### Anomaly Matching and Symmetry-protected Criticality in 1d Quantum Many-body Systems

### Chang-Tse Hsieh

Kavli Institute for the Physics and Mathematics of the Universe

**& Institute for Solid States Physics **
Univ. of Tokyo

### Chiral Matter and Topology, NTU CTS

December 7, 2018

**( )**

### Anomaly Matching and Symmetry-protected Criticality in 1d Quantum Many-body Systems

### Chang-Tse Hsieh

Kavli Institute for the Physics and Mathematics of the Universe

**& Institute for Solid States Physics **
Univ. of Tokyo

### Chiral Matter and Topology, NTU CTS

December 7, 2018

**( )**

### chiral

### discrete

**Collaborators**

Shinsei Ryu (Chicago)

Gil Young Cho (POSTECH)

G. Y. Cho^{*}**, C.-T. Hsieh**** ^{*}**, and S. Ryu, PRB 96, 195105 (2017); arXiv:1705.03892

Masaki Oshikawa (ISSP, U of Tokyo)

Yuan Yao (ISSP, U of Tokyo)

Y. Yao^{*}**, C.-T. Hsieh**** ^{*}**, and M. Oshikawa, arXiv:1805.06885

*Equal contributions

### Outline

•

### Introduction

•

### Example 1: 1d charged fermion systems

•

### Example 2: 1d SU(N) spin systems

•

### Conclusion

### Outline

•

### Introduction

•

### Example 1: 1d charged fermion systems

•

### Example 2: 1d SU(N) spin systems

•

### Conclusion

### Introduction

•

### Identifying the “phase” of a generic many-body system is an important but, in general, difficult problem

•

### Quite often, symmetries play an essential role in such a problem

•

### Various classes of phases of matter:

Ø

*Conventional Landau-Ginzburg-Wilson symm-breaking paradigm*

Ø

### Topological phases: Symm-protected top. (SPT) phases, etc

[Hasan-Kane 10; Qi-Zhang 11; Chiu-Teo-Schnyder-Ryu 16; Witten 16]

(all are Rev. Mod. Phys.)

### Today’s focus

•

### Phases associated with “symmetry-protected in-gap(p)-ability”

### trivial ó gappable nontrivial ó ingappable

### Symmetry protected critical phase classification of SU (N ) spin chains in rectangular Young tableaux representations, and global axial anomaly in 1+1 dimensions

### We derive the low-energy properties of SU (N ) spin systems in the presence of spin rotation and translation symmetries. Based on mixed P SU (N )- Z anomaly, the spin system cannot be gapped with a unique ground state if the number of Young-tableaux boxes per unit cell is not divisible by N . The SU (N ) WZW model are classified into N symmetry-protected critical classes and each spin system can only realize one of them at criticality. It implies a no-go theorem that an RG flow is possible between two critical points only if they belong to the same symmetry-protected critical class, as long as the underlying lattice spin models respect the imposed symmetries.

### Introduction.—

### Hom( ˜ ⌦ ^{spin} _{3}

^{c}

### (B Z), U(1)) ⇠ = U (1) ⇠ = R/Z (1)

### LSM index I = (total charges per unit cell) mod 1.

### (2)

### H ^{3} (P SU (N ) ⇥ Z, U(1))/H ^{3} (P SU (N ), U (1)) ⇠ = Z ^{N} (3)

### LSM index I ^{N} = (# of YT boxes per unit cell) mod N.

### (4)

### I ^{2} = 2s mod 2. (5)

### I ^{N} 6= 0 mod N. (6)

### GSD 2 N

### gcd( I ^{N} , N ) N, (7)

### U (1) : ! e ^{i}

### Z : ! e ^{ik}

^{F}

^{z}

### = e ^{i⇡⌫}

^{z}

### (8)

### Z(A _{U (1)} ) ^{axial} ! e ^{2⇡i⌫} ^{⇥integer} Z(A _{U (1)} ) (9)

### P SU (N ) : g ! wgw ^{1} , w 2 SU(N)

### Z ^{n} (trans) : g ! e ^{2⇡im/N} g, m 2 {0, 1, ..., N 1 } (10)

### n = N/ gcd(m, N ) (11)

### Z(A _{P SU (N )} ) ^{axial} ! e ^{2⇡i}

^{km}

^{N}

^{⇥integer} Z(A _{P SU (N )} ) (12)

### I ^{N} · N ^{0}

### gcd(N ^{0} , k ^{0} m ^{0} ) = 0 mod N. (13)

### H _{ULM} = X

### i

### S _{i} · S ^{i+1} + (S _{i} · S ^{i+1} ) ^{2} / X

### i

### X 8 A=1

### T _{i} ^{A} T _{i+1} ^{A} (14)

### N = 2 : = {2s} (15)

### N > 2 : = {4, 2, 1} (16)

### H _{0} + ↵H _{pert} (17) The classification of quantum phases is a central issue in condensed matter physics, where consider- able recent progresses were initiated in strongly in- teracting many-body systems. In particular, vari- ous topological phases with unbroken symmetries, e.g.

