Abstract
We point out that two classes of deformations of integrable models, developed completely independently, have deep connections and share the same algebraic origin. One class includes the \( T\overline{T} \)-deformation of 1+1 dimensional integrable quantum field theory and related solvable irrelevant deformations proposed recently. The other class is a specific type of long range integrable deformation of quantum spin chains introduced a decade ago, in the context of \( \mathcal{N} \) = 4 super-Yang-Mills theory. We show that the detailed structures of the two deformations are formally identical and therefore share many features. Both deformations preserve integrability and lead to non-local deformed theories, resulting in a change of the corresponding factorized S-matrices. We also prove a factorisation formula for the expectation value of the operators which trigger the deformation on the lattice; similar results in quantum field theory play an essential role in the solvability of such deformations. We point out that the long range deformation is a natural counterpart of the \( T\overline{T} \)-deformation for integrable spin chains, and argue that this observation leads to interesting new avenues to explore.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \)-deformed 2D quantum field theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].
G. Mussardo and P. Simon, Bosonic type S matrix, vacuum instability and CDD ambiguities, Nucl. Phys. B 578 (2000) 527 [hep-th/9903072] [INSPIRE].
A. Giveon, Comments on \( T\overline{T},J\overline{T} \) and string theory, arXiv:1903.06883 [INSPIRE].
Y. Jiang, Lectures on solvable irrelevant deformations of 2d quantum field theory, arXiv:1904.13376 [INSPIRE].
S. Dubovsky, V. Gorbenko and M. Mirbabayi, Asymptotic fragility, near AdS2 holography and \( T\overline{T} \) , JHEP 09 (2017) 136 [arXiv:1706.06604] [INSPIRE].
S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, \( T\overline{T} \) partition function from topological gravity, JHEP 09 (2018) 158 [arXiv:1805.07386] [INSPIRE].
A.J. Tolley, \( T\overline{T} \) deformations, massive gravity and non-critical strings, arXiv:1911.06142 [INSPIRE].
J. Cardy, The \( T\overline{T} \) deformation of quantum field theory as random geometry, JHEP 10 (2018) 186 [arXiv:1801.06895] [INSPIRE].
J. Cardy, \( T\overline{T} \) deformations of non-Lorentz invariant field theories, arXiv:1809.07849 [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, \( T\overline{T} \) and LST, JHEP 07 (2017) 122 [arXiv:1701.05576] [INSPIRE].
S. Datta and Y. Jiang, \( T\overline{T} \) deformed partition functions, JHEP 08 (2018) 106 [arXiv:1806.07426] [INSPIRE].
O. Aharony et al., Modular invariance and uniqueness of \( T\overline{T} \) deformed CFT, JHEP 01 (2019) 086 [arXiv:1808.02492] [INSPIRE].
M. Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, SciPost Phys. 5 (2018) 048 [arXiv:1710.08415] [INSPIRE].
B. Le Floch and M. Mezei, Solving a family of \( T\overline{T} \)-like theories, arXiv:1903.07606 [INSPIRE].
B. Le Floch and M. Mezei, KdV charges in \( T\overline{T} \) theories and new models with super-Hagedorn behavior, SciPost Phys. 7 (2019) 043 [arXiv:1907.02516] [INSPIRE].
S. Chakraborty, A. Giveon and D. Kutasov, \( T\overline{T},J\overline{T},T\overline{J} \) and string theory, J. Phys. A 52 (2019) 384003 [arXiv:1905.00051] [INSPIRE].
S. Frolov, \( T\overline{T},\tilde{J}J, JT\ and\ \tilde{J}T \) deformations, J. Phys. A 53 (2020) 025401 [arXiv:1907.12117] [INSPIRE].
A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
T. Bargheer, N. Beisert and F. Loebbert, Long-range deformations for integrable spin chains, J. Phys. A 42 (2009) 285205 [arXiv:0902.0956] [INSPIRE].
F. Loebbert, Recursion relations for long-range integrable spin chains with open boundary conditions, Phys. Rev. D 85 (2012) 086008 [arXiv:1201.0888] [INSPIRE].
N. Beisert, L. Fiévet, M. de Leeuw and F. Loebbert, Integrable deformations of the XXZ spin chain, J. Stat. Mech. 1309 (2013) P09028 [arXiv:1308.1584] [INSPIRE].
B. Pozsgay, Current operators in integrable spin chains: lessons from long range deformations, SciPost Phys. 8 (2020) 016 [arXiv:1910.12833] [INSPIRE].
O.A. Castro-Alvaredo, B. Doyon and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6 (2016) 041065 [arXiv:1605.07331] [INSPIRE].
B. Bertini, M. Collura, J. De Nardis and M. Fagotti, Transport in out-of-equilibrium X X Z chains: exact profiles of charges and currents, Phys. Rev. Lett. 117 (2016) 207201 [arXiv:1605.09790] [INSPIRE].
