Abstract
We determine both analytically and numerically the entanglement between chiral degrees of freedom in the ground state of massive perturbations of 1+1 dimensional conformal field theories quantised on a cylinder. Analytic predictions are obtained from a variational Ansatz for the ground state in terms of smeared conformal boundary states recently proposed by J. Cardy, which is validated by numerical results from the Truncated Conformal Space Approach. We also extend the scope of the Ansatz by resolving ground state degeneracies exploiting the operator product expansion. The chiral entanglement entropy is computed both analytically and numerically as a function of the volume. The excellent agreement between the analytic and numerical results provides further validation for Cardy’s Ansatz. The chiral entanglement entropy contains a universal O(1) term γ for which an exact analytic result is obtained, and which can distinguish energetically degenerate ground states of gapped systems in 1+1 dimensions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. Amico, R. Fazio, A. Osterloh and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80 (2008) 517 [quant-ph/0703044] [INSPIRE].
J. Eisert, M. Cramer and M.B. Plenio, Area laws for the entanglement entropy — a review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].
T. Nishioka, Entanglement entropy: holography and renormalization group, Rev. Mod. Phys. 90 (2018) 035007 [arXiv:1801.10352] [INSPIRE].
N. Laflorencie, Quantum entanglement in condensed matter systems, Phys. Rept. 646 (2016) 1 [arXiv:1512.03388] [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].
G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (2003) 227902 [quant-ph/0211074] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613].
H. Li and F.D.M. 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].
P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
A.M. Kaufman et al., Quantum thermalization through entanglement in an isolated many-body system, Science 353 (2016) 794 [arXiv:1603.04409].
V. Alba and P. Calabrese, Entanglement and thermodynamics after a quantum quench in integrable systems, Proc. Nat. Acad. Sci. 114 (2017) 7947 [arXiv:1608.00614].
M. Kormos, M. Collura, G. Takács and P. Calabrese, Real-time confinement following a quantum quench to a non-integrable model, Nature Phys. 13 (2017) 246 [arXiv:1604.03571].
M. Collura, M. Kormos and G. Takács, Dynamical manifestation of the Gibbs paradox after a quantum quench, Phys. Rev. A 98 (2018) 053610 [arXiv:1801.05817] [INSPIRE].
O. Pomponio, L. Pristyák and G. Takács, Quasi-particle spectrum and entanglement generation after a quench in the quantum Potts spin chain, arXiv:1810.05539 [INSPIRE].
L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, Quantum source of entropy for black holes, Phys. Rev. D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
S. Das and S. Shankaranarayanan, How robust is the entanglement entropy: Area relation?, Phys. Rev. D 73 (2006) 121701 [gr-qc/0511066] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
K.G. Wilson and J.B. Kogut, The Renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].
A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381 [INSPIRE].
J. Polchinski, Scale and Conformal Invariance in Quantum Field Theory, Nucl. Phys. B 303 (1988) 226 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
A. Cappelli, C. Itzykson and J.B. Zuber, Modular Invariant Partition Functions in Two-Dimensions, Nucl. Phys. B 280 (1987) 445 [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
A.B. Zamolodchikov, Thermodynamic Bethe Ansatz in Relativistic Models. Scaling Three State Potts and Lee-yang Models, Nucl. Phys. B 342 (1990) 695 [INSPIRE].
V.P. Yurov and A.B. Zamolodchikov, Truncated conformal space approach to scaling Lee-Yang model, Int. J. Mod. Phys. A 5 (1990) 3221 [INSPIRE].
J.L. Cardy, O.A. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129 [arXiv:0706.3384] [INSPIRE].
O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories, J. Phys. A 42 (2009) 504006 [arXiv:0906.2946] [INSPIRE].
O.A. Castro-Alvaredo, Massive Corrections to Entanglement in Minimal E8 Toda Field Theory, SciPost Phys. 2 (2017) 008 [arXiv:1610.07040] [INSPIRE].
O.A. Castro-Alvaredo, C. De Fazio, B. Doyon and I.M. Szécsényi, Entanglement Content of Quasiparticle Excitations, Phys. Rev. Lett. 121 (2018) 170602 [arXiv:1805.04948] [INSPIRE].
T. Pálmai, Entanglement Entropy from the Truncated Conformal Space, Phys. Lett. B 759 (2016) 439 [arXiv:1605.00444] [INSPIRE].
D. Gaiotto, Domain Walls for Two-Dimensional Renormalization Group Flows, JHEP 12 (2012) 103 [arXiv:1201.0767] [INSPIRE].
A. Konechny, RG boundaries and interfaces in Ising field theory, J. Phys. A 50 (2017) 145403 [arXiv:1610.07489] [INSPIRE].
X.-L. Qi, H. Katsura and A.W.W. Ludwig, General Relationship between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States, Phys. Rev. Lett. 108 (2012) 196402 [arXiv:1103.5437].
