Over the last decade tensor network states (TNS) have emerged as a powerful
tool for the study of quantum many body systems. The matrix product states
(MPS) are one particular class of TNS and are used for the simulation of
(1+1)-dimensional systems. In this proceeding we use MPS to determine the
elementary excitations of the Schwinger model in the presence of an electric
background field. We obtain an estimate for the value of the background field
where the one-particle excitation with the largest energy becomes unstable and
decays into two other elementary particles with smaller energy.Comment: Proceeding of talk presented at the 33rd International Symposium on
  Lattice Field Theory, 14-18 July 2015, Kobe, Japan; Proceeding of talk
  presented at The European Physical Society Conference on High Energy Physics,
  22-29 July 2015, Vienna, Austria (PoS(EPS-HEP2015)375

Buyens, Boye

Haegeman, Jutho

Van Acoleyen, Karel

Verstraete, Frank

English

arXiv

Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular class of TNS and are used for the simulation of (1+1)-dimensional systems. In this proceeding we use MPS to determine the elementary excitations of the Schwinger model in the presence of an electric background field. We obtain an estimate for the value of the background field where the one-particle excitation with the largest energy becomes unstable and decays into two other elementary particles with smaller energy

Ghent University Academic Bibliography

Tensor networks for gauge field theoriesB. Buyens∗a, J. Haegemana, F. Verstraeteab, K. Van AcoleyenaaDepartment of Physics and Astronomy, Ghent University, Krijgslaan 281, S9, 9000 Gent,BelgiumbVienna Center for Quantum Science and Technology, Faculty of Physics, University of Vienna,Boltzmanngasse 5, 1090 Vienna, AustriaE-mail: boye.buyens@ugent.be, jutho.haegeman@ugent.be,frank.verstaete@ugent.be, karel.vanacoleyen@ugent.beOver the last decade tensor network states (TNS) have emerged as a powerful tool for the studyof quantum many body systems. The matrix product states (MPS) are one particular class of TNSand are used for the simulation of (1+1)-dimensional systems. In this proceeding we use MPSto determine the elementary excitations of the Schwinger model in the presence of an electricbackground field. We obtain an estimate for the value of the background field where the one-particle excitation with the largest energy becomes unstable and decays into two other elementaryparticles with smaller energy.The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan*∗Speaker.c© Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/TN for gauge field theories B. Buyens1. IntroductionThe exponential scaling of the Hilbert space with the number of particles or sites of a quantummany body system makes any attempt of solving a realistic system by using exact diagonalizationimpossible. Fortunately, the study of entanglement has shown that the low energy states of localgapped Hamiltonians live in a tiny corner of Hilbert space. This tiny corner corresponds to statesfor which the entanglement entropy of region scales with the boundary of that region instead ofbeing extensive. Because they dominate the behavior of condensed matter systems at low tempera-tures, it makes sense to focus on the low energy states. Also for quantum field theories, independentof whether they are strongly coupled or weakly coupled, the low energy regime is of interest.Tensor network states (TNS) [1] are a variational class of states living in this tiny corner. Ide-ally, the number of parameters of these states is small and expectation values of local quantitiescan be computed efficiently in the number of sites in the system. In one spatial dimension themost famous example are the matrix product states (MPS). It is rigorously proven that they canefficiently approximate the low energy states of local gapped Hamiltonians [2]. The many suc-cessful simulations of many-body systems using MPS in the last decade showed that this result isnot only of theoretical interest. Furthermore, as MPS are formulated in the Hamiltonian frame-work, they enable the difficult simulation of out-of-equilibrium physics [3, 4]. In the last yearsMPS have proven to be relevant for studying gauge theories, e.g. [5, 6]. In particular for the mas-sive Schwinger model, QED2 with one flavor, different groups considered MPS simulations, e.g.