PHYSICAL REVIEW D 103, L071502 (2021) Letter Real-time dynamics of Chern-Simons fluctuations near a critical point † ‡ Kazuki Ikeda ,1,* Dmitri E. Kharzeev ,2,3,4, and Yuta Kikuchi 3, 1Research Institute for Advanced Materials and Devices, Kyocera Corporation, Soraku, Kyoto 619-0237, Japan 2Center for Nuclear Theory, Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA 3Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000 4RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000 (Received 16 December 2020; accepted 22 March 2021; published 21 April 2021) The real-time topological susceptibility is studied in ð1 þ 1Þ-dimensional massive Schwinger model with a θ-term. We evaluate the real-time correlation function of electric field that represents the topological Chern-Pontryagin number density in (1 þ 1) dimensions. Near the parity-breaking critical point located at θ ¼ π and fermion mass m to coupling g ratio of m=g ≈ 0.33, we observe a sharp maximum in the topological susceptibility. We interpret this maximum in terms of the growth of critical fluctuations near the critical point, and draw analogies between the massive Schwinger model, QCD near the critical point, and ferroelectrics near the Curie point. DOI: 10.1103/PhysRevD.103.L071502 I. INTRODUCTION Coleman [59] that the line of the first order phase transition terminates at some critical value mÃ, where the phase The Schwinger model [1] is quantum electrodynamics in transition is second order. The position of this critical point ð1 þ 1Þ space-time dimensions. For massless fermions, the was established at mà ≈ 0.33 g [60–62]; the resulting phase Schwinger model is analytically solvable and equivalent to diagram is shown in Fig. 1. The phase diagram of the theory the theory of a free massive boson field [1–8]; however the in the ðm=g; θÞ plane is thus reminiscent of the phase model with massive fermions presents a challenge for diagram of QCD in the ðT;μÞ plane of temperature T and analytical methods and has a rich dynamics. baryon chemical potential μ [63,64]. Recently, quantum algorithms have emerged as an To understand the physics behind this phase diagram, we efficient (and potentially superior) way to explore the need to recall the role of θ-angle in the model. The action of dynamics of quantum field theories, including the the massive Schwinger model with θ term in ð1 þ 1Þ- Schwinger model [9–56]. Previously, we have addressed dimensional Minkowski space is the real-time dynamics of vector current [47] (a ð1 þ 1Þ- dimensional analog of the chiral magnetic current [57,58]) Z “ ”— θ 2 1 μν gθ μν induced by the chiral quench an abrupt change of the - S ¼ d x − F Fμν þ ϵ Fμν þ ψ¯ ðiD − mÞψ ; ð1Þ angle, or the chiral chemical potential. In this paper, we will 4 4π explore the connection between the real-time topological μ fluctuations and criticality using Schwinger model as a with D ¼ γ ð∂μ − igAμÞ. Note that the gauge field Aμ and testing ground. the coupling constant g have mass dimensions 0 and 1, The massive Schwinger model possesses a quantum respectively. Upon a chiral transformation, ψ → eiγ5θψ and θ π γ θ 1 phase transition at ¼ between the phases with opposite ψ¯ → ψ¯ ei 5 , the action is transformed to, orientations of the electric field, see Fig. 1. For m ≫ g, this Z phase transition is first order. However, it was shown by 1 2 − μν ψ¯ γμ − iγ5θ ψ S ¼ d x 4 F Fμν þ ði Dμ me Þ : ð2Þ *kazuki.ikeda.gtATkyocera.jp † dmitri.kharzeevATstonybrook.edu ‡ ykikuchiATbnl.gov It is clear from (2) that the massive theory with a positive mass m>0 at θ ¼ π is equivalent to the theory at θ ¼ 0 Published by the American Physical Society under the terms of but with a negative mass −m. the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, 1The action is invariant under this transformation only up to and DOI. Funded by SCOAP3. the boundary term. 2470-0010=2021=103(7)=L071502(7) L071502-1 Published by the American Physical Society IKEDA, KHARZEEV, and KIKUCHI PHYS. REV. D 103, L071502 (2021) θ −μ φ ffiffiffi E ¼ þ 2pπ ; ð6Þ where the first term is the quantum, dynamical contribution and the second one is the classical background induced by θ − θ 2π the angle, Ecl ¼ g = . The physics is periodic as a function of θ in the absence of boundaries—as θ increases, the electric field becomes capable of producing a fermion-antifermion pair (or kink-antikink pair, in bosonic description) that screens it. At θ ¼ π the potential takes the form pffiffiffi μ2 π 2 pffiffiffi φ φ − μ 2 πφ Uð Þ¼ 2 þ 2 cm cos ð Þ: ð7Þ FIG. 1. Phase diagram of