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We show theoretically that the magnetic ions, randomly distributed in a two-dimensional (2D) semiconductor system, can generate a ferromagnetic long-range order via the RKKY interaction. The main physical reason is the discrete (rather than continuous) symmetry of the 2D Ising model of the spin-spin interaction mediated by the spin-orbit coupling of 2D free carriers, which precludes the validity of the Mermin-Wagner theorem. Further, the analysis clearly illustrates the crucial role of the molecular field fluctuations as opposed to the mean field. The developed theoretical model describes the desired magnetization and phase-transition temperature ${T}_{c}$ in terms of a single parameter, namely, the chemical potential $\ensuremath{\mu}$. Our results highlight a pathway to reach the highest possible ${T}_{c}$ in a given material as well as an opportunity to control the magnetic properties externally (e.g., via a gate bias). Numerical estimations show that magnetic impurities such as ${\mathrm{Mn}}^{2+}$ with spins $S=5/2$ can realize ferromagnetism with ${T}_{c}$ close to room temperature.

We propose a generalization of the coherent anomaly method to extract the critical exponents of a phase transition occurring in the steady-state of an open quantum many-body system. The method, originally developed by Suzuki [J. Phys. Soc. Jpn. {\bf 55}, 4205 (1986)] for equilibrium systems, is based on the scaling properties of the singularity in the response functions determined through cluster mean-field calculations. We apply this method to the dissipative transverse-field Ising model and the dissipative XYZ model in two dimensions obtaining convergent results already with small clusters., Accepted version, 9 pages, 7 figures

Recent experiments have discovered multipolar orders in a variety of $d$-orbital Mott insulators. Motivated by uncovering the exchange interactions which underlie octupolar order proposed in the osmate double perovskites, we study a two-site model using exact diagonalization on a five-orbital Hamiltonian, incorporating spin-orbit coupling (SOC) and interactions, and including both intra-orbital and inter-orbital hopping. Using an exact Schrieffer-Wolff transformation, we then extract an effective pseudospin Hamiltonian for the non-Kramers doublets, uncovering dominant ferrooctupolar coupling driven by the interplay of two distinct intra-orbital hopping terms. Using classical Monte Carlo simulations on the face-centered cubic lattice, we obtain a ferrooctupolar transition temperature which is in good agreement with experiments on the osmate double perovskites. We also explore the impact of uniaxial strain and dimensional tuning via ultrathin films, which are shown to induce a transverse field on the Ising octupolar order. This suppresses $T_c$ and potentially allows one to access octupolar Ising quantum critical points. We discuss possible implications of our results for a broader class of materials which may host such non-Kramers doublet ions., 11 pages, 8 figures

We investigate the low-energy physics of non-Hermitian quantum spin models with $PT$-symmetry. To this end we consider the one-dimensional Ising chain and the two-dimensional toric code in a non-Hermitian staggered field. For both systems dual descriptions in terms of non-Hermitian staggered Ising interactions in a conventional transverse field exist. We perform high-order series expansions about the high- and low-field limit for both systems to determine the ground-state energy per site and the one-particle gap. The one-dimensional non-Hermitian Ising chain is known to be exactly solvable. Its ground-state phase diagram consists of second-order quantum phase transitions, which can be characterized by logarithmic singularities of the second derivative of the ground-state energy and, in the symmetry-broken phase, the gap closing of the low-field gap. In contrast, the gap closing from the high-field phase is not accessible perturbatively due to the complex energy and the occurrence of exceptional lines in the high-field gap expression. For the two-dimensional toric code in a non-Hermitian staggered field we study the quantum robustness of the topologically ordered phase by the gap closing of the low-field gap. We find that the well-known second-order quantum phase transition of the toric code in a uniform field extends into a large portion of the non-Hermitian parameter space. However, the series expansions become unreliable for a dominant anti-Hermitian field. Interestingly, the analysis of the high-field gap reveals the potential presence of an intermediate region., Comment: 15 pages, 9 figures

The Ising magnet $\mathrm{Sm}{\mathrm{Pt}}_{2}{\mathrm{Si}}_{2}$ undergoes an antiferromagnetic (AFM) transition at ${T}_{\mathrm{N}1}$ = 5.6 K, but the magnetic moments of some Sm ions remain disordered even below ${T}_{\mathrm{N}1}$. Here, we report the results of measurements of thermal expansion, magnetostriction, and dc magnetization of $\mathrm{Sm}{\mathrm{Pt}}_{2}{\mathrm{Si}}_{2}$ in the temperature range from 0.26 to 7 K. The magnetization and thermal expansion results show clear irreversible phenomena at temperatures below ${T}_{\mathrm{g}}\phantom{\rule{4pt}{0ex}}\ensuremath{\approx}$ 2.4 K. These results suggest that the partially disordered state of the Sm ions becomes a cluster spin glass below ${T}_{\mathrm{g}}$ and coexists with the AFM long-range order. However, unlike the conventional coexistence of spin glass and antiferromagnetism, the isothermal magnetization process at 0.26 K shows a hysteresis curve with almost no remanent magnetization. This observation suggests that the cluster spin glass state is an unusual state with a short relaxation time at zero field.

Metallic quantum states coined strange metal (SM), with robust linear-$T$ resistivity, have been widely observed in many quantum materials under strong electron correlations, ranging from high-${T}_{c}$ cuprate superconductors and organic superconductors to twisted multilayer graphene and ${\mathrm{MoTe}}_{2}/{\mathrm{WSe}}_{2}$ superlattices. Despite decades of intensive studies, the mystery of SM still defies any sensible theoretical explanation and has been a key puzzle in modern condensed matter physics. Here, we solve a doped Mott insulator model called an Ising-Kondo lattice, which includes static spin fluctuation. With Monte Carlo simulation, the Ising-Kondo lattice unambiguously exhibits SM phenomena accompanied with quantum critical scaling in observables, e.g., resistivity, susceptibility, and specific heat. A closer look at SM reveals the breakdown of Landau's Fermi liquid theory without any symmetry breaking, i.e., the violation of Luttinger theorem. Our paper reveals that the long-overlooked static fluctuations in literature may play an essential role in non-Fermi liquid behaviors in correlated electron systems.

