Yuan-Ming Lu | Ohio State University (original) (raw)
Papers by Yuan-Ming Lu
Physical Review Letters, 2016
The anyonic excitations of a spin liquid can feature fractional quantum numbers under space group... more The anyonic excitations of a spin liquid can feature fractional quantum numbers under space group symmetries. Detecting these fractional quantum numbers, which are analogs of the fractional charge of Laughlin quasiparticles, may prove easier than the direct observation of anyonic braiding and statistics. Motivated by the recent numerical discovery of spin-liquid phases in the kagome Heisenberg antiferromagnet, we theoretically predict the pattern of space group symmetry fractionalization in the kagome lattice SO(3)-symmetric chiral spin liquid. We provide a method to detect these fractional quantum numbers in finite-size numerics which is simple to implement in the density matrix renormalization group. Applying these developments to the chiral spin liquid phase of a kagome Heisenberg model, we find perfect agreement between our theoretical prediction and numerical observations.
We show that gauge field fluctuations in the composite fermion field theory can be exactly integr... more We show that gauge field fluctuations in the composite fermion field theory can be exactly integrated out using a non-unitary transformation. An instantaneous statistical interaction is induced which makes the Fermi sea unstable to chiral p-wave pairing. We show that the paired state is a Moore-Read Pfaffian and discuss the effects of Coulomb interaction in connection to even-denominator fractional quantum Hall effect.
A theory is developed for the paired even-denominator fractional quantum Hall states in the lowes... more A theory is developed for the paired even-denominator fractional quantum Hall states in the lowest Landau level. We show that electrons bind to quantized vortices to form composite fermions, interacting through an exact instantaneous interaction that favors chiral p-wave pairing. There are two canonically dual pairing gap functions related by the bosonic Laughlin wavefunction (Jastraw factor) due to the correlation holes. Remarkably,we find a new ground state lower in energy than the Moore-Read pfaffian for intermediate Coulomb interactions: a pfaffian with an oscillatory pairing function. We explore the unusual features and experimental implications of such a state. Connections to recent experiments are also discussed.
Due to strong geometric frustration and quantum fluctuation, the S = 1/2 quantum Heisenberg antif... more Due to strong geometric frustration and quantum fluctuation, the S = 1/2 quantum Heisenberg antiferromagnet on the kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S = 1/2, kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2 [Z subscript 2] topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2 [Z subscript 2] SLs on the kagome lattice. Among them we found only one gapped Z2 [Z subscript 2] SL (which we call the Z2[0,π]β [Z subscript 2 [0,pi] Beta] state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117...
Recently, Z2 spin liquid was proposed as the ground state of the Kagome quantum antiferromagnet [... more Recently, Z2 spin liquid was proposed as the ground state of the Kagome quantum antiferromagnet [S. Yan, D.A. Huse, and S.R. White, Science, 332, 1173 (2011)]. We study proximate symmetry-broken phases that may appear on exiting the spin liquid phase, by tuning parameters such as further neighbor couplings. Given that the Dirac spin liquid is also a relatively low energy state, we consider models of Z2 spin liquids that are proximate to it. Specifically we consider the s-wave paired state of an algebraic spin liquid on Kagome lattice, Z2[0,π]β state of Y.-M Lu, Y. Ran, and P.A. Lee, Phys. Rev. B 83, 224413 (2011)] and examine its relations with other competing states. This allows us to characterize the proximate magnetically ordered and VBS phases and criticality between them and the quantum spin liquid.
While topological insulators of non-interacting fermions have been extensively studied, we know v... more While topological insulators of non-interacting fermions have been extensively studied, we know very little about topological insulators of bosons, whose realization necessitates strong interaction. In this work we apply Chern-Simons effective theory to classify and characterize interacting bosonic topological insulators in two spatial dimensions. These topological phases have a unique ground state on any closed manifold and no fractional excitations: yet they feature gapless edge states which are often protected by a symmetry. Examples include a bosonic analog of chiral superconductors, bosonic integer quantum Hall states (with Hall conductance quantized to even integers) and bosonic analog of the quantum spin Hall state. We show that these topological phases can be constructed in various ways: such as in arrays of coupled one-dimensional quantum wires. Our formulation also naturally applies to topological insulators of two-dimensional interacting fermions.