### symmetry-protected trivial (SPT) ordered phases [? ? ] and topological ordered phases enrich the phase diagram beyond conventional Landau-Ginzburg-Wilson spontaneously-symmetry-breaking paradigm. Therefore, the constraints in the appearance of symmetries on phase diagrams play essential roles since they rule out the ex- istence of a large class of forbidden gapped or gapless phases when the certain symmetry is respected. For gapped phases, the restriction on the ground-state degen- eracy allows a unified understanding on the low-energy spectral properties. For example, Lieb-Schultz-Mattis (LSM) theorem [? ] and its generalization LSMOH the- orem by Oshikawa and Hastings [? ? ] states that the ground states cannot be trivially gapped with a unique ground state if the particle number per unit cell is frac- tional and both the translation symmetry and charge U (1) conservation are preserved. As a part of gapless

*Ø Defined regarding “whether a system with symm has or can be * **gapped into a trivial – unique and symmetric – gapped ground ** state”

**gapped into a trivial – unique and symmetric – gapped ground**

symmetry-respecting

•

### Phases associated with “symmetry-protected in-gap(p)-ability”

8

Symmetry d

AZ ⇥ ⌅ ⇧ 1 2 3 4 5 6 7 8

A 0 0 0 0 Z 0 Z 0 Z 0 Z

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0

AI 1 0 0 0 0 0 Z 0 Z^{2} Z^{2} Z

BDI 1 1 1 Z 0 0 0 Z 0 Z^{2} Z^{2}

D 0 1 0 Z^{2} Z 0 0 0 Z 0 Z^{2}

DIII 1 1 1 Z2 Z2 Z 0 0 0 Z 0

AII 1 0 0 0 Z^{2} Z^{2} Z 0 0 0 Z

CII 1 1 1 Z 0 Z^{2} Z^{2} Z 0 0 0

C 0 1 0 0 Z 0 Z^{2} Z^{2} Z 0 0

CI 1 1 1 0 0 Z 0 Z^{2} Z^{2} Z 0

TABLE I Periodic table of topological insulators and super-
conductors. The 10 symmetry classes are labeled using the
notation of Altland and Zirnbauer (1997) (AZ) and are spec-
ified by presence or absence of T symmetry ⇥, particle-hole
symmetry ⌅ and chiral symmetry ⇧ = ⌅⇥. ±1 and 0 denotes
the presence and absence of symmetry, with ±1 specifying
the value of ⇥^{2} and ⌅^{2}. As a function of symmetry and space
dimensionality, d, the topological classifications (Z, Z^{2} and 0)
show a regular pattern that repeats when d! d + 8.

3. Periodic table

Topological insulators and superconductors fit to- gether into a rich and elegant mathematical structure that generalizes the notions of topological band theory described above (Schnyder, et al., 2008; Kitaev, 2009;

Schnyder, et al., 2009; Ryu, et al., 2010). The classes
of equivalent Hamiltonians are determined by specifying
the symmetry class and the dimensionality. The symme-
try class depends on the presence or absence of T sym-
metry (8) with ⇥^{2} = ±1 and/or particle-hole symmetry
(15) with ⌅^{2} = ±1. There are 10 distinct classes, which
are closely related to the Altland and Zirnbauer (1997)
classification of random matrices. The topological clas-
sifications, given by Z, Z2 or 0, show a regular pattern
as a function of symmetry class and dimensionality and
can be arranged into the periodic table of topological in-
sulators and superconductors shown in Table I.

The quantum Hall state (Class A, no symmetry; d =
2), the Z^{2} topological insulators (Class AII, ⇥^{2} = 1;

d = 2, 3) and the Z2 and Z topological superconductors
(Class D, ⌅^{2} = 1; d = 1, 2) described above are each
entries in the periodic table. There are also other non
trivial entries describing di↵erent topological supercon-
ducting and superfluid phases. Each non trivial phase is
predicted, via the bulk-boundary correspondence to have
gapless boundary states. One notable example is super-
fluid ^{3}He B (Volovik, 2003; Roy, 2008; Schnyder, et al.,
2008; Nagato, Higashitani and Nagai, 2009; Qi, et al.,
2009; Volovik, 2009), in (Class DIII, ⇥^{2} = 1, ⌅^{2} = +1;

d = 3) which has aZ classification, along with gapless 2D Majorana fermion modes on its surface. A generalization of the quantum Hall state introduced by Zhang and Hu

E

E_{F}

Conduction Band

Valence Band Quantum spin

Hall insulator ν=1 Conventional Insulator ν=0

(a) (b)

k 0

−π/a −π/a

FIG. 5 Edge states in the quantum spin Hall insulator. (a) shows the interface between a QSHI and an ordinary insula- tor, and (b) shows the edge state dispersion in the graphene model, in which up and down spins propagate in opposite directions.

(2001) corresponds to the d = 4 entry in class A or AII.

There are also other entries in physical dimensions that have yet to be filled by realistic systems. The search is on to discover such phases.

III. QUANTUM SPIN HALL INSULATOR

The 2D topological insulator is known as a quantum spin Hall insulator. This state was originally theorized to exist in graphene (Kane and Mele, 2005a) and in 2D semiconductor systems with a uniform strain gradient (Bernevig and Zhang, 2006). It was subsequently pre- dicted to exist (Bernevig, Hughes and Zhang, 2006), and was then observed (K¨onig, et al., 2007), in HgCdTe quan- tum well structures. In section III.A we will introduce the physics of this state in the model graphene system and describe its novel edge states. Section III.B will re- view the experiments, which have also been the subject of the review article by K¨onig, et al. (2008).

