Spin-rotation Symmetry Research Articles

Symmetry-protected topological phases in spin ladders with two-body interactions

Spin-1/2 two-legged ladders respecting interleg exchange symmetry $\ensuremath{\sigma}$ and spin rotation symmetry ${D}_{2}$ have new symmetry-protected topological (SPT) phases which are different from the Haldane phase. Three of the new SPT phases are ${t}_{x},{t}_{y},{t}_{z}$, which all have symmetry-protected twofold degenerate edge states on each end of an open chain. However, the edge states in different phases have different responses to magnetic field. For example, the edge states in the ${t}_{z}$ phase will be split by the magnetic field along the $z$ direction, but not by the fields in the $x$ and $y$ directions. We give the Hamiltonian that realizes each SPT phase and demonstrate a proof-of-principle quantum simulation scheme for Hamiltonians of the ${t}_{0}$ and ${t}_{z}$ phases based on the coupled-QED-cavity ladder.

Renormalization group flow for fermions into antiferromagnetically ordered phases: Method and mean-field models

We present a functional renormalization group (fRG) flow for many-fermion lattice models into phases with broken spin-rotational symmetry. The flow is expressed purely in terms of fermionic vertex functions. The symmetry breaking is seeded by a small initial anomalous self-energy which grows at the transition scale and which prevents a runaway flow at nonzero scales. Focusing on the case of commensurate antiferromagnetism, we discuss how the interaction vertex can be parametrized efficiently. For reduced models with long-range bare interactions, we show the results of standard mean-field theory are reproduced by the fRG and how anisotropies in the spin sector change the flows. We then describe a more efficient decomposition of the interaction vertex that should allow for the treatment of more general models.

Analytical solutions to the spin-1 Bose–Einstein condensates

We analytically solve the one-dimensional coupled Gross–Pitaevskii equations which govern the motion of the spinor-1 Bose–Einstein condensates. The nonlinear density–density interactions are decoupled by making use of the unique properties of the Jacobian elliptical functions. Several types of time-invariant solutions are deduced. Furthermore, exact time-evolving solutions to the time-dependent Gross–Pitaevskii equations are constructed by making use of the spin-rotational symmetry of the Hamiltonian. The spin-polarizations exhibit kinked configurations.

Antiferromagnetism in the Hubbard Model on the Bernal-Stacked Honeycomb Bilayer

Using a combination of quantum Monte Carlo simulations, functional renormalization group calculations and mean-field theory, we study the Hubbard model on the Bernal-stacked honeycomb bilayer at half-filling as a model system for bilayer graphene. The free bands consisting of two Fermi points with quadratic dispersions lead to a finite density of states at the Fermi level, which triggers an antiferromagnetic instability that spontaneously breaks sublattice and spin rotational symmetry once local Coulomb repulsions are introduced. Our results reveal an inhomogeneous participation of the spin moments in the ordered ground state, with enhanced moments at the threefold coordinated sites. Furthermore, we find the antiferromagnetic ground state to be robust with respect to enhanced interlayer couplings and extended Coulomb interactions.

Spin nematic ground state of the triangular latticeS=1biquadratic model

Motivated by the spate of recent experimental and theoretical interest in Mott insulating S=1 triangular lattice magnets, we consider a model S=1 Hamiltonian on a triangular lattice interacting with rotationally symmetric biquadratic interactions. We show that the partition function of this model can be expressed in terms of configurations of three colors of tightly-packed, closed loops with {\em non-negative} weights, which allows for efficient quantum Monte Carlo sampling on large lattices. We find the ground state has spin nematic order, i.e. it spontaneously breaks spin rotation symmetry but preserves time reversal symmetry. We present accurate results for the parameters of the low energy field theory, as well as finite-temperature thermodynamic functions.

Interacting One-Dimensional Fermionic Symmetry-Protected Topological Phases

In free fermion systems with given symmetry and dimension, the possible topological phases are labeled by elements of only three types of Abelian groups, 0, Z2, or Z. For example, noninteracting one-dimensional fermionic superconducting phases with S(z) spin rotation and time-reversal symmetries are classified by Z. We show that with weak interactions, this classification reduces to Z4. Using group cohomology, one can additionally show that there are only four distinct phases for such one-dimensional superconductors even with strong interactions. Comparing their projective representations, we find that all these four symmetry-protected topological phases can be realized with free fermions. Further, we show that one-dimensional fermionic superconducting phases with Z(n) discrete S(z) spin rotation and time-reversal symmetries are classified by Z4 when n is even and Z2 when n is odd; again, all these strongly interacting topological phases can be realized by noninteracting fermions. Our approach can be applied to systems with other symmetries to see which one-dimensional topological phases can be realized with free fermions.

