Ruprecht-Karls-Universität Heidelberg

Tilman Enss group | Ruprecht-Karls-Universität Heidelberg


Priv.-Doz. Dr. Tilman Enss

Institut für Theoretische Physik
Universität Heidelberg
Philosophenweg 19
69120 Heidelberg

Room 205
Phone: +49 6221 54 9449
Email: enss (at)

getting there (Google maps)
my entry in the university directory (LSF)
curriculum vitæ


Summer term 2019: Advanced Statistical Physics (lecture); Statistical Physics (seminar); Many-Body Theory (research seminar)
Winter term 2018/2019: Theory of Ultracold Atoms (lecture); Many-Body Theory (research seminar)
Summer term 2018: Advanced Quantum Theory (lecture); Solitonen (BSc seminar); Many-Body Theory (research seminar)
Winter term 2017/2018: Theory of Ultracold Atoms (lecture); Many-Body Theory (research seminar)
Summer term 2017: Statistical Physics (seminar)
Winter term 2016/2017: Theory of Ultracold Atoms (seminar)
Summer term 2016: Condensed Matter Theory
Winter term 2015/2016: Many-body physics with ultracold atoms
Summer term 2015: Condensed Matter Theory
Winter term 2014/2015: Theory of Ultracold Atoms (seminar)
Summer term 2014: Condensed Matter Theory
Winter term 2013/2014: Special topics in quantum mechanics (seminar)
Summer term 2013: Quantum mechanics
Winter term 2012/2013: Many-body methods in solid state physics
more on teaching

Team members

Dr. Nicolò Defenu (Postdoc)
M.Sc. Moritz Drescher (PhD student)
B.Sc. Hans Böhringer (Master student)
Till Johann (Bachelor student)

Former students and team members

Andrea Colcelli (visiting PhD student)
M.Sc. Volker Karle (Heidelberg 2018: Binary Bose mixtures in 2D)
M.Sc. Martin Braß (Heidelberg 2017: Phase transitions and stability in resonant Bose-Fermi mixtures)
M.Sc. Sergej Trenkenschu (Heidelberg 2017: Quench dynamics of a Fermi gas)
B.Sc. Dominik Lorenz (Heidelberg 2016: Quench dynamics in the Hubbard model)
B.Sc. Daniel Issing (Heidelberg 2015: Bethe ansatz and quench dynamics for 1D integrable quantum systems)
B.Sc. Bruno Faigle-Cedzich (Heidelberg 2015: Shear viscosity of two-component Fermi gas)

Research interests

My research focuses on many-body theory, mainly dynamics and transport in strongly correlated quantum systems, ranging from ultracold atomic gases to superconductors and quantum wires.
I work on the development of several modern and advanced theoretical tools including the functional renormalization group (fRG), Hamiltonian flows, Luttinger-Ward calculations, transfer-matrix DMRG and the newly developed Lightcone RG.
  • Quench dynamics in 1d chains and Many-body localization (MBL), using the Lightcone renormalization group
  • Many-body dynamics and transport in interacting ultracold atomic gases
  • Quantum critical behavior in disordered correlated electron systems
  • Nonequilibrium dynamics and transport in quantum dots and Luttinger liquids
  • Raman and optical spectroscopy in cuprates
  • Functional renormalization group method (fRG)
  • Aging behavior in stochastic systems without detailed balance
my articles on Google Scholar


Isolated quantum systems and universality under extreme conditions, 07/2016—06/2020

Recent work

Dynamical criticality and domain-wall coupling in long-range Hamiltonians

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Dynamical quantum phase transitions hold a deep connection to the underlying equilibrium physics of the quench Hamiltonian. In a recent study [J.~C.~Halimeh et al., arXiv:1810.07187], it has been numerically demonstrated that the appearance of anomalous cusps in the Loschmidt return rate coincides with the presence of bound domain walls in the spectrum of the quench Hamiltonian. Here, we consider transverse-field Ising chains with power-law and exponentially decaying interactions, and show that by removing domain-wall coupling via a truncated Jordan-Wigner transformation onto a Kitaev chain with long-range hopping and pairing, anomalous dynamical criticality is no longer present. This indicates that bound domain walls are necessary for anomalous cusps to appear in the Loschmidt return rate. We also calculate the dynamical phase diagram of the Kitaev chain with long-range hopping and pairing, which in the case of power-law couplings is shown to exhibit rich dynamical criticality including a doubly critical dynamical phase.