A. Model system: graphene

In section II.B.2 we argued that the degeneracy at the
Dirac point in graphene is protected by inversion and
T symmetry. That argument ignored the spin of the
electrons. The spin orbit interaction allows a new mass
term in (3) that respects all of graphene’s symmetries. In
the simplest picture, the intrinsic spin orbit interaction
commutes with the electron spin S_{z}, so the Hamiltonian
decouples into two independent Hamiltonians for the up
and down spins. The resulting theory is simply two copies
the Haldane (1988) model with opposite signs of the Hall
conductivity for up and down spins. This does not violate
T symmetry because time reversal flips both the spin and

xy. In an applied electric field, the up and down spins
have Hall currents that flow in opposite directions. The
Hall conductivity is thus zero, but there is a quantized
spin Hall conductivity, defined by J_{x}^{"} J_{x}^{#} = _{xy}^{s} E_{y} with

xys = e/2⇡ – a quantum spin Hall e↵ect. Related ideas were mentioned in earlier work on the planar state of

2

H_{MG}^{s=1/2} = X

i

✓

S_{i} · Si+1 + 1

2S_{i} · Si+2 + 3
8

◆

(24)

H_{edge} =
Z

dx ^{†}(x)( iv_{F}@_{x}) _{z} (x) (25)

P si(x) = ( _{R}(x), _{L}(x))^{T} (26)
The classification of quantum phases is a central is-
sue in condensed matter physics, where considerable
recent progresses were initiated in strongly interacting
many-body systems. In particular, various topologi-
cal phases with unbroken symmetries, e.g. symmetry-
protected trivial (SPT) ordered phases [1, 2] and topo-
logical ordered phases enrich the phase diagram beyond
conventional Landau-Ginzburg-Wilson spontaneously-
symmetry-breaking paradigm. Therefore, the constraints
in the appearance of symmetries on phase diagrams play
essential roles since they rule out the existence of a large
class of forbidden gapped or gapless phases when the cer-
tain symmetry is respected. For gapped phases, the re-
striction on the ground-state degeneracy allows a unified
understanding on the low-energy spectral properties. For
example, Lieb-Schultz-Mattis (LSM) theorem [3] and its
generalization LSMOH theorem by Oshikawa and Hast-
ings [4, 5] states that the ground states cannot be trivially
gapped with a unique ground state if the particle number
per unit cell is fractional and both the translation sym-
metry and charge U (1) conservation are preserved. As
a part of gapless phases, critical phase classifications is
a key ingredient in the understanding of universal criti-
cal behaviors of various phase transitions. However, the
classification of critical phases still remains open and the
related proposal is only given in SU (2) spin chains [6].

Furthermore, many stable critical phases have been found experimentally and numerically [7, 8], and the reason of such stability is not completely understood.

In reality, the constraints on phase diagrams with more general symmetries than U (1) and SU (2) are within intense interest. For example, SU (N ) “spin” systems with N > 2 are realized in ultracold atoms on op- tical lattices [9–16] and also in some spin-orbital sys- tems [? ], although SU (N ) systems was initially intro- duced as theoretical toy models to understand “physical”

SU (2) spins. Thus the study related with the phase di- agrams of SU (N ) spin systems is of realistic interest in its own. Furthermore, although spin-rotation and trans- lation symmetries are imposed, symmetry enhancements cannot be excluded, e.g. phases with emergent symme- tries. The symmetry can be enhanced on the lattice such as that the spin-1 chain has an explicit SU (3) symmetry in the Uimin-Lai-Sutherland model [17–19]. In addition,

higher symmetries can also emerge at the thermodynam- ical limit, e.g. the emergent SU (3) symmetry of critical spin-2 chain [20]. Therefore, the generalized SU (N ) spin models are necessary in understanding the symmetry en- hancement.

In this Letter, we focus on the fundamental constraints
on the phase diagrams of SU (N ) spin chains in gen-
eral representations with SU (N ) spin rotation symmetry
and lattice translation symmetry respected. More specif-
ically, we obtain the restriction of ground-state degener-
acy of gapped phases generalizing Lieb-Schultz-Mattis-
A✏eck (LSMA) theorem [21], and the classification of
critical phases, when the required symmetries are im-
posed. The compulsory degenerate ground states can
be understood by the concept of “ingappability” simi-
larly to the boundary states of nontrivial SPT phase,
which cannot be gapped with a specified symmetry un-
broken and bulk gap unclosed [1, 2]. On the SPT side,
the ingappability results from the quantum symmetry
anomaly of e↵ective boundary theory forbidding its lat-
tice realization [22]. CH: Should add more references
here. I will also do it later. Although the SU (N )
spin chain in our interest can be realized in its own di-
mension, the non-on-site nature of translation symme-
try enables us to understand its ingappability by consid-
ering the quantum anomaly of its e↵ective field theory
in the thermodynamical limit [23]. In a similar spirit,
the classification of critical phases can be done accord-
ing to their quantum anomaly. Such symmetry-protected
critical (SPC) classification has been proposed in SU (2)-
symmetric and translation-symmetric spin chains, where
the critical points are classified into two classes, one of
which can only take place in half-integer spin chains
while the other one can be realized only by integer spin
chains [6]. We generalize the SPC classification to SU (N )
symmetric critical phases of SU (N ) spin chains with-
out emergent symmetries. Moreover, we also obtain a
constraint on the candidates of emergent higher SU (N^{0})
symmetry with N^{0} > N by matching their symmetry
anomaly. As a special example, such restriction can ex-
plain that SU (3) symmetry enhancements have not (ac-
tually cannot) been found in SU (2) and translation sym-
metric half-integer spin chains.

