We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization (TEFR) approach that removes local entanglement and coarse grain the lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors $T_{inv}$ plus the symmetry group $G_{sym}$ of the tensors (i.e. the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by 2D statistical Ising model, 2D statistical loop gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a $(G_{sym}, T_{inv}) $ characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry protected topological phase. The $(G_{sym}, T_{inv})$ characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.
Symmetry-protected topological (SPT) phases are short-range entangled quantum phases with symmetry, which have gapped excitations in the bulk and gapless modes at the edge. In this paper, we study the SPT phases in the spin-1 Heisenberg chain with a single-ion anisotropy D, which has a quantum phase transition between a Haldane phase and a large-D phase. Using symmetric multiscale entanglement renormalization ansatz (MERA) tensor networks, we study the nonlocal order parameters for time-reversal and inversion symmetry. For the inversion symmetric MERA, we propose a brick-and-rope representatio…
A defining feature of a symmetry protected topological phase (SPT) in one-dimension is the degeneracy of the Schmidt values for any given bipartition. For the system to go through a topological phase transition separating two SPTs, the Schmidt values must either split or cross at the critical point in order to change their degeneracies. A renormalization group (RG) approach based on this splitting or crossing is proposed, through which we obtain an RG flow that identifies the topological phase transitions in the parameter space. Our approach can be implemented numerically in an efficient manne…
Entanglement renormalization is a real-space renormalization group (RG) transformation for quantum many-body systems. It generates the multi-scale entanglement renormalization ansatz (MERA), a tensor network capable of efficiently describing a large class of many-body ground states, including those of systems at a quantum critical point or with topological order. The MERA has also been proposed to be a discrete realization of the holographic principle of string theory. In this paper we propose the use of symmetric tensors as a mechanism to build a symmetry protected RG flow, and discuss two im…
We reexamine the one-dimensional spin-1 $XXZ$ model with on-site uniaxial single-ion anisotropy as to the appearance and characterization of the symmetry-protected topological Haldane phase. By means of large-scale density-matrix renormalization group (DMRG) calculations the central charge can be determined numerically via the von Neumann entropy, from which the ground-sate phase diagram of the model can be derived with high precision. The nontrivial gapped Haldane phase shows up in between the trivial gapped even Haldane and N{é}el phases, appearing at large single-ion and spin--exchange inte…
Symmetry protected topological (SPT) phases with gapless edge excitations have been shown to exist in principle in strongly interacting bosonic/fermionic systems and it is highly desirable to find practical systems to realize such phases through numerical calculation. A central question to be addressed is how to determine the SPT order in the system given the numerical simulation result while no local order parameter can be measured to distinguish the phases from a trivial one. In the tensor network approach to simulate strongly interacting systems, the quantum state renormalization algorithm …
Strong interactions can give rise to new fermionic symmetry protected topological phases which have no analogs in free fermion systems. As an example, we have systematically studied a spinless fermion model with $U(1)$ charge conservation and time reversal symmetry on a three-leg ladder using density-matrix renormalization group. In the non-interacting limit, there are no topological phases. Turning on interactions, we found two gapped phases. One is trivial and is adiabatically connected to a band insulator, while another one is a nontrivial symmetry protected topological phase resulting from…
Symmetry protected topological states cannot be deformed to a trivial state so long as the symmetry is preserved, yet there is no local order parameter that can distinguish them from a trivial state. We demonstrate how to detect whether a two dimensional ground state has symmetry protected topological order; the measurements play a similar role as the topological entanglement entropy does for detecting anyons. For any finite abelian onsite symmetry, the measurement completely determines the 3rd cohomology class that characterizes the order. The proposed measurement is validated numerically usi…
We introduce the subsystem symmetry-preserving real-space entanglement renormalization group and apply it to study bifurcating flows generated by linear and fractal subsystem symmetry-protected topological phases in two spatial dimensions. We classify all bifurcating fixed points that are given by subsystem symmetric cluster states with two qubits per unit cell. In particular, we find that the square lattice cluster state is a quotient-bifurcating fixed point, while the cluster states derived from Yoshida's first order fractal spin liquid models are self-bifurcating fixed points. We discuss th…
