Spin Glass Systems Research Articles

The matrix product approximation for the dynamic cavity method

Stochastic dynamics of classical degrees of freedom, defined on vertices of locally tree-like graphs, can be studied in the framework of the dynamic cavity method which is exact for tree graphs. Such models correspond for example to spin-glass systems, Boolean networks, neural networks, and other technical, biological, and social networks. The central objects in the cavity method are edge messages—conditional probabilities of two vertex variable trajectories. In this paper, we discuss a rather pedagogical derivation for the dynamic cavity method, give a detailed account of the novel matrix product edge message (MPEM) algorithm for the solution of the dynamic cavity equation as introduced in Barthel et al (2018 Phys. Rev. E 97 010104 (R)), and present optimizations and extensions. Matrix product approximations of the edge messages are constructed recursively in an iteration over time. Computation costs and precision can be tuned by controlling the matrix dimensions of the MPEM in truncations. Without truncations, the dynamics is exact. Data for Glauber–Ising dynamics shows a linear growth of computation costs in time. In contrast to Monte Carlo simulations, the approach has a much better error scaling. Hence, it gives for example access to low probability events and decaying observables like temporal correlations. We discuss optimized truncation schemes and an extension that allows to capture models which have a continuum time limit.

Intrinsic chiral-spin glass and spin glass transitions in compacted Ni5Al3/NiO core/shell nanoparticles

We report the first synthesis of the intermetallic compound ${\mathrm{Ni}}_{5}{\mathrm{Al}}_{3}$ in the nanocrystalline form. At the nanometer length scale, a stable ${\mathrm{Ni}}_{5}{\mathrm{Al}}_{3}$ compound (core) does not form without a NiO shell. Detailed structural and microstructural characterization as well as the compositional analysis indicate the presence of ${\mathrm{Ni}}_{5}{\mathrm{Al}}_{3}/\mathrm{NiO}$ core/shell nanoparticles and log-normal crystallite size distribution with mean size $d\phantom{\rule{4pt}{0ex}}\ensuremath{\simeq}$ 6 nm. ``Zero-field'' (dc magnetic field, $H=0$) linear (${\ensuremath{\chi}}_{1}$) and nonlinear (${\ensuremath{\chi}}_{n}$ with $n=2,3,4,5$) ac-magnetic susceptibilities have been measured as functions of temperature at various ac driving field amplitudes over three decades of frequency on the nanocrystalline samples ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$ of composition ${\mathrm{Ni}}_{5+x}{\mathrm{Al}}_{3\ensuremath{-}x}/\mathrm{NiO}$ ($x=0.014$) and ${\mathrm{Ni}}_{5\ensuremath{-}x}{\mathrm{Al}}_{3+x}/\mathrm{NiO}$ ($x=0.038$), respectively. ${\ensuremath{\chi}}_{1}(T)$ and ${\ensuremath{\chi}}_{n}(T)$ are shown to provide conclusive evidence for the existence of two spin glass (SG) thermodynamic phase transitions: one at ${T}_{i}(H=0)\ensuremath{\simeq}138$ K and the other at a lower temperature ${T}_{p}(H=0)\ensuremath{\simeq}108$ K in both ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$. Linear ac-susceptibility in superposed dc magnetic fields demonstrates that the thermodynamic nature of these transitions is preserved in finite fields and the $H\ensuremath{-}T$ phase diagrams for the samples ${\mathrm{S}}_{1}$ and ${\mathrm{S}}_{2}$ conform very well only with the $H\ensuremath{-}T$ phase diagram predicted by the chirality-driven spin glass (SG) ordering model for a three-dimensional nearest-neighbor Heisenberg SG system with weak random anisotropy. A detailed comparison between theory and experiment permits us to unambiguously identify various ``zero-field'' and ``in-field'' SG phase transitions as (i) the simultaneous paramagnetic (PM): chiral-spin glass (CG) and PM-SG phase transitions at ${T}_{i}(H)$ and (ii) the replica symmetry-breaking SG transition at ${T}_{p}(H)$.

