We report a new kagome quantum spin liquid candidate Cu$_3$Zn(OH)$_6$FBr, which does not experience any phase transition down to 50 mK, more than three orders lower than the antiferromagnetic Curie-Weiss temperature ($\sim$ 200 K). A clear gap opening at low temperature is observed in the uniform spin susceptibility obtained from $^{19}$F nuclear magnetic resonance measurements. We observe the characteristic magnetic field dependence of the gap as expected for fractionalized spin-1/2 spinon excitations. Our experimental results provide firm evidence for spin fractionalization in a topologically ordered spin system, resembling charge fractionalization in the fractional quantum Hall state.
Barlowite Cu$_4$(OH)$_6$FBr shows three-dimensional (3D) long-range antiferromagnetism, which is fully suppressed in Cu$_3$Zn(OH)$_6$FBr with a kagome quantum spin liquid ground state. Here we report systematic studies on the evolution of magnetism in the Cu$_{4-x}$Zn$_x$(OH)$_{6}$FBr system as a function of $x$ to bridge the two limits of Cu$_4$(OH)$_6$FBr ($x$=0) and Cu$_3$Zn(OH)$_6$FBr ($x$=1). Neutron-diffraction measurements reveal a hexagonal-to-orthorhombic structural change with decreasing temperature in the $x$ = 0 sample. While confirming the 3D antiferromagnetic nature of low-temper…
Kagome lattice Heisenberg antiferromagnets are known to be highly sensitive to perturbations caused by structural disorder. NMR is a local probe ideally suited for investigating such disorder-induced effects, but in practice large distributions in the conventional one-dimensional NMR data make it difficult to distinguish the intrinsic behavior expected for pristine kagome quantum spin liquids from disorder induced effects. Here we report the development of a two-dimensional NMR data acquisition scheme applied to Zn-barlowite (Zn$_{0.95}$Cu$_{0.05}$)Cu$_{3}$(OD)$_{6}$FBr kagome lattice, and suc…
We use $^{79}$Br nuclear quadrupole resonance (NQR) to demonstrate that ultra slow lattice dynamics set in below the temperature scale set by the Cu-Cu super-exchange interaction $J$~($\simeq160$~K) in the kagome lattice Heisenberg antiferromagnet Zn-barlowite. The lattice completely freezes below 50~K, and $^{79}$Br NQR lineshapes become twice broader due to increased lattice distortions. Moreover, the frozen lattice exhibits an oscillatory component in the transverse spin echo decay, a typical signature of pairing of nuclear spins by indirect nuclear spin-spin interaction. This indicates tha…
Local probe investigation of the spin dynamics in the kagome and inter-layers of orthorhombic barlowite Cu$_4$(OD)$_6$FBr: $^{79}$Br and $^{63}$Cu NQR study
Quantum spin liquid (QSL) represents a new class of condensed matter states characterized by the long-range many-body entanglement of topological orders. The most prominent feature of the elusive QSL state is the existence of fractionalized spin excitations. Subject to the strong quantum fluctuations, the spin-1/2 antiferromagnetic system on a kagome lattice is the promising candidate for hosting a QSL ground state, but the structurally ideal realization is rare. Here, we report Raman scattering on the single crystalline Cu$_3$Zn(OH)$_6$FBr, and confirm that the ideal kagome structure remains …
We report a comprehensive muon spectroscopy study of the Zn-barlowite series of $S={\frac{1}{2}}$ kagomé antiferromagnets, Zn$_x$Cu$_{4-x}$(OH)$_{6}$FBr, for $x=0.00$ to $0.99(1)$. By combining muon spin relaxation and rotation measurements with state-of-the-art density-functional theory muon-site calculations, we observe the formation of both $μ$--F and $μ$--OH complexes in Zn-barlowite. From these stopping sites, implanted muon spins reveal the suppression of long-range magnetic order into a possible quantum spin liquid state upon increasing concentration of Zn-substitution. In the parent co…
A Z2 quantum spin liquid hosts one of the simplest topological orders and exhibits many exotic properties due to long-range quantum entanglements. Its elementary excitations are anyons such as spinons carrying fractionalized spin quantum number and visons carrying emergent Z2 gauge flux. However, experimental detection of these anyons remains elusive. The difficulties lie not only in the fact that there exists few candidates for Z2 quantum spin liquids but also in that visons are magnetically inert hence immune to available experimental techniques. Here we have studied the spin excitations and…
Motivated by recent numerical and experimental studies of the spin-1/2 Heisenberg antiferromagnet on kagome, we formulate a many-body model for fermionic spinons introduced by us earlier [Phys. Rev. Lett. 103, 187203 (2009)]. The spinons interact with an emergent U(1) gauge field and experience strong short-range attraction in the S=0 channel. The ground state of the model is generically a $Z_2$ liquid. We calculate the edge of the two-spinon continuum and compare the theory to the slave-fermion approach to the Heisenberg model.
