Abstract
A symmetry-twisted boundary condition of the path integral provides a suitable framework for the semi-classical analysis of nonperturbative quantum field theories (QFTs), and we reinterpret it from the viewpoint of the Hilbert space. An appropriate twist with the unbroken symmetry can potentially produce huge cancellations among excited states in the state-sum, without affecting the ground states; we call this effect “quantum distillation”. Quantum distillation can provide the underlying mechanism for adiabatic continuity, by preventing a phase transition under S1 compactification. We revisit this point via the ’t Hooft anomaly matching condition when it constrains the vacuum structure of the theory on ℝd and upon compactification. We show that there is a precise relation between the persistence of the anomaly upon compactification, the Hilbert space quantum distillation, and the semi-classical analysis of the corresponding symmetry-twisted path integrals. We motivate quantum distillation in quantum mechanical examples, and then study its non-trivial action in QFT, with the example of the 2D Grassmannian sigma model Gr(N, M). We also discuss the connection of quantum distillation with large-N volume independence and flavor-momentum transmutation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G.V. Dunne and M. Ünsal, New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence, Ann. Rev. Nucl. Part. Sci. 66 (2016) 245 [arXiv:1601.03414] [INSPIRE].
G.V. Dunne and M. Ünsal, Resurgence and Trans-series in Quantum Field Theory: The CP N −1 Model, JHEP 11 (2012) 170 [arXiv:1210.2423] [INSPIRE].
M. Ünsal, Magnetic bion condensation: A new mechanism of confinement and mass gap in four dimensions, Phys. Rev. D 80 (2009) 065001 [arXiv:0709.3269] [INSPIRE].
M. Ünsal and L.G. Yaffe, Center-stabilized Yang-Mills theory: Confinement and large N volume independence, Phys. Rev. D 78 (2008) 065035 [arXiv:0803.0344] [INSPIRE].
M. Ünsal, Abelian Duality, Confinement, and Chiral-Symmetry Breaking in a SU(2) QCD-Like Theory, Phys. Rev. Lett. 100 (2008) 032005 [arXiv:0708.1772] [INSPIRE].
P. Kovtun, M. Ünsal and L.G. Yaffe, Volume independence in large N c QCD-like gauge theories, JHEP 06 (2007) 019 [hep-th/0702021] [INSPIRE].
M. Shifman and M. Ünsal, QCD-like Theories on R 3 × S 1 : A Smooth Journey from Small to Large r(S 1 ) with Double-Trace Deformations, Phys. Rev. D 78 (2008) 065004 [arXiv:0802.1232] [INSPIRE].
M. Shifman and M. Ünsal, Multiflavor QCD* on R 3 × S 1 : Studying Transition From Abelian to Non-Abelian Confinement, Phys. Lett. B 681 (2009) 491 [arXiv:0901.3743] [INSPIRE].
G. Cossu and M. D’Elia, Finite size phase transitions in QCD with adjoint fermions, JHEP 07 (2009) 048 [arXiv:0904.1353] [INSPIRE].
G. Cossu, H. Hatanaka, Y. Hosotani and J.-I. Noaki, Polyakov loops and the Hosotani mechanism on the lattice, Phys. Rev. D 89 (2014) 094509 [arXiv:1309.4198] [INSPIRE].
P.C. Argyres and M. Ünsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion and renormalon effects, JHEP 08 (2012) 063 [arXiv:1206.1890] [INSPIRE].
P. Argyres and M. Ünsal, A semiclassical realization of infrared renormalons, Phys. Rev. Lett. 109 (2012) 121601 [arXiv:1204.1661] [INSPIRE].
G.V. Dunne and M. Ünsal, Continuity and Resurgence: towards a continuum definition of the ℂℙ(N − 1) model, Phys. Rev. D 87 (2013) 025015 [arXiv:1210.3646] [INSPIRE].
E. Poppitz, T. Schäfer and M. Ünsal, Continuity, Deconfinement and (Super) Yang-Mills Theory, JHEP 10 (2012) 115 [arXiv:1205.0290] [INSPIRE].
M.M. Anber, S. Collier, E. Poppitz, S. Strimas-Mackey and B. Teeple, Deconfinement in \( \mathcal{N}=1 \) super Yang-Mills theory on ℝ3 ×\( \mathbb{S} \) 1 via dual-Coulomb gas and “affine” XY-model, JHEP 11 (2013) 142 [arXiv:1310.3522] [INSPIRE].
G. Basar, A. Cherman, D. Dorigoni and M. Ünsal, Volume Independence in the Large N Limit and an Emergent Fermionic Symmetry, Phys. Rev. Lett. 111 (2013) 121601 [arXiv:1306.2960] [INSPIRE].