LiZn2V3O8: a new geometrically frustrated cluster spin-glass

We have investigated the structural and magnetic properties of a new cubic spinel LiZn2V3O8 (LZVO) through x-ray diffraction, dc and ac susceptibility, magnetic relaxation, aging, memory effect, heat capacity and 7Li nuclear magnetic resonance (NMR) measurements. A Curie–Weiss fit of the dc susceptibility () yields a Curie–Weiss temperature K. This suggests strong antiferromagnetic (AFM) interactions among the magnetic vanadium ions. The dc and ac susceptibility data indicate the spin-glass behavior below a freezing temperature Tf 3 K. The frequency dependence of the Tf is characterized by the Vogel–Fulcher law and critical dynamic scaling behavior or power law. From both fitting, we obtained the value of the characteristic angular frequency 3.56 106 Hz, the dynamic exponent 2.65, and the critical time constant 1.82 10−6 s, which falls in the conventional range for typical cluster spin-glass (CSG) systems. The value of relative shift in freezing temperature 0.039 supports a CSG ground states. We also found aging phenomena and memory effects in LZVO. The asymmetric response of the magnetic relaxation below Tf supports the hierarchical model. Heat capacity data show no long-range or short-range ordering down to 2 K. Only about 25% magnetic entropy change signifies the presence of strong frustration in the system. The 7Li NMR spectra show a shift and broadening with decreasing temperature. The spin-lattice and spin–spin relaxation rates show anomalies due to spin freezing around 3 K as the bulk magnetization.

Dependence of Energy Product to the Shell Thickness in FePt/Fe3O4 Core/Shell Nanoparticles

A new model is presented to calculate energy product variation for FePt/Fe3O4 core/shell in the terms of Fe3O4 shell thickness. The spins at the core/shell interface and some parts of the core volume have a random orientation due to the presence of chemical disorder created by the experimental procedures. The mentioned random effects based on the chemical disorder and intermixing of core and shell produces random interactions, which according to the presented model can behave similar to the spin glass system. Due to the existence of a competition between exchange and anisotropy energies, the coercivity field contain two parts: In the first part, the effects of exchange interactions will dominate the effects of anisotropy. As a result, this section will include low anisotropy and frustrated spins, which is coupled to neighbor particles due to the strong exchange interactions. In the second part, which is the volume part of matter, the anisotropy is higher; in this case, the spins are frozen and remain fixed so that the neighbor particles do not have much effect on them. Since the nanoscale particles have a significant value of surface to volume ratio, there will be a competition between these two parts. Energy product variations for FePt/Fe3O4 core/shell increases when coherent rotation is dominant in the core/shell system. By increasing shell thickness, coherent rotation of spins disappears and the magnetization rotation will not be uniform, which leads to a decrease in energy product. To increase the maximum product energy, we need to increase the value of coercivity which is achieved by increasing the core thickness. If we can increase the magnetocrystalline anisotropy constant of the core, then the coercivity and thus the maximum energy product will increase.

Possible glass-like random singlet magnetic state in 1T-TaS2

Two-dimensional layered transition-metal-dichalcogenide compound 1T-TaS2 shows the rare coexistence of charge density wave (CDW) and electron correlation driven Mott transition. In addition, atomic-cluster spins on the triangular lattice of the CDW state of 1T-TaS2 give rise to the possibility of the exotic spin-singlet state in which quantum fluctuations of spins are strong enough to prevent any long range magnetic ordering down to the temperature absolute zero (0 K). We present here the evidences of a glass-like random singlet magnetic state in 1T-TaS2 at low temperatures through a study of temperature and time dependence of magnetization. Comparing the experimental results with a representative canonical spin-glass system Au(1.8%Mn), we show that this glass-like state is distinctly different from the well established canonical spin-glass state.

Towards artificial Ising spin glasses: Thermal ordering in randomized arrays of Ising-type nanomagnets

We explore the thermodynamics in two-dimensional arrays consisting of Ising-type nanomagnets lithographically arranged onto random sites and angular orientations. Introducing these basic spin-glass ingredients, we study the characteristic features of the low-energy states achieved, following thermal-annealing protocols. From direct visualization of real-time dynamics, we record relaxation timescales together with magnetic susceptibility variations over temperature, revealing trends towards short-range order as randomness is increased, but falling short of pure spin-glass behavior. Our work provides a route towards the realization of artificial Ising spin-glass systems.

