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Magnetic impurities at quantum critical points: Large- N expansion and connections to symmetry-protected topological states

Shang Liu, Hassan Shapourian, Ashvin Vishwanath, Max A. Metlitski

Published Jan 1, 2021• 31 citations
OpenAlex

Abstract

In symmetry protected topological (SPT) phases, the combination of symmetries and a bulk gap stabilizes protected modes at surfaces or at topological defects. Understanding the fate of these modes at a quantum critical point, when the protecting symmetries are on the verge of being broken, is an outstanding problem. This interplay of topology and criticality must incorporate both the bulk dynamics of critical points, often described by nontrivial conformal field theories, and SPT physics. Here, we study the simplest nontrivial setting - that of a 0+1 dimensional topological mode - a quantum spin - coupled to a 2+1D critical bulk. Using the large-$N$ technique we solve a series of models which, as a consequence of topology, demonstrate intermediate coupling fixed points. We compare our results to previous numerical simulations and find good agreement. We also point out intriguing connections to generalized Kondo problems and Sachdev-Ye-Kitaev (SYK) models. In particular, we show that a Luttinger theorem derived for the complex SYK models, that relates the charge density to particle-hole asymmetry, also holds in our setting. These results should help stimulate further analytical study of the interplay between SPT physics and quantum criticality.

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Gapless quantum phases can become distinct when internal symmetries are enforced, in analogy with gapped symmetry-protected topological (SPT) phases. However, this distinction does not always lead to protected edge modes, raising the question of how the bulk-boundary correspondence is generalized to gapless cases. We propose that the spatial interface between gapless phases -- rather than their boundaries -- provides a more robust fingerprint. We show that whenever two 1+1d conformal field theories (CFTs) differ in symmetry charge assignments of local operators or twisted sectors, any symmetry-preserving spatial interface between the theories must flow to a non-invertible defect. We illustrate this general result for different versions of the Ising CFT with $\mathbb{Z}_2 \times \mathbb{Z}_2^T$ symmetry, obtaining a complete classification of allowed conformal interfaces. When the Ising CFTs differ by nonlocal operator charges, the interface hosts 0+1d symmetry-breaking phases with finite-size splittings scaling as $1/L^3$, as well as continuous phase transitions between them. For general gapless phases differing by an SPT entangler, the interfaces between them can be mapped to conformal defects with a certain defect 't Hooft anomaly. This classification also gives implications for higher-dimensional examples, including symmetry-enriched variants of the 2+1d Ising CFT. Our results establish a physical indicator for symmetry-enriched criticality through symmetry-protected interfaces, giving a new handle on the interplay between topology and gapless phases.