A. Cherman, D. Dorigoni, G.V. Dunne and M. Ünsal, Resurgence in Quantum Field Theory: Nonperturbative Effects in the Principal Chiral Model, Phys. Rev. Lett. 112 (2014) 021601 [arXiv:1308.0127] [INSPIRE].
A. Cherman, D. Dorigoni and M. Ünsal, Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles, JHEP 10 (2015) 056 [arXiv:1403.1277] [INSPIRE].
T. Misumi and T. Kanazawa, Adjoint QCD on ℝ3 × S 1 with twisted fermionic boundary conditions, JHEP 06 (2014) 181 [arXiv:1405.3113] [INSPIRE].
T. Misumi, M. Nitta and N. Sakai, Neutral bions in the ℂP N −1 model, JHEP 06 (2014) 164 [arXiv:1404.7225] [INSPIRE].
T. Misumi, M. Nitta and N. Sakai, Classifying bions in Grassmann σ-models and non-Abelian gauge theories by D-branes, PTEP 2015 (2015) 033B02 [arXiv:1409.3444] [INSPIRE].
G.V. Dunne and M. Ünsal, Resurgence and Dynamics of O(N) and Grassmannian σ-models, JHEP 09 (2015) 199 [arXiv:1505.07803] [INSPIRE].
T. Misumi, M. Nitta and N. Sakai, Non-BPS exact solutions and their relation to bions in ℂP N −1 models, JHEP 05 (2016) 057 [arXiv:1604.00839] [INSPIRE].
A. Cherman, T. Schäfer and M. Ünsal, Chiral Lagrangian from Duality and Monopole Operators in Compactified QCD, Phys. Rev. Lett. 117 (2016) 081601 [arXiv:1604.06108] [INSPIRE].
T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai, Nonperturbative contributions from complexified solutions in ℂP N −1 models, Phys. Rev. D 94 (2016) 105002 [arXiv:1607.04205] [INSPIRE].
T. Sulejmanpasic, Global Symmetries, Volume Independence and Continuity in Quantum Field Theories, Phys. Rev. Lett. 118 (2017) 011601 [arXiv:1610.04009] [INSPIRE].
M. Yamazaki and K. Yonekura, From 4d Yang-Mills to 2d ℂℙN − 1 model: IR problem and confinement at weak coupling, JHEP 07 (2017) 088 [arXiv:1704.05852] [INSPIRE].
P.V. Buividovich and S.N. Valgushev, Lattice study of continuity and finite-temperature transition in two-dimensional SU(N) × SU(N) Principal Chiral Model, arXiv:1706.08954 [INSPIRE].
K. Aitken, A. Cherman, E. Poppitz and L.G. Yaffe, QCD on a small circle, Phys. Rev. D 96 (2017) 096022 [arXiv:1707.08971] [INSPIRE].
G. ’t Hooft, Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking, in Recent Developments in Gauge Theories. Proceedings, Nato Advanced Study Institute, Cargese, France, August 26 - September 8, 1979, vol. 59, pp. 135-157.
D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, Theta, Time Reversal and Temperature, JHEP 05 (2017) 091 [arXiv:1703.00501] [INSPIRE].
Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz, The Axial Anomaly and the Bound State Spectrum in Confining Theories, Nucl. Phys. B 177 (1981) 157 [INSPIRE].
S.R. Coleman and B. Grossman, ’t Hooft’s Consistency Condition as a Consequence of Analyticity and Unitarity, Nucl. Phys. B 203 (1982) 205 [INSPIRE].
A. Vishwanath and T. Senthil, Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect, Phys. Rev. X 3 (2013) 011016 [arXiv:1209.3058] [INSPIRE].
X.-G. Wen, Classifying gauge anomalies through symmetry-protected trivial orders and classifying gravitational anomalies through topological orders, Phys. Rev. D 88 (2013) 045013 [arXiv:1303.1803] [INSPIRE].
A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in three dimensions and group cohomology, Phys. Rev. Lett. 112 (2014) 231602 [arXiv:1403.0617] [INSPIRE].
A. Kapustin and R. Thorngren, Anomalies of discrete symmetries in various dimensions and group cohomology, arXiv:1404.3230 [INSPIRE].
G.Y. Cho, J.C.Y. Teo and S. Ryu, Conflicting Symmetries in Topologically Ordered Surface States of Three-dimensional Bosonic Symmetry Protected Topological Phases, Phys. Rev. B 89 (2014) 235103 [arXiv:1403.2018] [INSPIRE].
J.C. Wang, Z.-C. Gu and X.-G. Wen, Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond, Phys. Rev. Lett. 114 (2015) 031601 [arXiv:1405.7689] [INSPIRE].
