Quantum electronic circuit simulation of generalized sine-Gordon models

Investigation of strongly interacting, nonlinear quantum field theories (QFTs) remains one of the outstanding challenges of modern physics. Here, we describe analog quantum simulators for nonlinear QFTs using mesoscopic superconducting circuit lattices. Using the Josephson effect as the source of nonlinear interaction, we investigate generalizations of the quantum sine-Gordon model. In particular, we consider a two-field generalization, the double sine-Gordon model. In contrast to the sine-Gordon model, this model can be purely quantum integrable, when it does not admit a semiclassical description—a property that is generic to many multifield QFTs. The primary goal of this work is to investigate different thermodynamic properties of the double sine-Gordon model and propose experiments that can capture its subtle quantum integrability. First, we analytically compute the mass spectrum and the ground-state energy in the presence of an external “magnetic” field using Bethe ansatz and conformal perturbation theory. Second, we calculate the thermodynamic Bethe ansatz equations for the model and analyze its finite temperature properties. Third, we propose experiments to verify the theoretical predictions.

Figure 1

TBA diagram for the dSG model. The solid circles denote the physical massive particles: soliton ($s$), antisoliton ($a$), and breathers ($b_j, j=1,...,n$). The empty circles denote pseudoparticles ($j=1,...,n+2$) needed to diagonalize the dSG scattering matrix. The pseudoparticle 1 has a cross on it, which indicates that this particle has a mass term in its TBA equation. The connectivity of the diagram encodes which particles show up in the TBA equation of a given particle. The arrows on the links encode the sign of the term on the right-hand side (RHS) of the TBA equation. For instance, for a link connecting particles $p,q$, if there is an arrow incident on $q$ and none on $p$, then the RHS of the TBA equation for $p$ has a minus sign in front of the term involving $q$, while the term on the RHS for $q$ involving $p$ has a plus sign.

Figure 2

(a)–(d) Essential primitives of QEC lattices [(a) an inductance, $L$, (b) a capacitance, $C$, (c) a mutual inductance, $M$, and (d) a Josephson junction (junction energy $E_J$ and junction capacitance $C_J$)]. The magnetic fluxes going through the inductance ($L$), the mutual inductance ($M$), and the Josephson junction are denoted by $\varphi , {\varphi }_1,{\varphi }_2$, and $\varphi$ respectively. The charges on the capacitor plates of the capacitances $C$ and $C_J$ are denoted by $Q$. (e) One unit cell of the QEC for the dSG model. The top and bottom horizontal lines denote the electrical ground. The Josephson junctions (junction energy $E_{J,0}$ and junction capacitance $C_0$) on the vertical links give rise to the nonlinear interaction. The Josephson junctions (junction energy $E_J$ and junction capacitance $C_J$) give rise to a large impedance of the array, without any nonlinear effects. Finally, the mutual inductance $M$ and the capacitance $C$ provide the necessary inductive and capacitive coupling between the upper and lower parts of the array. The bosonic fields, ${\phi }_1,{\phi }_2$, are the node fluxes at the points indicated.

Figure 3

Effective central charge for $n=2$. The solid blue line is obtained by solving the TBA equations given by Eq. (B8). The black dots are obtained by performing a fit to Eq. (B10).

Figure 4

Unit cells for a free-boson quantum field theory (a), the quantum sine-Gordon model (b), and the double sine-Gordon model (c).