Pseudogap Phase Research Articles

Dynamic parallel spin stripes from the 1/8 anomaly to the end of superconductivity in La1.6−xNd0.4SrxCuO4

We have carried out neutron spectroscopic measurements on single crystals of ${\mathrm{La}}_{1.6\ensuremath{-}x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ from $0.12\ensuremath{\le}x\ensuremath{\le}0.26$ using time-of-flight techniques. These measurements allow us to follow the evolution of parallel spin stripe fluctuations with energies less than $\ensuremath{\sim}33$ meV, from $x=0.12$ to 0.26. Samples at these hole-doping levels are known to display static (on the neutron-scattering time scale) parallel spin stripes at low temperature, with onset temperatures and intensities which decrease rapidly with increasing $x$. Nonetheless, we report remarkably similar dynamic spectral weight for the corresponding dynamic parallel spin stripes, between 5 and 33 meV, from the 1/8 anomaly near $x=0.12$, to optimal doping near $x=0.19$ to the quantum critical point for the pseudogap phase near $x=0.24$, and finally to the approximate end of superconductivity near $x=0.26$. This observed dynamic magnetic spectral weight is structured in energy with a peak near 17 meV at all dopings studied. Earlier neutron and resonant x-ray scattering measurements on related cuprate superconductors have reported both a disappearance with increasing doping of magnetic fluctuations at ($\ensuremath{\pi}$, $\ensuremath{\pi}$) wave vectors characterizing parallel spin stripe structures and persistant paramagnon scattering away from this wave vector, respectively. Our results for ${\mathrm{La}}_{1.6\ensuremath{-}x}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ from 0.12 $\ensuremath{\le}x\ensuremath{\le}0.26$ clearly show persistent parallel spin stripe fluctuations at and around at ($\ensuremath{\pi}$, $\ensuremath{\pi}$), and across the full range of doping studied. These results are also compared to recent theory. Together with a rapidly declining $x$ dependence to the static parallel spin stripe order, the persistent parallel spin stripe fluctuations show a remarkable similarity to the expectations of a quantum spin glass, random t-J model, recently introduced to describe strong local correlations in cuprates.

Thermal Hall conductivity of electron-doped cuprates

Measurements of the thermal Hall conductivity in hole-doped cuprates have shown that phonons acquire chirality in a magnetic field, both in the pseudogap phase and in the Mott insulator state. The microscopic mechanism at play is still unclear. A number of theoretical proposals are being considered, including skew scattering of phonons by various defects, the coupling of phonons to spins, and a state of loop-current order with the appropriate symmetries, but more experimental information is required to constrain theoretical scenarios. Here we present our study of the thermal Hall conductivity $\kappa_{\rm {xy}}$ in the electron-doped cuprates Nd$_{2-x}$Ce$_x$CuO$_4$ and Pr$_{2-x}$Ce$_x$CuO$_4$, for dopings across the phase diagram, from $x$ = 0, in the insulating antiferromagnetic phase, up to $x$ = 0.17, in the metallic phase above optimal doping. We observe a large negative thermal Hall conductivity at all dopings, in both materials. Since heat conduction perpendicular to the CuO$_2$ planes is dominated by phonons, the large thermal Hall conductivity we observe in electron-doped cuprates for a heat current in that direction must also be due to phonons, as in hole-doped cuprates. Measurements with a heat current perpendicular to the CuO$_2$ planes confirm that phonons are responsible for this thermal Hall signal, as in hole-doped cuprates. However, the degree of chirality, measured as the ratio $|\kappa_{\rm {xy}}$ / $\kappa_{\rm {xx}} |$, where $\kappa_{\rm {xx}}$ is the longitudinal thermal conductivity, is much larger in the electron-doped cuprates. We discuss various factors that may be involved in the mechanism that confers chirality to phonons in cuprates, including short-range spin correlations.