Three dimensional topological insulator represents a class of novel quantum phases hosting robust... more Three dimensional topological insulator represents a class of novel quantum phases hosting robust gapless boundary excitations, which is protected by global symmetries such as time reversal, charge conservation and spin rotational symmetry. In this work we systematically study another class of topological phases of weakly interacting electrons protected by spatial inversion symmetry, which generally don't support stable gapless boundary states. We classify these inversion-symmetric topological insulators and superconductors in the framework of K-theory, and construct their lattice models. We also discuss quantized response functions of these inversion-protected topological phases, which serve as their experimental signatures.
Recent numerical studies have provided strong evidence for a gapped Z$_2$ quantum spin liquid in ... more Recent numerical studies have provided strong evidence for a gapped Z$_2$ quantum spin liquid in the kagome lattice Heisenberg model. A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of fractionalization imply distinct phases. The symmetry fractionalization pattern for the kagome spin liquid remains to be determined. A popular approach to studying spin liquids is to decompose the physical spin into partons which are either bosonic (Schwinger bosons) or fermionic (Abrikosov fermions), which are then treated within mean field theory. A longstanding question has been whether these two approaches are truly distinct or describe the same phase in complementary ways. Here we show that at least four spin liquid phases on the kagome lattice can be described both in terms of bosonic and fermionic partons, unifying pairs of theories that seem rather distinct. The key idea is that for kagome lattice states that admit a Schwinger boson mean field...
Physical Review B, 2014
AKLT state (or Haldane phase) in a one-dimensional spin-1 chain represents a large class of gappe... more AKLT state (or Haldane phase) in a one-dimensional spin-1 chain represents a large class of gapped featureless non-fractionalized paramagnets, which hosts symmetry-protected gapless excitations on the boundary. In this work we show how to realize this type of featureless spin-1 states on a generic two-dimensional lattice. These states have a gapped spectrum in the bulk but supports gapless excitations protected by spin rotational symmetry along a certain direction, and are featured by spin quantum Hall effect. Using fermion representation of integer-spins we show a concrete example of such spin-1 paramagnets on kagome lattice, and suggest a microscopic spin-1 Hamiltonian which may realize it.
ABSTRACT We study the electron doped cuprate superconductor Nd2-xCexCuO4+? using optical pump-pro... more ABSTRACT We study the electron doped cuprate superconductor Nd2-xCexCuO4+? using optical pump-probe spectroscopy over a range of dopings including both superconducting and underdoped antiferromagnetic samples. We focus on the pseudogap (PG) response, which is observed over the entire doping range, and its interaction with superconductivity (SC). The PG response onsets below values of T^* consistent with other probes, and its time dependence exhibits scaling consistent with critical fluctuations in samples near optimal doping. Furthermore, we observe laser fluence-dependent interaction between the PG and SC responses below Tc, indicative of a repulsive interaction between superconductivity and another fluctuating order.
Nature Physics, 2014
ABSTRACT Superconductivity in copper oxide (cuprate) high-transition-temperature superconductors ... more ABSTRACT Superconductivity in copper oxide (cuprate) high-transition-temperature superconductors follows from the chemical doping of an antiferromagnetic insulating state. The consensus that the wavefunction of the superconducting carrier, the Cooper pair, has d(x2-y2) symmetry(1,2) has long been reached. This pairing symmetry implies the existence of nodes in the superconducting energy gap. Recently, a series of angle-resolved photoemission spectroscopy experiments(3-9) have revealed that deeply underdoped cuprates exhibit a particle-hole symmetric(3) superconducting-like energy gap at the momentum-space locations where the d(x2-y2) gap nodes are expected. Here we discuss the possibility that this phenomenon is caused by a fully gapped topological superconducting state that coexists with the antiferromagnetic order. If experimentally confirmed, this result will completely change our view of how exactly the high-temperature superconductivity state evolves from the insulating antiferromagnet.