N. Defenu, T. Enss, and J.C. Halimeh,

Real-space dynamics of attractive and repulsive polarons in Bose-Einstein condensates

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We investigate the formation of a Bose polaron when a single impurity in a Bose-Einstein condensate is quenched from a non-interacting to an attractively interacting state in the vicinity of a Feshbach resonance. We use a beyond-Fröhlich Hamiltonian to describe both sides of the resonance and a coherent-state variational ansatz to compute the time evolution of boson density profiles in position space. We find that on the repulsive side of the Feshbach resonance, the Bose polaron performs long-lived oscillations, which is surprising given that the two-body problem has only one bound state coupled to a continuum. They arise due to interference between multiply occupied bound states and therefore can be only found with many-body approaches such as the coherent-state ansatz. Furthermore, an impurity introduced at non-zero velocity on the repulsive side is periodically slowed down or even arrested before speeding up again.

M. Drescher, M. Salmhofer, and T. Enss,
Phys. Rev. A 99, 023601 (2019).

Universal Spin Transport and Quantum Bounds for Unitary Fermions

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We review recent advances in experimental and theoretical understanding of spin transport in strongly interacting Fermi gases. The central new phenomenon is the observation of a lower bound on the (bare) spin diffusivity in the strongly interacting regime. Transport bounds are of broad interest for the condensed matter community, with a conceptual similarity to observed bounds in shear viscosity and charge conductivity. We discuss the formalism of spin hydrodynamics, how dynamics are parameterized by transport coefficients, the effect of confinement, the role of scale invariance, the quasi-particle picture, and quantum critical transport. We conclude by highlighting open questions, such as precise theoretical bounds, relevance to other phases of matter, and extensions to lattice systems.

T. Enss and J. H. Thywissen,
invited review for Annu. Rev. Condens. Matter Phys. 10, 85-106 (2019).

Quantum scale anomaly and spatial coherence in a 2D Fermi superfluid

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Quantum anomalies are violations of classical scaling symmetries caused by quantum fluctuations. Although they appear prominently in quantum field theory to regularize divergent physical quantities, their influence on experimental observables is difficult to discern. Here, we discovered a striking manifestation of a quantum anomaly in the momentum-space dynamics of a 2D Fermi superfluid of ultracold atoms. We measured the position and pair momentum distribution of the superfluid during a breathing mode cycle for different interaction strengths across the BEC-BCS crossover. Whereas the system exhibits self-similar evolution in the weakly interacting BEC and BCS limits, we found a violation in the strongly interacting regime. The signature of scale-invariance breaking is enhanced in the first-order coherence function. In particular, the power-law exponents that characterize long-range phase correlations in the system are modified due to this effect, indicating that the quantum anomaly has a significant influence on the critical properties of 2D superfluids.

P. A. Murthy, N. Defenu, L. Bayha, M. Holten, P. M. Preiss, T. Enss, and S. Jochim,

Dynamical critical scaling of long-range interacting quantum magnets

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Slow variations (quenches) of the magnetic field across the paramagnetic-ferromagnetic phase transition of spin systems produce heat. In systems with short-range interactions the heat exhibits universal power-law scaling as a function of the quench rate, known as Kibble-Zurek scaling. In this work we analyze slow quenches of the magnetic field in the Lipkin-Meshkov-Glick (LMG) model, which describes fully connected quantum spins. We analytically determine the quantum contribution to the residual heat...

