Supersymmetry (SUSY) interchanges bosons and fermions but no direct evidences of it have been revealed in nature yet. In this letter, we observe that fluctuating pair density waves (PDW) consist of two complex order parameters which can be superpartners of the unavoidably-doubled Weyl fermions in three-dimensional lattice models. We construct explicit fermionic lattice models featuring 3D Weyl fermions and show that PDW is the leading instability via a continuous phase transition as short-range interactions exceed a critical value. Using renormalization group, we theoretically show that N=2 space-time SUSY emerges at the continuous PDW transitions in 3D Weyl semimetals, which we believe is the first realization of emergent (3+1)D space-time SUSY in microscopic lattice models. We further discuss possible routes to realize such lattice models and experimental signatures of emergent SUSY at the PDW criticality.
Supersymmetry plays a crucial role in superstring theory and high-energy physics, but has never been observed in experiments. Recently, an effective space-time supersymmetry was argued to emerge in the low-energy region by tuning Dirac or Weyl semimetal to approach a superconducting quantum critical point, at which the Dirac or Weyl fermion and the bosonic order parameter are both massless. Here, we study under what circumstances can space-time supersymmetry be realized at a quantum critical point. We demonstrate that the Yukawa-type coupling between the massless fermion and massless boson can…
Topological states of electrons present new avenues to explore the rich phenomenology of correlated quantum matter. Topological insulators (TIs) in particular offer an experimental setting to study novel quantum critical points (QCPs) of massless Dirac fermions, which exist on the sample's surface. Here, we obtain exact results for the zero- and finite-temperature optical conductivity at the semimetal-superconductor QCP for these topological surface states. This strongly interacting QCP is described by a scale invariant theory with emergent supersymmetry, which is a unique symmetry mixing boso…
We study the effect of disorder on the spacetime supersymmetry that is proposed to emerge at the quantum critical point of pair density wave transition in (2+1)D Dirac semimetals and (3+1)D Weyl semimetals. In the (2+1)D Dirac semimetal, we consider three types of disorder, including random scalar potential, random vector potential and random mass potential, while the random mass disorder is absent in the (3+1)D Weyl semimetal. Via a systematic renormalization group analysis, we find that any type of weak random disorder is irrelevant due to the couplings between the disorder potential and the…
We investigate the emergence of ${\cal N}=1$ supersymmetry in the long-range behavior of three-dimensional parity-symmetric Yukawa systems. We discuss a renormalization approach that manifestly preserves supersymmetry whenever such symmetry is realized, and use it to prove that supersymmetry-breaking operators are irrelevant, thus proving that such operators are suppressed in the infrared. All our findings are illustrated with the aid of the $ε$-expansion and a functional variant of perturbation theory, but we provide numerical estimates of critical exponents that are based on the non-perturba…
The WZW models describe the dynamics of the edge modes of Chern-Simons theories in three dimensions. We explore the WZW models which can be mapped to supersymmetric theories via the generalized Jordan-Wigner transformation. Some of such models have supersymmetric Ramond vacua, but the others break the supersymmetry spontaneously. We also make a comment on recent proposals that the Read-Rezayi states at filling fraction $ν=1/2,~2/3$ are able to support supersymmetry.
