Continuum approach to real time dynamics of 1+1D gauge field theory: out of horizon correlations of the Schwinger model
CContinuum approach to real time dynamics of 1+1D gauge field theory:out of horizon correlations of the Schwinger model
Ivan Kukuljan
1, 2 Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-Str. 1, DE-85748 Garching, Germany Munich Center for Quantum Science and Technology, Schellingstr. 4, DE-80799 München, Germany
We develop a truncated Hamiltonian method to study the nonequilibrium real time dynamicsfollowing quenches in the Schwinger model - the quantum electrodynamics in D=1+1. This is apurely continuum method that captures reliably the invariance under local and global gauge trans-formations and does not require a discretisation of space-time. We show that the 1+1D quantumelectrodynamics admits the dynamical horizon violation effect which was recently discovered in thecase of the sine-Gordon model. Following a quench of a parameter of the model, oscillatory long-range correlations develop, manifestly violating the horizon bound. We find that the oscillationfrequencies of the out-of-horizon correlations correspond to twice the masses of the mesons presentin the spectrum of the model suggesting that the effect is mediated through correlated meson pairs,entangled by the quench. The results presented here reveal a novel nonequilibrium phenomenonin 1+1D quantum electrodynamics and make a first step to towards establishing that the horizonviolation effect is present in gauge field theory.
PACS numbers: 03.70.+k,11.15.-q,11.10.Ef
Introduction. -
Computing real time dynamics of aninteracting many-body quantum system is a notoriouslydifficult problem. It has been currently getting an over-whelming amount of attention due to the fast developingfield of nonequilibrium physics both in high energy [1–9]and condensed matter physics [10–12] on one side andrenewed interest in chaos and information scrambling onthe other side [13–17]. It is also becoming a matter ofincreased experimental importance [18–21] . The set oftools to deal with the problem has been greatly enrichedby developments and new insights in integrability the-ory [22–24], holography [25–29] and numerical algorithmssuch as density matrix renormalisation group (DMRG)[30, 31], tensor networks (TNS) [32–34] and lattice gaugetheory [35, 36]. Although in the present time, there isan abundance of excellent numerical methods availablefor discrete systems, the methods for the real time evo-lution directly in the continuum remain scarce and lessdeveloped.A powerful class of algorithms are the truncated Hamil-tonian methods (THM) [37–45]. They are numericalmethods for quantum field theories (QFT) that workin the continuum and do not require a discretisation ofspace-time. They can be applied to a wide set of tasks likecomputing spectra [37, 39–47] and level spacing statis-tics [48, 49], studying symmetry breaking [45], correla-tion functions [50, 51], real time dynamics [50–54] andalso gauge field theories [55, 56]. The class of methodsoriginates from the truncated conformal space approach (TCSA) introduced by Yurov and Zamolodchikov [37]. AQFT model on a compact domain is regarded as pointalong the renormalisation group (RG) flow from the ultraviolet (UV) fixed point generated by a relevant pertur-bation. The conformal field theory (CFT) algebraic ma-chinery is used to represent the Hamiltonian as a matrix in the basis of the fixed point CFT Hilbert space. Finally,an energy cutoff is introduced to obtain a finite matrixwhich enables numerical computation that indeed effi-ciently captures nonperturbative effects. More broadly,instead of CFT, any solvable QFT can be used as thestarting point for the expansion. H → H space ∝ cos(4Mt) t i m e - ε | x - y | → ∞ , t → ∞ ε < > ≠ < > < > Figure 1: Dynamical horizon violation as found in the sine-Gordon model [51]. The system is prepared in the groundstate of a gaped Hamiltonian H with short range correlations ∝ e −| x − y | /ξ . At time t = 0 the Hamiltonian is quenched to H .This generates cluster violating 4-point correlations of solitonsand antisolitons, eq. (2) (here symbolically pictured usingclassical solitons), which are not observable at t = 0 but resultin oscillating out-of-horizon correlations of local observables (cid:104) O ( − x ) O ( x ) (cid:105) at later times. The horizon is depicted herewith gray color and the horizon violating correlations withred. Asymptotically, the latter oscillate with a frequency 4times the soliton mass M . One of the central properties of quantum physicsout of equilibrium is the horizon effect introduced by a r X i v : . [ h e p - t h ] J a n Cardy and Calabrese [57–59]. A quantum system is ini-tially prepared in a short range correlated nonequilib-rium state, (cid:104) O ( x ) O ( y ) (cid:105) ∝ e −| x − y | /ξ for local observables O , the correlation length ξ , and let to evolve dynami-cally for t > - a protocol commonly termed a quan-tum quench . The horizon bound states that the con-nected correlations following the quench spread withinthe horizon: |(cid:104) O ( t, x ) O ( t, y ) (cid:105)| < κ e − max { ( | x − y |− ct ) /ξ h , } for some constant κ , where ξ h is called the horizon thick-ness and c is the maximal velocity of the theory - speedof light in quantum field theory and the Lieb-Robinson(LR) velocity in discrete systems [60]. The intuition isthat correlations spread by pairs of entangled particlescreated in initially correlated region | x − y | (cid:46) ξ and trav-eling to opposite directions. This bound has been rigor-ously proven in conformal field theory [57, 58, 61] anddemonstrated, analytically and numerically in a large setof