Physical Review B, 2013
We report the observation of coherent oscillations associated with charge density wave (CDW) orde... more We report the observation of coherent oscillations associated with charge density wave (CDW) order in the underdoped cuprate superconductor YBa 2 Cu 3 O 6+x by time-resolved optical reflectivity. Oscillations with frequency 1.87 THz onset at approximately 105 K and 130 K for dopings of x = 0.67 (ortho-VIII) and x = 0.75 (ortho-III), respectively. Upon cooling below the superconducting critical temperature (T c ), the oscillation amplitude is enhanced, the phase shifts by π, and the frequency softens by δ ν/ν ≈ 7%. A bi-quadratically coupled Landau-Ginzburg model qualitatively describes this behavior as arising from competition between superconducting and CDW orders.
An exotic feature of the fractional quantum Hall effect is the emergence of anyons, which are qua... more An exotic feature of the fractional quantum Hall effect is the emergence of anyons, which are quasiparticle excitations with fractional statistics. In the presence of a symmetry, such as U (1) charge conservation, it is well known that anyons can carry fractional symmetry quantum numbers. In this work we reveal a different class of symmetry realizations: i.e. anyons can " breed " in multiples under symmetry operation. We focus on the global Ising (Z2) symmetry and show examples of these unconventional symmetry realizations in Laughlin-type fractional quantum Hall states. One remarkable consequence of such an Ising symmetry is the emergence of anyons on the Ising symmetry domain walls. We also provide a mathematical framework which generalizes this phenomenon to any Abelian topological orders.
The nodal d x 2 −y 2 superconducting gap is a hallmark of the cuprate high Tc superconductors. Su... more The nodal d x 2 −y 2 superconducting gap is a hallmark of the cuprate high Tc superconductors. Surprisingly recent angle-resolved photoemission spectroscopy of deeply underdoped cuprates revealed a nodeless energy gap which is adhered to the Fermi surface. Importantly this phenomenon is observed for compounds across several different cuprate families. In this letter we propose an exciting possibility, namely the fully gapped state is a topological superconductor.
Symmetry protected topological (SPT) phases are gapped quantum phases which host symmetry-protect... more Symmetry protected topological (SPT) phases are gapped quantum phases which host symmetry-protected gapless edge excitations. On the other hand, the edge states can be gapped by spontaneously breaking symmetry. We show that topological defects on the symmetry-broken edge cannot proliferate due to their fractional statistics. A gapped symmetric boundary, however, can be achieved between an SPT phase and certain fractionalized phases by condensing the bound state of a topo-logical defect and an anyon. We demonstrate this by two examples in two dimensions: an exactly solvable model for the boundary between topological Ising paramagnet and double semion model, and a fermionic example about the quantum spin Hall edge. Such a hybrid structure containing both SPT phase and fractionalized phase generally support ground state degeneracy on torus.
We study 2+1 dimensional phases with topological order, such as fractional quantum Hall states an... more We study 2+1 dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry enriched topological (SET) phases. Here we present a K-matrix Chern-Simons approach to identify distinct phases with Abelian topological order, in the presence of unitary or anti-unitary global symmetries. A key step is the identification of an smooth edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method by classifying Z2 topological order (Z2 spin liquids), in the presence of an internal Z2 global symmetry for which we find six distinct phases. These include two phases with an unconventional action of symmetry that permutes anyons leading to symmetry protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry enriched Laughlin states and double semion theories are also discussed. Somewhat surprisingly we observe that : (i) gauging the global symmetry of distinct SET phases lead to topological orders with the same total quantum dimension, (ii) a pair of distinct SET phases can yield the same topological order on gauging the symmetry.
AKLT state (or Haldane phase) in a spin-1 chain represents a large class of gapped topological pa... more AKLT state (or Haldane phase) in a spin-1 chain represents a large class of gapped topological paramagnets, which hosts symmetry-protected gapless excitations on the boundary. In this work we show how to realize this type of featureless spin-1 states on a generic two-dimensional lattice. These states have a gapped spectrum in the bulk but supports gapless edge states protected by spin rotational symmetry along a certain direction, and are featured by spin quantum Hall effect. Using fermion representation of integer-spins we show a concrete example of such spin-1 topological paramagnets on kagome lattice, and suggest a microscopic spin-1 Hamiltonian which may realize it.
