Christian Schilling

Quantum Information Theory & Quantum Many-Body Physics

Ludwig-Maximilians-Universität München

Arnold-Sommerfeld Center for Theoretical Physics

Theresienstr. 37

80333 Munich

+49 89 2180 4594


Research Webpage

The beauty of theoretical physics makes our daily work with pencil and paper an inspiring endeavour.


Research focus: concept of fermionic entanglement, reduced density matrices, ground state methods.

Our group is carrying out research at the interface of Quantum Information Theory and Quantum Many-Body Physics. We resort to analytic approaches partly complemented/guided by computational studies to gain universal insights into interacting quantum many-body systems.

Foundation of Fermionic Correlation and Entanglement

Entanglement and correlation are some of the most fascinating concepts of modern physics. Yet, in the context of indistinguishable particles a solid foundation is still lacking. Inspired by resource theory we elaborate on the two most natural definitions of fermion entanglement and correlation. The first one describes how close our N-fermion state is to the set of particle-uncorrelated “Slater determinant”-states. The second one refers to the 2nd quantization and describes how strongly orbitals (rather than particles) are entangled and correlated. Corresponding entanglement and correlation measures follow from the underlying geometric picture of quantum states. We believe that the concept of such particle and orbital correlation provides a concise and operationally meaningful alternative to the concept of static and dynamic correlations, as used in chemistry and materials science.

Ground State Problem

Based on our interdisciplinary background and expertise in quantum many-body physics, quantum information theory and mathematical physics, we are working on a more systematic and more effective approach to the notoriously difficult ground state problem. To be more specific, let us recall that in realistic quantum many-body systems the electrons interact only by two-body forces and the interaction always exhibits some form of spatial locality. The consequences of exactly those two fundamental features of realistic systems shall be explored, quantified and exploited. In particular, we plan to show from a quite general perspective that it is the universal conflict between energy minimization and fermionic exchange symmetry which together with those two fundamental features enforces a significant reduction of the particle correlations and orbital correlations in ground states compared to generic quantum states.

Density Matrix Renormalization Group (DMRG) ansatz in quantum chemistry

The remarkable success of DMRG in lattice systems is based on the reduced spatial entanglement following from the locality of the interaction. Quite in contrast, the recent success of DMRG in quantum chemical systems is rather astonishing: Why should the respective molecular Hamiltonians exhibit any local structure on the underlying artificial lattice formed by the molecular orbitals? Our main goal is to explain the emergence of a local structure by tracing it back to a deeper origin, namely the universal conflict between energy minimization and fermionic exchange symmetry in systems of continuously confined fermions. Furthermore, limitations and shortcomings of the recent version of DMRG in quantum chemistry shall be overcome, such as its inability to recover dynamic correlations with sufficiently high precision.

Reduced Density Matrix Functional Theory

Reduced density matrix functional theory (RDMFT) is expected to replace in the future density functional theory as the workhorse of modern electronic structure calculations in physics, chemistry and materials science. By using concise tools from quantum information theory and mathematical physics we are working on a solid foundation for RDMFT. In particular, we are interested in deriving universal features of the exact functional. For instance, we recently succeeded in proving that the fermionic exchange symmetry manifests itself within RDMFT in the form of a universal “fermionic exchange force”, preventing fermionic occupation numbers from ever reaching the exact values 0 and 1. Its bosonic analogue, a “Bose Einstein Condensation”- force explains the absence of complete condensation in nature and in that sense provides a fundamental explanation for quantum depletion.


Foundation of One-Particle Reduced Density Matrix Functional Theory for Excited States

J. Liebert, F. Castillo, J.-P. Labbé, C. Schilling

Journal of Chemical Theory and Computation 18 (1), 124-140 (2022).

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In Phys. Rev. Lett. 2021, 127, 023001 a reduced density matrix functional theory (RDMFT) was proposed for calculating energies of selected eigenstates of interacting many-Fermion systems. Here, we develop a solid foundation for this so-called w-RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights w in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone's pioneering work on ground state RDMFT as the special case w = (1,0, ...) Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional's domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed.

DOI: 10.1021/acs.jctc.1c00561

Fermionic systems for quantum information people

S. Szalay, Z. Zimborás, M. Máté, G. Barcza, C. Schilling, Ö. Legeza

Journal of Physics a-Mathematical and Theoretical 54, 393001 (2021).