interacting systems [61–92, 92–97] as well as observedin experiments [98–100]. Due to strong evidence, thebound has been believed to be a universal property ofquantum physics.In a recent publication together with Sotiriadis andTakács [51], we have demonstrated that the horizonbound can be violated in QFT with nontrivial topolog-ical properties. We have proved this in the case of the sine-Gordon (SG) field theory, a prototypical example ofstrongly correlated QFT L SG = 12 ( ∂ µ Φ)( ∂ µ Φ) + µ β cos( β Φ) (1)Starting from short range correlated states, SG dynam-ics within a short time generates infinite range correla-tions oscillating in time and clearly violating the horizonbound. Quenches in the SG model create cluster vio-lating four-body correlations between solitons ( S ) andanti-solitons ( A ), the topological excitations of the the-ory, written schematically: lim | x − y |→∞ (cid:104) A ( x ) S ( x + a ) A ( y ) S ( y + b ) (cid:105)(cid:54) = (cid:104) A ( x ) S ( x + a ) (cid:105) (cid:104) A ( y ) S ( y + b ) (cid:105) (2)The dynamics of the model then converts these soli-tonic correlations into two-point correlations of lo-cal bosonic fields (cid:104) Φ( t, x )Φ( t, y ) (cid:105) , (cid:104) Π( t, x )Π( t, y ) (cid:105) and (cid:104) ∂ x Φ( t, x ) ∂ y Φ( t, y ) (cid:105) . There is no violation of relativisticcausality involved because the cluster violating correla-tions (2) are created by a quench, a global simultane-ous event and not by the unitary dynamics of the modelwhich is strictly causal. The mechanism of the effectsuggests that the horizon violation should be found inany QFTs with nontrivial field topologies, an importantclass of them being gauge field theories which lie in thecore of our understanding of elementary interactions andare also of high importance in condensed matter physics.The results presented in this Letter represent the firststeps towards establishing that. The Schwinger model. -
We focus here on the simplestexample of a gauge field theory, the (QED), i.e. the (massive)
Schwinger model : L = − F µν F µν + ¯Ψ ( iγ µ ∂ µ − eγ µ A µ − m ) Ψ , with Ψ = (Ψ − , Ψ + ) T the Dirac fermion, m the electronmass and e the electric charge. As a consequence of in-variance under large gauge transformations, the modelhas infinitely degenerate vacuum states, the θ vacua fora parameter θ ∈ [0 , π ) that enters the bosonised form ofthe Hamiltonian and plays the physical role of the con-stant background electric field [101, 102]. The Schwingermodel thus has two physical parameters, the ratio m/e and θ .The massless m = 0 version of the model was solvedexactly by Schwinger [103] and has a gap of e/ √ π corre-sponding to a meson, a bound state of a fermion and anantifermion. The full massive m > version of the modelis not integrable and has a rich phase diagram where thenumber of mesons depends on the values of the parame-ters m/e and θ [101, 102, 104–118]. The Schwinger modeldisplays confinement and has been extensively studied forpair creation and string breaking [101, 119–128].Finally, it is known that due to the vacuum degener-acy, the massless version of the Schwinger model exhibitscluster violation of correlators of chiral fermion densities ρ ± ( x ) = N (cid:104) ¯ ψ ( x ) ± γ ψ ( x ) (cid:105) , [129–131], (cid:104) ρ − ( x ) · · · ρ − ( x n ) ρ + ( y ) · · · ρ + ( y n ) (cid:105) , (3)closely related to the correlators from eq. (2). Clusterviolation in gauge field theory has recently been an activetopic and more general mathematical conditions for itsoccurrence have been established [132–135]. This makesthe model a good candidate for the horizon violation.Here we want to study general quenches of the mas-sive Schwinger model and focus on the spreading of thecurrent-current correlators: C µ ( t, x, y ) = (cid:104) J µ ( t, x ) J µ ( t, y ) (cid:105) . (4)We prepare the system in the ground state of the modelwith the prequench values of the parameters m /e , θ and at time t = 0 switch the parameters to theirpostquench values m/e , θ . The method. -
As a consequence of the Lieb-Robinsonbound [60, 136, 137] and the Araki theorem [138, 139],the horizon violation is expected not to be present in dis-crete systems with finite local Hilbert space dimensionand is likely a genuinely field theoretical phenomenon.Therefore discretising the model and simulating usingDMRG or TNS [35, 36, 104, 117, 124–128, 140] is notan option so methods working directly in the continuumare needed and THM seem to be the best candidate.Here we implement a THM for the Schwinger modelin finite volume L with periodic boundary conditions.Treating a gauge theory in Hamiltonian formalism re-quires a careful elimination of the gauge redundancy ofdegrees of freedom. For the Schwinger model, this isachieved alongside with the bosonisation of the model[141].Choosing the Weyl (time) gauge, A t = 0 , anddefining A ≡ A x , the Hamiltonian of the model is H = ´ L dx (cid:16) ˙ A − ¯Ψ (cid:2) γ ( i∂ x − eA ) − m (cid:3) Ψ (cid:17) . Ex-panding the fermion currents J σ ( x ) = Ψ † σ ( x )Ψ σ ( x ) = L (cid:104) Q σ − σ (cid:80) n> √ n (cid:16) b σ,n e − σin πL x + b † σ,n e σin πL x (cid:17)(cid:105) ,with the chirality σ = ± , its modes obey bosoniccanonical commutation relations. Further defin-ing the N σ vacua as | N − (cid:105) ≡ (cid:81) ∞ n = N − c †− ,n | (cid:105) , | N + (cid:105) ≡ (cid:81) N + − n = −∞ c † + ,n | (cid:105) , with c σ,n the fermion modeoperators, the Hilbert space spanned by acting withbosonic modes b † σ,n on top of | N − (cid:105) ⊗ | N + (cid:105) isequivalent to the Hilbert space spanned by c † σ,n on topof | (cid:105) . This is the foundation for the bosonisation ofthe model. Because of invariance