2000 character limit reached
A solvable non-unitary fermionic long-range model with extended symmetry (2404.10164v4)
Published 15 Apr 2024 in cond-mat.str-el, hep-th, math-ph, math.MP, and nlin.SI
Abstract: We define and study a long-range version of the XX model, arising as the free-fermion point of the XXZ-type Haldane--Shastry (HS) chain. It has a description via non-unitary fermions, based on the free-fermion Temperley--Lieb algebra, and may also be viewed as an alternating $\mathfrak{gl}(1|1)$ spin chain. Even and odd length behave very differently; we focus on odd length. The model is integrable, and we explicitly identify two commuting hamiltonians. While non-unitary, their spectrum is real by PT-symmetry. One hamiltonian is chiral and quadratic in fermions, while the other is parity-invariant and quartic. Their one-particle spectra have two linear branches, realising a massless relativistic dispersion on the lattice. The appropriate fermionic modes arise from 'quasi-translation' symmetry, which replaces ordinary translation symmetry. The model exhibits exclusion statistics, like the isotropic HS chain, with even more 'extended symmetry' and larger degeneracies.
- W. M. Koo and H. Saleur, Representations of the virasoro algebra from lattice models, Nucl. Phys. B 426, 459 (1994), hep-th/9312156 .
- A. M. Gainutdinov, N. Read, and H. Saleur, Continuum limit and symmetries of the periodic gl(1|1)𝑔𝑙conditional11gl(1|1)italic_g italic_l ( 1 | 1 ) spin chain, Nucl. Phys. B 871, 245 (2013), 1112.3403 .
- F. Calogero, Exactly solvable one-dimensional many-body problems, Lett. Nuov. Cim. (1971–1985) 13, 411 (1975).
- B. Sutherland, Exact ground-state wave function for a one-dimensional plasma, Phys. Rev. Lett. 34, 1083 (1975).
- F. D. M. Haldane, Exact Jastrow–Gutzwiller resonating-valence-bond groundstate of the spin-1/2 antiferromagnetic Heisenberg chain with 1/r21superscript𝑟21/r^{2}1 / italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT exchange, Phys. Rev. Lett. 60, 635 (1988).
- B. S. Shastry, Exact solution of an s=1/2𝑠12s=1/2italic_s = 1 / 2 Heisenberg antiferromagnetic chain with long-ranged interactions, Phys. Rev. Lett. 60, 639 (1988).
- V. I. Inozemtsev, On the connection between the one-dimensional s=1/2𝑠12s=1/2italic_s = 1 / 2 Heisenberg chain and Haldane–Shastry model, J. Stat. Phys. 59, 1143 (1990).
- F. Haldane, Physics of the ideal semion gas: spinons and quantum symmetries of the integrable Haldane–Shastry spin chain, in Correlation effects in low-dimensional electron systems, Vol. 118, edited by A. Okiji and N. Kawakami (Springer, 1994) cond-mat/9401001 .
- A. P. Polychronakos, Generalized statistics in one dimension, in Topological aspects of low dimensional systems: Session LXIX. 7–31 July 1998 (2002) p. 415, hep-th/9902157 .
- A. Rej, Review of AdS/CFT integrability, Chapter I.3: Long-range spin chains, Lett. Math. Phys. 99, 85 (2012), 1012.3985 .
- F. D. M. Haldane, “Spinon gas” description of the s=1/2𝑠12s=1/2italic_s = 1 / 2 Heisenberg chain with inverse-square exchange: Exact spectrum and thermodynamics, Phys. Rev. Lett. 66, 1529 (1991).
- D. Bernard, V. Pasquier, and D. Serban, Spinons in conformal field theory, Nucl. Phys. B 428, 612 (1994), hep-th/9404050 .
- D. Uglov, The trigonometric counterpart of the Haldane–Shastry model (1995), hep-th/9508145 .
- J. Lamers, Resurrecting the partially isotropic Haldane–Shastry model, Phys. Rev. B 97, 214416 (2018), 1801.05728 .
- See Supplemental Material for more details.
- V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theories through quantum groups, Nucl. Phys. B 330, 523 (1990).
- C. M. Bender, Making sense of non-hermitian hamiltonians, Rep. Prog. Phys. 70, 947 (2007), hep-th/0703096 .
- A. Dei and A. Sfondrini, Integrable spin chain for stringy Wess–Zumino–Witten models, JHEP 2018, 1, 1806.00422 .
- M. R. Gaberdiel, R. Gopakumar, and B. Nairz, Beyond the tensionless limit: Integrability in the symmetric orbifold, 2312.13288 (2023).
- C. Korff and R. Weston, PT symmetry on the lattice: the quantum group invariant XXZ spin chain, J. Phys. A: Math. Theor. 40, 8845 (2007), math-ph/0703085 .
- C. Korff, Turning the quantum group invariant XXZ spin-chain hermitian: a conjecture on the invariant product, J. Phys. A: Math. Theor. 41, 194013 (2008), 0709.3631 .
- C. Korff, PT-invariance and representations of the Temperley–Lieb algebra on the unit circle, in Proc. RAQIS’07, edited by L. Frappat and E. Ragoucy (2007) p. 99, 0712.2205 .
- R. Klabbers and J. Lamers, How coordinate Bethe ansatz works for Inozemtsev model, Commun. Math. Phys. 390, 827 (2022), 2009.14513 .
- V. Schomerus and H. Saleur, The GL(1|1)𝐺𝐿conditional11GL(1|1)italic_G italic_L ( 1 | 1 ) WZW-model: From supergeometry to logarithmic CFT, Nucl. Phys. B 734, 221 (2006), hep-th/0510032 .
- T. Creutzig and D. Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A: Math. Theor. 46, 494006 (2013).
- A. V. Razumov and Y. G. Stroganov, Spin chains and combinatorics, J. Phys. A: Math. Gen. 34, 3185 (2001), cond-mat/0012141 .
- R. Klabbers and J. Lamers, The deformed Inozemtsev spin chain, 2306.13066 (2023).
- M. Fowler and J. A. Minahan, Invariants of the Haldane–Shastry su(n)𝑠𝑢𝑛su(n)italic_s italic_u ( italic_n ) chain, Phys. Rev. Lett. 70, 2325 (1993), cond-mat/9208016 .
- J. C. Talstra and F. D. M. Haldane, Integrals of motion of the Haldane–Shastry model, J. Phys. A: Math. Gen. 28, 2369 (1995), cond-mat/9411065 .
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.