Seebeck Coefficient in a Cuprate Superconductor: Particle-Hole Asymmetry in the Strange Metal Phase and Fermi Surface Transformation in the Pseudogap Phase

We report measurements of the Seebeck effect in both the $ab$ plane ($S_{\rm a}$) and along the $c$ axis ($S_{\rm c}$) of the cuprate superconductor La$_{1.6-x}$Nd$_{0.4}$Sr$_{x}$CuO$_4$ (Nd-LSCO), performed in magnetic fields large enough to suppress superconductivity down to low temperature. We use the Seebeck coefficient as a probe of the particle-hole asymmetry of the electronic structure across the pseudogap critical doping $p^{\star} = 0.23$. Outside the pseudogap phase, at $p = 0.24 > p^{\star}$, we observe a positive and essentially isotropic Seebeck coefficient as $T \rightarrow 0$. That $S > 0$ at $p = 0.24$ is at odds with expectations given the electronic band structure of Nd-LSCO above $p^{\star}$ and its known electron-like Fermi surface. We can reconcile this observation by invoking an energy-dependent scattering rate with a particle-hole asymmetry, possibly rooted in the non-Fermi liquid nature of cuprates just above $p^{\star}$. Inside the pseudogap phase, for $ p < p^{\star}$, $S_{\rm a}$ is seen to rise at low temperature as previously reported, consistent with the drop in carrier density $n$ from $n \simeq 1 + p$ to $n \simeq p$ across $p^{\star}$ as inferred from other transport properties. In stark contrast, $S_{\rm c}$ at low temperature becomes negative below $p^{\star}$, a novel signature of the pseudogap phase. The sudden drop in $S_{\rm c}$ reveals a change in the electronic structure of Nd-LSCO upon crossing $p^{\star}$. We can exclude a profound change of the scattering across $p^{\star}$ and conclude that the change in the out-of-plane Seebeck coefficient originates from a transformation of the Fermi surface.

Model of charge triplets for high-[formula omitted] cuprates

Starting with a minimal model for the CuO2 planes with the on-site Hilbert space reduced to a charge triplet of the three effective valence centers [CuO4]7−,6−,5− (nominally Cu1+,2+,3+) with different conventional spin, different orbital symmetry, and different local lattice configuration, we develop a unified non-BCS spin–pseudospin model to describe the main phase states of doped cuprates. We argue that antiferromagnetic insulating, charge ordered, superconducting, and Fermi-liquid phases are possible phase states of a model parent cuprate, while typical phase state of a doped cuprate, in particular mysterious pseudogap phase, is a result of a phase separation. Superconductivity of cuprates is not a consequence of pairing of doped holes, but the result of quantum transport of on-site composite hole bosons, whereas main peculiarities of normal state can be related to an electron–hole interplay for unusual Fermi-liquid phase and features of the phase separation. Puzzlingly, but it is the electron–lattice interaction, which in the BCS model determines s-wave pairing, in the model of local composite bosons gives dx2−y2-symmetry of the superconducting order parameter, thus showing once again a substantial involvement of the lattice in the cuprate’s HTSC.

Unified energy law for fluctuating density wave orders in cuprate pseudogap phase

The quantum origin of the cuprate pseudogap is a central conundrum of condensed matter physics. Although many symmetry-broken scenarios were previously proposed, universal quantitative relationships have been rarely studied. Here, we report a unified energy law underlying the pseudogap, which determines the scattering rate, pseudogap energy, and its onset temperature, with a quadratic scaling of the wavevector of density wave order (DWO). The law is validated by data from over one hundred samples, and a further prediction that the master order of pseudogap transforms from fluctuating spin to charge DWO is also confirmed. Furthermore, the energy law enables our derivation of the well-known linear scalings for the resistivity of the strange metal phase and the transition temperature of the superconducting phase. Finally, it is concluded that fluctuating orders provide a critical bridge linking microscopic spectra to macroscopic transport, showing promise for the quantification of other strongly correlated materials.

