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.
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.
We develop a conditionally closed theoretical architecture for condensed-matter and solid-state physics from three physical axioms: microscopic ontic definiteness and finite localization, the physical reality of the spacetime-vacuum substrate, and persistent causal source–response coupling. Electrons, nuclei, atoms, and molecular units enter as finite topological carriers, whereas crystals, energy bands, phonons, quasiparticles, order parameters, defects, and topological phases are derived as spectral, variational, statistical, or homotopy properties of the resulting many-body system. We construct a sorted first-order language, a model-theoretic conservative extension of the underlying physical theory, and a constrained Legendre descent from a unified matter–substrate action to a self-adjoint, lower-bounded many-body Hamiltonian. A signed ownership ledger prevents double counting among self-energy, screening, correlation, electron–phonon, electromagnetic, historical-memory, and environmental contributions. For neutral stable branches, we establish a thermodynamic limit under translation covariance, two-sided almost-additivity, controlled Coulomb surface terms, and passive auxiliary realizations of causal memory. The framework derives crystal symmetry, Bloch and Wannier representations, lattice dynamics, phonon quantization, electronic bands, semiconductor statistics, Green functions, conserving Kadanoff–Baym dynamics, strong-correlation reductions, magnetism, polarons, superconductivity, disorder, localization, transport, critical phenomena, dielectric response, defects, interfaces, and low-dimensional systems. Conventional Bloch, tight-binding, nearly-free-electron, Debye, Landau, Fermi-liquid, Hubbard, Heisenberg, BCS/BdG, Boltzmann, and Kubo theories are recovered only as controlled limits with explicit validity domains and failure conditions. Fundamental-carrier topology, band topology, and interacting many-body topology are distinguished rigorously. Stability criteria are given for spectral and mobility gaps, Green-function invariants, Floquet and non-Hermitian phases, completely positive open dynamics, Liouvillian steady-state bundles, and finite-memory auxiliary systems. We also specify when thermodynamic, zero-frequency, zero-wave-number, long-time, symmetry-breaking, and regulator limits commute and when their ordering defines distinct physical observables. A terminal completion theorem identifies the conditions for pure-theory closure: semantic conservativity, action traceability, self-adjoint lower-bounded dynamics, thermodynamic-limit control, causality, positivity, conservation laws, controlled standard limits, valid topological domains, and falsifiable observable maps. This theoretical closure does not imply that the spectrum, phase diagram, or transport coefficients of any particular material have been calculated or experimentally confirmed; such tasks belong to separate material-application programs. Keywords **global realism; topological condensed matter; solid-state physics; many-body theory; causal memory; historical disturbance field; Green functions; strong correlations; superconductivity; topological phases; quantum transport; thermodynamic limit; open quantum systems; Liouvillian topology; controlled effective theories**
Non-Bloch band theory recovers the open-boundary spectrum of a non-Hermitian chain from its Bloch Hamiltonian evaluated along the generalised Brillouin zone rather than the unit circle. We develop its finite-size counterpart for banded Toeplitz matrices with real open-boundary spectrum. Reality of the spectrum forces the generalised Brillouin zone to be parametrised by an analytic polar curve, so the known sufficient condition for a real spectrum is also necessary under some mild non-degeneracy conditions. At finite size n, the eigenvalues of the original and the symmetrised Bloch Hamiltonians are given by samples at n quantised momenta from the same limiting band, although separated by a phase mismatch of order 1/n which we obtain in closed form. Iterating this correction K times yields a quantisation rule on the generalised Brillouin zone reproducing the finite spectrum with an error O(n−K−1) for any fixed K. At finite size, the generalised Brillouin zone itself is unchanged and only its parametrisation depends on n.
We formulate quantum geometry for non-Hermitian systems under open boundary conditions. By defining quantum-geometric quantities in both real-space and non-Bloch representations, we establish a unified framework beyond conventional Bloch band theory. Our central result is an exact equivalence between the real-space integrated quantum metric and a non-Bloch integrated quantum metric defined on the generalized Brillouin zone. We further introduce localized non-Bloch Wannier functions in the presence of the non-Hermitian skin effect and show that the non-Bloch integrated quantum metric gives the gauge-invariant part of their spread functional. These results establish quantum geometry as a natural framework for characterizing open-boundary non-Hermitian band structures and the localization properties encoded in skin modes.
The non-Hermitian skin effect describes the phenomenon of exponential localization of single-particle eigenstates near the boundary of the system. We consider its generalization to the many-body regime by investigating a general class of interacting fermion lattice models in Markovian open quantum systems. Therein, the elementary excitations from the "vacuum" (steady state) are given by two types of dissipative fermionic modes composed of single-fermion operators, which govern the long-time nonequilibrium dynamics. We perturbatively calculate the self-energy matrix of these bare modes in the presence of interactions, and utilize the non-Bloch band theory to derive an exact integral representation. By imposing complex momentum conservation, we obtain a simplified expression for corrections to Liouvillian spectrum that agrees well with numerical calculations to high precision. We further perform perturbative analysis of Liouvillian eigenstates and identify signatures of interaction-enhanced NHSE at the quasiparticle level, manifested as renormalization of the generalized Brillouin zone. Our results establish a diagrammatic framework for dissipative interacting fermions with non-Hermitian skin effect in a description of full-fledged Lindblad master equations, which resembles Fermi liquid theory in terms of interaction-dressed quasiparticles.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.
The subject of condensed matter physics is very rich — there are an infinite number of parameters producing a diversity of exciting phenomena. As a theorist, my goal is to distill general principles out of this complexity — to construct theories that can coherently explain many known examples altogether.This thesis is composed of several attempts to develop such theories in topics related to spontaneously symmetry breaking. A remarkable feature of many-body interacting systems is that although they are described by equations respecting various symmetries, they may spontaneously organize into a state that explicitly breaks symmetries. Examples are numerous: various types of crystalline and magnetic orders, Bose-Einstein condensates of cold atoms, superfluids of liquid helium, chiral symmetry in QCD, neutron stars, and cosmic inflation.These systems with spontaneously broken continuous symmetries have gapless excitations, so called Nambu-Goldstone bosons (NGBs). Although the properties of NGBs are well understood in Lorentz-invariant systems, surprisingly, some basic properties of NGBs such as their number and dispersion in nonrelativistic systems have not been discussed from a general perspective. In the first part of this thesis, we solve this issue by developing and analyzing an effective Lagrangian that coherently captures the low-energy, long-distance physics of many different symmetry-breaking states all at once.Next, we examine whether these NGBs originating from spontaneous symmetry breaking remain to be well-defined excitations inside a metal, where low-energy electrons near Fermi surface can collide with them. Our result is a one equation criterion that specifies whether the interactions between electrons and NGBs can be ignored, or whether it completely changes their character. In the latter case, unusual phases of matter such as non-Fermi liquids may arise; in that case, NGBs are overdamped and cannot form particle-like excitations in spite of the assumed symmetry breaking.In the last part of this thesis, we investigate the possibility of spontaneously breaking of Hamiltonian itself. The homogeneity of time is one of the most fundamental symmetries of nature, underlying the conservation of the energy. The question is whether this symmetry can be spontaneously broken, as suggested recently by Wilczek, in analogy with ordinary crystals. Contrary to his proposal that attracted a significant attention and stimulated many further studies, we prove a no-go theorem that rules out spontaneously breaking of time translation, in the ground state or in the canonical ensemble of a general Hamiltonian.