We introduce a systematic construction of a gapless symmetry protected topological phase in one dimension by "decorating" the domain walls of Luttinger liquids. The resulting strongly interacting phases provide a concrete example of a gapless symmetry protected topological (gSPT) phase with robust symmetry-protected edge modes. Using boundary conformal field theory arguments, we show that while the bulks of such gSPT phases are identical to conventional Luttinger liquids, their boundary critical behavior is controlled by a different, strongly-coupled renormalization group fixed point. Our resu…
We study different phases of the one-dimensional bond-alternating spin-$1/2$ Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on …
Symmetry-protected topological edge modes are one of the most remarkable phenomena in topological physics. Here, we formulate and quantitatively examine the effect of a quantum bath on these topological edge modes. Using the density matrix renormalization group method, we study the ground state of a composite system of spin-1 quantum chain, where the system and the bath degrees of freedom are treated on the same footing. We focus on the dependence of these edge modes on the global/partial symmetries of system-bath coupling and on the features of the quantum bath. It is shown that the time-reve…
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…
We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, $\mathbb Z_3\times\mathbb Z_2^T$, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy…
We study the phase diagram of the anisotropic spin-1 Heisenberg chain with single ion anisotropy (D) using a ground-state fidelity approach. The ground-state fidelity and its corresponding susceptibility are calculated within the quantum renormalization group scheme where we obtained the renormalization of fidelity preventing to calculate the ground state. Using this approach, the phase boundaries between the antiferromagnetic Néel, Haldane and large-D phases are obtained for the whole phase diagram, which justifies the application of quantum renormalization group to trace the symmetery protec…
By performing large-scale density-matrix renormalization group simulations, we investigate a one-dimensional correlated bosonic lattice model with a synthetic spin-orbit coupling realized in recent experiments. In the insulating regime, this model exhibits a symmetry-protected topological phase. This symmetry-protected topological phase is stabilized by time-reversal symmetry and it is identified as a Haldane phase. We confirm our conclusions further by analyzing the entanglement spectrum. In addition, we find four conventional phases: a Mott insulating phase with no long range order, a ferrom…
It is well known that the matrix product state (MPS) description of a gapped ground state with a global on-site symmetry can exhibit "symmetry fractionalization". Namely, even though the symmetry acts as a linear representation on the physical degrees of freedom, the MPS matrices---which act on some virtual degrees of freedom---can transform under a projective representation. This was instrumental in classifying gapped symmetry protected phases that manifest in one dimensional quantum many-body systems. Here we consider the multi-scale entanglement renormalization ansatz (MERA) description of …
Duality places an important constraint on the renormalization group flows and the phase diagrams. For self-dual theories, the self-duality can be promoted as a symmetry, this leads to the multi-criticalities. This work investigates a description of the deconfined quantum criticality, the $N_f=2$ QED$_3$, as an example of self-dual theories and its multi-critical behavior under perturbative deformations. The multi-criticality is described by the theory with Gross-Neveu couplings and falls in a different universality class than the standard deconfined quantum criticality. We systematically calcu…
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 …
We make a systematic study of symmetry-protected topological gapped phases of quantum spin chains in the presence of the frieze space groups in one dimension using matrix product states. Here, the spatial symmetries of the one-dimensional lattice are considered together with an additional 'vertical reflection', which we take to be an on-site $\mathbb{Z}_2$ symmetry. We identify seventeen distinct non-trivial phases, define canonical forms, and compare the topological indices obtained from the MPS analysis with the group cohomological predictions. We furthermore construct explicit renormalizati…
We investigate entanglement naturally present in the 1D topologically ordered phase protected with the on-site symmetry group of an octahedron as a potential resource for teleportation-based quantum computation. We show that, as long as certain characteristic lengths are finite, all its ground states have the capability to implement any unit-fidelity one-qubit gate operation asymptotically as a key computational building block. This feature is intrinsic to the entire phase, in that perfect gate fidelity coincides with perfect string order parameters under a state-insensitive renormalization pr…
An emerging insight is that ground states of symmetry-protected topological orders (SPTO's) possess latent computational complexity in terms of their many-body entanglement. By introducing a fractional symmetry of SPTO, which requires the invariance under 3-colorable symmetries of a lattice, we prove that every renormalization fixed-point state of 2D $(\mathbb{Z}_2)^m$ SPTO with fractional symmetry can be utilized for universal quantum computation using only Pauli measurements, as long as it belongs to a nontrivial 2D SPTO phase. Our infinite family of fixed-point states may serve as a base mo…