Компьютерное моделирование наноскопических фазово-неоднородных состояний и фазовых диаграмм ВТСП купратов и никелатов

More than 35 years of experience in the study of cuprate superconductors shows that the main characteristics of the phase diagram can only be obtained by taking into account mesoscopic static/dynamic phase inhomogeneity as a key property of these materials. Within a minimal model for the CuO2 planes with the on-site Hilbert space reduced to only three effective valence centers [CuO4]7-,6-,5- (nominally Cu1+,2+,3+) with different conventional spin and different orbital symmetry we propose a unified non-BCS model that allows one to describe the main features of the phase diagrams of doped cuprates within the framework of a simple effective field theory. Using Maxwell's construction, the global nature of the electronic phase separation in the CuO2 planes of HTSC cuprates is established, which makes it possible to understand and explain many fundamental features of the physics of the normal and superconducting state of cuprates, including the mechanism of formation of the HTSC and pseudogap phase. The features of phase-inhomogeneous states and their evolution with temperature and doping degree, including the special role of the impurity potential in cuprates/nickelates with nonisovalent substitution, are considered for particular examples of the charge triplet model in the framework of the classical Monte Carlo method.

Computer simulation of nanoscopic phase-inhomogeneous states and phase diagrams of HTSC cuprates and nickelates

More than 35 years of experience in the study of cuprate superconductors shows that the main characteristics of the phase diagram can only be obtained by taking into account mesoscopic static/dynamic phase inhomogeneity as a key property of these materials. Within a minimal model for the CuO2/NiO2 planes with the on-site Hilbert space reduced to only three effective valence centers [CuO4]7-,6-,5- (nominally Cu1+,2+,3+) with different conventional spin and different orbital symmetry we propose a unified non-BCS model that allows one to describe the main features of the phase diagrams of doped cuprates within the framework of a simple effective field theory. Using Maxwell's construction, the global nature of the electronic phase separation in the CuO2 planes of HTSC cuprates is established, which makes it possible to understand and explain many fundamental features of the physics of the normal and superconducting state of cuprates, including the mechanism of formation of the HTSC and pseudogap phase. The features of phase-inhomogeneous states and their evolution with temperature and doping degree, including the special role of the impurity potential in cuprates/nickelates with nonisovalent substitution, are considered for particular examples of the charge triplet model in the framework of the classical Monte Carlo method. Keywords: cuprates, nickelates, effective field theory, phase diagram, phase separation, Monte-Carlo.

Phenomenological analysis of the low temperature

The low-temperature phase diagram of a high-temperature cuprate superconductor consists of the superconducting, pseudogap and normal phases is analyzed in yttrium barium copper oxide systems by a phenomenological scheme using the shallow valence band electronic structure. The superconducting and pseudogap phases, which involve condensates, are assumed to be equilibrium systems of the condensate and normal subphases. The condition for phase transitions is given by equality of the total electronic energy gains. Using the mean strength of the pairing energy reproduces the experimental data well and the strength of the pairing energy decreases with increasing charge carrier concentration from the under-doped regime to the over-doped regime. We propose that this result can be explained by the screening effect.

Evidence for unconventional superconductivity in twisted bilayer graphene.

The emergence of superconductivity and correlated insulators in magic-angle twisted bilayer graphene (MATBG) has raised the intriguing possibility that its pairing mechanism is distinct from that of conventional superconductors1-4, as described by the Bardeen-Cooper-Schrieffer (BCS) theory. However, recent studies have shown that superconductivity persists even when Coulomb interactions are partially screened5,6. This suggests that pairing in MATBG might be conventional in nature and a consequence of the large density of states of its flat bands. Here we combine tunnelling and Andreev reflection spectroscopy with a scanning tunnelling microscope to observe several key experimental signatures of unconventional superconductivity in MATBG. We show that the tunnelling spectra below the transition temperature Tc are inconsistent with those of a conventional s-wave superconductor, but rather resemble those of a nodal superconductor with an anisotropic pairing mechanism. We observe a large discrepancy between the tunnelling gap ΔT, which far exceeds the mean-field BCS ratio (with 2ΔT/kBTc ~ 25), and the gap ΔAR extracted from Andreev reflection spectroscopy (2ΔAR/kBTc ~ 6). The tunnelling gap persists even when superconductivity is suppressed, indicating its emergence from a pseudogap phase. Moreover, the pseudogap and superconductivity are both absent when MATBG is aligned with hexagonal boron nitride. These findings and other observations reported here provide a preponderance of evidence for a non-BCS mechanism for superconductivity in MATBG.

