arXiv subjects

Ken Shiozaki

Publications and source records attributed to Ken Shiozaki.

At least 55 records · Page 3Linked to original sources

arXiv2020

$\mathbb{Z}_2$-enriched symmetry indicators for topological superconductors in the 1651 magnetic space groups

While the symmetry-based diagnosis of topological insulators and semimetals has enabled large-scale discovery of topological materials candidates, the extension of these approaches to the diagnosis of topological superconductors remains a major open question. One important new ingredient in the analysis of topological superconductivity is the presence of $\mathbb Z_2$-valued Pfaffian invariants associated with certain high-symmetry momenta. Such topological invariants lie beyond the conventional scope of symmetry representation theory for band structures, and as such they are nontrivial to incorporate into the systematic calculations of the symmetry indicators of band topology. Here, we overcome this challenge and report the full computation of the $\mathbb Z_2$-enriched symmetry indicators for superconductors in all symmetry settings. Our results indicate that incorporating the $\mathbb Z_2$ band labels enhance the diagnostic power of the scheme in roughly $60\%$ of the symmetry settings. Our framework can also be readily integrated with first-principles calculations to elucidate on the possible properties of unconventional superconductivity in a given compound. As a demonstration, we analyze explicitly the interplay between pairing symmetry and topological superconductivity in the recently discovered superconductors CaPtAs and CaSb$_2$.

cond-mat.supr-con↗

arXiv2020

Topological Origin of Non-Hermitian Skin Effects

A unique feature of non-Hermitian systems is the skin effect, which is the extreme sensitivity to the boundary conditions. Here, we reveal that the skin effect originates from intrinsic non-Hermitian topology. Such a topological origin not merely explains the universal feature of the known skin effect, but also leads to new types of the skin effects---symmetry-protected skin effects. In particular, we discover the $\mathbb{Z}_{2}$ skin effect protected by time-reversal symmetry. On the basis of topological classification, we also discuss possible other skin effects in arbitrary dimensions. Our work provides a unified understanding about the bulk-boundary correspondence and the skin effects in non-Hermitian systems.

cond-mat.mes-hall↗

arXiv2019

Symmetry and Topology in Non-Hermitian Physics

We develop a complete theory of symmetry and topology in non-Hermitian physics. We demonstrate that non-Hermiticity ramifies the celebrated Altland-Zirnbauer symmetry classification for insulators and superconductors. In particular, charge conjugation is defined in terms of transposition rather than complex conjugation due to the lack of Hermiticity, and hence chiral symmetry becomes distinct from sublattice symmetry. It is also shown that non-Hermiticity enables a Hermitian-conjugate counterpart of the Altland-Zirnbauer symmetry. Taking into account sublattice symmetry or pseudo-Hermiticity as an additional symmetry, the total number of symmetry classes is 38 instead of 10, which describe intrinsic non-Hermitian topological phases as well as non-Hermitian random matrices. Furthermore, due to the complex nature of energy spectra, non-Hermitian systems feature two different types of complex-energy gaps, point-like and line-like vacant regions. On the basis of these concepts and K-theory, we complete classification of non-Hermitian topological phases in arbitrary dimensions and symmetry classes. Remarkably, non-Hermitian topology depends on the type of complex-energy gaps and multiple topological structures appear for each symmetry class and each spatial dimension, which are also illustrated in detail with concrete examples. Moreover, the bulk-boundary correspondence in non-Hermitian systems is elucidated within our framework, and symmetries preventing the non-Hermitian skin effect are identified. Our classification not only categorizes recently observed lasing and transport topological phenomena, but also predicts a new type of symmetry-protected topological lasers with lasing helical edge states and dissipative topological superconductors with nonorthogonal Majorana edge states. Furthermore, our theory provides topological classification of Hermitian and non-Hermitian free bosons.

cond-mat.mes-hall↗

arXiv2019

Glide-symmetric magnetic topological crystalline insulators with inversion symmetry

