## The second workshop

# "Solving the Two-time Kadanoff-Baym Equations.

Status and Open Problems"

## held 12th - 13th October, 2011 in Kiel, Germany

Physikzentrum - Leibnizstrasse 15

From the hotel "Berliner Hof" use Bus 81 in the direction "Botanischer Garten", exit "Universitätsbibliothek" (your hotel card is valid for travel). Link to Google-Maps.

Lectures will be in building LS 15 (follow signs "ITAP" and "KBE 2011"), room 230.

# Workshop Program

### Arrival date: Tuesday evening, 11th October 2011

Tuesday 8.00 p.m.: Get together in restaurant "Brasserie Madeira", Sophienblatt 38 (located next to the hotel "Berliner Hof")

### Schedule:Wednesday, 12th October 2011

8.45 a.m.: Opening by Michael Bonitz.

9.00 a.m.: Arnau Rios Huguet (30 min.) ''Initial state preparation for 1D time-dependent Green's functions simulations of nuclear reactions''

9.30 a.m.: Brent W. Barker (30 min.) ''Practicalities of density matrix elements in quantum transport calculations''

#### Break

10.30 a.m.: Karsten Balzer (PhD defense) ''Solving the Two-time Kadanoff-Baym equations. Applications to model atoms and molecules''

#### 12.00 a.m.: Lunch buffet, room 229

2.00 p.m.: Vaclav Spicka (60 min.) - confirmed!

3.00 p.m.: Sebastian Hermanns (30 min.) ''Electron correlations in model systems and the generalized Kadanoff-Baym ansatz''

#### Break

4.00 p.m.: Robert van Leeuwen (45 min.) ''Kadanoff-Baym approach to isolated and contacted finite systems: linear response and quantum transport''

4.45 p.m.: Denes Sexty (30 min.) ''Real-time effective-action approach to the Anderson quantum dot''

5.15 p.m.: Discussion 1 (*). Initial state preparation, lattice vs. continuous models, finite size effects, pair distribution ...

#### Break

7.00 p.m.: Dinner at Restaurant Forstbaumschule, Düvelsbeker Weg 46, 24105 Kiel

### Schedule:Thursday, 13th October 2011

9.00 p.m.: Thomas Gasenzer (45 min.) ''Transport and relaxation of ultracold quantum gases in one spatial dimension''

9.45 a.m.: Mathias Garny (45 min.) ''Kadanoff-Baym Equations and Baryogenesis''

10.30 a.m.: Frank Michler (45 min.) ''Finite lifetime effects on the photon emission from a quark-gluon plasma''

#### Break

11.45 p.m.: Discussion 2 (*). Self-energies, parallelization, combinations with first-principle simulations, collaborations ...

12.15 a.m.: Closing.

(*) In the discussion everybody is invited to raise open questions of his own interest and present a few slides (5-10 min.). Additional topics are welcome.

# Abstracts

#### Karsten Balzer: ''Solving the Two-time Kadanoff-Baym equations. Applications to model atoms and molecules''

In this talk, I will present parts of my PhD thesis, which deals with the solution of the two-time Kadanoff-Baym equations (KBEs) for inhomogeneous, non-lattice systems. To overcome strong numerical challenges I will outline the finite element-discrete variable representation of the nonequilibrium Green function which greatly simplifies the treatment of many-body approximations. Further, I will omit the aspect of code parallelization which is also crucial for enabling efficient calculations. As a proof-of-principle, I will present results for (i) the ground states of atomic and molecular models, (ii) the model's short-time response to intense (laser) fields, and (iii), for a four-electron quantum dot, I will discuss on double excitations, the presence of which is a clear indicator of correlations in a quantum many-body system. Throughout, electron correlations are treated on the level of the second Born approximation. [1] S. Bauch, K. Balzer and M. Bonitz, Europhys. Lett. 91, 53001 (2010). [2] K. Balzer, S. Bauch and M. Bonitz, J. Phys. Conf. Ser. 220, 012020 (2010). [3] K. Balzer, S. Bauch and M. Bonitz, Phys. Rev. A 81, 022510 (2010); ibid. 82, 033427 (2010).

#### Brent W. Barker: ''Practicalities of density matrix elements in quantum transport calculations''

Efforts are on the way to develop a practical non-equilibrium Green's functions approach to central nuclear reactions. A truncation of the far off-diagonal elements of the spatial density matrix is implemented, resulting in speedup without affecting the evolution of the system close to the diagonal. The technique is applied to a simulation of collisions between one-dimensional nuclear slabs, as well as between quasi-1D Bose-Einstein condensates, using a modified Gross-Pitaevskii equation that allows inclusion of some transverse degrees of freedom in a one dimensional simulation.

