• 沒有找到結果。

4 (only at zero temperature), while the latter happens in

a system with degenerate gapped ground states with the periodicity larger than the one of the physical Hamilto-nian, e.g., dimerized phase of SU (2) spin chains.

(ii) Symmetry-respecting critical phases. If the whole symmetry is preserved, the system must be in a gapless phase. In simpler situations, a system in such a phase – at a critical point in particular – is described by a universality class, e.g. a WZW model. In this case, how-ever, the corresponding low-energy symmetry (that is, P SU (N ) ⇥ Z) would be anomalous.

The value of a nontrivial LSM index can further, quan-titatively, give a lower bound of the ground-state degen-eracy due to SSB of translation symmetry in case (i), and constrain, by anomaly-matching argument, the pos-sible WZW universality classes in case (ii). These lead to a natural generalization of the original LSMA theo-rem and a classification of symmetry-protected critical (SPC) phases for SU (N ) spin chains. In the following sections, we will discuss these constraints respectively in more detail.

Ground-state degeneracy associated with SSB of trans-lation symmetry — A physical implication of the LSM index is the ingappability when the required symmetry is respected by perturbations [? ]. In the following discus-sions, we take P SU (N ) ⇥Z

trans

as the protecting symme-try, where Z

trans

defines the unit cell of the system. Then, by considering a family of the LSM indices p I

N

associ-ated with lower translation symmetries p Z

trans

✓ Z

trans

for p 2 N of an SU(N) spin model, one can obtain a restriction on the ground-state degeneracy (G.S. deg) of gapped phases of this model [supple]. In general, the G.S.

deg must be a multiple of N/( I

N

, N ), namely N

deg

2 N

( I

N

, N ) N, (17) with I

N

defined in Eq. (??). In the first two rows of Ta-ble ??, we list the exactly solvaTa-ble SU (3) trimer and 8-VBS models – analogs of SU (2) dimer and ALKT models – that breaks and restorest a single-site translation sym-metry, respectively. Their G.S. degs in the fourth column are consistent with the G.S.-deg constraints by Eq. (??).

Constraints on the low-energy critical theories — The other possibility of an SU (N ) model with a nontrivial LSM index I

N

is the emergence of an infrared critical the-ory that preserves all the microscopic symmetries. How-ever, the corresponding low-energy e↵ective symmetries would enjoy an ’t Hooft anomaly associated to the value of I

N

. While the critical theory is described by a WZW CFT, such an anomaly is in relation to, besides the rep-resentation of P SU (N ) ⇥ Z, the affine (Lie) algebra of this CFT.

YY: does chiral algebra mean chiral anomaly?

We first focus on SU (N )

k

WZW CFTs – as they are the most concerned critical theories of SU (N ) spin

mod-els. The actions is kI(g) = k

8⇡

Z

M2

dtdxTr @

µ

g

1

@

µ

g + k

WZ

,

WZ

= 1

12⇡

Z

B:@B=M2

dtdxdyTr(dgg

1

)

3

, (18) where g is an SU (N ) matrix-valued field and “Tr” is the conventional matrix trace. The Wess-Zumino term

WZ

is defined as an extended integral onto an auxiliary manifold B whose boundary is M

2

, and a consistent CFT is independent on such extension. The lattice translation symmetry becomes a discrete “axial” symmetry in the continuum limit [? ]

g ! e

2⇡im/N

g, m 2 {0, 1, ..., N 1 }, (19) generating a Z

n

group, with n = N/(m, N ), which is a subgroup of the center of SU (N ). (Note that any e↵ec-tive symmetry at low-energy associated to the transla-tion symmetry must be a subgroup of Z.) On the other hand, the on-site spin-rotation symmetry corresponds to the vector P SU (N ) symmetry, a diagonal subgroup of a larger SU (N )

L

⇥SU(N)

R

symmetry, in the WZW model.

As P SU (N ) ⇥ Z

n

is exact as a global symmetry of the (quantum) WZW theory, there might be ’t Hooft anoma-lies of it, including the mixed one between P SU (N ) and Z

n

and the one for Z

n

itself [? ]. However, only the mixed anomaly is relevant to the physics characterized by the LSM index, while the the other one is an “emergent”

anomaly

which only characterizes ingappability against infinitesimal perturbations around criticality

[? ? ? ] that is irrelevant to the situation we consider here.