Luttinger's theorem connects a basic microscopic property of a given metallic crystalline material, the number of electrons per unit cell, to the volume, enclosed by its Fermi surface, which defines its low-energy observable properties. Such statements are valuable since, in general, deducing a low-energy description from microscopics, which may perhaps be regarded as the main problem of condensed matter theory, is far from easy. In this paper we present a unified framework, which allows one to discuss Luttinger theorems for ordinary metals, as well as closely analogous exact statements for to…
In contrast to ordinary symmetries, supersymmetry interchanges bosons and fermions. Originally proposed as a symmetry of our universe, it still awaits experimental verification. Here we theoretically show that supersymmetry emerges naturally in topological superconductors, which are well-known condensed matter systems. Specifically, we argue that the quantum phase transitions at the boundary of topological superconductors in both two and three dimensions display supersymmetry when probed at long distances and times. Supersymmetry entails several experimental consequences for these systems, suc…
No definitive evidence of spacetime supersymmetry (SUSY) that transmutes fermions into bosons and vice versa has been revealed in nature so far. Moreover, whether spacetime SUSY in 2+1 and higher dimensions can occur or emerge in generic microscopic models remains open. Here, we introduce a lattice realization of a \textit{single} Dirac fermion with attractive Hubbard interactions that preserves both time-reversal and chiral symmetries. By performing numerically-exact sign-problem-free determinant quantum Monte Carlo simulations, we show that the interacting single Dirac fermion in 2+1 dimensi…
When a Dirac semimetal is subject to a circularly polarized laser, it is predicted that the Dirac cone splits into two Weyl nodes and a nonequilibrium transient state called the Floquet Weyl semimetal is realized. We focus on the previously unexplored low-frequency regime, where the upper and lower Dirac bands resonantly couple with each other through multi-photon processes, which is a realistic situation in solid state ultrafast pump-probe experiments. We find a series of new Weyl nodes emerging in pairs when the Floquet replica bands hybridize with each other. The nature of the Floquet Weyl …
We show that supersymmetry emerges in a large class of models in 1+1 dimensions with both Z_2 and U(1) symmetry at the multicritical point where the Ising and Berezinskii-Kosterlitz-Thouless transitions coincide. To arrive at this result we perform a detailed renormalization group analysis of the multicritical theory including all perturbations allowed by symmetry. This analysis reveals an intricate flow with a marginally irrelevant direction that preserves part of the supersymmetry of the fixed point. The slow flow along this special line has significant consequences on the physics of the mul…
Inhomogeneous strain applied to lattice systems can induce artificial gauge fields for particles moving on this lattice. Here we demonstrate how to engineer a novel state of matter, namely an antiferromagnet with a Landau-level excitation spectrum of magnons. We consider a honeycomb-lattice Heisenberg model and show that triaxial strain leads to equally spaced pseudo-Landau levels at the upper end of the magnon spectrum, with degeneracies characteristic of emergent supersymmetry. We also present a particular strain protocol which induces perfectly quantized magnon Landau levels over the whole …
We show that a strongly interacting chain of Majorana zero modes exhibits a supersymmetric quantum critical point corresponding to the $c={7\over 10}$ tricritical Ising model, which separates a critical phase in the Ising universality class from a supersymmetric massive phase. We verify our predictions with numerical density-matrix-renormalization-group computations and determine the consequences for tunnelling experiments.
Proposed as a fundamental symmetry describing our universe, spacetime supersymmetry (SUSY) has not been discovered yet in nature. Nonetheless, it has been predicted that SUSY may emerge in low-energy physics of quantum materials such as topological superconductors and Weyl semimetals. Here, by performing state-of-the-art sign-problem-free Majorana quantum Monte Carlo (QMC) simulations of an interacting 2D topological superconductor, we show convincing evidences that the N=1 SUSY emerges at its edge quantum critical point (EQCP) while its bulk remains gapped and topologically nontrivial. Remark…
We propose a correspondence between the description of emergent Lorentz symmetry in condensed-matter systems and the established general effective field theory for Lorentz violation in fundamental theories of spacetime and matter. This correspondence has potential implications in both directions. We illustrate the proposal by investigating its consequences for the spectral and transport properties of Dirac and Weyl semimetals. Particular realizations of this framework give rise to Dirac nodal spectra with nodal lines and rings. We demonstrate a bulk-boundary correspondence between bulk topolog…
Emergent spacetime supersymmetry at 2D fractionalized quantum criticality