M. Borsi, B. Pozsgay and L. Pristyák, Current operators in Bethe Ansatz and generalized hydrodynamics: an exact quantum/classical correspondence, Phys. Rev. X 10 (2020) 011054 [arXiv:1908.07320] [INSPIRE].
G. Bonelli, N. Doroud and M. Zhu, \( T\overline{T} \)-deformations in closed form, JHEP 06 (2018) 149 [arXiv:1804.10967] [INSPIRE].
G. Arutyunov, S. Frolov and M. Zamaklar, The Zamolodchikov-Faddeev algebra for AdS5 × S5 superstring, JHEP 04 (2007) 002 [hep-th/0612229] [INSPIRE].
T. Anous and M. Guica, A general definition of JTa — deformed QFTs, arXiv:1911.02031 [INSPIRE].
V. Korepin, N. Bogoliubov and A. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, Cambridge U.K. (1993).
M.G. Tetelman, Lorentz group for two-dimensional integrable lattice systems, Sov. Phys. JETP 1981 (55) 306.
K. Sogo and M. Wadati, Boost operator and its application to quantum Gelfand-Levitan equation for Heisenberg-Ising chain with spin one-half, Prog. Theor. Phys. 69 (1983) 431.
H.B. Thacker, Corner transfer matrices and Lorentz invariance on a lattice, Physica D 18 (1986) 348.
M.P. Grabowski and P. Mathieu, Structure of the conservation laws in integrable spin chains with short range interactions, Annals Phys. 243 (1995) 299 [hep-th/9411045] [INSPIRE].
A. Pogrebkov, Hierarchy of quantum explicitly solvable and integrable models, in Bilinear integrable systems: from classical to quantum, continuous to discrete, L. Faddeev et al. eds., Springer, Germany (2006), nlin/0202043.
M. Nazarov and E. Sklyanin, Integrable hierarchy of the quantum Benjamin-Ono equation, SIGMA 9 (2013) 078 [arXiv:1309.6464].
T. Bargheer, N. Beisert and F. Loebbert, LETTER: boosting nearest-neighbour to long-range integrable spin chains, J. Stat. Mech. 0811 (2008) L11001 [arXiv:0807.5081] [INSPIRE].
E. Ilievski, M. Medenjak and T. Prosen, Quasilocal conserved operators in the isotropic Heisenberg spin-1/2 chain, Phys. Rev. Lett. 115 (2015) 120601 [arXiv:1506.05049] [INSPIRE].
E. Ilievski, M. Medenjak, T. Prosen and L. Zadnik, Quasilocal charges in integrable lattice systems, J. Stat. Mech. 1606 (2016) 064008 [arXiv:1603.00440] [INSPIRE].
H. Boos, M. Jimbo, T. Miwa, F. Smirnov and Y. Takeyama, Algebraic representation of correlation functions in integrable spin chains, Annales Henri Poincaŕe 7 (2006) 1395 [hep-th/0601132] [INSPIRE].
M. Jimbo, T. Miwa and F. Smirnov, Hidden Grassmann structure in the X X Z model III: introducing Matsubara direction, J. Phys. A 42 (2009) 304018 [arXiv:0811.0439] [INSPIRE].
J. Damerau et al., Density matrices for finite segments of Heisenberg chains of arbitrary length, J. Phys. A 40 (2007) 4439 [cond-mat/0701463].
J. Sato et al., Computation of static Heisenberg-chain correlators: control over length and temperature dependence, Phys. Rev. Lett. 106 (2011) 257201 [arXiv:1105.4447] [INSPIRE].
A. Rej, D. Serban and M. Staudacher, Planar N = 4 gauge theory and the Hubbard model, JHEP 03 (2006) 018 [hep-th/0512077] [INSPIRE].
D. Serban, A note on the eigenvectors of long-range spin chains and their scalar products, JHEP 01 (2013) 012 [arXiv:1203.5842] [INSPIRE].
N. Gromov and P. Vieira, Tailoring three-point functions and integrability IV. Theta-morphism, JHEP 04 (2014) 068 [arXiv:1205.5288] [INSPIRE].
Y. Jiang, I. Kostov, F. Loebbert and D. Serban, Fixing the quantum three-point function, JHEP 04 (2014) 019 [arXiv:1401.0384] [INSPIRE].
S. Datta, Y. Jiang and R. Tateo, \( T\overline{T} \) deformation on the lattice, to appear.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1911.11118
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Pozsgay, B., Jiang, Y. & Takács, G. \( T\overline{T} \)-deformation and long range spin chains. J. High Energ. Phys. 2020, 92 (2020). https://doi.org/10.1007/JHEP03(2020)092
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2020)092