N. Defenu, T. Enss, M. Kastner, and G. Morigi,
Phys. Rev. Lett. 121, 240403 (2018).

High-temperature pairing in a strongly interacting two-dimensional Fermi gas

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The nature of the normal phase of strongly correlated fermionic systems is an outstanding question in quantum many-body physics. We use spatially resolved radio-frequency spectroscopy to measure pairing energy of fermions across a wide range of temperatures and interaction strengths in a two-dimensional gas of ultracold fermionic atoms. We observed many-body pairing at temperatures far above the critical temperature for superfluidity. In the strongly interacting regime, the pairing energy in the normal phase considerably exceeds the intrinsic two-body binding energy of the system and shows a clear dependence on local density. This implies that pairing in this regime is driven by many-body correlations, rather than two-body physics. Our findings show that pairing correlations in strongly interacting two-dimensional fermionic systems are remarkably robust against thermal fluctuations.

Puneet A. Murthy, Mathias Neidig, Ralf Klemt, Luca Bayha, Igor Boettcher, Tilman Enss, Marvin Holten, Gerhard Zürn, Philipp M. Preiss, and Selim Jochim,
Science 359, 452-455 (2018).

Nonperturbative RG treatment of amplitude fluctuations for φ4 topological phase transitions

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The study of the Berezinskii-Kosterlitz-Thouless transition in two-dimensional φ4 models can be performed in several representations, and the amplitude-phase (AP) Madelung parametrization is a natural way to study the contribution of density fluctuations to nonuniversal quantities. We introduce a functional renormalization group scheme in AP representation where amplitude fluctuations are integrated first to yield an effective sine-Gordon model with renormalized superfluid stiffness...

Nicolò Defenu, Andrea Trombettoni, István Nándori, and Tilman Enss,
Phys. Rev. B 96, 174505 (2017).

Exploring the ferromagnetic behaviour of a repulsive Fermi gas through spin dynamics

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Ferromagnetism is a manifestation of strong repulsive interactions between itinerant fermions in condensed matter. Whether short-ranged repulsion alone is sufficient to stabilize ferromagnetic correlations in the absence of other effects, such as peculiar band dispersions or orbital couplings, is, however, unclear. Here, we investigate ferromagnetism in the minimal framework of an ultracold Fermi gas with short-range repulsive interactions tuned via a Feshbach resonance...

G. Valtolina, F. Scazza, A. Amico, A. Burchianti, A. Recati, T. Enss, M. Inguscio, M. Zaccanti, and G. Roati,
Nature Phys. 13, 704 (2017).

Observation of quantum-limited spin transport in strongly interacting two-dimensional Fermi gases

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We measure the transport and thermodynamic properties of two-dimensional ultracold Fermi gases through their demagnetization in a magnetic field gradient. Using a phase-coherent spin-echo sequence, we are able to distinguish bare spin diffusion from the Leggett-Rice effect, in which demagnetization is slowed by the precession of spin current around the local magnetization. When the two-dimensional scattering length is comparable to the inverse Fermi wave vector, we find that the bare transverse spin diffusivity reaches a minimum of 1.7(6)ℏ/m...

C. Luciuk, S. Smale, F. Böttcher, H. Sharum, B. A. Olsen, S. Trotzky, T. Enss, and J. H. Thywissen,
Phys. Rev. Lett. 118, 130405 (2017).

slides of talk in Trieste, 2017.

Many-body localization in infinite chains

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We investigate the phase transition between an ergodic and a many-body localized phase in infinite anisotropic spin-1/2 Heisenberg chains with binary disorder. Starting from the Neel state, we analyze the decay of antiferromagnetic order and the growth of entanglement entropy during unitary time evolution...

