Abstract
Critical 2D Ising model with a boundary magnetic field is arguably the simplest QFT that interpolates between two non-trivial fixed points. We use the diagonalising Bogolyubov transformation for this model to investigate two quantities. Firstly we explicitly construct an RG interface operator that is a boundary condition changing operator linking the free boundary condition with the one with a boundary magnetic field. We investigate its properties and in particular show that in the limit of large magnetic field this operator becomes the dimension 1/16 primary field linking the free and fixed boundary conditions. Secondly we use Schrieffer-Wolff method to construct effective Hamiltonians both near the UV and IR fixed points.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Ghoshal and A.B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841 [Erratum ibid. A 9 (1994) 4353] [hep-th/9306002] [INSPIRE].
R. Chatterjee and A.B. Zamolodchikov, Local magnetization in critical Ising model with boundary magnetic field, Mod. Phys. Lett. A 9 (1994) 2227 [hep-th/9311165] [INSPIRE].
R. Chatterjee, Exact partition function and boundary state of critical Ising model with boundary magnetic field, Mod. Phys. Lett. A 10 (1995) 973 [hep-th/9412169] [INSPIRE].
A. Konechny, Ising model with a boundary magnetic field: an example of a boundary flow, JHEP 12 (2004) 058 [hep-th/0410210] [INSPIRE].
G.Z. Toth, A study of truncation effects in boundary flows of the Ising model on the strip, J. Stat. Mech. 0704 (2007) P04005 [hep-th/0612256] [INSPIRE].
G.Z. Toth, Investigations in two-dimensional quantum field theory by the bootstrap and TCSA methods, Ph.D. thesis, Eotvos U., Budapest, Hungary (2007) [arXiv:0707.0015] [INSPIRE].
R. Chatterjee, Exact partition function and boundary state of 2D massive Ising field theory with boundary magnetic field, Nucl. Phys. B 468 (1996) 439 [hep-th/9509071] [INSPIRE].
O. Miroshnichenko, Differential equation for local magnetization in the boundary Ising model, Nucl. Phys. B 811 (2009) 385 [arXiv:0808.3808] [INSPIRE].
I. Brunner and D. Roggenkamp, Defects and bulk perturbations of boundary Landau-Ginzburg orbifolds, JHEP 04 (2008) 001 [arXiv:0712.0188] [INSPIRE].
S. Fredenhagen and T. Quella, Generalised permutation branes, JHEP 11 (2005) 004 [hep-th/0509153] [INSPIRE].
D. Gaiotto, Domain walls for two-dimensional renormalization group flows, JHEP 12 (2012) 103 [arXiv:1201.0767] [INSPIRE].
A. Konechny, Renormalization group defects for boundary flows, J. Phys. A 46 (2013) 145401 [arXiv:1211.3665] [INSPIRE].
A. Konechny, RG boundaries and interfaces in Ising field theory, J. Phys. A 50 (2017) 145403 [arXiv:1610.07489] [INSPIRE].
V.P. Yurov and A.B. Zamolodchikov, Correlation functions of integrable 2D models of relativistic field theory. Ising model, Int. J. Mod. Phys. A 6 (1991) 3419 [INSPIRE].
P. Fonseca and A. Zamolodchikov, Ising field theory in a magnetic field: analytic properties of the free energy, J. Statist. Phys. 110 (2003) 527 [hep-th/0112167] [INSPIRE].
F.A. Berezin, The method of second quantization, Academic Press, U.S.A. (1966).
J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
J. Cardy, Bulk renormalization group flows and boundary states in conformal field theories, SciPost Phys. 3 (2017) 011 [arXiv:1706.01568] [INSPIRE].
R.E. Behrend, P.A. Pearce, V.B. Petkova and J.-B. Zuber, Boundary conditions in rational conformal field theories, Nucl. Phys. B 570 (2000) 525 [hep-th/9908036] [INSPIRE].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
J. Armas and J. Tarrio, On actions for (entangling) surfaces and DCFTs, JHEP 04 (2018) 100 [arXiv:1709.06766] [INSPIRE].
J.R. Schrieffer and P.A. Wolff, Relation between the Anderson and Kondo Hamiltonians, Phys. Rev. 149 (1966) 491.
N. Datta, R. Fernández, J. Fröhlich and L. Rey-Bellet, Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy, Helv. Phys. Acta 69 (1996) 752.
S. Bravyi, D.P. DiVincenzo and D. Loss, Schrieffer-Wolff transformation for quantum many-body systems, Ann. Phys. 326 (2011) 2793 [arXiv:1105.0675].
A.B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B 358 (1991) 497 [INSPIRE].
A.B. Zamolodchikov, From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B 358 (1991) 524 [INSPIRE].
T.R. Klassen and E. Melzer, Spectral flow between conformal field theories in (1 + 1)-dimensions, Nucl. Phys. B 370 (1992) 511 [INSPIRE].
A. Berkovich, Conformal invariance, finite size effects and the exact correlators for the δ-function Bose gas, Nucl. Phys. B 356 (1991) 655 [INSPIRE].
G. Feverati, E. Quattrini and F. Ravanini, Infrared behavior of massless integrable flows entering the minimal models from ϕ 31, Phys. Lett. B 374 (1996) 64 [hep-th/9512104] [INSPIRE].
E.E. Burniston and C.E. Siewert, The use of Riemann problems in solving a class of transcendental equations, Math. Proc. Camb. Phil. Soc. 73 (1973) 111.
S.L. Lukyanov, Low energy effective Hamiltonian for the XXZ spin chain, Nucl. Phys. B 522 (1998) 533 [cond-mat/9712314] [INSPIRE].
S.L. Lukyanov and V. Terras, Long distance asymptotics of spin spin correlation functions for the XXZ spin chain, Nucl. Phys. B 654 (2003) 323 [hep-th/0206093] [INSPIRE].
G. Feverati, K. Graham, P.A. Pearce, G.Z. Toth and G. Watts, A renormalisation group for the truncated conformal space approach, J. Stat. Mech. 0803 (2008) P03011 [hep-th/0612203] [INSPIRE].
G.M.T. Watts, On the renormalisation group for the boundary truncated conformal space approach, Nucl. Phys. B 859 (2012) 177 [arXiv:1104.0225] [INSPIRE].
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. Konechny and D. McAteer, On asymptotic behaviour in TCSA, in preparation.
A. Konechny, Boundary renormalisation group interfaces, work in progress.
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: 1811.07599
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Konechny, A. Critical Ising model with boundary magnetic field: RG interface and effective Hamiltonians. J. High Energ. Phys. 2019, 1 (2019). https://doi.org/10.1007/JHEP04(2019)001
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2019)001