The quantum wetting transition in the one-dimensional transverse-field Ising model with random bonds is studied. Opposite boundary fields are applied at the two ends of the Ising chain. The interface between two domains with oppositely oriented magnetization is localized near the boundary or in the middle of the chain. The wetting transition refers to the phase transition of localization and delocalization of the interface. The tuning parameter ${h}_{L}$ is the boundary field at one end, e.g., the left end. At the transition point ${h}_{L}={h}_{w}$, the wetting transition occurs. First, we study the wetting transition with one defect bond. It is a first-order transition, and the interface jumps from the left end of the chain to the defect bond at the transition point. Second, we study the wetting transition with two defect bonds and find that the weaker defect bond dominates the phase transition. The wetting transition is still first order, and the interface is localized at the left end of the chain or at the weaker defect bond. Lastly, the random bond case is studied. The random bonds have a rectangular distribution. The wetting transition is still first order. The finite-size effects are studied. The statistics of the transition points and the energy gaps are obtained. We study lattices with sizes $N=100$, 150, 200, 250, 300, and 350. The deviation of the phase transition points ${h}_{w}$ does not decrease as the lattice size increases. It is argued that the transition point is sample dependent, i.e., the variation in the transition point ${h}_{w}$ does not approach zero, even at the thermodynamic limit.

Many magnetic materials have quantum disordered magnetic ground states, but it is not always clear what drives them. Here, the authors study the pyrochlore magnet Nd${}_{2}$ScNbO${}_{7}$ using neutron diffraction, heat capacity, and neutron-sampled nuclear hyperfine excitations. The static magnetic moments inferred from these three techniques are inconsistent, pointing to some fraction of spins remaining dynamic to the lowest temperatures -- an observation confirmed by the anomalous low-energy density of states in the specific heat. Similar behavior is seen in Nd${}_{2}$Zr${}_{2}$O${}_{7}$. It could indicate either quantum spin-liquid behavior or local disorder-induced spin singlets.

We show that a class of $\mathcal{PT}$ symmetric non-Hermitian Hamiltonians realizing the Yang-Lee edge singularity exhibits an entanglement transition in the long-time steady state evolved under the Hamiltonian. Such a transition is induced by a level crossing triggered by the critical point associated with the Yang-Lee singularity and hence is first-order in nature. At the transition, the entanglement entropy of the steady state jumps discontinuously from a volume-law to an area-law scaling. We exemplify this mechanism using a one-dimensional transverse field Ising model with additional imaginary fields, as well as the spin-1 Blume-Capel model and the three-state Potts model. We further make a connection to the forced-measurement induced entanglement transition in a Floquet non-unitary circuit subject to continuous measurements followed by post-selections. Our results demonstrate a new mechanism for entanglement transitions in non-Hermitian systems harboring a critical point., Updated to published version

Dynamical quantum phase transitions (DQPTs) feature singular temporal behavior in transient quantum states during nonequilibrium real-time evolution. In this work we show that DQPTs in random Ising chains exhibit critical behavior with nontrivial exponents that are not integer valued and not of mean-field type. By means of an exact renormalization group transformation we estimate the exponents with high accuracy eliminating largely any finite-size effects. We further discuss how the considered dynamical phenomena can be made accessible in current Rydberg atom platforms. In this context we explore signatures of the DQPTs in the statistics of spin configuration measurements available in such architectures. Specifically, we study the statistics of clusters of consecutively aligned spins and observe a marked influence of the DQPT on the corresponding distribution., Accepted version, 9 pages, 3 figures, journal article

We use large-scale exact diagonalization to study the quantum Ising chain in a transverse field with long-range power-law interactions decaying with exponent $\alpha$. We numerically study various probes for quantum chaos and eigenstate thermalization {on} the level of eigenvalues and eigenstates. The level-spacing statistics yields a clear sign towards a Wigner-Dyson distribution and therefore towards quantum chaos across all values of $\alpha>0$. Yet, for $\alpha, Comment: 17 pages, 15 figures

The NMR spectrum of FeSe shows a dramatic broadening on cooling towards the bulk nematic phase at $T_s=90$ K, due to the formation of a quasi-static, short-range-ordered nematic domain structure. However, a quantitative understanding of the NMR broadening and its relationship to the nematic susceptibility is still lacking. Here, we show that the temperature and pressure dependence of the broadening is in quantitative agreement with the mean-field Edwards-Anderson parameter of an Ising-nematic model in the presence of random-field disorder introduced by non-magnetic impurities. Furthermore, these results reconcile the interpretation of NMR and Raman spectroscopy data in FeSe under pressure., Comment: 9 pages, 3 figures

A coupled spin-electron model on a decorated square lattice formed by interconnected trigonal bipyramids is exactly solved by imposing two mobile electrons per each triangular plaquette with the help of a generalized decoration-iteration transformation, which establishes a precise mapping correspondence with the effective Ising model on a square lattice with temperature-dependent interaction. The investigated spin-electron model exhibits two different macroscopically degenerate ground states. The residual entropy of the first ground state, which shows a spontaneous ferromagnetic or ferrimagnetic long-range order depending on character of the exchange coupling, arises from chiral degrees of freedom of the mobile electrons. In contrast, the second ground state is disordered due to a kinetically driven frustration of the localized Ising spins triggered by the hopping term of the mobile electrons. The outstanding reentrant phase transitions connected with temperature induced formation of the spontaneous ferromagnetic or ferrimagnetic order can be found if the spin-electron model is driven sufficiently close to the ground-state phase boundary, but the disordered frustrated phase is the respective ground state. It is verified that the bipartite fermionic entanglement between two mobile electrons within the spontaneously ordered ferromagnetic or ferrimagnetic phase predominantly comes from their charge degrees of freedom, while the one within the disordered frustrated phase comes from both charge and spin degrees of freedom of these particles.

We study the realizations of topological defects in a one-dimensional quantum Ising model with an open boundary condition at criticality. Applying the construction discussed in M. Hauru, G. Evenbly, W. W. Ho, D. Gaiotto, and G. Vidal, Phys. Rev. B 94, 115125 (2016), we prove that the Ising model on an open chain with multiple topological defects can be transformed to the same model with boundary magnetic fields and noninteracting boundary degrees of freedom. This results in the appearance of a linear combination of Cardy states, which can be interpreted as an edge state of the spin or fermion chain. We show that this edge state with the large boundary entropy can be protected under bulk perturbation, whereas it is fragile to a boundary perturbation. Our formulation suggests the existence of nontrivial edge physics under the existence of topological defects and opens many interesting questions for future analysis related to boundary and bulk physics.