[7, 8, 9, 10, 11, 12]. For higher dimensions different gauge invariant TNS constructions have alsobeen developed [13, 14, 15, 16] with some first numerical applications on simple gauge theories.Here we continue our research on the Schwinger model [8, 9, 10] by investigating the one-particle spectrum in the presence of an electric background electric field gα . For α = 1/2 some-thing interesting happens. As the fermion mass increases there is a phase transition around (m/g)c≈0.33 related to the spontaneous breaking of the CT-symmetry [7, 17, 18]. The ground state is de-generate for m/g > (m/g)c and kinks ‘connecting’ the two vacua arise. This is different from thespectrum in the case of a zero background electric field where the elementary particle spectrumconsists of three or more stable particles for non-vanishing fermion mass [17]. In mass perturba-tion theory two stable particles survive for α < 1/4 and only one for 1/4 < α < 1/2. As a newstep in completing the phase diagram of the Schwinger model we will determine the elementaryexcitations for different values of α beyond mass perturbation theory. Earlier numerical studieson the elementary excitations in the non-perturbative regime of the Schwinger model exclusivelyfocussed on the cases α = 0 [7, 11, 19] and α = 1/2 [7]. An overview of the low energy spectrummight also help in a better understanding of the dynamics induced by a quench in the form of anelectric field. Indeed, in [9] we hinted that the behavior in the linear response regime can be under-stood by looking at the one-particle excitations of the Hamiltonian. Even beyond linear response,similar arguments explain the observations [20].2. Setup in the MPS-frameworkHamiltonian. The lattice Hamiltonian for the Schwinger model reads (see [10, 21] for details):H =g2√x(∑n∈ZE2(n)g2+√xmg ∑n∈Z(−1)nσz(n)+ x∑n∈Z(σ+(n)eiθ(n)σ−(n+1)+h.c.)). (1)2TN for gauge field theories B. BuyensHere we have introduced the parameter x ≡ 1/(g2a2) with a the lattice spacing, m the fermionmass and g the coupling constant. From now on we will work in units g = 1. The staggeredfermions are traded for a spin system on the sites: σz(n) |s〉n = s |s〉n (s = ±1),σ± = (1/2)(σx±iσy). The gauge fields live on the links and we have the operators θ(n) = agA1(na) and theirconjugate momenta E(n) ([θ(n),E(n′)] = iδn,n′), which correspond to the electric field. In anuniform electric background field E = gα in a compact formulation we have E(n) = g(L(n)+α)where L(n) has integer charge eigenvalues p ∈ Z and eiθ(n) and e−iθ(n) correspond to the laddersoperators: L(n) |p〉n = p |p〉n, e±iθ(n) |p〉n = |p±1〉n , p ∈ Z.The key-feature of the Hamiltonian (1) is that it has the gauge symmetry generated by G(n) =L(n)−L(n− 1)− (σz(n)+ (−1)n)/2. A gauge theory differs from other theories by the fact thatall physical states |Φ〉 have to satisfy G(n) |Φ〉 = 0,∀n ∈ Z. Furthermore, the Hamiltonian is in-variant under translations over an even number of sites. The fact that it is not invariant under asingle translation originates from the staggered formulation. Finally, for α = 0 and α = 1/2 theHamiltonian exhibits the CT symmetry which is the charge conjugation C (E →−E, σz→−σz)added with a translation T over one site. It is known that this CT symmetry is only spontaneousbroken for α = 1/2 and m/g& 0.33 [7, 17].Ground state and excitations. In our approach site n and link n are blocked into an effectivesite with local Hilbert space spanned by the kets |κ〉n where κ = (s, p)(s = ±1, p ∈ Z). In thethermodynamic limit (N = ∞) we proposed in [10] the following ansatz for the ground state|Ψ(A)〉= ∑{κn}v†L(∏n∈ZAκ2n−11 Aκ2n2)vR |κ 〉 , |κ 〉= {|κn〉}n∈Z,Aκn ∈ CDn×Dn+1 ,vL ∈ CD1 ,vR ∈ CD2(2a)[As,pn ](q,αq);(r,βr) = δp,q+(s+(−1)n)/2δr,p[as,pn ]αq,βr ;q,r ∈ Z,αq = 1 . . .Dnq,βr = 1 . . .Dn+1r . (2b)One observes immediately that (2a) is manifest invariant under T 2 while the constraint (2b) imposesgauge invariance. The variational freedom of this state thus lies within the matrices as,pn . For α = 0or 1/2 one can impose CT invariance by setting Aκ2 = Aκc1 with κc = (−s,−2α − p) [8]. Theoptimal approximation within this class of states for the ground state is obtained using the time-dependent variational principle [10, 22].For the elementary excitations with momentum k we take the ansatz [23, 24]:|Ψk(B)〉= ∑m∈Ze2ikn/√x ∑{κn}v†L(∏n≤mAκ2n−11 Aκ2n2)Bκ2m+1,...,κ2(m+M)k(∏n>m+MAκ2n−11 Aκ2n2)vR |κ 〉 , (3a)where A1 and A2 correspond to the ground state (2) and gauge invariance is imposed by[B(s1,p1),...,(s2M ,p2M)k ](q,αq);(r,βr) =(2M∏n=2δpn,pn−1+(s+(−1)n)/2)δp1,q+(s−1)/2δp2M ,r[bp1,s1,...,s2Mk ]αq,βr .