under large gaugetransformations, the true vacua of the system are theinfinitely degenerate θ vacua | θ (cid:105) = (cid:80) N ∈ Z e − iNθ | N (cid:105) for θ ∈ [0 , π ) . Gauge invariance further implies that theonly mode of the EM potential A that is not fixed bythe Gauss law is the zero mode α = L ´ L dx A ( x ) alongwith its its dual i∂ α = ´ L dx ˙ A ( x ) .By setting B = (cid:113) ML (cid:0) −√ π { Q + − Q − } + ∂∂α (cid:1) , B † = (cid:113) ML (cid:0) −√ π { Q + − Q − } − ∂∂α (cid:1) the part of theHamiltonian involving the zero modes transforms intoa harmonic oscillator with the mass M = e √ π . Com-plemented with a Bogoliubov transform of the nonzeromomentum modes into massive bosonic modes: B σ,n = (cid:16) √ E n √ k n + √ k n √ E n (cid:17) b σ,n − (cid:16) √ E n √ k n − √ k n √ E n (cid:17) b †− σ,n , with k n = πnL and E n = (cid:112) M + k n , the massless part ofthe Hamiltonian, ´ dx (cid:16) ˙ A − ¯Ψ (cid:2) γ ( i∂ x − eA ) (cid:3) Ψ (cid:17) , istransformed into the Hamiltonian of a massive free bo-son with the mass M . The mass term of the Hamil-tonian, ´ dx m ¯ΨΨ , is written in the bosonic form usingthe bosonisation relation Ψ σ ( x ) = √ L : e − σi √ π Φ σ ( x ) : F σ with ∂ x φ σ ( x ) = √ πJ σ ( x ) + σe √ π A ( x ) the chiral bosonfield and F σ the Klein factor. Then using F † σ F − σ | θ (cid:105) = e σiθ | θ (cid:105) , the Schwinger model Hamiltonian takes thebosonised form H = H M + U, (5) H M = M (cid:16) B † B (cid:17) + (cid:88) n> E n (cid:16) B † + ,n B + ,n + B †− ,n B − ,n (cid:17) ,U = − mM π e γ ˆ L dx :cos (cid:16) √ π Φ( x ) + θ (cid:17) : M . with : • : M denoting normal ordering w.r.t. the mass M and γ the Euler-Mascheroni constant. Notice that this form of the model is related to the SG model (1)at the free fermion point β = √ π with an additionalmass term with the mass M . Quenches from the Klein-Gordon model to the SG model solved in [51] can thusbe seen as a special case of Schwinger model quenches,from m /e = 0 to m/e = ∞ .The bosonised form of the Hamiltonian (5) offersa natural THM splitting into the massive free partand the cosine potential. To implement the numeri-cal method, the cosine potential and the observables,are represented as matrices in the Hilbert space of thefree part - the Fock space generated by applying the B † σ,n modes on the θ vacuum. Finally, an energy cut-off (cid:104) Ψ | H M | Ψ (cid:105) ≤ E cut is imposed on the states | Ψ (cid:105) ofthe THM Hilbert space. Translation invariance of themodel implies that the total momentum of the states p tot = (cid:80) σ = ± σ (cid:80) ∞ n =1 k n (cid:68) B † k,σ B k,σ (cid:69) = 0 is conserved.Together with the the decoupling of the B mode fromthe rest of the modes and the parity symmetry σ → σ this can be used to diagonalise the model in each of thesectors separately and thus reduce the dimension of theHilbert space kept in the computer’s memory. For theresults presented here, we use the Hilbert spaces with upto 15000 states per sector. The full details of the methodcan be found in the Supplementary Material [142]. Figure 2: The THM spectrum the Schwinger model at m/e =0 . in dependence of the system size L in the p tot = 0 , , , (cid:68) B † B (cid:69) = 0 , . The spectral lines are compared with the L → ∞ results of the MPS computations [116] for the vectorand the scalar particles and the TNS [118] for the heavy vectorparticle. On top of the spectrum, the dominant frequency ofthe oscillations of the out-of-horizon correlations are plotted.Due simulation times limited to t ≤ L/ , the frequencies haveconsiderable error bars. Results. -
As demonstrated in fig. 2, our THM im-plementation of the Schwinger model recovers the resultsfrom the literature for the masses of the bound statesof the model and gives a region of highly dense statesabove them, referred to as the continuum in the L → ∞ Figure 3: Time dependent (cid:104) J x ( t, x ) J x ( t, y ) (cid:105) and (cid:10) J t ( t, x ) J t ( t, y ) (cid:11) correlations for different type of quenches in the Schwingermodel. The quenches are from left to right: 1.) Quench in m/e with m = 0 . , m = 0 . , θ = θ = 0 ; 2.) Quench in θ with θ = π , θ = 0 , m = m = 0 . ; 3.) Quenches from the massless Schwinger model with m = 0 , m = 0 . , θ = θ = 0 ; 4.)Quenches to the massless Schwinger model with m = 0 . , m = 0 , θ = θ = 0 . All the quenches are simulated for e = e = 1 , L = 40 . The overall sign of the correlations changes depending whether the quenched parameter is increased or decreased asis usual in QFT quenches. limit. This serves as a sanity check of the method. Weare able to get the masses of the vector meson very pre-cisely, while our THM method seems to be slightly lessprecise for the scalar meson mass. We have been able tosimulate large system sizes L (cid:29) M where the finite sizeeffects are exponentially suppressed.The results shown in fig. 3 indeed confirm that theSchwinger model exhibits the horizon violation effect -the correlation functions C x ( t, x, y ) are nonzero and os-cillating for | x − y | > t . The effect is found in quenchesof any of the two parameters of the system, e/m and θ aswell as naturally expected in quenches to and from themassless Schwinger model. As is expected for periodicboundary conditions, the effect is present in the C x andnot present in the C t channel. Figure 4: Fourier spectrum of the time series of the out-of-horizon correlations for a θ quench θ = π/ , θ = 0 at m =0 . , e = 1 , L = 65 . The peaks are compared with themultiples of meson masses [116, 118]. To shed light on the mechanism of the effect, we haveanalysed the oscillation frequencies of the out-of-horizoncorrelations using the discrete Fourier