Physical Review Letters, 2016
The anyonic excitations of a spin liquid can feature fractional quantum numbers under space group... more The anyonic excitations of a spin liquid can feature fractional quantum numbers under space group symmetries. Detecting these fractional quantum numbers, which are analogs of the fractional charge of Laughlin quasiparticles, may prove easier than the direct observation of anyonic braiding and statistics. Motivated by the recent numerical discovery of spin-liquid phases in the kagome Heisenberg antiferromagnet, we theoretically predict the pattern of space group symmetry fractionalization in the kagome lattice SO(3)-symmetric chiral spin liquid. We provide a method to detect these fractional quantum numbers in finite-size numerics which is simple to implement in the density matrix renormalization group. Applying these developments to the chiral spin liquid phase of a kagome Heisenberg model, we find perfect agreement between our theoretical prediction and numerical observations.
We show that gauge field fluctuations in the composite fermion field theory can be exactly integr... more We show that gauge field fluctuations in the composite fermion field theory can be exactly integrated out using a non-unitary transformation. An instantaneous statistical interaction is induced which makes the Fermi sea unstable to chiral p-wave pairing. We show that the paired state is a Moore-Read Pfaffian and discuss the effects of Coulomb interaction in connection to even-denominator fractional quantum Hall effect.
A theory is developed for the paired even-denominator fractional quantum Hall states in the lowes... more A theory is developed for the paired even-denominator fractional quantum Hall states in the lowest Landau level. We show that electrons bind to quantized vortices to form composite fermions, interacting through an exact instantaneous interaction that favors chiral p-wave pairing. There are two canonically dual pairing gap functions related by the bosonic Laughlin wavefunction (Jastraw factor) due to the correlation holes. Remarkably,we find a new ground state lower in energy than the Moore-Read pfaffian for intermediate Coulomb interactions: a pfaffian with an oscillatory pairing function. We explore the unusual features and experimental implications of such a state. Connections to recent experiments are also discussed.
Due to strong geometric frustration and quantum fluctuation, the S = 1/2 quantum Heisenberg antif... more Due to strong geometric frustration and quantum fluctuation, the S = 1/2 quantum Heisenberg antiferromagnet on the kagome lattice has long been considered as an ideal platform to realize a spin liquid (SL), a phase exhibiting fractionalized excitations without any symmetry breaking. A recent numerical study (Yan et al., e-print arXiv:1011.6114) of the Heisenberg S = 1/2, kagome lattice model (HKLM) shows, in contrast to earlier results, that the ground state is a singlet-gapped SL with signatures of Z2 [Z subscript 2] topological order. Motivated by this numerical discovery, we use the projective symmetry group to classify all 20 possible Schwinger fermion mean-field states of Z2 [Z subscript 2] SLs on the kagome lattice. Among them we found only one gapped Z2 [Z subscript 2] SL (which we call the Z2[0,π]β [Z subscript 2 [0,pi] Beta] state) in the neighborhood of the U(1) Dirac SL state. Since its parent state, i.e., the U(1) Dirac SL, was found [Ran et al., Phys. Rev. Lett. 98, 117...
Recently, Z2 spin liquid was proposed as the ground state of the Kagome quantum antiferromagnet [... more Recently, Z2 spin liquid was proposed as the ground state of the Kagome quantum antiferromagnet [S. Yan, D.A. Huse, and S.R. White, Science, 332, 1173 (2011)]. We study proximate symmetry-broken phases that may appear on exiting the spin liquid phase, by tuning parameters such as further neighbor couplings. Given that the Dirac spin liquid is also a relatively low energy state, we consider models of Z2 spin liquids that are proximate to it. Specifically we consider the s-wave paired state of an algebraic spin liquid on Kagome lattice, Z2[0,π]β state of Y.-M Lu, Y. Ran, and P.A. Lee, Phys. Rev. B 83, 224413 (2011)] and examine its relations with other competing states. This allows us to characterize the proximate magnetically ordered and VBS phases and criticality between them and the quantum spin liquid.
While topological insulators of non-interacting fermions have been extensively studied, we know v... more While topological insulators of non-interacting fermions have been extensively studied, we know very little about topological insulators of bosons, whose realization necessitates strong interaction. In this work we apply Chern-Simons effective theory to classify and characterize interacting bosonic topological insulators in two spatial dimensions. These topological phases have a unique ground state on any closed manifold and no fractional excitations: yet they feature gapless edge states which are often protected by a symmetry. Examples include a bosonic analog of chiral superconductors, bosonic integer quantum Hall states (with Hall conductance quantized to even integers) and bosonic analog of the quantum spin Hall state. We show that these topological phases can be constructed in various ways: such as in arrays of coupled one-dimensional quantum wires. Our formulation also naturally applies to topological insulators of two-dimensional interacting fermions.
