Fixed point annihilation for a spin in a fluctuating field

The annihilation of a stable with an unstable fixed point is a generic possibility in renomalization group (RG) flows when a parameter such as the spatial dimensionality, which does not flow, is varied Nienhuis et al. (1979); Cardy et al. (1980); Newman et al. (1984); Zumbach (1993); Gies and Jaeckel (2006); Kaplan et al. (2009); Gukov (2017). When this happens it leads to an interesting regime just beyond the annihilation point. No physical fixed point exists in this regime (though “annihilation” really means that the real fixed points disappear into the complex plane, where they may correspond to nonunitary conformal field theories Gorbenko et al. (2018a)). Nevertheless the RG flows become very slow. This can yield particles with anomalously small masses, or weakly first-order phase transitions with extremely long correlation lengths Nienhuis et al. (1979) that show quasiuniversal Zumbach (1993); Wang et al. (2017) behaviour below this scale.

One generic class of examples includes field theories with cubic terms that have continuous transitions in low dimensions, which become first order (as predicted by mean-field theory) in high dimensions. These include the Potts model Newman et al. (1984) (which also undergoes annilation in 2D as a function of the number of states Nienhuis et al. (1979); Cardy et al. (1980); Gorbenko et al. (2018b); Iino et al. (2019); Ma and He (2019)) as well as Landau theories for order parameters on complex or real projective space Nahum et al. (2013); Serna and Nahum .

This note is motivated by a fixed point annihilation phenomenon that was proposed to resolve debates about Monte Carlo results for deconfined criticality Senthil et al. (2004) in 2+1D antiferromagnets Nahum et al. (2015a); Wang et al. (2017). In Refs. Ma and Wang (2020); Nahum (2020) this was put in terms of a dimensional hierarchy of nonlinear sigma models in $d$ spacetime dimensions with ${\mathrm{SO}(d+2)}$ global symmetry Abanov and Wiegmann (2000). These sigma models have a topological Wess-Zumino-Witten (WZW) term in the action. The case ${d=2}$ is the well-known WZW theory with a conformal fixed point Witten (1984), and ${d=3}$ is an effective field theory for various competing order parameters in 2+1D magnets Tanaka and Hu (2005); Senthil and Fisher (2006). It was argued that fixed point annihilation occurs between two and three dimensions.

Unfortunately, this example of fixed point annihilation, like the others mentioned above, requires an integer-valued parameter (here $d$) to be treated as continuously variable. An annihilation that takes place at a noninteger critical dimensionality may be useful conceptually for understanding nearby values of $d$, but it cannot be realized physically (and there may be ambiguities in defining the continuous $d$ theory). It would be instructive to have a toy model that retained basic features of the WZW example, without the unphysical feature of noninteger $d$.

Here we show that the simplest member of the “WZW” hierarchy, in $d=1$, provides such a model if we augment it with a long-ranged interaction. This is a model of a spin impurity in a gapless environment Sengupta (2000); Smith and Si (1999); Sachdev and Ye (1993); Vojta (2006); Sachdev (2004), and was suggested as a model for fixed point annihilation in Nahum (2020). We find that many key features of the higher-dimensional example are retained (fixed point annihilation, quasiuniversality, emergent symmetry). But since the fixed point annihilation occurs in $d=1$ the model is accessible to numerical simulations and perhaps to experiment. The model is analytically tractable at large spin.

The $d=1$ theory without a long-range interaction is simply the quantum mechanics of a spin-$S$ (or more generally a rotor), described using the spin path integral with its well-known Berry phase term Altland and Simons (2010). The version with a power-law interaction ${\sim 1/|t-t^{\prime}|^{2-\delta}}$ describes a spin with a retarded interaction, physically representing an interaction with a gapless zero-temperature bath that has been integrated out. This is known as the “Bose-Kondo” model Sengupta (2000); Smith and Si (1999); Sachdev and Ye (1993); Vojta (2006); Sachdev (2004); Sachdev et al. (1999); Vojta et al. (2000); Si et al. (2001); Zaránd and Demler (2002); Zhu and Si (2002); Vojta and Kirćan (2003); Zhu et al. (2004); Novais et al. (2005); Pixley et al. (2013); Chowdhury et al. (2021); Joshi et al. (2020). It falls into the larger family of quantum impurity models describing a local quantum-mechanical degree of freedom interacting with a bath of critical fluctuations Vojta (2006); Leggett et al. (1987); Sachdev (2004); Hewson (1997); Allais and Sachdev (2014).