The simple update (SU) and full update (FU) are the two paradigmatic time evolution algorithms for a tensor network known as the infinite projected entangled pair state (iPEPS). They differ by an error measure that is either, respectively, local or takes into account full infinite tensor environment. In this paper we test an intermediate neighborhood tensor update (NTU) accounting for the nearest neighbor environment. This small environment can be contracted exactly in a parallelizable way. It provides an error measure that is Hermitian and non-negative down to machine precision. In the 2D quantum Ising model NTU is shown to yield stable unitary time evolution following a sudden quench. It also yields accurate thermal states despite correlation lengths that reach up to 20 lattice sites. The latter simulations were performed with a manifestly Hermitian purification of a thermal state. Both were performed with reduced tensors that do not include physical (and ancilla) indices. This modification naturally leads to two other schemes: a local SVD update (SVDU) and a full tensor update (FTU) being a variant of FU., Comment: 11 pages, 12 figures

We study dynamical properties of the finite-size fully connected Ising model with a transverse field at zero temperature. In a quench dynamics, we study the time period and the first critical time, which play important roles in the dynamical phase transitions, based on a dynamical order parameter and the Loschmidt rate, respectively. When all the spins are initially polarized in the direction of their mutual interaction, we show that both the time period and critical time diverge logarithmically with the system size at the dynamical critical point. When all the spins are initially in the direction of transverse field, both the time period and critical time exhibit logarithmic or power-law divergences depending on the final field strength. In the case of convergence, we provide estimates for the finite-size scaling and converged value. We also investigate the equilibrium phase transition, presenting approximate ground and first excited states away from the criticality, and compare their energy gap and bipartite and multipartite entanglements with the exact eigenstates.

We show that the observed time-reversal symmetry breaking (TRSB) of the superconducting state in Sr2RuO4 can be understood as originating from inhomogeneous strain fields near edge dislocations of the crystal. Specifically, we argue that, without strain inhomogeneities, Sr2RuO4 is a single-component, time-reversal symmetric superconductor, likely with dx2−y2 symmetry. However, due to the strong strain inhomogeneities generated by dislocations, a slowly decaying subleading pairing state contributes to the condensate in significant portions of the sample. As it phase winds around the dislocation, time-reversal symmetry is locally broken. Global phase locking and TRSB occur at a sharp Ising transition that is not accompanied by a change of the single-particle gap and yields a very small heat capacity anomaly. Our model thus explains the puzzling absence of a measurable heat capacity anomaly at the TRSB transition in strained samples and the dilute nature of the time-reversal symmetry broken state probed by muon spin rotation experiments. We propose that plastic deformations of the material may be used to manipulate the onset of broken time-reversal symmetry.

We study a generalized Blume-Capel model on the simple cubic lattice. In addition to the nearest neighbor coupling there is a next to next to nearest neighbor coupling. In order to quantify spatial anisotropy, we determine the correlation length in the high temperature phase of the model for three different directions. It turns out that the spatial anisotropy depends very little on the dilution parameter $D$ of the model and is essentially determined by the ratio of the nearest neighbor and the next to next to nearest neighbor coupling. This ratio is tuned such that the leading contribution to the spatial anisotropy is eliminated. Next we perform a finite size scaling (FSS) study to tune $D$ such that also the leading correction to scaling is eliminated. Based on this FSS study, we determine the critical exponents $\nu=0.62998(5)$ and $\eta=0.036284(40)$, which are in nice agreement with the more accurate results obtained by using the conformal bootstrap method. Furthermore we provide accurate results for fixed point values of dimensionless quantities such as the Binder cumulant and for the critical couplings. These results provide the groundwork for broader studies of universal properties of the three-dimensional Ising universality class., Comment: 45 pages, 15 figures; Changes: Estimate of $\eta$ corrected to 0.036284(40) [Before 0.036244(40) due to trivial but annoying mistake]. Text rewritten at several places. New appendix B, several Figures changed. Core results unchanged

Disorder or interactions can turn metals into insulators. One of the simplest settings to study this physics is given by the Falikov-Kimball model, which describes itinerant fermions interacting with a classical Ising background field. Despite the translational invariance of the model, inhomogenous configurations of the background field give rise to effective disorder physics which lead to a rich phase diagram in two (or more) dimensions with finite temperature charge density wave (CDW) transitions and interaction-tuned Anderson versus Mott localized phases. Here, we propose a generalised Falikov-Kimball model in one dimension with long-range interactions which shows a similarly rich phase diagram. We use an exact Markov Chain Monte Carlo method to map the phase diagram and compute the energy resolved localisation properties of the fermions. We compare the behaviour of this transitionally invariant model to an Anderson model of uncorrelated binary disorder about a background CDW field which confirms that the fermionic sector only fully localizes for very large system sizes., 7 pages, 4 figures

By developing models of increasing complexity, we show that a model without single-ion anisotropy (SIA) cannot explain the magnetic properties of ${J}_{\mathrm{eff}}=1/2 {\mathrm{Tb}}^{3+}$ moments in the orthorhombically distorted, honeycomb material ${\mathrm{Tb}}_{2}{\mathrm{Ir}}_{3}{\mathrm{Ga}}_{9}$. In four different models for the magnetization of a single honeycomb layer, the only sources of anisotropy are symmetric exchange interactions ${J}_{n\ensuremath{\alpha}\ensuremath{\beta}}={J}_{n\ensuremath{\beta}\ensuremath{\alpha}}$ along three different bonds $n$, an anisotropic $\underline{g}$ tensor, and a Dzyalloshinskii-Moriya interaction (asymmetric exchange) that produces the observed canted moment along $\mathbf{b}$. With 21 parameters, the best such model yields ${\ensuremath{\chi}}^{2}=0.065$, which is substantially smaller than ${\ensuremath{\chi}}^{2}=0.112$ obtained using a Heisenberg model containing six parameters including easy-axis anisotropy. However, models without SIA fail to reproduce the linear dependence of the magnetization with a field perpendicular to the Ising axis while predicting a saturation magnetization that is far too low. Due to the complex crystal-field environments, we argue that SIA is necessary to study low-symmetry, three-dimensional ${J}_{\mathrm{eff}}=1/2$ materials containing ${\mathrm{Tb}}^{3+}$ ions.