We study one-dimensional systems of two-orbital SU(4) fermionic cold atoms. In particular, we focus on an SU(4) spin model [named SU(4) $e$-$g$ spin model] that is realized in a low-energy state in the Mott insulator phase at the filling $n_g=3, n_e=1$ ($n_g, n_e$: numbers of atoms in ground and excited states, respectively). Our numerical study with the infinite-size density matrix renormalization group shows that the ground state of SU(4) $e$-$g$ spin model is a nontrivial symmetry protected topological (SPT) phase protected by $Z_4 \times Z_4$ symmetry. Specifically, we find that the ground…
Understanding non-Fermi liquids in dimensions higher than one, has been a subject of great interest. Such phases may serve as parent states for other unconventional phases of quantum matter, in a similar manner that conventional broken symmetry states can be understood as instabilities of the Fermi liquid. In this work, we investigate the emergence of a novel non-Fermi liquid in two dimensions, where the fermions with quadratic band-touching dispersion interact with the Bose metal. The bosonic excitations in the Bose metal possess an extended nodal-line spectrum in momentum space, which arises…
We study a symmetrical double quantum dot (DD) system with strong capacitive inter-dot coupling using renormalization group methods. The dots are attached to separate leads, and there can be a weak tunneling between them. In the regime where there is a single electron on the DD the low-energy behavior is characterized by an SU(4)-symmetric Fermi liquid theory with entangled spin and charge Kondo correlations and a phase shift $π/4$. Application of an external magnetic field gives rise to a large magneto-conductance and a crossover to a purely charge Kondo state in the charge sector with SU(2) …
For most chiralities, semiconducting nanotubes display topologically protected end states of multiple degeneracies. We demonstrate using density matrix renormalization group based quantum chemistry tools that the presence of Coulomb interactions induces the formation of robust end spins. These are the close analogues of ferromagnetic edge states emerging in graphene nanoribbons. The interaction between the two ends is sensitive to the length of the nanotube, its dielectric constant, as well as the size of the end spins: for $S=1/2$ end spins their interaction is antiferromagnetic, while for $S…
According to the classification using projective representations of the SO(3) group, there exist two topologically distinct gapped phases in spin-1 chains. The symmetry-protected topological (SPT) phase possesses half-integer projective representations of the SO(3) group, while the trivial phase possesses integer linear representations. In the present work, we implement non-Abelian symmetries in the density matrix renormalization group (DMRG) method, allowing us to keep track of (and also control) the virtual bond representations, and to readily distinguish the SPT phase from the trivial one b…
We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single particle excitations in the b…
We study a bosonic model with correlated hopping on a honeycomb lattice, and show that its ground state is a bosonic integer quantum Hall (BIQH) phase, a prominent example of a symmetry protected topological (SPT) phase. By using the infinite density matrix renormalization group method, we establish the existence of the BIQH phase by providing clear numerical evidence: (i) a quantized Hall conductance with $|σ_{xy}|= 2$ (ii) two counter propagating gapless edge modes. Our simple model is an example of a novel class of systems that can stabilize SPT phases protected by a continuous symmetry on …
We propose an exotic scenario that topological superconductivity can emerge by doping strongly interacting fermionic systems whose spin degrees of freedom form bosonic symmetry protected topological (SPT) state. Specifically, we study a 1-dimensional (1D) example where the spin degrees of freedom form a spin-1 Haldane phase. {Before doping, the charge and spin degrees of freedom are both gapped.} Upon doping, {the charge channel becomes gapless and is described by a $c=1$ compactified bosonic conformal field theory (CFT), while the spin channel remains gapped and still form a bosonic SPT state…
We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by $(\times Z_3)^3\equiv Z_3\times Z_3\times Z_3$ symmetry and smaller $Z_3$ symmetry. We derive microscopic models for the gapless edge, uncover their symmetries, and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary f…
We study $\mathbb{Z}_3$ symmetry-protected topological (SPT) phases in one-dimensional spin systems with $Z_3 \times Z_3$ symmetry. We construct ground-state wave functions of the matrix product form for nontrivial $\mathbb{Z}_3$ phases and their parent Hamiltonian from a cocycle of the group cohomology $H^2(Z_3\times Z_3,U(1))$. The Hamiltonian is an SU(3) version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, consisting of bilinear and biquadratic terms of su(3) generators in the adjoint representation. A generalization to the SU($N$) case, the SU($N$) AKLT Hamiltonian, is also presented w…