J. Dubail, N. Read and E.H. Rezayi, Edge state inner products and real-space entanglement spectrum of trial quantum Hall states, Phys. Rev. B 86 (2012) 245310 [arXiv:1207.7119] [INSPIRE].
L.A. Pando Zayas and N. Quiroz, Left-Right Entanglement Entropy of Boundary States, JHEP 01 (2015) 110 [arXiv:1407.7057] [INSPIRE].
D. Das and S. Datta, Universal features of left-right entanglement entropy, Phys. Rev. Lett. 115 (2015) 131602 [arXiv:1504.02475] [INSPIRE].
L.A. Pando Zayas and N. Quiroz, Left-Right Entanglement Entropy of Dp-branes, JHEP 11 (2016) 023 [arXiv:1605.08666] [INSPIRE].
M.P. Zaletel and R.S.K. Mong, Exact matrix product states for quantum Hall wave functions, Phys. Rev. B 86 (2012) 245305 [arXiv:1208.4862].
B. Estienne, Z. Papić, N. Regnault and B.A. Bernevig, Matrix product states for trial quantum Hall states, Phys. Rev. B 87 (2013) 161112 [arXiv:1211.3353].
J. Cardy, Bulk Renormalization Group Flows and Boundary States in Conformal Field Theories, SciPost Phys. 3 (2017) 011 [arXiv:1706.01568] [INSPIRE].
A.J.A. James, R.M. Konik, P. Lecheminant, N.J. Robinson and A.M. Tsvelik, Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods, Rept. Prog. Phys. 81 (2018) 046002 [arXiv:1703.08421] [INSPIRE].
H. Casini and M. Huerta, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
H. Casini, Relative entropy and the Bekenstein bound, Class. Quant. Grav. 25 (2008) 205021 [arXiv:0804.2182] [INSPIRE].
N. Lashkari, Relative Entropies in Conformal Field Theory, Phys. Rev. Lett. 113 (2014) 051602 [arXiv:1404.3216] [INSPIRE].
E. Witten, APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory, Rev. Mod. Phys. 90 (2018) 045003 [arXiv:1803.04993] [INSPIRE].
B.-Q. Jin and V.E. Korepin, Quantum Spin Chain, Toeplitz Determinants and the Fisher-Hartwig Conjecture, J. Stat. Phys. 116 (2004) 79 [quant-ph/0304108].
G.Y. Cho, A.W.W. Ludwig and S. Ryu, Universal entanglement spectra of gapped one-dimensional field theories, Phys. Rev. B 95 (2017) 115122 [arXiv:1603.04016] [INSPIRE].
J.J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].
J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
P. Calabrese and J.L. Cardy, Time-dependence of correlation functions following a quantum quench, Phys. Rev. Lett. 96 (2006) 136801 [cond-mat/0601225] [INSPIRE].
J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [INSPIRE].
E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
D.C. Lewellen, Sewing constraints for conformal field theories on surfaces with boundaries, Nucl. Phys. B 372 (1992) 654 [INSPIRE].
I. Runkel, Boundary structure constants for the A series Virasoro minimal models, Nucl. Phys. B 549 (1999) 563 [hep-th/9811178] [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ’ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
A.B. Zamolodchikov, Conformal Symmetry and Multicritical Points in Two-Dimensional Quantum Field Theory. (In Russian), Sov. J. Nucl. Phys. 44 (1986) 529 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York U.S.A. (1997).
F.Y. Wu, The Potts model, Rev. Mod. Phys. 54 (1982) 235 [Erratum ibid. 55 (1983) 315] [INSPIRE].
D. Friedan, Z.-a. Qiu and S.H. Shenker, Superconformal Invariance in Two-Dimensions and the Tricritical Ising Model, Phys. Lett. 151B (1985) 37 [INSPIRE].
P. Christe and G. Mussardo, Integrable Systems Away from Criticality: The Toda Field Theory and S Matrix of the Tricritical Ising Model, Nucl. Phys. B 330 (1990) 465 [INSPIRE].
I. Affleck, Edge critical behavior of the two-dimensional tricritical Ising model, J. Phys. A 33 (2000) 6473 [cond-mat/0005286] [INSPIRE].
J.L. Cardy, Scaling and renormalization in statistical physics, Cambridge Lecture Notes in Physics, Cambridge University Press, Cambridge U.K. (1996).
A. Feiguin et al., Interacting anyons in topological quantum liquids: The golden chain, Phys. Rev. Lett. 98 (2007) 160409 [cond-mat/0612341] [INSPIRE].
V.A. Fateev, The Exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories, Phys. Lett. B 324 (1994) 45 [INSPIRE].
A.B. Zamolodchikov, Resonance factorized scattering and roaming trajectories, J. Phys. A 39 (2006) 12847.
