Physical Review B

Macroscopic quantum systems would reach thermodynamic equilibrium if in contact with a bath. If isolated, a subtle argument known as Eigenstate Thermalization Hypothesis (ETH) would still allow for thermalization despite the strictly unitary evolution of the closed system whereby the latter essentially serves as its own bath. However, if such an isolated system is sufficiently disordered, it won’t obey the ETH. As Anderson demonstrated for a very large class of noninteracting systems in 1958, “absence of diffusion”, aka Anderson localization, sets in if the randomness is sufficiently strong [1]. However, the natural and ubiquitous situation in a system of particles would be that they interact. The fate of localized systems with interactions was probed early on by Fleishman and Anderson, with the conclusion that localization survives in the weak-interaction regime [2].

The huge difficulties in handling disorder and interaction in a many-body system shifted the burden of exploring many-body localization (MBL) - the localization of closed, disordered, interacting systems - to this century. The prototypical system in question would, but need not, be interacting electrons in a random potential where localization corresponds to an insulating phase. In this context, Basko et al. argued convincingly that such electrons would have an insulating (many-body localized) phase even at nonzero temperature [3].

On the pages of Physical Review B, David Huse and collaborators have contributed substantially and repeatedly to the deeper understanding of MBL. Thus, Oganesyan and Huse suggested in 2007 that if there is MBL for strongly disordered and weakly interacting electrons, then MBL will also occur when both disorder and interactions are strong, even if the temperature is effectively infinitely high. In this latter case, they demonstrated that the MBL transition may be studied through exact diagonalization of small systems [4]. The same method was then implemented to study the MBL transition in another paradigmatic system, the random-field spin-½ chain. The localization transition and its finite-size scaling were explored with the highly nontrivial finding that this unusual, finite-temperature quantum phase transition shows infinite-randomness scaling with an infinite dynamical critical exponent. On a physical level, it is the competition between the Thouless energy (Planck constant times relaxation rate) and the level spacing that decides the outcome for ergodicity, if the former is much larger than that latter, or for MBL if the opposite is true [5].

In the PRB Milestone at hand, Huse and collaborators make the stunning prediction that many-body localized quantum systems, which do not equilibrate even if prepared with macroscopic amount of energy above their ground state, may still order in the sense that individual many-body eigenstates can display broken symmetry or topological order – and this at energy densities where the corresponding thermalized system would be disordered. To put it differently, MBL protects order. The authors go on to demonstrate that nonthermodynamic transitions between ordered and disordered phases, seen as transitions in the properties of the many-body eigenstates will occur and may proceed via localized critical points. The analysis is extended beyond one-dimensional systems. The big promise of such a type of MBL is that it may allow experimental manipulation of macroscopic quantum states since it provides protection against decoherence.

[1] P. W. Anderson, Phys. Rev. 109, 1492 (1958).
[2] L. Fleishman and P. W. Anderson, Phys. Rev. B 21, 2366 (1980).
[3] D. M. Basko, I. L. Aleiner, and B. L. Altshuler, Ann. Phys. (NY) 321, 1126 (2006).
[4] Vadim Oganesyan and David A. Huse, Phys. Rev. B 75, 155111 (2007).
[5] Arjeet Pal and David A. Huse, Phys. Rev. B 82, 174411 (2010).

In DFT calculations with a plane-wave basis set, the strong Coulomb potential of the nucleus and the effects of the core electrons are usually combined together and replaced with a much softer effective ionic potential, or pseudopotential. As a result, the valence electrons can be described by pseudo wave functions that are smooth in the core region, thus requiring fewer Fourier modes and making plane-wave basis sets practical to use.

The two PRB papers highlighted together in this Milestone put forward one of the most common pseudopotentials, the ultrasoft pseudopotential (USPP), and a powerful generalization of the pseudopotential method, the projector augmented-wave method (PAW).

In the first paper, Vanderbilt proposed the concept of USPP as an alternative to the norm-conserving pseudopotentials, which are computationally demanding for strongly localized orbitals, such as those in first-row atoms and transition-metal systems, due to the small core radius of the pseudo wave function and the norm-conserving constraint which requires that the norm of the pseudo wave function inside the core matches that of the all-electron wave function. In the USPP method, the norm-conserving constraint is relaxed, which makes it possible to construct pseudo wave functions that are as smooth as possible, therefore reducing the size of the required plane wave basis set. The resulting deficit of valence charge in the core region is compensated by introducing an auxiliary function around each ion core that represents the rapidly varying part of the electron density. The widely popular USPP method has provided a computationally efficient scheme for accurate description of the electronic structure, especially for systems with strongly localized orbitals.

In DFT calculations with a plane-wave basis set, the strong Coulomb potential of the nucleus and the effects of the core electrons are usually combined together and replaced with a much softer effective ionic potential, or pseudopotential. As a result, the valence electrons can be described by pseudo wave functions that are smooth in the core region, thus requiring fewer Fourier modes and making plane-wave basis sets practical to use.

The two PRB papers highlighted together in this Milestone put forward one of the most common pseudopotentials, the ultrasoft pseudopotential (USPP), and a powerful generalization of the pseudopotential method, the projector augmented-wave method (PAW).

In the second Milestone paper, Blöchl developed the PAW method, which is a generalization of ideas of the Vanderbilt-type USPP from the first paper and of the linearized augmented-plane-wave (LAPW) method. Like the USPP method, it introduces projectors and auxiliary localized functions, and defines the total energy as a functional that involves the compensation charge density. On the other hand, similar to the LAPW method, the PAW method provides the full all-electron wave functions and all-electron potentials that are not directly accessible with the pseudopotential approach. The PAW method bridges the gap between the existing augmented-wave methods and the pseudopotential methods. It is capable of handling many complex cases, such as magnetic materials, with exceptional precision and efficiency.

The application of the PAW method was greatly boosted by a subsequent paper by G. Kresse and D. Joubert [1] that established the formal relationship between Vanderbilt-type USPP and Blöchl’s PAW method. These authors provided a simpler way to implement the PAW method in existing plane-wave codes, leading to its widespread use in modern electronic structure calculations.

[1] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999)

In this excitingly interdisciplinary PRB Milestone, Richard Davison and collaborators compute the thermodynamic properties of higher-dimensional generalizations of the Sachdev-Ye-Kitaev (SYK) model of fermions with a conserved fermion number. In these generalizations, the researchers uncover disordered metallic states without quasiparticle excitations. Remarkably, the low-temperature thermodynamics and the transport behavior are found to be in precise agreement with the holographic gravity dual of the SYK models.

Read and Green consider the simplest model of a chiral superconductor realized by pairing spinless fermions in the p+ip channel. Using the BCS mean-field approach, they show that the system can be in either the weak- or the strong-pairing phase and that these phases are separated by a topological phase transition. In the weak-pairing phase, the ground state of the system is equivalent to that of the Moore-Read Pfaffian fractional quantum Hall state which supports non-Abelian excitations while the strong-pairing phase is topologically trivial. In the weak-coupling phase, the non-Abelian excitations are Majorana fermions hosted by vortices, which show non-Abelian statistics when braided.

While reproducing a number of already known results, this study gave a strong boost to the identification of classes of topological phase transitions with the properties of the weak-coupling paired phases.