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The operator algebra of fermionic modes is isomorphic to that of qubits, the difference between them is twofold: the embedding of subalgebras corresponding to mode subsets and multiqubit subsystems on the one hand, and the parity superselection in the fermionic case on the other. We discuss these two fundamental differences extensively, and illustrate these through the Jordan-Wigner representation in a coherent, self-contained, pedagogical way, from the point of view of quantum information theory. Our perspective leads us to develop useful new tools for the treatment of fermionic systems, such as the fermionic (quasi-)tensor product, fermionic canonical embedding, fermionic partial trace, fermionic products of maps and fermionic embeddings of maps. We formulate these by direct, easily applicable formulas, without mode permutations, for arbitrary partitionings of the modes. It is also shown that fermionic reduced states can be calculated by the fermionic partial trace, containing the proper phase factors. We also consider variants of the notions of fermionic mode correlation and entanglement, which can be endowed with the usual, local operation based motivation, if the parity superselection rule is imposed. We also elucidate some other fundamental points, related to joint map extensions, which make the parity superselection inevitable in the description of fermionic systems.

DOI: 10.1088/1751-8121/ac0646

Ensemble Reduced Density Matrix Functional Theory for Excited States and Hierarchical Generalization of Pauli's Exclusion Principle

C. Schilling, S. Pittalis

Physical Review Letters 127, 023001 (2021).

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We propose and work out a reduced density matrix functional theory (RDMFT) for calculating energies of eigenstates of interacting many-electron systems beyond the ground state. Various obstacles which historically have doomed such an approach to be unfeasible are overcome. First, we resort to a generalization of the Ritz variational principle to ensemble states with fixed weights. This in combination with the constrained search formalism allows us to establish a universal functional of the one-particle reduced density matrix. Second, we employ tools from convex analysis to circumvent the too involved N-representability constraints. Remarkably, this identifies Valone's pioneering work on RDMFT as a special case of convex relaxation and reveals that crucial information about the excitation structure is contained in the functional's domain. Third, to determine the crucial latter object, a methodology is developed which eventually leads to a generalized exclusion principle. The corresponding linear constraints are calculated for systems of arbitrary size.

DOI: 10.1103/PhysRevLett.127.023001

Functional Theory for Bose-Einstein Condensates

J. Liebert, C. Schilling

Physical Review Research 3, 13282 (2021).

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One-particle reduced density matrix functional theory would potentially be the ideal approach for describing Bose-Einstein condensates. It namely replaces the macroscopically complex wave function by the simple one-particle reduced density matrix, and therefore provides direct access to the degree of condensation and still recovers quantum correlations in an exact manner. We initiate and establish this theory by deriving the respective universal functional F for homogeneous Bose-Einstein condensates with arbitrary pair interaction. Most importantly, the successful derivation necessitates a particle-number conserving modification of Bogoliubov theory and a solution of the common phase dilemma of functional theories. We then illustrate this approach in several bosonic systems such as homogeneous Bose gases and the Bose-Hubbard model. Remarkably, the general form of F reveals the existence of a universal Bose-Einstein condensation force which provides an alternative and more fundamental explanation for quantum depletion.

DOI: 10.1103/PhysRevResearch.3.013282

How creating one additional well can generate Bose-Einstein condensation

M. Máté, Ö. Legeza, R. Schilling, M. Yousif, C. Schilling

Communications Physics 4, 29 (2021).

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The realization of Bose-Einstein condensation in ultracold trapped gases has led to a revival of interest in this fascinating quantum phenomenon. This experimental achievement necessitated both extremely low temperatures and sufficiently weak interactions. Particularly in reduced spatial dimensionality even an infinitesimal interaction immediately leads to a departure to quasi-condensation. We propose a system of strongly interacting bosons, which overcomes those obstacles by exhibiting a number of intriguing related features: (i) The tuning of just a single control parameter drives a transition from quasi-condensation to complete condensation, (ii) the destructive influence of strong interactions is compensated by the respective increased mobility, (iii) topology plays a crucial role since a crossover from one- to ‘infinite’-dimensionality is simulated, (iv) a ground state gap opens, which makes the condensation robust to thermal noise. Remarkably, all these features can be derived by analytical and exact numerical means despite the non-perturbative character of the system.

DOI: 10.1038/s42005-021-00533-3

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