Watanabe, Haruki A proof of the Bloch theorem for lattice models. (English) Zbl 1447.81229 J. Stat. Phys. 177, No. 4, 717-726 (2019). Summary: The Bloch theorem is a powerful theorem stating that the expectation value of the U(1) current operator averaged over the entire space vanishes in large quantum systems. The theorem applies to the ground state and to the thermal equilibrium at a finite temperature, irrespective of the details of the Hamiltonian as far as all terms in the Hamiltonian are finite ranged. In this work we present a simple yet rigorous proof for general lattice models. For large but finite systems, we find that both the discussion and the conclusion are sensitive to the boundary condition one assumes: under the periodic boundary condition, one can only prove that the current expectation value is inversely proportional to the linear dimension of the system, while the current expectation value completely vanishes before taking the thermodynamic limit when the open boundary condition is imposed. We also provide simple tight-binding models that clarify the limitation of the theorem in dimensions higher than one. Cited in 6 Documents MSC: 81V70 Many-body theory; quantum Hall effect 81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices 81T25 Quantum field theory on lattices 81T27 Continuum limits in quantum field theory 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 82B30 Statistical thermodynamics 82D25 Statistical mechanics of crystals Keywords:Bloch theorem; persistent current; many-body systems PDFBibTeX XMLCite \textit{H. Watanabe}, J. Stat. Phys. 177, No. 4, 717--726 (2019; Zbl 1447.81229) Full Text: DOI arXiv References: [1] Bohm, D., Note on a theorem of Bloch concerning possible causes of superconductivity, Phys. Rev., 75, 502 (1949) · Zbl 0032.37803 [2] Ohashi, Y.; Momoi, T., On the Bloch theorem concerning spontaneous electric current, J. Phys. Soc. Jpn., 65, 3254 (1996) [3] Yamamoto, N., Generalized Bloch theorem and chiral transport phenomena, Phys. Rev. D, 92, 085011 (2015) [4] Hikihara, T.; Kecke, L.; Momoi, T.; Furusaki, A., Vector chiral and multipolar orders in the spin-1 2 frustrated ferromagnetic chain in magnetic field, Phys. Rev. B, 78, 144404 (2008) [5] Tada, Y.; Koma, T., Two No-Go theorems on superconductivity, J. Stat. Phys., 165, 455 (2016) · Zbl 1360.82099 [6] Bachmann, S., Bols, A., Roeck, W.D., Fraas, M.: A many-body index for quantum charge transport. Commun. Math. Phys. (2019). 10.1007/s00220-019-03537-x · Zbl 1436.81159 [7] Lieb, E.; Schultz, T.; Mattis, D., Two soluble models of an antiferromagnetic chain, Ann. Phys., 16, 407 (1961) · Zbl 0129.46401 [8] Jaynes, Et, Information theory and statistical mechanics. II, Phys. Rev., 106, 620 (1957) · Zbl 0084.43701 [9] Takahiro, S.; Nakahara, M., Second law-like inequalities with quantum relative entropy: an introduction, Lectures on Quantum Computing, Thermodynamics and Statistical Physics, 125-190 (2013), Singapore: World scientific, Singapore [10] Affleck, I.; Lieb, Eh, A proof of part of Haldane’s conjecture on spin chains, Lett. Math. Phys., 12, 57 (1986) [11] Yamanaka, M.; Oshikawa, M.; Affleck, I., Nonperturbative approach to Luttinger’s theorem in one dimension, Phys. Rev. Lett., 79, 1110 (1997) [12] Koma, T., Spectral gaps of quantum Hall systems with interactions, J. Stat. Phys., 99, 313 (2000) · Zbl 1002.81055 [13] Cheung, H-F; Gefen, Y.; Riedel, Ek; Shih, W-H, Persistent currents in small one-dimensional metal rings, Phys. Rev. B, 37, 6050 (1988) [14] Oshikawa, M.; Yamanaka, M.; Affleck, I., Magnetization plateaus in spin chains: “Haldane gap” for half-integer spins, Phys. Rev. Lett., 78, 1984 (1997) [15] Kapustin, A.; Spodyneiko, L., Absence of energy currents in an equilibrium state and chiral anomalies, Phys. Rev. Lett., 123, 060601 (2019) [16] Watanabe, H.; Oshikawa, M., Inequivalent Berry phases for the bulk polarization, Phys. Rev. X, 8, 021065 (2018) [17] Jackson, Jd, Classical Electrodynamics (1999), New York: Wiley, New York · Zbl 0920.00012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.