Spin-polarized injection into semiconductors became a field of major interest In the 1990’s, following the successful implementations of ferromagnet (FM)—nonmagnetic-metal—ferromagnet and FM-insulator-FM heterostructures for giant magnetoresistance and magnetic tunnel-junction devices, respectively. The novel device where semiconductors (S) would be indispensable was in the form of a theoretical proposal of a spin transistor by Datta and Das. Various modifications soon abounded. While theory remained free to speculate on the basis of assumed high degree of spin-polarized injection, experiment stumbled over and over again on the miserably low degree of polarization that could be injected from a ferromagnetic metal into a semiconductor, keeping in mind the need to inject the barely polarized current back into the collecting FM detector (electrode).

Schmidt et al. zoomed into the reality of spin injection from a FM metal into a diffusive semiconductor. Given that spin scattering was much slower than competing scattering effects, distinct electrochemical potentials for each spin polarization could be defined at any point of the FM-S-FM prospective device. The latter was then examined straightforwardly as a two-branch network of resistors, representing the two independent spin channels in all three parts of the device. The simultaneous consideration of Ohm’s law and the diffusion equation under natural boundary conditions led to an explicit expression for the degree of spin polarization of the current density in the semiconductor part of the device. The formula was quite remarkable in that it depended only on the degree of bulk spin polarization in the FM electrodes and on a compound dimensionless parameter. While somewhat more complicated, the expression for the magnitude of the overall magnetoresistance effect shared the nice feature of depending only on the same pair of parameters. And there came the moment of truth: Even with the most optimistic assumptions about the microscopic parameters, built into the dimensionless ratio, the spin polarization in the semiconductor remained below 0.1% even if the bulk FM polarization would be as high as 99%.

Physically, the authors’ demonstration could be subsumed by stating that the success of the FM-S-FM device was doomed because of the large conductivity mismatch between the ferromagnetic metal and the semiconductor. Beside the clarity of thought and execution, this Milestone left its mark not just as a no-go result of heavy consequence, but also as outlining the truly promising avenues to advance the agenda that would give the fancy word spintronics some substance beyond what was already known at the turn of the century.

The discovery of iron-based superconductivity in a doped oxypnictide in 2006 was a major step in advancing the field of high-temperature superconductivity.

In a Rapid Communication published in Physical Review B, Marianne Rotter and colleagues reported the discovery of a new iron-based compound that could potentially host superconductivity upon doping in the absence of oxygen. With a simpler crystal structure that makes it easier to synthesize it in a single-crystal form, BaFe₂As₂ was shown to exhibit a spin-density wave and a structural transition at 140 K.

Subsequent work would go on to show that BaFe₂As₂ would indeed become a high-temperature superconductor with a maximum T_c of 38 K upon doping with potassium.

Over the last few decades, there has been an explosion in the realm of nanotechnology, nanodevices, and nanomaterials, where anything ‘nano’ has become part of the condensed matter and materials physics lexicon. As has often been restated, this is due to the great potential for technological applications. However, this potential produced the need for a better understanding of the fundamental physics at the atomic scale not just for molecular modeling, but for device and application purposes as well. In the early 2000s, the use of density functional theory (DFT) and ab initio modeling continued to revolutionize the way we understand materials. Two outstanding PRB papers reported on the incursion of DFT into the arena of quantum electron transport properties by means of nonequilibrium Green’s functions. They helped pave the way for the advancement of device modeling at the atomistic level.

The PRB work by J. Taylor et al. in 2001 focused on the determination of ballistic conductance through device interfaces. The authors applied DFT in combination with a nonequilibrium Green’s-function approach to model a device system made of a carbon nanotube and an Al metallic electrode. They used a screening approximation to reduce the calculation to a defined region of space.

In the 2002 PRB, M. Brandbyge et al. used a similar method on systems including single-atom carbon or gold wires and large carbon nanotubes with point defects using a simulated electrical bias. The importance of this double Milestone derives from the development of reliable nanoscale simulations for device applications. The pair of papers have been widely recognized as foundational studies in this branch of computational condensed matter and materials physics.

Continuing the quest for new phases of matter that go beyond the Landau theory with its symmetry-breaking paradigm, Levin and Wen introduce a class of two-dimensional exactly solvable lattice spin models whose degrees of freedom are described by extended objects, dubbed string-nets. Depending on the ratio of kinetic and potential energies, the string-nets can either be confined to a few small regions corresponding to a normal state or proliferate through the whole system filling up all space. In the latter case, the ground state of the system is string-net condensed and gives rise to a number of exotic topological orders, including those described by non-Abelian anyon theories. The authors generalize the model to three dimensions and find that in this setting its emergent collective excitations are gauge bosons and fermions, demonstrating that such distinct objects can have common origin.

Van der Waals (vdW) interactions are nonlocal, long-ranged interactions originating from electromagnetic fluctuations of the electrons. This ubiquitous force is present between any type of objects and plays a crucial role in determining the interactions and stability of materials of different kinds, including rare-gas dimers, organic and biological systems, low-dimensional materials, and the emerging classes of two-dimensional materials and heterostructures. Despite the significance of vdW interactions in fundamental and applied science of materials, state-of-the-art vdW-corrected density functional theory methods have only become available in the last two decades. The two PRB articles that are highlighted together here stand out for their major relevance to the field.

In the first paper, Lee and colleagues follow up the pioneering vdW density functional (vdW-DF) approach by Dion et al. and propose an enhanced version by replacing the revPBE exchange functional with the less repulsive PW86 functional and employing a large-N asymptote gradient correction. The performance of the new functional (vdW-DF2) is assessed by comparison with accurate quantum chemical calculations of the S22 duplexes and demonstrates remarkable agreement in the predicted binding energy, equilibrium separation, and potential-energy curve shape. Applications of vdW-DF2 to extended solid systems such as graphite and H₂ adsorption in metal-organic frameworks also show substantial improvement over vdW-DF, making the former a promising scheme for the accurate and efficient treatment of dispersion bonded systems.

Following the success of the vdW-DF approach, several variations of vdW-corrected density functions have been proposed, including a group of “opt” family of vdW approaches by Klimeš, Bowler, and Michaelides, where the original revPBE exchange functional is replaced with ones that are optimized to S22 data of intermolecular interaction energies.

In the second Milestone paper at hand, Klimeš, Bowler, and Michaelides thoroughly test the performance of the original vdW-DF, vdW-DF2, and several “opt” variants. They compare the predicted lattice constants, bulk moduli, and atomization energies of a wide range of solids in order to establish the general applicability of the vdw-DF method. This work provides valuable reference data and a broad overview of the strengths and deficiencies of the vdW-DF family of functionals; it has thus become an indispensable guide for further development of the method.

The arrival of graphene has engendered a renaissance in the study of electronic properties of two-dimensional systems, including the cherished two-dimensional electron gas (2DEG) model system. The key difference between the 2DEG and two-dimensional graphene lies in their electronic energy dispersion. In the former, the electron energy depends quadratically on the momentum, while in graphene this dependence is linear in momentum near the Fermi level, similar to the dispersion of a relativistic particle. This intrinsic distinction has brought about intense activity in an effort to elucidate the electronic response of graphene, much like the polarizability of the 2DEG was studied in the seminal work of Frank Stern (1967).

In this double Milestone, two important PRB papers, highlighted jointly here, have had substantial impact in understanding the properties of plasmons in graphene.