Scattering interference signature of a pair density wave state in the cuprate pseudogap phase

An unidentified quantum fluid designated the pseudogap (PG) phase is produced by electron-density depletion in the CuO2 antiferromagnetic insulator. Current theories suggest that the PG phase may be a pair density wave (PDW) state characterized by a spatially modulating density of electron pairs. Such a state should exhibit a periodically modulating energy gap {Delta }_{{{{{{rm{P}}}}}}}({{{{{boldsymbol{r}}}}}}) in real-space, and a characteristic quasiparticle scattering interference (QPI) signature {Lambda }_{{{{{{rm{P}}}}}}}({{{{{boldsymbol{q}}}}}}) in wavevector space. By studying strongly underdoped Bi2Sr2CaDyCu2O8 at hole-density ~0.08 in the superconductive phase, we detect the 8a0-periodic {Delta }_{{{{{{rm{P}}}}}}}({{{{{boldsymbol{r}}}}}}) modulations signifying a PDW coexisting with superconductivity. Then, by visualizing the temperature dependence of this electronic structure from the superconducting into the pseudogap phase, we find the evolution of the scattering interference signature Lambda ({{{{{boldsymbol{q}}}}}}) that is predicted specifically for the temperature dependence of an 8a0-periodic PDW. These observations are consistent with theory for the transition from a PDW state coexisting with d-wave superconductivity to a pure PDW state in the Bi2Sr2CaDyCu2O8 pseudogap phase.

Translation-Invariant Bipolarons and Charge Density Waves in High-Temperature Superconductors

A correlation is established between the theories of superconductivity based on the concept of charge density waves (CDWs) and the translation invariant (TI) bipolaron theory. It is shown that CDWs are originated from TI-bipolaron states in the pseudogap phase due to the Kohn anomaly and form a pair density wave (PDW) for wave vectors corresponding to nesting. Emerging in the pseudogap phase, CDWs coexist with superconductivity at temperatures below those of superconducting transition, while their wave amplitudes decrease as a Bose condensate is formed from TI bipolarons, vanishing at zero temperature.

Phase diagram for hole-doped Kitaev systems on the honeycomb lattice

In extended Kitaev models on the honeycomb lattice, off-diagonal interactions (e.g., the $\mathrm{\ensuremath{\Gamma}},\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ terms) can give rise to non-Kitaev quantum spin liquids and several magnetically ordered phases. In the present paper, we dope holes to the system and study the resultant $t\text{\ensuremath{-}}K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ model using mean-field theory. The interplay between the charge and spin degrees of freedom results in a rich phase diagram. Similar to doped cuprates, superconductors, pseudogap phases, Fermi liquid, strange metal and paramagnetic phase are generated. What is different is that, we obtain more than one superconducting phase (including a topological one) and more than one pseudogap phase no matter what the original spin state is. The Chern number of the topological superconductor is either $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}2$ or $\ensuremath{\nu}=\ifmmode\pm\else\textpm\fi{}1$, depending on the ratio $\mathrm{\ensuremath{\Gamma}}/|K|$ in the spin channel. We further find that an intermediate in-plane magnetic field can slightly enlarge the size of the topological superconducting phase.

Data associated with "Scattering Interference Signature of a Pair Density Wave State in the Cuprate Pseudogap Phase"

Data associated with "Scattering Interference Signature of a Pair Density Wave State in the Cuprate Pseudogap Phase"

Pseudogap Isotope Effect as a Probe of Bipolaron Mechanism in High Temperature Superconductors.

A theory of a pseudogap phase of high-temperature superconductors where current carriers are translation invariant bipolarons is developed. A temperature of a transition from a pseudogap phase to a normal one is calculated. For the temperature of a transition to the pseudogap phase, the isotope coefficient is found. It is shown that the results obtained, in particular, the possibility of negative values of the isotope coefficient, are consistent with the experiment. New experiments on the influence of the magnetic field on the isotope coefficient are proposed.

