Skip to main content Accessibility help
×
Home
Hostname: page-component-54cdcc668b-gpwtm Total loading time: 0.595 Render date: 2021-03-09T05:56:21.840Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

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

Published online by Cambridge University Press:  17 July 2014

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
E-mail address:
Get access

Abstract

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

Type
Research Article
Copyright
© EDP Sciences, 2014

Access options

Get access to the full version of this content by using one of the access options below.

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), 116. CrossRefGoogle Scholar
Koma, T., Nachtergaele, B.. The complete set of ground states of the ferromagnetic XXZ chains. Adv. Theor. Math. Phys., 2 (1998), 533558. CrossRefGoogle Scholar
Nachtergaele, B., Starr, S.. Droplet states in the XXZ Heisenberg chain. Comm. Math. Phys., 218 (2001), 569607. 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), 165201. 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.

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 21 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 9th March 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The Infinite XXZ Quantum Spin Chain Revisited: Structure of Low Lying Spectral Bands and Gaps
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

The Infinite XXZ Quantum Spin Chain Revisited: Structure of Low Lying Spectral Bands and Gaps
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

The Infinite XXZ Quantum Spin Chain Revisited: Structure of Low Lying Spectral Bands and Gaps
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *