Supplementary material comes first and main text is followed. 12 pages, 3 figures and 1 table for supplementary material. 13 pages and 4 figures for main text
Investigation of materials that exhibit quantum phase transition provides valuable insights into fundamental problems in physics. We present neutron scattering under pressure in a triangular-lattice antiferromagnet which has a quantum disorder in the low-pressure phase and a noncollinear structure in the high-pressure phase. The neutron spectrum continuously evolves through the critical pressure; a single mode in the disordered state becomes soft with the pressure, and it splits into gapless and gapped modes in the ordered phase. Extended spin-wave theory reveals that the longitudinal and transverse fluctuations of spins are hybridized in the modes because of the noncollinearity, and novel magnetic excitations are formed. We report a new hybridization of the phase and amplitude fluctuations of the order parameter in a spontaneously symmetry-broken state.
Using the density-matrix renormalization group, we determine the phase diagram of the spin 1/2 Heisenberg antiferromagnet on a honeycomb lattice with a nearest neighbor interaction J1 and a frustrating, next-neighbor exchange J2. As frustration increases, the ground state exhibits Neel, plaquette and dimer orders, with critical points at J2/J1 = 0.22 and 0.35. We observe that both the spin gap and the corresponding order parameters vanish continuously at both the critical points, indicating the presence of deconfined quantum criticality.
Recently it was argued that quantum phase transitions can be radically different from classical phase transitions with as a highlight the 'deconfined critical points' exhibiting fractionalization of quantum numbers due to Berry phase effects. Such transitions are supposed to occur in frustrated ('$J_1$-$J_2$') quantum magnets. We have developed a novel renormalization approach for such systems which is fully respecting the underlying lattice structure. According to our findings, another profound phenomenon is around the corner: a fluctuation induced (order-out-of-disorder) first order transiti…
Ca2Y2Cu5O10 is build up from edge-shared CuO4 plaquettes forming spin chains. From inelastic neutron scattering data we extract an in-chain nearest neighbor exchange J1 approximately -170 K and the frustrating next neighbor J2 approximately 32 K interactions, both significantly larger than previous estimates. The ratio alpha= J2/J1 approximately 0.19 places the system very close to the critical point alpha_c=0.25 of the J1-J2 chain, but in the ferromagnetic regime. We establish that the vicinity to criticality only marginally affects the dispersion and coherence of the elementary spin-wave-lik…
The antiferromagnetic Heisenberg model on the two-leg ladder with exchange interactions along the chains, rungs, and diagonals is studied using the Jordan-Wigner transformation and bond-mean-field theory. The inclusion of all three couplings introduces frustration to the system and depending on their relative strengths the ladder can adopt one of three possible magnetically-disordered gapped states. The phase diagram found in this mean-field approach is in very good agreement with the one calculated by Weihong and colleagues using the Lanczos exact diagonalization method. By analyzing the grou…
We study the ground-state (gs) phase diagram of the frustrated spin-1/2 $J_{1}$-$J_{2}$-$J_{3}$ antiferromagnet with $J_{2} = J_{3} =κJ_1$ on the honeycomb lattice, using coupled-cluster theory and exact diagonalization methods. We present results for the gs energy, magnetic order parameter, spin-spin correlation function, and plaquette valence-bond crystal (PVBC) susceptibility. We find a Néel antiferromagnetic (AFM) phase for $κ< κ_{c_{1}} \approx 0.47$, a collinear striped AFM phase for $κ> κ_{c_{2}} \approx 0.60$, and a paramagnetic PVBC phase for $κ_{c_{1}} \lesssim κ\lesssim κ_{c_{2}}$. …
Antiferromagnetic quantum spin systems can exhibit a transition between collinear and spiral ground states, driven by frustration. Classically this is a smooth crossover and the crossover point is termed a Lifshitz point. Quantum fluctuations change the nature of the transition. In particular it has been argued previously that in the two-dimensional (2D) case a spin liquid (SL) state is developed in the vicinity of the Lifshitz point, termed a Lifshitz SL. In the present work, using a field theory approach, we solve the Lifshitz quantum phase transition problem for the 2D frustrated XY-model. …
When magnetic order is suppressed by frustrated interactions, spins form a highly correlated fluctuating "spin liquid" state down to low temperatures. Magnetic order of local moments can also be suppressed when they are fully screened by conduction electrons through the Kondo effect. Thus, the combination of strong geometrical frustration and Kondo screening may lead to novel types of quantum phase transitions. We report low-temperature thermodynamic measurements on the frustrated Kondo lattice Pr$_2$Ir$_2$O$_7$, which displays a chiral spin liquid state below 1.5 K due to the frustrated inter…
