Weak Order and Disorder of Quantum States Invariant Under Onsite Action by an Arbitrary Finite Group on a Spin Chain

Maxim Zelenko

Dean’s Scholars Honors Thesis In partial fulﬁllment of the requirements for graduation with the Bachelor of Science in Mathematics, Honors

Department of Mathematics The University of Texas at Austin December 14, 2019 Contents

1 Introduction 1

2 Setup of the Problem 2

3 Mathematical Preliminaries 5

4 Generalized Deﬁnitions of Weak Order and Disorder 8 4.1 Weak Order ...... 8 4.2 Weak Disorder ...... 10

5 Main Result and Proof 11

6 Discussion of Future Work 15

7 Acknowledgments 16

References 17 Abstract In Ref. [4], Michael Levin derived general constraints on order and disorder parameters in Ising symmetric quantum spin chains. Levin’s main result in his paper was a theorem showing that in a circular spin chain, any local Hamiltonian that has a non-degenerate ground state and is translationally invariant and Ising symmetric must at least have either a nonzero order parameter or a nonzero disorder parameter. In the process of proving the theorem, he also proved a lemma that made a general statement about correlation and symmetry defect properties of any state in a circular spin chain. These properties namely had to do with notions of order and disorder that are weaker than long-range order and disorder. In this thesis, I prove an extension of the lemma to any simultaneous eigenstate of an onsite representation of an arbitrary finite group. Based on this generalization, we discuss some possible implications regarding how Levin’s theorem could be generalized to arbitrary finite symmetries as well. It is important to note that my generalization only applies to representations with one- dimensional invariant subspaces. While this condition is satisfied by any representation of an abelian group, it only holds for a subset of non-abelian group representations.

1 Introduction One of the objectives of condensed matter physics is to study emergent phenomena in quantum many-body systems. Often, systems differing in microscopic detail, such as different interaction potentials and chemical makeup, display some common collective behavior. For this reason, condensed matter physicists are interested in the unifying principles of these common emergent behaviors rather than in peculiar details of very specific systems [1]. Some examples of condensed-matter systems with interesting emergent phenomena include superconductors, superfluids, ferromagnets, anti-ferromagnets, and graphene. This thesis explores some theoretical properties of quantum spin chains. Though spin chains are toy models, they are inspired by real physical examples of 1D condensed matter systems, such as thin wires, optical lattices, carbon nanotubes, and one-dimensional arrays of quantum dots, vortices, and other confined quantum systems. Another important physical origin of spin chains is the ion lattice of some typical electronic material like a metal or an insulating solid [1]. In this thesis, I prove an extension of Michael Levin’s theoretical results in Ref. [4]. While Levin derived general constraints on order and disorder parameters in Ising (i.e. onsite Z2) symmetric spin chains, we will discuss how some of these constraints could be extended to spin chains with arbitrary finite, onsite symmetries. In section 2, we provide a brief back- ground in quantum mechanics, rigorously define the spin chain Hilbert space, and indicate the properties of the Hamiltonians we consider. In section 3, we provide some mathematical preliminaries necessary for understanding the main definitions and results. In section 4, we generalize Ref. [4]’s definitions of weak order and weak disorder to any finite symmetry group. In section 5, we exploit these definitions to formulate and prove the main result of this thesis, which is a generalization of Lemma 1 from Ref. [4]. This lemma was originally a statement that if one takes an Ising symmetric state and some arbitrary pair of disjoint intervals, the state must either be weakly-ordered on those intervals or weakly-disordered on

1 the complementary intervals with some nonzero strength. Here, we generalize this lemma by proving an analogous claim for any simultaneous eigenstate of an onsite representation of a ﬁnite group. Along with two other lemmas, Levin used Lemma 1 to prove Theorems 1 and 2, the main results of his paper [4]. Theorem 1 claimed that in a circular spin chain, any local Hamiltonian that has a non-degenerate ground state and is translationally invariant and Ising symmetric must at least have either a nonzero order parameter or a nonzero disorder parameter. Theorem 2 is a weaker statement but for Hamiltonians that are not required to be translationally invariant. In section 6, we suggest how Theorems 1 and 2 from Ref. [4] could be generalized based on our generalization of Lemma 1.

2 Setup of the Problem In quantum mechanics, any problem is characterized by defining a complete inner product space, or Hilbert space, over the field of complex numbers. In a Hilbert space, every one- dimensional subspace corresponds to a so-called pure state, or wavefunction. Each pure state is represented by a vector of unit norm lying in its corresponding complex line. As we shall see soon, this ensures that the inner product has a direct relationship with the probability that a measurement of an observable in the system will yield a particular result. In our quantum problem, the Hilbert space is a finite quantum spin chain consisting of L spin sites forming a circular lattice. As shown in Figure 1, we label the L sites by 1, 2, 3,..., L. For every i ∈ {1, 2, ..., L}, the spin at site i forms a d-dimensional inner-product L space Hi, where d is a finite nonzero integer. As a whole, the quantum spin chain is the d dimensional Hilbert space L O H := Hi. i=1 As usual in physics, we denote the inner product associated with H by Dirac’s bra-ket h·|·i.

Figure 1: A spin chain made up of L spins arranged in a ring lattice. Figure taken from Ref. [4].

2 Before we move on, let us brieﬂy discuss two of the postulates of quantum mechanics (this is not the full list of postulates!):

1. Associated with any collection of particles forming a closed quantum system, there is a wavefunction that contains all of the physical information that can be known about the system.

2. For every physical observable q (e.g. energy and angular momentum), there is an associated linear Hermitian operator Q acting1 on the Hilbert space.

In the first postulate, the so-called “physical information” about the system’s state is what is acquired by performing measurements of physical observables. The second postulate ex- plains the outcome of the measurement. By the spectral theorem, a Hermitian operator is always orthogonally diagonalizable in a finite-dimensional Hilbert space like our quantum spin chain above2. Together with the second postulate, this implies that one can decompose any wavefunction into an orthonormal eigenbasis of an operator Q associated with an observable q. The spectral theorem also tells us that all of the eigenvalues of a Hermitian operator are real. This is important, because the spectrum of operator Q is the set of all possible measurements that can be made for the observable q. Thus, every measurement yields a real number. In addition, by decomposing a wavefunction onto an eigenbasis of Q, we can assign to it a probability distribution defined over the spectrum of Q, where the probability of measuring each eigenvalue is equal to the modulus squared of the projection of the state onto the eigenvalue’s eigenspace. (Notice that if we do not represent the wavefunction by a vector of unit norm, the sum of probabilities taken over the whole spectrum would not add up to one, because the sum of the squared moduli of projections equals to the norm of the vector. This is the reason for choosing a unit vector to represent the state). Such an ability of a quantum mechanical state to assign a probability distribution to any observable is precisely what the first postulate means, since these probability distributions represent all of the “physical information” that one can acquire about a quantum system. Hermitian operators corresponding to energy are called Hamiltonians, and the spectrum of a Hamiltonian is called the energy spectrum. Thus far, we have indicated that the Hilbert space we consider is the quantum spin chain H, but to fully characterize the physical problem, we also need to specify our Hamiltonians of interest. Before doing so, however, we need the following definition:

Deﬁnition 1. Let X ⊂ {1, 2, ..., L} be a subset of lattice sites on the spin chain. A linear operator A acting on the spin-chain H is said to be supported on X if and only if one can decompose O A = AX ⊗ 1j, (1) j∈ /X

1When we say that a linear operator acts on a vector space V , we are implying that it is an element of End(V ) 2For more general Hilbert spaces, we can guarantee that a linear Hermitian operator is orthogonally diagonalizable if it is compact as well.

3 N 1 where AX is an operator acting on i∈X Hi and j is the identity operator acting on Vj. The intersection of all subsets X for which decomposition (1) holds is called the support of A, which we denote by supp(A).

We are interested in Hamiltonians on H that describe nearest neighbor interactions with bounded operator supnorms. In other words, we consider Hamiltonians of form

L X H = Hi,i+1, (2) i=1 where Hi,i+1 are linear operators acting on H satisfying

supp(Hi,i+1) ⊂ {i, i + 1} and kHi,i+1k ≤ 1.