The quantum-critical properties of the transverse-field Ising model with algebraically decaying interactions are investigated by means of stochastic series expansion quantum Monte Carlo, on both the one-dimensional linear chain and the two-dimensional square lattice. We extract the critical exponents $\nu$ and $\beta$ as a function of the decay exponent of the long-range interactions. For ferromagnetic Ising interactions, we resolve the limiting regimes known from field theory, ranging from the nearest-neighbor Ising to the long-range Gaussian universality classes, as well as the intermediate regime with continuously varying critical exponents. In the long-range Gaussian regime, we treat the effect of dangerous irrelevant variables on finite-size scaling forms. For antiferromagnetic and therefore competing Ising interactions, the stochastic series expansion algorithm displays growing auto-correlation times leading to a reduced performance. Nevertheless, our results are consistent with the nearest-neighbor Ising universality for all investigated interaction ranges both on the linear chain and the square lattice., Comment: 11 pages, 4 figures

We present the detailed structural and magnetic properties of ${\mathrm{Li}}_{2}{\mathrm{Mn}}_{3}{\mathrm{O}}_{7}$ from powder x-ray diffraction (XRD), dc susceptibility, heat capacity, ac susceptibility, thermoremanent magnetization, magnetic memory, and exchange bias effect. Rietveld refinement of XRD data reveals that this compound has a rhombhohedral structure composed of a layered triangular lattice. Onset of spin-glass transition was confirmed by dc magnetization and ac susceptibility measurements. Dynamic scaling laws were used to analyze and classify the glassy behavior of the compound. Magnetic field dependence of irreversible temperature follows the Almeida-Thouless line, which is characteristic for an Ising spin-glass system. Fitting of the frequency-dependent freezing temperature with a power law results in $z{\ensuremath{\nu}}^{\ensuremath{'}}=4.06\ifmmode\pm\else\textpm\fi{}0.06$, which indicates the critical exponent of the sluggish spin dynamics and ${\ensuremath{\tau}}_{0}=4.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$ s is a characteristic time scale for a single spin-flip. Further evidence of cluster-glass behavior comes from the frequency dependence of the freezing temperature fitted with the Vogel-Fulcher law, which considers interaction between bigger magnetic entities. Values of fitting parameters are ${E}_{a}/{k}_{B}=27.62$ K and ${T}_{0}=9.57$ K, which confirm cluster-glass behavior. The presence of magnetic relaxation below freezing temperature and the magnetic memory effect confirms the nonequilibrium dynamics of the system through a number of metastable states. Moreover, observation of the exchange bias effect reflects the presence of intrinsic phase inhomogeneity. These results indicate that the triangular lattice causes a disordered ground state as a result of competing exchange interactions.

We discover two distinct topological pathways through which the pentagonal Cairo tiling (P5), a structural model for single-layer $A{B}_{2}$ pyrite materials, respectively transforms into a crystalline rhombus-hexagon (C46) and random rhombus-pentagon-hexagon (R456) tilings, by continuously introducing Stone-Wales (SW) topological defects. We find these topological transformations are controlled by the orientation correlations among neighboring $B\ensuremath{-}B$ bonds and exhibit a phenomenological analogy of the (anti)ferromagnetic-to-paramagnetic transition in two-state Ising systems. Unlike the SW defects in hexagonal two-dimensional (2D) materials such as graphene, which cause distortions, the defects in pentagonal 2D materials preserve the shape and symmetry of the fundamental cell of P5 tiling and are associated with a minimal energy cost, making the intermediate R456 tilings realizable metastable states at room temperature. Moreover, the intermediate structures along the two pathways are neither crystals nor quasicrystals, and yet these random tilings preserve the hyperuniformity of the P5 or C46 crystal (i.e., the infinite-wavelength normalized density fluctuations are completely suppressed) and can be viewed as 2D analogs of disordered Barlow packings in three dimensions. The resulting 2D materials possess metallike electronic properties, making them promising candidates for forming Schottky barriers with the semiconducting P5 material.

We study phase transitions in the two-dimensional Heisenberg model with the Dzyaloshinskii-Moriya interaction, the Ising anisotropy ($\eta$), and the dipolar interaction under zero and finite magnetic fields ($H$). For three typical strengths (zero, weak, and strong) of the dipolar interaction, we present the $H$-$\eta$ phase diagrams by estimating order parameters for skyrmion-lattice and helical phases and in-plane magnetization by using a Monte Carlo method with an $O(N)$ algorithm. We find in the phase diagrams three types of skyrmion-lattice phases, i.e., two square lattices and a triangular lattice, helical phases with diagonal and vertical (or horizontal) stripes, canted ferromagnetic phase and polarized ferromagnetic phase. The effect of the dipolar interaction varies the types of the skyrmion and helical phases in a complex manner. The dipolar interaction also expands the regions of the ordered phases accompanying shifts of the phase boundaries to the positive $H$ and $\eta$ directions, and causes increase of the density of skyrmions and shortening of the pitch length (stripe width) of helical structures. We discuss the details of the features of the phase transitions., Comment: 10 pages, 12 figures

Noncentrosymmetric PrAlGe is considered to be a magnetic Weyl semimetal that breaks both time-reversal and inversion symmetry, in which the Weyl nodes can be moved by magnetization. However, the magnetic interaction of this system remains poorly understood. In this work we perform a systematical investigation on magnetic properties and critical behaviors in single-crystal PrAlGe. The temperature, field, and angle dependence of magnetization reveal a strong magnetic anisotropy along the $c$ axis and absolute isotropic characteristic in the $ab$ plane. By fitting to the magnetic entropy change $[\mathrm{\ensuremath{\Delta}}{S}_{M}(T,H)]$ around the critical temperature ${T}_{C}=16$ K, critical exponents $\ensuremath{\beta}=0.136(6), \ensuremath{\gamma}=1.801(7)$, and $\ensuremath{\delta}=14.2(6)$ are obtained. Based on the obtained critical exponents, $\mathrm{\ensuremath{\Delta}}{S}_{M}(T,H)$ and $M(T,H)$ curves are scaled into universality curves under the framework of universality principle. The critical behaviors and exponents suggest that the magnetic interaction in PrAlGe is of a two-dimensional Ising type, revealing a uniaxial magnetic interaction along the $c$ axis. However, the ordering moments are tilted from the $c$ axis, which causes antiferromagnetism in the $ab$ plane. The clarification of the magnetic interaction and magnetic structure is of significance for unveiling its interplay with topological properties in this system.