Three dimensional topological insulator represents a class of novel quantum phases hosting robust... more Three dimensional topological insulator represents a class of novel quantum phases hosting robust gapless boundary excitations, which is protected by global symmetries such as time reversal, charge conservation and spin rotational symmetry. In this work we systematically study another class of topological phases of weakly interacting electrons protected by spatial inversion symmetry, which generally don't support stable gapless boundary states. We classify these inversion-symmetric topological insulators and superconductors in the framework of K-theory, and construct their lattice models. We also discuss quantized response functions of these inversion-protected topological phases, which serve as their experimental signatures.
Recent numerical studies have provided strong evidence for a gapped Z$_2$ quantum spin liquid in ... more Recent numerical studies have provided strong evidence for a gapped Z$_2$ quantum spin liquid in the kagome lattice Heisenberg model. A special feature of spin liquids is that symmetries can be fractionalized, and different patterns of fractionalization imply distinct phases. The symmetry fractionalization pattern for the kagome spin liquid remains to be determined. A popular approach to studying spin liquids is to decompose the physical spin into partons which are either bosonic (Schwinger bosons) or fermionic (Abrikosov fermions), which are then treated within mean field theory. A longstanding question has been whether these two approaches are truly distinct or describe the same phase in complementary ways. Here we show that at least four spin liquid phases on the kagome lattice can be described both in terms of bosonic and fermionic partons, unifying pairs of theories that seem rather distinct. The key idea is that for kagome lattice states that admit a Schwinger boson mean field...
Physical Review B, 2014
AKLT state (or Haldane phase) in a one-dimensional spin-1 chain represents a large class of gappe... more AKLT state (or Haldane phase) in a one-dimensional spin-1 chain represents a large class of gapped featureless non-fractionalized paramagnets, which hosts symmetry-protected gapless excitations on the boundary. In this work we show how to realize this type of featureless spin-1 states on a generic two-dimensional lattice. These states have a gapped spectrum in the bulk but supports gapless excitations protected by spin rotational symmetry along a certain direction, and are featured by spin quantum Hall effect. Using fermion representation of integer-spins we show a concrete example of such spin-1 paramagnets on kagome lattice, and suggest a microscopic spin-1 Hamiltonian which may realize it.
ABSTRACT We study the electron doped cuprate superconductor Nd2-xCexCuO4+? using optical pump-pro... more ABSTRACT We study the electron doped cuprate superconductor Nd2-xCexCuO4+? using optical pump-probe spectroscopy over a range of dopings including both superconducting and underdoped antiferromagnetic samples. We focus on the pseudogap (PG) response, which is observed over the entire doping range, and its interaction with superconductivity (SC). The PG response onsets below values of T^* consistent with other probes, and its time dependence exhibits scaling consistent with critical fluctuations in samples near optimal doping. Furthermore, we observe laser fluence-dependent interaction between the PG and SC responses below Tc, indicative of a repulsive interaction between superconductivity and another fluctuating order.
Nature Physics, 2014
ABSTRACT Superconductivity in copper oxide (cuprate) high-transition-temperature superconductors ... more ABSTRACT Superconductivity in copper oxide (cuprate) high-transition-temperature superconductors follows from the chemical doping of an antiferromagnetic insulating state. The consensus that the wavefunction of the superconducting carrier, the Cooper pair, has d(x2-y2) symmetry(1,2) has long been reached. This pairing symmetry implies the existence of nodes in the superconducting energy gap. Recently, a series of angle-resolved photoemission spectroscopy experiments(3-9) have revealed that deeply underdoped cuprates exhibit a particle-hole symmetric(3) superconducting-like energy gap at the momentum-space locations where the d(x2-y2) gap nodes are expected. Here we discuss the possibility that this phenomenon is caused by a fully gapped topological superconducting state that coexists with the antiferromagnetic order. If experimentally confirmed, this result will completely change our view of how exactly the high-temperature superconductivity state evolves from the insulating antiferromagnet.