the massive Schwinger model in the ðθ; m=gÞ plane. At θ ¼ π and large masses m>mÃ, the ferro- When m ≫ μ, the potential has two well separated minima at electric phases with opposite orientations of electric field are pffiffiffi φ ≈ 0; − π, associated with spontaneous symmetry break- separated by the line of the first order phase transition. This line pffiffiffi terminates at mà ≈ 0.33 g at the critical point, where the phase ing. As follows from (6), the minima φ ¼ 0; − π correspond transition is second order. For small masses m ≪ mÃ, the electric − 2 2 to the electric fields E ¼ g= and E ¼ g= , respectively.pffiffiffi field is screened by the production of light fermion-antifermion When m ≪ μ, there is a single minimum at φ ¼ − π=2, pairs. which according to (6) corresponds to the phase with no electric field. This is because at small m, the electric field is II. TOPOLOGICAL FLUCTUATIONS NEAR THE easily screened by the production of fermion-antifermion CRITICAL POINT pairs—so we are dealing with the screened phase. At some critical value m ≈ mÃ, the effective potential becomes flat— Let us now discuss topological fluctuations in massive this corresponds to the critical point with a second order Schwinger model. For that purpose, it will be convenient to phase transition. use the Hamiltonian formalism with the temporal gauge, For m>mÃ, the minima are separated by a potential 0 A0 ¼ . From the action (2) the canonical momentum barrier, and the transition between them (corresponding to _ conjugate to A1 can be read off as Π ¼ A1. The corre- the change in the direction of the electric dipole moment of sponding Hamiltonian is then given by the system) is first order—for example, having a domain Z with an opposite orientation of the electric dipole moment 1 “ 2 1 iγ5θ would cost an additional energy due to the surface H ¼ dx Π − ψ¯ ðiγ D1 − me Þψ ; ð3Þ 2 tension.” Due to this extra energy, the fluctuations of the electric dipole moment at m ≫ mà are suppressed. At the à with commutation relations ½A1ðxÞ; ΠðyÞ ¼ iδðx − yÞ, critical point, where m ≈ m , the potential barrier between and fψðxÞ; ψ¯ ðyÞg ¼ γ0δðx − yÞ. the two minima disappears, and the “surface tension” of the In the bosonized description of the theory, the domains with opposite orientations of the electric dipole Hamiltonian is given by (see Appendix A for details) moments vanishes. Because of this, the fluctuations of the Z electric dipole moment near the critical point are strongly 1 1 μ2 θ 2 enhanced. Here one can draw a useful analogy to the φ_ 2 ∂ φ 2 φ ffiffiffi H ¼ dx 2 þ 2 ð 1 Þ þ 2 þ 2pπ physics of ferroelectrics, where the electric susceptibility exhibits critical behavior near the Curie point [65]. pffiffiffi − cmμ cosð2 πφÞ : ð4Þ When the ratio m=g becomes very small, the electric field is easily screened by the production of fermion- antifermion pairs (or kink-antikink pairs in the bosonized The potential of the model is thus given by description), and the fluctuations of electric dipole moment again become small. Basing on this qualitative picture (that μ2 θ 2 pffiffiffi UðφÞ¼ φ þ pffiffiffi − cmμ cos ð2 πφÞ; ð5Þ we will confirm below with a more formal treatment), we 2 2 π expect to see a maximum in the electric susceptibility near pffiffiffi the critical point. We will show that this is indeed the case. where μ ¼ g= π is the mass of the scalar boson, and the In order to characterize the topological fluctuation we dimensionless coefficient c is given by c ¼ eγ=2π with the compute the static topological susceptibility. It is the zero Euler constant γ ¼ 0.5774. The electric field is related to frequency and wavelength limit of the real-time two-point the boson field as correlation function of the topological charge: L071502-2 REAL-TIME DYNAMICS OF CHERN-SIMONS FLUCTUATIONS … PHYS. REV. D 103, L071502 (2021) Z Xn 1 − −1 i χ 2 iωt−ikx 0 − 0 † ð Þ CS ¼ lim lim d xe ½hQðxÞQð Þi hQðxÞihQð Þi L ¼ χ χ − ; ð11Þ ω→0 k→0 n i i 2 Z i¼1 ¼ 2Re d2xΘðtÞ½hQðxÞQð0Þi − hQðxÞihQð0Þi; ð8Þ where we have fixed the boundary electric field, L0 ¼ 0. By enforcing the relation (11), the states are automatically μ where ΘðtÞ is the Heaviside’sstepfunction.QðxÞ¼∂μK is restricted to the physical ones. We furthermore eliminate the density of the Chern-Pontryagin number given by the the link fields Un by the gauge transformation, divergence of the Chern-Simons current Kμ.Inð1 þ 1Þ μ μν μν χ → χ χ† → χ† † → † dimensions, K ¼ðg=2πÞϵ Aν,andQ ¼ðg=4πÞϵ Fμν, n gn n; n ngn;Un gnþ1Ungn; ð12Þ where Fμν is the electromagnetic field strength tensor.