The operators Hi,i+1 correspond to interactions between neighboring spins. The bound on their operator supnorm represents a bound on the strength of the spin interactions. Another constraint we impose on H is that it is invariant under the action of a group G of ﬁnite order n. That is, there exists a faithful unitary operator representation R of G on H such that H commutes with every element of the image R(G). For notational convenience, we will treat the image R(G) as the group G itself, so for any g ∈ G, R(g) will be denoted by g whenever there is no danger of ambiguity. Just as in Ref [4], we further assume that the G-invariance of H is onsite, which means that we can decompose L O R = ri, i=1 where every ri is a faithful unitary operator representation of G on Hi. We also deﬁne O gi := ri(g) ⊗ 1j. j6=i

This allows us to express an operator R(g) (which, as mentioned above, we denote by g) in the form

g = g1 ◦ g2 ◦ ... ◦ gL (3) For notational convenience, if we take a composition of operators that commute pairwise, we will use Q notation. With this notation, we write Eq. (3) as

L Y g = gi i=1 Lastly, we require R to have a one-dimensional invariant subspace. If G is abelian, this is guaranteed to hold. Nonetheless, this becomes a strict condition when G is non-abelian.

4 3 Mathematical Preliminaries

For each subset X ⊂ {1, 2, ..., L}, let RX be the representation of G on H with the action by R restricted to X, i.e. O O RX := ri ⊗ 1j, i∈X j∈ /X where 1j denotes the trivial representation of G on Vj. Furthermore, let GX stand for the image RX (G), and ∀g ∈ G, let gX denote the element of GX corresponding to g, i.e. Y gX := gi. i∈X

∗ We are also interested in the representation RX ⊗ RX acting on the space of linear operators H ⊗ H∗. What this representation does is conjugate operators acting on H by

elements of GX . Let D be the set of all isomorphism classes of irreducible representations of G. By Maschke’s theorem, ∃ α : D → N (where N includes zero) s.t.

α(ρ) ∗ M M (j) H ⊗ H = Vρ , (4) ρ∈Dα j=1

−1 (j) where Dα = D − α (0) and for every ρ ∈ Dα, {Vρ : j ∈ {1, 2, ..., α(ρ)}} is a chosen collection of mutually orthogonal G-invariant subspaces of H ⊗ H∗ s.t. ∀j, the restriction of ∗ (j) Lα(ρ) (j) RX ⊗ RX to Vρ equals to an element of the class ρ. The subspace j=1 Vρ is called the ∗ isotypical component of RX ⊗ RX corresponding to ρ. We are about to show how to construct an orthogonal projection onto an isotypical component. Proposition 2. Let P be a representation of G on some ﬁnite-dimensional inner product

space W, ρ be an isomorphism class of G, and πρ(P ) denote the projection operator onto the isotypical component of P corresponding to ρ. Then, dim ρ X π (P ) = χ (g)P (g). (5) ρ n ρ g∈G Proof. Suppose we had two irreducible representations ρ and ρ0 of G on vector spaces V and V 0, respectively. Consider the linear map 1 X φ (ρ0) := χ (g)ρ0 ∈ End(V 0), (6) ρ n ρ g g∈G

0 0 where n is the order of G, ρg denotes ρ (g), and χρ is the character of ρ. We want to show 0 0−1 0 0 0 that φρ(ρ ) is G-invariant, i.e. that ∀h ∈ G, ρ h φρ(ρ )ρ h = φρ(ρ ). First, we make the following three observations: 1. Let g00 = h−1gh. Since conjugation by h is an automorphism of G, we have that as g runs over the whole group G, so does g00.

5 0−1 0 0 0 0 2. ρ h ρgρh = ρ g00 , because ρ is a homomorphism.

00 3. Characters must be constant over conjugacy classes, so χρ(g) = χρ(g ). Using these facts, we derive

0−1 0 0 1 X 0−1 0 0 1 X 00 0 1 X 0 0 ρ φ (ρ )ρ = χ (g)ρ ρ ρ = χ (g )ρ 00 = χ (g)ρ = φ (ρ ), h ρ h n ρ h g h n ρ g n ρ g ρ g∈G g∈G g∈G

0 0 so indeed, φρ(ρ ) is G-invariant. Since ρ is an irreducible representation, Schur’s Lemma implies that φ = λ1 for some scalar λ, so