We study a two-dimensional frustrated Heisenberg antiferromagnet on the windmill lattice consisting of triangular and dual honeycomb lattice sites. In the classical ground state the spins on different sublattices are decoupled, but quantum and thermal fluctuations drive the system into a coplanar state via an "order from disorder" mechanism. We obtain the finite temperature phase diagram using renormalization group approaches. In the coplanar regime, the relative U$(1)$ phase between the spins on the two sublattices decouples from the remaining degrees of freedom, and is described by a six-sta…
We report detailed measurements of the low temperature magnetic phase diagram of Er$_2$Ti$_2$O$_7$. Heat capacity and time-of-flight neutron scattering studies of single crystals, subject to magnetic fields applied along the crystallographic [110] direction, reveal unconventional low energy states. Er$^{3+}$ magnetic ions reside on a pyrochlore lattice in Er$_2$Ti$_2$O$_7$, where local XY anisotropy and antiferromagnetic interactions give rise to a unique frustrated system. In zero field, the ground state exhibits coexisting short and long range order, accompanied by soft collective spin excit…
We study the spin-1 honeycomb lattice magnets with frustrated exchange interactions. The proposed microscopic spin model contains first and second neighbor Heisenberg interactions as well as the single-ion anisotropy. We establish a rich phase diagram that includes a featureless quantum paramagnet and various spin spiral states induced by the mechanism of order by quantum disorder. Although the quantum paramagnet is dubbed featureless, it is shown that, the magnetic excitations develop a contour degeneracy in the reciprocal space at the band minima. These contour degenerate excitations are res…
Quantum phase transitions have captured the interest of a large community in condensed-matter and atom physics research. The common feature of these very different material classes lies in the fact that the competition between low-energy scales can be tuned by a nonthermal parameter, such as pressure, magnetic or electric field, and chemical composition for the condensed-matter systems. In heavy-fermion materials, the strong exchange J between f-electrons and conduction electrons can lead to quenching of the f-electron-derived (nearly) localized magnetic moments via the Kondo effect or, if J b…
We study the challenging thermal phase transition to stripe order in the frustrated square-lattice Ising model with couplings J1<0 (nearest-neighbor, ferromagnetic) and J2>0 (second-neighbor, antiferromagnetic) for g=J2/|J1|>1/2. Using Monte Carlo simulations and known analytical results, we demonstrate Ashkin-Teller criticality for g>= g*, i.e., the critical exponents vary continuously between those of the 4-state Potts model at g=g* and the Ising model for g -> infinity. Thus, stripe transitions offer a route to realizing a related class of conformal field theories with conformal charge c=1 …
A recent study revealed the dynamics of the charge sector of a one-dimensional quarter-filled electronic system with extended Hubbard interactions to be that of an effective pseudospin transverse-field Ising model (TFIM) in the strong coupling limit. With the twin motivations of studying the co-existing charge and spin order found in strongly correlated chain systems and the effects of inter-chain couplings, we investigate the phase diagram of coupled effective (TFIM) systems. A bosonisation and RG analysis for a two-leg TFIM ladder yields a rich phase diagram showing Wigner/Peierls charge ord…
Low-temperature magnetization curves and thermodynamics of a frustrated spin-1/2 Heisenberg triangular bilayer with the antiferromagnetic intradimer interaction and either ferromagnetic or antiferromagnetic interdimer interaction are investigated in a highly frustrated parameter region, where localized many-magnon eigenstates provide the most dominant contribution to magnetothermodynamics. Low-energy states of the highly frustrated spin-1/2 Heisenberg triangular bilayer can be accordingly found from a mapping correspondence with an effective triangular-lattice spin-1/2 Ising model in a field. …
Motivated by the proposal of topological quantum paramagnet in the diamond lattice antiferromagnet NiRh$_2$O$_4$, we propose a minimal model to describe the magnetic interaction and properties of the diamond material with the spin-one local moments. Our model includes the first and second neighbor Heisenberg interactions as well as a local single-ion spin anisotropy that is allowed by the spin-one nature of the local moment and the tetragonal symmetry of the system. We point out that there exists a quantum phase transition from a trivial quantum paramagnet when the single-ion spin anisotropy i…