In the first one, Hwang and Das Sarma calculate the dielectric function of doped graphene at an arbitrary wave vector q and frequency ω. Within the random-phase approximation (RPA), they chart out the dispersion of the plasmon mode and the electrostatic screening of the Coulomb interaction in the graphene layer. At long wavelengths, the plasmon dispersion exhibits classical behavior, in which the frequency scales quadratically with q; however, the plasma frequency scales with the two-dimensional carrier density as n^1/4, in contrast with that of the 2DEG where it scales as n^1/2. The RPA plasmon dispersion is seen as the solid line in the figure. The solid diagonal lines represent the boundaries of the Landau damping regimes for intraband and interband electron-hole excitations. Landau damping in graphene occurs due to the loss of collective motion to the excitation of electron-hole pairs, specifically during interband excitation. In contrast, Landau damping for conventional plasmons in 2DEG occurs in the intraband regime. The plasma frequency scales as q^1/2 in the long-wavelength limit, a behavior also expected for conventional 2D plasmons (dashed line). At larger values of q, however, the plasma frequency departs from the conventional-plasmon $q$ dependence, and becomes more linear with q. A similar set of results was also obtained by Wunsch et al. Experimental verification of the Hwang–Das Sarma theory was demonstrated by Liu et al., Phys. Rev. B 78, 201403(R) (2008).

In the second paper of this Milestone pair, Jablan, Buljan, and Soljacic show that for sufficiently large doping values graphene supports plasmons that simultaneously have low losses and display strong wave localization for photon frequencies ω below that of the optical phonon branch ω_Oph ≈ 0.2 eV. By increasing the electron-type doping, the Fermi level goes up with the region of interband plasmonic losses moving towards higher frequencies. However, at sufficiently large doping, the interband threshold frequency ω_inter can be larger than graphene’s optical phonon frequency. Thus, there are three regimes of interest: ω < ω_Oph, ω_Oph < ω < ω_inter, and ω < ω_inter. In the second one, it is possible to excite an electron-hole pair accompanied by an emission of an optical phonon, leading to a different kind of damping associated with interband excitation due to the finite lifetime of the phonon. For ω < ω_Oph, this decay channel becomes inactive. To calculate the loss, degree of localization, and group velocity in these regimes, the authors implement the RPA and number-conserving relaxation-time approximation with the relevant input parameters, taken from experiment or from theoretical estimates. Here, the accompanying figure demonstrates the high doping case for the three quantities in the three regions of interest. Such calculations reveal that at certain specific wavelengths, mostly in the infrared regime, plasmons may have low losses and long propagation lengths. The implications of this work in nanophotonics include the reduction in size and operational power of photonic devices.

One of the cornerstones of the Bardeen–Cooper–Schrieffer theory of superconductivity (1957) was the prediction of the superconducting transition temperature in terms of just a handful of material parameters – the Debye phonon frequency Ω_D, the electron density of states at the Fermi level N₀, and the effective phonon-mediated attractive interaction between the electrons V. The result they obtained in the limit of weak interactions reads: k_BT_c = 1.14 ħ Ω_D e^-1/λ with the dimensionless interaction parameter λ = N₀ V. Even this idealized and simple form was sufficient to qualitatively describe the isotope effect for many superconducting materials.

The generalization of this celebrated relation to strongly coupled superconductors involved solving a set of coupled integral equations within the Migdal-Eliashberg theory (1960). The ingredients that entered these equations were more realistic, but also more complicated since one needed to know the Coulomb repulsion as well as the product of the phonon density of states F(ω) and the averaged electron-phonon interaction α² (ω) as a function of frequency - that information was beyond reach for any first-principles calculations back then.

A major practical attempt to reduce these equations to an approximate analytical solution that involved only a few parameters was made by William L. McMillan in 1967. Based on numerical solutions of the Eliashberg equations, he proposed an empirical formula that was valid for λ ≲ 1 and included only one additional parameter μ^* – a renormalized Coulomb repulsion between the electrons. Quite pessimistically, when extrapolated to λ > 1 McMillan’s formula predicted that the superconducting transition temperatures were limited from above for any class of materials by a maximum value that was attained for λ = 2. The T_c observed in experiments had been below 20 K for a number of years by then (the so-called “Matthias limit”), and McMillan’s upper bound appeared to be perfectly reasonable.

In a decisive step forward, Philip B. Allen and Robert C. Dynes performed a rigorous analysis of the Eliashberg equations in the limit of large λ and proposed a modification of McMillan’s formula that was valid for any coupling strength. Their results proved that the earlier prediction of T_c, bounded below a maximum possible value, had been an artefact of extrapolating McMillan’s formula beyond its range of validity. Moreover, Allen and Dynes demonstrated that there was no theoretical limiting maximum for the transition temperature, and in the strong-coupling limit T_c would grow as λ^1/2. In the decades following their work, the capabilities for first-principles calculations have improved dramatically, and the Allen-Dynes formula has been central to a great number of computational studies that predict and explore novel superconducting materials. As a remarkable example for the success of the theory, may it be mentioned here that recent theoretical predictions of room-temperature superconductivity under very high pressure in sulfur and lanthanum hydrides have been confirmed experimentally.

With G for Green’s function and W for screened Coulomb interaction, the GW approximation to the electron self-energy operator in many-body perturbation theory provides the basis for one of the most powerful ab initio methods for predicting electronic band structures ​of real materials. It was originally put forward by Lars Hedin in his seminal work more than fifty years ago. Due to computational limitations, its application to realistic materials had to wait until much later.

The present PRB Milestone gives a full exposition of the methodology for an original first-principles approach to electronic excitation energies, incorporating the GW approximation directly into solid-state electronic structure theory and demonstrating its advantages for semiconductors and insulators.

The authors build on the norm-conserving pseudopotential method combined with linear response theory to develop a feasible framework for the computation of quasiparticle energies in solids. In the formulation of the electron self-energy operator, key physical effects include dynamical and local-field effects in the screening of the Coulomb interaction and dynamical renormalization. Hybertsen and Louie unleash the new approach to find the quasiparticle energies of the homopolar materials diamond, silicon, and germanium as well as of the ionic compound LiCl, demonstrating excellent agreement with experimental data and overcoming the systematic underestimation of band gaps that resulted from density functional theory applications.

Since 1986, the GW approximation has been successfully implemented for the computation of electronic excitation energies in a broad array of materials, including bulk solids, surfaces, and nanostructures such as carbon nanotubes. Many-body perturbation theory has become a mainstream approach in the electronic structure community. The large-scale efforts that have been invested into the development, improvement, and implementation of the GW method have transformed it into the state-of-the-art approach for the accurate description of excitations in weakly correlated molecular systems and solids.

Quantum computing is an exciting frontier of research that promises not only revolutionary technological advancements but also a potential demonstration of quantum entanglement on a macroscopic scale. Scalability is at the core of the challenge to realize quantum computation. One of the promising proposals for a quantum computer is to encode qubits into states of topological phases of matter which are protected from decoherence through coupling to the environment.

In a major development that appeared in Physical Review B as a regular article, Karzig and colleagues proposed a few scalable architectures for topological quantum computation using a network of Majorana nanowires. They also devised strategies to optimize robustness to environmental temperature and noise, size of the network, and computational efficiency. The practicality of these proposed architectures depends on overcoming several experimental challenges to reliably generate Majorana nanowires in the topological phase.

This Milestone study builds on the publication by the same group of authors in Science 357, 61 (2017), which extended to quadrupole and octupole electric moments earlier work by Vanderbilt, King-Smith, Resta, Martin, and others on modern, Berry-phase formulation of electric polarization in crystals. As a byproduct, the trio found that some types of three-dimensional topological insulator generically support the existence of corner states with fractional charges, leading naturally to the notion of the higher-order topological insulator. Unlike the more conventional topological states which appear on the boundary of the system and have their dimensionality reduced by one (as, for instance, in going from volume to surface and from plane to edge), the corner states in higher-order topological insulators appear on the boundary of the boundary and have a dimensionality reduced by two from where the host system lives in.

