algorithms, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods.
• Brief introduction to projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction.
• Tensor-network representations of mixed states, quantum channels, and Lindblad dynamics, with applications to thermal states and open quantum systems.

Based on https://arxiv.org/abs/2606.24803.

Website: courses.ipht.fr
Livestream: youtube.com/IPhT-TV (free access)
Videoconference (non-IPhT attendees): to receive the links join the course newsletter, see courses.ipht.fr/subscriptions

Abstract: 

These lectures will provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS).

• Basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation.
• Main MPS algorithms, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods.
• Brief introduction to projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction.
• Tensor-network representations of mixed states, quantum channels, and Lindblad dynamics, with applications to thermal states and open quantum systems.

Based on https://arxiv.org/abs/2606.24803.

Website: courses.ipht.fr
Livestream: youtube.com/IPhT-TV (free access)
Videoconference (non-IPhT attendees): to receive the links join the course newsletter, see courses.ipht.fr/subscriptions

Abstract: 

These lectures will provide an introduction to tensor-network methods in quantum many-body physics, with an emphasis on matrix-product states (MPS).

• Basic tensor-network language, including graphical notation, virtual indices, bond dimensions, gauge freedom, canonical forms, QR and singular-value decompositions, and the role of entanglement in controlling the efficiency of the representation.
• Main MPS algorithms, including contractions, correlation functions, matrix-product operators, DMRG, and time-evolution methods.
• Brief introduction to projected entangled-pair states (PEPS) as a higher-dimensional generalization of MPS, together with the basic ideas behind approximate PEPS contraction.
• Tensor-network representations of mixed states, quantum channels, and Lindblad dynamics, with applications to thermal states and open quantum systems.

Based on https://arxiv.org/abs/2606.24803.

Website: courses.ipht.fr
Livestream: youtube.com/IPhT-TV (free access)
Videoconference (non-IPhT attendees): to receive the links join the course newsletter, see courses.ipht.fr/subscriptions