Systems with low mechanical dissipation are extensively used in precision measurements such as gravitational wave detection, atomic force microscopy, and quantum control of mechanical oscillators via optomechanics and electromechanics. The mechanical quality factor ($Q$) of these systems determines the thermomechanical force noise and the thermal decoherence rate of mechanical quantum states. While the dissipation rate is typically set by the bulk acoustic properties of the material, by exploiting dissipation dilution, mechanical $Q$ can be engineered through geometry and increased by many orders of magnitude Recently, soft clamping in combination with strain engineering has enabled room temperature quality factors approaching ${10}^9$ in millimeter-scale resonators. Here we demonstrate a new approach to soft clamping which exploits vibrations in the perimeter of polygon-shaped resonators tethered at their vertices. In contrast to previous approaches, which rely on cascaded elements to achieve soft clamping, perimeter modes are soft clamped due to symmetry and the boundary conditions at the polygon vertices. Perimeter modes reach $Q$’s of $3.6\times {10}^9$—a record at room temperature—while spanning only two acoustic wavelengths. We demonstrate thermal-noise-limited force sensitivity of $1.3\mathrm{aN}/\sqrt{\mathrm{Hz}}$ for a 226 kHz perimeter mode with quality factor of $1.5\times {10}^9$ at room temperature. The small size of our devices makes them well suited for near-field integration with microcavities for quantum optomechanical experiments. Moreover, their compactness allows the realization of phononic lattices. We demonstrate a one-dimensional Su-Schrieffer-Heeger chain of high-$Q$ perimeter modes coupled via nearest-neighbour interaction and characterize the localized edge modes.

Photoinduced ultrafast phase transitions can generate quasiequilibrium states with novel emergent properties modulated by the interplay of electronic and lattice degrees of freedoms. Therefore, accurately probing transient atomic structures and their dynamics is crucial to understand and control the interaction of electrons and lattice but remains a central challenge of ultrafast science. Using MeV ultrafast electron diffraction on single crystals, we quantitatively reveal the photoinduced lattice distortion of the monoclinic M1 phase of ${\mathrm{VO}}_2$. Our results resolve previous controversies concerning decoupled distortion components, as well as a proposed M2 intermediate phase. Further, we emphasize the importance of quantifying the transformed phase fraction into the metallic rutile phase, beyond previously reported analyses, and we also clarify the importance of thermal heating in assisting the insulator-metal transition. Our complementary ab initio molecular dynamics calculations support the experimental findings and identify the primary $A_g$ phonon mode coupling to photoexcitation. Our study provides the critical and previously missing precise 3D and time-resolved structural evolution of the crystal structure of photoexcited M1 ${\mathrm{VO}}_2$. These results provide the basis upon which to rationalize this archetypal photoinduced transition.

A general mechanism is presented by which topological physics arises in strongly correlated systems without flat bands. Starting from a charge transfer insulator, topology emerges when the charge transfer energy between the cation and anion is reduced to invert the lower Hubbard band and the spin-degenerate charge transfer band. A universal low-energy theory is developed for the inversion of the charge transfer gap in a quantum antiferromagnet. The inverted state is found to be a quantum anomalous Hall (QAH) insulator with noncoplanar magnetism. Interactions play two essential roles in this mechanism: producing the insulating gap and quasiparticle bands prior to the band inversion, and causing the change of magnetic order necessary for the QAH effect after inversion. Our theory explains the electric-field-induced transition from a correlated insulator to a QAH state in $AB$-stacked transition-metal-dichalcogenides bilayer ${\mathrm{MoTe}}_2/{\mathrm{WSe}}_2$.

As a promising avenue to obtain a new extreme ultraviolet light source and detect electronic properties, high-harmonic generation (HHG) has been actively developed in both theory and experiment. In solids lacking inversion symmetry, when electrons undergo a nonadiabatic transition, a directional charge shift occurs and is characterized by a shift vector, which measures the real-space shift of the photoexcited electron and hole. For the first time, we reveal that the shift vector plays a prominent role in the real-space tunneling mechanism of the three-step model for electrons under strong laser fields. Since the shift vector is determined by the topological properties of related wave functions, we expect that the contribution of HHG can provide direct knowledge of the band topology in noncentrosymmetric topological insulators (TIs). In both the Kane-Mele model and the realistic material BiTeI, we find that the shift vector reverses when band inversion happens during the topological phase transition between the normal insulator and the TI. Under oscillating strong laser fields, the reversal of the shift vector leads to completely opposite radiation time of high-order harmonics. This makes HHG a feasible, all-optical, strong-field method to directly identify the band inversion in noncentrosymmetric TIs.