We have systematically studied the magnetic properties of Cu$_{4-x}$Zn$_x$(OH)$_6$FBr by the neutron diffraction and muon spin rotation and relaxation ($μ$SR) techniques. Neutron-diffraction measurements suggest that the long-range magnetic order and the orthorhombic nuclear structure in the $x$ = 0 sample can persist up to $x$ = 0.23 and 0.43, respectively. The temperature dependence of the zero-field (ZF) $μ$SR spectra provide two characteristic temperatures, $T_{A0}$ and $T_λ$. Comparison between $T_{A0}$ and $T_M$ from previously reported magnetic-susceptibility measurements suggest that t…
Spin dynamics of the spin-1/2 kagome lattice antiferromagnet Cs$_2$Cu$_3$SnF$_{12}$ was studied using high-resolution, time-of-flight inelastic neutron scattering. The flat mode, a characteristic of the frustrated kagome antiferromagnet, and the low-energy dispersive mode, which is dominated by magnons, can be well described by the linear spin-wave theory. However, the theory fails to describe three weakly dispersive modes between 9 and 14 meV. These modes could be attributed to two-spinon bound states, which decay into free spinons away from the zone center and at a high temperature, giving r…
Temperature- and field- dependent $^1$H-, $^{19}$F-, and $^{79,81}$Br- NMR measurements together with zero - field $^{79,81}$Br-NQR measurements on polycrystalline samples of barlowite, Cu$_4$(OH)$_6$FBr are conducted to study the magnetism and possible structural distortions on a microscopic level. The temperature dependence of the $^{79,81}$Br- NMR spin-lattice relaxation rates 1/$T_1$ indicate a phase transition at $T_{\rm N}\simeq$15 K which is of magnetic origin, but with an unusually weak slowing down of fluctuations below $T_{\rm N}$. Moreover, 1/$T_1T$ scales linear with the bulk susce…
The ground state of the S=1/2 Heisenberg antiferomagnet on kagome can be viewed as a collection of fermionic spinons bound into small, heavy singlet pairs. Low-energy magnetic excitations in this system correspond to breaking the pairs into individual spinons. We calculate the structure factor for inelastic neutron scattering from independent spinon pairs.
We develop a theory for the thermal Hall coefficient in a spin-$\frac{1}{2}$ system on a strip of Kagome lattice, where a chiral spin-interaction term is present. To this end, we model the Kagome strip as a three-leg $XXZ$ spin-ladder, and use Bosonization to derive a low-energy theory for the spinons in this system. Introducing further a Dzyaloshinskii-Moriya interaction ($D$) and a tunable magnetic field ($B$), we identify three distinct $B$-dependent quantum phases: a valence-bond crystal (VBC), a "metallic" spin liquid (MSL) and a chiral spin liquid (CSL). In the presence of a temperature …
The search for exotic quantum spin liquid states in simple yet realistic spin models remains a central challenge in the field of frustrated quantum magnetism. Here we consider the canonical nearest-neighbor kagome Heisenberg antiferromagnet restricted to a quasi-1D strip consisting entirely of corner-sharing triangles. Using large-scale density matrix renormalization group calculations, we identify in this model an extended gapless quantum phase characterized by central charge $c=2$ and power-law decaying spin and bond-energy correlations which oscillate at tunably incommensurate wave vectors.…
The spin-$\frac{1}{2}$ kagome antiferromagnet is considered an ideal host for a quantum spin liquid ground state. We find that when the bonds of the kagome lattice are modulated with a periodic pattern, new quantum ground states emerge. Newly synthesized crystalline barlowite (Cu$_4$(OH)$_6$FBr) and Zn-substituted barlowite demonstrate the delicate interplay between singlet states and spin order on the spin-$\frac{1}{2}$ kagome lattice. Comprehensive structural measurements demonstrate that our new variant of barlowite maintains hexagonal symmetry at low temperatures with an arrangement of dis…
We study dynamical spin structure factor (DSSF) of S = 1/2 Heisenberg model on the kagome lattice (KAFM) by means of density-matrix renormalization group. By comparison with the well-defined magnetic ordered state and chiral spin liquid sitting nearby in the phase diagram, the KAFM with the nearest-neighbor interaction shows distinct dynamical response behaviors. First of all, the DSSF displays important spectral intensity predominantly at low frequency region around Q = M point in momentum space, and shows a broad spectral distribution at high frequency region for momenta along the boundary o…