0 ! Tr(φρ(ρ )) 1 1 X λ = = χ (g)χ 0 (g) . dim V 0 dim V 0 n ρ ρ g∈G Notice that what is inside the parentheses on the right-hand side is the inner product of characters standardly used in representation theory. For any two characters χ and χ0 of G, their standard inner product is 1 X hχ, χ0i = χ(g)χ0(g). n g∈G

0 This means that λ = hχρ, χρ0 i/dim V , so by orthogonality relations among irreducible characters,  1 1  0 0 0 = for ρ = ρ φρ(ρ ) = dim V dim V (7) 0 for ρ0 6= ρ Now, instead of an irreducible ρ0, let us consider the representation P on W from the statement of our proposition. By analogy with equation (6), we can deﬁne

1 X φ (P ) := χ (g)P (g) ∈ End(W ) (8) ρ n ρ g∈G Again, let D be the set of isomorphism classes of irreducible representations of G. By Maschke’s theorem, ∃α : D → N s.t.

α(ρ) M M (j) W = Wρ , (9) ρ∈Dα j=1

−1 (j) where where Dα = D − α (0) and for every ρ ∈ Dα, {Wρ : j ∈ {1, 2, ..., α(ρ)}} is a chosen collection of mutually orthogonal G-invariant subspaces of W s.t. ∀j the restriction of P to (j) (j) (j) Wρ equals to an element of the class ρ. Let ρ denote the restriction of P to Wρ . Then, by Eq. (9), we can write α(ρ) M M P = ρ(j) (10) ρ∈D j=1

6 Substituting Eq. (10) into Eq. (8) yields

α(ρ) α(ρ) 1 M M X M M φ (P ) = χ (g)˜ρ(j)(g) = φ (˜ρ(j)), (11) ρ n ρ ρ ρ˜∈D j=1 g∈G ρ˜∈D j=1 which by Eq. (7), is equal to the projection operator πρ(P ) divided by dim ρ. Notice that

πρ(P ) = dim ρ · φρ(P ) is precisely Eq. (5). Lα(ρ) (j) Recall how in the paragraph after Eq. (4), we wrote j=1 Vρ to denote the isotypical ∗ ∗ component of RX ⊗ RX corresponding to ρ. Using projection πρ(RX ⊗ RX ), we can now ∗ ∗ write this isotypical component as the image πρ(RX ⊗ RX )(H ⊗ H ). This next claim will be used directly to prove the main theorem in section 5. Lemma 3. ∀ρ, ρ0 ∈ D and for all subsets X,Y ⊂ {1, 2, ..., L} s.t. X ⊂ Y , we have

∗ ∗ ∗ ∗ πρ(RY ⊗ RY ) ◦ πρ0 (RX ⊗ RX ) = πρ0 (RX ⊗ RX ) ◦ πρ(RY ⊗ RY ). (12) Proof. By Proposition 2, ∀T ∈ H ⊗ H∗, 0 ∗ ∗ dim ρ dim ρ X X ∗ ∗ π (R ⊗ R ) ◦ π 0 (R ⊗ R )(T ) = χ (a)χ 0 (b)a b T b a , (13) ρ Y Y ρ X X n2 ρ ρ Y X X Y a∈G b∈G where we use the fact that R is a unitary representation to write a∗ and b∗ instead of a−1 and −1 (a) ∗ ∗ b . For every a, b ∈ G, let b := a ba. Since X ⊂ Y , we have that bX aY = aY aY bX aY = ∗ (a) aY aX bX aX = aY ( b)X , so we can rewrite Eq. (13) as 0 ∗ ∗ dim ρ dim ρ X X (a) ∗ ∗ (a) π (R ⊗ R ) ◦ π 0 (R ⊗ R ) = χ 0 (b)χ (a)( b ) a T a ( b ). (14) ρ Y Y ρ X X n2 ρ ρ X Y Y X a∈G b∈G Since conjugation by a is an automorphism of G, we get that as b runs over the whole group (a) (a) G, so does b. Also, note that χρ0 (b) = χρ0 ( b). Therefore, ∀a ∈ G,