Here, Benalcazar, Bernevig, and Hughes lay out all the nitty-gritty details of the theory. They introduce a new kind of state, the hinge state that extends along a line on the surface of a three-dimensional object, and propose a new topological invariant which can be computed with the help of nested Wilson loops. In addition and in extension of the notion of Thouless charge pumping for adiabatically modified dipole moments, the authors show the existence of two electronic topological pumping processes when the higher multipole moments are adiabatically modified.

The topic of higher-order topological insulators has since gained a tremendous momentum, bringing new excitement to the field.

The authors elucidate the subtleties, involved in field-theoretical calculations of the critical exponents of the n-vector model. The coefficients of the renormalization functions of the φ⁴ theory are expanded in (divergent) series of the coupling constant to sufficiently high order. This allows us to apply resummation techniques to the Borel-transformed asymptotic series, so that reliable values of the critical exponents for the continuous model are obtained directly in the dimensionality of space of interest as opposed to the extrapolation of ε-expansion results to large values. The study stands out as one of the most authoritative expositions of the fast-growing body of knowledge about the field-theoretical handling of critical phenomena.

Research on black phosphorus dates back to over a century ago when it was first synthesized by P. W. Bridgman in 1914. As a two-dimensional material, black phosphorus exhibits a layered structure similar to graphite, permitting mechanical exfoliation of a few layers all the way to monolayer phosphorene, a structurally puckered version of graphene. Its anisotropic structure gives rise to extraordinary anisotropic electronic, optical, thermal, and transport properties.

A flurry of activity at the beginning of 2014 has ushered several significant experimental and theoretical advances, with the two Milestone PRB papers standing out.

In the first one, Tran, Soklaski, Liang, and Yang performed first-principles GW Bethe-Salpeter-equation simulations to obtain the quasiparticle band gap and exciton binding energies of few-layer and bulk black phosphorus. The band gap was shown to be direct and thickness‐dependent, decreasing with the increase of the number of layers. The calculated band gap for monolayer black phosphorus (phosphorene) was in good agreement with the concomitant experimental measurement by H. Liu et al.. Tran et al. observed enhanced many-electron effects, which resulted from the effectively quasi-one-dimensional band dispersion of phosphorene, leading to enhanced self-energy corrections and excitonic effects. For few-layer black phosphorus, the excitonic effects modulated substantially the optical spectra. Tran et al. found that the highly anisotropic optical response would make phosphorene ideal as a linear optical polarizer over a wide spectral range, with tunability as a function of the number of layers.

In the second PRB Milestone paper here, Rudenko and Katsnelson calculated the GW band structure of few-layer and bulk black phosphorene. They analyzed systematically their ab initio results by constructing a tight-binding parametrization based on the GW band structure. This offered physical insight into the mechanism of the band gap formation. For example, phosphorene involves two parameters corresponding to the in-plane and out-of-plane nearest-neighbor hopping parameters. The latter parameter is a consequence of the puckered nature of the black phosphorus structure, and is largely responsible for the wide band gap of phosphorene. The decrease in energy gap with increasing number of layers arises mostly from the increased interlayer hopping.

The interface of a normal metal (N) with a superconductor (S) separated by a thin barrier has served as a host for a number of dramatic effects, beginning with the pioneering experiments of Giaever in 1960. The initial measurements that involved weak-coupling superconductors were in agreement with the BCS theory, and the subsequent tunneling experiments played an important role in the development of the theory of strong-coupling superconductors.

Unlike the high-barrier tunnel junctions, the properties of N/S systems with a low barrier proved to be difficult to explain. Among other peculiarities, their I-V curves displayed the region of high conductance for applied bias below the gap Δ and excess current at higher applied bias V > Δ, suggesting the ability to transmit more current in the superconducting state.

In the late 1970s, Artemenko, Volkov, and Zaitsev realized that the key to this puzzle was the Andreev–Saint-James reflection (1964): when approaching the N/S interface, a normal electron enters the superconductor as a Cooper pair, with a simultaneous reflection of a hole on the normal side [1].

With this in mind, G.E. Blonder, M.Tinkham, and T.M. Klapwijk (BTK) formulated a complete treatment of N/S interfaces with an intuitively simple physical description. Their work provided a unified description for barriers of different transparency via an effective barrier parameter that could describe the high-barrier tunneling limit as well as the transmissive barriers. The BTK model became a standard tool in point-contact spectroscopy by allowing to extract the superconducting gap and the effective transparency of the interface from the fitting of I-V curves.

The original BTK model was restricted to s-wave superconductors and normal nonmagnetic metals, but during the decades that followed it was extended to describe cuprates, iron-based, and other unconventional superconductors as well as the effects of spin-orbit coupling, topological N/S junctions, and others. It remains as relevant as ever.

[1] See T. M. Klapwijk’s account in Journal of Superconductivity 17, 593 (2004) (special issue on the occasion of Professor Tinkham’s 75^th birthday).

Insulating in the bulk yet metallic on the surface, topological insulators are possible in certain crystals with strong spin-orbit coupling and unbroken time-reversal symmetry.

Can topological physics manifest itself even in materials where time-reversal symmetry is broken by magnetic order? This question was answered in the affirmative in the groundbreaking work by Xiangang Wan and colleagues appearing in Physical Review B in 2011. Employing LDA+U calculations, Wan et al. demonstrated that rare-earth iridium pyrochlores have an antiferromagnetic ground state, in which the conduction and valence bands touch at a discrete number of momenta resulting in a zero-gap semimetal.

These so-called diabolical points are topologically protected, and the electrons in their vicinity can be described by a two-component Dirac equation, that is, as Weyl fermions. In addition, certain surfaces have bound states at the Fermi energy that form open arcs rather than the usual closed Fermi surfaces. The necessary conditions of observing this new state of quantum matter are rather general, and the Weyl semimetal has since been experimentally observed.

Soon after the realization of graphene as the first two-dimensional crystal, single layers of transition metal dichalcogenides emerged as a new class of two-dimensional semiconductor with unique and interesting properties. To advance the fundamental understanding of these new materials, measurement of their linear optical response was essential. The authors of this paper made a significant contribution by experimentally determining the complex in-plane dielectric functions for four transition metal dichalcogenide monolayers: MoS₂, MoSe₂, WS₂, and WSe₂. They were able to establish the presence of strong light-matter interaction in these materials. They also studied the dielectric response of the spin-split exciton transitions and furnished valuable insights into the behavior of two-dimensional excitons. The experimental exciton splittings were found to be in good agreement with the values predicted by theoretical calculations. Comparing the optical response of dichalcogenide monolayers with that of the corresponding bulk materials, the authors were able to show how reduced dimensionality influences optical properties in these systems. Due to the importance of these results, this paper has also been serving as a major reference for further investigation of two-dimensional transition metal dichalcogenides.

