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arXiv:2202.08431 · Physical review. B./Physical review. B

Fixed point annihilation for a spin in a fluctuating field

Adam Nahum

Published Aug 9, 2022• 16 citations
arXivHTMLPDF

Abstract

A quantum spin impurity coupled to a gapless free field (the Bose-Kondo model) can be represented as a $(0+1)$-dimensional field theory with long-range-in-time interactions that decay as $|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}(2\ensuremath{-}\ensuremath{\delta})}$. This theory is a simpler analog of nonlinear $\ensuremath{\sigma}$ models with topological Wess-Zumino-Witten terms in higher dimensions. In this Letter we show that the renormalization group (RG) flows for the impurity problem exhibit an annihilation between two nontrivial RG fixed points at a critical value ${\ensuremath{\delta}}_{c}$ of the interaction exponent. The calculation is controlled at large spin $S$. This clarifies the phase diagram of the Bose-Kondo model and shows that it serves as a toy model for phenomena involving fixed point annihilation and ``quasiuniversality'' in higher dimensions.

DOI
10.1103/physrevb.106.l081109

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A quantum spin impurity coupled to a gapless free field (the Bose-Kondo model) can be represented as a (0+1)(0+1)(0+1)-dimensional field theory with long-range-in-time interactions that decay as ∣t\ensuremath−t\ensuremath′∣\ensuremath−(2\ensuremath−\ensuremathδ)|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}(2\ensuremath{-}\ensuremath{\delta})}∣t\ensuremath−t\ensuremath′∣\ensuremath−(2\ensuremath−\ensuremathδ). This theory is a simpler analog of nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ models with topological Wess-Zumino-Witten terms in higher dimensions. In this Letter we show that the renormalization group (RG) flows for the impurity problem exhibit an annihilation between two nontrivial RG fixed points at a critical value \ensuremathδc{\ensuremath{\delta}}_{c}\ensuremathδc​ of the interaction exponent. The calculation is controlled at large spin SSS. This clarifies the phase diagram of the Bose-Kondo model and shows that it serves as a toy model for phenomena involving fixed point annihilation and ``quasiuniversality'' in higher dimensions.
The annihilation of two intermediate-coupling renormalization-group (RG) fixed points is of interest in diverse fields from statistical mechanics to high-energy physics, but has so far only been studied using perturbative techniques. Here we present high-accuracy quantum Monte Carlo results for the SU(2)-symmetric S=1/2 spin-boson (or Bose-Kondo) model. We study the model with a power-law bath spectrum ∝ω^{s} where, in addition to a critical phase predicted by perturbative RG, a stable strong-coupling phase is present. Using a detailed scaling analysis, we provide direct numerical evidence for the collision and annihilation of two RG fixed points at s^{*}=0.6540(2), causing the critical phase to disappear for s<s^{*}. In particular, we uncover a surprising duality between the two fixed points, corresponding to a reflection symmetry of the RG beta function, which we utilize to make analytical predictions at strong coupling which are in excellent agreement with numerics. Our work makes phenomena of fixed-point annihilation accessible to large-scale simulations, and we comment on the consequences for impurity moments in critical magnets.
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Spin-boson models are simple examples of quantum dissipative systems, but also serve as effective models in quantum magnetism and exhibit nontrivial criticality. Recently, they have been established as a platform to study the nontrivial renormalization-group (RG) scenario of fixed-point annihilation, in which two intermediate-coupling RG fixed points collide and generate an extremely slow RG flow near the collision. For the Bose Kondo model, a single S=1/2S=1/2S=1/2 spin where each spin component couples to an independent bosonic bath with power-law spectrum ∝ωs\propto ω^s∝ωs via dissipation strengths αiα_iαi​, i∈{x,y,z}i\in\{x,y,z\}i∈{x,y,z}, such phenomena occur sequentially for the U(1)-symmetric model at αz=0α_z=0αz​=0 and the SU(2)-symmetric case at αz=αxyα_z = α_{xy}αz​=αxy​, as the bath exponent s<1s<1s<1 is tuned. Here we use an exact wormhole quantum Monte Carlo method for retarded interactions to explore how this nontrivial fixed-point structure affects the phase diagram and phase