T. Enss, F. Andraschko, and J. Sirker,
Phys. Rev. B 95, 045121 (2017).

Equation of state of ultracold fermions in the 2D BEC-BCS crossover region
PRL Editors' suggestion
Viewpoint in Physics: Journey from Classical to Quantum in Two Dimensions, by Meera M. Parish

synopsis image We report the experimental measurement of the equation of state of a two-dimensional Fermi gas with attractive s-wave interactions throughout the crossover from a weakly coupled Fermi gas to a Bose gas of tightly bound dimers as the interaction strength is varied. We demonstrate that interactions lead to a renormalization of the density of the Fermi gas by several orders of magnitude. We compare our data near the ground state and at finite temperature with predictions for both fermions and bosons from quantum Monte Carlo simulations and Luttinger-Ward theory. Our results serve as input for investigations of close-to-equilibrium dynamics and transport in the two-dimensional system.

Boettcher, L. Bayha, D. Kedar, P. A. Murthy, M. Neidig, M. G. Ries, A. N. Wenz, G. Zürn, S. Jochim, and T. Enss,
Phys. Rev. Lett. 116, 045303 (2016).

Viewpoint in Physics: Journey from Classical to Quantum in Two Dimensions, by Meera M. Parish.
Ultrakalte Atome in zwei Dimensionen (popular article in German, Phys. Unserer Zeit 47, 113 (2016)).

Nonlinear spin diffusion and spin rotation in a trapped Fermi gas

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Transverse spin diffusion in a polarized, interacting Fermi gas leads to the Leggett-Rice effect, where the spin current precesses around the local magnetization. With a spin-echo sequence both the transverse diffusivity and the spin-rotation parameter γ are obtained; the sign of γ reveals the repulsive or attractive character of the effective interaction. In a trapped Fermi gas the spin diffusion equations become nonlinear, and their numerical solution exhibits an inhomogeneous spin state even at the spin echo time. While the microscopic diffusivity and γ increase at weak coupling, their apparent values inferred from the trap-averaged magnetization saturate in agreement with a recent experiment for a dilute ultracold Fermi gas.

Tilman Enss, Phys. Rev. A 91, 023614 (2015).

Observation of the Leggett-Rice effect in a unitary Fermi gas
PRL Editors' suggestion

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We observe that the diffusive spin current in a strongly interacting degenerate Fermi gas of 40K precesses about the local magnetization. As predicted by Leggett and Rice, precession is observed both in the Ramsey phase of a spin-echo sequence, and in the nonlinearity of the magnetization decay. At unitarity, we measure a Leggett-Rice parameter γ=1.08(9) and a bare transverse spin diffusivity D0=2.3(4)ℏ/m for a normal-state gas initialized with full polarization and at one fifth of the Fermi temperature, where m is the atomic mass. For a unitary gas, γ→0 as temperature is increased to the Fermi temperature. Tuning the scattering length a, we find that a sign change in γ occurs in the range 0<(kFa)-1≤1.3, where kF is the Fermi momentum. We discuss how γ reveals the effective interaction strength of the gas, such that a change in γ indicates a switching of branch, between a repulsive and an attractive Fermi gas.

S. Trotzky, S. Beattie, C. Luciuk, S. Smale, A. B. Bardon, T. Enss, E. Taylor, S. Zhang, and J. H. Thywissen,
Phys. Rev. Lett. 114, 015301 (2015).

Purification and many-body localization in cold atomic gases

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We propose to observe many-body localization in cold atomic gases by realizing a Bose-Hubbard chain with binary disorder and studying its non-equilibrium dynamics. In particular, we show that measuring the difference in occupation between even and odd sites, starting from a prepared density-wave state, provides clear signatures of localization. As hallmarks of the many-body localized phase we confirm, furthermore, a logarithmic increase of the entanglement entropy in time and Poissonian level statistics. Our numerical density-matrix renormalization group calculations for infinite system size are based on a purification approach which allows to perform the disorder average exactly, thus producing data without any statistical noise, and with maximal simulation times of up to a factor 10 longer than in the clean case.

Felix Andraschko, Tilman Enss, and Jesko Sirker, Phys. Rev. Lett. 113, 217201 (2014).