One way to visualize the mixed anomaly in the WZW model is to make use of the equivalence between SU (N )

k

WZW model to U (kN )

1

/U (k) constrained Dirac fermion (CDF) theory [? ? ] and then compute the Z

n

axial anomaly in the CDF theory coupled to a background P SU (N ) gauge field. More explicitly, we show that [supple] there is a phase ambiguity of the WZW/CDF partition function with a background P SU (N ) field under the Z

n

transformation (??), which takes the form exp ⇣

2⇡ikm N

R

M2

w(P ) ⌘

, where P is the underlying P SU (N ) bundle (over M

2

) and w(P ) 2 H

2

(M

2

, Z

N

) ⇠ = Z

N

. From this fact, we deduce that the mixed P SU (N ) Z

n

anomaly for any SU (N )

k

WZW theory with m de-fined by Eq. (??) is characterized by a mod-N integer

km mod N. (20)

Note that the anomaly computed here is actually the mixed P SU (N ) Z anomaly discussed before, since the value of the mixed anomaly is unchanged when Z

n

is extended to Z (by nZ); see similar discussions in [? ? ].

Then, by the definition of the LSM index (??) and also the way we represent it, we conclude that an SU (N )

symm: vector axial

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( ˜⌦spin3 c(BZ), U(1)) ⇠= U (1) ⇠= R/Z (1)

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

(2)

H3(P SU (N ) ⇥ Z, U(1))/H3(P SU (N ), U (1)) ⇠= ZN (3)

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

(4)

I2 = 2s mod 2. (5)

IN 6= 0 mod N. (6)

GSD 2 N

gcd(IN, N )N, (7)

U (1) : ! ei

Z : ! eikF z = ei⇡⌫ z (8)

Z(A) axial! e2⇡i⌫⇥integerZ(A) (9)

P SU (N ) : g ! wgw 1, w 2 SU(N)

Zn(trans) : g ! e2⇡im/Ng, m 2 {0, 1, ..., N 1} (10)

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

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 ) syssys-tems 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 critthermodynam-ical spin-2 chain [20]. Therefore, the generalized SU (N ) spin 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( ˜spin3 c(BZ), U(1)) ⇠= U (1) ⇠= R/Z (1)

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

(2)

H3(P SU (N )⇥ Z, U(1))/H3(P SU (N ), U (1)) ⇠=ZN (3)

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

(4)

I2 = 2s mod 2. (5)

IN 6= 0 mod N. (6)

GSD 2 N

gcd(IN, N )N, (7)

U (1) : ! ei

Z : ! eikF z = ei⇡⌫ z (8)

Z(A) axial! e2⇡i⌫⇥integerZ(A) (9)

P SU (N ) : g ! wgw 1, w 2 SU(N)

Zn(trans) : g ! e2⇡im/Ng, m 2 {0, 1, ..., N 1} (10)

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

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 ) syssys-tems 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 critthermodynam-ical spin-2 chain [20]. Therefore, the generalized SU (N ) spin

level

characterized by #$/&, or #$ mod &

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( ˜ ⌦

spin3 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

ikF 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⇡ikmN ⇥integer

Z(A

P SU (N )

) (12) 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 ) syssys-tems 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

Our prediction agrees with known examples in previous studies of SU(N) models.

Y. Yao, C.-T. Hsieh, and M. Oshikawa, arXiv:1805.06885

Greiter et al. 07

Takhtajan; Babujian82

Andrei-Johannesson84 Johanness 86

Rachel et al. 09 Dufour et al. 15 Lecheminant 15

3

Model YT IN GSD IR CFT; m Mixed anomaly

SU(3) trimer model [43] 1 mod 3 32 3N -

-SU(3) 10-VBS model [43] 0 mod 3 12 1N -

-SU(6) 70-VBS model [44] 3 mod 6 22 2N -

-S-3/2 TB model[45, 46] 1 mod 2 - SU (2)3 WZW; 1 1 mod 2 H [3,2] AJ model[47, 48] 2 mod 3 - SU (3)2 WZW; 1 2 mod 3 SU(3) 1⇥2-YT HAF[49, 50] 2 mod 3 - SU (3)1 WZW; 2 2 mod 3 SU(9) 2⇥1-YT HAF[51] 2 mod 9 - SU (9)1 WZW; 2 2 mod 9 SU(3) 2-leg ladder [52] 2 mod 3 - SU (3)1 WZW; 2 2 mod 3

TABLE I. Examples of gapped and critical SU (N ) spin systems. For the first two gapped exactly solvable models, the actual GSDs are consistent with our constraint. For the following critical models, the numerically proposed IR CFTs in the fifth column obey SPC classification specified by Eq. (10). VBS: Valence-bond-solid; TB: Takhtajan-Babujian; AJ: Andrei-Johannesson.

the sum of the associated indices of these spin chains, which equals the total number of YT boxes P

ib i per unit cell in the original model, namely, Eq. (5). In Ta-ble I (the third column), we list the LSM indices for sev-eral SU (N ) models with given YT reps. per unit cell.