Low-temperature properties of two-dimensional ideal ferromagnets

The manifestation of the spin-wave interaction in the low-temperature series of the partition function has been investigated extensively over more than seven decades in the case of the three-dimensional ferromagnet. Surprisingly, the same problem regarding ferromagnets in two spatial dimensions, to the best of our knowledge, has never been addressed in a systematic way so far. In the present paper the low-temperature properties of two-dimensional ideal ferromagnets are analyzed within the model-independent method of effective Lagrangians. The low-temperature expansion of the partition function is evaluated up to two-loop order and the general structure of this series is discussed, including the effect of a weak external magnetic field. Our results apply to two-dimensional ideal ferromagnets which exhibit a spontaneously broken spin rotation symmetry O(3) $\to $ O(2) and are defined on a square, honeycomb, triangular or Kagom\'e lattice. Remarkably, the spin-wave interaction only sets in at three-loop order. In particular, there is no interaction term of order $T^3$ in the low-temperature series for the free energy density. This is the analog of the statement that, in the case of three-dimensional ferromagnets, there is no interaction term of order $T^4$ in the free energy density. We also provide a careful discussion of the implications of the Mermin-Wagner theorem in the present context and thereby put our low-temperature expansions on safe grounds.

Quantum correlations in nanostructured two-impurity Kondo systems

We study the ground-state entanglement properties of nanostructured Kondo systems consisting of a pair of impurity spins coupled to a background of confined electrons. The competition between the Ruderman-Kittel-Kasuya-Yosida-like coupling and the Kondo effect determines the development of quantum correlations between the different parts of the system. A key element is the electronic filling due to confinement. An even electronic filling leads to results similar to those found previously for extended systems, where the properties of the reduced impurity spin subsystem are uniquely determined by the spin-correlation function defining a one-dimensional phase space. An odd filling, instead, breaks spin rotation symmetry unfolding a two-dimensional phase space showing rich entanglement characteristics as, e.g., the requirement of a larger amount of entanglement for the development of nonlocal correlations between impurity spins. We illustrate these results by numerical simulations of elliptic quantum corrals with magnetic impurities at the foci as a case study.

Competing nematic, antiferromagnetic, and spin-flux orders in the ground state of bilayer graphene

We analyze the phase diagram of bilayer graphene (BLG) at zero temperature and zero doping. Assuming that at high energies the electronic system of BLG can be described within a weak-coupling theory (consistent with the experimental evidence), we systematically study the evolution of the couplings with going from high to low energies. The divergences of the couplings at some energies indicate the tendency towards certain symmetry breakings. Carrying out this program, we found that the phase diagram is determined by microscopic couplings defined on the short distances (initial conditions). We explored all plausible space of these initial conditions and found that the three states have the largest phase volume of the initial couplings: nematic, antiferromagnetic, and spin flux (a.k.a. quantum spin Hall). In addition, ferroelectric and two superconducting phases appear only near the very limits of the applicability of the weak-coupling approach. The paper also contains the derivation and analysis of the renormalization group equations and the group theory classification of all the possible phases which might arise from the symmetry breakings of the lattice, spin rotation, and gauge symmetries of graphene.

Photoinduced spin Chern number change in a two-dimensional quantum spin Hall insulator with broken spin rotational symmetry

We propose a scheme by which the spin Chern number of two-dimensional insulators with generic spin-orbit interaction can be externally controlled by optical methods. When submitting the system to a circularly polarized laser field, and then adiabatically sweeping the amplitude of the latter, the various hopping integrals associated with an electron system effectively turn into tunable variables. This in turn transforms the spin Chern number, which governs the topological properties of the system into a function of the laser amplitude. This scheme can steer a conventional insulator into the quantum spin Hall state, and vice versa. We anticipate that our findings, which provide a firmly grounded connection between optical techniques and spin transport, will open up the possibility of developing spintronics devices that build on the physics of the photosteering of spin currents.

