**PHY ****632: ****Quantum Entanglement in Many-Body Systems**** (4)**

**Prerequisites: **PHY 303 : Quantum Mechanics I

*Learning Objectives*:

The course will develop the notion of entanglement with the aim of exploring its consequences in the field of many-body physics. The ideas of pure and mixed states, and various appropriate measures of entanglement will be introduced. Second quantization formalism will be developed for free fermions and free bosons. Techniques for computing entanglement in such systems will then be described. An attempt will be made to make contact with current research where familiar phenomena of many-body physics are being studied afresh from a quantum entanglement perspective.

*Course Contents*:

*Quantum Theory: *Linear algebra, the postulates of quantum mechanics, measurement.

*Entanglement: *Density operators, Schmidt decomposition, Einstein- Podolsky-Rosen, Bell inequality.

*Measures of entanglement: *Entropy of entanglement, Concurrence, Tangle, Positive Partial Transpose, Mutual information, Quantum discord etc.

*Second Quantization: *States of many-particle system, Fock space: creation and destruction operators, Identity of particles and the Pauli principle, Representation of Symmetric operators in Fock space, Independent particle states, Coherent states, Canonical Transformations, Bogolyubov transformation, Canonical form of the generalized density matrix, Diagonalization of quadratic Hamiltonians.

*Entanglement in Solvable Many-Particle Models: *Reduced density matrices, Note on application to DMRG, Entanglement entropy: von Neumann, and Renyi entropy, Free-particle models: fermionic hopping models, coupled oscillator models, correlation functions, Entanglement hamiltonians, correlator matrix approach to computing entanglement entropy, Schmidt form for fermions, Integrable models: Transverse field Ising models, Relation to a 2D partition function, Entanglement entropies for various chain models.

*Outlook: *Recent and current activity in the field of many body physics, open questions, hints on possible directions for research. Other models and questions being investigated.

*Suggested Books*:

- Nielsen and Chuang,
*Quantum Computation and Quantum Information* - Blaizot and Ripka,
*Quantum Theory of Finite Systems.* - J. J. Sakurai,
*Modern Quantum Mechanics*(Pearson) - D. J. Griffiths,
*Introduction of Quantum Mechanics*, 2nd Ed. (Pearson) - Ingo Peschel,
*Special Review: Entanglement in Solvable Many- Particle Models*(Braz. J. Phys (2012) 42:267-291) - Vlatko Vedral,
*Entanglement in the Second Quantization Formalism*(CEJP 2 (2003) 289-306).

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