We study the model in a large spin limit that allows the RG equation to be obtained to all orders in the coupling. Using the background field method, the calculation is a simple extension (to include the Berry phase term) of the analysis by Kosterlitz of the long-range classical Heisenberg model in one dimension Kosterlitz (1976). The beta function shows an interesting structure, with two nontrivial fixed points that annihilate with each other when the interaction power law ${2-\delta}$ is varied through a critical value. The flows as a function of $\delta$ are qualitatively like those suggested for the WZW model as a function of $\epsilon$ (in $2+\epsilon$ dimensions) Ma and Wang (2020); Nahum (2020), except in the behaviour of one of the nontrivial fixed points (the stable one) when ${\delta\rightarrow 0}$.

Model. We consider a Euclidean action for a spin of size $S$ with a long-ranged temporal interaction:

Here ${\vec{n}=(n_{1},n_{2},n_{3})}$, with ${\vec{n}^{2}=1}$, is the field appearing in the coherent states path integral. This is a formulation of the $\mathrm{SO}(3)$-symmetric Bose-Kondo model, in which the spin is coupled to a local magnetization ${\vec{m}}$ (associated with additional “bulk” degrees of freedom) via a Hamiltonian ${H_{\text{int}}=J{\vec{S}}.{\vec{m}}}$ Sengupta (2000); Smith and Si (1999); Sachdev and Ye (1993); Vojta (2006); Sachdev (2004). If $\vec{m}$ has $\mathrm{SO}(3)$-invariant autocorrelations obeying Wick’s theorem, then integrating ${\vec{m}}$ out yields (1) with ${g^{-1}\propto J^{2}S^{2}}$ and with a kernel ${K(t-t^{\prime})}$ that is proportional to the autocorrelator of ${\vec{m}}$. We assume this to be a power law at large ${\tau=t-t^{\prime}}$, ${K(t-t^{\prime})\propto|t-t^{\prime}|^{-(2-\delta)}}$ with ${-1<\delta<1}$. For convenience we normalize $K$ as

The constant $C$ is chosen so that the Fourier transform of $K(\tau)$ has a simple normalization (App. A) Kosterlitz (1976), and a power of the UV cutoff frequency $\Lambda$ is included in $K(\tau)$ so that $g$ is dimensionless. Finally, the Berry phase term $\Omega[\vec{n}]$ is the solid angle on the sphere traced out by the trajectory, written in terms of an extension of the field $\vec{n}(t)$ to a field $\vec{n}(t,u)$ defined on a strip with $u\in[0,1]$ as Altland and Simons (2010):

or more simply as ${\Omega[\vec{n}]=\int\mathrm{d}t(1-\sin\psi)\dot{\phi}}$ in the coordinates ${\vec{n}=(\cos\psi\cos\phi,\cos\psi\sin\phi,\sin\psi)}$.

Before calculating the beta function, let us ask what we can expect from stability considerations.

The action (1) has two trivial fixed points, at $g=0$ and at $g=\infty$. That at $g=0$ is a perfectly ordered state, with no local fluctuations in $\vec{n}(t)$. The fixed point at $g=\infty$ describes a quantum spin with ${2S+1}$ degenerate ground states.

By counting dimensions we see that when $\delta$ is negative the ordered fixed point at ${g=0}$ is unstable and the ${g=\infty}$ fixed point is stable. The simplest expectation (confirmed in the large $S$ calculation below) is that in this ${\delta<0}$ regime the model flows, for any positive $g$, to ${g=\infty}$. On the other hand when $\delta$ is positive the ordered fixed point becomes stable, so the model is in a stable ordered phase for small enough $g$. At the same time the ${g=\infty}$ fixed point becomes unstable.

The flows for infinitesimal $g$ are similar to those in a classical 1D model without the Berry phase term, because the Berry phase term in the action is subleading in the limit $g\rightarrow 0$. As in the classical model, the change in stability of the ordered fixed point is accompanied by the appearance of a nontrivial unstable fixed point, representing a phase transition, at a coupling $g_{u}$ that is of order $\delta$ for small positive $\delta$ Kosterlitz (1976).