(3b)For α = 0 and α = 1/2 one can further constrain Bk to classify the states according to their CT -number, see [8] for an example. In this case the excitations with CT =−1 are referred to as ‘vector’particles and excitations with CT = 1 are referred to as ‘scalar’ particles.3TN for gauge field theories B. Buyens1/√x0 0.05 0.1 0.15 0.20.9850.990.99511.0051.011.0151.021.0251.03E2(x) (m/g = 0.25,α = 0.45)linear: x1 − x6linear: x2 − x6linear: x3 − x6linear: x4 − x6quadratic: x1 − x6quadratic: x2 − x6quadratic: x3 − x6×10-30 7.50.9870.9880.989α0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.10.250.50.751E10.1250.25m/gFigure 1: Left (a): m/g = 0.25,α = 0.45. Extrapolation of the energy E2 of the second excited stateto x = ∞. We perform several linear and quadratic fits in 1/√x through the points with x = (x1, . . . ,x6) =(25,50,60,75,90,100) (see legend). Inset: our continuum estimate is the mean of all the fits. The error isthe standard deviation. Right (b): Comparision of E1 obtained by our numerical simulations (full line) withmass perturbation theory (dashed line).By minimizing 〈Ψk(B¯k)|H|Ψk(Bk)〉/〈Ψk(B¯k)|Ψk(Bk)〉 with respect to bk one finds the opti-mal approximations |Ψk(Bk)〉 for the excitations. For sufficiently large bond dimension this ansatzshould converge exponentially fast as M increases to an elementary particle with momentum k [23].The speed of convergence depends on how far this excitation is separated from the other excitationsin the same momentum sector (in units of the Lieb-Robinson velocity). Note that the computationtime scales as O(4M maxp D3p) allowing only simulations for small M.3. Results and discussionFor a fixed value of x we approximate the excited states using the ansatz (3) for M = 2. By com-paring these energies with simulations for other values of the bond dimension Dp and for M = 1we obtain an error for truncating the bond dimension and truncating M. Continuum estimates forthe excitation energies are obtained similar as in [10, 11, 18]: we compute excitation energies forx = 25,50,60,75,90,100 and perform linear fits in 1/√x through the points corresponding to thelargest three, four, five and six x-values, see fig. 1 (a) . Furthermore we fit the points correspondingto the largest four, five and six x−values against a quadratic function in 1/√x. All these fits giveus an estimate of the energy in the continuum limit. To have some robustness against the choice offitting interval and the fitting function we take the mean of all these energies as our final estimate.The standard deviation of this mean serves as an error on this value. In our simulations this stan-dard deviation is not larger than of order 10−3 and dominates over the errors of truncating Dp andtruncating M. More accurate results can be achieved by taking larger x−values, although this willrequire a larger bond dimension and thus longer computation time.Note that physics is periodic in α with period 1 and the excitations for α ∈ [1/2,1] can be4TN for gauge field theories B. Buyensα0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.10.511.522.5E (m/g = 0.125)E1E22E1E2,vα0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.10.511.522.5E (m/g = 0.25)Figure 2: Left (a): m/g= 0.125. Energy of the elementary particles for different values of α . For α . 