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arXiv:2111.11428 · Physical review. B./Physical review. B

Counting edge modes via dynamics of boundary spin impurities

Umar Javed, Jamir Marino, Vadim Oganesyan, Michael Kolodrubetz

Published Oct 13, 2023• 7 citations
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Abstract

We study the dynamics of the one-dimensional Ising model in the presence of a static symmetry-breaking boundary field via the two-time autocorrelation function of the boundary spin. We find oscillatory correlations that decay as power laws. We uncover a phase diagram of dynamical responses where, upon tuning the strength of the boundary field, we observe distinct power laws that directly correspond to changes in the number of edge modes as the boundary and bulk magnetic field are varied. We suggest how the universal physics can be demonstrated in current experimental setups, such as Rydberg chains.

DOI
10.1103/physrevb.108.l140301

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We study the dynamics of the one-dimensional Ising model in the presence of a static symmetry-breaking boundary field via the two-time autocorrelation function of the boundary spin. We find oscillatory correlations that decay as power laws. We uncover a phase diagram of dynamical responses where, upon tuning the strength of the boundary field, we observe distinct power laws that directly correspond to changes in the number of edge modes as the boundary and bulk magnetic field are varied. We suggest how the universal physics can be demonstrated in current experimental setups, such as Rydberg chains.
Semi-infinite quantum spin chains display spin autocorrelations near the boundary with power-law exponents that are given by boundary conformal field theories. We show that NMR measurements on spinless impurities that break a quantum spin chain lead to a spin-lattice relaxation rate 1/T1edge1/{T}_{1}^{\mathrm{edge}}1/T1edge​ that has a temperature dependence which is a direct probe of the anomalous boundary exponents. For the antiferromagnetic S=1/2S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2S=1/2 spin chain, we show that $1/{T}_{1}^{\mathrm{edge}}\ensuremath{\propto}T{log}^{2}T$ instead of log1/2T{log}^{1/2}Tlog1/2T for a bulk measurement. We show that, for a one-dimensional conductor described by a Luttinger liquid, a similar measurement leads to a rate $1/{T}_{1}^{\mathrm{edge}}\ensuremath{\propto}T$, independent of the exponent ${K}_{\ensuremath{\rho}}$.
The edge of a quantum critical system can exhibit multiple distinct types of boundary criticality. We use a numerical real-space renormalization group (RSRG) to study the boundary criticality of a 2d quantum Ising model with random exchange couplings and transverse fields, whose bulk exhibits an infinite randomness critical point. This approach enables an asymptotically numerically exact extraction of universal scaling data from very large systems with many thousands of spins that cannot be efficiently simulated directly. We identify three distinct classes of boundary criticality, and extract key scaling exponents governing boundary-boundary and boundary-bulk correlations and dynamics. We anticipate that this approach can be generalized to studying a broad class of (disordered) boundary criticality, including symmetry-enriched criticality and edge modes of gapless symmetry-protected topological states, in contexts were other numerical methods are restricted to one-dimensional chains.
We study the influence of reflective boundaries on time-dependent responses of one-dimensional quantum fluids at zero temperature beyond the low-energy approximation. Our analysis is based on an extension of effective mobile impurity models for nonlinear Luttinger liquids to the case of open boundary conditions. For integrable models, we show that boundary autocorrelations oscillate as a function of time with the same frequency as the corresponding bulk autocorrelations. This frequency can be identified as the band edge of elementary excitations. The amplitude of the oscillations decays as a power law with distinct exponents at the boundary and in the bulk, but boundary and bulk exponents are determined by the same coupling constant in the mobile impurity model. For nonintegrable models, we argue that the power-law decay of the oscillations is generic for autocorrelations in the bulk, but turns into an exponential decay at the boundary. Moreover, there is in general a nonuniversal shift of the boundary frequency in comparison with the band edge of bulk excitations. The predictions of our effective field theory are compared with numerical results obtained by time-dependent density matrix renormalization group (tDMRG) for both integrable and nonintegrable critical spin-SSS chains with S=1/2S=1/2S=1/2, 1, and 3/23/23/2.
We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance r as 1 / r α , as well as a related class of fermionic Hamiltonians that generalize the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents α , based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for α ≲ 1 , while connected correlation functions can decay with a hybrid exponential and power-law behavior, with a power that is α -dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every α . Along the critical lines, for all models breaking of conformal symmetry is argued at low enough α . For the fermionic models we show that the edge modes, massless for α ≳ 1 , can acquire a mass for α < 1 . The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.
We determine dynamical response functions in the scaling limit of the quantum Ising chain on the half-line in the presence of a boundary magnetic field. Using a spectral representation in terms of infinite volume form factors and a boundary state, we derive an expansion for the correlator that is found to be rapidly convergent as long as , where ξ is the correlation length. At sufficiently late times we observe oscillatory behaviour of the correlations arbitrarily far away from the boundary. We investigate the effects of the boundary bound state that is present for a range of boundary magnetic fields.
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.