X (a) ∗ ∗ (a) X (a) (a) ∗ ∗ (a) χρ0 (b)χρ(a)( bX ) aY T aY ( bX ) = χρ0 ( b)χρ(a)( bX ) aY T aY ( bX ) b∈G b∈G X ∗ ∗ = χρ0 (b)χρ(a)bX aY T aY bX . (15) b∈G Combining Eqs. (14) and (15) yields 0 ∗ ∗ dim ρ dim ρ X X ∗ ∗ π (R ⊗ R ) ◦ π 0 (R ⊗ R ) = χ 0 (b)χ (a)b a T a b ρ Y Y ρ X X n2 ρ ρ X Y Y X a∈G b∈G 0 ! dim ρ dim ρ X ∗ X ∗ = χ 0 (b)b χ (a)a T a b n2 ρ X ρ Y Y X b∈G a∈G ∗ ∗ = πρ0 (RX ⊗ RX ) ◦ πρ(RY ⊗ RY ). (16)

7 4 Generalized Deﬁnitions of Weak Order and Disorder We are about to introduce a classiﬁcation of states in H into those that are “ordered” and those that are “disordered.”

4.1 Weak Order In a spin chain, a state is considered ordered if it exhibits a correlation between measurements performed in two disjoint neighborhoods of the chain. This notion of order is called long-range order, which is what Levin calls strong order in Ref. [4]. Now, we are going to deﬁne a weaker notion of order.

Deﬁnition 4. Let δ ∈ [0, 1] and I1 and I2 be disjoint subintervals of {1, 2, ..., L}. A state |ψi is δ weakly-ordered on the ordered pair (I1,I2) with respect to g ∈ G − {1} if and only if there exists an operator B ∈ End(H) such that

1. supp(B) ⊂ I1 ∪ I2. ∗ ∗ ∗ 2. ∀h ∈ G, h Bh = B. This signiﬁes that B ∈ πρ0 (R ⊗ R )(H ⊗ H ), where ρ0 is the trivial irreducible representation. 1 ∗ ∗ 3. ∃ρ ∈ D s.t. ρ(g) 6= and B ∈ πρ(RI2 ⊗ RI2 )(H ⊗ H ). 4. |hψ| B |ψi |≥ δ. 5. kBk ≤ 1.

∗ ∗ For the rest of this section, we will denote the isotypical πρ0 (R ⊗ R )(H ⊗ H ) by M0. ∗ By Lemma 3, ∀ρ ∈ D and ∀X ⊂ {1, 2, ..., L}, projection πρ(RX ⊗ RX ) commutes with ∗ πρ0 (R ⊗ R ), so ∀T ∈ M0, we have

∗ ∗ ∗ πρ(RX ⊗ RX )(T ) = πρ(RX ⊗ RX ) ◦ πρ0 (R ⊗ R )(T ) ∗ ∗ = πρ0 (R ⊗ R ) ◦ πρ(RX ⊗ RX )(T ) ∈ M0. (17)

∗ Thus, M0 is an invariant subspace of πρ(RX ⊗ RX ), so it is appropriate to talk about the ∗ ∗ operator πρ(RX ⊗ RX )|M0 ∈ End(M0) denoting πρ(RX ⊗ RX ) restricted to M0.

∗ ∗ Proposition 5. Suppose ρ ∈ D and operator B ∈ M0∩πρ(RX ⊗RX )(H⊗H ) s.t. X ⊂ Z and ∗ ∗ supp(B) ⊂ Z. Then, B ∈ πρ(RY ⊗ RY )(H ⊗ H ), where Y = Z − X and ρ is representation that is the complex conjugate of ρ.

∗ ∗ c Proof. We begin by proving that πρ(RX ⊗ RX )|M0 = πρ(RX ⊗ RXc )|M0 . By Proposition 2, ∀T ∈ H ⊗ H∗, dim ρ X π (R ⊗ R∗ )(T ) = χ (h)h∗ T h . (18) ρ X X n ρ X X h∈G ∗ Suppose T ∈ M0. Then, ∀h ∈ G, hT h = T , so Eq. (18) can be rewritten as

dim ρ X π (R ⊗ R∗ )(T ) = χ (h)h∗ hT h∗h . (19) ρ X X n ρ X X h∈G

8 ∗ ∗ ∗ ∗ Notice that h hX = hXc hX hX = hXc , so

∗ dim ρ X ∗ π (R ⊗ R )(T ) = χ (h)h c T h c ρ X X n ρ X X h∈G

dim ρ X −1 ∗ = χ (h )h c T h c n ρ X X h∈G

dim ρ X ∗ = χ (h)h c T h c n ρ X X h∈G

dim ρ X ∗ = χ (h)h c T h c n ρ X X h∈G ∗ = πρ(RXc ⊗ RXc )(T ) (20)