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In symmetry protected topological (SPT) phases, the combination of symmetries and a bulk gap stabilizes protected modes at surfaces or at topological defects. Understanding the fate of these modes at a quantum critical point, when the protecting symmetries are on the verge of being broken, is an outstanding problem. This interplay of topology and criticality must incorporate both the bulk dynamics of critical points, often described by nontrivial conformal field theories, and SPT physics. Here, we study the simplest nontrivial setting - that of a 0+1 dimensional topological mode - a quantum spin - coupled to a 2+1D critical bulk. Using the large-NNN technique we solve a series of models which, as a consequence of topology, demonstrate intermediate coupling fixed points. We compare our results to previous numerical simulations and find good agreement. We also point out intriguing connections to generalized Kondo problems and Sachdev-Ye-Kitaev (SYK) models. In particular, we show that a Luttinger theorem derived for the complex SYK models, that relates the charge density to particle-hole asymmetry, also holds in our setting. These results should help stimulate further analytical study of the interplay between SPT physics and quantum criticality.
Motivated by recent studies of symmetry protected topological states that are stabilized even in the absence of an energy gap, the authors study analytically the interaction of a (0+1)DDD impurity spin, which in some cases may be viewed as a topological state bound to a lattice defect, with a (2+1)DDD strongly interacting bulk, composed of bosonic excitations tuned to a quantum critical point. The question of whether the spin is (partially) screened is addressed within the framework of four different models, using mainly the large-NNN technique. The authors also point out intriguing connections with the physics of the Sachdev-Ye-Kitaev model.
We describe recent progress in our understanding of the interplay between interactions, symmetry, and topology in states of quantum matter. We focus on a minimal generalization of the celebrated topological band insulators (TBIs) to interacting many-particle systems known as symmetry-protected topological (SPT) phases. As with the TBIs, these states have a bulk gap and no exotic excitations but have nontrivial surface states that are protected by symmetry. We describe the various possible phases and their properties in three-dimensional systems with realistic symmetries. We develop many key ideas for the theory of these states using simple examples. The emphasis is on physical rather than mathematical properties. We survey insights obtained from the study of SPT phases for a number of other theoretical problems.
Decades of research have revealed a deep understanding of topological quantum matter with protected edge modes. We report that even richer physics emerges when tuning between two topological phases of matter whose respective edge modes are incompatible. The frustration at the edge leads to novel boundary physics, such as symmetry-breaking phases with exotic non-Landau transitions---even when the edge is zero-dimensional. As a minimal case study, we consider spin chains with ${\mathbb{Z}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{3}$ symmetry, exhibiting two nontrivial symmetry-protected topological (SPT) phases. At the bulk 1+1D critical transition between these SPT phases, we find two stable 0+1D boundary phases, each spontaneously breaking one of the Z3{\mathbb{Z}}_{3}Z3​ symmetries. Furthermore, we find that a single boundary parameter tunes a non-Landau boundary critical transition between these two phases. This constitutes a 0+1D version of an exotic phenomenon driven by charged vortex condensation known as deconfined quantum criticality. This work highlights the rich unexplored physics of criticality between nontrivial topological phases and provides insights into the burgeoning field of gapless topological phases.
We propose a series of simple two-dimensional (2D) lattice interacting fermion models that we demonstrate at low energy describe bosonic symmetry-protected topological (SPT) states and quantum phase transitions between them. This is because due to interaction, the fermions are gapped both at the boundary of the SPT states and at the bulk quantum phase transition, thus these models at low energy can be described completely by bosonic degrees of freedom. We show that the bulk of these models is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a topological $\mathrm{\ensuremath{\Theta}}$ term, whose boundary is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a Wess-Zumino-Witten term at level 1. The quantum phase transition between SPT states in the bulk is tuned by a particular interaction term, which corresponds to tuning $\mathrm{\ensuremath{\Theta}}$ in the field theory, and the phase transition occurs at $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$. The simplest version of these models with N=1N=1N=1 is equivalent to the familiar O(4) nonlinear sigma model (NLSM) with a topological term, whose boundary is a (1+1)D(1+1)\text{D}(1+1)D conformal field theory with central charge c=1c=1c=1. After breaking the O(4) symmetry to its subgroups, this model can be viewed as bosonic SPT states with U(1), or Z2{Z}_{2}Z2​ symmetries, etc. All of these fermion models, including the bulk quantum phase transitions, can be simulated with the determinant quantum Monte Carlo method without the sign problem. Recent numerical results strongly suggest that the quantum disordered phase of the O(4) NLSM with precisely $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$ is a stable (2+1)D(2+1)\text{D}(2+1)D conformal field theory with gapless bosonic modes.
Symmetry, topology and entanglement are three indispensable concepts for our understanding of quantum many-body systems. In this dissertation, we demonstrate the use and interplay of these concepts with four distinct topics. The first part focuses on two-dimensional (2D) symmetry protected topological (SPT) phases with lattice rotation symmetries. We start by discussing a prototypical example that exhibits interesting topological responses, and then investigate general aspects of rotation protected topology. In the second part, we report on a Hartree-Fock calculation of possible symmetry-broken phases in magic angle bilayer graphene at charge neutrality, assuming that the valley U(1) symmetry is preserved. Our solutions include both correlated insulators and nematic topological semimetals, whose relative stability can be tuned by weak strains. In the third part, inspired by the recent studies on gapless SPT states, we examine the fate of magnetic impurities at 2+1D quantum critical points. We introduced a series of large-N solvable impurity models which, as a consequence of topology, demonstrate intermediate coupling fixed points. In the last part, we develop a diagrammatic method to compute the entanglement properties of random mixed states which are obtained by partial tracing random pure states and are relevant for a variety of problems ranging from chaotic quantum dynamics to black hole physics.
Gapless quantum phases can become distinct when internal symmetries are enforced, in analogy with gapped symmetry-protected topological (SPT) phases. However, this distinction does not always lead to protected edge modes, raising the question of how the bulk-boundary correspondence is generalized to gapless cases. We propose that the spatial interface between gapless phases -- rather than their boundaries -- provides a more robust fingerprint. We show that whenever two 1+1d conformal field theories (CFTs) differ in symmetry charge assignments of local operators or twisted sectors, any symmetry-preserving spatial interface between the theories must flow to a non-invertible defect. We illustrate this general result for different versions of the Ising CFT with Z2×Z2T\mathbb{Z}_2 \times \mathbb{Z}_2^TZ2​×Z2T​ symmetry, obtaining a complete classification of allowed conformal interfaces. When the Ising CFTs differ by nonlocal operator charges, the interface hosts 0+1d symmetry-breaking phases with finite-size splittings scaling as 1/L31/L^31/L3, as well as continuous phase transitions between them. For general gapless phases differing by an SPT entangler, the interfaces between them can be mapped to conformal defects with a certain defect 't Hooft anomaly. This classification also gives implications for higher-dimensional examples, including symmetry-enriched variants of the 2+1d Ising CFT. Our results establish a physical indicator for symmetry-enriched criticality through symmetry-protected interfaces, giving a new handle on the interplay between topology and gapless phases.