Revealing magnetic ground state of a layered cathode material by muon spin relaxation and neutron scattering experiments

Although the majority of studies on battery materials have focused on enhancing their electrochemical performance, the understanding of their magnetic and atomic structures as well as the intimate connections between these structures and properties is significant. Muon spin relaxation (μSR) spectroscopy and neutron scattering techniques at low temperature have been used to explore the nature of the magnetic state of Li(Ni0.4Mn0.4Co0.2)O2 cathode materials. Interestingly, the μSR experiment reveals that the spins of Ni/Mn/Co ions turn glassy at a low temperature of 19 K, while the unprecedented spin dynamics survives until the base temperature of 5.6 K. Moreover, the complementary neutron scattering measurements show magnetic diffuse scattering signals in the form of a broad peak at Q = 1.6 Å−1 attributed to short-range spin correlation which establishes below ∼120 K. Upon cooling to as low as 480 mK, long-range magnetic order still could not be established. The inelastic neutron scattering intensities demonstrate clearly the existence of dynamic spin fluctuations. In contrast to the conventional spin glass system, the coexistence of both dynamic and static components in the ground state suggests an unconventional spin glass state in magnetically frustrated Li(Ni0.4Mn0.4Co0.2)O2 cathode materials for Li-ion batteries.

Market basket analysis by solving the inverse Ising problem: Discovering pairwise interaction strengths among products

Large datasets containing the purchasing information of thousands of consumers are difficult to analyze because the possible number of different combinations of products is huge. Thus, market baskets analysis to obtain useful information and find interesting pattern of buying behavior could be a daunting task. Based on the maximum entropy principle, we build a probabilistic model that explains the probability of occurrence of market baskets which is equivalent to Ising models. This type of model allows us to understand and to explore the functional interactions among products that make up the market offer. Additionally, the parameters of the model inferred using Boltzmann learning, allow us to suggest that the buying behavior is very similar to the spin-glass physical system. Moreover, we show that the resulting parameters of the model could be useful to describe the hierarchical structure of the system which leads to interesting information about the different market baskets.

Cluster-glass dynamics of the Griffiths phase in Tb5−xLaxSi2Ge2

The static magnetization and dynamic susceptibility responses of the cluster system within a Griffiths phase of the magnetocaloric compound ${\mathrm{Tb}}_{5\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{Si}}_{2}{\mathrm{Ge}}_{2}$ ($x=0.075$) have been investigated. A novel cluster-glass state within the Griffiths phase is formed at a characteristic freezing temperature where short-range ferromagnetic correlations set in the paramagnetic regime. Ferromagneticlike correlations are built up at around 155 K, which suddenly become frozen at a lower temperature $\ensuremath{\sim}140\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, thus in analogy with a reentrant spin glass behavior. The ac susceptibility near the freezing temperature follows a critical slowing down process characterized by ${\ensuremath{\tau}}_{0}={10}^{\ensuremath{-}13}s$ and dynamic exponents $\mathrm{z}\ensuremath{\nu}\ensuremath{\sim}6$ and $\ensuremath{\beta}\ensuremath{\sim}0.4$, similar to well-known spin glass systems. The nonlinear ac susceptibility analysis shows clearly the existence of a transition associated to the reentrant behavior. The origin of the intermediate cluster-glass phase inside the Griffiths phase is proposed to be the result of a combination of short-ranged RKKY intralayer positive exchange interactions between rare-earth ${\mathrm{Tb}}^{3+}$ ions and antiferromagnetic exchange between adjacent interlayers involving Si and Ge atoms in connection to the ${\mathrm{Tb}}^{3+}$ atoms.

(Fe,Sn)4N alloy as a model spin-glass system with short-range competing interactions on a nonfrustrated simple cubic lattice

The origin of the spin-glass state in (Fe,Sn)4N alloys is studied on the basis of a Heisenberg Hamiltonian with parameters derived from first principles within the magnetic force theorem applied in the framework of the disordered local moments method and local spin-density approximation. We show that in the alloy concentration range where the spin-glass state is stable only one Fe sublattice is intrinsically magnetic and the interatomic exchange magnetic interactions are essentially short ranged due to effects of chemical and magnetic disorder. The magnetic Fe atoms with well-localized spin moments are randomly distributed over the nongeometrically frustrated simple cubic lattice. The magnetic frustration, which generally is believed to be an essential ingredient of the spin-glass state formation condition, may occur only due to the competition of the two nearest-neighbor interactions. We thus argue that (Fe,Sn)4N is a rare example of a spin-glass system where the mechanism of spin-glass state formation might be studied in the framework of the minimal random-site model on a simple cubic lattice with competing interactions, while the effects of the geometrical frustration can be excluded.

