Non-perturbative breakdown of Bloch's theorem in bosonic BdG systems
Bloch's theorem,which is one of the cornerstones of condensed matter physics,has played a fundamental role in the development of many theories,such as the band theory,Fermi liquid theory,and BCS theory.It states that when the system has discrete translational symmetry,the corresponding eigenstate can be labeled by a conserved quantity,or a good quantum number-Crystal momentum.Based on Bloch's theorem,many important physical quantities can be expressed as the integral over the entire Brillouin zone or the Fermi surface.In realistic macroscopic materials,the translational symmetry is explicitly broken down due to the existence of boundaries.It is natural to ask why we can still use Bloch's theorem to understand the physical properties of real materials?In traditional textbooks of solid-state physics,a thermodynamic limit argument is used to explain the above question.Since the lattice size of the macroscopic material is very large,e.g.N～1023,its asymptotic behaviors can be described by the thermodynamic limit N → ∞.Therefore,if we fix the lattice constant,i.e.,a,the system length will extend to infinity,i.e.L=aN → ∞.Since wave functions,e.g.Ψ(x)∝ eikx,should be bounded at the infinity,e.g.|Ψ(x→±∞)|＜∞,the corresponding momentum is restricted to be real numbers.This revives Bloch's theorem.Physically,for a finite-size system with open boundary conditions,the boundary only acts the role of scattering potential and can be regarded as a perturbation to the Bloch Hamiltonian.Therefore,the eigenstate of the OBC Hamiltonian is a superposition of all the scattered Bloch waves with the same energy.In this sense,one can say that,although the existence of boundary definitely breaks the translational symmetry,Bloch's theorem is perturbatively broken down,and one can still use Bloch's theorem to understand the physical properties in real materials.Even for interacting systems,Bloch's theorem is also approximately preserved.Indeed,in a many-body system,the elementary excitations(or quasiparticles)are referred to the eigenmodes with well-defined energy,momentum,and dispersion relation.This means the excitations of the system are extended Bloch-like states in the bulk,which is consistent with the physical intuition in a disorder-free system.In this paper,we show that all the above arguments are challenged in some Hermitian bosonic systems,in which all the eigenstates,or quasiparticles are localized at the boundary,indicating that Bloch's theorem is nonperturbatively broken down.This phenomenon generalizes the concept of the non-Hermitian skin effect from dissipative systems to non-dissipative systems,and can be understood as the Hermitian skin effect in general.Here the physical origin of the Hermitian skin effect is that although the bosonic Hamiltonian is Hermitian,its elementary excitations are determined by a non-Hermitian matrix.In order to understand the quasiparticles in such systems,the so-called generalized Brillouin zone theory is necessary.Finally,based on the Bogoliubov theory,the quench dynamic is also studied to illustrate the non-perturbative breakdown of Bloch's theorem.
skin effectBloch's theoremthe generalized Brillouin zone theoryelementary excitations
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