E. Witten, Fermion Path Integrals And Topological Phases, Rev. Mod. Phys. 88 (2016) 035001 [arXiv:1508.04715] [INSPIRE].
N. Seiberg and E. Witten, Gapped Boundary Phases of Topological Insulators via Weak Coupling, PTEP 2016 (2016) 12C101 [arXiv:1602.04251] [INSPIRE].
E. Witten, The “Parity” Anomaly On An Unorientable Manifold, Phys. Rev. B 94 (2016) 195150 [arXiv:1605.02391] [INSPIRE].
Y. Tachikawa and K. Yonekura, On time-reversal anomaly of 2+1d topological phases, PTEP 2017 (2017) 033B04 [arXiv:1610.07010] [INSPIRE].
Y. Tachikawa and K. Yonekura, More on time-reversal anomaly of 2+1d topological phases, Phys. Rev. Lett. 119 (2017) 111603 [arXiv:1611.01601] [INSPIRE].
C. Wang, A. Nahum, M.A. Metlitski, C. Xu and T. Senthil, Deconfined quantum critical points: symmetries and dualities, Phys. Rev. X 7 (2017) 031051 [arXiv:1703.02426] [INSPIRE].
Y. Tanizaki and Y. Kikuchi, Vacuum structure of bifundamental gauge theories at finite topological angles, JHEP 06 (2017) 102 [arXiv:1705.01949] [INSPIRE].
Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, Comments on Abelian Higgs Models and Persistent Order, arXiv:1705.04786 [INSPIRE].
Z. Komargodski, T. Sulejmanpasic and M. Ünsal, Walls, anomalies and deconfinement in quantum antiferromagnets, Phys. Rev. B 97 (2018) 054418 [arXiv:1706.05731] [INSPIRE].
G.Y. Cho, S. Ryu and C.-T. Hsieh, Anomaly Manifestation of Lieb-Schultz-Mattis Theorem and Topological Phases, Phys. Rev. B 96 (2017) 195105 [arXiv:1705.03892] [INSPIRE].
H. Shimizu and K. Yonekura, Anomaly constraints on deconfinement and chiral phase transition, Phys. Rev. D 97 (2018) 105011 [arXiv:1706.06104] [INSPIRE].
J. Wang, X.-G. Wen and E. Witten, Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions, arXiv:1705.06728 [INSPIRE].
M.A. Metlitski and R. Thorngren, Intrinsic and emergent anomalies at deconfined critical points, arXiv:1707.07686 [INSPIRE].
Y. Kikuchi and Y. Tanizaki, Global inconsistency, ’t Hooft anomaly and level crossing in quantum mechanics, PTEP 2017 (2017) 113B05 [arXiv:1708.01962] [INSPIRE].
D. Gaiotto, Z. Komargodski and N. Seiberg, Time-reversal breaking in QCD 4 , walls and dualities in 2 + 1 dimensions, JHEP 01 (2018) 110 [arXiv:1708.06806] [INSPIRE].
Y. Tanizaki, T. Misumi and N. Sakai, Circle compactification and ’t Hooft anomaly, JHEP 12 (2017) 056 [arXiv:1710.08923] [INSPIRE].
Y. Tanizaki, Y. Kikuchi, T. Misumi and N. Sakai, Anomaly matching for the phase diagram of massless ℤN -QCD, Phys. Rev. D 97 (2018) 054012 [arXiv:1711.10487] [INSPIRE].
M. Guo, P. Putrov and J. Wang, Time reversal, SU(N ) Yang-Mills and cobordisms: Interacting topological superconductors/insulators and quantum spin liquids in 3+1D, Annals Phys. 394 (2018) 244 [arXiv:1711.11587] [INSPIRE].
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
E. Witten, Supersymmetry and Morse theory, J. Diff. Geom. 17 (1982) 661, http://projecteuclid.org/euclid.jdg/1214437492.
E. Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109 (1987) 525 [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09): Prague, Czech Republic, August 3-8, 2009, pp. 265-289, arXiv:0908.4052 [INSPIRE].
F. Verstraete, V. Murg and J. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys. 57 (2008) 143.
M. Ünsal and L.G. Yaffe, Large-N volume independence in conformal and confining gauge theories, JHEP 08 (2010) 030 [arXiv:1006.2101] [INSPIRE].
A. Cherman, S. Sen, M.L. Wagman and L.G. Yaffe, Exponential reduction of finite volume effects with twisted boundary conditions, Phys. Rev. D 95 (2017) 074512 [arXiv:1612.00403] [INSPIRE].