A chiral spin liquid state is discovered in the highly frustrated, noncentrosymmetric swedenborgite compound ${\mathrm{YBaCo}}_3{\mathrm{FeO}}_7$, a layered kagome system of hexagonal symmetry, by advanced polarized neutron scattering from a single domain crystalline sample. The observed diffuse magnetic neutron scattering has an antisymmetric property that relates to its specific chirality, which consists of three cycloidal waves perpendicular to the $c$ axis, forming an entity of cylindrical symmetry. Chirality and symmetry agree with relevant antisymmetric exchanges arising from broken spatial parity. Applying a Fourier analysis to the chiral interference pattern, with distinction between kagome sites and the connecting trigonal interlayer sites of threefold symmetry, the chiral spin correlation function is determined. Characteristic chiral waves originate from the trigonal sites and extend over several periods in the kagome planes. The chiral spin liquid is remarkably stable at low temperatures despite strong antiferromagnetic spin exchange. The observation raises a challenge, since the commonly accepted ground states in condensed matter either have crystalline long-range order or form a quantum liquid. We show that, within the classical theory of magnetic order, a disordered ground state may arise from chirality. The present scenario, with antisymmetric exchange acting as a frustrating gauge background that stabilizes local spin lumps, is similar to the avoided phase transition in coupled gauge and matter fields for subnuclear particles.

The material $\alpha \text{−}{\mathrm{RuCl}}_3$ has been the subject of intense scrutiny as a potential Kitaev quantum spin liquid, predicted to display Majorana fermions as low-energy excitations. In practice, $\alpha \text{−}{\mathrm{RuCl}}_3$ undergoes a transition to a state with antiferromagnetic order below a temperature $T_N\approx 7\mathrm{K}$, but this order can be suppressed by applying an external in-plane magnetic field of $H_{\parallel }=7\mathrm{T}$. Whether a quantum spin liquid phase exists just above that field is still an open question, but the reported observation of a quantized thermal Hall conductivity at $H_{\parallel }>7\mathrm{T}$ by Kasahara and co-workers [Nature (London) 559, 227 (2018)] has been interpreted as evidence of itinerant Majorana fermions in the Kitaev quantum spin liquid state. In this study, we reexamine the origin of the thermal Hall conductivity ${\kappa }_{xy}$ in $\alpha \text{−}{\mathrm{RuCl}}_3$. Our measurements of ${\kappa }_{xy}(T)$ on several different crystals yield a temperature dependence very similar to that of the phonon-dominated longitudinal thermal conductivity ${\kappa }_{xx}(T)$, for which the natural explanation is that ${\kappa }_{xy}$ is also mostly carried by phonons. Upon cooling, ${\kappa }_{xx}$ peaks at $T\simeq 20\mathrm{K}$, then drops until $T_N$, whereupon it suddenly increases again. The abrupt increase below $T_N$ is attributed to a sudden reduction in the scattering of phonons by low-energy spin fluctuations as these become partially gapped when the system orders. The fact that ${\kappa }_{xy}$ also increases suddenly below $T_N$ is strong evidence that the thermal Hall effect in $\alpha \text{−}{\mathrm{RuCl}}_3$ is also carried predominantly by phonons. This implies that any quantized signal from Majorana edge modes would have to come on top of a sizable—and sample-dependent—phonon background.

The study of ${\mathrm{CoNb}}_2{\mathrm{O}}_6$ sits at the confluence of simplicity and complexity: on one hand, the model for Ising chains—the building blocks of ${\mathrm{CoNb}}_2{\mathrm{O}}_6$—in a transverse field can be exactly solved and, thus, serves as an archetype of quantum criticality; on the other hand, the weak but nonzero interchain coupling adds geometric frustration to the stage, substantially complicating the phase diagram. Here we utilize low-temperature specific heat and thermal conductivity measurements to study the low-lying magnetic excitations in ${\mathrm{CoNb}}_2{\mathrm{O}}_6$ and its spin-1 analog ${\mathrm{NiNb}}_2{\mathrm{O}}_6$. The thermal conductivity is found to be suppressed around the quantum critical point, where the specific heat is enhanced due to gapless magnetic excitations, pointing to the localized nature of the latter. These results highlight the predominant role of frustration in determining the quantum critical magnetic excitations of spin chains, which may furthermore underlie the remarkable similarities between the phenomenology of these spin-$1/2$ and spin-1 systems.