transforms (FFT)of the time series. The periodic boundary conditionslimit our time evolution to times t ≤ L before the prop-agating signal interferes with itself which is only enoughto see a few oscillations and limits our frequency preci-sion to ∆ ω ≈ π/L (half of the frequency bin). Fig. 2shows how the dominant frequencies of the oscillationscompare to the spectrum of the model. Since the peakfrequencies can only take bin values of the FFT frequen-cies, the one nearest to the actual value, the frequenciesshift with L in a chain saw manner. The measured fre-quencies with their error bars compare to both the scalarmeson mass and twice the vector meson mass. However,looking also at the subdominant peaks in the Fourierspectrum, an example is displayed in fig. 4, clearly showsthat the peaks correspond in fact to twice meson masses.This is also confirmed with computations at higher val-ues of m/e , where the convergence of the method at suchlarge system sizes is worse, but the scalar meson massand twice the vector meson mass can be better discrim-inated. This suggests that the horizon violation in theSchwinger model is mediated through correlated mesonpairs which is consistent with the mechanism of the ef-fect in the sine-Gordon model [51]. The understanding isthat a quench of a system parameter entangles pairs ofparticles available in the spectrum which is reflected inlong range current-current correlations. Discussion. -
We have implemented a truncatedHamiltonian method for the real time dynamics of the1+1D quantum electrodynamics, the Schwinger modeland used it to study quenches of the model. We havedemonstrated that the connected correlations of the spa-tial component of the fermionic current (cid:104) J x ( t, x ) J x ( t, y ) (cid:105) spread outside of the horizon in the way that has beenpreviously observed in the sine-Gordon (SG) model [51].In the initial state, (cid:104) J x J x (cid:105) is short-range correlated whilea short time later, correlations extend throughout the en-tire system manifestly violating the horizon bound.We stress again that the observed phenomenon is in nocontradiction with relativistic causality as guaranteed bythe Lorentz invariance of the model the micro causality ofthe fields. Rather it is expected that, as has been analyti-cally explained in case of the SG model [51], the violationof horizon can be traced back to the cluster violation ofcertain correlators generated by the quench. They do notcontribute to local observables at t = 0 but the interact-ing dynamics of the model mixes the degrees of freedomin the way that they show up also in correlators of lo-cal observables. In the Schwinger model, these are likelyclosely related to the known cluster violating correlatorsof chiral fermion densities [129–131]. Thus, the horizonviolation effect implies no faster than light transport butrather represents a novel mechanism for generating longrange correlations in QFT through quenches.Using the simplest representative, we have herebydemonstrated that the horizon violation occurs in gaugefield theory. In the future, it would be interesting to ex-plore higher gauge theories like SU(2) or SU(3) or studythe Wess–Zumino–Witten models. It would be of crucialimportance to answer whether the effect is present also in D > . Gauge fields are not dynamical in D = 1 + 1 because there are no transverse directions so the physicscould be drastically different in higher dimensions. Aswell as that, analytical approaches should be found toget a better understanding of the effect in the Schwingermodel.The horizon violation presented in this work isat the same time a novel phenomenon in quantum-electrodynamics. It is reasonable to expect that it couldhave interesting physical implications, in particular if itturns out that the effect is present also in higher dimen-sions. In condensed matter physics, phase transitionsare an ubiquitous phenomenon and could serve as a trig-ger for horizon violation generating quenches. For thispurpose already the D = 1 + 1 case of QED could bean interesting candidate since at the present day thereare numerous experiments available for probing one di-mensional physics [10]. An especially important class areultra cold atoms in atom chips, where one dimensionalQFTs are directly realised and correlation functions canbe measure both in equilibrium states and nonequilib-rium dynamics [143]. It cosmology, there are also can-didates for quenches like the end of inflation, the QCDand the electroweak transitions and topological symme-try breaking in grand unified theories [144–147]. Con-sider also the following example illustrated in fig. 5: a toyuniverse is created with an anisotropic initial condition -a nonzero background electric field. This is a possibilitysince the zero background field case is a special, fine-tuned, value. In D = 1 + 1 the background electric field is stable and therefore appears in the Schwinger modelwhile in D = 1 + 3 , it is extremely unstable which is whyit is not regularly included in the QED Lagrangian [102].The rapid decay of the background electric field wouldindeed trigger a quench that starts a horizon violationeffect in the QED degrees of freedom as we have seenhere in the θ (cid:54) = 0 → θ = 0 quenches. This transformsthe initial anisotropy of the toy universe into long rangecorrelations. It would be interesting for the future workto explore the possible predictions for traces of this effectin the cosmic microwave background. E Ψ † Ψ Ψ † Ψ E → Figure 5: Decay of the anisotropic initial condition in a toyuniverse as a quench that generates long range correlationsthrough the horizon violation effect. Long range correla-tions are the price that the toy universe has to pay for initialanisotropy.