0.35we detect two stable particles with energies E1 and E2. For α & 0.38 the particle with energy E2 hasdisappeared in the continuum spectrum (yellow). Right (b): The same as (a) but for m/g = 0.25. Now thesecond particle is unstable for α & 0.47.obtained from the excitations for α ∈ [0,1/2] by a CT transformation. Therefore we can restrictour computation to α ∈ [0,1/2]. Also, as the Schwinger model is a relativistic theory, the energyEk of a particle with momentum k can be obtained from the energy E0 of this excitation withmomentum zero by the Einstein dispersion relation Ek =√k2+E0. As a consequence we onlyneed to compute the excitations with momentum zero. In fig 1 (b) we compare our numericalresults for the energy E1 of the first excited state with mass perturbation theory [25]:E1 = µ0√1+3.5621mµ0cos(2piα)+5.4807(mµ0)2−2.0933(mµ0)2cos(4piα)+O([mµ0]3).(4)where µ0 = g√pi . The plot shows that our numerical results converge towards (4) when m/g→ 0.For α = 0, we found in [8] two elementary excitations with CT =−1 and energy E1,v and E2,v andone elementary excitation with CT = 1 and energy E1,s. For these energies we had E2,v <E1,v+E1,sbut E2,v > 2E1,v, which means that the decay of E2,v into two elementary particles is only preventedby the CT symmetry. When 0 < α < 1/2 the CT symmetry is broken and this decay is no longerforbidden. This is indeed what we observe in the one-particle spectrum: for α > 0 only the exci-tations with energy E1 resp. E2 corresponding to E1,v and E1,s for α → 0 remain stable, see fig. 2.Furthermore, we observe that the binding energy Ebind = 2E1−E2 decreases as α tends towards1/2. When the binding energy becomes small, the convergence rate of the ansatz (3) as a functionof M to the excited state with energy E2 is rather slow.For m/g = 0.125, see fig 2 (a), the second particle is stable until α . 0.35. For α = 0.38our estimates are E1 = 0.4784(5) and E2 = 0.965(2), indicating that the second excited state is5TN for gauge field theories B. Buyensunstable, E2 > 2E1. When α > 0.38 we have E2(x) > 2E1(x) for all the x−values we used. Weconclude that there are two stable particles for α . 0.35 and only one stable particle for α & 0.38.For m/g = 0.25, see fig 2 (b), our estimates for the energy E2 were unstable against variationof the bond dimension D and M for α ≥ 0.47. The errors on E2 are too large and prevent anextrapolation towards x = ∞. Nevertheless, in our simulations we have E2(x) < 2E1(x) for x =(25,50,60,75,90,100) and the fact that E2(x) decreases as the bond dimension and M increasemight suggest that this particle is still stable but with very small binding energy.For α = 1/2 the ground state is CT invariant. We computed the excitation energies with andwithout classifying the states according to their CT−number. In both cases, we found only oneelementary particle. In the vector sector (CT = −1) all other states had energies that were largerthan 3E1 and in the scalar sector (CT = 1) the energies were larger than 2E1. This correspondsto a theory with one stable particle. Therefore we estimate the value of the electric backgroundfield where the second elementary particle disappears to be larger than 0.47 but smaller than 0.5for m/g = 0.25.In mass perturbation theory it is shown that there are two stable particles for α ≤ 1/4 and onestable particle for 1/4 < α ≤ 1/2 [17]. Together with our results in the non-perturbative regime,we conclude that the value of αc, where one particle disappears, tends to 1/2 for increasing massof the fermions. Note however that the mass gap also decreases for larger m/g and eventually thesystem becomes gapless for m/g = (m/g)c ≈ 0.33 and α = 1/2 [7].In [7] DMRG was used on a finite lattice to investigate the Schwinger model for α = 1/2 and adegeneracy for the first elementary excitation was reported. Contrary, our simulations do not showany traces of that.4. ConclusionIn this proceeding we continued the exploration of the Schwinger model as a testbed for MPSsimulation of Hamiltonian lattice gauge theories. We investigated the elementary particle spec-trum for a non-zero background electric field. For m/g = 0.125,0.25 and small values of thisbackground field we detected two stable particles, but as α → 1/2 one particle disappears in thecontinuum of the spectrum. This mechanism is best understood as the binding energy of the sec-ond excited state becoming too small to be stable against a decay into two elementary particles withsmaller energy. It would definitely be interesting to observe what happens closer to the critical point(m/g,α) = (0.33,1/2).Looking further afield a logic step is the simulations of non-abelian