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We study free fermion systems with the sine-square deformation (SSD), in which the energy scale of local Hamiltonians is modified according to the scaling function f(x)=sin^2[\pi(x-1/2)/L], where x is the position of the local Hamiltonian and L is the length of the system in the x direction.
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The commutator of ℋ {\cal H} and ℋ ( ± ) {\cal H}^{(\pm)} is related to the energy current, which is valid in arbitrary dimensions. For a detailed derivation, see e.g., H. Katsura, P. A. Lee, and N. Nagaosa, Phys. Rev. Lett. 104
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H. Katsura, J. Phys. A: Math. Theor. 44
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H. Ueda, H. Nakano, K. Kusakabe, and T. Nishino, arXiv:0812.4513
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To be precise, in some rare cases that there is a quantized momentum ( k x , k y ) (k_{x},k_{y}) being very close to the Fermi surface at a = 0 a=0 , there can be a one-particle state for a = 1 a=1 which is extended to the bulk and has small | E i | |E_{i}| . However, such a state is present already at a = 0 a=0 and does not concern the change of the ground-state properties induced by the SSD. We have selected the parameters in Fig. 4
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H. Katsura, e-print arXiv:1110.2459
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M. Moliner and P. Schmitteckert, arXiv:1106.4505
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