Abstract
Defects in two-dimensional conformal field theories (CFTs) contain signatures of their characteristics. In this work, we analyze entanglement properties of subsystems in the presence of energy and duality defects in the Ising CFT using the density matrix renormalization group (DMRG) technique. In particular, we compute the entanglement entropy (EE) and the entanglement negativity (EN) in the presence of defects. For the EE, we consider the cases when the defect lies within the subsystem and at the edge of the subsystem. We show that the EE for the duality defect exhibits fundamentally different characteristics compared to the energy defect due to the existence of localized and delocalized zero energy modes. Of special interest is the nontrivial ‘finite-size correction’ in the EE obtained recently using free fermion computations [1]. These corrections arise when the subsystem size is appreciable compared to the total system size and lead to a deviation from the usual logarithmic scaling characteristic of one-dimensional quantum-critical systems. Using matrix product states with open and infinite boundary conditions, we numerically demonstrate the disappearance of the zero mode contribution for finite subsystem sizes in the thermodynamic limit. Our results provide further support to the recent free fermion computations, but clearly contradict earlier analytical field theory calculations based on twisted torus partition functions. Subsequently, we compute the logarithm of the EN (log-EN) between two disjoint subsystems separated by a defect. We show that the log-EN scales logarithmically with the separation of the subsystems. However, the coefficient of this logarithmic scaling yields a continuously-varying effective central charge that is different from that obtained from analogous computations of the EE. The defects leave their fingerprints in the subleading term of the scaling of the log-EN. Furthermore, the log-EN receives similar ‘finite size corrections’ like the EE which leads to deviations from its characteristic logarithmic scaling.
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References
A. Roy and H. Saleur, Entanglement Entropy in the Ising Model with Topological Defects, Phys. Rev. Lett. 128 (2022) 090603 [arXiv:2111.04534] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 2004 (2004) P06002 [hep-th/0405152] [INSPIRE].
H. Casini, Geometric entropy, area, and strong subadditivity, Class. Quant. Grav. 21 (2004) 2351 [hep-th/0312238] [INSPIRE].
H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys. Rev. A 78 (2008) 032329 [arXiv:0806.3059].
L. Tagliacozzo, T.R. de Oliveira, S. Iblisdir and J.I. Latorre, Scaling of entanglement support for Matrix Product States, Phys. Rev. B 78 (2008) 024410 [arXiv:0712.1976] [INSPIRE].
F. Pollmann, S. Mukerjee, A.M. Turner and J.E. Moore, Theory of finite-entanglement scaling at one-dimensional quantum critical points, Phys. Rev. Lett. 102 (2009) 255701 [arXiv:0812.2903].
M.B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. 08 (2007) P08024 [arXiv:0705.2024] [INSPIRE].
S.R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69 (1992) 2863 [INSPIRE].
U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals Phys. 326 (2011) 96 [arXiv:1008.3477].
A. Roy and H. Saleur, Entanglement entropy in critical quantum spin chains with boundaries and defects, arXiv:2111.07927 [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
I. Affleck, Conformal field theory approach to the Kondo effect, Acta Phys. Polon. B 26 (1995) 1869 [cond-mat/9512099] [INSPIRE].
H. Saleur, Lectures on nonperturbative field theory and quantum impurity problems, cond-mat/9812110 [INSPIRE].
M.R. Gaberdiel, Lectures on nonBPS Dirichlet branes, Class. Quant. Grav. 17 (2000) 3483 [hep-th/0005029] [INSPIRE].
I. Affleck, N. Laflorencie and E.S. Sørensen, Entanglement entropy in quantum impurity systems and systems with boundaries, J. Phys. A 42 (2009) 504009 [arXiv:0906.1809] [INSPIRE].
J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 2016 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
A. Roy, F. Pollmann and H. Saleur, Entanglement Hamiltonian of the 1+1-dimensional free, compactified boson conformal field theory, J. Stat. Mech. 2008 (2020) 083104 [arXiv:2004.14370] [INSPIRE].
J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
J. Belletête, A.M. Gainutdinov, J.L. Jacobsen, H. Saleur and T.S. Tavares, Topological defects in lattice models and affine Temperley-Lieb algebra, arXiv:1811.02551 [INSPIRE].
J. Belletête, A.M. Gainutdinov, J.L. Jacobsen, H. Saleur and T.S. Tavares, Topological defects in periodic RSOS models and anyonic chains, arXiv:2003.11293.
D. Aasen, P. Fendley and R.S.K. Mong, Topological defects on the lattice: Dualities and degeneracies, arXiv:2008.08598 [INSPIRE].
M. Henkel, A. Patkos and M. Schlottmann, The Ising Quantum Chain With Defects. 1. The Exact Solution, Nucl. Phys. B 314 (1989) 609 [INSPIRE].
M. Baake, P. Chaselon and M. Schlottmann, The Ising Quantum Chain With Defects. 2. The SO(2n) Kac-Moody Spectra, Nucl. Phys. B 314 (1989) 625 [INSPIRE].