Led by the pioneer of the field Shou-Cheng Zhang, the trio of authors work out the effective Chern-Simons theory description of four-dimensional time-reversal invariant topological insulators. The description is adapted to the more natural cases of two and three dimensions by means of a dimensional-reduction procedure, furnishing along the way a framework to describe the response of a topological insulator to an electromagnetic field. It turns out that such a response is governed by axion electrodynamics which had been proposed to explain anomalies of the strong interaction in high-energy physics but was eventually discarded. Compared to the conventional Maxwell Lagrangian, axion electrodynamics involves a term which couples the electric and the magnetic field in analogy to the coupling in multiferroic materials. This leads to a number of fascinating effects, in which the electric field generates transverse currents at the boundaries that produce net magnetization, while the magnetic field induces nonzero charge polarization. However, unlike multiferroics, in the case of topological insulators the applied electric and magnetic fields produce universal quantized topological contributions to the magnetization and to the charge polarization, respectively. Physically, the magnetization is quantized because the surface current associated with it is the quantized Hall current on the boundary induced by the electric field.

These findings paved the way to a number of follow-up studies, unraveling further highly counterintuitive consequences of axion electrodynamics as applied to topological materials.

The 1990s saw the emergence of photonic crystals as a promising and rapidly expanding area of research. Photonic crystals are artificial media with periodically modulated refractive index at sufficient contrast to allow previously unseen band structure phenomena with photons in the lattice, in analogy with electronic band structure effects in semiconductors. One of the early demonstrations of such exotic behavior at optical wavelengths is the superprism phenomenon, discovered and so labeled by Hideo Kosaka and colleagues, whereby a three-dimensional photonic crystal fabricated on a silicon substrate causes an extremely large and negative refraction to an incident beam of light, at least a hundred times stronger than that in a conventional prism. The phenomenon arises from the strong modification of the group velocity in the photonic crystal in stark contrast to conventional bending of light in a prism wherein the phase velocity is modified.

Steven White expounds with great clarity a new variety of renormalization group approach to strongly correlated many-body quantum systems. If one ever needs a concrete example of the proverbial paradigm shift, his idea is it. It takes the guise of a shift in focus: In real-space block renormalization, instead of tracking down the lowest-lying states for the effective Hamiltonian, one should focus on the most significant eigenstates of the block density matrix.

The highly successful methodology – that of the density-matrix renormalization group or DMRG – is conceptually complete and operationally demonstrated in two complementary papers, a Milestone Letter and the present PRB Milestone article. The method, initially applied to finite and infinite Heisenberg spin chains, has been used for almost any one-dimensional quantum lattice system of interest, much like White stated as possible without any exaggeration. Further generalizations have allowed to deploy DMRG in studies of dynamic and finite-temperature properties as well as to reach out to quantum lattice systems in higher dimensions.

Multiferroic behavior and magnetoelectric coupling have been of great interest to the condensed matter and materials physics communities for decades, especially since such coupling would allow for a state with ferromagnetic polarization to be controlled by electric polarization or vice versa. To make multiferroics viable for technological purposes, one hopes to identify a mechanism for a reversible and controllable change in a property that would act as a switch for the manipulation and storage of information. Clearly, one needs to find materials that would harbor the requisite behavior.

Perovskite materials are known to exhibit a number of interesting and exciting phenomena ranging from various structural and electronic transitions to multiferroicity and magnetoelectric coupling. While transition-metal oxides have dominated the perovskite landscape over the years, manganites have become some of the best materials in this class of compounds. In order to determine the magnetoelectric coupling in such materials, it is important to measure the change in the capacitance with magnetic field (magnetocapacitance effect). In this Milestone, T. Kimura and colleagues report for the first time a large negative magnetocapacitance effect in the vicinity of the ferromagnetic transition, demonstrating the possibility for reliable control of dielectric properties by magnetic fields in multiferroics.

In a landmark PRB paper published in 1976 John Hertz developed a renormalization group approach to phase transitions and critical phenomena for quantum mechanical systems at zero temperature. Hertz showed that at zero temperature the statics and dynamics are interlinked via the dynamical critical exponent z so that for a large class of critical systems the simple but far-reaching underlying principle of the quantum-to-classical crossover is valid. This is because the Landau-Ginzburg-Wilson functional of a d-dimensional quantum system can be cast as a (d + z)-dimensional classical system with a finite imaginary time dimension. Accordingly, the upper critical dimension of space that separates classical from nonclassical scaling behavior is not d_c = 4 as for classical phase transitions, but 4-z. In the particular case when the (Matsubara) frequency enters the functional in the same way as the wave vector, which is to say that time and space are present in the same way in that functional, the crossover is of the especially simple and appealing type d → d+1, bringing down the upper critical dimension to d = 3 - the physical dimensionality of greatest immediate interest.

In a significant development in 1993, Andrew Millis explored magnetic phase transitions in systems of itinerant fermions at zero temperature in two and three dimensions by renormalization-group techniques and found that for such systems important modifications of the 1976 scheme were needed. Notably, he determined the conditions that allow one to integrate out the fermions whereby the critical region in the disordered phase can be described using an effective bosonic theory for the fluctuations of the ordering field only. The scaling equations and the resulting phase diagrams were found yet again to depend crucially upon both the dimension d as well as the dynamical critical exponent z of the zero-temperature transition.

Peter B. Johnson and Robert W. Christy measure the optical parameters of noble metals (copper, silver, and gold) for photon energies ranging from 0.5 - 6.5 eV. In this work, the authors use a new technique for accurately extracting optical constants at a given photon energy by first carrying out a combination of three transmission and reflection measurements to determine the thickness of the thin-film metal sample, followed by an appropriate pair of measurements to determine the real and imaginary parts of the dielectric constant. Optical constants are important not only because they govern how light is scattered, transmitted, and absorbed, but also because they elucidate the electronic structure of matter. Thus, the authors are able to deduce from their data the free-electron optical effective masses and relaxation times as well as the interband contribution to the imaginary part of the dielectric constant.

As a basis for comparison, the study has become the “gold standard” for further optical characterization of noble metals. In addition, the paper has become an indispensable reference that continues to be relevant for the development of various fields of optics including plasmonics, metamaterials, optoelectronics, and nonlinear optics. At this time, the article is the highest cited experimental paper in the half-century history of Physical Review B.

Sadly, Robert W. Christy passed away at the age of 97 on June 16, 2020 while this PRB Milestone presentation was in preparation.

MoS₂ is a layered material which can be exfoliated to produce from few-layer flakes down to a monolayer two-dimensional crystal. It has attracted a great deal of research interest ever since graphite was successfully exfoliated to obtain its single layer, now well known as graphene. The experimental realization of graphene opened the door for the discovery of other two-dimensional materials, among which MoS₂ soon gained wider interest. Graphene turned out to be a semimetal with a zero band overlap, and the community began actively looking for a semiconducting two-dimensional system. In this regard, MoS₂ was a promising material since its bulk was known to be an indirect band gap semiconductor, making it likely that it might yield a semiconducting two-dimensional system. Researchers ultimately showed that both single and bilayer MoS₂ preserve the semiconducting character, yet there remained ambiguity about whether both of these systems were indirect gap semiconductors.

The paper by Tawinan Cheiwchanchamnangij and Walter Lambrecht made a significant contribution in settling this ambiguity. They performed ab initio band structure calculations at a quasiparticle self-consistent GW level with spin-orbit coupling and they confirmed that bilayer MoS₂ was an indirect gap semiconductor which transitioned to a direct gap semiconductor in the single-layer form. They also calculated the exciton binding energy in the monolayer and bilayer systems, and found that the absolute value of the exciton gap and its splitting agreed well with the experimentally measured optical absorption spectra. They established that in the monolayer system the splitting arose entirely due to spin-orbit coupling whereas in the bilayer system it was due to the combined effect of spin-orbit and interlayer coupling.

