Finite density QED 1 + 1 near Lefschetz thimbles

This work examines non-perturbative dynamics of a 2-dimensional QFT by using discrete 't Hooft anomaly, semi-classics with circle compactification and bosonization. We focus on charge-q N -flavor Schwinger model, and also Wess-Zumino-Witten model. We first apply the recent developments of discrete 't Hooft anomaly matching to theories on R 2 and its compactification to R × S 1 L . We then compare the 't Hooft anomaly with dynamics of the models by explicitly constructing eigenstates and calculating physical quantities on the cylinder spacetime with periodic and flavor-twisted boundary conditions. We find different boundary conditions realize different anomalies. Especially under the twisted boundary conditions, there are N q vacua associated with discrete chiral symmetry breaking. Chiral condensates for this case have fractional θ dependence e iθ/N q , which provides the N q-branch structure with soft fermion mass. We show that these behaviors at a small circumference cannot be explained by usual instantons but should be understood by "quantum" instantons, which saturate the BPS bound between classical action and quantum-induced effective potential. The effects of the quantum-instantons match the exact results obtained via bosonization within the region of applicability of semi-classics. We also argue that large-N limit of the Schwinger model with twisted boundary conditions satisfy volume independence. arXiv:1905.05781v3 [hep-th]

Section: Symmetry Of Charge-q N -Flavor Schwinger Modelmentioning

confidence: 99%

Fractional θ angle, ’t Hooft anomaly, and quantum instantons in charge-q multi-flavor Schwinger model

Misumi

Tanizaki

Ünsal

2019

“…We have provided a calculation of quantum mechanical tunnelling using the generalized thimble approach [28][29][30], and compared it to the full Schrödinger equation computation, as well as the classical-statistical approximation [37]. The thimble method presented by the authors in [33] explicitly breaks up the path integral into initial conditions (external) and a dynamic (internal) part, and so is ideally suited to understanding the classical-statistical approximation, where the path integral is approximated by summing over solutions to the classical equations of motion.…”

Section: Resultsmentioning

confidence: 99%

“…In this way, each point X on the real surface is flowed into the complex plane. The surface M is called a generalized thimble [28][29][30], and its shape depends on how long one chooses to flow the equations. In the limit of infinite flow time one recovers the Lefschetz thimbles.…”

Section: Lefschetz Thimble and Generalized Thimble Methodsmentioning

confidence: 99%

“…It is clear from these expressions that the exponent I is purely imaginary, and so the exponential term in (2.6) is purely a phase, leading to a highly oscillatory integral. Dealing with this oscillatory behaviour is the main role of Picard-Lefschetz thimbles [17,45], and in this paper we use the generalized thimbles of [28][29][30] to compute the integral.…”

Section: The Closed-time Path Integralmentioning

confidence: 99%

“…This effective doubling of the degrees of freedom helps to significantly improve convergence and one may show that under sensible assumptions, the correct physical result is achieved .Another remedy for the sign problem is to again complexify the real integration variables φ, but rather than doubling the dimensionality of field space R n → C n , we constrain the field variables to live on a specific manifold in the complex plane of real dimension n. The task is then to find a manifold where the integral is better behaved, and we can expect convergence with a manageable amount of importance sampling. This manifold could be the Lefschetz thimbles [17][18][19][20][21][22], or other manifolds such as generalized thimbles [23][24][25][26][27][28][29][30][31], and in each case, we need to provide a numerical algorithm to find the given manifold. The upshot is that as a consequence of Cauchy's theorem (generalised to n dimensions), the path integral on any sensible deformation of R n will give the same result.Since the sampling is performed along the thimble, the method works best when the thimble is unique, or when there is a way of including each thimble systematically one by one.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Quantum tunnelling, real-time dynamics and Picard-Lefschetz thimbles