Revelation of spin glass behavior in Ru doped MnNiGe: experiment and theory

We report on the nature of the magnetism in Ru substituted MnNiGe using the combined results of x-ray diffraction, dc-magnetization, ac-susceptibility and ab initio calculations. Mn0.7Ru0.3NiGe crystallizes in Ni2In-type hexagonal structure (P63/mmc) at room temperature with lattice parameters a = b = 4.099 and c = 5.367 . From the dc-magnetization; a broad peak around 46.55 K, separation between zero-field cooled and field-cooled warming state and non-saturating isothermal magnetization with typical S-type hysteresis indicate glassy behavior. A cusp in is observed to shift toward high temperatures with increasing frequency. Mydosh parameter (), single-relaxation time ( s) obtained through critical slowing-down analysis, from the Vogel–Fulcher law and Tholence criterion , confirm that Mn0.7Ru0.3NiGe belongs to the short-range interaction spin-glass systems with strong coupling between the magnetic clusters. LSDA+U calculations confirmed the competing exchange interactions between large magnetic moments of the Mn ions in Mn0.7Ru0.3NiGe compound resulting in the formation of spin-glassy characteristics.

Specific heat and nonlinear susceptibility in spin glasses with random fields

We study magnetic properties of spin glass (SG) systems under a random field (RF), based on the suggestion that RFs can be induced by a weak transverse field in the compound ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$. We consider a cluster spin model that allows long-range disordered interactions among clusters and short-range interactions inside the clusters, besides a local RF for each spin following a Gaussian distribution with standard deviation $\mathrm{\ensuremath{\Delta}}$. We adopt the one-step replica symmetry breaking approach to get an exactly solvable single-cluster problem. We discuss the behavior of order parameters, specific heat ${C}_{m}$, nonlinear susceptibility ${\ensuremath{\chi}}_{3}$, and phase diagrams for different disorder configurations. In the absence of RF, the ${\ensuremath{\chi}}_{3}$ exhibits a divergence at ${T}_{f}$, while the ${C}_{m}$ shows a broad maximum at a temperature ${T}^{**}$ around $30%$ above ${T}_{f}$, as expected for conventional SG systems. The presence of RF changes this scenario. The ${C}_{m}$ still shows the maximum at ${T}^{**}$ that is weakly dependent on $\mathrm{\ensuremath{\Delta}}$. However, the ${T}_{f}$ is displaced to lower temperatures, enhancing considerably the ratio ${T}^{**}/{T}_{f}$. Furthermore, the divergence in ${\ensuremath{\chi}}_{3}$ is replaced by a rounded maximum at a temperature ${T}^{*}$, which becomes increasingly higher than ${T}_{f}$ as $\mathrm{\ensuremath{\Delta}}$ is enhanced. As a consequence, the paramagnetic phase is unfolded in three regions: (i) a conventional paramagnetism ($T>{T}^{**}$); (ii) a region with formation of short-range order with frozen spins (${T}^{*}<T<{T}^{**}$); (iii) a region with slow growth of free-energy barriers slowing down the spin dynamics before the SG transition (${T}_{f}<T<{T}^{*}$) suggesting an intermediate Griffiths phase before the SG state. Our results reproduce qualitatively some findings of ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ as the rounded maximum of ${\ensuremath{\chi}}_{3}$ behavior triggered by RF and the deviation of the conventional relationship between the ${T}_{f}$ and ${T}^{**}$.

Replica-Exchange Methods for Biomolecular Simulations.

In this study, a replica-exchange method was developed to overcome conformational sampling difficulties in computer simulations of spin glass or other systems with rugged free-energy landscapes. This method was then applied to the protein-folding problem in combination with molecular dynamics (MD) simulation. Owing to its simplicity and sampling efficiency, the replica-exchange method has been applied to many other biological problems and has been continuously improved. The method has often been combined with other sampling techniques, such as umbrella sampling, free-energy perturbation, metadynamics, and Gaussian accelerated MD (GaMD). In this chapter, we first summarize the original replica-exchange molecular dynamics (REMD) method and discuss how new algorithms related to the original method are implemented to add new features. Heterogeneous and flexible structures of an N-glycan in a solution are simulated as an example of applications by REMD, replica exchange with solute tempering, and GaMD. The sampling efficiency of these methods on the N-glycan system and the convergence of the free-energy changes are compared. REMD simulation protocols and trajectory analysis using the GENESIS software are provided to facilitate the practical use of advanced simulation methods.