Finally, it would be interesting to use THM to explorethe confinement and string breaking phenomena in theSchwinger model and to use THM implementations [56]to study dynamics of higher gauge theories.
Acknowledgments
This work was supported by the Max-Planck-HarvardResearch Center for Quantum Optics (MPHQ). The au-thor wants to thank Mari Carmen Bañuls, Peter Lowdon,Jernej Fesel Kamenik and Sašo Grozdanov for useful dis-cussions. Special thanks to Spyros Sotiriadis for manyof our valuable discussions and Gabor Takács for use-ful discussions and feedback to the first version of themanuscript that helped improve this work. [1] A. Kamenev,
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Details of the THM for the Schwinger model
Here we discuss the details of the truncated Hamiltonian method (THM) implementation of the Schwinger modeldefined on an interval of length L with periodic boundary conditions. Bosonisation
The Schwinger model, quantum electrodynamics in D = 1 + 1 is define by the Lagrangian density L = − F µν F µν + ¯Ψ( iγ µ ∂ µ − eγ µ A µ − m )Ψ (6)where Ψ = (cid:18) Ψ − Ψ + (cid:19) is the Dirac fermion field, A µ the electromagnetic (EM) potential, F µν = ∂ µ A ν − ∂ ν A µ the EMtensor, m is the electron mass and e the electric charge. Choosing the Weyl (time) gauge, A = 0 , defining A ≡ A x ,the Hamiltonian density takes the form: H = H EM + H F + H m , H EM = 12 ˙ A , H F = − ¯Ψ γ ( i∂ x − eA )Ψ , H m = m ¯ΨΨ . (7)We use here the metric g = diag (1 , − and the gamma matrices γ = − σ , γ = iσ .In order to treat a gauge field theory in Hamiltonian formalism, one has to remove the redundancy in the degreesof freedom coming from gauge invariance. For the Schwinger model this goes hand in hand with bosonisation.Bosonisation is an exact duality between fermionic and bosonic theories in 1+1D relativistic QFT, developed byTomonaga, Mattis, Lieb, Mandelstam, Coleman, Haldane and others [101, 148–154]. We bosonise the Schwingermodel here using the operatorial (constructive) approach of Iso and Murayama [141].The eigenfunctions and eigenvalues of the massless fermion part of the Hamiltonian H F ( x ) = (cid:80) σ = ± σ Ψ † ( x )( i∂ x − eA )Ψ( x ) are ( i∂ x − eA ) ψ n = (cid:15) n ψ n : ψ n ( x ) = 1 √ L e − i ( (cid:15) n x + e ´ x dx A ) ,(cid:15) n = 2 πL (cid:18) n + 12 δ b − eαL π (cid:19) . (8)The eigenfunctions satisfy periodic boundary conditions (Ramond sector) for δ b = 0 and anti-periodic (Neveu-Schwarzsector) for δ b = 1 . The bosonised form of the Hamiltonian will not depend on δ b reflecting the fact that the bosonisationis an equivalence between bosons and fermions up to Z . We quantise the fermion field by expansion Ψ( x ) = (cid:88) n ∈ Z c − ,n ψ n ( x ) (cid:18) (cid:19) + c + ,n ψ n ( x ) (cid:18) (cid:19) (9)with the canonical anticommutation relation (cid:8) c σ,n , c † ρ,m (cid:9) = δ σ,ρ δ n,m . Then H F = H + + H − with H σ = σ (cid:80) n ∈ Z (cid:15) n c † σ,n c σ,n .1We expand the EM potential and its conjugate dual as: A ( x ) = α + (cid:88) n (cid:54) =0 A n e i πL nx , ˙ A ( x ) = i δδA ( x ) = iL ∂∂α + (cid:88) n (cid:54) =0 δδA n e − i πL nx , (10)and we shall see that as a consequence of the gauge invariance only the zero modes of the EM field α and i ∂∂α aredynamical [141, 155].Expanding the fermion currents J σ ( x ) = Ψ † σ ( x )Ψ σ ( x ) = 1 L (cid:34) Q σ − σ (cid:88) n> √ n (cid:16) b σ,n e − σin πL x + b † σ,n e σin πL x (cid:17)(cid:35) , (11)its modes b σ,n = − σ √ n (cid:80) k ∈ Z c † σ,k c σ,k + σn obey canonical commutation relations (cid:2) b σ,n , b † ρ,m (cid:3) = δ σ,ρ δ n,m . Furtherdefining the N σ vacua as | N − (cid:105) ≡ ∞ (cid:89) n = N − c †− ,n | (cid:105) , | N + (cid:105) ≡ N + − (cid:89) n = −∞ c † + ,n | (cid:105) (12)it can be shown that the Hilbert space spanned by excitations with all possible combinations of b † σ,n on top of | N − (cid:105) ⊗ | N + (cid:105) is equivalent to the Hilbert space spanned by all the possible combinations of c † σ,n on top of | (cid:105) .This is the core of bosonisation.The fermion number operators take the following expectation values on the N σ vacua, (cid:104) Q σ (cid:105) N σ = σ (cid:18) N σ − eαL π + 12 ( δ b − (cid:19) , (13)as can be shown by regularisation by Hurwitz zeta resummation. Similarly, (cid:104) H σ (cid:105) N σ = 2 πL (cid:20) (cid:104) Q σ (cid:105) N σ − (cid:21) . (14)for H = ´ L dx H . Gauge invariance. -
The fermionic Hilbert space combined with the Hilbert space generated by the modes ofthe EM modes display a large redundancy of degrees of freedom as is characteristic for gauge invariant theories andwe have to eliminate this redundancy. QED is invariant under the transformations A µ ( x ) → A µ ( x ) − ∂ µ λ ( x ) , ψ ( x ) → e ieλ ( x ) ψ ( x ) . (15)For systems defined on the circle (and other topologies with nontrivial homotopic groups), the gauge transformationscan be divided into small gauge transformations where both λ ( x ) and e ieλ ( x ) are single valued and large gaugetransformations where e ieλ ( x ) is single valued but λ ( x ) is not. Mathematically speaking, small gauge transformationsare homotopic to the identity of the Lie group and large gauge transformations are not.Let’s begin with small gauge transformations. As a consequence of the Dirac conjecture [156, 157] these arerepresented in the Hilbert space by the operator U ( λ ) = exp( − i ´ L dx G ( x ) λ ( x )) with the Gauss law generator G ( x ) = ∂ x (cid:18) − i δδA ( x ) (cid:19) − eJ ( x ) (16)with J = J + + J − . Requiring that the physical states are invariant under G using the expansions (11), (10) gives Q | physical state (cid:105) = 0 (cid:26) δδA n + eL √ n π (cid:16) b †− ,n − b + ,n (cid:17)(cid:27) | physical state (cid:105) = 0 (cid:26) δδA − n + eL √ n π (cid:16) b † + , | n | − b − , | n | (cid:17)(cid:27) | physical state (cid:105) = 0 (17)2For Q = Q + + Q − . Taking into account (13), the first constraint means that N − = N + = N in physical states andwe can define the N vacua | N (cid:105) ≡ | N − (cid:105) ⊗ | N + (cid:105) . (18)The second and the third constrain mean that all the nonzero momentum modes of the EM field are fixed by gaugeinvariance and the only dynamical modes of the EM field are the zero modes α and i ∂∂α [141, 155]. It is also easy tosee that under small gauge transformations the wave functions (8) transform as ψ n ( x ) → e ieλ ( x ) − ieλ (0) ψ n ( x ) and thus c σ,n → e ieλ (0) c σ,n . It is clear that the currents and its momentum modes, including the charges are invariant underall gauge transformations.The homotopy grout of U (1) symmetry is π ( U (1)) = Z and large gauge transformations are generated by λ ( x ) = 2 πeL wx, w ∈ Z (19)The wave functions ψ n ( x ) are invariant under those thus the fermion operators transform as c n → c σ,n + w . Con-sequently, the N vacua transform as | N (cid:105) → | N + w (cid:105) . The large gauge transformations commute with theHamiltonian so they can be diagonalised in the same basis. The eigenstates of large gauge transformations are the θ vacua | θ (cid:105) = (cid:88) N ∈ Z e − iNθ | N (cid:105) , θ ∈ [0 , π ) (20)which form a continuous degenerate family of ground states of the fermionic parts of the Schwinger model Hamiltonian.A ground state of the full Hamiltonian is obtained as a tensor product of the ground state of the EM part times a θ vacuum. Hamiltonian. -
Following from the gauge invariance constraints (17) we have H EM = − L (cid:34)(cid:18) ∂∂α (cid:19) + 2 (cid:18) eL π (cid:19) (cid:88) n> n (cid:16) b †− ,n − b + ,n (cid:17) (cid:16) b † + ,n − b − ,n (cid:17)(cid:35) . (21)Taking into account that (cid:2) H F , b † σ,n (cid:3) = πL nb † σ,n and deducing its zero mode content from (14), the Hamiltonian H F can only take the form H F = 2 πL (cid:88) σ = ± (cid:34) Q σ −
124 + (cid:88) n> nb † σ,n b σ,n (cid:35) . (22)We can then split the massless part of the Schwinger model Hamiltonian into a part with zero modes and a part withnonzero momentum modes: H EM + H F = H + (cid:88) n> H n − π LH = 2 πL (cid:18) Q + Q (cid:19) − L (cid:18) ∂∂α (cid:19) H n = 2 πL n (cid:16) b † + ,n b + ,n + b †− ,n b − ,n (cid:17) − e L π n (cid:16) b † + ,n − b − ,n (cid:17) (cid:16) b †− ,n − b + ,n (cid:17) . (23)with Q = Q + − Q − which takes the value Q = 2 N − ecLπ + δ b − on physical states and we keep in mind that Q = 0 on physical states.The zero mode Hamiltonian H is the Hamiltonian of a massive harmonic oscillator with mass M = e √ π (24)and can be written in the canonical form as H = M (cid:18) B † B + 12 (cid:19) (25)3with B = (cid:113) ML (cid:0) −√ πQ + ∂∂α (cid:1) , B † = (cid:113) ML (cid:0) −√ πQ − ∂∂α (cid:1) .The nonzero momentum Hamiltonians H n can be diagonalised with a Bogoliubov transformation B σ,n = cosh( t n ) b σ,n + sinh( t n ) b †− σ,n cosh( t n ) = 12 (cid:18) √ E n √ k n + √ k n √ E n (cid:19) sinh( t n ) = − (cid:18) √ E n √ k n − √ k n √ E n (cid:19) (26)with k n = πnL and E n = (cid:112) M + k n . Then H n = E n (cid:16) B † + ,n B + ,n + B †− ,n B − ,n + 1 (cid:17) (27)and H EM + H F becomes the Hamiltonian of the free massive boson with the mass M . This reproduces the Schwinger’sresult that the QED in D = 1 + 1 is gaped even if the bare mass of the fermion is zero. The Bogoliubov operator ofthis transformation, U n b σ,n U † n = B σ,n , is the squeezing operator U n = exp (cid:104) − t n (cid:16) B † + ,n B †− ,n − B + ,n B − ,n (cid:17)(cid:105) meaningthat the vacua annihilated by the massive modes B σ,n are the squeezed coherent θ vacua | θ (cid:105) M = (cid:32) (cid:89) n> U n (cid:33) | θ (cid:105) . (28)It remains to treat the mass term in the Hamiltonian, H m . We can express it in terms of the bosonic momentummodes of the currents, b σ,n using the relation Ψ σ ( x ) = 1 √ L : e − σi √ π Φ σ ( x ) : F σ (29)with Φ σ ( x ) = 1 √ π (cid:40) πL Q σ x − i (cid:88) n> √ n (cid:16) b σ,n e − σin πL x − b † σ,n e σin πL x (cid:17) + σe ˆ x dx (cid:48) A ( x (cid:48) ) (cid:41) (30)and with the normal ordering with respect to the modes b σ,n , which is the bosonisation relation for a fermion coupledto the EM field. Here, F σ are the Klein factors satisfying [ F σ , A m ] = (cid:20) F σ , δδA m (cid:21) = 0 , [ F σ , b ρ,m ] = (cid:2) F σ , b † ρ,m (cid:3) = 0 , (cid:2) Q σ , F † ρ (cid:3) = δ σ,ρ F † ρ , [ Q σ , F ρ ] = − δ σ,ρ F ρ , (cid:8) F † σ , F ρ (cid:9) = 2 δ σ,ρ , F † σ F σ = 1 . (31)Since a function of b σ,n and b † σ,n can never alter the fermion number, the Klein factors make sure that Ψ σ ( x ) as definedabove has the true fermionic character. Some authors prefer to use exponentials of the zero modes of the compactifiedmassless boson field in place of the Klein factors and the two conventions are fully equivalent. In particular, it canbe shown that { Ψ σ ( x ) , Ψ ρ ( x ) } = δ σ,ρ δ ( x − y ) . Using the relations (cid:104) δδA n , b † + ,n (cid:105) = (cid:104) δδA − n , b + ,n (cid:105) = (cid:104) δδA − n , b †− ,n (cid:105) = (cid:104) δδA n , b − ,n (cid:105) = eL π √ n which follow from (17) it’s easy to see that (cid:104) δδA ( x ) , Ψ σ ( y ) (cid:105) = 0 . Finally, considering that F σ → e ieλ (0) F σ under gauge transformations, if follows that Ψ σ ( x ) transforms as the fermion field. We also have, asfollows from the second line of (31) that F † σ F − σ | N (cid:105) = | N + σ (cid:105) and thus: F † σ F − σ | θ (cid:105) M = e σiθ | θ (cid:105) M . (32)Using the definition (30) we can also read out the fermionisation relation for the fermion field coupled to the EMfield, the inverse of the bosonisation relation: ∂ x φ σ ( x ) = √ πJ σ ( x ) + σe √ π A ( x ) . (33)4We can use the bosonisation relation (29) to express the mass term in the Hamiltonian as H m = − m L e (cid:80) n> n ( − knEn ) ˆ L dx (cid:88) σ = ± e − σi πL : e σi √ π Φ( x ) : M F † σ F − σ , (34)where we have defined Φ( x ) ≡ Φ + ( x ) + Φ − ( x ) and : • : M denotes normal ordering with respect to the massive modes B σ,n . The prefactor e − σi πL comes from commuting F † σ past e iσ πL Q σ x using (31). The prefactor e (cid:80) n> n ( − knEn ) comesfrom substituting the Bogoliubov transform (26) into Φ σ ( x ) and then rearranging the expression for H m into thenormal ordered form w.r.t. M . In the L → ∞ limit these prefactors take the value M π e γ where γ = 0 . . . . is theEuler-Mascheroni constant.Finally, putting all the terms together, with the L → ∞ expression for the prefactor in the mass term, the Schwingermodel Hamiltonian takes the form H = M (cid:16) B † B (cid:17) + (cid:88) n> E n (cid:16) B † + ,n B + ,n + B †− ,n B − ,n (cid:17) + const − mM π e γ L ˆ L dx (cid:88) σ = ± : e σi √ π Φ( x ) : M F † σ F − σ ”=” ˆ L (cid:20) (cid:0) Π + ( ∂ x Φ) + M Φ (cid:1) − mM π e γ :cos (cid:16) √ π Φ( x ) + θ (cid:17) : M (cid:21) (35)where const = (cid:80) n> E n + M only affects the ground state energy and will be irrelevant to us. The last "equality"is to be understood only up to the details of the modes captured through the above bosonisation procedure and hastaken into account (32) and the fact that all the physical states are created on top of the θ vacuum. The parameter θ thus appears in the Hamiltonian and plays the role of the constant background electric field as first pointed outby Coleman [101, 102]. As is manifest in the first line, the zero mode B does not enter in the cosine term and is aharmonic oscillator decoupled from the other degrees of freedom. Hilbert space. -
As has been made explicit in the above discussion, the Hilbert space of the Schwinger modelafter eliminating the gauge redundancy takes the form of the tensor product of the Hilbert space of the zero modeswith the Hilbert space generated by all the possible bosonic excitations on top of the theta vacuum. All together wecan write any state in the Hilbert space in the form | (cid:126)r (cid:105) ≡ N (cid:126)r (cid:16) B † (cid:17) r ∞ (cid:89) n =1 (cid:16) B †− ,n (cid:17) r − ,n (cid:16) B † + ,n (cid:17) r + ,n | (cid:105) ⊗ | θ (cid:105) M (36)where (cid:126)r ≡ ( r , r − , , r − , , . . . , r + , , r + , , . . . ) is a vector of occupation numbers and | (cid:105) is the vacuum of the B mode.The normalisation is N (cid:126)r = ( r !) (cid:81) ∞ k =1 ( r k, − !)( r k, + !) . Truncated Hamiltonian method
The truncated Hamiltonian method (THM) consists of splitting the Hamiltonian into an analytically solvable andan unsolvable part, the perturbing potential. Then, the perturbing operator and the observables are expressed as amatrix in the eigenbasis of the solvable part. Finally, an energy cutoff is introduced which renders the matrices finiteand enables numerical diagonalisation which is the key to nonperturbative treatment of a strong interaction with theTHM. The above procedure of eliminating the redundant degrees of freedom and bosonising the model suggests anatural splitting of the Hamiltonian (35) into the quadratic part H EM + H F and the cosine potential H m .In the following we first list the matrix elements in the Hilbert space of the quadratic part of the Hamiltonian forof all the required operators and then discuss how to implement the THM. Matrix elements. -
The matrix elements are computed between general states of the Hilbert space | (cid:126)r (cid:105) and | (cid:126)r (cid:48) (cid:105) , defined in eq. (36)). The required matrix elements are:5Boson mode operators: (cid:104) (cid:126)r (cid:48) | B † σ,n | (cid:126)r (cid:105) = (cid:89) ρ,k (cid:54) = σ,n δ r (cid:48) ρ,k ,r ρ,k (cid:113) ( r σ,n + 1) δ r (cid:48) σ,n − ,r σ,n (37) (cid:104) (cid:126)r (cid:48) | B σ,n | (cid:126)r (cid:105) = (cid:89) ρ,k (cid:54) = σ,n δ r (cid:48) ρ,k ,r ρ,k √ r σ,n δ r (cid:48) σ,n +1 ,r σ,n (38)Boson number operator: (cid:104) (cid:126)r (cid:48) | B † σ,n B σ,n | (cid:126)r (cid:105) = r σ,n δ (cid:126)r (cid:48) ,(cid:126)r (39)Vertex operator: To implement the cosine potential we need the matrix elements (cid:104) (cid:126)r (cid:48) | : e ρi √ π Φ( x ) : M F † ρ F − ρ | (cid:126)r (cid:105) == e ρiθ (cid:104) (cid:126)r (cid:48) | (cid:89) σ = ± ∞ (cid:89) n =1 e − ρ √ πL √ En B † σ,n e iσknx e ρ √ πL √ En B σ,n e − iσknx | (cid:126)r (cid:105) = e ρiθ δ r (cid:48) ,r (cid:89) σ = ± ∞ (cid:89) n =1 (cid:112) r (cid:48) σ,n ! r σ,n ! e iσk n x ( r (cid:48) σ,n − r σ,n ) ·· ∞ (cid:88) j (cid:48) σ,n =0 ∞ (cid:88) j σ,n =0 ( − j (cid:48) σ,n j σ,n ! j (cid:48) σ,n ! (cid:32)(cid:114) πL ρ √ E k (cid:33) j σ,n + j (cid:48) σ,n (cid:68) ( B σ,n ) r (cid:48) σ,n (cid:0) B † σ,n (cid:1) j (cid:48) σ,n ( B σ,n ) j σ,n (cid:0) B † σ,n (cid:1) r σ,n (cid:69) (40)with (cid:68) ( B σ,n ) r (cid:48) σ,n (cid:0) B † σ,n (cid:1) j (cid:48) σ,n ( B σ,n ) j σ,n (cid:0) B † σ,n (cid:1) r σ,n (cid:69) = (cid:18) r (cid:48) σ,n j (cid:48) σ,n (cid:19) (cid:18) r σ,n j σ,n (cid:19) j (cid:48) σ,n ! j σ,n !