The interplay of interactions and disorder in low-dimensional superconductors supports the formation of multiple quantum phases, as possible instabilities of the Superconductor-Insulator Transition (SIT) at a singular quantum critical point. We explore a one-dimensional model which exhibits such variety of phases in the strongly quantum fluctuations regime. Specifically, we study the effect of weak disorder on a two-leg Josephson ladder with comparable Josephson and charging energies ($E_J$~$E_C$). An additional key feature of our model is the requirement of perfect $\mathbb{Z}_2$-symmetry, respected by all parameters including the disorder. Using a perturbative renormalization-group (RG) analysis, we derive the phase diagram and identify at least one intermediate phase between a full-fledged superconductor and a disorder-dominated insulator. Most prominently, for repulsive interactions on the rungs we identify two distinct mixed phases: in both of them the longitudinal charge mode is a gapless superconductor, however one phase exhibits a dipolar charge density order on the rungs, while the other is disordered. This latter phase is characterized by coexisting superconducting (phase-locked) and charge-ordered rungs, and encompasses the potential of evolving into a Griffith's phase characteristic of the random-field Ising model in the strong disorder limit., Comment: 18 pages, 3 figures

Random measurements have been shown to induce a phase transition in an extended quantum system evolving under chaotic unitary dynamics, when the strength of measurements exceeds a threshold value. Below this threshold, a steady state with a sub-thermal volume law entanglement emerges, which is resistant to the disentangling action of measurements, suggesting a connection to quantum error-correcting codes. Here we quantify these notions by identifying a universal, subleading logarithmic contribution to the volume law entanglement entropy: $S^{(2)}(A)=\kappa L_A+\frac{3}{2}\log L_A$ which bounds the mutual information between a qudit inside region $A$ and the rest of the system. Specifically, we find the power law decay of the mutual information $I(\{x\}:\bar{A})\propto x^{-3/2}$ with distance $x$ from the region's boundary, which implies that measuring a qudit deep inside $A$ will have negligible effect on the entanglement of $A$. We obtain these results by mapping the entanglement dynamics to the imaginary time evolution of an Ising model, to which we can apply field-theoretic and matrix-product-state techniques. Finally, exploiting the error-correction viewpoint, we assume that the volume-law state is an encoding of a Page state in a quantum error-correcting code to obtain a bound on the critical measurement strength $p_{c}$ as a function of the qudit dimension $d$: $p_{c}\log[(d^{2}-1)({p_{c}^{-1}-1})]\le \log[(1-p_{c})d]$. The bound is saturated at $p_c(d\rightarrow\infty)=1/2$ and provides a reasonable estimate for the qubit transition: $p_c(d=2) \le 0.1893$., Comment: 26 pages, 10 figures

A quantum spin-$\frac{1}{2}$ antiferromagnetic Heisenberg trimerized chain with strong intradimer and weak monomer-dimer coupling constants is studied using the many-body perturbation expansion, which is developed from the exactly solved spin-$\frac{1}{2}$ Ising-Heisenberg diamond chain preserving correlations between all interacting spins of the trimerized chain. It is evidenced that this perturbation approach is superior with respect to the standard perturbation scheme developed from a set of noninteracting spin monomers and dimers, and its accuracy even coincides, up to a moderate ratio of the coupling constants, with state-of-the-art numerical techniques. The Heisenberg trimerized chain shows the intermediate one-third plateau, which was also observed in the magnetization curve of the polymeric compound ${\mathrm{Cu}}_{3}{({\mathrm{P}}_{2}{\mathrm{O}}_{6}\mathrm{OH})}_{2}$ affording its experimental realization. Within the modified strong-coupling method we have obtained the effective Hamiltonians for the magnetic field range from zero to the one-third plateau, and from the one-third plateau to the saturation magnetization. The unconventional second-order perturbation theory provides extremely accurate results for both critical fields of the intermediate one-third plateau up to a moderate ratio of the coupling constants as convincingly evidenced through a comparison with numerical density-matrix renormalization group data. It is shown that the derived effective Hamiltonian also provides at low enough temperatures sufficiently accurate results for magnetization curves and thermodynamic properties as corroborated through a comparison with quantum Monte Carlo simulations. Using the results for the effective Hamiltonian, we additionally suggest a straightforward procedure for finding the microscopic parameters of one-dimensional trimerized magnetic compounds with strong intradimer and weak monomer-dimer couplings. We found the refined values for the coupling constants of ${\mathrm{Cu}}_{3}{({\mathrm{P}}_{2}{\mathrm{O}}_{6}\mathrm{OH})}_{2}$ by matching the theoretical results with the available experimental data for the magnetization and magnetic susceptibility in a wide range of temperatures and magnetic fields.

Building on quantum Monte Carlo simulations, we study the phase diagram of a one-parameter Hamiltonian interpolating between trivial and topological Ising paramagnets in two dimensions, which are dual to the toric code and the double semion. We discover an intermediate phase with stripe order which spontaneously breaks the protecting Ising symmetry. Remarkably, we find evidence that this intervening phase is gapless due to the incommensurability of the stripe pattern and that it is dual to a $U(1)$ gauge theory exhibiting Cantor deconfinement., Comment: 7 pages, 4 figures, supplemental material included (7 pages, 10 figures)

We report neutron scattering and thermodynamic measurements of the quasi-one-dimensional Ising magnet $\mathrm{Sr}{\mathrm{Co}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$ under transverse fields along the tetragonal $a/b$ direction. Our experiments reveal a N\'eel-type magnetic order in zero field, which successively changes into a coplanar antiferromagnetic order in applied fields. The detailed evolution of the noncollinear magnetic order in magnetic fields can be understood by considering the competition between the intrinsic magnetic interactions, uniform transverse field, and effective staggered field. Moreover, a series of gapped discrete confined-spinon modes are observed in zero field. With increasing field, the spin gap progressively softens, reaching a minimum value at $\ensuremath{\sim}7\phantom{\rule{0.16em}{0ex}}\mathrm{T}$ where the N\'eel-type magnetic ordering moment is completely suppressed. This corresponds to the three-dimensional quantum critical point (QCP). The implications for multiple QCPs in this class of materials are discussed.

