Periodic striped ground states in Ising models with competing interactions

A. Giuliani, R. Seiringer, Communications in Mathematical Physics 347 (2016) 983–1007.

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Abstract
We consider Ising models in two and three dimensions, with short range ferromagnetic and long range, power-law decaying, antiferromagnetic interactions. We let J be the ratio between the strength of the ferromagnetic to antiferromagnetic interactions. The competition between these two kinds of interactions induces the system to form domains of minus spins in a background of plus spins, or vice versa. If the decay exponent p of the long range interaction is larger than d + 1, with d the space dimension, this happens for all values of J smaller than a critical value Jc(p), beyond which the ground state is homogeneous. In this paper, we give a characterization of the infinite volume ground states of the system, for p > 2d and J in a left neighborhood of Jc(p). In particular, we prove that the quasi-one-dimensional states consisting of infinite stripes (d = 2) or slabs (d = 3), all of the same optimal width and orientation, and alternating magnetization, are infinite volume ground states. Our proof is based on localization bounds combined with reflection positivity.
Publishing Year
Date Published
2016-11-01
Journal Title
Communications in Mathematical Physics
Acknowledgement
Open access funding provided by Institute of Science and Technology (IST Austria). The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme ERC Starting Grant CoMBoS (Grant Agreement No. 239694), from the Italian PRIN National Grant Geometric and analytic theory of Hamiltonian systems in finite and infinite dimensions, and the Austrian Science Fund (FWF), project Nr. P 27533-N27. Part of this work was completed during a stay at the Erwin Schrödinger Institute for Mathematical Physics in Vienna (ESI program 2015 “Quantum many-body systems, random matrices, and disorder”), whose hospitality and financial support is gratefully acknowledged.
Volume
347
Issue
3
Page
983 - 1007
IST-REx-ID

Cite this

Giuliani A, Seiringer R. Periodic striped ground states in Ising models with competing interactions. Communications in Mathematical Physics. 2016;347(3):983-1007. doi:10.1007/s00220-016-2665-0
Giuliani, A., & Seiringer, R. (2016). Periodic striped ground states in Ising models with competing interactions. Communications in Mathematical Physics, 347(3), 983–1007. https://doi.org/10.1007/s00220-016-2665-0
Giuliani, Alessandro, and Robert Seiringer. “Periodic Striped Ground States in Ising Models with Competing Interactions.” Communications in Mathematical Physics 347, no. 3 (2016): 983–1007. https://doi.org/10.1007/s00220-016-2665-0.
A. Giuliani and R. Seiringer, “Periodic striped ground states in Ising models with competing interactions,” Communications in Mathematical Physics, vol. 347, no. 3, pp. 983–1007, 2016.
Giuliani A, Seiringer R. 2016. Periodic striped ground states in Ising models with competing interactions. Communications in Mathematical Physics. 347(3), 983–1007.
Giuliani, Alessandro, and Robert Seiringer. “Periodic Striped Ground States in Ising Models with Competing Interactions.” Communications in Mathematical Physics, vol. 347, no. 3, Springer, 2016, pp. 983–1007, doi:10.1007/s00220-016-2665-0.
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