#### Mathias Garny: ''Kadanoff-Baym Equations and Baryogenesis''

The Standard Model of particle physics has proven to describe a huge amount of high-energy experimental data in an excellent way. Nevertheless, there are several theoretical and experimental questions that the Standard Model cannot answer, and the LHC collider currently searches for answers to these questions. One of the most profound experimental hints for physics beyond the Standard Model comes from astro-physical observations. Namely, a plethora of different observations points to the existence of a form of dark matter, which could originate from a weakly interacting particle with mass in the region accessible to LHC. In addition, the Standard Model cannot explain the observed excess of matter over antimatter in our cosmos. The creation of this excess could be explained, however, by an extension of the Standard Model that has been introduced in order to explain the observed smallness of the neutrino mass scale via the so-called seesaw mechanism. The Majorana nature of neutrino masses introduces a violation of lepton number. In turn, this means that an excess of leptons over antileptons can be generated in the Early Universe, when the temperatures are high enough to allow for processes that violate lepton and baryon number to occur at a sizeable rate. In order to relate the model parameters related to neutrino physics with the observed cosmic matter-antimatter asymmetry, it is necessary to describe the relevant non-equilibrium evolution in the primordial plasma. In this talk, we review recent developments to derive appropriate kinetic equations from first principles, based on Kadanoff-Baym equations. In addition, we report on progress and certain open problems in the context of a two-time approach.

#### Thomas Gasenzer: ''Transport and relaxation of ultracold quantum gases in one spatial dimension''

Non-equilibrium transport and relaxation dynamics of ultracold quantum gases is described in terms of Kadanoff-Baym dynamic equations derived from the 2PI effective action to next-to-leading in a 1/N expansion. For a bosonic Bose-Hubbard model the results are quantitatively compared to those simulations of the classical field equation. Furthermore, the thermalisation of a Fermi gas is studied both in the single-particle momentum distribution and in the spectral function. Non-thermal equilibration is found for final states in the degenerate regime, indicated by a violation of the fluctuation-dissipation relation.

#### Sebartian Hermanns, Karsten Balzer and Michael Bonitz: ''Electron correlations in model systems and the generalized Kadanoff-Baym ansatz''

For the description of many-particle systems at zero and finite temperatures the nonequilibrium Green's function (NEGF) method has become widely used in the last decade. The equations of motion for this quantity are the Keldysh-Kadanoff-Baym equations. However, finding solutions is numerically very involved when correlations are being included in a beyond-Hartree-Fock treatment. In this regard, the application of the generalized Kadanoff-Baym (GKB) ansatz [1], which reconstructs the two-time NEGF from its time diagonal, promises calculations with greatly increased spatial and temporal resolution. In this talk, we report on the current status of applying the GKB ansatz to inhomogeneous model systems, compare with previously achieved results [2] and discuss on single and double excitations in the second Born approximation [3]. In addition, we address the problem of preparing a consistent initial state, e.g., by using the method of adiabatic switching [4]. [1] P. Lipavsky, V. Spicka and B. Velicky Phys. Rev. B 34, 6933 (1986). [2] K. Balzer, S. Bauch and M. Bonitz, Phys. Rev. A 81, 022510 (2010); ibid. 82, 033427 (2010). [3] S. Hermanns, K. Balzer and M. Bonitz, to be published (2011). [4] A. Rios, B. Barker, M. Buchler and P Danielewicz, Annals of Physics 326, 1274 (2011).

#### Arnau Rios Huguet: ''Initial state preparation for 1D time-dependent Green's functions simulations of nuclear reactions''

Time-dependent simulations of collisions provide a realistic description of central nuclear reactions with minimal assumptions. In this context, non-equilibrium Green's functions techniques have the potential to include consistently the effect of many-body correlations in the static description and that of memory effects in the dynamical evolution of the system [1, 2]. As a first theoretical application of these techniques, we have considered collisions of one-dimensional nuclear slabs [3]. In this talk, I will discuss the preparation of the initial state of such collisions. In the mean-field approximation, an adiabatic switch-on of the interactions has been implemented. The potential of a similar approach for the correlated ground state will be assessed. [1] L. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962). [2] G. Baym, Phys. Rev. 127, 1391 (1962). [3] A. Rios, B. Barker, M. Buchler and P. Danielewicz, Ann. Phys. 326, 1274 (2011).