It is known that three-dimensional magnetic systems with glide symmetry can be characterized by a $Z_2$ topological invariant together with the Chern number associated with the normal vector of the glide plane, and they are expressed in terms of integrals of the Berry curvature. In the present paper, we study the fate of this topological invariant when inversion symmetry is added while time-reversal symmetry is not enforced. There are two ways to add inversion symmetry, leading to space groups No.~13 and No.~14. In space group No.~13, we find that the glide-$Z_2$ invariant is expressed solely from the irreducible representations at high-symmetry points in $k$-space. It constitutes the $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry-based indicator for this space group, together with another $\mathbb{Z}_2$ representing the Chern number modulo 2. In space group No.~14, we find that the symmetry-based indicator $\mathbb{Z}_2$ is given by a combination of the glide-$Z_2$ invariant and the Chern number. Thus, in space group No.~14, from the irreducible representations at high-symmetry points we can only know possible combinations of the glide-$Z_2$ invariant and the Chern number, but in order to know each value of these topological numbers, we should calculate integrals of the Berry curvature. Finally, we show that in both cases, the symmetry-based indicator $\mathbb{Z}_4$ for inversion symmetric systems leading to the higher-order topological insulators is directly related with the glide-$Z_2$ invariant and the Chern number. As an independent approach to these results, we also construct all invariants from the layer construction for these space groups, and we show complete agreement with the above results for the topological invariants constructed from $k$-space topology.

cond-mat.mes-hall↗

arXiv2019

Variants of the symmetry-based indicator

The symmetry-based indicator [H. C. Po, A. Vishwanath, H. Watanabe, Nat. Commun. 8, 50 (2017)] is a practical tool to diagnose topological materials in the band theory. In this note, we present two directions to generalize the symmetry-based indicator for other classes of topological materials. The one is for superconductors. The careful definition of the atomic insulators and the trivial vacuum Hamiltonian yields the symmetry-based indicators specific to superconductors. The other is for ingap boundary states. The quotient of the group of atomic insulators by a subset of atomic insulators such as those localized at the interior of the unit cell gives us the symmetry-based indicator for detecting ingap corner, hinge, and surface states.

cond-mat.mes-hall↗

arXiv2019

Classification of topological crystalline superconducting nodes on high-symmetry lines: Point nodes, line nodes, and Bogoliubov Fermi surfaces

Recent development in exact classification of a superconducting gap has elucidated various unconventional gap structures, which have not been predicted by the classification of order parameter based on the point group. One of the important previous results is that all symmetry-protected line nodes are characterized by nontrivial topological numbers. Another intriguing discovery is the gap structures depending on the angular momentum $j_z$ of normal Bloch states on threefold and sixfold rotational-symmetric lines in the Brillouin zone. Stimulated by these findings, we classify irreducible representations of the Bogoliubov-de Gennes Hamiltonian at each $\boldsymbol{k}$ point on a high-symmetry $n$-fold ($n = 2$, $3$, $4$, and $6$) axis for centrosymmetric and paramagnetic superconductors, by using the combination of group theory and $K$ theory. This leads to the classification of all crystal symmetry-protected nodes (including $j_z$-dependent nodes) on the axis that crosses a normal-state Fermi surface. As a result, it is shown that the classification by group theory completely corresponds with the topological classification. Based on the obtained results, we discuss superconducting gap structures in SrPtAs, CeCoIn$_5$, UPt$_3$, and UCoGe.

cond-mat.supr-con↗

arXiv2019

Many-Body Order Parameters for Multipoles in Solids

We propose many-body order parameters for bulk multipoles in crystalline systems, which originate from the non-trivial topology of the nested Wilson loop spectrum. The many-body order parameters are designed to measure multipolar charge distribution in a crystalline unit cell, and they match the localized corner charge originating from the multipoles. We provide analytic arguments and numerical proof for the order parameters. Furthermore, we show that the many-body order parameters faithfully measure the physical multipole moments even when the symmetries quantizing multipoles are lost and thus the nested Wilson loop spectrum does not exactly reproduce the physical multipole moments.

cond-mat.str-el↗

arXiv2019

Topological classification under nonmagnetic and magnetic point group symmetry: Application of real-space Atiyah-Hirzebruch spectral sequence to higher-order topology

We classify time-reversal breaking (class A) spinful topological crystalline insulators with crystallographic non-magnetic (32 types) and magnetic (58 types) point groups. The classification includes all possible magnetic topological crystalline insulators protected by point group symmetry. Whereas the classification of topological insulators is known to be given by the $K$-theory in the momentum space, computation of the $K$-theory has been a difficult task in the presence of complicated crystallographic symmetry. Here we consider the $K$-homology in the real space for this problem, instead of the $K$-theory in the momentum space, both of which give the same topological classification. We apply the Atiyah-Hirzebruch spectral sequence (AHSS) for computation of the $K$-homology, which is a mathematical tool for generalized (co)homology. In the real space picture, the AHSS naturally gives the classification of higher-order topological insulators at the same time. By solving the group extension problem in the AHSS on the basis of physical arguments, we completely determine possible topological phases including higher-order ones for each point group. Relationships among different higher-order topological phases are argued in terms of the AHSS in the $K$-homology. We find that in some nonmagnetic and magnetic point groups, a stack of two $\mathbb{Z}_2$ second-order topological insulators can be smoothly deformed into non-trivial fourth-order topological insulators, which implies non-trivial group extensions in the AHSS.

