Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T06:28:58.517Z Has data issue: false hasContentIssue false
  • English
  • Français

The Infinite XXZ Quantum Spin Chain Revisited: Structure of LowLying Spectral Bands and Gaps

Published online by Cambridge University Press:  17 July 2014

C. Fischbacher  and
G. Stolz
C. Fischbacher
Affiliation:
School of Mathematics, Statistics and Actuarial Science University of Kent Canterbury, Kent CT2 7NF, UK
G. Stolz*
Affiliation:
Department of Mathematics, University of Alabama at Birmingham Birmingham, AL 35294, USA
*
⋆Corresponding author. E-mail: [email protected]
Get access

Abstract

We study the structure of the spectrum of the infinite XXZ quantum spin chain, ananisotropic version of the Heisenberg model. The XXZ chain Hamiltonian preserves thenumber of down spins (or particle number), allowing to represent it as a direct sum ofN-particleinteracting discrete Schrödinger-type operators restricted to the fermionic subspace. Inthe Ising phase of the model we use this representation to give a detailed determinationof the band and gap structure of the spectrum at low energy. In particular, we show thatat sufficiently strong anisotropy the so-called droplet bands are separated from higherspectral bands uniformly in the particle number. Our presentation of all necessarybackground is self-contained and can serve as an introduction to the mathematical theoryof the Heisenberg and XXZ quantum spin chains.

Keywords

XXZ spin chainHeisenberg modeldroplet spectrum
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 9 , Issue 5: Spectral problems , 2014 , pp. 44 - 72
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aizenman, M., Warzel, S.. Localization bounds for multiparticle systems. Comm. Math. Phys., 290 (2009), 903-934. CrossRefGoogle Scholar
Babbitt, D., Thomas, L.. Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. II. An explicit Plancherel formula. Comm. Math. Phys., 54 (1977), 255-278. CrossRefGoogle Scholar
Babbitt, D., Thomas, L.. Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. III. Scattering theory. J. Math. Phys., 19 (1978), 1699-1704. CrossRefGoogle Scholar
Babbitt, D., Thomas, L.. Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. IV. A completely integrable quantum system. J. Math. Anal. Appl., 72 (1979), 305-328. CrossRefGoogle Scholar
Babbitt, D., Gutkin, E.. The plancherel formula for the infinite XXZ Heisenberg spin chain. Lett. Math. Phys., 20 (1990), 91-99. CrossRefGoogle Scholar
Bethe, H.. Theorie der Metalle I: Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Phys., 71 (1931), 205-226. CrossRefGoogle Scholar
A. Borodin, I. Corwin, L. Petrov, T. Sasamoto. Spectral theory for the q -boson particle system. arXiv:1308.3475.
R. Carmona, J. Lacroix. Spectral theory of random Schrödinger operators. Probability Theory and its Applications, Birkhäuser, Boston, 1990.
Chulaevsky, V., Suhov, Y.. Multi-particle Anderson localisation: induction on the number of particles. Math. Phys. Anal. Geom., 12 (2009), 117-139. CrossRefGoogle Scholar
C. Fischbacher. On the spectrum of the XXZ spin chain, Master Thesis, Ludwig-Maximilians-Universität München, 2013. http://www.kent.ac.uk/smsas/maths/our-people/resources/thesis-cf299.pdf.
E. Gutkin. Plancherel formula and critical spectral behaviour of the infinite XXZ chain. Quantum symmetries (Clausthal, 1991), 84-98, World Sci. Publ., River Edge, NJ, 1993.
E. Gutkin. Heisenberg-Ising spin chain: Plancherel decomposition and Chebyshev polynomials. In Calogero-Moser-Sutherland Models (Montréal, QC, 1997), 177-192, CRM Ser. Math. Phys., Springer, New York, 2000.
S. Haeseler, M. Keller. Generalized solutions and spectrum for Dirichlet forms on graphs, Random walks, boundaries and spectra. 181–199, Progr. Prob., 64, Birkhäuser, Basel, 2011.
Hao, W., Nepomechie, R. I., Sommese, A. J.. On the completeness of solutions of Bethe’s equations. Phys. Rev. E, 88 (2013), 052113. CrossRefGoogle ScholarPubMed
Koma, T., Nachtergaele, B.. The spectral gap of the ferromagnetic XXZ chain. Lett. Math. Phys., 40 (1997), 1–16. CrossRefGoogle Scholar
Koma, T., Nachtergaele, B.. The complete set of ground states of the ferromagnetic XXZ chains. Adv. Theor. Math. Phys., 2 (1998), 533–558. CrossRefGoogle Scholar
Nachtergaele, B., Starr, S.. Droplet states in the XXZ Heisenberg chain. Comm. Math. Phys., 218 (2001), 569–607. CrossRefGoogle Scholar
Nachtergaele, B., Spitzer, W., Starr, S.. Droplet excitations for the spin-1/2 XXZ chain with kink boundary conditions. Ann. Henri Poincaré 8 (2007), 165–201. CrossRefGoogle Scholar
M. Reed, B. Simon. Methods of modern mathematical physics, IV. Analysis of operators. Academic Press, New York, 1978.
S. Starr. Some properties for the low-lying spectrum of the ferromagnetic, quantum XXZ spin system. PhD Thesis, UC Davis, 2001.
G. Stolz. An introduction to the mathematics of Anderson localization. Entropy and the Quantum II (Tucson, AZ, 2010), 71-108, Contemp. Math., 552 Amer. Math. Soc., Providence, RI, 2011.
Thomas, L.. Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. I, J. Math. Anal. Appl., 59 (1977), 392-414. CrossRefGoogle Scholar
J. Weidmann. Linear Operators in Hilbert Spaces, Graduate Texts in Mathematics. Volume 68, Springer, New York-Heidelberg-Berlin, 1980.