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arXiv:2010.07947 · Physical Review Research

Spectroscopic fingerprints of gapped quantum spin liquids, both conventional and fractonic

Rahul Nandkishore, Wonjune Choi, Yong Baek Kim

Published Mar 19, 2021• 52 citations
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Abstract

We explain how gapped quantum spin liquids, both conventional and "fractonic," may be unambiguously diagnosed experimentally using the technique of multidimensional coherent spectroscopy."Conventional" nonchiral gapped quantum spin liquids (e.g., Z 2 spin liquid) do not have sharp signatures in linear response, but they do have clear fingerprints in nonlinear response, accessible through the already existing experimental technique of two-dimensional coherent spectroscopy.Type I fracton phases (e.g., X-cube) are (surprisingly) even easier to distinguish, with strongly suggestive features even in linear response, and unambiguous signatures in nonlinear response.Type II fracton systems, like Haah's code, are most subtle, and they may require consideration of high-order nonlinear response for unambiguous diagnosis.

DOI
10.1103/physrevresearch.3.013254

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We explain how gapped quantum spin liquids, both conventional and "fractonic," may be unambiguously diagnosed experimentally using the technique of multidimensional coherent spectroscopy."Conventional" nonchiral gapped quantum spin liquids (e.g., Z 2 spin liquid) do not have sharp signatures in linear response, but they do have clear fingerprints in nonlinear response, accessible through the already existing experimental technique of two-dimensional coherent spectroscopy.Type I fracton phases (e.g., X-cube) are (surprisingly) even easier to distinguish, with strongly suggestive features even in linear response, and unambiguous signatures in nonlinear response.Type II fracton systems, like Haah's code, are most subtle, and they may require consideration of high-order nonlinear response for unambiguous diagnosis.
Unambiguous identification of fractionalized excitations in quantum spin liquids has been a long-standing issue in correlated topological phases. Conventional spectroscopic probes, such as the dynamical spin structure factor, can only detect composites of fractionalized excitations, leading to a broad continuum in energy. Lacking a clear signature in conventional probes has been the biggest obstacle in the field. In this work, we theoretically investigate what kinds of distinctive signatures of fractionalized excitations can be probed in two-dimensional nonlinear spectroscopy by considering the exactly solvable Kitaev spin liquids. We demonstrate the existence of a number of salient features of the Majorana fermions and fluxes in two-dimensional nonlinear spectroscopy, which provide crucial information about such excitations.
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We study how sharp signatures of fractionalization emerge in nonlinear spectroscopy experiments on spin liquids with separated energy scales. Our model is that of dipolar-octupolar rare earth pyrochlore materials, prime candidates for realizing quantum spin ice. This family of three-dimensional quantum spin liquids exhibits fractionalization of spin degrees of freedom into spinons charged under an emergent U(1) gauge field. We show that the technique of two-dimensional coherent spectroscopy can identify clear signatures of fractionalized spinon dynamics in dipolar-octupolar quantum spin ices. However, at intermediate temperatures, spinon dynamics are heavily constrained in the presence of an incoherent spin background, leading to a broad two-dimensional coherent spectroscopy response. At lower temperatures, a sharp signal emerges as the system enters a coherent spin liquid state. This lower temperature signal can in turn distinguish between zero-flux and π-flux forms of quantum spin ice.
We introduce hybrid fracton orders: three-dimensional gapped quantum phases that exhibit the phenomenology of both conventional three-dimensional topological orders and fracton orders. Hybrid fracton orders host both (i) mobile topological quasiparticles and loop excitations, as well as (ii) pointlike topological excitations with restricted mobility, with nontrivial fusion rules and mutual braiding statistics between the two sets of excitations. Furthermore, hybrid fracton phases can realize either conventional three-dimensional topological orders or fracton orders after undergoing a phase transition driven by the condensation of certain gapped excitations. Therefore they serve as parent orders for both long-range-entangled quantum liquid and nonliquid phases. We study the detailed properties of hybrid fracton phases through exactly solvable models in which the resulting orders hybridize a three-dimensional Z2{\mathbb{Z}}_{2}Z2​ topological order with (i) the X-cube fracton order or (ii) Haah's code. The hybrid orders presented here can also be understood as the deconfined phase of a gauge theory whose gauge group is given by an Abelian global symmetry GGG and subsystem symmetries of a normal subgroup NNN along lower-dimensional subregions.
We demonstrate how pump-probe techniques enable specific spectroscopic diagnostics of fracton phases of matter by studying lineon-planon braiding in the paradigmatic X-cube model. Our discussion builds on previous works probing anyonic exchange statistics in conventional spin liquids, but the extension to fracton phases reveals qualitatively new phenomena due to the restricted mobility of fractionalized excitations. A key feature is that nearby planons can form an emergent bound state by accessing different planes via alternative pairing configurations. We show that this bound state qualitatively modifies the long-time linear and nonlinear responses, leading to an asymptotic linear-in-ttt behavior for the pump-probe signal χZZXχ_\mathit{ZZX}χZZX​. By contrast, swapping the pump and probe polarizations produces a nonlinear response χXXZχ_\mathit{XXZ}χXXZ​ that is asymptotically ttt-independent. This asymmetry reflects the fact that the two species of fractionalized excitations live in different spatial dimensions. Thus, the pump-probe signals studied here are sensitive to (i) nontrivial braiding statistics in three dimensions, (ii) the existence of bound states among fractionalized excitations, and (iii) the one-dimensional mobility of lineons. Our results therefore provide spectroscopic signatures that distinguish fracton phases from conventional topologically ordered spin liquids.
Identifying quantum spin liquids, magnon breakdown, or fractionalized excitations in quantum magnets is an ongoing challenge due to the ambiguity in interpreting excitation continua occurring in linear-response probes. Recently, it was proposed that techniques measuring higher-order response, such as two-dimensional coherent spectroscopy (2DCS), could resolve such ambiguities. Numerically simulating nonlinear response functions can, however, be computationally very demanding. We present an efficient Lanczos-based method to compute second-order susceptibilities χ 2 ( ω t , ω τ ) directly in the frequency domain. Applying this to extended Kitaev models describing α − RuCl 3 , we find qualitatively different nonlinear responses between intermediate magnetic field strengths and the high-field regime. To put these results into context, we derive the general 2DCS response of partially polarized magnets within the linear spin-wave approximation, establishing that χ 2 ( ω t , ω τ ) is restricted to a distinct universal form if the excitations are conventional magnons. Deviations from this form, as predicted in our Lanczos-based simulations for α − RuCl 3 , can hence serve in 2DCS experiments as direct criteria to determine whether an observed excitation continuum is of conventional two-magnon type or of different nature.