# P5: Edge state correlations with matrix product states

## Project Description

The computational effort for numerical studies of interacting electron systems increases dramatically with system size. Within this project, effective low-energy theories for edge states ^{(1),(2)} will be combined with a matrix-product-state (MPS)-based density matrix renormalization group (DMRG)^{(3)}. This will enable the study of realistically large systems. In particular, we will be interested in two-dimensional (2D) systems where strong correlations appear exclusively at the edges. This is (a) ordinary graphene with zigzag edge magnetism^{(4)-(6)} and (b) 2D topological insulators with enhanced density of states at the edge^{(7),(8)} . In conventional approaches the edge correlations are described by solving the complete 2D system with brute-force numerics. In our approach we will first reduce the number of degrees of freedom to those relevant for correlation physics. The emerging effective theory is then solved by a numerical method---DMRG in the present case. We have shown recently^{(1)} that a similar approach can describe edge correlations in systems with millions of lattice sites practically with QMC-precision, where conventional approaches are restricted to \( \le 10^3\) sites.

The states that are confined to edges of a 2D system have 1D character. However, they also show special features that are not shared by usual 1D modes (e.g., restricted Brillouin zone, breaking of the Fermion doubling theorem, etc.), and which have been shown to give rise to exotic behavior^{(9),(10)}. Thus, it is obviously not allowed to describe those edge states by conventional chain-like models. The first part of this project will therefore be concerned with the derivation of specialized interacting edge state models, which are especially suitable for DMRG. For half-filling and for special edge geometries, one may derive an effective Heisenberg theory for the edge states^{(1)}, which is well accessible by QMC methods (see P6). DMRG, however, is more versatile than QMC, so that we will be interested in effective edge state theories beyond the QMC realm. These are fermionic theories (a) away from half-filling and (b) with contributions that go beyond the Hubbard model with nearest-neighbor hopping on the honeycomb lattice (spin-orbit interaction^{(11)}, effective kinetic edge state energy^{(12)}). Furthermore, the quality of the Heisenberg approximation needed for the QMC-based treatment of edge state correlations will be assessed within the present project.

The various effective theories derived in the first part of the project will be studied by DMRG methods in the second part. In order to cope with the long-range interaction terms of the edge state theories, we will develop a specialized high-performance DMRG code, based on both finite and infinite MPS, using Matrix Product Operator (MPO) encodings of the Hamiltonian^{(13)}. Capturing the resulting long-range correlations accurately will require very large bond dimensions, which necessitates the incorporation of fermionic superselection rules^{(14),(15)} as well as symmetries of the theory^{(16)}. In a first step we will then study the static properties of edge states, such as the correlation functions and the magnetic response.

Much of the literature about interaction effects at graphene edges is based on mean-field methods, suggesting a simple picture based on static spin polarizations^{(5)}. This picture, however, is in contradiction with rigorous results^{
(17)
} asserting the true ground state to be a spin-singlet. With the methodology described above we are in a position to go beyond the mean-field approximation not only for the static properties, but also for the quantum dynamics^{(18)}. Based on the MPS formulation of the edge states we will study the real-time evolution of a quantum quench (see also projects P12 and P13). For this we will use a commutator-free Magnus expansion in combination with Krylov subspace propagation^{(19)}, as well as improved methods for avoiding the entropy barrier in the MPS simulation of time evolution^{(20)}. There is a strong connection to the project P6 in which another type of effective edge state theories, which are especially suitable for very large systems, is studied. Furthermore, there are overlaps with projects P1, P2, P4, and P14.

(1) M.J. Schmidt, M. Golor, T.C. Lang and S. Wessel, Phys. Rev. B **87**, 245431 (2013)

(2) M. Golor, C. Koop, T.C. Lang, S. Wessel and M. J. Schmidt, Phys. Rev. Lett. **111**, 085504 (2013)

(3) U. Schollwöck, Ann. of Phys. **326**, 96 (2011)

(4) M. Fujita, K. Wakabayashi, K. Nakada and K. Kusakabe, J. Phys. Soc. Jpn. **65**, 1920 (1996)

(5) O. V. Yazyev and M. I. Katsnelson, Phys. Rev. Lett. **100**, 047209 (2008)

(6) H. Karimi and I. Affleck, Phys. Rev. B **86**, 115446 (2012)

(7) M. J. Schmidt, Phys. Rev. B **86**, 161110(R) (2012)

(8) M. Sitte, A. Rosch and L. Fritz, arXiv:1305.1788

(9) D. J. Luitz, F. F. Assaad and M. J. Schmidt, Phys. Rev. B **83** , 195432 (2011)

(10) M. J. Schmidt, Phys. Rev. B **86** , 075458 (2012)

(11) C. L. Kane and E. J. Mele, Phys. Rev. Lett. **95**, 226801 (2005)

(12) M. J. Schmidt and D. Loss, Phys. Rev. B **82**, 085422 (2010)

(13) F. Fröwis, V. Nebendahl and W. Dür, Phys. Rev. A **81**, 062337 (2010)

(14) C. V. Kraus, N. Schuch, F. Verstraete and J. I. Cirac, Phys. Rev. A **81**, 052338 (2010)

(15) P. Corboz, G. Evenbly, F. Verstraete and G. Vidal, Phys. Rev. A **81**, 010303(R) (2010)

(16) D. Perez-Garcia, M. Wolf, M. Sanz, F. Verstraete and J. Cirac, Phys. Rev. Lett. **100**, 167202 (2008)

(17) E. H. Lieb, Phys. Rev. Lett. **62**, 1201 (1989)

(18) G. Vidal, Phys. Rev. Lett.** 93**, 040502 (2004)

(19) M. L. Wall and L. D. Car, New J. Phys. **14**, 125015 (2012)

(20) M. C. Bañuls, M. B. Hastings, F. Verstraete and J. I. Cirac, Phys. Rev. Lett.** 102**, 240603 (2009)