Based on.... anomaly, we come to the conclusion that.... End of intro(?)

Translationally invariant SU (N ) spin system in 1 + 1
dimensions and the LSM index — We consider a generic
(1 + 1)d SU (N ) spin system described by a Hamilto-
nian H_{SU (N )} with the lattice translation symmetry and
a global spin-rotation symmetry specified by the projec-
tive special unitary group P SU (N ). Here the system is
subject to periodic boundary condition and the trans-
lation, for generality, defines the unit cell consisting of
multiple sites and forms a discrete group Z^{trans} in the
thermodynamic limit. A typical example of such a sys-
tem is the SU (N ) Heisenberg antiferromagnetic (HAF)

2

H_{MG}^{s=1/2} = X

i

✓

S_{i} · S^{i+1} + 1

2S_{i} · S^{i+2} + 3
8

◆

(24)

H_{edge} =
Z

dx ^{†}(x)( iv_{F}@_{x}) _{z} (x) (25)

(x) = ( _{R}(x), _{L}(x))^{T} (26)
The classification of quantum phases is a central is-
sue in condensed matter physics, where considerable
recent progresses were initiated in strongly interacting
many-body systems. In particular, various topologi-
cal phases with unbroken symmetries, e.g. symmetry-
protected trivial (SPT) ordered phases [1, 2] and topo-
logical ordered phases enrich the phase diagram beyond
conventional Landau-Ginzburg-Wilson spontaneously-
symmetry-breaking paradigm. Therefore, the constraints
in the appearance of symmetries on phase diagrams play
essential roles since they rule out the existence of a large
class of forbidden gapped or gapless phases when the cer-
tain symmetry is respected. For gapped phases, the re-
striction on the ground-state degeneracy allows a unified
understanding on the low-energy spectral properties. For
example, Lieb-Schultz-Mattis (LSM) theorem [3] and its
generalization LSMOH theorem by Oshikawa and Hast-
ings [4, 5] states that the ground states cannot be trivially
gapped with a unique ground state if the particle number
per unit cell is fractional and both the translation sym-
metry and charge U (1) conservation are preserved. As
a part of gapless phases, critical phase classifications is
a key ingredient in the understanding of universal criti-
cal behaviors of various phase transitions. However, the
classification of critical phases still remains open and the
related proposal is only given in SU (2) spin chains [6].

Furthermore, many stable critical phases have been found experimentally and numerically [7, 8], and the reason of such stability is not completely understood.

In reality, the constraints on phase diagrams with more general symmetries than U (1) and SU (2) are within intense interest. For example, SU (N ) “spin” systems with N > 2 are realized in ultracold atoms on op- tical lattices [9–16] and also in some spin-orbital sys- tems [? ], although SU (N ) systems was initially intro- duced as theoretical toy models to understand “physical”

SU (2) spins. Thus the study related with the phase di- agrams of SU (N ) spin systems is of realistic interest in its own. Furthermore, although spin-rotation and trans- lation symmetries are imposed, symmetry enhancements cannot be excluded, e.g. phases with emergent symme- tries. The symmetry can be enhanced on the lattice such as that the spin-1 chain has an explicit SU (3) symmetry in the Uimin-Lai-Sutherland model [17–19]. In addition,

higher symmetries can also emerge at the thermodynam- ical limit, e.g. the emergent SU (3) symmetry of critical spin-2 chain [20]. Therefore, the generalized SU (N ) spin models are necessary in understanding the symmetry en- hancement.

In this Letter, we focus on the fundamental constraints
on the phase diagrams of SU (N ) spin chains in gen-
eral representations with SU (N ) spin rotation symmetry
and lattice translation symmetry respected. More specif-
ically, we obtain the restriction of ground-state degener-
acy of gapped phases generalizing Lieb-Schultz-Mattis-
A✏eck (LSMA) theorem [21], and the classification of
critical phases, when the required symmetries are im-
posed. The compulsory degenerate ground states can
be understood by the concept of “ingappability” simi-
larly to the boundary states of nontrivial SPT phase,
which cannot be gapped with a specified symmetry un-
broken and bulk gap unclosed [1, 2]. On the SPT side,
the ingappability results from the quantum symmetry
anomaly of e↵ective boundary theory forbidding its lat-
tice realization [22]. CH: Should add more references
here. I will also do it later. Although the SU (N )
spin chain in our interest can be realized in its own di-
mension, the non-on-site nature of translation symme-
try enables us to understand its ingappability by consid-
ering the quantum anomaly of its e↵ective field theory
in the thermodynamical limit [23]. In a similar spirit,
the classification of critical phases can be done accord-
ing to their quantum anomaly. Such symmetry-protected
critical (SPC) classification has been proposed in SU (2)-
symmetric and translation-symmetric spin chains, where
the critical points are classified into two classes, one of
which can only take place in half-integer spin chains
while the other one can be realized only by integer spin
chains [6]. We generalize the SPC classification to SU (N )
symmetric critical phases of SU (N ) spin chains with-
out emergent symmetries. Moreover, we also obtain a
constraint on the candidates of emergent higher SU (N^{0})
symmetry with N^{0} > N by matching their symmetry
anomaly. As a special example, such restriction can ex-
plain that SU (3) symmetry enhancements have not (ac-
tually cannot) been found in SU (2) and translation sym-
metric half-integer spin chains.

Based on.... anomaly, we come to the conclusion that.... End of intro(?)