Vanishing nematic order beyond the pseudogap phase in overdoped cuprate superconductors

During the last decade, translational and rotational symmetry-breaking phases-density wave order and electronic nematicity-have been established as generic and distinct features of many correlated electron systems, including pnictide and cuprate superconductors. However, in cuprates, the relationship between these electronic symmetry-breaking phases and the enigmatic pseudogap phase remains unclear. Here, we employ resonant X-ray scattering in a cuprate high-temperature superconductor [Formula: see text] (Nd-LSCO) to navigate the cuprate phase diagram, probing the relationship between electronic nematicity of the Cu 3d orbitals, charge order, and the pseudogap phase as a function of doping. We find evidence for a considerable decrease in electronic nematicity beyond the pseudogap phase, either by raising the temperature through the pseudogap onset temperature T* or increasing doping through the pseudogap critical point, p*. These results establish a clear link between electronic nematicity, the pseudogap, and its associated quantum criticality in overdoped cuprates. Our findings anticipate that electronic nematicity may play a larger role in understanding the cuprate phase diagram than previously recognized, possibly having a crucial role in the phenomenology of the pseudogap phase.

Transport signatures of the pseudogap critical point in the cuprate superconductor Bi2Sr2−xLaxCuO6+δ

Five transport coefficients of the cuprate superconductor ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$ were measured in the normal state down to low temperature, reached by applying a magnetic field (up to 66 T) large enough to suppress superconductivity. The electrical resistivity, Hall coefficient, thermal conductivity, Seebeck coefficient, and thermal Hall conductivity were measured in two overdoped single crystals, with La concentration $x=0.2$ (${T}_{\mathrm{c}} =18$ K) and $x=0.0$ (${T}_{\mathrm{c}} =10$ K). The samples have dopings $p$ very close to the critical doping ${p}^{★}$ where the pseudogap phase ends. The resistivity displays a linear dependence on temperature whose slope is consistent with Planckian dissipation. The Hall number ${n}_{\mathrm{H}}$ decreases with reduced $p$, consistent with a drop in carrier density from $n=1+p$ above ${p}^{★}$ to $n=p$ below ${p}^{★}$. This drop in ${n}_{\mathrm{H}}$ is concomitant with a sharp drop in the density of states inferred from prior NMR Knight shift measurements. The thermal conductivity satisfies the Wiedemann-Franz law, showing that the pseudogap phase at $T=0$ is a metal whose fermionic excitations carry heat and charge as do conventional electrons. The Seebeck coefficient diverges logarithmically at low temperature, a signature of quantum criticality. The thermal Hall conductivity becomes negative at low temperature, showing that phonons are chiral in the pseudogap phase. Given the observation of these same properties in other, very different cuprates, our study provides strong evidence for the universality of these five signatures of the pseudogap phase and its critical point.

Pauli-limit violation and re-entrant superconductivity in moiré graphene.

Moiré quantum matter has emerged as a materials platform in which correlated and topological phases can be explored with unprecedented control. Among them, magic-angle systems constructed from two or three layers of graphene have shown robust superconducting phases with unconventional characteristics1-5. However, direct evidence of unconventional pairing remains to be experimentally demonstrated. Here we show that magic-angle twisted trilayer graphene exhibits superconductivity up to in-plane magnetic fields in excess of 10 T, which represents a large (2-3 times) violation of the Pauli limit for conventional spin-singlet superconductors6,7. This is an unexpected observation for a system that is not predicted to have strong spin-orbit coupling. The Pauli-limit violation is observed over the entire superconducting phase, which indicates that it is not related to a possible pseudogap phase with large superconducting amplitude pairing. Notably, we observe re-entrant superconductivity at large magnetic fields, which is present over a narrower range of carrier densities and displacement fields. These findings suggest that the superconductivity in magic-angle twisted trilayer graphene is likely to be driven by a mechanism that results in non-spin-singlet Cooper pairs, and that the external magnetic field can cause transitions between phases with potentially different order parameters. Our results demonstrate the richness of moiré superconductivity and could lead to the design of next-generation exotic quantum matter.