However the Berry phase term plays a role for non-infinitesimal $g$. In particular, the $g=\infty$ fixed point is unstable for $\delta>0$, unlike a simple classical disordered fixed point. The simplest consistent hypothesis is therefore that for sufficiently small positive $\delta$ there is another nontrivial fixed point $g_{s}$, with $g_{s}>g_{u}$, which is stable. This fixed point governs a stable large-$g$ phase with power-law correlations. [Heuristically, the Berry phase term has prevented $\vec{n}(t)$ from being trivially disordered at large $g$, leading instead to a stable “critical phase”.] At small $\delta$, with $S$ fixed, this stable fixed point can be studied by perturbative RG in the strength of the impurity-spin coupling Vojta (2006).

What happens to these fixed points as $\delta$ is increased? A simple guess (in analogy to the higher-dimensional problem) is that at some critical value $\delta_{c}$ they merge and annihilate, meaning that for a sufficiently long range interaction the model is always in the ordered phase (Fig. 1). We will confirm this directly when $S$ is large.

RG results. At large $S$ the interesting regime is where the coupling $g$ and the exponent $\delta$ are both of order $1/S$, so we will write

This scaling of the coupling ensures that the two terms in the action are of comparable size in the limit of large $S$. (If this is not the case, then one of the two terms dominates the action for the “fast” modes that we integrate out in the RG step, leading to a more trivial RG equation.) The spin size $S$ itself is quantised and does not flow, but it serves as a large parameter that justifies a one-loop calculation Witten (1984). This calculation can be done with the background field method Kosterlitz (1976); Polyakov (1975) and is described in App. A.

Our basic result is the RG equation

where the RG time $\tau$ is the logarithm of a physical timescale. The topology of the associated flows is shown in Fig. 1.

The value

for the interaction exponent separates two regimes. For larger $\delta$, all flows lead to the ordered phase as noted above, but for ${0<\delta<\delta_{c}}$ there is stable nontrivial phase (governed by the fixed point at $g_{s}$), separated by a second-order phase transition (governed by $g_{u}$) from the ordered phase. The RG eigenvalue of the coupling at $g_{u,s}$ is ${y_{g}=\pm\delta\sqrt{1-\pi^{2}\widetilde{\delta}^{2}}}$, with the $+$ sign for $g_{u}$.

The scaling dimension of the field $\vec{n}$ is $x_{1}=\delta/2$ at both nontrivial fixed points, so that the spin autocorrelator decays as ${|t-t^{\prime}|^{-\delta}}$. This exponent value is expected to be exact, as for other long range models, since the two local operators appearing in the long-range term renormalize independently when ${t-t^{\prime}}$ is large Fisher et al. (1972); Kosterlitz (1976); Brézin et al. (1976); Paulos et al. (2016); Zhu and Si (2002). Below we will also need the scaling dimension $x_{k}$ of the symmetric $k$-index tensor $X^{(k)}_{a_{1},\ldots,a_{k}}$ that is obtained as the traceless part of the operator $n_{a_{1}}\ldots n_{a_{k}}$. At one-loop order this obeys Brezin et al. (1976) (App. A)

We conjecture that the topology of the flows found here at large $S$ applies for all values of the spin, including ${S=1/2}$. It would be interesting to study this numerically. The partition function for the spin has a Monte-Carlo-sign-free diagrammatic formulation, with propagators of $\vec{m}$ represented as arcs connecting points $t$, $t^{\prime}$ on the spin’s worldline Weber (2021), and the model may also be studied with numerical RG Bulla et al. (2005); Glossop and Ingersent (2005).

Let us return to the analogy with higher-dimensional models for deconfined criticality and competing orders. In the WZW hierarchy, two key features are (1) quasiuniversality in the regime just beyond the fixed point annihilation ($\epsilon\gtrsim\epsilon_{c}$ in $2+\epsilon$ dimensions); and (2) the emergence of the full symmetry of the sigma model from a smaller microscopic symmetry group, thanks to the irrelevance of operators analogous to $X^{(k)}$ for large enough $k$. We examine analogues of these phenomena in the present system.

Quasiuniversality. The quasiuniversality phenomenon will occur in this 1D model when $\delta\gtrsim\delta_{c}$. The spin will ultimately be ordered even if the bare coupling $h$ is large, but this will not be apparent until a timescale $\xi$ that diverges exponentially with ${(\delta-\delta_{c})^{-1/2}}$, because the flows spend a large amount of RG time close to ${h=1}$ Nienhuis et al. (1979); Cardy et al. (1980).