This review article is devoted to the interplay between frustrated magnetism and quantum critical phenomena, covering both theoretical concepts and ideas as well as recent experimental developments in correlated-electron materials. The first part deals with local-moment magnetism in Mott insulators and the second part with frustration in metallic systems. In both cases, frustration can either induce exotic phases accompanied by exotic quantum critical points or lead to conventional ordering with unconventional crossover phenomena. In addition, the competition of multiple phases inherent to fru…
We report detailed dc and ac magnetic susceptibilities, specific heat, and thermal conductivity measurements on the frustrated magnet ZnCr$_2$Se$_4$. At low temperatures, with increasing magnetic field, this spinel material goes through a series of spin state transitions from the helix spin state to the spiral spin state and then to the fully polarized state. Our results indicate a direct quantum phase transition from the spiral spin state to the fully polarized state. As the system approaches the quantum criticality, we find strong quantum fluctuations of the spins with the behaviors such as …
CePdAl with Ce $4f$ moments forming a distorted kagomé network is one of the scarce materials exhibiting Kondo physics and magnetic frustration simultaneously. As a result, antiferromagnetic (AF) order setting in at $T_{\mathrm{N}} = 2.7$~K encompasses only two thirds of the Ce moments. We report measurements of the specific heat, $C$, and the magnetic Grüneisen parameter, $Γ_{\rm mag}$, on single crystals of CePd$_{1-x}$Ni$_x$Al with $x\leq 0.16$ at temperatures down to 0.05~K and magnetic fields $B$ up to $8$~T. Field-induced quantum criticality for various concentrations is observed with th…
FeF$_3$, with its half-filled Fe$^{3+}$ $3d$ orbital, hence zero orbital angular momentum and $S=5/2$, is often put forward as a prototypical highly-frustrated classical Heisenberg pyrochlore antiferromagnet. By employing {\it ab initio} density functional theory (DFT), we obtain an effective spin Hamiltonian for this material. This Hamiltonian contains nearest-neighbor antiferromagnetic Heisenberg, bi-quadratic and Dzyaloshinskii-Moriya interactions as dominant terms and we use Monte Carlo simulations to investigate the nonzero temperature properties of this minimal model. We find that upon d…
Near quantum Lifshitz points the stiffness of the lifetime of the spin fluctuations is very small. This gives rise to a regime where the critical fluctuations are local in space but extended along the time directions. The Fermi liquid properties in this local quantum regime is governed by a single energy scale, the coherence temperature. These results are relevant for heavy fermions and frustrated systems whenever there is a competition between different instabilities in $q$-space.
The phase diagram of the one-dimensional Su-Schrieffer-Heeger model of spinless fermions coupled to quantum phonons is determined by quantum Monte Carlo simulations. It differs significantly from previous work. In addition to Luttinger liquid and bond-order-wave (BOW) phases, we find an extended charge-density-wave (CDW) phase. Because of different broken symmetries, BOW and CDW phases are connected by a retardation-driven phase transition. Our results are consistent with the theory of the frustrated $XXZ$ chain, including unconventional power-law exponents at criticality, and an interpretatio…
Building on recent neutron and NMR experiments, we investigate the field-induced exotic criticality observed in the frustrated spin-1/2 dimer compound BaCuSi2O6 using a frustrated model with two types of bilayers. A semiclassical treatment of the effective hard-core boson model shows that perfect inter-layer frustration leads to a 2D-like critical exponent phi=1 without logarithmic corrections and to a 3D low temperature phase with different but non vanishing triplet populations in both types of bilayers. These results further suggest a simple phenomenology in terms of a field-dependent transv…
This talk was about the frustration-induced criticality in the antiferromagnetic Heisenberg model on the two-leg ladder with exchange interactions along the chains, rungs, and diagonals, and also about the effect of thermal fluctuations on this criticlity. The method used is the bond mean-field theory, which is based on the Jordan-Wigner transformation in dimensions higher than one. In this paper, we will summarize the main results presented in this talk, and report on new results about the couplings and temperature dependences of the spin susceptibility.
Several recent works on quantum criticality beyond the Landau-Ginzburg-Wilson paradigm have led to a number of field theories, potentially important for certain two-dimensional magnetic insulating systems, where criticality is not very well understood. This situation highlights the need for non-perturbative information about criticality in two spatial dimensions (three space-time dimensions), which is a longstanding challenge. As a step toward addressing these issues, we present evidence that the O(4) vector model is dual to a theory of Dirac fermions coupled to both SU(2) and U(1) gauge field…
The classical Ising model on the frustrated 3d swedenborgite lattice has disordered spin liquid ground states for all ratios of inter- and intra-planar couplings. Quantum fluctuations due to a transverse field give rise to several exotic quantum phenomena. In the limit of weakly coupled Kagomé layers we find a 3d version of disorder by disorder. For large out-of-plane couplings 1d macro-spins are formed which realize a disordered macro-spin liquid on an emerging triangular lattice. Signatures of this dimensional reduction are also found in critical exponents of the quantum phase transition out…
We study the problem of partially ordered phases with periodically arranged disordered (paramagnetic) sites on the pyrochlore lattice, a network of corner-sharing tetrahedra. The periodicity of these phases is characterized by one or more wave vectors k=(1/2 1/2 1/2). Starting from a general microscopic Hamiltonian including anisotropic nearest-neighbor exchange, long-range dipolar interactions and second- and third-nearest neighbor exchange, we identify using standard mean-field theory (s-MFT) an extended range of interaction parameters that support partially ordered phases. We demonstrate th…
Although initially introduced to mimic the spin-ice pyrochlores, no artificial spin ice has yet exhibited the expected degenerate ice-phase with critical correlations similar to the celebrated Coulomb phase in the pyrochlore lattice. Here we study a novel artificial spin ice based on a vertex-frustrated rather than pairwise frustrated geometry and show that it exhibits a quasicritical ice phase of extensive residual entropy and, significantly, algebraic correlations. Interesting in its own regard as a novel realization of frustration in a vertex system, our lattice opens new pathways to study …
We study the critical breakdown of two-dimensional quantum magnets in the presence of algebraically decaying long-range interactions by investigating the transverse-field Ising model on the square and triangular lattice. This is achieved technically by combining perturbative continuous unitary transformations with classical Monte Carlo simulations to extract high-order series for the one-particle excitations in the high-field quantum paramagnet. We find that the unfrustrated systems change from mean-field to nearest-neighbor universality with continuously varying critical exponents, while the …
Frustrated spin-$1/2$ XXZ zigzag chains relevant to Rb$_2$Cu$_2$Mo$_3$O$_{12}$ are revisited in the light of symmetry-protected topological (SPT) phases. Using a density-matrix renormalization group method for infinite systems, we identify projective representations for four distinct time-reversal invariant SPT phases; two parity-symmetric dimer phases near the Heisenberg and XX limits and two parity-broken vector-chiral (VC) dimer phases in between. A small bond alternation in the nearest-neighbor ferromagnetic exchange coupling induces a direct SPT transition between the two distinct VC dime…
The von Neumann entanglement entropy is studied with the density-matrix renormalization group technique. We propose a simple approach to calculate the central charge using the entanglement entropy for one-dimensional (1D) quantum system. This approach is applied to a couple of quantum systems: (i) 1D frustrated spin model and (ii) 1D half-filled spinless fermions with nearest-neighbor repulsion; and, it is confirmed that the central charge is estimated very accurately for the both systems. Also, a new method to determine the critical point between TL-liquid and gapped (or ordered) phases from …
Bad-Metal (BM) behavior featuring linear temperature dependence of the resistivity extending to well above the Mott-Ioffe-Regel (MIR) limit is often viewed as one of the key unresolved signatures of strong correlation. Here we associate the BM behavior with the Mott quantum criticality by examining a fully frustrated Hubbard model where all long-range magnetic orders are suppressed, and the Mott problem can be rigorously solved through Dynamical Mean-Field Theory. We show that for the doped Mott insulator regime, the coexistence dome and the associated first-order Mott metal-insulator transiti…