We systematically study the low-temperature specific heats for the two-dimensional kagome antiferromagnet, Cu$_{3}$Zn(OH)$_6$FBr. The specific heat exhibits a $T^{1.7}$ dependence at low temperatures and a shoulder-like feature above it. We construct a microscopic lattice model of $Z_2$ quantum spin liquid and perform large-scale quantum Monte Carlo simulations to show that the above behaviors come from the contributions from gapped anyons and magnetic impurities. Surprisingly, we find the entropy associated with the shoulder decreases quickly with grain size $d$, although the system is parama…
Motivated by spin-wave continuum (SWC) observed in recent neutron scattering experiments in Herbertsmithite, we use Gutzwiller-projected wave functions to study dynamic spin structure factor $S(\mathbf{q},ω)$ of spin liquid states on the kagome lattice. Spin-1 excited states in spin liquids are represented by Gutzwiller-projected two-spinon excited wave functions. We investigate three different spin liquid candidates, spinon Fermi-surface spin liquid (FSL), Dirac spin liquid (DSL) and random-flux spin liquid (RSL). FSL and RSL have low energy peaks in $S(\mathbf{q},ω)$ at $K$ points in the ext…
We show that the best variational ground state of the spin-$\frac{1}{2}$ Kagome antiferromagnetic Heisenberg model with nearest-neighboring exchange coupling(NN-KAFH) is a $Z_{2}$ spin liquid state, rather than the widely believed $U(1)$ Dirac spin liquid state. The spinon excitation in the $Z_{2}$ spin liquid state has a small gap of about $1/40$ of the spinon band width. We find that while the $Z_{2}$ and the $U(1)$ spin liquid state have a large overlap on finite clusters and are thus very close in energy, they host totally different spinon excitation spectrum. The strength of the RVB theor…
Motivated by recent experiments on the quantum-spin-liquid candidate material LiZn2Mo3O8, we study a single-band extended Hubbard model on an anisotropic Kagome lattice with the 1/6 electron filling. Due to the partial filling of the lattice, the inter-site repulsive interaction is necessary to generate Mott insulators, where electrons are localized in clusters, rather than at lattice sites. It is shown that these cluster Mott insulators are generally U(1) quantum spin liquids with spinon Fermi surfaces. The nature of charge excitations in cluster Mott insulators can be quite different from co…
We combine the pseudofermion functional renormalization group (PFFRG) method with a self-consistent Fock-like mean-field scheme to calculate low-energy effective theories for emergent spinon excitations in spin-1/2 quantum spin liquids. Using effective spin interactions from PFFRG as an input for the Fock equation and allowing for the most general types of free spinon ansätze as classified by the projective symmetry group (PSG) method, we are able to systematically determine spinon band structures for spin-liquid candidate systems beyond mean-field theory. We apply this approach to the antifer…
Motivated by recent experiments on the quantum-spin-liquid candidate material LiZn2Mo3O8, we study a single-band extended Hubbard model on an anisotropic Kagome lattice with the 1/6 electron filling. Due to the partial filling of the lattice, the inter-site repulsive interaction is necessary to generate Mott insulators, where electrons are localized in clusters, rather than at lattice sites. We provide examples of such cluster Mott insulators and study the phase transitions between metallic states and cluster Mott insulators on an anisotropic Kagome lattice. It is shown that these cluster Mott…
We analyse the antiferromagnetic spin-$1/2$ XXZ model on the kagome lattice at finite external magnetic field with the help of a nonperturbative zero-temperature renormalization group (RG) technique. Following the work of Kumar \emph{et al} (Phys. Rev. B {\bf 90}, 174409 (2014)), we use a Jordan-Wigner transformation to map the spin problem into one of spinless fermions (spinons) in the presence of a statistical gauge field, and with nearest-neighbour interactions. While the work of Kumar \emph{et al} was confined mostly to the plateau at $1/3$-filling (magnetisation per site) in the XY regime…
We present thermodynamic studies of a new spin-1/2 antiferromagnet containing undistorted kagome lattices---barlowite Cu$_{4}$(OH)$_{6}$FBr. Magnetic susceptibility gives $θ_{CW}$ = $-$136 K, while long-range order does not happen until $T_{N}$ = 15 K with a weak ferromagnetic moment $μ$ $<$ 0.1$μ_{B}$/Cu. A 60 T magnetic field induces a moment less than 0.5$μ_{B}$/Cu at $T$ = 0.6 K. Specific-heat measurements have observed multiple phase transitions at $T \ll$ $\mid$$θ_{CW}$$\mid$. The magnetic entropy of these transitions is merely 18% of $k_{B}$ln2 per Cu spin. These observations suggest th…