At small ${\delta-\delta_{c}}$ we can continue to classify operators as relevant or irrelevant, and the long RG time spent close to $h=1$ means that irrelevant perturbations, which will be present in a generic microscopic model with the appropriate symmetry, become exponentially small in ${(\delta-\delta_{c})^{-1/2}}$ Wang et al. (2017). This exponential suppression of differences between bare models underlies quasiuniversality. For example, we will have approximate universality in the functional form of the spin autocorrelator ${\left\langle\vec{n}(t).\vec{n}(0)\right\rangle}$, despite the fact that it is not a power law for ${\delta>\delta_{c}}$.

In fact, a simplifying feature of RG for the long-range model is that the flow of the renormalized coupling $h(\tau)$ — obtained from running the RG up to a physical timescale ${\Lambda^{-1}e^{\tau}}$ — can be plotted simply by plotting the spin autocorrelator, at least within the present large $S$ approximation. This is because the RG equation (5) can be expressed in terms of the running scaling dimension $x(h)$ of the sigma model field $\vec{n}$, as ${\dot{h}=(-\delta+2x(h))h}$ (see App. A, Eq. 26). RG for the correlator then gives:

It would be interesting to use the correlator (8) to obtain a proxy for the beta function from Monte Carlo simulations in the quasiuniversal regime.

As an aside, a curious feature of the model is that the two point function (8) tends to a constant at large times both in the ordered (${h\rightarrow 0}$) phase, which is stable for ${\delta>0}$, and also in the free spin (${h\rightarrow\infty}$) phase that exists for $\delta<0$. However the two fixed points are different.

One concrete way to see the difference is in connected two-point functions $G^{(k)}(t)$ of operators with higher spin, ${k>2S}$. For a completely free spin, nonvanishing operators only exist with spin ${k\leq 2S}$. However, in the microscopic theory of a spin coupled to a bath we can construct nonvanishing operators with any spin $k$, as discussed in App. A. Let $G^{(k)}(t)$ be a connected 2-point function for such operators. In the free spin phase $\lim_{t\rightarrow\infty}G^{(k)}(t)$ is nonzero for $k\leq 2S$ and vanishes for $k>2S$ (because the spin and bath decouple at the governing IR fixed point, see App. A). In contrast, in the ordered phase we expect that ${\lim_{t\rightarrow\infty}G^{(k)}(t)}$ is nonzero for all $k$, because the corresponding continuum operators $X^{(k)}$, defined above, have nonvanishing long-distance correlations at the ordered fixed point. (Correlation functions at the ordered fixed point are simple since only the zero mode of $\vec{n}(t)$ needs to be averaged over.)

Emergent symmetry. We can construct a simple toy model for the emergent symmetries [$\mathrm{SO}(4)$ in 1+1D or $\mathrm{SO}(5)$ in 2+1D] that arise in various higher-dimensional microscopic models for which the WZW sigma models serve as effective field theories Tanaka and Hu (2005); Senthil and Fisher (2006); Fradkin (2013); Nahum et al. (2015b); Wang et al. (2017); Sreejith et al. (2019); Ma et al. (2019); Tsvelik (2007); Patil et al. (2018). In these examples, the $N$-component sigma model field ${\vec{n}}$ is viewed as the concatenation of two separate fields, ${\vec{n}=(\vec{\Phi}^{A},\vec{\Phi}^{B})}$. $\vec{\Phi}^{A}$ and $\vec{\Phi}^{B}$ are not related by microscopic symmetry, but may be related by an emergent $SO(N)$ symmetry at a critical point. In 2+1D, for example, $\vec{\Phi}^{A}$ and $\vec{\Phi}^{B}$ could be the Néel and VBS order parameters, with the critical point of interest separating Néel and VBS phases.

Here we take ${\Phi^{A}=n_{z}}$ and ${\vec{\Phi}^{B}=(n_{x},n_{y})}$. That is, we think of Eq. 1 as an effective field theory for a phase transition in an anisotropic microscopic Hamiltonian, with only ${\mathrm{O}(2)=\mathbb{Z}_{2}\ltimes\mathrm{U}(1)}$ symmetry, which is promoted to emergent ${\mathrm{SO}(3)}$ at a critical point. The critical point lies at the boundary of a phase with easy-axis order for $n_{z}$.

