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Draft:The c-d conjecture

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In an arXiv preprint[1], José Ignacio Latorre and Germán Sierra made the following conjecture about the upper bound of the central charge for one-dimensional quantum critical lattice Hamiltonians with nearest-neighbor interactions:

  • If the local Hilbert space dimension of the lattice model is , the maximal central charge that the model can reach is .

Intuitions

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Examples

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The currently known examples are consistent with this conjecture.

The upper bound is saturated for the SU() Uimin-Lai-Sutherland model[2][3], whose low-energy effective theory is the SU() level 1 Wess-Zumino-Witten model[4]. The local Hilbert space dimension of the lattice model is , and the SU() level 1 Wess-Zumino-Witten model has central charge .

The Reshetikhin models[5] are a family of integrable models with SO() symmetric nearest-neighbor interactions. The local Hilbert space dimension of the Reshetikhin models is , which transforms under the vector representation of SO(). These models are critical and their low-energy effective theory is the SO() level 1 Wess-Zumino-Witten model with central charge [6].

The spin- XXX models[7][8][9] are a family of integrable models with SU() symmetric nearest-neighbor interactions. The local Hilbert space dimension of the spin- XXX models is , which transforms under the spin- irreducible representation of SU(). These models are critical and their low-energy effective theory is the SU() level Wess-Zumino-Witten model[10][11] with central charge .

The parafermion models[12][13][14] are a family of integrable self-dual models with symmetric nearest-neighbor interactions. The local Hilbert space dimension of the parafermion models is . These models are critical and their low-energy effective theory is the parafermion conformal field theory[15][14] with central charge . For and , the models correspond to the critical Ising model and the critical Potts model, respectively. For , the model corresponds to a special point of the critical Ashkin-Teller model[16][17].

References

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  1. ^ Latorre, José I.; Sierra, Germán (2024). "The c-d conjecture". arXiv:2403.17242 [cond-mat.stat-mech].
  2. ^ Lai, C. K. (1974-10-01). "Lattice gas with nearest-neighbor interaction in one dimension with arbitrary statistics". Journal of Mathematical Physics. 15 (10): 1675–1676. Bibcode:1974JMP....15.1675L. doi:10.1063/1.1666522. ISSN 0022-2488.
  3. ^ Sutherland, Bill (1975-11-01). "Model for a multicomponent quantum system". Physical Review B. 12 (9): 3795–3805. Bibcode:1975PhRvB..12.3795S. doi:10.1103/PhysRevB.12.3795.
  4. ^ Affleck, Ian (December 1988). "Critical behaviour of SU(n) quantum chains and topological non-linear σ-models". Nuclear Physics B. 305 (4): 582–596. Bibcode:1988NuPhB.305..582A. doi:10.1016/0550-3213(88)90117-4. ISSN 0550-3213.
  5. ^ Reshetikhin, N. Yu. (1983-05-01). "A method of functional equations in the theory of exactly solvable quantum systems". Letters in Mathematical Physics. 7 (3): 205–213. Bibcode:1983LMaPh...7..205R. doi:10.1007/BF00400435. ISSN 1573-0530.
  6. ^ Tu, Hong-Hao; Orús, Román (2011-08-09). "Effective Field Theory for the $\mathrm{SO}(n)$ Bilinear-Biquadratic Spin Chain". Physical Review Letters. 107 (7): 077204. arXiv:1104.0494. doi:10.1103/PhysRevLett.107.077204. PMID 21902426.
  7. ^ Kulish, P. P.; Reshetikhin, N. Yu.; Sklyanin, E. K. (1981-09-01). "Yang-Baxter equation and representation theory: I". Letters in Mathematical Physics. 5 (5): 393–403. Bibcode:1981LMaPh...5..393K. doi:10.1007/BF02285311. ISSN 1573-0530.
  8. ^ Takhtajan, L. A. (1982-02-08). "The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins". Physics Letters A. 87 (9): 479–482. Bibcode:1982PhLA...87..479T. doi:10.1016/0375-9601(82)90764-2. ISSN 0375-9601.
  9. ^ Babujian, H. M. (1982-08-09). "Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spins S". Physics Letters A. 90 (9): 479–482. Bibcode:1982PhLA...90..479B. doi:10.1016/0375-9601(82)90403-0. ISSN 0375-9601.
  10. ^ Affleck, Ian (1986-03-10). "Exact critical exponents for quantum spin chains, non-linear σ-models at θ=π and the quantum hall effect". Nuclear Physics B. 265 (3): 409–447. Bibcode:1986NuPhB.265..409A. doi:10.1016/0550-3213(86)90167-7. ISSN 0550-3213.
  11. ^ Affleck, Ian; Haldane, F. D. M. (1987-10-01). "Critical theory of quantum spin chains". Physical Review B. 36 (10): 5291–5300. Bibcode:1987PhRvB..36.5291A. doi:10.1103/PhysRevB.36.5291. PMID 9942166.
  12. ^ Fateev, V. A.; Zamolodchikov, A. B. (1982-10-18). "Self-dual solutions of the star-triangle relations in ZN-models". Physics Letters A. 92 (1): 37–39. doi:10.1016/0375-9601(82)90736-8. ISSN 0375-9601.
  13. ^ Alcaraz, Francisco C.; Lima Santos, A. (1986-11-24). "Conservation laws for Z(N) symmetric quantum spin models and their exact ground state energies". Nuclear Physics B. 275 (3): 436–458. Bibcode:1986NuPhB.275..436A. doi:10.1016/0550-3213(86)90608-5. ISSN 0550-3213.
  14. ^ a b Alcaraz, F C (1987-06-21). "The critical behaviour of self-dual Z(N) spin systems: finite-size scaling and conformal invariance". Journal of Physics A: Mathematical and General. 20 (9): 2511–2526. Bibcode:1987JPhA...20.2511A. doi:10.1088/0305-4470/20/9/035. ISSN 0305-4470.
  15. ^ "Journal of Experimental and Theoretical Physics". www.jetp.ras.ru. Retrieved 2024-09-27.
  16. ^ Ashkin, J.; Teller, E. (1943-09-01). "Statistics of Two-Dimensional Lattices with Four Components". Physical Review. 64 (5–6): 178–184. Bibcode:1943PhRv...64..178A. doi:10.1103/PhysRev.64.178.
  17. ^ Kohmoto, Mahito; den Nijs, Marcel; Kadanoff, Leo P. (1981-11-01). "Hamiltonian studies of the $d=2$ Ashkin-Teller model". Physical Review B. 24 (9): 5229–5241. Bibcode:1981PhRvB..24.5229K. doi:10.1103/PhysRevB.24.5229.
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