cond-mat.mes-hall↗

arXiv2019

Anomaly indicator of rotation symmetry in (3+1)D topological order

We examine (3+1)D topological ordered phases with $C_k$ rotation symmetry. We show that some rotation symmetric (3+1)D topological orders are anomalous, in the sense that they cannot exist in standalone (3+1)D systems, but only exist on the surface of (4+1)D SPT phases. For (3+1)D discrete gauge theories, we propose anomaly indicator that can diagnose the $\mathbb{Z}_k$ valued rotation anomaly. Since (3+1)D topological phases support both point-like and loop-like excitations, the indicator is expressed in terms of the symmetry properties of point and loop-like excitations, and topological data of (3+1)D discrete gauge theories.

cond-mat.str-el↗

arXiv2018

Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models

The Lieb-Schultz-Mattis theorem dictates that a trivial symmetric insulator in lattice models is prohibited if lattice translation symmetry and $U(1)$ charge conservation are both preserved. In this paper, we generalize the Lieb-Schultz-Mattis theorem to systems with higher-form symmetries, which act on extended objects of dimension $n > 0$. The prototypical lattice system with higher-form symmetry is the pure abelian lattice gauge theory whose action consists only of the field strength. We first construct the higher-form generalization of the Lieb-Schultz-Mattis theorem with a proof. We then apply it to the $U(1)$ lattice gauge theory description of the quantum dimer model on bipartite lattices. Finally, using the continuum field theory description in the vicinity of the Rokhsar-Kivelson point of the quantum dimer model, we diagnose and compute the mixed 't Hooft anomaly corresponding to the higher-form Lieb-Schultz-Mattis theorem.

cond-mat.stat-mech↗

arXiv2018

Anomalous helical edge states in a non-Hermitian Chern insulator

A non-Hermitian extension of a Chern insulator and its bulk-boundary correspondence are investigated. It is shown that in addition to the robust chiral edge states that reflect the nontrivial topology of the bulk (nonzero Chern number), anomalous helical edge states localized only at one edge can appear, which are unique to the non-Hermitian Chern insulator.

cond-mat.mes-hall↗

arXiv2018

Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries

We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single particle wave functions. For example, we construct the many-body $\mathbb{Z}_2$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele $\mathbb{Z}_2$ invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermionic partial transpose which is basically the standard partial transpose equipped with a sign structure to account for anti-commuting property of fermion operators. We also report some basic results on various kinds of pin structures -- a key concept behind our strategy for constructing many-body topological invariants -- such as the obstructions, isomorphism classes, and Dirac quantization conditions.

cond-mat.str-el↗

arXiv2018

Universal Relation among Many-Body Chern Number, Rotation Symmetry, and Filling

Understanding the interplay between the topological nature and the symmetry property of interacting systems has been a central matter of condensed matter physics in recent years. In this Letter, we establish nonperturbative constraints on the quantized Hall conductance of many-body systems with arbitrary interactions. Our results allow one to readily determine the many-body Chern number modulo a certain integer without performing any integrations, solely based on the rotation eigenvalues and the average particle density of the many-body ground state.

cond-mat.str-el↗

arXiv2017

Bosonic topological phases of matter: bulk-boundary correspondence, SPT invariants and gauging

We analyze $2+1d$ and $3+1d$ Bosonic Symmetry Protected Topological (SPT) phases of matter protected by onsite symmetry group $G$ by using dual bulk and boundary approaches. In the bulk we study an effective field theory which upon coupling to a background flat $G$ gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat $G$ gauge fields to obtain Dijkgraaf-Witten topological $G$ gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoils the gauge symmetry. This mechanism is related to anyon condensation in $2+1d$ and condensing bosonic gauge charges in $3+1d$. In the dual boundary approach, we study $1+1d$ and $2+1d$ quantum field theories that have $G$ 't-Hooft anomalies that can be precisely cancelled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further we sum over boundary partition functions with different background gauge fields to construct $G$-characters that generate topological data for the bulk topological gauge theory. Finally, we study a $2+1d$ quantum field theory with a mixed $\mathbb{Z}_2^{T/R} \times U(1)$ anomaly where $\mathbb{Z}_2^{T/R}$ is time-reversal/reflection symmetry, and the $U(1)$ could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in $3+1d$ that cancels this anomaly. This signals the existence of SPTs in $3+1d$ protected by 0,1-form $U(1)\times \mathbb{Z}_{2}^{T,R}$.