G. Basar, A. Cherman and D.A. McGady, Bose-Fermi Degeneracies in Large N Adjoint QCD, JHEP 07 (2015) 016 [arXiv:1409.1617] [INSPIRE].
D.J. Gross and Y. Kitazawa, A Quenched Momentum Prescription for Large N Theories, Nucl. Phys. B 206 (1982) 440 [INSPIRE].
A. Gonzalez-Arroyo and M. Okawa, The Twisted Eguchi-Kawai Model: A Reduced Model for Large N Lattice Gauge Theory, Phys. Rev. D 27 (1983) 2397 [INSPIRE].
A. Gonzalez-Arroyo and M. Okawa, A Twisted Model for Large N Lattice Gauge Theory, Phys. Lett. B 120 (1983) 174 [INSPIRE].
A. Gonzalez-Arroyo and M. Okawa, Large N reduction with the Twisted Eguchi-Kawai model, JHEP 07 (2010) 043 [arXiv:1005.1981] [INSPIRE].
A.J. Macfarlane, Solution of the Schrödinger equation of the complex manifold CP n, J. Phys. A 36 (2003) 9689.
A.M. Perelomov, Chiral models: geometrical aspects, Phys. Rept. 146 (1987) 135 [INSPIRE].
W.J. Zakrzewski, Classical solutions of two-dimensional Grassmannian models, J. Geom. Phys. 1 (1984) 39 [INSPIRE].
A. Yu. Morozov, A.M. Perelomov and M.A. Shifman, Exact Gell-Mann-Low Function of supersymmetric Kähler σ-models, Nucl. Phys. B 248 (1984) 279 [INSPIRE].
E. Witten, Instantons, the Quark Model and the 1/n Expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].
I. Affleck, The Role of Instantons in Scale Invariant Gauge Theories, Nucl. Phys. B 162 (1980) 461 [INSPIRE].
T. Sulejmanpasic and Y. Tanizaki, C-P-T anomaly matching in bosonic quantum field theory and spin chains, Phys. Rev. B 97 (2018) 144201 [arXiv:1802.02153] [INSPIRE].
A. Kapustin and N. Seiberg, Coupling a QFT to a TQFT and Duality, JHEP 04 (2014) 001 [arXiv:1401.0740] [INSPIRE].
N.D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 (1966) 1133 [INSPIRE].
S.R. Coleman, There are no Goldstone bosons in two-dimensions, Commun. Math. Phys. 31 (1973) 259 [INSPIRE].
X. Chen, Z.-C. Gu and X.-G. Wen, Classification of gapped symmetric phases in one-dimensional spin systems, Phys. Rev. B 83 (2011) 035107 [arXiv:1008.3745].
F.D.M. Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnetic identification with the O(3) nonlinear σ-model, Phys. Lett. A 93 (1983) 464 [INSPIRE].
F.D.M. Haldane, Nonlinear field theory of large spin Heisenberg antiferromagnets. Semiclassically quantized solitons of the one-dimensional easy Axis Neel state, Phys. Rev. Lett. 50 (1983) 1153 [INSPIRE].
A. Cherman, S. Sen, M. Ünsal, M.L. Wagman and L.G. Yaffe, Order parameters and color-flavor center symmetry in QCD, Phys. Rev. Lett. 119 (2017) 222001 [arXiv:1706.05385] [INSPIRE].
D. Tong, The moduli space of BPS domain walls, Phys. Rev. D 66 (2002) 025013 [hep-th/0202012] [INSPIRE].
F. Bruckmann, Instanton constituents in the O(3) model at finite temperature, Phys. Rev. Lett. 100 (2008) 051602 [arXiv:0707.0775] [INSPIRE].
W. Brendel, F. Bruckmann, L. Janssen, A. Wipf and C. Wozar, Instanton constituents and fermionic zero modes in twisted CP**n models, Phys. Lett. B 676 (2009) 116 [arXiv:0902.2328] [INSPIRE].
I. Affleck and E.H. Lieb, A Proof of Part of Haldane’s Conjecture on Spin Chains, Lett. Math. Phys. 12 (1986) 57 [INSPIRE].
I. Affleck, T. Kennedy, E.H. Lieb and H. Tasaki, Rigorous Results on Valence Bond Ground States in Antiferromagnets, Phys. Rev. Lett. 59 (1987) 799 [INSPIRE].
F.D.M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the ‘Parity Anomaly’, Phys. Rev. Lett. 61 (1988) 2015 [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: 1803.02430
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
Dunne, G.V., Tanizaki, Y. & Ünsal, M. Quantum distillation of Hilbert spaces, semi-classics and anomaly matching. J. High Energ. Phys. 2018, 68 (2018). https://doi.org/10.1007/JHEP08(2018)068
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)068