Supersolids are a fundamental quantum phase of matter where the global phase and translational symmetries are spontaneously broken. The recently discovered supersolidity in quantum gases of strongly magnetic atoms gives the opportunity to explore in depth how superfluidity and crystalline order are mixed in this fascinating phase. The dipolar supersolid is usually created from a Bose-Einstein condensate, i.e., a standard superfluid, crossing a quantum phase transition that is related to the crystallization transitions of ordinary matter. In this work, we assess experimentally and theoretically the character of the superfluid-supersolid quantum phase transition. We find that one-row supersolids can have already two types of phase transitions, discontinuous and continuous, that are reminiscent of the first- and second-order transitions predicted in the thermodynamic limit in 2D and 1D, respectively. The dimensional crossover is peculiar to supersolids, is controlled via the transverse confinement and the atom number, and can be justified on the general ground of the Landau theory of phase transitions. The quasiadiabatic crossing of a continuous phase transition opens new directions of investigation for supersolids.

Motivated by recent experiments indicating strong superconductivity and intricate correlated insulating and flavor-polarized physics in mirror-symmetric twisted-trilayer graphene, we study the effects of interactions in this system close to the magic angle, using a combination of analytical and numerical methods. We identify asymptotically exact correlated many-body ground states at all integer filling fractions $\nu$ of the flat bands. To determine their fate when moving away from these fine-tuned points, we apply self-consistent Hartree-Fock numerics and analytic perturbation theory, with good agreement between the two approaches. This allows us to construct a phase diagram for the system as a function of $\nu$ and the displacement field, the crucial experimental tuning parameter of the system, and study the spectra of the different phases. The phase diagram is dominated by a correlated semimetallic intervalley coherent state and an insulating sublattice-polarized phase around charge neutrality $\nu =0$, with additional spin polarization being present at quarter ($\nu =-2$) or three-quarter ($\nu =+2$) fillings of the quasiflat bands. We further study the superconducting instabilities emerging from these correlated states, both in the absence and in the additional presence of electron-phonon coupling, also taking into account possible Wess-Zumino-Witten terms. In the experimentally relevant regime, we find triplet pairing to dominate, possibly explaining the observed violation of the Pauli limit. Our results have several consequences for experiments as well as future theoretical work and illustrate the rich physics resulting from the interplay of almost-flat bands and dispersive Dirac cones in twisted-trilayer graphene.

Symmetry formulated by group theory plays an essential role with respect to the laws of nature, from fundamental particles to condensed-matter systems. Here, by combining symmetry analysis and model calculations, we elucidate that the crystallographic symmetry groups of a vast number of magnetic materials with light elements, in which the neglect of relativistic spin-orbit coupling (SOC) is an appropriate approximation, are considerably larger than the conventional magnetic groups. Thus, a symmetry description that involves partially decoupled spin and spatial rotations, dubbed spin group, is required. We derive the classifications of spin point groups describing coplanar and collinear magnetic structures, and the irreducible corepresentations of spin space groups illustrating more energy degeneracies that are disallowed by magnetic groups. One consequence of the spin group is the new antiunitary symmetries that protect SOC-free $Z_2$ topological phases with unprecedented surface-node structures. Our work not only manifests the physical reality of materials with weak SOC, but also sheds light on the understanding of all solids with and without SOC by a unified group theory.