( r σ,n − j σ,n )! δ r (cid:48) σ,n − j (cid:48) σ,n ,r σ,n − j σ,n Θ( r σ,n ≥ j σ,n ) In the first line we have substituted in the Bogoliubov transformation (26) and used (32) to evaluate the ex-pectation value of the Klein factors. Upon the integration ´ L dx (cid:104) (cid:126)r (cid:48) | : e ρi √ π Φ( x ) : M F † ρ F − ρ | (cid:126)r (cid:105) , the factor (cid:81) σ = ± (cid:81) ∞ n =1 e iσk n x ( r (cid:48) σ,n − r σ,n ) gives the momentum conservation δ (cid:0)(cid:80) σ = ± σ (cid:80) ∞ n =1 n ( r (cid:48) σ,n − r σ,n ) (cid:1) . This is a mani-festation of translation invariance and means that we can diagonalise different total momentum sectors separatelyand compute the dynamics only in the sector where the initial state resides, the total momentum zero sector, (cid:80) σ = ± σ (cid:80) ∞ n =1 nr σ,n = 0 . The expression for the matrix elements of the vertex operator is a product of termscorresponding to the two chiralities which is another property that facilitates the implementation. Furthermore, alsothe parity symmetry, σ → − σ , is manifest here as well as the fact that the cosine potential does not mix differentsectors of the B mode.Observables:In the zero sector of the total momentum where the quench dynamics resides, only those quadratic terms of bosonicmodes in C µ ( t, x, y ) give nonzero contributions which preserve the momentum. Thus, the expectation values are: (cid:104) Ψ | : J ( x ) J ( y ): | Ψ (cid:105) = 1 πL ∞ (cid:88) n =1 k n E n cos ( k n ( x − y )) (cid:32) (cid:88) σ = ± (cid:10) B † σ,n B σ,n (cid:11) Ψ − (cid:104) B − ,n B + ,n (cid:105) Ψ − (cid:68) B †− ,n B † + ,n (cid:69) Ψ (cid:33) (cid:104) Ψ | : J ( x ) J ( y ): | Ψ (cid:105) = 1 πL ∞ (cid:88) n =1 E k cos ( k n ( x − y )) (cid:32) (cid:88) σ = ± (cid:10) B † σ,n B σ,n (cid:11) Ψ + (cid:104) B − ,n B + ,n (cid:105) Ψ + (cid:68) B †− ,n B † + ,n (cid:69) Ψ (cid:33) (41)where we used the mode expansion of the currents (11), expressed the charges in terms of the B modes usingthe equations below (25), abbreviated (cid:104)•(cid:105) Ψ ≡ (cid:104) Ψ | • | Ψ (cid:105) and dropped the diverging (cid:80) ∞ n =1 k n E n cos ( k n ( x − y )) and (cid:80) ∞ n =0 E n cos ( k n ( x − y )) by normal ordering. The expectation values of the quadratic terms on a state can becomputed using the matrix elements (37), (38) and (39).6 Truncation. -
We preform the THM truncation by choosing a value for the cutoff energy E cut and keeping onlythose states of the Hilbert space | (cid:126)r (cid:105) for which (cid:104) (cid:126)r | H EM + H F | (cid:126)r (cid:105) ≤ E cut . This results in a better converging code thanfor example if truncating by keeping a fixed number of momentum modes. The truncation criterium depends on thecharge e and the system size L (as E n = (cid:113) k n + e π = πL (cid:113) n + L (2 π ) e π ) and for fixed E cut the number of states inthe THM Hilbert space decreases with increasing e and L . Therefore, in practice the truncation is done by choosing adesired number of states in the THM Hilbert space and then for a given e and L finding E cut that gives us a Hilbertspace size closest to the desired one. In that way we can assure that results obtained at different e and L are achievedwith comparable Hilbert space sizes.The size of the Hilbert space that has to be kept in the computer’s memory can be reduced by taking into accountthe symmetries of the model. Since the zero mode B is decoupled from the rest of the modes, we can diagonalisethe Hamiltonian in each of it’s sectors separately. In particular, for real time dynamics following quenches it isenough to keep the (cid:68) B † B (cid:69) = 0 sector where the initial states, the ground states, reside. Furthermore, becauseof the translation invariance of the model, the ground states are in the the zero total momentum sector ( p tot = (cid:80) σ = ± σ (cid:80) ∞ n =1 k n (cid:10) B † σ,n B σ,n (cid:11) = 0 ) of the Hilbert space, which drastically reduces the number of states that have tobe kept in the computer’s memory in order to compute the quench dynamics. We do, however, have to diagonalisethe Hamiltonian also in the sectors with other values of the total momentum in order to compute the full spectrumof the model (excited states). Finally, we can further reduce the Hilbert space dimension by taking into account theparity symmetry ( σ → − σ ) of the Hamiltonian. For the results presented in this Letter, we use up to 15000 statesper sector.In case of truncated conformal space approach (TCSA) methods, where the expansion is around a CFT, therenormalisation group theory guarantees that for relevant perturbing operators, the cut-away high energy part of theHilbert space is only very weakly coupled to the low energy part and therefore does not modify the low energy physicsthat one studies with such methods [38, 46]. In a more general expansion like we use here, we cannot directly rely onthe RG theory and have to establish convergence by extensive tests. We have therefore tested that all our results haveconverged with the THM cutoff. We have also tested that the scalar particle mass computed with our method agreeswith matrix product states (MPS) and tensor network (TN) computations with a discretised version of the Schwingermodel [116–118] (fig. 2 in the main text) and that in the e → limit we recover the spectrum of the sine-Gordonmodel. Quench protocol. -
In order to study the quench dynamics, one takes for the initial state the ground state | Ψ (cid:105) of the prequench Hamiltonian H ( m /e , θ , L ) which can be found by numerical diagonalisation of the Hamiltonian.At t = 0 , the parameters are quenched to the postquench values H ( m/e, θ, L ) . The dynamics is computed using thenumerical exponentiation of the postquench Hamiltonian: | Ψ( t ) (cid:105) = e − itH | Ψ (cid:105) . (42)Finally, correlators are computed as expectation values on these states C µ ( t, x, y ) = (cid:104) J µ ( t, x ) J µ ( t, y ) (cid:105) ..