Recent experiments indicate that Na$_4$Ir$_3$O$_8$, a material in which s=1/2 iridium local moments form a three dimensional network of corner-sharing triangles, may have a quantum spin liquid ground state with gapless spin excitations. Using a combination of exact diagonalization, symmetry analysis of fermionic mean field ground states and Gutzwiller projected variational wavefunction studies, we propose a quantum spin liquid with spinon Fermi surfaces as a favorable candidate for the ground state of the Heisenberg model on the hyper-kagome lattice of Na$_4$Ir$_3$O$_8$. We present a renormali…
We investigate the zero-temperature phase diagram of the nearest-neighbor kagome antiferromagnet in the presence of Dzyaloshinksii-Moriya interaction. We develop a theory for the transition between Z2 spin liquids with bosonic spinons and a phase with antiferromagnetic long-range order. Connections to recent numerical studies and experiments are discussed.
Experimental evidence for a plethora of low energy spin excitations in the spin-1/2 kagome antiferromagnet ZnCu3(OH)6Cl2 may be understandable in terms of an extended Fermi surface of spinons coupled to a U(1) gauge field. We carry out variational calculations to examine the possibility that such a state may be energetically viable. A Gutzwiller-projected wavefunction reproduces the dimerization of a kagome strip found previously by DMRG. Application to the full kagome lattice shows that the inclusion of a small ferromagnetic next-nearest-neighbor interaction favors a ground state with a spino…
The doping of quantum Heisenberg antiferromagnets on the kagome lattice by non-magnetic impurities is investigated within the framework of a generalized quantum dimer model (QDM) describing a) the valence bond crystal (VBC), b) the dimer liquid and c) the critical region on equal footing. Following the approach by Ralko et al. [Phys. Rev. Lett. 101, 117204 (2008)] for the square and triangular lattices, we introduce the (minimal) extension of the QDM on the Kagome lattice to account for spontaneous creation of mobile S=1/2 spinons at finite magnetic field. Modulations of the dimer density (at …
The kagome Heisenberg antiferromagnet formed by frustrated spins arranged in a lattice of corner-sharing triangles is a prime candidate for hosting a quantum spin liquid (QSL) ground state consisting of entangled spin singlets. But the existence of various competing states makes a convincing theoretical prediction of the QSL ground state difficult, calling for experimental clues from model materials. The kagome lattice materials Zn-barlowite ZnCu$_{3}$(OD)$_{6}$FBr and herbertsmithite ZnCu$_{3}$(OD)$_{6}$Cl$_2$ do not exhibit long range order, and they are considered the best realizations of t…
Based on large-scale quantum Monte Carlo simulations, we examine the dynamical spin structure factor of the Balents-Fisher-Girvin kagome lattice quantum spin-$1/2$ model, which is known to harbor an extended $Z_2$ quantum spin liquid phase. We use a correlation-matrix sampling scheme combined with a stochastic analytic continuation method to resolve the spectral functions of this anisotropic quantum spin model with a three-site unit-cell. Based on this approach, we monitor the spin dynamics throughout the phase diagram of this model, from the XY-ferromagnetic region to the $Z_2$ quantum spin l…
The confinement of a spinon liberated by doping two-dimensional frustrated quantum antiferromagnets with a non-magnetic impurity or a mobile hole is investigated. For a static vacancy, an intermediate behavior between complete deconfinement (kagome) and strong confinement (checkerboard) is identified in the J_1{-}J_2{-}J_3 model on the square lattice, with the emergence of two length scales, a spinon confinement length larger than the magnetic correlation length. For mobile holes, this translates into an extended spinon-holon boundstate allowing one to bridge momentum (ARPES spectral function)…
Gapped Z_2 spin liquids have been proposed as candidates for the ground-state of the S=1/2 quantum antiferromagnet on the Kagome lattice. We extend the use of Projected Entangled Pair States to construct (on the cylinder)Resonating Valence Bond (RVB) states including both nearest-neighbor and next-nearest neighbor singlet bonds. Our ansatz -- dubbed "simplex spin liquid" -- allows for an asymmetry between the two types of triangles (of order 2-3% in the energy density after optimization) leading to the breaking of inversion symmetry. We show that the topological Z_2 structure is still preserve…