∗ ∗ c This proves that πρ(RX ⊗ RX )|M0 = πρ(RX ⊗ RXc )|M0 . ∗ ∗ Now, let us consider our operator B ∈ M0 ∩ πρ(RX ⊗ RX )(H ⊗ H ) s.t. X ⊂ supp(B). ∗ ∗ c Since we just proved that πρ(RX ⊗ RX )|M0 = πρ(RX ⊗ RXc )|M0 , we know that

∗ ∗ πρ(RXc ⊗ RXc )(B) = πρ(RX ⊗ RX )(B) = B, (21)

∗ ∗ where the last equality holds because B ∈ πρ(RX ⊗ RX )(H ⊗ H ). Since supp(B) ⊂ Z, we further have that Xc ∩ supp(B) ⊂ Z − X = Y , so

∗ dim ρ X ∗ π (R c ⊗ R c )(B) = χ (h)h c Bh c ρ X X n ρ X X h∈G

dim ρ X ∗ = χ (h)h c Bh c n ρ X ∩supp(B) X ∩supp(B) h∈G dim ρ X = χ (h)h∗ Bh n ρ Y Y h∈G ∗ = πρ(RY ⊗ RY )(B) (22) Combining Eqs. (21) and (22), we obtain

∗ πρ(RY ⊗ RY )(B) = B, (23)

∗ ∗ so B ∈ πρ(RY ⊗ RY )(H ⊗ H ) as desired.

Corollary 6. If ∃g ∈ G − {1} s.t. |ψi is δ weakly-ordered on (I1,I2) with respect to g, then |ψi is also δ weakly-ordered on (I2,I1) with respect to g.

Proof. By taking Proposition 5 and substituting Z = I1 ∪ I2 and X = I2, we get that ∗ ∗ conditions 1, 2, and 3 for operator B in Deﬁnition 4 imply that B ∈ πρ(RY ⊗RY )(H⊗H ) = ∗ ∗ πρ(RI1 ⊗ RI1 )(H ⊗ H ).

Remark 7. This is an important corollary, because it allows us to treat I1 and I2 as an unordered pair of disjoint intervals. From now on, if |ψi is δ weakly-ordered on (I1,I2) with respect to g, we can say instead that it is δ weakly-ordered on {I1,I2} with respect to g.

9 The physical intuition behind Definition 4 goes as follows: For simplicity, suppose B = ∗ O1O2, where supp(O1) ⊂ I1 and supp(O2) ⊂ I2. Then, condition 4 implies that |hO1ψ|O2ψi |≥ δ, which signifies that there is a lower bound on the correlation between |ψi modified by ∗ O1 on I1 and |ψi modified by O2 on I2. This is the special case when |ψi is considered δ strongly-ordered (or long-range ordered) on I1,I2 with respect to g. In general, however, B is not necessarily decomposable into a pair of operators O1, O2. Instead, B is a sum P α O1αO2α, where for every α, supp(O1α) ⊂ I1 and supp(O2α) ⊂ I2. Therefore, the gen- ∗ eral case is when each individual |hO1αψ|O2αψi | is not subject to a lower bound, because P ∗ condition 4 only requires | α hO1αψ|O2αψi |≥ δ. Hence the name weakly-ordered.

4.2 Weak Disorder A state is disordered if it is not severely altered by so-called “symmetry defects”. An

operator is called a symmetry defect on interval J if it is a non-identity element of group GJ . Again, there is a notion of long-range disorder, which Levin refers to as strong disorder, that characterizes how a state is not sensitive to all symmetry defects on an interval separating two neighborhoods where measurements are being performed. By analogy with the ordered case, we are interested in a weaker notion of disorder. To deﬁne it, we will use the representation ∗ ∗ (RX ⊗ RX )|H , which is the representation RX ⊗ RX restricted to a subgroup H in G. In ∗ particular, for a given g ∈ G − {1}, we will need the representation (RX ⊗ RX )|Z(g), where Z(g) is the centralizer of g.