A system with a nonzero LSM index (IN 6= 0 mod N) must exhibit nontrivial low-energy behaviors in connec-tion with the ingappability menconnec-tioned earlier, as we will elaborate in the following.

Ground-state degeneracy associated with a spontaneous broken translation symmetry — First let us assume that the system has a non-zero gap above the ground state(s).

In this case, a non-zero LSM index implies GSD, as in the case of the existing LSM-type theorems. Here we derive the degeneracy based on the mixed anomaly. By considering a family of the LSM indices pIN associated with lower translation symmetries pZtrans ✓ Ztrans for p 2 N of an SU(N) spin model, we obtain a restriction on the GSD of any gapped phase of this model [54]

GSD 2 N

gcd(IN, N )N, (6) and the translation symmetry is spontaneously broken to at least N/gcd(IN, N ) of unit cells, realizable by exactly solvable models [44]. This indeed corresponds to the LSM theorem for the SU (N ) spin chain [2], with an explicit statement on GSD. In the first two rows of Table I, we list the exactly solvable SU (3) trimer and 8-VBS models – analogs of SU (2) dimer and AKLT models – whose ground states break and respect a single-site translation symmetry, respectively. Their GSDs (shown in the fourth column) are consistent with the constraint (6).

Constraint on critical WZW SU (N )k universality classes — Next we consider the other possibility, namely when the system is gapless. While the usual LSM-type theorems do not give any further restriction in this case,

the anomaly-based approach leads to constraints on the possible universality class of the gapless critical phase.

The most natural universality classes of a critical SU (N ) spin model is the SU (N ) WZW theories, with the global SU (N ) and conformal symmetries. Their action is given as

kI(g) = k 8⇡

Z

M2

dtdxTr @µg 1@µg + k WZ,

WZ = 1 12⇡

Z

B:@B=M2

dtdxdyTr(dgg 1)3, (7) where k is an integer called level, g is an SU (N ) matrix-valued field, and Tr is the conventional matrix trace. The level k characterizes the SU (N ) WZW theory, and we denote the SU (N ) WZW theory of level k as SU (N )k WZW. The Wess-Zumino term WZ is defined as an extended integral onto an auxiliary manifold B whose boundary is M2, and a consistent CFT is independent on such extension. The lattice translation symmetry in the infrared becomes a discrete “axial” symmetry [55]

g ! e2⇡im/Ng, (8)

which forms a Zn group with n = N/gcd(m, N ) and the integer m defined modulo N . (Note that any low-energy e↵ective symmetry associated to the translation symme-try must be a subgroup of Z.) On the other hand, the on-site spin-rotation symmetry corresponds to the vec-tor P SU (N ) symmetry, a diagonal subgroup of a larger [SU (N )L⇥SU(N)R]/ZN symmetry, in the WZW theory.

As P SU (N )⇥ Zn is exact as a global symmetry of the (quantum) WZW theory, there might be ’t Hooft anoma-lies of it, including the mixed one between P SU (N ) and Zn and the one forZn itself [56]. However, only the mixed anomaly is relevant to the physics characterized by the LSM index, while the the other one is an “emergent”

Greiter-Rachel 07

In summary, if a spin model with an exact SU(N) spin-rotation and transl symm has a nontrivial LSM index, i.e., the total

umber of Young-tableau boxes per unit cell is not divisible by N, the system must have either

degenerate gapped ground states, with the multiplicity (1), or

gapless excitations / symm-protected critical states (SPC). If the

low-energy SPC is given by an SU(N) WZW theory, its level is

constrained by (2).

Outline

Introduction

Example 1: 1d charged fermion systems

Example 2: 1d SU(N) spin systems

Conclusion

Conclusion

We apply the idea of (’t Hooft) anomaly matching to study 1d condensed matter systems – many-body systems in general – in the presence of both lattice transl and some on-site symm.

H

lattice

G

site

trans

UV

H

IREFT

G

vector

axial

There is a potential disc chiral anomaly at IR

It can be traced back to the non-on-site nature of (part of) the lattice symm

By “matching” the IR anomaly we identify a top. index, the LSM index, for any lattice

system to characterize its phase

Such an anomaly can diagnose the ingappability of the system!

[Hsieh et al. 14]

相關文件