In the eta-carbide-type correlated-electron metal Fe3Mo3N, ferromagnetism is abruptly induced from a nonmagnetic non-Fermi-liquid ground state either when a magnetic field (~14 T) applied to it or when it is doped with a slight amount of impurity (~5% Co). We observed a peak in the paramagnetic neutron scattering intensity at finite wave vectors, revealing the presence of the antiferromagnetic (AF) correlation hidden in the magnetic measurements. It causes a new type of geometrical frustration in the stellla quadrangula lattice of the Fe sublattice. We propose that the frustrated AF correlatio…
Exchange interactions between $S=\frac{1}{2}$ sites in piperazinium hexachlorodicuprate produce a frustrated bilayer magnet with a singlet ground state. We have determined the field-temperature phase diagram by high field magnetization and neutron scattering experiments. There are two quantum critical points: $H_{c1}=7.5$ T separates a quantum paramagnet phase from a three dimensional, antiferromagnetically-ordered state while $H_{c2}=37$ T marks the onset of a fully polarized state. The ordered phase, which we describe as a magnon Bose-Einstein condensate (BEC), is embedded in a quantum criti…
The antiferromagnetic Heisenberg two-leg ladder in the presence of frustration and an external magnetic field is a system that is characterized by two sorts of quantum criticalities, not only one. One criticality is the consequence of intrinsic frustration, and the other one is a result of the external magnetic field. So the behaviour of each of them in the presence of the other deserves to be studied. Using the Jordan-Wigner transformation in dimensions higher than one and bond-mean-field theory we examine the interplay between the field-induced and frustration-induced quantum criticalities i…
We construct and analyze a two-dimensional frustrated quantum spin model with plaquette order, in which the low-energy dynamics is controlled by spin singlets. At a critical value of frustration the singlet spectrum becomes gapless, indicating a quantum transition to a phase with dimer order. This T=0 transition belongs to the 3D Ising universality class, while at finite temperature a 2D Ising critical line separates the plaquette and dimerized phases. The magnetic susceptibility has an activated form throughout the phase diagram, whereas the specific heat exhibits a rich structure and a power…
Spontaneous current orders due to odd-parity order parameters attract increasing attention in various strongly correlated metals. Here, we discover a novel spin-fluctuation-driven charge loop current (cLC) mechanism based on the functional renormalization group (fRG) theory. The present mechanism leads to the ferro-cLC order in a simple frustrated chain Hubbard model. The cLC appears between the antiferromagnetic and $d$-wave superconducting ($d$SC) phases. While the microscopic origin of the cLC has a close similarity to that of the $d$SC, the cLC transition temperature $T_{\rm cLC}$ can be h…
Extended Ashkin-Teller transition in two coupled frustrated Haldane chains
Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学卡弗里纳米科学研究所）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·派尔斯理论物理中心）
Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in $t-J$ chains coupled by frustrated interactions. This study is built on an earlier finding that at the maximally frustrated point, the ground state of weakly-coupled Heisenberg chains is disordered, the transverse exchanges being irrelevant. This result is extended here to transverse couplings up to $J_{\perp}=0.6$, and we argue that it may also be valid at the isotropic point. A finite size analysis of coupled …
We study the thermodynamic phase transition of a spin Hamiltonian comprising two 3D magnetic sublattices. Each sublattice contains XY spins coupled by the usual bilinear exchange, while spins in different sublattices only interact via biquadratic exchange. This Hamiltonian is an effective model for XY magnets on certain frustrated lattices such as body centered tetragonal. By performing a cluster Monte Carlo simulation, we investigate the crossover from the 3D-XY fixed point (decoupled sublattices) and find a systematic flow toward a first-order transition without a separatrix or a new fixed p…
Magneto-acoustic investigations of the frustrated triangular-lattice antiferromagnet Cs2CuCl4 were performed for the longitudinal modes c11 and c33 in magnetic fields along the a-axis. The temperature dependence of the sound velocity at zero field shows a mild softening at low temperature and displays a small kink-like anomaly at TN. Isothermal measurements at T < TN of the sound attenuation reveal two closely-spaced features of different character on approaching the materials quantum-critical point (QCP) around Bs = 8.5 T for B // a. The peak at slightly lower fields remains sharp down to the…