The ${\mathrm{Ce}}^{3+}$ pseudospin-$1/2$ degrees of freedom in the pyrochlore magnet ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ are known to possess dipole-octupole character, making it a candidate for novel quantum spin liquid ground states at low temperatures. We report new polarized neutron diffraction at low temperatures, as well as heat capacity ($C_p$) measurements on single crystal ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$. The former bears both similarities and differences with that measured from the canonical dipolar spin ice compound ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, while the latter rises sharply at low temperatures, initially plateauing near 0.08 K, before falling off toward a high temperature zero beyond 3 K. $\text{Above}\sim 0.5\mathrm{K}$, the $C_p$ dataset can be fit to the results of a quantum numerical linked cluster calculation, carried out to fourth order, that allows estimates for the terms in the near-neighbor XYZ Hamiltonian expected for such dipole-octupole pyrochlore systems. Fits of the same theory to the temperature dependence of the magnetic susceptibility and unpolarized neutron scattering complement this analysis. A comparison between the resulting best-fit numerical linked cluster calculation and the polarized neutron diffraction shows both agreement and discrepancies, mostly in the form of zone-boundary diffuse scattering in the non-spin-flip channel, which are attributed to interactions beyond near neighbors. The lack of an observed thermodynamic anomaly and the constraints on the near-neighbor XYZ Hamiltonian suggest that ${\mathrm{Ce}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ realizes a ${\mathrm{U}(1)}_{\pi }$ quantum spin liquid state at low temperatures, and one that likely resides near the boundary between dipolar and octupolar character.

Novel phenomena in magnetically intercalated graphite have been the subject of much research, pioneered and promoted by M. S. and G. Dresselhaus and many others in the 1980s. Among the most enigmatic findings of that era was the dramatic, roller-coaster-like behavior of the magnetoresistivity in a ${\mathrm{EuC}}_6$ compound, in which magnetic ${\mathrm{Eu}}^{2+}$ ions form a triangular lattice that is commensurate to graphite honeycomb planes. In this study, we provide a long-awaited microscopic explanation of this behavior, demonstrating that the resistivity of ${\mathrm{EuC}}_6$ is dominated by spin excitations in Eu planes and their highly nontrivial evolution with the magnetic field. Together with our theoretical analysis, the present study showcases the power of the synthetic 2D materials as a source of potentially significant insights into the nature of exotic spin excitations.

One manifestation of quantum chaos is a random-matrix-like fine-grained energy spectrum. Prior to the inverse level spacing time, random matrix theory predicts a “ramp” of increasing variance in the connected part of the spectral form factor. However, in realistic quantum chaotic systems, the finite-time dynamics of the spectral form factor is much richer, with the pure random matrix ramp appearing only at sufficiently late time. In this article, we present a hydrodynamic theory of the connected spectral form factor prior to the inverse level spacing time. We first derive a general formula for the spectral form factor of a system with almost-conserved sectors in terms of return probabilities and spectral form factors within each sector. Next we argue that the theory of fluctuating hydrodynamics can be adapted from the usual Schwinger-Keldysh contour to the periodic time setting needed for the spectral form factor, and we show explicitly that the general formula is recovered in the case of energy diffusion. We also initiate a study of interaction effects in this modified hydrodynamic framework and show how the Thouless time defined as the time required for the spectral form factor to approach the pure random matrix result is controlled by the slow hydrodynamics modes. We then extend the formalism to Floquet systems, where a ramp is expected but with a different coefficient, and we derive a crossover formula from the Hamiltonian ramp to the Floquet ramp when the Floquet drive is weak. Taken together, these results establish an effective field theory of chaotic spectral correlations which predicts the random matrix ramp at late time and computes corrections to it at earlier times.

The low-energy physics of fractional quantum Hall (FQH) states—a paradigm of strongly correlated topological phases of matter—to a large extent is captured by weakly interacting quasiparticles known as composite fermions. In this paper, based on numerical simulations and effective field theory, we argue that some high-energy states in the FQH spectra necessitate a different description based on parton quasiparticles. We show that Jain states at filling factor $\nu =n/(2pn\pm 1)$ with integers $n$, $p\ge 2$ support two kinds of collective modes: In addition to the well-known Girvin-MacDonald-Platzman (GMP) mode, they host a high-energy collective mode, which we interpret as the GMP mode of partons. We elucidate observable signatures of the parton mode in the dynamics following a geometric quench. We construct a microscopic wave function for the parton mode and demonstrate agreement between its variational energy and exact diagonalization. Using the parton construction, we derive a field theory of the Jain states and show that the previously proposed effective theories follow from our approach. Our results point to partons being “real” quasiparticles which, in a way reminiscent of quarks, become observable only at sufficiently high energies.