We show how angular momentum conservation can stabilise a symmetry-protected quasi-topological phase of matter supporting Majorana quasi-particles as edge modes in one-dimensional cold atom gases. We investigate a number-conserving four-species Hubbard model in the presence of spin-orbit coupling. The latter reduces the global spin symmetry to an angular momentum parity symmetry, which provides an extremely robust protection mechanism that does not rely on any coupling to additional reservoirs. The emergence of Majorana edge modes is elucidated using field theory techniques, and corroborated b…
We analyze the $\mathcal{N}=1$ supersymmetric Wess-Zumino model dimensionally reduced to the $\mathcal{N}=2$ supersymmetric model in three Euclidean dimensions. As in the original model in four dimensions and the $\mathcal{N}=(2,2)$ model in two dimensions the superpotential is not renormalized. This property puts severe constraints on the non-trivial fixed-point solutions, which are studied in detail. We admit a field-dependent wave function renormalization that in a geometric language relates to a Kähler metric. The Kähler metric is not protected by supersymmetry and we calculate its explici…
We study 't Hooft anomalies of discrete groups in the framework of (1+1)-dimensional multiscale entanglement renormalization ansatz states on the lattice. Using matrix product operators, general topological restrictions on conformal data are derived. An ansatz class allowing for optimization of MERA with an anomalous symmetry is introduced. We utilize this class to numerically study a family of Hamiltonians with a symmetric critical line. Conformal data is obtained for all irreducible projective representations of each anomalous symmetry twist, corresponding to definite topological sectors. It…
We study the renormalization of a single impurity potential in one-dimensional interacting electron systems in the presence of magnetic field. Using the bosonization technique and Bethe ansatz solutions, we determine the renormalization group flow diagram for the amplitudes of scattering of up- and down-spin electrons by the impurity in a quantum wire at low electron density and in the Hubbard model at less than half filling. In the absence of magnetic field the repulsive interactions are known to enhance backscattering and make the impurity potential impenetrable in the low-energy limit. On t…
Motivated by recent studies of symmetry protected topological (SPT) phases, we explore the possible gapless quantum disordered phases in the $(2+1)d$ nonlinear sigma model defined on the Grassmannian manifold $\frac{U(N)}{U(n)\times U(N - n)}$ with a Wess-Zumino-Witten (WZW) term at level $k$, which is the effective low energy field theory of the boundary of certain $(3+1)d$ SPT states. With $k = 0$, this model has a well-controlled large-$N$ limit, $i.e.$ its renormalization group equations can be computed exactly with large-$N$. However, with the WZW term, the large-$N$ and large-$k$ limit a…
We classify the different ways in which matrix product states (MPSs) can stay invariant under the action of matrix product operator (MPO) symmetries. This is achieved through a local characterization of how the MPSs, that generate a ground space, remain invariant under a global MPO symmetry. This characterization yields a set of quantities satisfying the coupled pentagon equations, associated with a module category over the fusion category that describes the MPO symmetry. Equivalence classes of these quantities provide complete invariants for an MPO symmetry protected phase: they are robust un…
Topological states may be protected by a lattice symmetry in a class of topological semi-metals. In three spatial dimensions, the Berry flux around gapless excitations in momentum space defines a chirality concretely, so a protecting symmetry may be referred to as a chiral symmetry. Prime examples include Dirac semi-metal (DSM) in a distorted spinel, BiZnSiO$_4$, protected by a mirror symmetry and DSM in Na$_3$Bi, protected by a rotational symmetry. In these states, topology and a chiral symmetry are intrinsically tied. In this work, we investigate characteristics interplay between a chiral sy…
We examine the pairing structure of holes injected into two \emph{distinct} spin backgrounds: a short-range antiferromagnetic phase versus a symmetry protected topological phase. Based on density matrix renormalization group (DMRG) simulation, we find that although there is a strong binding between two holes in both phases, \emph{phase fluctuations} can significantly influence the pair-pair correlation depending on the spin-spin correlation in the background. Here the phase fluctuation is identified as an intrinsic string operator nonlocally controlled by the spins. We show that while the pair…
We study one-dimensional, interacting, gapped fermionic systems described by variants of the Peierls-Hubbard model and characterize their phases via a topological invariant constructed out of their Green's functions. We demonstrate that the existence of topologically protected, zero-energy states at the boundaries of these systems can be tied to the values of their topological invariant, just like when working with the conventional, noninteracting topological insulators. We use a combination of analytical methods and the numerical density matrix renormalization group method to calculate the va…