Institute for Advanced Study, Tsinghua University（清华大学高等研究院）；Center for Computational Quantum Physics, Flatiron Institute（平顿研究所计算量子物理中心）；Department of Physics and Engineering Physics, Tulane University（杜兰大学物理与工程物理系）
We present a real-space view of one-dimensional (1D) to three-dimensional (3D) topological materials with 13 representative samples selected from each class, including 1D trans-polyacetylene, two-dimensional (2D) graphene, and 3D topological insulators, Dirac semimetals, Weyl semimetals, and nodal-line semimetals. This review is not intended to present a complete up-to-date list of publications on topological materials, nor to provide a progress report on the theoretical concepts and experimental advances, but rather to focus on an analysis based on the valence-bond model to help the readers g…
We study conformal field theories with Yukawa interactions in dimensions between 2 and 4; they provide UV completions of the Nambu-Jona-Lasinio and Gross-Neveu models which have four-fermion interactions. We compute the sphere free energy and certain operator scaling dimensions using dimensional continuation. In the Gross-Neveu CFT with $N$ fermion degrees of freedom we obtain the first few terms in the $4-ε$ expansion using the Gross-Neveu-Yukawa model, and the first few terms in the $2+ε$ expansion using the four-fermion interaction. We then apply Pade approximants to produce estimates in $d…
We study a simple model of three-dimensional fermions close to a quadratic band touching point, built from the celebrated Luttinger single-particle Hamiltonian and an attractive contact interaction between the particles. Such a system displays a novel quantum critical point between the semimetallic and an s-wave superconducting phase at which the low-energy "Luttinger fermions" are inextricably coupled to the order parameter fluctuations. The quantum critical point is perturbatively accessible near four spatial dimensions, where it features nontrivial scaling with dynamical exponent 1<z<2 and …
Topological semimetals with superconducting properties provide an emergent platform to explore the properties of topological superconductors. We report magnetization, and magneto-transport measurements on high quality single crystals of transition metal dichalcogenide SnTaS2. It is a nodal line semimetal with superconducting transition below Tc = 2.9 K. Moderate anisotropy (3.1) is observed in upper critical fields along H||c and H||ab plane. In the normal state we observe large magneto-resistance and weak anti-localization effect that provide unambiguous confirmation of topological features i…
Magnetic Weyl semimetals are predicted to host emergent electromagnetic fields at heterogeneous strained phases or at the magnetic domain walls. Tunability and control of the topological and magnetic properties is crucial for revealing these phenomena, which are not well understood or fully realized yet. Here, we use a scanning SQUID microscope to image spontaneous magnetization and magnetic susceptibility of CeAlSi, a noncentrosymmetric ferromagnetic Weyl semimetal candidate. We observe large metastable domains alongside stable ferromagnetic domains. The metastable domains most likely embody …
We investigate the nature of the magnetic phase transition induced by the short-ranged electron-electron interactions in a Weyl semimetal by using the perturbative renormalization-group method. We find that the critical point associated with the quantum phase transition is characterized by a Gaussian fixed point perturbed by a dangerously irrelevant operator. Although the low-energy and long-distance physics is governed by a free theory, the velocities of the fermionic quasiparticles and the magnetic excitations suffer from nontrivial renormalization effects. In particular, their ratio approac…
Interface and Strain Control of Emergent Weyl Semimetallic Phase in SrNbO$_{3}$/LaFeO$_{3}$ Heterostructures
Institute of Physics, Bhubaneswar（布巴内斯瓦尔物理研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Center for Atomistic Modelling and Materials Design, Indian Institute of Technology Madras（马德拉斯印度理工学院原子建模与材料设计中心）；Condensed Matter Theory and Computational Lab, Department of Physics, Indian Institute of Technology Madras（马德拉斯印度理工学院物理系凝聚态理论与计算实验室）；Laboratoire Albert Fert, CNRS, Thales, Université Paris-Saclay（阿尔伯特·费特实验室，法国国家科学研究中心，泰雷兹，巴黎萨克雷大学）；School of Basic Sciences, Indian Institute of Technology Bhubaneswar（布巴内斯瓦尔印度理工学院基础科学学院）；School of Physical Sciences, National Institute of Science Education and Research（国立科学教育与研究学院物理科学学院）；Peter Grünberg Institute, Forschungszentrum Jülich（于利希研究中心彼得·格林伯格研究所）；Department of Physics, Indian Institute of Science Bangalore（班加罗尔印度理工学院物理系）
We examine the appearance of superconductivity in the strong-coupling limit of the Hubbard model on the pyrochlore lattice. We focus upon the limit of half filling, where the normal-state band structure realizes a $j=3/2$ semimetal. Introducing doping, we show that the pairing is favored in a $J=2$ quintet $E_g$ state. The attractive interaction in this channel relies on the fact that $E_g$ pairing on the pyrochlore lattice avoids the detrimental on-site repulsion. Our calculations show that a time-reversal symmetry-breaking superconducting phase is favored, which displays Bogoliubov Fermi sur…