The technological importance of the interface between silicon and its oxide, SiO₂, has made it one of the most studied interfaces in condensed matter physics. Already back in 1988, a large array of experimental techniques had been brought to bear on the atomistic understanding of its structure. Unfortunately, this had only led to numerous inconsistent results and contradictory interfacial models.

In this state of conflicting ideas, F. J. Himpsel and colleagues at IBM and G. Hollinger in Lyon utilized the forefront second generation synchrotron light source facility at Brookhaven National Laboratory to perform what has since become a classic study of the microscopic structure of the SiO₂/Si interface.

They measured the Si 2p core level intensities on oxidized Si(111) and Si(100) surfaces with different depth sensitivity using a varying photon energy, and carefully analyzed the intensities of the series of peaks originating from Si oxidation states intermediate between Si⁰ from silicon and Si⁴⁺ from SiO₂. By quantitatively determining the escape depth of the photoelectrons from, and the photoionization cross section for, the Si 2p core level electrons in both Si and SiO₂ as well as the oxidation states in-between, they were able to extract the density of Siⁿ⁺ atoms at the interfaces. This allowed them to construct microscopic structural models of the interface that extended beyond idealized atomically abrupt models.

The complete, comprehensive, and quantitative analysis put forward in this work has withstood the test of time, and has become the standard for core level spectroscopic analyses of other buried interfaces.

John E. Littlewood famously asserted that in mathematics a dissertation could be condensed into two lines. The work of Robert B. Laughlin on the theory of the Quantum Hall Effect proves that a groundbreaking, seminal idea in theoretical physics can be laid out on two pages. In 1980 Klaus von Klitzing and his collaborators made a puzzling discovery – the Hall conductivity of high-quality two-dimensional electron systems was found to be quantized in units of e²/h to an astonishing precision, far exceeding what one would expect given the presence of impurities and other imperfections in the system. The robust nature of the quantization suggested that there should be a fundamental universal argument behind it. The enigma was resolved by Laughlin who provided an elegant explanation of the effect using general principles of gauge invariance.

Lee, Yang, and Parr recast the original Colle-Salvetti correlation-energy formula into a density functional form with the second order gradient expansion, leading to the so-called LYP correlation functional. Its combination with the Becke-88 exchange functional, delivered in another remarkable Milestone paper in a PRB sister journal, Phys. Rev. A 38, 3098 (1988), yields the widely used GGA BLYP functional and the most popular hybrid density functional B3LYP. This PRB Milestone paper is among the top 100 most cited papers in science of all time and has contributed to a highly effective and accurate recipe for carrying out density functional calculations in computational chemistry, physics, and materials science.

• Word cloud image by Peter Elliott

Electric polarization properties of materials and related critical phenomena, such as the paraelectric to ferroelectric phase transition, have been in the focus of much interest due to an ever expanding range of potential technological applications.

Developments in the theoretical understanding of the paraelectric to ferroelectric phase transition predicted that for materials where it was expected to occur at low temperatures, quantum-mechanical fluctuations would act to suppress the emergence of the ordered, ferroelectric phase. The would-be ferroelectric would remain a paraelectric all the way down to T = 0 K.

This Milestone paper reports the experimental discovery of such a quantum paraelectric, SrTiO₃. Müller and Burkard set up measurements of the dielectric constant at temperatures down to 35 millikelvin. Instead of developing a peak at the temperature of the anticipated phase transition to a ferroelectric state, the dielectric constant remains very high and constant for temperatures below 4 K, demonstrating the predicted suppression of ferroelectric ordering and, hence, the quantum paraelectric effect.

Müller’s deep understanding of the promise of perovskites such as SrTiO₃ led to the discovery with J. G. Bednorz in 1986 of a new class of superconductors, which earned them the Nobel Prize for Physics in the year 1987.

Luc Berger demonstrates that a significant electron-magnon coupling is at play at the interface between a normal or nonmagnetic (N) and a ferromagnetic (F) metal. For an F-N-F plane geometry sandwich, he examines in great detail the circumstances and consequences of a dc current, traversing the structure at normal incidence and transferring vector spin between the ferromagnetic layers (for N thinner than its spin diffusion length and some other experimentally feasible conditions). The current excites precession of the F-layer magnetizations, whose attenuation (damping) is significantly enhanced near the interfaces. Berger provides a rare quantitative insight into the microscopical origin of Gilbert damping, which even with the transparent physical model at hand depends on a fundamental constant, eleven parameters, and an average over the active half of the Fermi surface.

Berger observes that the dc current effectively counteracts the spin damping to an extent such that for a sufficiently large current density the damping turns negative. Hence, a spontaneous precession of the magnetization vector is predicted to arise above a threshold. The F-N-F device then acts as a spin-wave emitting diode, quite analogous to an injection laser. Accordingly, Berger dubs the process spin-wave amplification by stimulated emission of radiation, that is, SWASER. Additionally, he demonstrates the possibility for high-frequency switching with potential technological applications to high-speed, high-density storage and memory that have since been pursued with an ever increasing intensity.

In the same year, similar ideas were developed and reported independently by J. C. Slonczewski in JMMM as duly noted in Reference [16] of this PRB Milestone.

Bosonic fluids such as helium-4 at ultralow temperatures exhibit a superfluid state wherein the flow is dissipationless and the viscosity goes to zero. What happens when helium-4 is absorbed into a porous disordered medium such as porous Vycor glass?

In this PRB Milestone work, Fisher and colleagues address this question by exploring the phase diagrams and phase transitions of bosons with short-ranged repulsive interactions, moving in periodic and/or random external potentials at zero temperature.

Within a periodic potential, bosons at T=0 exhibit a superfluid phase and Mott insulating phases characterized by zero compressibility, integer (or commensurate) boson densities, and the existence of a gap for particle-hole excitations. In the presence of disorder, a new phase is found to exist in between the superfluid and Mott insulating phases. This insulating “Bose glass” phase is quite peculiar, being characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility.

Ferrari and Robertson introduce a three-stage classification scheme of disorder that allows assessment of all visible Raman spectra of carbons during amorphization of graphite. The evolution of this amorphization process occurs in the following stages indicated by the three shaded regions on the left side of the ternary phase diagram of amorphous carbons shown in the figure: (i) graphite to nanocrystalline graphite, (ii) nanocrystalline graphite to amorphous carbon with low sp³ content, and (iii) amorphous carbon with low sp³ content to amorphous carbon with high sp³ content. Through this study, the authors are able to classify all the available visible Raman data, in which disorder is introduced into graphite in three stages and the Raman spectra are measured. Furthermore, they show how this description can be extended to hydrogenated amorphous carbons.

The quantification of disorder or amorphization using Raman spectroscopy in the large family of sp² carbon structures that includes graphite and graphene has laid down a firm foundation for the analysis of their fundamental properties and applications. In graphene, for example, this knowledge has provided a crucial step towards the understanding of the limits to its mobility. While Raman studies of graphite began half a century ago, the arrival of graphene has triggered a far-reaching resurgence of Raman spectroscopy for applications to two-dimensional materials, including other single or few-layer van der Waals materials, such as transition metal dichalcogenides.

This PRB Milestone paper comes at an opportune time to commemorate also the 50^th anniversary of Raman spectroscopy of graphite that began with the work of F. Tuinstra and J. L. Koenig, published in J. Chem. Phys. 53, 1126 (1970).