A spin glass state in Ba3TiRu2O9

The magnetic properties of Ba3TiRu2O9 , whose crystal structure is based on stacked triangular planar lattices of MO6 dimers and single MO6 octahedra, are reported. The system is magnetically disturbed by a substantial amount of Ti/Ru chemical disorder. The Weiss temperature and effective magnetic moment were found to be -29.5 K and 1.82 {\mu}B/f.u. respectively, and a bifurcation in the zero field cooled and field cooled magnetic susceptibility is observed below 4.7 K, suggesting that this is a compositionally-disordered spin-glass system. The material is a semiconductor with an activation energy for charge transport of approximately 0.14 eV.

Spin-glass behavior in KxRu4−yNiyO8 hollandite materials

We report the synthesis and comprehensive ac and dc susceptibility measurements of ${\mathrm{K}}_{x}\mathrm{R}{\mathrm{u}}_{4\ensuremath{-}y}\mathrm{N}{\mathrm{i}}_{y}{\mathrm{O}}_{8}$ hollandite. The value of the relative frequency shift, $\ensuremath{\delta}{T}_{f}$, has been determined as 0.025 which is within the range expected for spin-glass systems (0.005--0.06). Additionally, the characteristic flipping time of a single spin flip, ${\ensuremath{\tau}}_{0}$, and the dynamical critical exponent, $\ensuremath{-}zv$, were determined to have values of $5.82\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}\phantom{\rule{0.16em}{0ex}}\mathrm{s}$ and 6.1(3), respectively from the power law. While the value of ${\ensuremath{\tau}}_{0}$ is comparatively very large, $\ensuremath{-}zv$ is consistent with what is expected for spin-glass systems. Field-cooled hysteresis behavior demonstrates a small increase in the remnant magnetization (at 2 K) on increasing the strength of the cooling field, suggesting that the degree of short-range correlations increases consistent with the formation of larger spin clusters. Thermoremnant magnetization data indicate an exponential-like decay of the magnetization as a function of time with the remnant magnetization remaining nonzero. However, it is clear from these data that multiple components contribute to the decay behavior. Collectively, these data confirm spin-glass character for ${\mathrm{K}}_{0.73(3)}\mathrm{N}{\mathrm{i}}_{1.9(5)}\mathrm{R}{\mathrm{u}}_{2.1(5)}{\mathrm{O}}_{8}$ and clearly demonstrate that the magnetic behavior of this material is far from simplistic.

Magnetoelastic coupling at spin-glass-like transition in Sr3NiSb2O9

We report a characteristic spin-glass-like behavior in Sr3NiSb2O9, as confirmed from the low-field dc magnetometry and ac susceptibility measurements. The memory effects in the dc magnetometry exhibit similar characteristic features of the spin-glass systems. The ac susceptibility measurements provide that the frequency dependent peak-shift follows the conventional Vogel-Fulcher and dynamical scaling laws with the characteristic relaxation time to be ∼ 10−14 s. This relaxation time fits in the recommended range of ∼10−12−10−14 s for the classical spin-glasses. Low temperature synchrotron diffraction studies point to the strong magnetoelastic coupling close to the spin-glass-like transition. The transition is found to be associated with the step-like anomalous lattice contraction. Possible origin of magnetic frustration leading to the spin-glass ground state is discussed by correlating the microstructural results, as obtained from the analysis of synchrotron diffraction studies.

Sparse Hardware Embedding of Spiking Neuron Systems for Community Detection

We study the applicability of spiking neural networks and neuromorphic hardware for solving general opti- mization problems without the use of adaptive training or learning algorithms. We leverage the dynamics of Hopfield networks and spin-glass systems to construct a fully connected spiking neural system to generate synchronous spike responses indicative of the underlying community structure in an undirected, unweighted graph. Mapping this fully connected system to current generation neuromorphic hardware is done by embedding sparse tree graphs to generate only the leading-order spiking dynamics. We demonstrate that for a chosen set of benchmark graphs, the spike responses generated on a current generation neuromorphic processor can improve the stability of graph partitions and non-overlapping communities can be identified even with the loss of higher-order spiking behavior if the graphs are sufficiently dense. For sparse graphs, the loss of higher-order spiking behavior improves the stability of certain graph partitions but does not retrieve the known community memberships.