Translationally invariant SU (N ) spin system in 1 + 1
dimensions and the LSM index — We consider a generic
(1 + 1)d SU (N ) spin system described by a Hamilto-
nian H_{SU (N )} with the lattice translation symmetry and
a global spin-rotation symmetry specified by the projec-
tive special unitary group P SU (N ). Here the system is
subject to periodic boundary condition and the trans-
lation, for generality, defines the unit cell consisting of
multiple sites and forms a discrete group Z^{trans} in the
thermodynamic limit. A typical example of such a sys-
tem is the SU (N ) Heisenberg antiferromagnetic (HAF)

Figure from Hasan-Kane 10

### ØIngappability (stability) of edge states of 2d SPT phases has been well *studied, e.g. helical edge states of 2d QSHE*

### Symmetry protected critical phase classification of SU (N ) spin chains in rectangular Young tableaux representations, and global axial anomaly in 1+1 dimensions

We derive the low-energy properties of SU (N ) spin systems in the presence of spin rotation and translation symmetries. Based on mixed P SU (N )-Z anomaly, the spin system cannot be gapped with a unique ground state if the number of Young-tableaux boxes per unit cell is not divisible by N . The SU (N ) WZW model are classified into N symmetry-protected critical classes and each spin system can only realize one of them at criticality. It implies a no-go theorem that an RG flow is possible between two critical points only if they belong to the same symmetry-protected critical class, as long as the underlying lattice spin models respect the imposed symmetries.

### Introduction.—

### Hom( ˜ ⌦

^{spin}

_{3}

^{c}

### (B Z), U(1)) ⇠ = U (1) ⇠ = R/Z (1)

### LSM index I = (total charges per unit cell) mod 1.

### (2)

### H

^{3}

### (P SU (N ) ⇥ Z, U(1))/H

^{3}

### (P SU (N ), U (1)) ⇠ = Z

^{N}

### (3)

### LSM index I

^{N}

### = (# of YT boxes per unit cell) mod N.

### (4)

### I

^{2}

### = 2s mod 2. (5)

### I

^{N}

### 6= 0 mod N. (6)

### GSD 2 N

### gcd( I

^{N}

### , N ) N, (7)

### U (1) : ! e

^{i}

### Z : ! e

^{ik}

^{F}

^{z}

### = e

^{i⇡⌫}

^{z}

### (8)

### Z(A

_{U (1)}

### )

^{axial}

### ! e

^{2⇡i⌫}

^{⇥integer}

### Z(A

_{U (1)}

### ) (9)

### P SU (N ) : g ! wgw

^{1}

### , w 2 SU(N)

### Z

^{n}

### (trans) : g ! e

^{2⇡im/N}

### g, m 2 {0, 1, ..., N 1 } (10)

### n = N/ gcd(m, N ) (11)

### Z(A

_{P SU (N )}

### )

^{axial}

### ! e

^{2⇡i}

^{km}

^{N}

^{⇥integer}

### Z(A

_{P SU (N )}

### ) (12)

### I

^{N}

### · N

^{0}

### gcd(N

^{0}

### , k

^{0}

### m

^{0}

### ) = 0 mod N. (13)

### H

_{ULM}

### = X

i

### S

_{i}

### · S

^{i+1}

### + (S

_{i}

### · S

^{i+1}

### )

^{2}

### / X

i

### X

8 A=1### T

_{i}

^{A}

### T

_{i+1}

^{A}

### (14)

### N = 2 : = {2s} (15)

### N > 2 : = {4, 2, 1} (16)

### H

_{0}

### + ↵H

_{pert}

### (17)

### M =

_{R}

^{†}

_{L}

### + h.c (18)

### I

_{fw}

### = g

_{1}

_{R}

^{†}

_{R}

_{L}

^{†}

_{L}

### (19)

### I

_{Umkl}

### = g

_{2}

### e

^{i4k}

^{F}

^{x}

_{R}

^{†}

### (x)

_{R}

^{†}

### (x + a)

### ⇥

^{L}

### (x + a)

_{L}

### (x) + h.c (20) The classification of quantum phases is a central is- sue in condensed matter physics, where considerable recent progresses were initiated in strongly interacting many-body systems. In particular, various topologi- cal phases with unbroken symmetries, e.g. symmetry- protected trivial (SPT) ordered phases [1, 2] and topo- logical ordered phases enrich the phase diagram beyond conventional Landau-Ginzburg-Wilson spontaneously- symmetry-breaking paradigm. Therefore, the constraints in the appearance of symmetries on phase diagrams play

mass (magnetic impurity)

•

### Phases associated with “symmetry-protected in-gap(p)-ability”

8

Symmetry d

AZ ⇥ ⌅ ⇧ 1 2 3 4 5 6 7 8

A 0 0 0 0 Z 0 Z 0 Z 0 Z

AIII 0 0 1 Z 0 Z 0 Z 0 Z 0

AI 1 0 0 0 0 0 Z 0 Z^{2} Z^{2} Z

BDI 1 1 1 Z 0 0 0 Z 0 Z^{2} Z^{2}

D 0 1 0 Z^{2} Z 0 0 0 Z 0 Z^{2}

DIII 1 1 1 Z2 Z2 Z 0 0 0 Z 0

AII 1 0 0 0 Z^{2} Z^{2} Z 0 0 0 Z

CII 1 1 1 Z 0 Z^{2} Z^{2} Z 0 0 0

C 0 1 0 0 Z 0 Z^{2} Z^{2} Z 0 0

CI 1 1 1 0 0 Z 0 Z^{2} Z^{2} Z 0

TABLE I Periodic table of topological insulators and super-
conductors. The 10 symmetry classes are labeled using the
notation of Altland and Zirnbauer (1997) (AZ) and are spec-
ified by presence or absence of T symmetry ⇥, particle-hole
symmetry ⌅ and chiral symmetry ⇧ = ⌅⇥. ±1 and 0 denotes
the presence and absence of symmetry, with ±1 specifying
the value of ⇥^{2} and ⌅^{2}. As a function of symmetry and space
dimensionality, d, the topological classifications (Z, Z^{2} and 0)
show a regular pattern that repeats when d! d + 8.