We investigate the unconventional quantum phase transitions in Weyl semimetals. The emergent boson fields, coupling with the Weyl fermion bilinears, contain a Wess-Zumino-Witten term or topological $Θ$ term inherited from the momentum space monopoles carried by Weyl points. Three types of unconventional quantum critical points will be studied in order: (1) The transition between two distinct symmetry breaking phases whose criticality is beyond Landau's paradigm. (2) The transition between a symmetry breaking state to a topological ordered state. (3) The transition between $3d$ topological orde…
When a metal undergoes continuous quantum phase transition, the correlation length diverges at the critical point and the quantum fluctuation of order parameter behaves as a gapless bosonic mode. Generically, the coupling of this boson to fermions induces a variety of unusual quantum critical phenomena, such as non-Fermi liquid behavior and various emergent symmetries. Here, we perform a renormalization group analysis of the semimetal-superconductor quantum criticality in a three-dimensional anisotropic Weyl semimetal. Surprisingly, distinct from previously studied quantum critical systems, th…
We introduce (3+1) dimensional models of short-range-interacting electrons that form a strongly correlated many-body state whose low-energy excitations are relativistic neutral fermions coupled to an emergent gauge field, $\text{QED}_{4}$. We discuss the properties of this critical state and its instabilities towards exotic phases such as a gapless `composite' Weyl semimetal and fully gapped topologically ordered phases that feature anyonic point-like as well as line-like excitations. These fractionalized phases describe electronic insulators. They may be further enriched by symmetries which r…
We discuss the modified quantum electrodynamics from a time-reversal-breaking Weyl semimetal coupled with a $U(1)$ gauge (electromagnetic) field. A key role is played by the soft dispersion of the photons in a particular direction, say $\hat{z}$, due to the Hall conductivity of the Weyl semimetal. Due to the soft photon, the fermion velocity in $\hat{z}$ is logarithmically reduced under renormalization group flow, together with the fine structure constant. Meanwhile, fermions acquire a finite lifetime from spontaneous emission of the soft photon, namely the Cherenkov radiation. At low energy $…
Nodal line semimetals are characterized by nontrivial bulk-band crossings, giving rise to almost flat drumhead-like surface states (DSS), which provide an attractive playground where interaction can induce symmetry-broken states and potential emergent phases. Here, we show that electronic interaction causes Stoner ferromagnetic instability in the DSS while the bulk remains non-magnetic, which together with spin-orbit coupling drive the surface states into a 2D Chern insulator. We show that each DSS carries a half-monopole charge, which for systems containing two pieces of DSS yield a net Chern…
We introduce a model of Dirac fermions in 2+1 dimensions with a semimetallic, a quantum spin-Hall insulating (QSHI), and an s-wave superconducting (SSC) phase. The phase diagram features a multicritical point at which all three phases meet as well as a QSHI-SSC deconfined critical point. The QSHI and SSC orders correspond to mutually anti-commuting mass terms of the Dirac Hamiltonian. Based on this algebraic property, SO(5) symmetric field theories have been put forward to describe both types of critical points. Using quantum Monte Carlo simulations, we directly study the operator that rotates…
We discover three-dimensional intertwined Weyl phases, by developing a theory to create topological phases. The theory is based on intertwining existing topological gapped and gapless phases protected by the same crystalline symmetry. The intertwined Weyl phases feature both unconventional Weyl semimetallic (monopole charge>1) and higher-order topological phases, and more importantly, their exotic intertwining. While the two phases are independently stabilized by the same symmetry, their intertwining results in the specific distribution of them in the bulk. The construction mechanism allows us…
We report full vector mapping of local magnetization in CeAlSi, a Weyl semimetal in which both inversion and time-reversal symmetries are broken. The vector maps reveal unanticipated features both within domains and at their boundaries. Boundaries between domains form two kinds of walls with distinct topology and therefore different interactions with Weyl fermions. Domain walls aligned along the tetragonal axes, e.g. (100), exhibit emergent chirality forbidden by the bulk space group, while diagonal walls are non-chiral. Within the domains, we observe that the previously reported set of four e…
Quantum spin liquids (QSL) are exotic phases of matter that host fractionalized excitations. It is difficult for local probes to characterize QSL, whereas quantum entanglement can serve as a powerful diagnostic tool due to its non-locality. The kagome antiferromagnetic Heisenberg model is one of the most studied and experimentally relevant models for QSL, but its solution remains under debate. Here, we perform a numerical Aharonov-Bohm experiment on this model and uncover universal features of the entanglement entropy. By means of the density-matrix renormalization group, we reveal the entangl…