Mou

Saffin

Tranberg

2019

We follow up the work, where in light of the Picard-Lefschetz thimble approach, we split up the real-time path integral into two parts: the initial density matrix part which can be represented via an ensemble of initial conditions, and the dynamic part of the path integral which corresponds to the integration over field variables at all later times. This turns the path integral into a two-stage problem where, for each initial condition, there exits one and only one critical point and hence a single thimble in the complex space, whose existence and uniqueness are guaranteed by the characteristics of the initial value problem. In this paper, we test the method for a fully quantum mechanical phenomenon, quantum tunnelling in quantum mechanics. We compare the method to solving the Schrödinger equation numerically, and to the classical-statistical approximation, which emerges naturally in a well-defined limit. We find that the Picard-Lefschetz result matches the expectation from quantum mechanics and that, for this application, the classical-statistical approximation does not.-1 -There are a number of ways to try and sidestep this problem. One is to solve for the time evolution of the fields, or correlators of the fields, through some effective, systematically truncated, evolution equations. Good examples include the classical statistical approximation, where an ensemble mimicking the quantum initial density matrix are evolved using straightforward classical equations of motion. In cases where the occupation numbers are large, and the system is far from thermal equilibrium, this is a powerful and easily implementable approximation (see for instance [3][4][5][6][7]). Truncated Schwinger-Dyson equations for two-or higher-point correlators have been used to good effects for the approach to equilibrium, as well as in cosmological applications involving scalar and fermion fields (see [8][9][10][11]). Stochastic equations have also proven a valuable tool for concrete cases, that allow for a mapping of the full dynamics to such a description in a systematic way (see for instance [12,13]).In some cases, however, one must return to the direct computation of the path integral (1.2), but standard Monte-Carlo techniques, routinely and effectively applied to euclidean systems in QCD, fall short. A number of ways to try and resolve this sign problem have been proposed. Stochastic quantization is particularly promising, providing a Langevin type equation to sample the field space [14-16]. For a real-time path integral, the procedure leads to complex Langevin equations, driving the field φ away from the real axis sampling the entire complex plane. This effective doubling of the degrees of freedom helps to significantly improve convergence and one may show that under sensible assumptions, the correct physical result is achieved .Another remedy for the sign problem is to again complexify the real integration variables φ, but rather than doubling the dimensionality of field space R n → C n , we constrain the field variables to live on a...

Real-time quantum dynamics, path integrals and the method of thimbles

Mou

Saffin

Tranberg

et al. 2019

Direct numerical evaluation of the real-time path integral has a well-known sign problem that makes convergence exponentially slow. One promising remedy is to use Picard-Lefschetz theory to flow the domain of the field variables into the complex plane, where the integral is better behaved. By Cauchy's theorem, the final value of the path integral is unchanged. Previous analyses have considered the case of real scalar fields in thermal equilibrium, employing a closed Schwinger-Keldysh time contour, allowing the evaluation of the full quantum correlation functions. Here we extend the analysis by not requiring a closed time path, instead allowing for an initial density matrix for out-of-equilibrium initial value problems. We are able to explicitly implement Gaussian initial conditions, and by separating the initial time and the later times into a two-step Monte-Carlo sampling, we are able to avoid the phenomenon of multiple thimbles. In fact, there exists one and only one thimble for each sample member of the initial density matrix. We demonstrate the approach through explicitly computing the real-time propagator for an interacting scalar in 0+1 dimensions, and find very good convergence allowing for comparison with perturbation theory and the classical-statistical approximation to real-time dynamics.∂I ∂φ i p = 0, and det ∂ 2 I ∂φ i ∂φ j p = 0.(2.3)By Morse theory/Picard-Lefschetz theory, the matrix of second order derivatives of Re[I] has n positive eigenvalues and n negative ones, and near the critical point, we can approximate the Lefschetz thimble with the manifold generated by these n positive eigenvalues/eigenvectors. The "sign problem" is milder on the Lefschetz thimbles than on the real space. On each thimble, Im[I] is constant, and the only varying complex phase comes from the Jacobian of the transformation that maps the complex integration variables into real ones [16]. More