transitions of the anisotropic model. In particular, we show how fixed-point annihilation within a symmetry-enhanced critical manifold leads to (i) a continuous order-to-order transition beyond the Landau paradigm, (ii) a symmetry-enhanced first-order transition, and (iii) pseudocriticality, which can be tuned into each other via the bath exponent sss. We extract critical exponents at the continuous transition, but also find scaling behavior at the symmetry-enhanced first-order transition, for which the inverse correlation-length exponent is given by the bath exponent sss. Moreover, we provide direct numerical evidence for pseudocritical scaling on both sides of the fixed-point collision, which manifests in an extremely slow drift of the correlation-length exponent. We also study the crossover away from the SU(2)-symmetric case and determine the phase boundary of an extended U(1)-symmetric critical phase.
The numerical renormalization group method is used to investigate zero-temperature phase transitions in quantum impurity systems, in particular in the particle–hole symmetric soft-gap Anderson model. The model displays two stable phases whose fixed points can be built up of non-interacting single-particle states. In contrast, the quantum phase transitions turn out to be described by interacting fixed points, and their excitations cannot be described in terms of free particles. We show that the structure of the many-body spectrum of these critical fixed points can be understood using renormalized perturbation theory close to certain values of the bath exponents which play the role of critical dimensions. Contact is made with perturbative renormalization group calculations for the soft-gap Anderson and Kondo models. A complete description of the quantum critical many-particle spectra is achieved using suitable marginal operators; technically this can be understood as epsilon-expansion for full many-body spectra.
We use the fuzzy-sphere approach to study the Bose-Kondo impurity problem, namely a spin-SSS impurity coupled to the (2+1)(2+1)(2+1)-dimensional O(3)O(3)O(3) Wilson-Fisher CFT (Heisenberg universality class). We demonstrate that for S=1/2,1,3/2S=1/2,1,3/2S=1/2,1,3/2 the impurity flows to a distinct stable interacting conformal defect for each SSS. Using large-scale exact diagonalization and density-matrix renormalization group methods, we observe integer-spaced defect spectrum consistent with defect conformal symmetry and compute several low-lying defect primary operators as well as the RG monotonic ggg-function. Our findings show that despite sharing the same symmetry and anomaly, Bose-Kondo impurities flow to distinct stable infrared conformal fixed points, which we refer to as \emph{fortuitous universality}. We expect this fortuitous universality to persist for all SSS, extending to S→∞S\rightarrow\inftyS→∞, with each spin-SSS impurity flowing to its own stable infrared conformal fixed point.
We study a system consisting of a junction of NNN wires, where the junction is characterized by a scalar SSS-matrix, and an impurity spin is coupled to the electrons near the junction. The wires are modeled as weakly interacting Tomonaga-Luttinger liquids. We derive the renormalization group (RG) equations for the Kondo couplings of the spin to the electrons on different wires. We analyze the RG flows and fixed points for different values of the initial Kondo couplings and of the junction SSS-matrix, such as the decoupled SSS-matrix and the Griffiths (connected) SSS-matrix. We find that the Kondo couplings flow either towards a ferromagnetic (FM) fixed point or towards large and antiferromagnetic (AFM) values in one of two ways. For the Griffiths SSS-matrix, one of the strong coupling flows is towards a FM fixed point with decoupled wires; this is seen by a perturbative analysis. Thus if we start with a system of connected wires with an AFM coupling to the spin impurity, the flow at large distances is towards a system of disconnected wires at the FM fixed point. For the decoupled SSS-matrix, the flow is either to a FM fixed point or to one of two strong coupling fixed points in which all the channels are strongly coupled to each other through the impurity spin. Strong interactions between the electrons with ${K}_{\ensuremath{\rho}}<N∕(N+2)$ can stabilize a multichannel fixed point in which the coupling between different channels goes to zero. We also study the temperature dependence of the conductance at the decoupled and Griffiths SSS-matrices.