U. Grimm, The Quantum Ising Chain With a Generalized Defect, Nucl. Phys. B 340 (1990) 633 [hep-th/0310089] [INSPIRE].
M. Oshikawa and I. Affleck, Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line, Nucl. Phys. B 495 (1997) 533 [cond-mat/9612187] [INSPIRE].
U. Grimm, Spectrum of a duality twisted Ising quantum chain, J. Phys. A 35 (2002) L25 [hep-th/0111157] [INSPIRE].
V.B. Petkova and J.B. Zuber, Generalized twisted partition functions, Phys. Lett. B 504 (2001) 157 [hep-th/0011021] [INSPIRE].
C. Bachas, J. de Boer, R. Dijkgraaf and H. Ooguri, Permeable conformal walls and holography, JHEP 06 (2002) 027 [hep-th/0111210] [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Kramers-Wannier duality from conformal defects, Phys. Rev. Lett. 93 (2004) 070601 [cond-mat/0404051] [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [INSPIRE].
D. Aasen, R.S.K. Mong and P. Fendley, Topological Defects on the Lattice I: The Ising model, J. Phys. A 49 (2016) 354001 [arXiv:1601.07185] [INSPIRE].
H.A. Kramers and G.H. Wannier, Statistics of the two-dimensional ferromagnet. Part 1, Phys. Rev. 60 (1941) 252 [INSPIRE].
R. Savit, Duality in Field Theory and Statistical Systems, Rev. Mod. Phys. 52 (1980) 453 [INSPIRE].
M. Buican and A. Gromov, Anyonic Chains, Topological Defects, and Conformal Field Theory, Commun. Math. Phys. 356 (2017) 1017 [arXiv:1701.02800] [INSPIRE].
H. Saleur, Lectures on nonperturbative field theory and quantum impurity problems: Part 2, cond-mat/0007309 [INSPIRE].
M. Gutperle and J.D. Miller, A note on entanglement entropy for topological interfaces in RCFTs, JHEP 04 (2016) 176 [arXiv:1512.07241] [INSPIRE].
K. Sakai and Y. Satoh, Entanglement through conformal interfaces, JHEP 12 (2008) 001 [arXiv:0809.4548] [INSPIRE].
E.M. Brehm and I. Brunner, Entanglement entropy through conformal interfaces in the 2D Ising model, JHEP 09 (2015) 080 [arXiv:1505.02647] [INSPIRE].
V. Eisler and I. Peschel, Entanglement in fermionic chains with interface defects,Annalen Phys. 522 (2010) 679.
I. Peschel and V. Eisler, Exact results for the entanglement across defects in critical chains, J. Phys. A 45 (2012) 155301 [arXiv:1201.4104].
P. Calabrese, M. Mintchev and E. Vicari, Entanglement Entropy of Quantum Wire Junctions, J. Phys. A 45 (2012) 105206 [arXiv:1110.5713] [INSPIRE].
J. Hauschild and F. Pollmann, Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy), SciPost Phys. Lect. Notes 2018 (2018) 5 [arXiv:1805.00055].
I. Klich, D. Vaman and G. Wong, Entanglement Hamiltonians for chiral fermions with zero modes, Phys. Rev. Lett. 119 (2017) 120401 [arXiv:1501.00482] [INSPIRE].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].
F. Iglói and I. Peschel, On reduced density matrices for disjoint subsystems, Europhys. Lett. 89 (2010) 40001 [arXiv:0910.5671].
L.P. Kadanoff and H. Ceva, Determination of an opeator algebra for the two-dimensional Ising model, Phys. Rev. B 3 (1971) 3918 [INSPIRE].
J.L. Cardy, Continuously varying exponents and the value of the central charge, J. Phys. A 20 (1987) L891.
F. Iglói, I. Peschel and L. Turban, Inhomogeneous systems with unusual critical behaviour, Adv. Phys. 42 (1993) 683 [cond-mat/9312077].
U. Grimm and G.M. Schutz, The Spin 1/2 XXZ Heisenberg chain, the quantum algebra U(q)[sl(2)], and duality transformations for minimal models, J. Statist. Phys. 71 (1993) 921 [hep-th/0111083] [INSPIRE].
J.L. Cardy, Finite-size scaling in strips: antiperiodic boundary conditions, J. Phys. A 17 (1984) L961.
C.P. Herzog and T. Nishioka, Entanglement Entropy of a Massive Fermion on a Torus, JHEP 03 (2013) 077 [arXiv:1301.0336] [INSPIRE].
C. Bachas, I. Brunner and D. Roggenkamp, Fusion of Critical Defect Lines in the 2D Ising Model, J. Stat. Mech. 1308 (2013) P08008 [arXiv:1303.3616] [INSPIRE].
L. Lewin, Polylogarithms and associated functions, North Holland, Amsterdam, The Netherlands (1981).
L. Amico, R. Fazio, A. Osterloh and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80 (2008) 517 [quant-ph/0703044] [INSPIRE].