Physical Review B, 2013
We report the observation of coherent oscillations associated with charge density wave (CDW) orde... more We report the observation of coherent oscillations associated with charge density wave (CDW) order in the underdoped cuprate superconductor YBa 2 Cu 3 O 6+x by time-resolved optical reflectivity. Oscillations with frequency 1.87 THz onset at approximately 105 K and 130 K for dopings of x = 0.67 (ortho-VIII) and x = 0.75 (ortho-III), respectively. Upon cooling below the superconducting critical temperature (T c ), the oscillation amplitude is enhanced, the phase shifts by π, and the frequency softens by δ ν/ν ≈ 7%. A bi-quadratically coupled Landau-Ginzburg model qualitatively describes this behavior as arising from competition between superconducting and CDW orders.
An exotic feature of the fractional quantum Hall effect is the emergence of anyons, which are qua... more An exotic feature of the fractional quantum Hall effect is the emergence of anyons, which are quasiparticle excitations with fractional statistics. In the presence of a symmetry, such as U (1) charge conservation, it is well known that anyons can carry fractional symmetry quantum numbers. In this work we reveal a different class of symmetry realizations: i.e. anyons can " breed " in multiples under symmetry operation. We focus on the global Ising (Z2) symmetry and show examples of these unconventional symmetry realizations in Laughlin-type fractional quantum Hall states. One remarkable consequence of such an Ising symmetry is the emergence of anyons on the Ising symmetry domain walls. We also provide a mathematical framework which generalizes this phenomenon to any Abelian topological orders.
The nodal d x 2 −y 2 superconducting gap is a hallmark of the cuprate high Tc superconductors. Su... more The nodal d x 2 −y 2 superconducting gap is a hallmark of the cuprate high Tc superconductors. Surprisingly recent angle-resolved photoemission spectroscopy of deeply underdoped cuprates revealed a nodeless energy gap which is adhered to the Fermi surface. Importantly this phenomenon is observed for compounds across several different cuprate families. In this letter we propose an exciting possibility, namely the fully gapped state is a topological superconductor.
Symmetry protected topological (SPT) phases are gapped quantum phases which host symmetry-protect... more Symmetry protected topological (SPT) phases are gapped quantum phases which host symmetry-protected gapless edge excitations. On the other hand, the edge states can be gapped by spontaneously breaking symmetry. We show that topological defects on the symmetry-broken edge cannot proliferate due to their fractional statistics. A gapped symmetric boundary, however, can be achieved between an SPT phase and certain fractionalized phases by condensing the bound state of a topo-logical defect and an anyon. We demonstrate this by two examples in two dimensions: an exactly solvable model for the boundary between topological Ising paramagnet and double semion model, and a fermionic example about the quantum spin Hall edge. Such a hybrid structure containing both SPT phase and fractionalized phase generally support ground state degeneracy on torus.
We study 2+1 dimensional phases with topological order, such as fractional quantum Hall states an... more We study 2+1 dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry enriched topological (SET) phases. Here we present a K-matrix Chern-Simons approach to identify distinct phases with Abelian topological order, in the presence of unitary or anti-unitary global symmetries. A key step is the identification of an smooth edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method by classifying Z2 topological order (Z2 spin liquids), in the presence of an internal Z2 global symmetry for which we find six distinct phases. These include two phases with an unconventional action of symmetry that permutes anyons leading to symmetry protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry enriched Laughlin states and double semion theories are also discussed. Somewhat surprisingly we observe that : (i) gauging the global symmetry of distinct SET phases lead to topological orders with the same total quantum dimension, (ii) a pair of distinct SET phases can yield the same topological order on gauging the symmetry.
AKLT state (or Haldane phase) in a spin-1 chain represents a large class of gapped topological pa... more AKLT state (or Haldane phase) in a spin-1 chain represents a large class of gapped topological paramagnets, which hosts symmetry-protected gapless excitations on the boundary. In this work we show how to realize this type of featureless spin-1 states on a generic two-dimensional lattice. These states have a gapped spectrum in the bulk but supports gapless edge states protected by spin rotational symmetry along a certain direction, and are featured by spin quantum Hall effect. Using fermion representation of integer-spins we show a concrete example of such spin-1 topological paramagnets on kagome lattice, and suggest a microscopic spin-1 Hamiltonian which may realize it.