Majorana spin liquids and projective realization of SU(2) spin symmetry

We revisit the fermionic parton approach to $S=1/2$ quantum spin liquids with $\mathrm{SU}(2)$ spin-rotation symmetry, and the associated projective symmetry group (PSG) classification. We point out that the existing PSG classification is incomplete; upon completing it, we find spin-liquid states with $S=1$ and $S=0$ Majorana fermion excitations coupled to a deconfined ${Z}_{2}$ gauge field. The crucial observation leading us to this result is that, like space group and time-reversal symmetries, spin rotations can act projectively on the fermionic partons; that is, a spin rotation may be realized by simultaneous $\mathrm{SU}(2)$ spin and gauge rotations. We show that there are only two realizations of spin rotations acting on fermionic partons: the familiar naive realization where spin rotation is not accompanied by any gauge transformation, and a single type of projective realization. We discuss the PSG classification for states with projective spin rotations. To illustrate these results, we show that there are four such PSGs on the two-dimensional square lattice. We study the properties of the corresponding states, finding that one---with gapless Fermi points---is a stable phase beyond mean-field theory. In this phase, depending on parameters, a small Zeeman magnetic field can open a partial gap for the Majorana fermion excitations. Moreover, there are nearby gapped phases supporting ${Z}_{2}$ vortex excitations obeying non-Abelian statistics. We conclude with a discussion of various open issues, including the challenging question of where such $S=1$ Majorana spin liquids may occur in models and in real systems.

Symmetry-protected topological phases in noninteracting fermion systems

Symmetry protected topological (SPT) phases are gapped quantum phases with a certain symmetry, which can all be smoothly connected to the same trivial product state if we break the symmetry. For non-interacting fermion systems with time reversal (T), charge conjugation (C), and/or U(1) (N) symmetries, the total symmetry group can depend on the relations between those symmetry operations, such as T N T^{-1}= N or T N T^{-1}= -N. As a result, the SPT phases of those fermion systems with different symmetry groups have different classifications. In this paper, we use Kitaev's K-theory approach to classify the gapped free fermion phases for those possible symmetry groups. In particular, we can view the U(1) as a spin rotation. We find that superconductors with the S_z spin rotation symmetry are classified by Z in even dimensions, while superconductors with the time reversal plus the S_z spin rotation symmetries are classified by Z in odd dimensions. We show that all 10 classes of gapped free fermion phases can be realized by electron systems with certain symmetries. We also point out that to properly describe the symmetry of a fermionic system, we need to specify its full symmetry group that includes the fermion number parity transformation (-)^N. The full symmetry group is actually a projective symmetry group.

Preemptive nematic order, pseudogap, and orbital order in the iron pnictides

Starting from a microscopic itinerant model, we derive and analyze the effective low-energy model for collective magnetic excitations in the iron pnictides. We show that the stripe magnetic order is generally preempted by an Ising-nematic order which breaks $C_{4}$ lattice symmetry but preserves O(3) spin-rotational symmetry. This leads to a rich phase diagram as function of doping, pressure, and elastic moduli, displaying split magnetic and nematic tri-critical points. The nematic transition may instantly bring the system to the verge of a magnetic transition, or it may occur first, being followed by a magnetic transition at a lower temperature. In the latter case, the preemptive nematic transition is accompanied by either a jump or a rapid increase of the magnetic correlation length, triggering a pseudogap behavior associated with magnetic precursors. Furthermore, due to the distinct orbital character of each Fermi pocket, the nematic transition also induces orbital order. We compare our results to various experiments, showing that they correctly address the changes in the character of the magneto-structural transition across the phase diagrams of different compounds, as well as the relationship between the orthorhombic and magnetic order parameters.

Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors

One of the defining properties of the conventional three-dimensional (``${\mathbb{Z}}_{2}$'' or ``spin-orbit'') topological insulator is its characteristic magnetoelectric effect, as described by axion electrodynamics. In this paper, we discuss an analog of such a magnetoelectric effect in the thermal (or gravitational) and magnetic dipole responses in all symmetry classes that admit topologically nontrivial insulators or superconductors to exist in three dimensions. In particular, for topological superconductors (or superfluids) with time-reversal symmetry, which lack $\text{SU}(2)$ spin rotation symmetry (e.g., due to spin-orbit interactions), such as the B phase of ${}^{3}$He, the thermal response is the only probe that can detect the nontrivial topological character through transport. We show that, for such topological superconductors, applying a temperature gradient produces a thermal- (or mass-) surface current perpendicular to the thermal gradient. Such charge, thermal, or magnetic dipole responses provide a definition of topological insulators and superconductors beyond the single-particle picture. Moreover, we find, for a significant part of the ``tenfold'' list of topological insulators found in previous work in the absence of interactions, that in general dimensions, the effective field theory describing the space-time responses is governed by a field theory anomaly. Since anomalies are known to be insensitive to whether the underlying fermions are interacting, this shows that the classification of these topological insulators is robust to adiabatic deformations by interparticle interactions in general dimensionality. In particular, this applies to symmetry classes DIII, CI, and AIII in three spatial dimensions, and to symmetry classes D and C in two spatial dimensions.