Resonant Inelastic X-Ray Scattering Spectra of Cuprate Superconductors Predicted by Model of Fractionalized Fermions

We theoretically analyze the intensity of the resonant inelastic X-ray scattering (RIXS) based on Ansatz that the electron is fractionalized into two components. By fitting the angle resolved photoemission spectroscopy (ARPES) data of a cuprate high-$T_{\rm c}$ superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ and the accompanied machine learning results obtained before by Yamaji {\it et al.} (arXiv:1903.08060), we determine the paprameters of the two-component fermion model which describes the electron fractionalization. The RIXS spectra are predicted by using this description. We find that the intensity is enhanced in the superconducting phase relative to the normal or pseudogap phase. Since the enhancement is unusual, we propose this can be used as a critical test whether the unconventional electron fractionalization captures the essence of the cuprate superconductivity. This type of combined analyses using multiple independent spectroscopy data, with the help of theoretical and machine learning insights, regarded as {\it integrated spectroscopy} opens a new route to study difficult open issues in strongly correlated electron systems.

Stripes, Antiferromagnetism, and the Pseudogap in the Doped Hubbard Model at Finite Temperature

The interplay between thermal and quantum fluctuations controls the competition between phases of matter in strongly correlated electron systems. We study finite-temperature properties of the strongly coupled two-dimensional doped Hubbard model using the minimally-entangled typical thermal states (METTS) method on width $4$ cylinders. We discover that a phase characterized by commensurate short-range antiferromagnetic correlations and no charge ordering occurs at temperatures above the half-filled stripe phase extending to zero temperature. The transition from the antiferromagnetic phase to the stripe phase takes place at temperature $T/t \approx 0.05$ and is accompanied by a step-like feature of the specific heat. We find the single-particle gap to be smallest close to the nodal point at $\mathbf{k}=(\pi/2, \pi/2)$ and detect a maximum in the magnetic susceptibility. These features bear a strong resemblance to the pseudogap phase of high-temperature cuprate superconductors. The simulations are verified using a variety of different unbiased numerical methods in the three limiting cases of zero temperature, small lattice sizes, and half-filling. Moreover, we compare to and confirm previous determinantal quantum Monte Carlo results on incommensurate spin-density waves at finite doping and temperature.

Normal state specific heat in the cuprate superconductors La2−xSrxCuO4 and Bi2+ySr2−x−yLaxCuO6+δ near the critical point of the pseudogap phase

The specific heat $C$ of the cuprate superconductors La$_{2-x}$Sr$_x$CuO$_4$ and Bi$_{2+y}$Sr$_{2-x-y}$La$_x$CuO$_{6+\delta}$ was measured at low temperature (down to $0.5~{\rm K}$), for dopings $p$ close to $p^\star$, the critical doping for the onset of the pseudogap phase. A magnetic field up to $35~{\rm T}$ was applied to suppress superconductivity, giving direct access to the normal state at low temperature, and enabling a determination of $C_e$, the electronic contribution to the normal-state specific heat, at $T \to 0$. In La$_{2-x}$Sr$_x$CuO$_4$ at $x=p = 0.22$, $0.24$ and $0.25$, $C_e / T = 15-16~{\rm mJmol}^{-1}{\rm K}^{-2}$ at $T = 2~{\rm K}$, values that are twice as large as those measured at higher doping ($p > 0.3$) and lower doping ($p < 0.15$). This confirms the presence of a broad peak in the doping dependence of $C_e$ at $p^\star\simeq 0.19$, as previously reported for samples in which superconductivity was destroyed by Zn impurities. Moreover, at those three dopings, we find a logarithmic growth as $T \to 0$, such that $C_e / T \sim {\rm B}\ln(T_0/T)$. The peak vs $p$ and the logarithmic dependence vs $T$ are the two typical thermodynamic signatures of quantum criticality. In the very different cuprate Bi$_{2+y}$Sr$_{2-x-y}$La$_x$CuO$_{6+\delta}$, we again find that $C_e / T \sim {\rm B}\ln(T_0/T$) at $p \simeq p^\star$, strong evidence that this $\ln(1/T)$ dependence - first discovered in the cuprates La$_{1.8-x}$Eu$_{0.2}$Sr$_x$CuO$_4$ and La$_{1.6-x}$Nd$_{0.4}$Sr$_x$CuO$_4$ - is a universal property of the pseudogap critical point. All four materials display similar values of the $\rm B$ coefficient, indicating that they all belong to the same universality class.