For concreteness, consider simple ${\mathrm{O}(2)}$-invariant Hamiltonians for spin-1/2 and spin-1. Nontrivial examples require at least two anisotropic couplings in the microscopic Hamiltonian, as will be clear below. For a spin-1 we could consider single-ion anisotropy and an anisotropic bath coupling: ${H_{\text{aniso}}=J\left(S_{x}m_{x}+S_{y}m_{y}+\gamma S_{z}m_{z}\right)-\Delta S_{z}^{2}}$. For a spin-1/2 the $S_{z}^{2}$ term trivializes, but we could consider a local anisotropy for the bath, ${H_{\text{aniso}}^{\prime}=J\left(S_{x}m_{x}+S_{y}m_{y}+\gamma S_{z}m_{z}\right)-\Delta m_{z}^{2}}$. We assume that $\delta\leq\delta_{c}$, and that $J$ is small enough that the isotropic models (${\Delta=0,\gamma=1}$) flow to the fixed point at $g_{s}$, which is stable in the absence of anisotropy (Fig. 1).

Microscopic $\mathrm{O}(2)$ symmetry allows the perturbations ${\delta\mathcal{S}=\int\mathrm{d}t\big{(}uX^{(2)}_{33}+vX^{(4)}_{3333}+\ldots\big{)}}$ to the continuum action. Here ${X^{(2)}_{33}\propto n_{z}^{2}-(n_{x}^{2}+n_{y}^{2})/2}$ is the leading anisotropy which will drive the transition, and $X^{(4)}_{3333}$ is a subleading anisotropy. The scaling dimension formula Eq. 7 is reliable only at large $S$, but it suggests that for small spin there is a range of positive $\delta$ where $X^{(2)}$ is the only relevant anisotropy, $X^{(4)}$ and higher anisotropies being irrelevant. We assume $\delta$ is in this range.

Then, the $\mathrm{SO}(3)$-invariant fixed point governs a phase transition line in the $(\Delta/J,\gamma)$ plane. One point on this line, at $(0,1)$, has microscopic $\mathrm{SO}(3)$ symmetry, but at other points on the line $\mathrm{SO}(3)$ emerges only in the IR. One adjacent phase is the easy-axis phase, where $\mathbb{Z}_{2}$ is broken. The nature of the other phase will depend on the spin. For spin-1/2 it is likely a power-law phase Sengupta (2000) in which the the easy-plane order parameter dominates.

It may be interesting to check for symmetry enhancement starting from other microscopic symmetry groups. For example $S_{4}$ (tetrahedral) symmetry allows the perturbation $X^{(3)}_{123}$. We may argue that the field theory with this symmetry breaking, and with $S=1$, is an effective theory for a long-range 4-state Potts model in which the partition sum is weighted by $(-1)$ for each domain wall.

Conclusions. The impurity model can be seen as the simplest member of a dimensional hierarchy of sigma models with topological terms Abanov and Wiegmann (2000). We have argued that some interesting features of the RG flows in higher dimensions are also present in 0+1D, giving a rich phase diagram for the Bose-Kondo model. The model yields an example of fixed point annihilation that is tractable both analytically and in simulations, and also shows analogues of phenomena from higher-dimensional “non-Landau” phase transitions. It would be interesting to examine other variations — for example models in large $N$ limits, with other symmetric spaces for the target space, or with coupling to fermions — and to explore physical realizations of the tunable interaction exponent $\delta$ (perhaps via bosonic bath whose hopping parameters varied with distance from the impurity). Finally it would also be interesting to look for the annihilation phenomenon in models relevant to impurities in critical magnets in which the bath is not Gaussian Sachdev et al. (1999), or settings where the impurity arises as a self-consistent description of an interacting many-body system Chowdhury et al. (2021).

Related work: The impurity in the large $S$ limit has also been analyzed recently in two other papers, Ref. Cuomo et al. (2022) and Ref. Beccaria et al. (2022), with results for the beta function consistent with those above. In addition, these papers make interesting connections with Wilson lines and line defects in conformal field theory. Quantum Monte Carlo results for spin-1/2 are now also available Weber and Vojta (2022), and are consistent with the phase diagram obtained here.

Acknowledgements: I am grateful to D. Bernard, X. Cao, M. Metlitski, and T. Senthil for useful discussions, and to D. Bernard also for comments on the draft. I thank the authors of Ref. Cuomo et al. (2022) for correspondence. I acknowledge support from a Royal Society University Research Fellowship during part of this work.