Motivated by a search for experimental probes to access the physics of fractionalized excitations called spinons in a spin liquids, we study the interaction of spinons with lattice vibrations. We consider the case of algebraic spin liquid, when spinons have fermionic statistics and a Dirac-like dispersion. We establish the general procedure for deriving spinon-phonon interactions which is based on a symmetry considerations. The procedure is illustrated for four different algebraic spin liquids: pi-flux and staggered-flux phases on a square lattice, pi-flux phase on a kagome lattice, and zero f…
Perfect deporalization of the Loudon-Fleury inelastic visible-light scattering in the Kitaev honeycomb model is well known. Though it happens in Heisenberg Kagome and triangular antiferromagnets as well, yet we prove it to be of geometric origin rather than peculiar to quantum spin liquids. A Kitaev spin liquid in the square planar geometry indeed exhibits polarized Raman spectra containing defferent symmetry species, each brought by symmetry-compatible spinon geminate excitations, i.e. arising from symmetry-compatible direct-product representations made of double-valued irreducible representa…
A symmetry based analysis (Projective Symmetry Group) is used to study spin liquid phases on the triangular and Kagomé lattices in the Schwinger boson framework. A maximum of eight distinct $Z_2$ spin liquid states are found for each lattice, which preserve all symmetries. Out of these only a few have nonvanishing nearest neighbor amplitudes which are studied in greater detail. On the triangular lattice, only two such states are present - the first (zero-flux state) is the well known state introduced by Sachdev, which on condensation of spinons leads to the 120 degree ordered state. The other …
Using Schwinger-boson mean-field theory, we calculate the dynamic spin structure factor at low temperatures $0<T\ll J$ for the spin-$1/2$ antiferromagnetic Heisenberg kagome model, within the gapped $\mathbb{Z}_2$ spin liquid phase Ansatz. We find that the spectral gap rapidly fills with temperature, with robust low-energy spectral weight developing by a temperature of $Δ/3$, where the spin gap is $2Δ$ (i.e., $Δ$ is the spinon gap), before any appreciable rise in spinon density or change in zero-temperature mean-field parameters. This is due to deconfinement of spinons which leads to terms sup…
We study the origin of Lorentz force on the spinons in a U(1) spin liquid. We are inspired by the previous observation of gauge field correlation in the pairwise spin correlation using the neutron scattering measurement when the Dzyaloshinskii-Moriya interaction intertwines with the lattice geometry. We extend this observation to the Lorentz force that exerts on the (neutral) spinons. The external magnetic field, that polarizes the spins, effectively generates an internal U(1) gauge flux for the spinons and twists the spinon motion through the Dzyaloshinskii-Moriya interaction. Such a mechanis…
The ground state of the quantum kagome antiferromagnet Zn-brochantite, ZnCu$_3$(OH)$_6$SO$_4$, which is one of only a few known spin-liquid (SL) realizations in two or three dimensions, has been described as a gapless SL with a spinon Fermi surface. Employing nuclear magnetic resonance in a broad magnetic-field range down to millikelvin temperatures, we show that in applied magnetic fields this enigmatic state is intrinsically unstable against a SL with a full or a partial gap. A similar instability of the gapless Fermi-surface SL was previously encountered in an organic triangular-lattice ant…
Motivated by recent experiments on the Heisenberg S=1/2 quantum spin liquid candidate material kapellasite, we classify all possible chiral (time-reversal symmetry breaking) spin liquids with fermionic spinons on the kagome lattice. We obtain the phase diagram for the physically relevant extended Heisenberg model, comparing the energies of a wide range of microscopic variational wave functions. We propose that, at low temperature, kapellasite exhibits a gapless chiral spin liquid phase with spinon Fermi surfaces. This two-dimensional state inherits many properties of the nearby one-dimensional…
Spin excitations in an ordered Heisenberg magnet are magnons--bosons with spin 1. That may change when frustration and quantum fluctuations suppress the order and restore the spin-rotation symmetry. We show that spin excitations in the $S=1/2$ Heisenberg antiferromagnet on kagome are spinons--fermions with spin 1/2. In the ground state the system can be described as a collection of small, heavy pairs of spinons with spin 0. A magnetic excitation of lowest energy amounts to breaking up a pair into two spinons at a cost of $0.06 J$.