Supersymmetry is a symmetry between a boson and a fermion. Although there is no apparent supersymmetry in nature, its mathematical consistency and appealing property have led many people to believe that supersymmetry may exist in nature in the form of a spontaneously broken symmetry. In this paper, we explore an alternative possibility by which supersymmetry is realized in nature, that is, supersymmetry dynamically emerges in the low energy limit of a non-supersymmetric condensed matter system. We propose a 2+1D lattice model which exhibits an emergent space-time supersymmetry at a quantum cri…
We study a new class of topological charge density wave states exhibiting monopole harmonic symmetries. The density-wave ordering is equivalent to pairing in the particle-hole channel due to Fermi surface nesting under interactions. When electron and hole Fermi surfaces carry different Chern numbers, the particle-hole pairing exhibits a non-trivial Berry phase inherited from band structure topology independent of concrete density-wave ordering mechanism. The associated density-wave gap functions become nodal, and the net nodal vorticity is determined by the monopole charge of the pairing Berry…
We consider a model of Dirac fermions in $2+1$ dimensions with dynamically generated, anticommuting SO(3) Néel and Z$_2$ Kekulé mass terms that permits sign-free quantum Monte Carlo simulations. The phase diagram is obtained from finite-size scaling and includes a direct and continuous transition between the Néel and Kekulé phases. The fermions remain gapped across the transition, and our data support an emergent SO(4) symmetry unifying the two order parameters. While the bare symmetries of our model do not allow for spinon-carrying Z$_3$ vortices in the Kekulé mass, the emergent SO(4) invaria…
We study the physics of 3d supersymmetric abelian gauge theories (with small supersymmetry breaking perturbations) at finite density. Using mirror symmetry, which provides a natural generalization of the duality between the XY model and the abelian Higgs model but now including fermionic fields, we see many dynamical phenomena conjectured to be of relevance in condensed matter systems. In particular, we find examples of the emergence of a Fermi surface at low energies from hybridization of fermions localized at magnetic defects at high energies, as well as fractionalization of charged fermions…
We investigate the pressure evolution of the anomalous Hall effect in magnetic topological semimetal Co3Sn2S2 in diamond anvil cells with pressures up to 44.9-50.9 GPa. No evident trace of structural phase transition is detected through synchrotron x-ray diffraction over the measured pressure range of 0.2-50.9 GPa. We find that the anomalous Hall resistivity and the ferromagnetism are monotonically suppressed as increasing pressure and almost vanish around 22 GPa. The anomalous Hall conductivity varies non-monotonically against pressure at low temperatures, involving competition between origin…
We introduce and analyze a quantum spin/Majorana chain with a tricritical Ising point separating a critical phase from a gapped phase with order-disorder coexistence. We show that supersymmetry is not only an emergent property of the scaling limit, but manifests itself on the lattice. Namely, we find explicit lattice expressions for the supersymmetry generators and currents. Writing the Hamiltonian in terms of these generators allows us to find the ground states exactly at a frustration-free coupling. These confirm the coexistence between two (topologically) ordered ground states and a disorde…
Nd2Ir2O7 is a correlated semimetal with the pyrochlore structure, in which competing spin-orbit coupling and electron-electron interactions are believed to induce a time-reversal symmetry broken Weyl semimetal phase characterized by pairs of topologically protected Dirac points at the Fermi energy. However, the emergent properties in these materials are far from clear, and exotic new states of matter have been conjectured. Here we demonstrate optically that at low temperatures the free carrier spectral weight is proportional to T^2 where T is the temperature, as expected for massless Dirac ele…
The emergent Weyl fermions in condensed matter generally break the Lorentz invariance resulting in a tilted (type-I) or over-tilted (type-II) energy dispersion. The tilting energy spectrums can lead to exotic quantum interference effects in a junction set up. Here, we theoretically investigate the Josephson current in a Weyl superconductor-Weyl (semi)metal-Weyl superconductor junction of a time-reversal (TR) broken type-I Weyl semimetal. We demonstrate that the Cooper pairs of BCS-like pairing acquire a finite momentum in case of inversion symmetric tilt. Consequently, the system exhibits tilt…
Topology and strong electron correlations are crucial ingredients in emerging quantum materials, yet their intersection in experimental systems has been relatively limited to date. Strongly correlated Weyl semimetals, particularly when magnetism is incorporated, offer a unique and fertile platform to explore emergent phenomena in novel topological matter and topological spintronics. The antiferromagnetic Weyl semimetal Mn3Sn exhibits many exotic physical properties such as a large spontaneous Hall effect and has recently attracted intense interest. In this work, we report synthesis of epitaxia…