cond-mat.str-el↗

arXiv2017

Möbius topological superconductivity in UPt$_3$

Intensive studies for more than three decades have elucidated multiple superconducting phases and odd-parity Cooper pairs in a heavy fermion superconductor UPt$_3$. We identify a time-reversal invariant superconducting phase of UPt$_3$ as a recently proposed topological nonsymmorphic superconductivity. Combining the band structure of UPt$_3$, order parameter of $E_{\rm 2u}$ representation allowed by $P6_3/mmc$ space group symmetry, and topological classification by $K$-theory, we demonstrate the nontrivial $Z_2$-invariant of three-dimensional DIII class enriched by glide symmetry. Correspondingly, double Majorana cone surface states appear at the surface Brillouin zone boundary. Furthermore, we show a variety of surface states and clarify the topological protection by crystal symmetry. Majorana arcs corresponding to tunable Weyl points appear in the time-reversal symmetry broken B-phase. Majorana cone protected by mirror Chern number and Majorana flat band by glide-winding number are also revealed.

cond-mat.supr-con↗

arXiv2017

Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries

We propose the definitions of many-body topological invariants to detect symmetry-protected topological phases protected by point group symmetry, using partial point group transformations on a given short-range entangled quantum ground state. Partial point group transformations $g_D$ are defined by point group transformations restricted to a spatial subregion $D$, which is closed under the point group transformations and sufficiently larger than the bulk correlation length $ξ$. By analytical and numerical calculations,we find that the ground state expectation value of the partial point group transformations behaves generically as $\langle GS | g_D | GS \rangle \sim \exp \Big[ i θ+ γ- α\frac{{\rm Area}(\partial D)}{ξ^{d-1}} \Big]$. Here, ${\rm Area}(\partial D)$ is the area of the boundary of the subregion $D$, and $α$ is a dimensionless constant. The complex phase of the expectation value $θ$ is quantized and serves as the topological invariant, and $γ$ is a scale-independent topological contribution to the amplitude. The examples we consider include the $\mathbb{Z}_8$ and $\mathbb{Z}_{16}$ invariants of topological superconductors protected by inversion symmetry in $(1+1)$ and $(3+1)$ dimensions, respectively, and the lens space topological invariants in $(2+1)$-dimensional fermionic topological phases. Connections to topological quantum field theories and cobordism classification of symmetry-protected topological phases are discussed.

cond-mat.str-el↗

arXiv2017

Topological Crystalline Materials - General Formulation, Module Structure, and Wallpaper Groups -

We formulate topological crystalline materials on the basis of the twisted equivariant $K$-theory. Basic ideas of the twisted equivariant $K$-theory is explained with application to topological phases protected by crystalline symmetries in mind, and systematic methods of topological classification for crystalline materials are presented. Our formulation is applicable to bulk gapful topological crystalline insulators/superconductors and their gapless boundary and defect states, as well as bulk gapless topological materials such as Weyl and Dirac semimetals, and nodal superconductors. As an application of our formulation, we present a complete classification of topological crystalline surface states, in the absence of time-reversal invariance. The classification works for gapless surface states of three-dimensional insulators, as well as full gapped two-dimensional insulators. Such surface states and two-dimensional insulators are classified in a unified way by 17 wallpaper groups, together with the presence or the absence of (sublattice) chiral symmetry. We identify the topological numbers and their representations under the wallpaper group operation. We also exemplify the usefulness of our formulation in the classification of bulk gapless phases. We present a new class of Weyl semimetals and Weyl superconductors that are topologically protected by inversion symmetry.

cond-mat.mes-hall↗

arXiv2017

Many-body topological invariants for fermionic symmetry-protected topological phases

We define and compute many-body topological invariants of interacting fermionic symmetry-protected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a non-local operation, "partial" reflection or transpose. As an application of our scheme, we study the $\mathbb{Z}_8$ and $\mathbb{Z}_{16}$ classification of topological superconductors in one and three dimensions.

cond-mat.str-el↗