Layered transition-metal dichalcogenides (TMDCs) host a plethora of interesting physical phenomena ranging from charge order to superconductivity. By introducing magnetic ions into $2H\text{−}T{A}_2$ ($T=\mathrm{Nb}$, Ta; $A=\mathrm{S}$, Se), the material forms a family of magnetic intercalated TMDCs $M_{x}T{A}_2$ ($M=3d$ transition metal). Recently, ${\mathrm{Fe}}_{1/3+\delta }{\mathrm{NbS}}_2$ has been found to possess intriguing resistance switching and magnetic memory effects coupled to the Néel temperature of $T_N\sim 45\mathrm{K}$ [Maniv et al., Nat. Phys. 17, 525 (2021); Sci. Adv. 7, eabd8452 (2021)]. We present comprehensive single crystal neutron diffraction measurements on underintercalated ($\delta \sim -0.01$), stoichiometric, and overintercalated ($\delta \sim 0.01$) samples. Magnetic defects are usually considered to suppress magnetic correlations and, concomitantly, transition temperatures. Instead, we observe highly tunable magnetic long-ranged states as the Fe concentration is varied from underintercalated to overintercalated, that is, from Fe vacancies to Fe interstitials. The under- and overintercalated samples reveal distinct antiferromagnetic stripe and zigzag orders, associated with wave vectors $k_1=(0.5,0,0)$ and $k_2=(0.25,0.5,0)$, respectively. The stoichiometric sample shows two successive magnetic phase transitions for these two wave vectors with an unusual rise-and-fall feature in the intensities connected to $k_1$. We ascribe this sensitive tunability to the competing next-nearest neighbor exchange interactions and the oscillatory nature of the Ruderman-Kittel-Kasuya-Yosida mechanism. We discuss experimental observations that relate to the observed intriguing switching resistance behaviors. Our discovery of a magnetic defect tuning of the magnetic structure in bulk crystals ${\mathrm{Fe}}_{1/3+\delta }{\mathrm{NbS}}_2$ provides a possible new avenue to implement controllable antiferromagnetic spintronic devices.

A single-hole ground state Ansatz for the two-dimensional $t\text{−}J$ model has been recently studied by the variational Monte Carlo (VMC) method. Such a doped hole behaves like a “twisted” non-Landau quasiparticle characterized by an emergent quantum number in agreement with exact numerics. In this work, we further investigate the ground state of two holes by VMC. It is found that the two holes strongly attract each other to form a pairing state with a new quantum number the same as obtained by the numerical exact diagonalization and density matrix renormalization group (DMRG) calculations. A unique feature of this pairing state is a dichotomy in the pairing symmetry, i.e., a d wave in terms of the electron $c$ operators and an s wave in terms of the new quasiparticles, as explicitly illustrated in the ground state wave function. A similar VMC study of a two-hole wave function for the $t\text{−}J$ two-leg ladder also yields a good agreement with the DMRG result. We demonstrate that the pairing mechanism responsible for the strong binding here is not due to the long-range antiferromagnetic order nor the resonating-valence-bound pairing in the spin background but is the consequence of the quantum phase strings created by the hopping of holes. The resulting spin-current pattern mediating the pairing force is explicitly illustrated in the VMC calculation. Physical implications to superconductivity at finite doping are also discussed.

The rich phenomenology of twisted bilayer graphene (TBG) near the magic angle is believed to arise from electron correlations in topological flat bands. An unbiased approach to this problem is highly desirable, but also particularly challenging, given the multiple electron flavors, the topological obstruction to defining tight-binding models, and the long-ranged Coulomb interactions. While numerical simulations of realistic models have thus far been confined to zero temperature, typically excluding some spin or valley species, analytic progress has relied on fixed point models away from the realistic limit. Here, we present unbiased Monte Carlo simulations of realistic models of magic-angle TBG at charge neutrality. We establish the absence of a sign problem for this model in a momentum-space approach and describe a computationally tractable formulation that applies even on breaking chiral symmetry and including band dispersion. Our results include (i) the emergence of an insulating Kramers intervalley coherent ground state in competition with a correlated semimetal phase, (ii) detailed temperature evolution of order parameters and electronic spectral functions that reveal a “pseudogap” regime, in which gap features are established at a higher temperature than the onset of order, and (iii) predictions for electronic tunneling spectra and their evolution with temperature. Our results pave the way towards uncovering the physics of magic-angle graphene through exact simulations of over a hundred electrons across a wide temperature range.