Magnetic impurity bands in Ga1−xMnxS : Towards understanding the anomalous spin-glass transition

We report on the magnetic and electronic properties of single-crystalline ${\mathrm{Ga}}_{0.91}{\mathrm{Mn}}_{0.09}\mathrm{S}$, which is a quasi-two-dimensional diluted magnetic semiconductor (DMS). Through an analysis of magnetization data, we show the existence of an anomalously high spin-glass transition temperature at 11.2 K. Using density functional theory (DFT), we characterize the properties contributing to the spin-glass transition through an examination of the electronic and magnetic properties for ${\mathrm{Ga}}_{1\ensuremath{-}x}{\mathrm{Mn}}_{x}\mathrm{S}$ with $x$ varying from 0.00 to 0.18 by randomly substituting Mn atoms into the gallium (Ga) lattice sites. We show that the presence of magnetic atoms produces impurity bands in the electronic structure, where an analysis of the density of states shows an increase in magnetic impurity bands at the Fermi level that lowers the semiconducting gap and is consistent with diluted magnetic semiconductors. Furthermore, this indicates that the spin-glass transition in ${\mathrm{Ga}}_{0.91}{\mathrm{Mn}}_{0.09}\mathrm{S}$ is similar to other DMS materials, where the primary mechanism is likely through magnetic exchange. However, the increased electron density in the system with Mn doping could explain the anomalously higher spin-glass transition temperature in ${\mathrm{Ga}}_{0.91}{\mathrm{Mn}}_{0.09}\mathrm{S}$. In comparison with the substantially lower transition temperatures in related II-VI based systems (i.e., ${\mathrm{Zn}}_{1\ensuremath{-}x}\mathrm{Mn}{}_{x}\mathrm{Te}$), the high transition temperature is associated with more metallic spin-glass systems that interact through RKKY exchange, which leads to the conclusion that there may be a combination of interactions occurring in these systems. Further measurements on the other substitution percentages will hopefully clarify these interactions.

Lower lower-critical spin-glass dimension from quenched mixed-spatial-dimensional spin glasses

By quenched-randomly mixing local units of different spatial dimensionalities, we have studied Ising spin-glass systems on hierarchical lattices continuously in dimensionalities 1 =< d =< 3. The global phase diagram in temperature, antiferromagnetic bond concentration, and spatial dimensionality is calculated. We find that, as dimension is lowered, the spin-glass phase disappears to zero temperature at the lower-critical dimension d_c=2.431. Our system being a physically realizable system, this sets an upper limit to the lower-critical dimension in general for the Ising spin-glass phase. As dimension is lowered towards d_c, the spin-glass critical temperature continuously goes to zero, but the spin-glass chaos fully sustains to the brink of the disappearance of the spin-glass phase. The Lyapunov exponent, measuring the strength of chaos, is thus largely unaffected by the approach to d_c and shows a discontinuity to zero at d_c.

Physical foundations of biological complexity

Biological systems reach hierarchical complexity that has no counterpart outside the realm of biology. Undoubtedly, biological entities obey the fundamental physical laws. Can today's physics provide an explanatory framework for understanding the evolution of biological complexity? We argue that the physical foundation for understanding the origin and evolution of complexity can be gleaned at the interface between the theory of frustrated states resulting in pattern formation in glass-like media and the theory of self-organized criticality (SOC). On the one hand, SOC has been shown to emerge in spin-glass systems of high dimensionality. On the other hand, SOC is often viewed as the most appropriate physical description of evolutionary transitions in biology. We unify these two faces of SOC by showing that emergence of complex features in biological evolution typically, if not always, is triggered by frustration that is caused by competing interactions at different organizational levels. Such competing interactions lead to SOC, which represents the optimal conditions for the emergence of complexity. Competing interactions and frustrated states permeate biology at all organizational levels and are tightly linked to the ubiquitous competition for limiting resources. This perspective extends from the comparatively simple phenomena occurring in glasses to large-scale events of biological evolution, such as major evolutionary transitions. Frustration caused by competing interactions in multidimensional systems could be the general driving force behind the emergence of complexity, within and beyond the domain of biology.