Some features of the phase diagram of the two-dimensional axial next-nearest-neighbor Ising model have long been debated. The extended structural correlations and long relaxation times associated with its Kosterlitz-Thouless phase indeed result in analytical and numerical treatments making contradictory predictions. Here, we introduce a numerical transfer matrix approach that bypasses these problems and thus clears up various ambiguities. In particular, we confirm the transition temperatures and the order of the transition to the floating incommensurate phase. Our approach motivates considering transfer matrices for solving long-standing problems in related models.

We provide and analyze a periodic Anderson model for studying magnetism and superconductivity in UTe$_2$, a recently-discovered candidate for a topological spin-triplet superconductor. The 24-band tight-binding model reproduces the band structure obtained from a DFT$+U$ calculation consistent with an angle-resolved photoemission spectroscopy. The Coulomb interaction of $f$-electrons enhances Ising ferromagnetic fluctuation along the $a$-axis and stabilizes spin-triplet superconductivity of either $B_{3u}$ or $A_{u}$ symmetry. When effects of pressure are taken into account in hopping integrals, the magnetic fluctuation changes to antiferromagnetic one, and accordingly spin-singlet superconductivity of $A_{g}$ symmetry is stabilized. Based on the results, we propose pressure-temperature and magnetic field-temperature phase diagrams revealing multiple superconducting phases as well as an antiferromagnetic phase. In particular, a mixed-parity superconducting state with spontaneous inversion symmetry breaking is predicted., Comment: 9 pages, 9 figures

We report results of magnetization and $^{31}\mathrm{P}$ NMR measurements under high pressure up to 6.4 GPa on ${\mathrm{RbMoOPO}}_{4}\mathrm{Cl}$, which is a frustrated square-lattice antiferromagnet with competing nearest-neighbor and next-nearest-neighbor interactions. Anomalies in the pressure dependencies of the NMR shift and the transferred hyperfine coupling constants indicate a structural phase transition at 2.6 GPa, which is likely to break mirror symmetry and triggers significant change of the exchange interactions. In fact, the NMR spectra in magnetically ordered states reveal a change from the columnar antiferromagnetic (CAF) order below 3.3 GPa to the N\'eel antiferromagnetic (NAF) order above 3.9 GPa. The spin lattice relaxation rate $1/{T}_{1}$ also indicates a change of dominant magnetic fluctuations from CAF-type to NAF-type with pressure. Although the NMR spectra in the intermediate pressure region between 3.3 and 3.9 GPa show coexistence of the CAF and NAF phases, a certain component of $1/{T}_{1}$ shows paramagnetic behavior with persistent spin fluctuations, leaving a possibility for a quantum disordered phase. The easy-plane anisotropy of spin fluctuations with unusual nonmonotonic temperature dependence at ambient pressure gets reversed to the Ising anisotropy at high pressures. This unexpected anisotropic behavior for a spin 1/2 system may be ascribed to the strong spin-orbit coupling of Mo-$4d$ electrons.

Two-dimensional (2D) materials possessing intrinsic superconductivity with high transition temperature and unconventional pairing are highly desired, but their realizations are few and far between. Recently, ${\mathrm{W}}_{2}{\mathrm{N}}_{3}$ nanosheets down to three layers were successfully prepared [Yu et al., Adv. Mater. 30, 1805655 (2018)]. By performing solid ab initio calculations based on the anisotropic Migdal-Eliashberg theory, we predict that monolayer ${\mathrm{W}}_{2}{\mathrm{N}}_{3}$ is an unexplored intrinsic (without the assistance of external gating, strain, or a special substrate) 2D superconductor with large electron-phonon coupling (EPC) and high critical temperature ${T}_{c}=38\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ accompanied with a single and broad superconducting gap \ensuremath{\sim}7.5 meV. The ratio between the gap and the critical temperature is much larger than the value derived from Bardeen-Cooper-Schrieffer (BCS) theory, further confirming the strong coupling feature of monolayer ${\mathrm{W}}_{2}{\mathrm{N}}_{3}$. The extremely strong EPC originates from the large deformation potential of low-frequency acoustic phonons rather than Fermi-surface nesting. Due to the partially filled $d$ orbitals, electron-electron correlation leads to remarkable enhancement of EPC based on frozen phonon analysis. Here, ${T}_{c}$ can be further enhanced via hydrogen passivation based on the McMillian-Allen-Dynes formula. In addition, the symmetry-restricted spin-orbit coupling (SOC) brings forth exotic type-I Ising pairing whose in-plane upper critical field is far beyond the Pauli paramagnetic limit. Our predictions provide a fascinating and highly feasible platform for realizing high-temperature 2D superconductivity and studying the interplay between electron-phonon coupling, electron-electron correlation, and SOCs.

We determined the crystal structure of ${\mathrm{Cs}}_{2}{\mathrm{LiRuCl}}_{6}$, which was synthesized in this work, and investigated its magnetic properties. ${\mathrm{Cs}}_{2}{\mathrm{LiRuCl}}_{6}$ has a hexagonal structure composed of linear chains of face-sharing ${\mathrm{RuCl}}_{6}$ and ${\mathrm{LiCl}}_{6}$ octahedra. In two thirds of the structural chains, ${\mathrm{Ru}}^{3+}$ and ${\mathrm{Li}}^{+}$ sites are almost ordered, while in the other chains their sites are disordered. This situation is analogous to the ground state of the antiferromagnetic Ising model on a triangular lattice. Using electron paramagnetic resonance, we evaluated the $g$ factors of ${\mathrm{Ru}}^{3+}$ with effective spin-$\frac{1}{2}$ as ${g}_{c}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}2.72$ and ${g}_{ab}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1.50$ for magnetic fields $H$ parallel and perpendicular to the $c$ axis, respectively. Magnetization curves for $H\ensuremath{\parallel}c$ and $H\phantom{\rule{0.16em}{0ex}}\ensuremath{\perp}\phantom{\rule{0.16em}{0ex}}c$ are highly anisotropic. However, these magnetization curves approximately coincide when normalized by the $g$ factors. It was found from the magnetization and specific heat results that ${\mathrm{Cs}}_{2}{\mathrm{LiRuCl}}_{6}$ can be described as a coupled one-dimensional $S\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1/2$ Heisenberg-like antiferromagnet with $J/{k}_{\mathrm{B}}\phantom{\rule{0.16em}{0ex}}\ensuremath{\simeq}\phantom{\rule{0.16em}{0ex}}3.7$ K. Three-dimensional ordering occurs at ${T}_{\mathrm{N}}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}0.48$ K. A magnetic phase diagram for $H\ensuremath{\parallel}c$ is also presented.

