# Project B4:

Numerical time evolution of strongly correlated quantum systems out of equilibrium

## Summary

The aim of this project is to develop and apply numerical techniques to study strongly correlated quantum systems out of equilibrium. In contrast to systems in equilibrium, the field of non-equilibrium quantum dynamics is much less developed the major difficulty being that the density matrix giving the weight for the states during the evolution, is not known apriori. In the first period of this project, we plan to extend the Lanczos time evolution with almost exponentially fast convergence with the Density Matrix Renormalization Group, to be able to deal with system sizes of experimental relevance in quantum gases and condensed matter physics. Non-equilibrium phenomena are becoming a common denominator of mesoscopic systems and quantum gases, since in both kind of systems, external potentials as well as interactions can be readily controlled. It is in fact the high degree of control that opens new perspectives in dealing with quantum phase transitions, an aspect of non-equilibrium phenomena, on which we will focus here. A particularly impressive example for the systems showing such transitions was given recently by the time evolution of a gas of 87Rb atoms, where the initial state is in the superfluid phase, whereas the evolution takes place in a deep optical lattice whose ground-state corresponds to a Mott-insulator. It was observed that the superfluid phase collapses as expected but it is recovered again after a certain time. The experiments showed furthermore, that the collaps and revival of the superfluid phase repeats itself over several periods. It can be anticipated, that in view of the experience with correlated quantum systems in equilibrium, numerical techniques will be central to the understanding of such systems out of equilibrium. For the numerical simulation of situations like the one described above, a Lanczos time evolution is planned. The Lanczos process recursively generates an orthonormal basis by repeated action of the Hamiltonian on a given state, giving rise to the so-called Krylov subspace, whose dimension is usually chosen much smaller than the actual dimension of the problem. The dimension of Krylov subspace determines the convergence to the exact solution as the exponent of the minimal time step used for the evolution. Hence, an almost exponentially fast convergence can be achieved. Due to the limitations in system size imposed by a Hilbert space whose dimension increases exponentially with the number of degrees of freedom, this technique should be extendend by the Density Matrix Renormalization Group (DMRG), while preserving the fast convergence properties of the Lanczos process. In this way we expect to theoretically predict how to dynamically tailor cooperative quantum states and possibly find new states of matter.

## Project leaders

Dr. Benjamin Deissler, Institut für Quantenmaterie, Universität Ulm

Prof. Dr. Johannes Hecker Denschlag, Institut für Quantenmaterie, Universität Ulm

Prof. Dr. Alejandro Muramatsu, Institut für Theoretische Physik III, Universität Stuttgart

## Refs & Publications

D. K. Hoffmann, B. Deissler, W. Limmer, and J. Hecker Denschlag*"Holographic method for site-resolved detection of a 2D array of ultracold atoms"*

*Appl. Phys. B,*

**122**: 227 (2016)*;*doi: 10.1007/s00340-016-6501-1

T. C. Lang, Z. Y. Meng, M. M. Scherer, S. Uebelacker, F. F. Assaad, A. Muramatsu, C. Honerkamp, and S. Wessel

*"Antiferromagnetism in the Hubbard Model on the Bernal-Stacked Honeycomb Bilayer"*

*Phys. Rev. Lett.*

**109**, 126402 (2012)*;*doi: 10.1103/PhysRevLett.109.126402

M. Hohenadler, Z. Y. Meng, T. C. Lang, S. Wessel, A. Muramatsu, and F. F. Assaad

*"Quantum phase transitions in the Kane-Mele-Hubbard model"*

*Phys. Rev. B*

**85**, 115132 (2012)*;*doi: 10.1103/PhysRevB.85.115132

L. Fritz, R. L. Doretto, S. Wessel, S. Wenzel, S. Burdin, and M. Vojta

*"Cubic interactions and quantum criticality in dimerized antiferromagnets"*

*Phys. Rev. B*

**83**, 174416 (2011)*;*doi: 10.1103/PhysRevB.83.174416

L. Bonnes and S. Wessel

*"Pair Superfluidity of Three-Body Constrained Bosons in Two Dimensions"*

*Phys. Rev. Lett.*

**106**, 185302 (2011)*;*doi: 10.1103/PhysRevLett.106.185302

A. Foussats, A. Greco, and A. Muramatsu,

*"Path integrals for dimerized quantum spin systems"*

*Nucl. Phys. B*

**842**, 225 (2011)*;*doi: 10.1016/j.nuclphysb.2010.09.001

T. Fabritius, N. Laflorencie, and S. Wessel

*"Finite-temperature ordering of dilute graphene antiferromagnets"*

*Phys. Rev. B*

**82**, 035402 (2010)*;*doi: 10.1103/PhysRevB.82.035402

Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad, and A. Muramatsu

*"Quantum spin liquid emerging in two-dimensional correlated Dirac fermions"*

*Nature*

**464**, 847 (2010)F. Heidrich-Meisner, M. Rigol, A. Muramatsu, A. E. Feiguin, and E. Dagotto

*"Ground-state reference systems for expanding correlated fermions in one dimension"*

*Phys. Rev. A*;

**78**, 013620 (2008)*arXiv: 0801.4454*

A. Muramatsu and T. Pfau (eds.)

*"Focus on Quantum Correlations in Tailored Matter"*

*New J. Phys.*

**10**, 045001 (2008)S.R. Manmana, S. Wessel, R.M. Noack, and A. Muramatsu

*"Strongly correlated fermions after a quantum quench"*

*Phys. Rev. Lett.*

**98**, 210405 (2007)M. Rigol, A. Muramatsu, and M. Olshanii

*"Hard-core bosons on optical lattices: Dynamics and relaxation in the superfluid and insulating regimes"*

*Phys. Rev. A*

**74**, 053616 (2006)K. Rodriguez, S.R. Manmana, M. Rigol, R.M. Noack, and A. Muramatsu

*"Coherent matter waves emerging from Mott-insulators"*

*New J. Phys.*

**8**, 169 (2006)M. Rigol and A. Muramatsu

*"Nonequilibrium Dynamics of Tonks-Girardeau Gases on Optical Lattices"*

*Laser Physics*

**16**, 348 (2006)S.R. Manmana, A. Muramatsu, and R.M. Noack

*"Time evolution of one-dimensional quantum many-body systems"*

*AIP Conf. Proc.*

**789**, 269 (2005)M. Rigol and A. Muramatsu

*"Free expansion of impenetrable bosons on one-dimensional optical lattices"*

*Mod. Phys. Lett. B*

**19**, 861 (2005)M. Rigol and A. Muramatsu

*"Fermionization in an expanding 1D gas of hard-core bosons"*

*Phys. Rev. Lett.*

**94**, 240403 (2005)