Universal equation of state and pseudogap in the two-dimensional Fermi gas

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We determine the thermodynamic properties and the spectral function for a homogeneous two- dimensional Fermi gas in the normal state using the Luttinger-Ward, or self-consistent T-matrix, approach. The density equation of state deviates strongly from that of the ideal Fermi gas even for moderate interactions, and our calculations suggest that temperature has a pronounced effect on the pressure in the crossover from weak to strong coupling, consistent with recent experiments. We also compute the superfluid transition temperature for a finite system in the crossover region. There is a pronounced pseudogap regime above the transition temperature: the spectral function shows a Bogoliubov-like dispersion with backbending, and the density of states is significantly suppressed near the chemical potential. The contact density at low temperatures increases with interaction and compares well with both experiment and zero-temperature Monte Carlo results.

Marianne Bauer, Meera M. Parish, and Tilman Enss, Phys. Rev. Lett. 112, 135302 (2014).

Viewpoint: Crossing a Quantum Fluid Divide

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Ultracold atoms provide a platform for studying the transition between bosonic and fermionic superfluidity in 2D quantum fluids.

Tilman Enss, Physics 7, 9 (2014).

Damping of the quadrupole mode in a two-dimensional Fermi gas

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In a recent experiment [E. Vogt et al., Phys. Rev. Lett. 108, 070404 (2012)], quadrupole and breathing modes of a two-dimensional Fermi gas were studied. We model these collective modes by solving the Boltzmann equation via the method of phase-space moments up to fourth order, including in-medium effects on the scattering cross section. In our analysis, we use a realistic Gaussian potential deformed by the presence of gravity and magnetic field gradients. We conclude that the origin of the experimentally observed damping of the quadrupole mode, especially in the weakly interacting (or even non-interacting) case, cannot be explained by these mechanisms.

Silvia Chiacchiera, Dany Davesne, Tilman Enss, and Michael Urban, Phys. Rev. A 88, 053616 (2013).
M. Urban, S. Chiacchiera, D. Davesne, T. Enss, and P.-A. Pantel, J. Phys.: Conf. Ser. 497, 012028 (2014).

Transverse spin diffusion in strongly interacting Fermi gases

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We compute spin diffusion in a dilute Fermi gas at arbitrary temperature, polarization and strong interaction in the normal phase. While the longitudinal spin diffusivity depends weakly on polarization and diverges for small temperatures, the transverse spin diffusivity D has a strong polarization dependence and approaches a finite value for T→0 in the Fermi liquid phase. For a 3D unitary Fermi gas at infinite scattering length the diffusivities reach a minimum near the quantum limit of diffusion ℏ/m in the quantum degenerate regime and are strongly suppressed by medium scattering, and we discuss the importance of the spin-rotation effect. In two dimensions, D attains a minimum at strong coupling -1 ≤ ln(kFa2D) ≤ 1 and reaches D ~ 0.2...0.3 ℏ/m at large polarization. These values are consistent with recent measurements of two-dimensional ultracold atomic gases in the strong coupling regime...

Tilman Enss, Phys. Rev. A 88, 033630 (2013).
slides of talk at conference "Cold Atoms and beyond", Aarhus 2014
slides of talk at conference "Recent Progress in Many-Body Theories", Rostock 2013

Quantum mechanical limitations to spin diffusion in the unitary Fermi gas

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We compute spin transport in the unitary Fermi gas using the strong-coupling Luttinger-Ward theory. In the quantum degenerate regime the spin diffusivity attains a minimum value of Ds ~ 1.3 ℏ/m approaching the quantum limit of diffusion for a particle of mass m. Conversely, the spin drag rate reaches a maximum value of Γsd ~ 1.2 kBTF/ℏ in terms of the Fermi temperature TF...