Electric polarization is an area of great interest to the condensed matter community. The ability to produce a shift in electronic density in a solid material such that an electric dipole is formed has many potential uses. There are various ways to obtain a state with electric polarization. However, technological applications require not just the presence of one state, but the ability to switch between at least two such states. Therefore, it is not so much the mere presence of electric polarization but the ability to switch it that makes a material technologically viable. Ferroelectrics are known to exhibit the desired combination of properties, whereby the switching mechanism comes with the application of an external field.

While ferroelectrics have been known experimentally for many decades, the ability to understand and calculate their properties from scratch – that is, from quantum mechanical first principles – had made very little progress until this groundbreaking contribution by King-Smith and Vanderbilt that derives “simple formulas for calculating finite changes in the polarization of a crystalline solid” as the authors modestly put it. In this Rapid Communication, they usher in firm guidance for theoretical investigations of potentially ferroelectric materials through the use of density functional theory. In the years to follow, the understanding of polarization in crystalline solids that ensued has allowed researchers to canvas many materials and to provide exciting avenues for the experimental realization of ferroelectrics.

The discovery of conducting polymers in 1976 by Alan Heeger, Alan MacDiarmid, and Hideki Shirakawa was a breakthrough of lasting impact in chemistry, applied physics, and condensed matter physics that was awarded the Nobel Prize in Chemistry in the year 2000. Soon after the initial discovery, one of the recipients of that prize, Alan Heeger, in collaboration with Wu-Pei Su and John Robert Schrieffer (of BCS fame) explored the unusual electronic properties of polyacetylene. Within a framework, nowadays known as the SSH model, they reduced the problem to a quasi-one-dimensional tight-binding description of electrons that hop from one carbon site to another along the zigzag carbon chain, while the vibrational degrees of freedom in the problem were treated within the Born-Oppenheimer approximation.

The seemingly simple setting proves to be the source of a number of counterintuitive and exciting phenomena. The chain undergoes a Peierls transition that results in spontaneous dimerization, with an alternating pattern of short and long bonds between the carbon atoms. Importantly, there are two such equivalent patterns that can form domains and correspond to topologically distinct electronic states. By analyzing the electronic states corresponding to the domain walls, SSH reveal in a lucid form both spin-charge separation and topological zero modes – to use more recent terminology for two of the centerpieces of modern condensed matter physics.

Laying down another massive brick into the foundation of topological band theory, Fu and Kane extend the latter to three dimensions and give a precise and simple way to calculate topological invariants in systems with inversion symmetry. They show that despite a potentially complex global band structure the topological nature of the system can be deduced from the knowledge of energy states at eight time-reversal and parity invariant points in the Brillouin zone. Analyzing several specific materials as candidates, they predict the Bi_1-xSb_x alloy within a certain range of x to be a three-dimensional topological insulator. This study enabled a number of ab initio methods to efficiently identify materials realization of topological insulators.

The study of phase transitions in two dimensions has a rich history. The exact solution to the 2D Ising model by Onsager revealed a phase transition to an ordered phase. On the other hand, Mermin-Wagner and, independently, Hohenberg work ruled out a transition to such a phase in 2D isotropic models with continuous symmetry like the Heisenberg one.

It then came to Halperin and Nelson who, building on the Nobel Prize winning work of Kosterlitz and Thouless, showed in two papers published in Physical Review Letters and in Physical Review B (this very Milestone) that the melting of a 2D solid involves an intermediate phase with both liquidlike and solidlike properties. The discovery of this so-called hexatic phase, which was later observed in experiments and simulations, demonstrated that melting in 2D systems, such as thin films, is fundamentally different from that in 3D materials.

At about the same time, Allan Peter Young contributed independently to the theory of 2D melting on the triangular lattice, publishing his work also in Physical Review B and, among other things, providing the correct critical exponent for the melting of the solid. The finalized authoritative and influential theory of low-dimensional melting has since been known as the KTHNY theory with the initials of Kosterlitz, Thouless, Halperin, Nelson, and Young to remind us of the substantial contribution of each of these authors to the solution of a fundamental problem of both condensed matter and statistical physics.

While tunneling in planar junctions allowing one-dimensional descriptions was well-understood, the development of the scanning tunneling microscope (STM) in the early 1980’s required new approaches to be added to the theoretical toolbox. In this study, the authors present an approach to calculate simulated STM images. Using a complete description of the surface electronic structure in terms of the local density of states as well as an idealized model for the STM tip, it allows for a quantitative comparison with newly available atomically resolved experimental STM images. The computationally tractable Tersoff-Hamann method has since been instrumental in elucidating a wide variety of surface structures observed with scanning tunneling microscopy.

The authors derive exact scaling theory for the electronic conduction of a generic random chain, based on the quantum scattering approach in the pragmatic and lucid form, suggested by Landauer. The broader conceptual framework of the effect of quantum interference in random media for any type of wave propagation was laid out by the leading author in his Nobel Prize winning work “Absence of Diffusion in Certain Random Lattices”, Phys. Rev. 109, 1492 (1958). Nowadays, it is widely referred to as Anderson localization.

Sir Neville Mott who shared with Anderson and van Vleck the 1977 Nobel Prize and, independently, Thouless had shown that one-dimensional electronic systems are always localized, while the presence of extended states in two dimensions remained a matter of controversy until the 1979 work by the “Gang of the Four” came to light (E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979)).

Here, the authors develop their new method by exploring the competition between the two relevant physical length scales in the problem, the mean free path of the electrons and the coherence length characterizing the degree of randomness in the system. Guided by deep physical intuition, they identify the appropriate scaling variable as the logarithm of unity plus the resistance in dimensionless units. A complete solution of the one-dimensional problem is then developed. The exact scaling function β that derives from it is remarkably compact and simple, confirming the perturbative form hypothesized a year earlier.

We dedicate this Milestone to the memory of the condensed matter physics legend Phil Anderson (1923-2020).

Two-dimensional (2D) carbon, now more popularly known as graphene, is a honeycomb lattice in which one carbon atom sits at each vertex of the underlying hexagonal building block. Long-range ordering of such 2D structures at a finite temperature was believed to be impossible because of the detrimental effect of thermal fluctuations. This expectation was altered by the experimental realization of graphene in 2004 that motivated researchers to unearth new 2D materials, physics, and applications.

One of the most intriguing features of a perfect graphene plane is that the electronic energy-momentum relation in this system is linear, which leads to zero effective mass of carriers and vanishing density of states close to the Fermi energy. In contrast, the energy-momentum relations in normal metals and semiconductors are quadratic, leading to a finite effective mass and density of states at the Fermi level. This fundamental difference gives rise to the unusual and interesting physical properties of graphene.

In reality, the positional ordering of carbon atoms in a two-dimensional honeycomb lattice is not perfect above zero temperature, and the 2D systems are bound to have lattice defects such as vacancies. For the advancement of 2D research, it was imperative to understand the nontrivial physical effects of lattice imperfections on the properties of graphene. This is where the paper by Peres, Guinea, and Castro Neto made its lasting mark: The authors presented a comprehensive study of the electronic properties of disordered graphene in the presence of electron-electron interaction and as a function of external parameters such as temperature, gate voltage, and magnetic field. They showed that in graphene the response of massless fermions to disorder is significantly different compared to the response of fermions to disorder in metals and semiconductors. Their theory was able to explain the relevant contemporary experimental results, and its predictions were verified by subsequent experiments. This study has continued to serve as a “go to” reference in the pursuit and understanding of other two-dimensional materials.