R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81 (2009) 865 [quant-ph/0702225] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: a field theoretical approach, J. Stat. Mech. 2013 (2013) P02008 [arXiv:1210.5359] [INSPIRE].
P. Calabrese, L. Tagliacozzo and E. Tonni, Entanglement negativity in the critical Ising chain, J. Stat. Mech. 1305 (2013) P05002 [arXiv:1302.1113] [INSPIRE].
O. Blondeau-Fournier, O.A. Castro-Alvaredo and B. Doyon, Universal scaling of the logarithmic negativity in massive quantum field theory, J. Phys. A 49 (2016) 125401 [arXiv:1508.04026] [INSPIRE].
M. Hoogeveen and B. Doyon, Entanglement negativity and entropy in non-equilibrium conformal field theory, Nucl. Phys. B 898 (2015) 78 [arXiv:1412.7568] [INSPIRE].
V. Eisler and Z. Zimborás, Entanglement negativity in the harmonic chain out of equilibrium, New J. Phys. 16 (2014) 123020 [arXiv:1406.5474].
R. Haag, Local Quantum Physics: Fields, Particles, Algebras, Theoretical and Mathematical Physics, Springer, Heidelberg, Germany (2012).
H. Li and F. Haldane, Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States, Phys. Rev. Lett. 101 (2008) 010504 [arXiv:0805.0332] [INSPIRE].
V. Alba, P. Calabrese and E. Tonni, Entanglement spectrum degeneracy and the Cardy formula in 1+1 dimensional conformal field theories, J. Phys. A 51 (2018) 024001 [arXiv:1707.07532] [INSPIRE].
M. Mintchev and E. Tonni, Modular Hamiltonians for the massless Dirac field in the presence of a defect, JHEP 03 (2021) 205 [arXiv:2012.01366] [INSPIRE].
P. Ruggiero, V. Alba and P. Calabrese, Negativity spectrum of one-dimensional conformal field theories, Phys. Rev. B 94 (2016) 195121 [arXiv:1607.02992] [INSPIRE].
S. Murciano, V. Vitale, M. Dalmonte and P. Calabrese, Negativity Hamiltonian: An Operator Characterization of Mixed-State Entanglement, Phys. Rev. Lett. 128 (2022) 140502 [arXiv:2201.03989] [INSPIRE].
R. Verresen, N.G. Jones and F. Pollmann, Topology and Edge Modes in Quantum Critical Chains, Phys. Rev. Lett. 120 (2018) 057001 [arXiv:1709.03508] [INSPIRE].
P. Ruggiero, V. Alba and P. Calabrese, Entanglement negativity in random spin chains, Phys. Rev. B 94 (2016) 035152 [arXiv:1605.00674] [INSPIRE].
H. Wichterich, J. Molina-Vilaplana and S. Bose, Scaling of entanglement between separated blocks in spin chains at criticality, Phys. Rev. A 80 (2009) 010304 [arXiv:0811.1285] [INSPIRE].
R.Z. Bariev, Effect of Linear Defects on the Local Magnetization of a Plane Ising Lattice, JETP 50 (1979) 613.
A. Bayat, P. Sodano and S. Bose, Negativity as the Entanglement Measure to Probe the Kondo Regime in the Spin-Chain Kondo Model, Phys. Rev. B 81 (2010) 064429 [arXiv:0904.3341] [INSPIRE].
A. Bayat, S. Bose, P. Sodano and H. Johannesson, Entanglement probe of two-impurity Kondo physics in a spin chain, Phys. Rev. Lett. 109 (2012) 066403 [arXiv:1201.6668] [INSPIRE].
B. Alkurtass, Entanglement Structure of the Two-Channel Kondo Model, Phys. Rev. B 93 (2016) 081106 [arXiv:1509.02949].
M. Gruber and V. Eisler, Time evolution of entanglement negativity across a defect, J. Phys. A 53 (2020) 205301 [arXiv:2001.06274] [INSPIRE].
L.-F. Ko, H. Au-Yang and J.H.H. Perk, Energy-Density Correlation Functions in the Two-Dimensional Ising Model with a Line Defect, Phys. Rev. Lett. 54 (1985) 1091.
B.M. McCoy and J.H.H. Perk, Two Spin Correlation Functions of an Ising Model With Continuous Exponents, Phys. Rev. Lett. 44 (1980) 840 [INSPIRE].
I. Peschel and V. Eisler, Exact Results for the Entanglement across Defects in Critical Chains, J. Phys. A 45 (2012) 155301 [arXiv:1201.4104].
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Rogerson, D., Pollmann, F. & Roy, A. Entanglement entropy and negativity in the Ising model with defects. J. High Energ. Phys. 2022, 165 (2022). https://doi.org/10.1007/JHEP06(2022)165
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DOI: https://doi.org/10.1007/JHEP06(2022)165