3. Periodic table

Topological insulators and superconductors fit to- gether into a rich and elegant mathematical structure that generalizes the notions of topological band theory described above (Schnyder, et al., 2008; Kitaev, 2009;

Schnyder, et al., 2009; Ryu, et al., 2010). The classes
of equivalent Hamiltonians are determined by specifying
the symmetry class and the dimensionality. The symme-
try class depends on the presence or absence of T sym-
metry (8) with ⇥^{2} = ±1 and/or particle-hole symmetry
(15) with ⌅^{2} = ±1. There are 10 distinct classes, which
are closely related to the Altland and Zirnbauer (1997)
classification of random matrices. The topological clas-
sifications, given by Z, Z2 or 0, show a regular pattern
as a function of symmetry class and dimensionality and
can be arranged into the periodic table of topological in-
sulators and superconductors shown in Table I.

The quantum Hall state (Class A, no symmetry; d =
2), the Z^{2} topological insulators (Class AII, ⇥^{2} = 1;

d = 2, 3) and the Z2 and Z topological superconductors
(Class D, ⌅^{2} = 1; d = 1, 2) described above are each
entries in the periodic table. There are also other non
trivial entries describing di↵erent topological supercon-
ducting and superfluid phases. Each non trivial phase is
predicted, via the bulk-boundary correspondence to have
gapless boundary states. One notable example is super-
fluid ^{3}He B (Volovik, 2003; Roy, 2008; Schnyder, et al.,
2008; Nagato, Higashitani and Nagai, 2009; Qi, et al.,
2009; Volovik, 2009), in (Class DIII, ⇥^{2} = 1, ⌅^{2} = +1;

d = 3) which has aZ classification, along with gapless 2D Majorana fermion modes on its surface. A generalization of the quantum Hall state introduced by Zhang and Hu

E

E_{F}

Conduction Band

Valence Band Quantum spin

Hall insulator ν=1 Conventional Insulator ν=0

(a) (b)

k 0

−π/a −π/a

FIG. 5 Edge states in the quantum spin Hall insulator. (a) shows the interface between a QSHI and an ordinary insula- tor, and (b) shows the edge state dispersion in the graphene model, in which up and down spins propagate in opposite directions.

(2001) corresponds to the d = 4 entry in class A or AII.

There are also other entries in physical dimensions that have yet to be filled by realistic systems. The search is on to discover such phases.

III. QUANTUM SPIN HALL INSULATOR

The 2D topological insulator is known as a quantum spin Hall insulator. This state was originally theorized to exist in graphene (Kane and Mele, 2005a) and in 2D semiconductor systems with a uniform strain gradient (Bernevig and Zhang, 2006). It was subsequently pre- dicted to exist (Bernevig, Hughes and Zhang, 2006), and was then observed (K¨onig, et al., 2007), in HgCdTe quan- tum well structures. In section III.A we will introduce the physics of this state in the model graphene system and describe its novel edge states. Section III.B will re- view the experiments, which have also been the subject of the review article by K¨onig, et al. (2008).

A. Model system: graphene

In section II.B.2 we argued that the degeneracy at the
Dirac point in graphene is protected by inversion and
T symmetry. That argument ignored the spin of the
electrons. The spin orbit interaction allows a new mass
term in (3) that respects all of graphene’s symmetries. In
the simplest picture, the intrinsic spin orbit interaction
commutes with the electron spin S_{z}, so the Hamiltonian
decouples into two independent Hamiltonians for the up
and down spins. The resulting theory is simply two copies
the Haldane (1988) model with opposite signs of the Hall
conductivity for up and down spins. This does not violate
T symmetry because time reversal flips both the spin and

xy. In an applied electric field, the up and down spins
have Hall currents that flow in opposite directions. The
Hall conductivity is thus zero, but there is a quantized
spin Hall conductivity, defined by J_{x}^{"} J_{x}^{#} = _{xy}^{s} E_{y} with

xys = e/2⇡ – a quantum spin Hall e↵ect. Related ideas were mentioned in earlier work on the planar state of

2

H_{MG}^{s=1/2} = X

i

✓

S_{i} · Si+1 + 1

2S_{i} · Si+2 + 3
8

◆

(24)

H_{edge} =
Z

dx ^{†}(x)( iv_{F}@_{x}) _{z} (x) (25)