Kresse and Furthmüller present an iterative matrix diagonalization scheme and a charge density mixing scheme for the determination of the electronic ground state of a condensed matter system within the Kohn-Sham density functional approach. Based on this and three other major publications by Kresse and colleagues from 1993 and 1996, the proposed techniques have been expertly built into the computational software now widely known and used as the Vienna Ab initio Simulation Package (VASP). The resulting flexible tool has enabled the reliable prediction of the electronic structure and other important ground-state properties of an impressive range of materials. The package is by far the most popular and widely used computer program for atomic-scale materials modeling.

In the early, explosive years of metamaterials research, the quest for electromagnetic media with negative refractive index galvanized a large and diverse community of physicists, chemists, engineers, and materials scientists. In this paper, the authors demonstrate that the traditional procedure of obtaining material parameters (electric permittivity ε and magnetic permeability μ) from transmission and reflection coefficients can be successfully applied to metamaterials so as to assign unambiguous values for ε and μ. Furthermore, for a particular configuration of periodic arrangements of wires and split ring resonators, they reveal negative refractive index at frequencies where both the real parts of ε and μ are simultaneously negative. The technique has become the standard method for retrieving effective material parameters from experimental transmission/reflection data across a range of frequencies of practical importance.

The complex oxide BiFeO₃, also known as BFO, has become one of the most pivotal multiferroic materials due to its large ferroelectricity and spiral magnetic order. This Milestone paper implements density functional theory to shed light on the structural and electronic properties. The authors demonstrate the strength of the first-principles approach. They achieve an excellent agreement with experiment by predicting the insulating and antiferromagnetic behavior of BFO, while also revealing its large electric polarization. This study has proven to be indispensable as density functional theory methodology as well as execution for much of the progress with the prediction and understanding of multiferroic materials over the fifteen years since its publication in PRB.

The discovery in 1987 of high-T_c superconductivity above liquid nitrogen temperatures in layered copper-oxide compounds opened up a vast new field of research that remains as vigorous as it is challenging. Superconductivity in these materials emerges when a Mott insulator with long-range antiferromagnetic order is doped with either holes or electrons.

In this Rapid Communication, Fuchun Zhang and Thomas Rice derive a single-band effective Hamiltonian description of superconductivity, according to which an oxygen hole forms a local singlet with the spin of a neighboring copper ion and then moves on through the lattice carrying the supercurrent. This widely studied model is one of the simplest descriptions of the Coulomb interaction between holes confined in copper-oxide planes that is crucial to explaining superconductivity in the cuprates.

The discovery in 2006 of the first iron-based superconductor by the group of Hideo Hosono was followed by explosive expansion of the field, with several families of iron-based pnictide and chalcogenide superconductors discovered within several years. One of the structurally simplest iron-based compounds to exhibit superconductivity was iron selenide, but application of extremely high pressure was required to obtain a high superconducting transition temperature.

In this Rapid Communication, Jiangang Guo and collaborators report on the resolution of this problem by synthesizing potassium-intercalated FeSe, a superconductor with a critical temperature above 30 K at ambient pressure. Intercalated FeSe compounds proved to be unique in their electronic properties among iron-based superconductors and became a prototypical platform for the exciting, ongoing exploration of the interplay between superconductivity and nematicity.

Recognizing the fundamental role of time-reversal and particle-hole symmetries, Altland and Zirnbauer analyze the structure of quadratic Hamiltonians of fermionic systems. They show that, despite the apparent multitude of options, the symmetry classification can be narrowed down to ten symmetry classes (the tenfold way). These provide the basis for what is now known as the periodic table of topological insulators and superconductors. This study led the way to the exploding field of classification and scrutiny of topological phases in terms of symmetry and number of spatial dimensions.

Developing new functionalities in materials, built with atomic precision via layer-by-layer deposition, is a significant endeavor of materials research today. One of the earliest successes of this concept has been the discovery of the giant magnetoresistance (GMR) effect in 1988.

In a landmark paper published as a Rapid Communication in Physical Review B, Peter Grünberg and colleagues showed that electrical resistance along a sample of nanometer-thick nonmagnetic chromium sandwiched between two layers of magnetized iron is significantly different depending upon whether the iron layers are magnetized parallel or antiparallel to each other. When the two iron layers are magnetized parallel to each other, conduction electrons with spin opposite to that of the iron layers are scattered less producing an overall lower resistance than in the antiparallel case where both spin-up and spin-down conduction electrons are scattered equally.

Around the same time, this phenomenon was independently discovered by Albert Fert and colleagues who published their results in Physical Review Letters. The discovery of the GMR effect not only brought the 2007 Nobel prize for Physics to Grünberg and Fert, but also led to a revolutionary advance in data storage technology. Within a few years, GMR-based magnetic disk drive technology became the industry standard in what is arguably one of the fastest journeys from fundamental discovery to consumer applications.

Manipulation of electron spin in nanostructures to discover new phenomena and to advance device technology remains a burgeoning field in condensed matter physics, materials research, and solid state quantum information.

What happens to the quantum mechanical motion of electrons in a crystalline lattice when an external magnetic field is applied? This problem exhibits a remarkable and puzzling property. The general nature of the energy spectrum depends on whether the magnetic field, expressed in natural units, is a rational number. This defies common wisdom, as the energy of an electron is expected to be a smooth function of external parameters, such as the magnetic field. In this Milestone paper, Hofstadter reveals that the electron spectrum plotted as a function of the magnetic field exhibits a recursive, self-replicating structure. It is widely known as Hofstadter’s butterfly – one of the most impressive and iconic images in condensed matter physics. The fractal nature of Hofstadter’s butterfly is very delicate. It can be destroyed by impurities or magnetic field fluctuations, leading its creator to express some reservations about the possibility for its experimental observation. Yet, it has been detected in recent years in a number of systems as diverse as microwave guides, patterned GaAs/AlGaAs heterostructures, moiré superlattices, and cold atom systems in optical lattices. No less importantly, this type of analysis has played a major role in the understanding of the integer quantum Hall effect.

Known today as the Hicks-Dresselhaus model, one of quite a few theories named after Mildred Dresselhaus, the paper outlines a way to improve the most important parameter of thermoelectric materials via the implementation of low-dimensional, quantum-well structures. This truly groundbreaking study became central to her induction into The National Inventors Hall of Fame in 2014. To honor her remarkable scientific career and inspiring community legacy, the American Physical Society has recently established the Millie Dresselhaus Fund for Science & Society.

Kenneth G. Wilson outlines a profound framework for conceptual and quantitative understanding of critical behavior in most diverse condensed matter systems, in which the relevant correlation length close to a phase transition grows to span the whole system. Wilson’s renormalization group theory revolutionizes the way scientists handle a great variety of problems across all of physics, where fluctuations at all scales matter and have to be dealt with. Kondo physics, fully developed turbulence, and elementary particle physics fall in this category. This development led to Wilson’s Nobel Prize for Physics award in 1982.

Kenneth G. Wilson outlines a profound framework for conceptual and quantitative understanding of critical behavior in most diverse condensed matter systems, in which the relevant correlation length close to a phase transition grows to span the whole system. Wilson’s renormalization group theory revolutionizes the way scientists handle a great variety of problems across all of physics, where fluctuations at all scales matter and have to be dealt with. Kondo physics, fully developed turbulence, and elementary particle physics fall in this category. This development led to Wilson’s Nobel Prize for Physics award in 1982.