lattice gauge theories in1+1 and the simulations of higher dimensional gauge theories. The latter will be more challengingas the present algorithms for PEPS, the two-dimensional analogue of MPS, are restricted to rela-tively small bond dimension. Nevertheless, in the last years there has been some progress in PEPSalgorithms [26, 27, 28]. Furthermore, as this approach is free of any sign problem and enablesreal-time simulations, it is certainly worthwhile to further explore in this direction.Acknowledgements. We acknowledge very interesting discussions with Laurens Vanderstraeten.This work is supported by an Odysseus grant from the FWO, a PhD-grant from the FWO (B.B), apost-doc grant from the FWO (J.H.), the FWF grants FoQuS and Vicom, the ERC grant QUERGand the EU grant SIQS.6TN for gauge field theories B. BuyensReferences[1] R. Orús, Annals of Physics 349, 117 (2014).[2] M. B. Hastings, Journal of Statistical Mechanics: Theory and Experiment 8, 24 (2007).[3] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003).[4] M. C. Bañuls, M. B. Hastings, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 102, 240603 (2009).[5] S. Kühn, E. Zohar, J. I. Cirac, and M. C. Bañuls, Journal of High Energy Physics 7, 130 (2015), arXiv1505.04441.[6] T. Pichler, M. Dalmonte, E. Rico, P. Zoller, and S. Montangero, (2015), arXiv 1505.04440.[7] T. Byrnes, Density Matrix Renormalization Group: A New Approach to Lattice Gauge Theory(University of New South Wales, 2003).[8] B. Buyens, J. Haegeman, K. Van Acoleyen, H. Verschelde, and F. Verstraete, Physical Review Letters113 (2014).[9] B. Buyens, K. Van Acoleyen, J. Haegeman, and F. Verstraete, (2014), arXiv 1411.0020.[10] B. Buyens, J. Haegeman, H. Verschelde, F. Verstraete, and K. Van Acoleyen, (2015), arXiv1509.00246.[11] M. C. Bañuls, K. Cichy, J. I. Cirac, and K. Jansen, Journal of High Energy Physics 11, 158 (2013).[12] M. C. Bañuls, K. Cichy, J. I. Cirac, K. Jansen, and H. Saito, Phys. Rev. D 92, 034519 (2015).[13] L. Tagliacozzo and G. Vidal, Phys. Rev. B 83, 115127 (2011).[14] L. Tagliacozzo, A. Celi, and M. Lewenstein, Physical Review X 4, 041024 (2014).[15] J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, Physical Review X 5,011024 (2015).[16] E. Zohar, M. Burrello, T. Wahl, and J. I. Cirac, (2015), arXiv 1507.08837.[17] S. R. Coleman, Annals Phys. 101 (1976).[18] C. Hamer, J. Kogut, D. Crewther, and M. Mazzolini, Nuclear Physics B 208, 413 (1982).[19] M. Szyniszewski, (2013), arXiv 1303.0441.[20] B. Buyens, J. Haegeman, H. Verschelde, F. Verstraete, and K. Van Acoleyen, in preparation .[21] J. Kogut and L. Susskind, Phys. Rev. D 11 (1975).[22] J. Haegeman et al., Physical Review Letters 107 (2011).[23] J. Haegeman et al., Physical Review Letters 111 (2013).[24] J. Haegeman, T. J. Osborne, and F. Verstraete, Physical Review B 88 (2013).[25] C. Adam, Annals of Physics 259, 1 (1997), hep-th/9704064.[26] P. Corboz, R. Orús, B. Bauer, and G. Vidal, Physical Review B 81, 165104 (2010), 0912.0646.[27] J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, Physical Review Letters 101, 250602(2008), cond-mat/0703788.[28] L. Vanderstraeten, F. Verstraete, and J. Haegeman, (2015), arXiv 1507.02151.7

Tensor networks for gauge field theories

Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular class of TNS and are used for the simulation of (1+1)-dimensional systems. In this proceeding we use MPS to determine the elementary excitations of the Schwinger model in the presence of an electric background field. We obtain an estimate for the value of the background field where the one-particle excitation with the largest energy becomes unstable and decays into two other elementary particles with smaller energy.Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular class of TNS and are used for the simulation of (1+1)-dimensional systems. In this proceeding we use MPS to determine the elementary excitations of the Schwinger model in the presence of an electric background field. We obtain an estimate for the value of the background field where the one-particle excitation with the largest energy becomes unstable and decays into two other elementary particles with smaller energy.C

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