P si(x) = ( _{R}(x), _{L}(x))^{T} (26)
The classification of quantum phases is a central is-
sue in condensed matter physics, where considerable
recent progresses were initiated in strongly interacting
many-body systems. In particular, various topologi-
cal phases with unbroken symmetries, e.g. symmetry-
protected trivial (SPT) ordered phases [1, 2] and topo-
logical ordered phases enrich the phase diagram beyond
conventional Landau-Ginzburg-Wilson spontaneously-
symmetry-breaking paradigm. Therefore, the constraints
in the appearance of symmetries on phase diagrams play
essential roles since they rule out the existence of a large
class of forbidden gapped or gapless phases when the cer-
tain symmetry is respected. For gapped phases, the re-
striction on the ground-state degeneracy allows a unified
understanding on the low-energy spectral properties. For
example, Lieb-Schultz-Mattis (LSM) theorem [3] and its
generalization LSMOH theorem by Oshikawa and Hast-
ings [4, 5] states that the ground states cannot be trivially
gapped with a unique ground state if the particle number
per unit cell is fractional and both the translation sym-
metry and charge U (1) conservation are preserved. As
a part of gapless phases, critical phase classifications is
a key ingredient in the understanding of universal criti-
cal behaviors of various phase transitions. However, the
classification of critical phases still remains open and the
related proposal is only given in SU (2) spin chains [6].

Furthermore, many stable critical phases have been found experimentally and numerically [7, 8], and the reason of such stability is not completely understood.

In reality, the constraints on phase diagrams with more general symmetries than U (1) and SU (2) are within intense interest. For example, SU (N ) “spin” systems with N > 2 are realized in ultracold atoms on op- tical lattices [9–16] and also in some spin-orbital sys- tems [? ], although SU (N ) systems was initially intro- duced as theoretical toy models to understand “physical”

SU (2) spins. Thus the study related with the phase di- agrams of SU (N ) spin systems is of realistic interest in its own. Furthermore, although spin-rotation and trans- lation symmetries are imposed, symmetry enhancements cannot be excluded, e.g. phases with emergent symme- tries. The symmetry can be enhanced on the lattice such as that the spin-1 chain has an explicit SU (3) symmetry in the Uimin-Lai-Sutherland model [17–19]. In addition,

higher symmetries can also emerge at the thermodynam- ical limit, e.g. the emergent SU (3) symmetry of critical spin-2 chain [20]. Therefore, the generalized SU (N ) spin models are necessary in understanding the symmetry en- hancement.

In this Letter, we focus on the fundamental constraints
on the phase diagrams of SU (N ) spin chains in gen-
eral representations with SU (N ) spin rotation symmetry
and lattice translation symmetry respected. More specif-
ically, we obtain the restriction of ground-state degener-
acy of gapped phases generalizing Lieb-Schultz-Mattis-
A✏eck (LSMA) theorem [21], and the classification of
critical phases, when the required symmetries are im-
posed. The compulsory degenerate ground states can
be understood by the concept of “ingappability” simi-
larly to the boundary states of nontrivial SPT phase,
which cannot be gapped with a specified symmetry un-
broken and bulk gap unclosed [1, 2]. On the SPT side,
the ingappability results from the quantum symmetry
anomaly of e↵ective boundary theory forbidding its lat-
tice realization [22]. CH: Should add more references
here. I will also do it later. Although the SU (N )
spin chain in our interest can be realized in its own di-
mension, the non-on-site nature of translation symme-
try enables us to understand its ingappability by consid-
ering the quantum anomaly of its e↵ective field theory
in the thermodynamical limit [23]. In a similar spirit,
the classification of critical phases can be done accord-
ing to their quantum anomaly. Such symmetry-protected
critical (SPC) classification has been proposed in SU (2)-
symmetric and translation-symmetric spin chains, where
the critical points are classified into two classes, one of
which can only take place in half-integer spin chains
while the other one can be realized only by integer spin
chains [6]. We generalize the SPC classification to SU (N )
symmetric critical phases of SU (N ) spin chains with-
out emergent symmetries. Moreover, we also obtain a
constraint on the candidates of emergent higher SU (N^{0})
symmetry with N^{0} > N by matching their symmetry
anomaly. As a special example, such restriction can ex-
plain that SU (3) symmetry enhancements have not (ac-
tually cannot) been found in SU (2) and translation sym-
metric half-integer spin chains.

Based on.... anomaly, we come to the conclusion that.... End of intro(?)

Translationally invariant SU (N ) spin system in 1 + 1
dimensions and the LSM index — We consider a generic
(1 + 1)d SU (N ) spin system described by a Hamilto-
nian H_{SU (N )} with the lattice translation symmetry and
a global spin-rotation symmetry specified by the projec-
tive special unitary group P SU (N ). Here the system is
subject to periodic boundary condition and the trans-
lation, for generality, defines the unit cell consisting of
multiple sites and forms a discrete group Z^{trans} in the
thermodynamic limit. A typical example of such a sys-
tem is the SU (N ) Heisenberg antiferromagnetic (HAF)

2

H_{MG}^{s=1/2} = X

i

✓

S_{i} · S^{i+1} + 1

2S_{i} · S^{i+2} + 3
8

◆

(24)

H_{edge} =
Z

dx ^{†}(x)( iv_{F}@_{x}) _{z} (x) (25)

(x) = ( _{R}(x), _{L}(x))^{T} (26)
The classification of quantum phases is a central is-
sue in condensed matter physics, where considerable
recent progresses were initiated in strongly interacting
many-body systems. In particular, various topologi-
cal phases with unbroken symmetries, e.g. symmetry-
protected trivial (SPT) ordered phases [1, 2] and topo-
logical ordered phases enrich the phase diagram beyond
conventional Landau-Ginzburg-Wilson spontaneously-
symmetry-breaking paradigm. Therefore, the constraints
in the appearance of symmetries on phase diagrams play
essential roles since they rule out the existence of a large
class of forbidden gapped or gapless phases when the cer-
tain symmetry is respected. For gapped phases, the re-
striction on the ground-state degeneracy allows a unified
understanding on the low-energy spectral properties. For
example, Lieb-Schultz-Mattis (LSM) theorem [3] and its
generalization LSMOH theorem by Oshikawa and Hast-
ings [4, 5] states that the ground states cannot be trivially
gapped with a unique ground state if the particle number
per unit cell is fractional and both the translation sym-
metry and charge U (1) conservation are preserved. As
a part of gapless phases, critical phase classifications is
a key ingredient in the understanding of universal criti-
cal behaviors of various phase transitions. However, the
classification of critical phases still remains open and the
related proposal is only given in SU (2) spin chains [6].

