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arXiv:2007.06874 · Nuclear Physics B

The quantum sine-Gordon model with quantum circuits

Ananda Roy, Dirk Schuricht, Johannes Hauschild, Frank Pollmann, Hubert Saleur

Published May 25, 2021• 61 citations
arXivHTMLPDF

Abstract

Analog quantum simulation has the potential to be an indispensable technique in the investigation of complex quantum systems. In this work, we numerically investigate a one-dimensional, faithful, analog, quantum electronic circuit simulator built out of Josephson junctions for one of the paradigmatic models of an integrable quantum field theory: the quantum sine-Gordon (qSG) model in 1+1 space-time dimensions. We analyze the lattice model using the density matrix renormalization group technique and benchmark our numerical results with existing Bethe ansatz computations. Furthermore, we perform analytical form-factor calculations for the two-point correlation function of vertex operators, which closely agree with our numerical computations. Finally, we compute the entanglement spectrum of the qSG model. We compare our results with those obtained using the integrable lattice-regularization based on the quantum XYZ chain and show that the quantum circuit model is less susceptible to corrections to scaling compared to the XYZ chain. We provide numerical evidence that the parameters required to realize the qSG model are accessible with modern-day superconducting circuit technology, thus providing additional credence towards the viability of the latter platform for simulating strongly interacting quantum field theories.

DOI
10.1016/j.nuclphysb.2021.115445

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Analog quantum simulation has the potential to be an indispensable technique in the investigation of complex quantum systems. In this work, we numerically investigate a one-dimensional, faithful, analog, quantum electronic circuit simulator built out of Josephson junctions for one of the paradigmatic models of an integrable quantum field theory: the quantum sine-Gordon (qSG) model in 1+1 space-time dimensions. We analyze the lattice model using the density matrix renormalization group technique and benchmark our numerical results with existing Bethe ansatz computations. Furthermore, we perform analytical form-factor calculations for the two-point correlation function of vertex operators, which closely agree with our numerical computations. Finally, we compute the entanglement spectrum of the qSG model. We compare our results with those obtained using the integrable lattice-regularization based on the quantum XYZ chain and show that the quantum circuit model is less susceptible to corrections to scaling compared to the XYZ chain. We provide numerical evidence that the parameters required to realize the qSG model are accessible with modern-day superconducting circuit technology, thus providing additional credence towards the viability of the latter platform for simulating strongly interacting quantum field theories.
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.
Multicritical Ising models and their perturbations are paradigmatic models of statistical mechanics. In two space-time dimensions, these models provide a fertile testbed for investigation of numerous nonperturbative problems in strongly-interacting quantum field theories. In this work, analog superconducting quantum electronic circuit simulators are described for the realization of these multicritical Ising models. The latter arise as perturbations of the quantum sine-Gordon model with ppp-fold degenerate minima, p=2,3,4,...p=2,3,4,...p=2,3,4,.... The corresponding quantum circuits are constructed with Josephson junctions with $cos(n\ensuremath{\phi}+{\ensuremath{\delta}}_{n})$ potential with $1\ensuremath{\le}n\ensuremath{\le}p$ and ${\ensuremath{\delta}}_{n}\ensuremath{\in}[\ensuremath{-}\ensuremath{\pi},\ensuremath{\pi}]$. The simplest case, p=2p=2p=2, corresponds to the quantum Ising model and can be realized using conventional Josephson junctions and the so-called $0\text{\ensuremath{-}}\ensuremath{\pi}$ qubits. The lattice models for the Ising and tricritical Ising models are analyzed numerically using the density matrix renormalization group technique. Evidence for the multicritical phenomena are obtained from computation of entanglement entropy of a subsystem and correlation functions of relevant lattice operators. The proposed quantum circuits provide a systematic approach for controlled numerical and experimental investigation of a wide range of nonperturbative phenomena occurring in low-dimensional quantum field theories.
Abstract Entanglement or modular Hamiltonians play a crucial role in the investigation of correlations in quantum field theories. In particular, in 1 + 1 space–time dimensions, the spectra of entanglement Hamiltonians of conformal field theories (CFTs) for certain geometries are related to the spectra of the physical Hamiltonians of corresponding boundary CFTs. As a result, conformal invariance allows exact computation of the spectra of the entanglement Hamiltonians for these models. In this work, we perform this computation of the spectrum of the entanglement Hamiltonian for the free compactified boson CFT over a finite spatial interval. We compare the analytical results obtained for the continuum theory with numerical simulations of a lattice-regularized model for the CFT using density matrix renormalization group technique. To that end, we use a lattice regularization provided by superconducting quantum electronic circuits, built out of Josephson junctions and capacitors. Up to non-universal effects arising due to the lattice regularization, the numerical results are compatible with the predictions of the exact computations.
The sine-Gordon field theory emerges as the low-energy description in a wealth of quantum many-body systems. Recent efforts have been directed towards realizing quantum simulators of the model, by interfering two weakly coupled one-dimensional cold atomic gases. The weak interactions within the atomic clouds provide a sine-Gordon realization in the semiclassical regime. Furthermore, the intricate microscopic dynamics prevents a quantitative understanding of the effective sine-Gordon validity realm. In this work, we focus on a spin-ladder realization and observe the emergent sine-Gordon dynamics deep in the quantum regime. We use matrix-product state techniques to numerically characterize the low-energy sector of the system and compare it with the exact field-theory predictions. From this comparison, we obtain quantitative boundaries for the validity of the sine-Gordon description. We provide encompassing evidence for the emergent field theory by probing its rich spectrum and by observing the signatures of integrable dynamics in scattering events.
Building on the established methods for superconducting circuit quantization, we present a new theoretical framework for approximate numerical simulation of Josephson quantum circuits. Simulations based on this framework provide access to a degree of complexity and circuit size heretofore inaccessible to quantitative analysis, including fundamentally new kinds of superconducting quantum devices. This capability is made possible by two improvements over previous methods: first, physically-motivated choices for the canonical circuit modes and physical basis states which allow a highly-efficient matrix representation; and second, an iterative method in which subsystems are diagonalized separately and then coupled together, at increasing size scales with each iteration, allowing diagonalization of Hamiltonians in extremely large Hilbert spaces to be approximated using a sequence of diagonalizations in much smaller spaces.
We present exact results for the susceptibility of the interacting resonant level model in equilibrium. Detailed simulations using both the numerical renormalization group and density matrix renormalization group were performed in order to compare with closed analytical expressions. By first bosonizing the model and then utilizing the integrability of the resulting boundary sine-Gordon model, one finds an analytic expression for the relevant energy scale TK{T}_{K}TK​ with excellent agreement to the numerical results. On the other hand, direct application of the Bethe ansatz of the interacting resonant level mode does not correctly reproduce TK{T}_{K}TK​---however, if the bare parameters in the model are renormalized, then quantities obtained via the direct Bethe ansatz such as the occupation of the resonant level as a function of the local chemical potential do match the numerical results. The case of one lead is studied in the most detail, with many results also extending to multiple leads, although there still remain open questions in this case.