#### Frank Michler, Bjoern Schenke, Carsten Greiner, Stefan Leupold and Hendrik van Hees: ''Finite lifetime effects on the photon emission from a quark-gluon plasma''

Direct photons play an important role as electromagnetic probes from a quark-gluon plasma created in heavy-ion collisions. After being once produced, they leave the medium almost undisturbed and thus provide direct insight into the early stage of the collision. Here the question arises how non-equilibrium effects such as a finite lifetime influence the resulting photon spectra. One main result has been the occurrence of first order contributions which are kinematically forbidden in equilibrium. Describing this phenomenon in the real time Keldysh formalism is mainly accompanied by two problems, namely the consistent renormalization of the divergent contribution from the vacuum polarization and getting the resulting photon spectra convergent in the ultraviolet domain. We present two attempts to handle these issues. Our first ansatz of modelling the finite lifetime via time dependent occupation numbers allows for the renormalization of the divergent vacuum contribution. It also renders the photon spectra UV-finite if the medium is turned on and off again as to mimic a heavy ion collision. However, the problem with the UV-finiteness is not fully solved yet. We conjecture that this shortcoming results from a violation of the Ward Takahshi identities. Therefore, we follow a second approach where we introduce a Yukawa-like source term in the Lagrangian. The source field is assumed to be classical and time dependent only which assigns a time dependent mass to the fermions. This leads to the spontaneous pair creation of quarks and antiquarks and we investigate the photon emission arising from this creation process. Similar to the first approach, this ansatz allows for a consistent renormalization of the vacuum polarization. However, the control of the UV-behavior of the resulting photon spectra still reveals itself as difficult and we currently investigate if our approach allows for an accordant modification.

#### Denes Sexty: ''Real-time effective-action approach to the Anderson quantum dot''

The non-equilibrium time evolution of an Anderson quantum dot is investigated. The quantum dot is coupled between two leads forming a chemical-potential gradient. We use Kadanoff-Baym dynamic equations within a non-perturbative resummation of the s-channel bubble chains. The effect of the resummation leads to the introduction of a frequency-dependent 4-point vertex. The tunneling to the leads is taken into account exactly. The method allows the determination of the transient as well as stationary transport through the quantum dot, and results are compared with different schemes discussed in the literature (fRG, ISPI, tDMRG and QMC).

#### Vaclav Spicka: (to be announced)

...

#### Robert van Leeuwen: ''Kadanoff-Baym approach to isolated and contacted finite systems: linear response and quantum transport''

We will give an overview of a number of applications of the Kadanoff-Baym equations (KBE) in the study of electronic properties of finite systems. These systems are either isolated or connected to macroscopic leads. In the isolated systems we investigate the doubly excited states by linear reponse using the KBE at 2nd Born level. In the quantum transport we investigate the effects of Coulomb interactions in the leads and image charge effects. We further make comparison with benchmark data. In all cases the inclusion of many-body effects beyond time-dependent Hartree-Fock turns out to be crucial.

# Contacts

**Karsten Balzer**

University of Kiel, Germany

Email: balzer@theo-physik.uni-kiel.de

**Brent W. Barker**

NSCL, Michigan State University, USA

Email: barker@nscl.msu.edu

**Mathias Garny**

Technical University of Munich/DESY, Germany

Email: mathias.garny@tum.de

**Thomas Gasenzer**

University of Heidelberg, Germany

Email: T.Gasenzer@thphys.uni-heidelberg.de

**Sebastian Hermanns**

University of Kiel, Germany

Email: hermanns@theo-physik.uni-kiel.de

**Kay Kobusch**

University of Kiel, Germany

Email: kobusch@theo-physik.uni-kiel.de

**Frank Michler**

University of Frankfurt, Germany

Email: michler@th.physik.uni-frankfurt.de

**Arnau Rios Huguet**

University of Surrey, United Kingdom

Email: A.Rios@surrey.ac.uk

**Denes Sexty**

University of Heidelberg, Germany

Email: sexty.denes@gmail.com

**Vaclav Spicka**

University of Prague, Czech Republic

Email: spicka@fzu.cz

**Robert van Leeuwen**

University of Jyväskylä, Finland

Email: roleeuwe@cc.jyu.fi