The power of machine learning algorithms to automatically classify different phases of matter and detect quantum phase transitions without necessity to characterize phases by various quantities like local order parameters or topological invariants as in conventional approaches defined machine learning phases of matter as a new research frontier and basic research tool in condensed matter and statistical physics. We study quantum criticality in the spin-1/2 square-lattice J1-J2 model with additional plaquette structure by combination of reinforcement and supervised machine learning techniques. …
Using the density-matrix renormalization group method for the ground state and excitations of the Shastry- Sutherland spin model, we demonstrate the existence of a narrow quantum spin liquid phase between the previously known plaquette-singlet and antiferromagnetic states. Our conclusions are based on finite-size scaling of excited level crossings and order parameters. Together with previous results on candidate models for deconfined quantum criticality and spin liquid phases, our results point to a unified quantum phase diagram where the deconfined quantum-critical point separates a line of f…
Unifying Dirac Spin Liquids on Square and Shastry-Sutherland Lattices via Fermionic Deconfined Criticality
Universität Würzburg（维尔茨堡大学）；Flatiron Institute（平顿研究所）；Massachusetts Institute of Technology（麻省理工学院）；Indian Institute of Technology Madras（马德拉斯印度理工学院）；Harvard University（哈佛大学）
We discuss the effects of exponential fragmentation of the Hilbert space on phase transitions in the context of coupled ferromagnetic Ising models in arbitrary dimension with special emphasis on the one dimensional case. We show that the dynamics generated by quantum fluctuations is bounded within spatial partitions of the system and weak mixing of these partitions caused by global transverse fields leads to a zero temperature phase with ordering in the local product of both Ising copies but no long range order in either species. This leads to a natural connection with the Ashkin-Teller univer…
Unified gauge-theory description of quantum spin liquids on square-based frustrated lattices
Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Würzburg-Dresden Cluster of Excellence ctd.qmat（维尔茨堡-德累斯顿卓越集群 ctd.qmat）；Center for Computational Quantum Physics, Flatiron Institute（计算量子物理中心，西蒙斯基础研究所）；Department of Physics, Indian Institute of Technology Madras（马德拉斯印度理工学院物理系）；Department of Physics, Harvard University（哈佛大学物理系）
To gain a better understanding of the interplay between frustrated long-range interactions and zero-temperature quantum fluctuations, we investigate the ground-state phase diagram of the transverse-field Ising model with algebraically-decaying long-range Ising interactions on quasi one-dimensional infinite-cylinder triangular lattices. Technically, we apply various approaches including low- and high-field series expansions. For the classical long-range Ising model, we investigate cylinders with an arbitrary even circumference. We show the occurrence of gapped stripe-ordered phases emerging out…
We investigate quantum impurity problems, where a local magnetic moment is coupled to the spin density of a bosonic environment, leading to bosonic versions of the standard Kondo and Anderson impurity models. In a physical situation, these bosonic environments can correspond either to deconfined spinons in certain classes of Z_2 frustrated antiferromagnets, or to particles in a multicomponent Bose gase (in which case the spin degree of freedom is attributed to hyperfine levels). Using renormalization group techniques, we establish that our impurity models, which feature an exchange interaction…
We study spontaneous dimerization and emergent criticality in a spin-3/2 chain with antiferromagnetic nearest-neighbor $J_1$, next-nearest-neighbor $J_2$ and three-site $J_3$ interactions. In the absence of three-site interaction $J_3$, we provide evidence that the model undergoes a remarkable sequence of three phase transitions as a function of $J_2/J_1$, going successively through a critical commesurate phase, a partially dimerized gapped phase, a critical floating phase with quasi-long-range incommensurate order, to end up in a fully dimerized phase at very large $J_2/J_1$. In the field the…
We study a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions. We call this model an anisotropic triangular-strip (ATS) model. We numerically and field-theoretically show that its ground state belongs to a gapless symmetry-protected topological (SPT) phase. The numerical approach is based on density-matrix renormalization group analyses of the entanglement entropy and the entanglement spectrum. Whereas the entanglement entropy exhibits a critical behavior, the entanglement spectrum is nontrivially degenerate. These entanglement properties imply that the ground state …