W.M. Koo and H. Saleur, Representations of the Virasoro algebra from lattice models, Nucl. Phys. B 426 (1994) 459 [hep-th/9312156] [INSPIRE].
A. Milsted and G. Vidal, Extraction of conformal data in critical quantum spin chains using the Koo-Saleur formula, Phys. Rev. B 96 (2017) 245105 [arXiv:1706.01436] [INSPIRE].
T. Rakovszky, M. Mestyán, M. Collura, M. Kormos and G. Takács, Hamiltonian truncation approach to quenches in the Ising field theory, Nucl. Phys. B 911 (2016) 805 [arXiv:1607.01068] [INSPIRE].
D.X. Horváth and G. Takács, Overlaps after quantum quenches in the sine-Gordon model, Phys. Lett. B 771 (2017) 539 [arXiv:1704.00594] [INSPIRE].
K. Hódsági, M. Kormos and G. Takács, Quench dynamics of the Ising field theory in a magnetic field, SciPost Phys. 5 (2018) 027 [arXiv:1803.01158] [INSPIRE].
I. Kukuljan, S. Sotiriadis and G. Takács, Correlation Functions of the Quantum sine-Gordon Model in and out of Equilibrium, Phys. Rev. Lett. 121 (2018) 110402 [arXiv:1802.08696] [INSPIRE].
D.X. Horváth, I. Lovas, M. Kormos, G. Takács and G. Zaránd, Non-equilibrium time evolution and rephasing in the quantum sine-Gordon model, arXiv:1809.06789 [INSPIRE].
D. Schuricht and F.H.L. Essler, Dynamics in the Ising field theory after a quantum quench, J. Stat. Mech. 1204 (2012) P04017 [arXiv:1203.5080] [INSPIRE].
B. Bertini, D. Schuricht and F.H.L. Essler, Quantum quench in the sine-Gordon model, J. Stat. Mech. 1410 (2014) P10035 [arXiv:1405.4813] [INSPIRE].
A. Cortés Cubero and D. Schuricht, Quantum quench in the attractive regime of the sine-Gordon model, J. Stat. Mech. 1710 (2017) 103106 [arXiv:1707.09218] [INSPIRE].
G. Delfino, Quantum quenches with integrable pre-quench dynamics, J. Phys. A 47 (2014) 402001 [arXiv:1405.6553] [INSPIRE].
G. Delfino and J. Viti, On the theory of quantum quenches in near-critical systems, J. Phys. A 50 (2017) 084004 [arXiv:1608.07612] [INSPIRE].
M. Kormos and G. Zaránd, Quantum quenches in the sine-Gordon model: a semiclassical approach, Phys. Rev. E 93 (2016) 062101 [arXiv:1507.02708] [INSPIRE].
D.X. Horváth, S. Sotiriadis and G. Takács, Initial states in integrable quantum field theory quenches from an integral equation hierarchy, Nucl. Phys. B 902 (2016) 508 [arXiv:1510.01735] [INSPIRE].
D.X. Horváth, M. Kormos and G. Takács, Overlap singularity and time evolution in integrable quantum field theory, JHEP 08 (2018) 170 [arXiv:1805.08132] [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Operator Algebra of Two-Dimensional Conformal Theories with Central Charge C ≤ 1, Phys. Lett. B 154 (1985) 291 [INSPIRE].
G. Feverati, K. Graham, P.A. Pearce, G.Z. Tóth and G. Watts, A Renormalisation group for the truncated conformal space approach, J. Stat. Mech. 0803 (2008) P03011 [hep-th/0612203] [INSPIRE].
R.M. Konik and Y. Adamov, A Numerical Renormalization Group for Continuum One-Dimensional Systems, Phys. Rev. Lett. 98 (2007) 147205 [cond-mat/0701605] [INSPIRE].
P. Giokas and G. Watts, The renormalisation group for the truncated conformal space approach on the cylinder, arXiv:1106.2448 [INSPIRE].
M. Lencsés and G. Takács, Excited state TBA and renormalized TCSA in the scaling Potts model, JHEP 09 (2014) 052 [arXiv:1405.3157] [INSPIRE].
M. Hogervorst, S. Rychkov and B.C. van Rees, Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?, Phys. Rev. D 91 (2015) 025005 [arXiv:1409.1581] [INSPIRE].
S. Rychkov and L.G. Vitale, Hamiltonian truncation study of the φ 4 theory in two dimensions, Phys. Rev. D 91 (2015) 085011 [arXiv:1412.3460] [INSPIRE].
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: 1811.06500
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
Lencsés, M., Viti, J. & Takács, G. Chiral entanglement in massive quantum field theories in 1+1 dimensions. J. High Energ. Phys. 2019, 177 (2019). https://doi.org/10.1007/JHEP01(2019)177
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2019)177