Magnetic frustration can produce exotic spin configurations and dynamics. Here, the authors explore the seemingly simple competition between ferromagnetic nearest- and antiferromagnetic next-nearest neighbor interactions along Ising spin chains, as realized in the magnetoelectric compound Lu${}_{2}$CoMnO${}_{6}$. Commonly, this situation is described by the axial next-nearest-neighbor Ising model. For the first time, the authors measure and calculate its correlated magnetic dynamics, resulting from a characteristic fractal set of first-order phase boundaries. Experiments and Monte Carlo simulations reveal a dynamics slowdown while approaching the phase transition regime.

The two-dimensional infinite projected entangled pair state tensor network is evolved in imaginary time with the full update (FU) algorithm to simulate the Shastry-Sutherland model in a magnetic field at finite temperature directly in the thermodynamic limit. We focus on the phase transition into the $m=\frac{1}{2}$ magnetization plateau, which was observed in experiments on ${\mathrm{SrCu}}_{2}{({\mathrm{BO}}_{3})}_{2}$. For the largest simulated bond dimension, the early evolution in the high-temperature regime is simulated with the simple update (SU) scheme and then, as the correlation length increases, continued with the FU scheme towards the critical regime. We apply a small symmetry-breaking bias field and then extrapolate towards zero bias using a simple scaling theory in the bias field. The combined SU $+$ FU scheme provides an accurate estimate of the critical temperature, even though the results could not be fully converged in the bond dimension in the vicinity of the transition. The critical temperature estimate is improved with a generalized scaling theory that combines two divergent length scales: One due to the bias, and the other due to the finite bond dimension. The obtained results are consistent with the transition being in the universality class of the two-dimensional classical Ising model. The estimated critical temperature is 3.5(2) K, which is well above the temperature 2.1 K used in the experiments.

A tensor network renormalization algorithm with global optimization based on the corner transfer matrix is proposed. Since the environment is updated by the corner transfer matrix renormalization group method, the forward-backward iteration is unnecessary, which is a time-consuming part of other methods with global optimization. In addition, a further approximation reducing the order of the computational cost of contraction for the calculation of the coarse-grained tensor is proposed. The computational time of our algorithm in two dimensions scales as the sixth power of the bond dimension while the higher-order tensor renormalization group and the higher-order second renormalization group methods have the seventh power. We perform benchmark calculations in the Ising model on the square lattice and show that the time-to-solution of the proposed algorithm is faster than that of other methods., 6 pages, 9 figures

Control of quantum coherence in a many-body system is one of the key issues in modern condensed matter, and conventional wisdom is that lattice vibration is an innate source of decoherence. Much research has been conducted to eliminate lattice effects. Challenging this wisdom, we show that lattice vibration may not be a decoherence source but an impetus of a novel coherent quantum many-body state. We demonstrate the possibility by studying the transverse-field Ising model on a chain with renormalization group and density-matrix renormalization group methods and theoretically discover a stable $\mathcal{N}=1$ supersymmetric quantum criticality with central charge $c=3/2$. Thus, we propose an Ising spin chain with strong spin-lattice coupling as a candidate to observe supersymmetry. Generic precursor conditions of novel quantum criticality are obtained by generalizing the Larkin-Pikin criterion of thermal transitions. Our work provides the perspective that lattice vibration may be a knob for exotic quantum many-body states.

We develop a ``self-avoiding walks'' approximation and use it to calculate the spectral function of a single hole introduced into a two-dimensional square lattice Ising antiferromagnet. The local spectral function obtained qualitatively agrees with the exact diagonalization result and is largely incoherent. Such a result stays in contrast with the spectrum obtained on a Bethe lattice, which consists of the well-separated quasiparticle-like peaks and stems from the motion of a hole in an effective linear potential. We determine that this onset of the incoherent spectrum on a square lattice (i) is not triggered by the so-called Trugman loops but (ii) originates in the warping of the linear potential by the interactions between magnons created along the tangential paths of the moving hole.

The one-dimensional transverse field Ising model with boundary fields is studied analytically and numerically. The phase diagram in the ordered state is obtained. We find that there exist two types of phase transitions. The longitudinal boundary fields applied to the left and right ends of the Ising chain are ${h}_{L}$ and ${h}_{R}$, respectively. For $|{h}_{L}|,|{h}_{R}|l\sqrt{1\ensuremath{-}g}$, where $g$ is the transverse field and ${h}_{L}{h}_{R}l0$, a first-order phase transition occurs with the changing ${h}_{L}$ or ${h}_{R}$. The energy gap and boundary magnetization are solved exactly. The analytical expressions of the finite-size scaling for the first-order phase transition are obtained. For $|{h}_{R}|g\sqrt{1\ensuremath{-}g}$ and ${h}_{L}{h}_{R}l0$, there exists a continuous phase transition with changing ${h}_{L}$ and vice versa. This transition is identified as a quantum wetting transition. The singularity of the boundary magnetization in this phase transition is explicitly shown. A simple computational procedure with high accuracy and efficiency is proposed to calculate the magnetization. The magnetization profile, correlation functions, and wetting layer thickness are studied numerically.

Based on the matrix product state (MPS) formalism, we introduce an ansatz for capturing excited states in finite systems with open boundary conditions, providing a very efficient method for computing, e.g., the spectral gap of quantum spin chains. This method can be straightforwardly implemented on top of an existing density-matrix renormalization group or MPS ground-state code. Although this approach is built on open-boundary MPSs, we also apply it to systems with periodic boundary conditions. Despite the explicit breaking of translation symmetry by the MPS representation, we show that momentum emerges as a good quantum number and can be exploited for labeling excitations on top of MPS ground states. We apply our method to the critical Ising chain on a ring and the classical Potts model on a cylinder. Finally, we apply the same idea to compute excitation spectra for 2D quantum systems on infinite cylinders. Again, despite the explicit breaking of translation symmetry in the periodic direction, we recover momentum as a good quantum number for labeling excitations. We apply this method to the 2D transverse-field Ising model and the half-filled Hubbard model; for the latter, we obtain accurate results for, e.g., the hole dispersion for cylinder circumferences up to eight sites.