Tilman Enss and Rudolf Haussmann, Phys. Rev. Lett. 109, 195303 (2012).
slides of talk at TU Darmstadt
slides of talk at Delta 2013 workshop

Shear viscosity and spin diffusion in a two-dimensional Fermi gas

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We investigate the temperature dependence of the shear viscosity and spin diffusion in a two-dimensional Fermi gas with contact interactions, as realized in ultra-cold atomic gases. We describe the transport coefficients in terms of a Boltzmann equation and present a full numerical solution for the degenerate gas. In contrast to previous works we take the medium effects due to finite density fully into account. This effect reduces the viscosity to entropy ratio, η/s, by a factor of three, and similarly for spin diffusion.

Tilman Enss, Carolin Küppersbusch, and Lars Fritz, Phys. Rev. A 86, 013617 (2012).

Quantum critical transport in the unitary Fermi gas

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The thermodynamic and transport properties of the unitary Fermi gas at finite temperature T are governed by a quantum critical point at T=0 and zero density. We compute the universal shear viscosity to entropy ratio η/s in the high-temperature quantum critical regime T≫|μ| and find that this strongly coupled quantum fluid comes close to perfect fluidity η/s=ℏ/4πkB. Using a controlled large-N expansion we show that already at the first non-trivial order the equation of state and the Tan contact density C agree well with the most recent experimental measurements and theoretical Luttinger-Ward and Bold Diagrammatic Monte Carlo calculations.

Tilman Enss, Phys. Rev. A 86, 013616 (2012).
slides of talk at ERG 2012 conference

Fermi polarons in two dimensions

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We theoretically analyze inverse radiofrequency (rf) spectroscopy experiments in two-component Fermi gases. We consider a small number of impurity atoms interacting strongly with a bath of majority atoms. In two-dimensional geometries we find that the main features of the rf spectrum correspond to an attractive polaron and a metastable repulsive polaron. Our results suggest that the attractive polaron has been observed in a recent experiment [Phys. Rev. Lett. 106, 105301 (2011)].

Richard Schmidt, Tilman Enss, Ville Pietilš, and Eugene Demler, Phys. Rev. A 85, 021602(R) (2012).

Lightcone renormalization and quantum quenches in one-dimensional Hubbard models

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The Lieb-Robinson bound implies that the unitary time evolution of an operator can be restricted to an effective light cone for any Hamiltonian with short-range interactions. Here we present a very efficient renormalization group algorithm based on this light cone structure to study the time evolution of prepared initial states in the thermodynamic limit in one-dimensional quantum systems...

Tilman Enss and Jesko Sirker, New J. Phys. 14, 023008 (2012).
Light cone renormalization group (LCRG) source code
Movie of impurity dynamics

Excitation spectra and rf-response near the polaron-to-molecule transition from the functional renormalization group

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A light impurity in a Fermi sea undergoes a transition from a polaron to a molecule for increasing interaction. We develop a new method to compute the spectral functions of the polaron and molecule in a unified framework based on the functional renormalization group with full self-energy feedback. We discuss the energy spectra and decay widths of the attractive and repulsive polaron branches as well as the molecular bound state and confirm the scaling of the excited state decay rate near the transition. The quasi-particle weight of the polaron shifts from the attractive to the repulsive branch across the transition, while the molecular bound state has a very small residue characteristic for a composite particle. We propose an experimental procedure to measure the repulsive branch in a Li6 Fermi gas using rf-spectroscopy and calculate the corresponding spectra.

Richard Schmidt and Tilman Enss, Phys. Rev. A 83, 063620 (2011).
slides of talk at ERG 2014 conference

Viscosity and Scale Invariance of the Unitary Fermi Gas

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The unitary Fermi gas, a scale invariant quantum fluid, appears to be the most perfect non-relativistic fluid that exists in nature: the ratio between its shear viscosity and the entropy density exhibits a minimum near the superfluid transition temperature, whose value is larger than the string theory bound ℏ/4πkB by a factor of about seven.

Tilman Enss, Rudolf Haussmann, and Wilhelm Zwerger, Annals of Physics 326, 770–796 (2011).
slides of viscosity talk in Nancy
Konzept: Carmen Enss, Büro für Denkmalforschung und Denkmalpflege