The light-matter interaction in bulk semiconductors is in the strong-coupling regime with hybrid eigenstates, the so-called exciton polaritons and phonon polaritons. In two-dimensional (2D) systems, the translational invariance is broken in the direction perpendicular to the plane of the 2D system. The light-matter interaction switches to the weak-coupling regime with a finite radiative lifetime of the matter excitations in 2D. Radiative phenomena have been extensively studied for 2D excitons in quantum wells and 2D crystals, but their counterpart has never been addressed for optical phonons in 2D. Here we present a parallel study of the exciton and phonon radiative linewidths in atomically thin layers of hexagonal boron nitride ($h$-BN), epitaxially grown on graphite. Reflectivity experiments are performed either in the deep ultraviolet for the excitonic resonance or in the midinfrared for the phononic one. A quantitative interpretation is implemented in the framework of a transfer matrix approach generalized to the case of monolayers with the inclusion of Breit-Wigner resonances of either excitonic or phononic nature. For the exciton we find a giant radiative broadening in comparison to other 2D crystals, with a value of $\sim 25\mathrm{meV}$ related to the strong excitonic effects in $h$-BN. For the phonon we provide the first estimation of the radiative linewidth of a 2D phonon, with a value of $\sim 0.2\mathrm{meV}$ in monolayer $h$-BN. Our results are found to be in good agreement with first-principles calculations. Our study unravels the existence of radiative states for optical phonons in 2D, with numerous perspectives for fundamental physics, optoelectronic applications in the midinfrared spectral range, and advanced thermal management, and $h$-BN is emerging as a model system in this novel topic.

The discovery of superconductivity in square-planar low valence nickelates has ignited a vigorous debate regarding their essential electronic properties: Do these materials have appreciable oxygen charge-transfer character akin to the cuprates, or are they in a distinct Mott-Hubbard regime where oxygen plays a minimal role? Here, we resolve this question using O $K$-edge resonant inelastic x-ray scattering (RIXS) measurements of the low valence nickelate ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ and a prototypical cuprate ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ ($x=0.35$). As expected, the cuprate lies deep in the charge-transfer regime of the Zaanen-Sawatzky-Allen (ZSA) scheme. The nickelate, however, is not well described by either limit of the ZSA scheme and is found to be of mixed charge-transfer–Mott-Hubbard character with the Coulomb repulsion $U$ of similar size to the charge-transfer energy $\mathrm{\Delta }$. Nevertheless, the transition-metal-oxygen hopping is larger in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ than in ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$, leading to a significant superexchange interaction and an appreciable hole occupation of the ligand O orbitals in ${\mathrm{La}}_4{\mathrm{Ni}}_3{\mathrm{O}}_8$ despite its larger $\mathrm{\Delta }$. Our results clarify the essential characteristics of low valence nickelates and put strong constraints on theoretical interpretations of superconductivity in these materials.

We study the phenomenon of Hilbert space fragmentation in isolated Hamiltonian and Floquet quantum systems using the language of commutant algebras, the algebra of all operators that commute with each local term that appears in the Hamiltonian or each local gate of the circuit. We provide a precise definition of Hilbert space fragmentation in this formalism as the case where the dimension of the commutant algebra grows exponentially with the system size. Fragmentation can, hence, be distinguished from systems with conventional symmetries such as U(1) or SU(2), where the dimension of the commutant algebra grows polynomially with the system size. Furthermore, the commutant algebra language also helps distinguish between “classical” and “quantum” Hilbert space fragmentation, where the former refers to fragmentation in the product state basis. We explicitly construct the commutant algebra in several systems exhibiting classical fragmentation, including the $t-J_z$ model and the spin-1 dipole-conserving model, and we illustrate the connection to previously studied “statistically localized integrals of motion.” We also revisit the Temperley-Lieb spin chains, including the spin-1 biquadratic chain widely studied in the literature, and show that they exhibit quantum Hilbert space fragmentation. Finally, we study the contribution of the full commutant algebra to the Mazur bounds in various cases. In fragmented systems, we use expressions for the commutant to analytically obtain new or improved Mazur bounds for autocorrelation functions of local operators that agree with previous numerical results. In addition, we are able to rigorously show the localization of the on-site spin operator in the spin-1 dipole-conserving model.