Magnetization isotherms of the $5f$-electron ferromagnets URhGa, UCoGa, and $\mathrm{U}{\mathrm{Co}}_{0.98}{\mathrm{Ru}}_{0.02}\mathrm{Al}$ were measured at temperatures in the vicinity of their Curie temperature to investigate the critical behavior near the ferromagnetic phase transition. These compounds adopt the layered hexagonal ZrNiAl-type structure and exhibit huge uniaxial magnetocrystalline anisotropy. The critical \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\delta} exponents were determined by analyzing Arrott-Noakes plots, Kouvel-Fisher plots, critical isotherms, scaling theory, and Widom scaling relations. The values obtained for URhGa and UCoGa can be explained by the results of the renormalization group theory for a two-dimensional (2D) Ising system with long-range interactions similar to URhAl reported by other investigators. On the other hand, the critical exponents determined for $\mathrm{U}{\mathrm{Co}}_{0.98}{\mathrm{Ru}}_{0.02}\mathrm{Al}$ are characteristic of a three-dimensional (3D) Ising ferromagnet with short-range interactions suggested in previous studies also for the itinerant $5f$-electron paramagnet UCoAl situated near a ferromagnetic transition. The change from the 2D to the 3D Ising system is related to the gradual delocalization of $5f$ electrons in the series of the URhGa, URhAl, and UCoGa to $\mathrm{U}{\mathrm{Co}}_{0.98}{\mathrm{Ru}}_{0.02}\mathrm{Al}$ and UCoAl compounds and appears close to the strongly itinerant nonmagnetic limit. This indicates possible new phenomena that may be induced by the change of dimensionality in the vicinity of the quantum critical point.

The few-layer transition metal dichalcogenides (TMDs) have been recently suggested as a platform for controlled unconventional superconductivity. We study the manifestations of unconventional triplet pairing in the density of states of a disordered TMD based monolayer. The conventional singlet pairing attraction is assumed to be the dominant pairing interaction. We map the phase diagrams of disordered Ising superconductors in the plane of temperature and the in-plane magnetic field. The latter suppresses singlet and promote triplet correlations. The triplet order parameters of a trivial (non-trivial) symmetry compete (cooperate) with the singlet order parameter which gives rise to a rich phase diagram. We locate the model-dependent phase boundaries and compute the order parameters in each of the distinct phases. With this information, we obtain the density of states by solving the Gorkov equation. The triplet components of the order parameters may change an apparent width of the density of states by significantly increasing the critical field. The triplet components of the order parameters lead to the density of states broadening significantly exceeding the broadening induced by magnetic field and disorder in the singlet superconductor., 20 pages, 9 figures

We demonstrate that two toric code layers on the square lattice coupled by an Ising interaction display two distinct phases with intrinsic topological order. The second-order quantum phase transition between the weakly-coupled $\mathbb{Z}_2\times\mathbb{Z}_2$ and the strongly-coupled $\mathbb{Z}_2$ topological order can be described by the condensation of bosonic quasiparticles from both sides and belongs to the 3d Ising$^*$ universality class. This can be shown by an exact duality transformation to the transverse-field Ising model on the square lattice, which builds on the existence of an extensive number of local $\mathbb{Z}_2$ conserved parities. These conserved quantities correspond to the product of two adjacent star operators on different layers. Notably, we show that the low-energy effective model derived about the limit of large Ising coupling is given by an effective single-layer toric code in terms of the conserved quantities of the Ising toric code bilayer. The two topological phases are further characterized by the topological entanglement entropy which serves as a non-local order parameter., 6 pages, 3 figures

We study the spin-$1/2$ ferromagnetic Heisenberg-Kitaev-$\Gamma$ model in the anisotropic (Toric code) limit to reveal the nature of the quantum phase transition between the gapped $Z_2$ quantum spin liquid and a spin ordered phase (driven by Heisenberg interactions) as well as a trivial paramagnet (driven by pseudo-dipolar interactions, $\Gamma$). The transitions are obtained by a simultaneous condensation of the Ising electric and magnetic charges-- the fractionalized excitations of the $Z_2$ quantum spin liquid. Both these transitions can be continuous and are examples of deconfined quantum critical points. Crucial to our calculations are the symmetry implementations on the soft electric and magnetic modes that become critical. In particular, we find strong constraints on the structure of the critical theory arising from time reversal and lattice translation symmetries with the latter acting as an anyon permutation symmetry that endows the critical theory with a manifestly self-dual structure. We find that the transition between the quantum spin liquid and the spin-ordered phase belongs to a self-dual modified Abelian Higgs field theory while that between the spin liquid and the trivial paramagnet belongs to a self-dual $Z_2$ gauge theory. We also study the effect of an external Zeeman field to show an interesting similarity between the polarised paramagnet obtained due to the Zeeman field and the trivial paramagnet driven the pseudo-dipolar interactions. Interestingly, both the spin liquid and the spin ordered phases have easily identifiable counterparts in the isotropic limit and the present calculations may shed insights into the corresponding transitions in the material relevant isotropic limit., Comment: 28 pages, 14 figures

Driving a homogeneous system across a quantum phase transition in a quench-time $\tau_Q$ generates excitations on wavelengths longer than the Kibble-Zurek (KZ) length $\hat\xi\propto\tau_Q^{\nu/(1+z\nu)}$ within the KZ time window $\hat t\propto\tau_Q^{z\nu/(1+z\nu)}$, where $z$ and $\nu$ are the critical exponents. Quenches designed with local time-dependent inhomogeneity can introduce a gap in the spectrum. For a variety of setups with short-range interactions, they have been shown to suppress excitations if the spatial velocity of the inhomogenous front is below the characteristic KZ velocity $\hat v \propto \hat\xi/\hat t$. Ising-like models with long-range interactions can have no sonic horizon, spreading information instantaneously across the system. Usually, this should imply that inhomogenous transitions will render the dynamics adiabatic regardless of the front velocity. However, we show that we get an adiabatic transition with no defects only when the inhomogeneous front moves slower than a characteristic crossover velocity $\tilde v \propto \theta^{(z-1)\nu/(1+\nu)}$, where $\theta$ is the slope of the inhomogeneous front at the critical point. The existence of this crossover velocity and adiabaticity of the model is a consequence of the energy gap in the quasiparticle spectrum that is opened by the inhomogeneity. This effect can be employed for efficient adiabatic quantum state preparation in systems with long-range interactions., Comment: 12 pages, 10 figures (closed to published version)