Projective studies of spin nematics in a quantum frustrated ferromagnet

We study the ground-state properties of the spin-$\frac{1}{2}$ frustrated ferromagnetic ${J}_{1}$-${J}_{2}$ Heisenberg model on the square lattice, employing projected BCS wave functions with spin-triplet pairings of the spinon fields as trial wave functions. Based on the variational Monte Carlo analysis, we argue that, in the competing coupling regime, a certain type of the projected BCS wave function, dubbed the projected ${Z}_{2}$ planar state, achieves the best optimal energy among the other competing states such as the ferromagnetic state and collinear antiferromagnetic state. As in quantum spin liquids, the projected ${Z}_{2}$ planar state preserves the translational symmetry of the square lattice. However, it is also accompanied by a $d$-wave ordering of the quadrupole moments, breaking the spin rotational symmetry. The state thus describes a quantum spin analog of the nematic liquid crystals. The calculated static correlation functions also reveal that the projected ${Z}_{2}$ planar state has a strong collinear antiferromagnetic fluctuation.

RKKY interaction in the spin-density-wave phase of iron-based superconductors

Using the multiband model we analyze the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between the magnetic impurities in layered ferropnictide superconductors. In the normal state the interaction is spin isotropic and is dominated by the nesting features of the electron and hole bands separated by the antiferromagnetic momentum, ${\bf Q}_{AF}$. In the AF state the RKKY interaction maps into an effective anisotropic XXZ-type Heisenberg exchange model. The anisotropy originates from the breaking of the spin-rotational symmetry induced by the AF order and its strength depends on the size of the AF gap and the structure of the folded Fermi surface. We discuss our results in connection to the recent experiments.

Multicritical behavior ofZ2×O(2)Gross-Neveu-Yukawa theory in graphene

Multi-critical behavior of interacting fermions in graphene's honeycomb lattice is presented. In particular, we considered the spin triplet insulating orders, where the spin rotational symmetry of the order parameter is explicitly broken. By casting the problem in terms of Gross-Neveu-Yukawa theory we show that such symmetry-breaking terms are irrelevant near the metal-insulator critical point. A finite Yukawa coupling among bosons and fermions improves the stability of such critical point against the symmetry-breaking perturbations. Physical sources of such symmetry-breaking terms are pointed out. Critical exponents are calculated near the transitions as well.

Composite pairing in a mixed-valent two-channel Anderson model

Using a two-channel Anderson model, we develop a theory of composite pairing in the 115 family of heavy fermion superconductors that incorporates the effects of f-electron valence fluctuations. Our calculations introduce "symplectic Hubbard operators": an extension of the slave boson Hubbard operators that preserves both spin rotation and time-reversal symmetry in a large N expansion, permitting a unified treatment of anisotropic singlet pairing and valence fluctuations. We find that the development of composite pairing in the presence of valence fluctuations manifests itself as a phase-coherent mixing of the empty and doubly occupied configurations of the mixed valent ion. This effect redistributes the f-electron charge within the unit cell. Our theory predicts a sharp superconducting shift in the nuclear quadrupole resonance frequency associated with this redistribution. We calculate the magnitude and sign of the predicted shift expected in CeCoIn_5.

SU(2)-invariant spin liquids on the triangular lattice with spinful Majorana excitations

We describe a new class of spin liquids with global SU(2) spin rotation symmetry in spin 1/2 systems on the triangular lattice, which have real Majorana fermion excitations carrying spin S = 1. The simplest translationally-invariant mean-field state on the triangular lattice breaks time-reversal symmetry and is stable to fluctuations. It generically possesses gapless excitations along 3 Fermi lines in the Brillouin zone. These intersect at a single point where the excitations scale with a dynamic exponent z = 3. An external magnetic field has no orbital coupling to the SU(2) spin rotation-invariant fermion bilinears that can give rise to a transverse thermal conductivity, thus leading to the absence of a thermal Hall effect. The Zeeman coupling is found to gap out two-thirds of the z = 3 excitations near the intersection point and this leads to a suppression of the low temperature specific heat, the spin susceptibility and the Wilson ratio. We also compute physical properties in the presence of weak disorder and discuss possible connections to recent experiments on organic insulators.

Chiral spin liquid in a correlated topological insulator

In this paper, we investigate the topological Hubbard model, the spinful Haldane model with on-site interaction on honeycomb lattice with spin-rotation symmetry by using the slave-rotor approach, and find that chiral spin liquid exists in such a correlated electron system of the intermediate coupling region. By considering the anyon nature of excitations, chiral spin liquid may be the ground state of the topological Hubbard model. The low-energy physics is basically determined by its Chern-Simons gauge theory.