In quantum spin liquid states, the fractionalized spinon excitations can carry fractional crystal symmetry quantum numbers, and this symmetry fractionalization distinguishes different topologically ordered spin liquid states. In this work we propose a simple way to detect signatures of such crystal symmetry fractionalizations from the crystal symmetry representations of the ground state wave function. We demonstrate our method on projected $\mathbb Z_2$ spin liquid wave functions on the kagome lattice, and show that it can be used to classify generic wave functions. Particularly our method can…
Quantum spin ice, modeled for magnetic rare-earth pyrochlores, has attracted great interest for hosting a U(1) quantum spin liquid, which involves spin-ice monopoles as gapped deconfined spinons, as well as gapless excitations analogous to photons. However, the global phase diagram under a [111] magnetic field remains open. Here we uncover by means of unbiased quantum Monte-Carlo simulations that a supersolid of monopoles, showing both a superfluidity and a partial ionization, intervenes the kagomé spin ice and a fully ionized monopole insulator, in contrast to classical spin ice where a direc…
We investigate the nature of quantum spin liquid (QSL) phase of the spin-$1/2$ nearest-neighbor XXZ antiferromagnet on the kagome lattice. Recent numerical calculations suggest that such a kagome spin liquid is insensitive to the XXZ anisotropy and is adjacent to a chiral spin liquid phase. Reformulating the problem in terms of a $U(1)$ lattice gauge theory with dynamical bosonic spinons, we propose that the kagome spin liquid can be understood as a deconfined critical phase extended by the $U(1)$ gauge fluctuation from a deconfined critical point between a symmetry-protected topological phase…
We provide strong numerical evidence, using improved variational wave functions, for a ground state with vanishing spin gap in the spin-$1/2$ quantum Heisenberg model on the kagome lattice. Starting from the algebraic $U(1)$ Dirac spin liquid state proposed by Y. Ran $et al.$ [Phys. Rev. Lett. $98$, $117205$ ($2007$)] and iteratively applying a few Lanczos steps, we compute the lowest $S=2$ excitation constructed by exciting spinons close to the Dirac nodes. Our results are compatible with a vanishing spin gap in the thermodynamic limit and in consonance with a power-law decay of long distance…
We calculate dynamical spin structure factors for gapped chiral spin liquid states in the spin-1/2 Heisenberg antiferromagnet on the kagome lattice using Schwinger-boson mean-field theory. In contrast to static (equal-time) structure factors, the dynamical structure factor shows clear signatures of time-reversal symmetry breaking for chiral spin liquid states. In particular, momentum inversion $\bm{k} \to -\bm{k}$ symmetry as well as the six-fold rotation symmetry around the $Γ$-point are lost. We highlight other interesting features, such as a relatively flat onset of the two-spinon continuum…
We study the ground state phase diagram of a two-dimensional kagome lattice spin-1/2 XY model (J) with a four-site ring exchange interaction (K) using quantum Monte Carlo simulations. We find that the superfluid phase, existing in the regime of small ring exchange, undergoes a direct transition to a Z_2 quantum spin liquid phase at (K/J)_c ~ 22, which is related to the phase proposed by Balents, Girvin and Fisher [Phys. Rev. B, 65 224412 (2002)]. The quantum phase transition between the superfluid and the spin liquid phase has exponents z and νfalling in the 3D XY universality class, making it…
