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arXiv:2408.10304 · Physical review. B./Physical review. B

Quantum Zeno effect in noisy integrable quantum circuits for impurity models

Yicheng Tang, Pradip Kattel, J. H. Pixley, Natan Andrei

Published Feb 26, 2025• 10 citations
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Abstract

We theoretically study the open quantum system dynamics (in the Trotterized limit) of integrable quantum circuits in the presence of on-site dephasing noise with a spin-1/2 ``impurity'' interacting at the edge. Using a combination of Bethe ansatz (BA) and exact diagonalization (ED), we study the dynamics of both the bulk and the impurity for the XXX (Heisenberg) and the XX qubit chains with and without bulk noise. In the absence of noise, we show that the impurity exhibits two distinct phases: the bound-mode phase, where the impurity undergoes persistent oscillations, and the Kondo phase, where it decays with Kondo time ${t}_{K}$. Turning on the bulk dephasing noise, we find for the two models that in the long-time limit in both regimes, the quantum Zeno effect takes place where the dynamics of the impurity magnetization slows down as the noise strength $\ensuremath{\gamma}$ increases. In contrast, the impurity magnetization in the bound-mode regime shows the opposite effect, decaying faster as the noise strength increases at short times $(t\ensuremath{\ll}1/\ensuremath{\gamma})$. We show that the Zeno effect slows down the impurity dynamics in the long-time limit, driven by the transition in bulk dynamics from ballistic (KPZ) universality class in the XX (XXX) chain to diffusive dynamics in the presence of noise.

DOI
10.1103/physrevb.111.054313

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We investigate the boundary effects that arise when spin-12\frac{1}{2}21​ impurities interact with the edges of the antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg chain through spin exchange interactions. We consider both cases when the couplings are ferromagnetic or antiferromagnetic. We find that in the case of antiferromagnetic interaction, when the impurity coupling strength is much weaker than that in the bulk, the impurity is screened in the ground state via the Kondo effect. The Kondo phase is characterized by the Lorentzian density of states and a dynamical scale, the Kondo temperature TK{T}_{K}TK​, is generated. As the impurity coupling strength increases, TK{T}_{K}TK​ increases until it reaches its maximum value ${T}_{0}=2\ensuremath{\pi}J$ which is the maximum energy carried by a single spinon. When the impurity coupling strength is increased further, we enter another phase, the bound mode phase, where the impurity is screened in the ground state by a single particle bound mode exponentially localized at the edge to which the impurity is coupled. We find that, in contrast to the Kondo phase, the impurity can be unscreened by unoccupying the single particle bound mode. This costs an energy Eb{E}_{b}Eb​ that is greater than T0{T}_{0}T0​. There exists a boundary eigenstate phase transition between the Kondo and the bound-mode phases, a transition which is characterized by the change in the number of towers of the Hilbert space. The transition also manifests itself in local thermodynamic quantities---local impurity density of states and the local impurity magnetization in the ground state. When the impurity coupling is ferromagnetic, the impurity is unscreened in the ground state; however, when the absolute value of the ratio of the impurity and bulk coupling strengths is greater than 45\frac{4}{5}54​, the impurity can be screened by adding a bound mode that costs energy greater than T0{T}_{0}T0​. When two impurities are considered, the phases exhibited by each impurity remain unchanged in the thermodynamic limit, but nevertheless the system exhibits a rich phase diagram.
In few-qubit systems, the quantum Zeno effect arises when measurement occurs sufficiently frequently that the spins are unable to relax between measurements. This can compete with Hamiltonian terms, resulting in interesting relaxation processes which depend nonmonotonically on the ratio of measurement rate to coherent oscillations. While Zeno physics for a single qubit is well understood, an interesting open question is how the Zeno effect is modified by coupling the measured spin to a nontrivial bulk. In this work, we study the effect of coupling a one-dimensional transverse field Ising to a Zeno spin which lives at the boundary. We find that sharp singularities occur in the boundary relaxation dynamics, which can be tied to the emergence or destruction of edge modes that can be found analytically. Finally, we provide numerical indications that the dynamical singularities are stable in the presence of integrability-breaking interactions.
Abstract We investigate the boundary phenomena that arise in a finite-size XX spin chain interacting through an XX interaction with a spin − 1 2 impurity located at its edge. Upon Jordan–Wigner transformation, the model is described by a quadratic Fermionic Hamiltonian. Our work displays, within this ostensibly simple model, the emergence of the Kondo effect, a quintessential hallmark of strongly correlated physics. We also show how the Kondo cloud shrinks and turns into a single particle bound state as the impurity coupling increases beyond a critical value. In more detail, using both Bethe Ansatz and exact diagonalization techniques, we show that the local moment of the impurity is screened by different mechanisms depending on the ratio of the boundary and bulk coupling J imp J . When the ratio falls below the critical value 2 , the impurity is screened via the Kondo effect. However, when the ratio between the coupling exceeds the critical value 2 an exponentially localized bound mode is formed at the impurity site which screens the spin of the impurity in the ground state. We show that the boundary phase transition is reflected in local ground state properties by calculating the spinon density of states, the magnetization at the impurity site in the presence of a global magnetic field, and the finite temperature susceptibility of the impurity. We find that the spinon density of states in the Kondo phase has the characteristic Lorentzian peak that moves from the Fermi level to the maximum energy of the spinon as the impurity coupling is increased and becomes a localized bound mode in the bound mode phase. Moreover, the impurity magnetization and the finite temperature impurity susceptibility behave differently in the two phases. When the boundary coupling J imp exceeds the critical value 2 J , the model is no longer boundary conformal invariant as a massive bound mode appears at the impurity site.
This thesis focuses on the study of dissipative quantum impurity models. Quantum impurity models describe a localized quantum system, known as the impurity, that interacts with a large quantum bath, enabling the exploration of essential phenomena such as the Kondo effect in magnetic impurities and the transport properties in quantum dots. An intriguing extension of these models is the exploration of the interplay between strong correlations and dissipation. Dissipation, often described by the Lindblad master equation in its Markovian limit, introduces challenges and opportunities as it interacts with coherent many-body effects. In this thesis, we want to investigate the non-equilibrium dynamics of quantum impurity systems influenced by both strong correlations and dissipation across four main parts. We begin by developing numerical methods, specifically a diagrammatic Monte Carlo technique, to analyze the real-time dynamics of dissipative quantum impurity models. Our findings demonstrate that local Markovian dissipation enhances the convergence of Monte Carlo sampling by constraining the process to a reduced subspace of diagrams. This effectively mitigates the sign problem and facilitates the exploration of longer time scales compared to conventional unitary approaches. In the second part, we derive a self-consistent dynamical map using the Non-Crossing Approximation to study the long-term behavior of dissipative quantum impurity models. This method effectively resums diagrams into a self-consistent Dyson equation, capturing essential non-Markovian environmental effects, including the Kondo effect. In a third part, we examine the relationship between Kondo physics and Markovian dissipation by investigating an interacting quantum dot coupled to a metallic bath under continuous charge monitoring. We identify a crossover from Kondo screening to the Quantum Zeno effect, effectively described by a non-Hermitian Kondo model in the weak dissipation regime. Finally, the last part explores the effects of particle losses on the Kondo effect and the Kondo-Zeno crossover. Preliminary results indicate that while single-particle losses reduce many-body effects, two-body losses introduce intriguing phenomena, such as spin decay and hints of the Kondo-Zeno crossover, along with an unusual spectral function exhibiting a low-frequency lineshape.