The effect of a local Kondo coupling and Hubbard interaction on the topological phase of the one-dimensional topological Kondo insulator (TKI) is numerically investigated using the infinite matrix-product state density-matrix renormalization group algorithm. The groundstate of the TKI is a symmetry-protected topological (SPT) phase protected by inversion symmetry. It is found that on its own, the Hubbard interaction that tends to force fermions into a one-charge per site order is insufficient to destroy the SPT phase. However when the local Kondo Hamiltonian term that favors a topologically tr…
The interplay of symmetry, topology, and many-body effects in the classification of possible phases of matter poses a formidable challenge that is attracting great attention in condensed-matter physics. Such many-body effects are typically induced by inter-particle interactions involving an action at a distance, such as the Coulomb interaction between electrons in a symmetry-protected topological (SPT) phase. In this work, we show that similar phenomena also appear in high-energy physics, where inter-particle interactions are mediated by gauge bosons, and constrained by a local gauge symmetry.…
We study the bond-alternating Heisenberg model using the finite-size density-matrix renormalization group (DMRG) technique and analytical arguments based on the matrix product state, where we pay particular attention to the boundary-condition dependence on the entanglement spectrum of the system. We show that, in the antiperiodic boundary condition (APBC), the parity quantum numbers are equivalent to the topological invariants characterizing the topological phases protected by the bond-centered inversion and $π$ rotation about $z$ axis. We also show that the odd parity in the APBC, which chara…
We use a nonperturbative extended contractor renormalization (ENCORE) method for engineering quantum devices for the implementation of topologically protected quantum bits described by an effective quantum dimer model on the triangular lattice. By tuning the couplings of the device, topological protection might be achieved if the ratio between effective two-dimer interactions and flip amplitudes lies in the liquid phase of the phase diagram of the quantum dimer model. For a proposal based on a quantum Josephson junction array [L. B. Ioffe {\it et al.}, Nature (London) {\bf 415}, 503 (2002)] ou…
We derive universal constraints on $(1+1)d$ rational conformal field theories (CFTs) that can arise as transitions between topological theories protected by a global symmetry. The deformation away from criticality to the trivially gapped phase is driven by a symmetry preserving relevant deformation and under renormalization group flow defines a conformal boundary condition of the CFT. When a CFT can make a transition between distinct trivially gapped phases the spectrum of the CFT quantized on an interval with the associated boundary conditions has degeneracies at each energy level. Using tech…
Fate of average symmetry-protected topological states under symmetry-preserving quantum operations
Graduate School of Engineering Science, Akita University（秋田大学工学研究科）；Department of Physics, College of Humanities and Sciences, Nihon University（日本大学文理学部物理学科）
We present a detailed study of the fidelity, the entanglement entropy, and the entanglement spectrum, for a dimerized chain of spinless fermions---a simplified Su-Schrieffer-Heeger (SSH) model---with open boundary conditions which is a well-known example for a model supporting a symmetry protected topological (SPT) phase. In the non-interacting case the Hamiltonian matrix is tridiagonal and the eigenvalues and -vectors can be given explicitly as a function of a single parameter which is known analytically for odd chain lengths and can be determined numerically in the even length case. From a s…
Using a perturbative renormalization group approach, we show that the extended ($J_1$-$J_2$-$J_d$) Heisenberg model on the kagome lattice with a staggered chiral interaction ($J_χ$) can exhibit a gapless chiral quantum spin liquid phase. Within a coupled-chains construction, this phase can be understood as a chiral sliding Luttinger liquid with algebraic decay of spin correlations along the chain directions. We calculate the low-energy properties of this gapless chiral spin liquid using the effective field theory and show that they are compatible with the predictions from parton mean-field the…
We study boundary conditions of topological sigma models with the goal of generalizing the concepts of anomalous symmetry and symmetry protected topological order. We find a version of 't Hooft's anomaly matching conditions on the renormalization group flow of boundaries of invertible topological sigma models and discuss several examples of anomalous boundary theories. We also comment on bulk topological transitions in dynamical sigma models and argue that one can, with care, use topological data to draw sigma model phase diagrams.
A scenario for the formation of symmetry protected non-Fermi-liquid (NFL) Kondo effect (KE) with spin variable enumerating Kondo channels is suggested and worked out. In doubly occupied symmetric triple quantum dot within parallel geometry, the NFL low-energy regime arises provided the device possesses both source-drain and left-right parity. Kondo screening follows a multistage renormalization group mechanism: reduction of the energy scale is accompanied by the change of the relevant symmetry group from SO(8) to SO(5). At low energy, three phases compete: 1) under-screening spin triplet (conv…