^{0})
symmetry with N^{0} > N by matching their symmetry
anomaly. As a special example, such restriction can ex-
plain that SU (3) symmetry enhancements have not (ac-
tually cannot) been found in SU (2) and translation sym-
metric half-integer spin chains.

Based on.... anomaly, we come to the conclusion that.... End of intro(?)

_{SU (N )} with the lattice translation symmetry and
a global spin-rotation symmetry specified by the projec-
tive special unitary group P SU (N ). Here the system is
subject to periodic boundary condition and the trans-
lation, for generality, defines the unit cell consisting of
multiple sites and forms a discrete group Z^{trans} in the
thermodynamic limit. A typical example of such a sys-
tem is the SU (N ) Heisenberg antiferromagnetic (HAF)

Figure from Hasan-Kane 10

### ØIngappability (stability) of edge states of 2d SPT phases has been well *studied, e.g. helical edge states of 2d QSHE*

### Symmetry protected critical phase classification of SU (N ) spin chains in rectangular Young tableaux representations, and global axial anomaly in 1+1 dimensions

We derive the low-energy properties of SU (N ) spin systems in the presence of spin rotation and translation symmetries. Based on mixed P SU (N )-Z anomaly, the spin system cannot be gapped with a unique ground state if the number of Young-tableaux boxes per unit cell is not divisible by N . The SU (N ) WZW model are classified into N symmetry-protected critical classes and each spin system can only realize one of them at criticality. It implies a no-go theorem that an RG flow is possible between two critical points only if they belong to the same symmetry-protected critical class, as long as the underlying lattice spin models respect the imposed symmetries.

### Introduction.—

### Hom( ˜ ⌦

^{spin}

_{3}

^{c}

### (B Z), U(1)) ⇠ = U (1) ⇠ = R/Z (1)

### LSM index I = (total charges per unit cell) mod 1.

### (2)

### H

^{3}

### (P SU (N ) ⇥ Z, U(1))/H

^{3}

### (P SU (N ), U (1)) ⇠ = Z

^{N}

### (3)

### LSM index I

^{N}

### = (# of YT boxes per unit cell) mod N.

### (4)

### I

^{2}

### = 2s mod 2. (5)

### I

^{N}

### 6= 0 mod N. (6)

### GSD 2 N

### gcd( I

^{N}

### , N ) N, (7)

### U (1) : ! e

^{i}

### Z : ! e

^{ik}

^{F}

^{z}

### = e

^{i⇡⌫}

^{z}

### (8)

### Z(A

_{U (1)}

### )

^{axial}

### ! e

^{2⇡i⌫}

^{⇥integer}

### Z(A

_{U (1)}

### ) (9)

### P SU (N ) : g ! wgw

^{1}

### , w 2 SU(N)

### Z

^{n}

### (trans) : g ! e

^{2⇡im/N}

### g, m 2 {0, 1, ..., N 1 } (10)

### n = N/ gcd(m, N ) (11)

### Z(A

_{P SU (N )}

### )

^{axial}

### ! e

^{2⇡i}

^{km}

^{N}

^{⇥integer}

### Z(A

_{P SU (N )}

### ) (12)

### I

^{N}

### · N

^{0}

### gcd(N

^{0}

### , k

^{0}

### m

^{0}

### ) = 0 mod N. (13)

### H

_{ULM}

### = X

i

### S

_{i}

### · S

^{i+1}

### + (S

_{i}

### · S

^{i+1}

### )

^{2}

### / X

i

### X

8 A=1### T

_{i}

^{A}

### T

_{i+1}

^{A}

### (14)

### N = 2 : = {2s} (15)

### N > 2 : = {4, 2, 1} (16)

### H

_{0}

### + ↵H

_{pert}

### (17)

### M =

_{R}

^{†}

_{L}

### + h.c (18)

### I

_{fw}

### = g

_{1}

_{R}

^{†}

_{R}

_{L}

^{†}

_{L}

### (19)

### I

_{Umkl}

### = g

_{2}

### e

^{i4k}

^{F}

^{x}

_{R}

^{†}

### (x)

_{R}

^{†}

### (x + a)

### ⇥

^{L}

### (x + a)

_{L}

### (x) + h.c (20) The classification of quantum phases is a central is- sue in condensed matter physics, where considerable recent progresses were initiated in strongly interacting many-body systems. In particular, various topologi- cal phases with unbroken symmetries, e.g. symmetry- protected trivial (SPT) ordered phases [1, 2] and topo- logical ordered phases enrich the phase diagram beyond conventional Landau-Ginzburg-Wilson spontaneously- symmetry-breaking paradigm. Therefore, the constraints in the appearance of symmetries on phase diagrams play

breaks TR symm

mass (magnetic impurity)