Deﬁnition 8. Let δ : G − {1} → [0, 1] be a function and I1 and I2 be disjoint subintervals of {1, 2, ..., L}. A state |ψi is δ weakly-disordered on the ordered pair (I1,I2) if and only if ∀g ∈ G − {1}, there exists an operator C(g) such that (g) 1. supp(C ) ⊂ I1 ∪ I2. ∗ (g) (g) (g) ∗ ∗ 2. ∀h ∈ Z(g), h C h = C . This signiﬁes that C ∈ πρ0 ((R ⊗ R )|Z(g))(H ⊗ H ), where ρ0 is the trivial irreducible representation. (g) ∗ 3. ∃ρ ∈ D s.t. C ∈ πρ((RI2 ⊗ RI2 )|Z(g)). (g) 4. |hψ| C gJ |ψi |≥ δ(g), where J is an interval between I1 and I2.

5. C(g) ≤ 1.

For every g ∈ G − {1}, we can simply adopt Proposition 5 for the representation R|Z(g). Hence, by analogy with the weak-order case, ∀g ∈ G − {1}, conditions 1, 2, and 3 for (g) (g) ∗ operator C in Deﬁnition 8 imply that C ∈ πρ((RI1 ⊗ RI1 )|Z(g)), so Proposition 5 has another corollary:

Corollary 9. If |ψi is δ weakly-disordered on (I1,I2), then it is also δ weakly-disordered on (I2,I1).

Remark 10. From now on, if |ψi is δ weakly-disordered on (I1,I2), we can say instead that it is δ weakly-disordered on {I1,I2}. Notice that unlike in the ordered case, weak disorder is not a notion deﬁned relative to a single non-identity element g of G. If a state is δ weakly-ordered, we must specify with

10 respect to which g ∈ G − {1} it is weakly-ordered, but if it is δ weakly-disordered, we don’t have to. On the other hand, while δ in Definition 4 is simply a number, the δ in Definition 8 is a function that assigns a number to every g ∈ G − {1}. This choice for constructing Definitions 2 and 3 is based on physical intuition. Suppose that ∀g ∈ G − {1}, we can (g) (g) (g) (g) (g) decompose C = O1 O2 , where supp(O1 ) ⊂ I1 and supp(O2 ) ⊂ I2. By analogy, this is the situation where we call |ψi a state that is δ strongly-disordered on I1, I2. Then, condition 4 implies that D E (g)∗ (g) O1 ψ O2 gJ ψ ≥ δ(g).

What this tells us is that for every defect gJ , states |ψi and |gJ ψi can be forced by operators (g) (g) (g) (g) O1 and O2 to overlap. Due to conditions 2 and 3, O1 and O2 are not just an arbitrary ∗ pair of operators. Instead, O1 belongs belongs to an isotypical of (RI1 ⊗ RI1 )|Z(g) and O2 ∗ belongs to an isotypical of (RI2 ⊗RI2 )|Z(g). Thus, the overlap between |ψi and |gJ ψi is forced by operators subject to a mathematical constraint imposed by the action of group G. The physical meaning of all this is that δ measures the extent by which a state |ψi altered by a symmetry defect can be reverted back to itself by a single measurement on I1 ∪ I2. In the more general situation when the δ weakly-disordered state |ψi is not δ strongly disordered, (g) (g) (g) not all C are decomposable into O1 and O2 . Now, δ provides the extent by which |gJ ψi is reverted back |ψi by a series of measurements on I1 ∪ I2. In essence, the reason why weak-disorder is not deﬁned relative to a single g ∈ G − {1} is that we want a disordered state to have some degree of reversibility regardless of what symmetry defect it gets altered by.

5 Main Result and Proof First, we need some notation: for any ﬁnite group H, we use #Conj[H] to denote the number of conjugacy classes in H. Also, for any g ∈ G, let C(g) denote the conjugacy class of g. Note that the smallest normal subgroup of G containing g is hC(g)i, the subgroup generated by the conjugacy class of g. This is true for two reasons:

1. Any normal subgroup of G containing g must contain C(g), so by closure, it must also contain hC(g)i.

2. hC(g)i is a normal subgroup, because any conjugation by an element of G is a homomorphism that permutes the generators in C(g).

We are now ready to state our generalization of Lemma 1 from Ref. [4].

11 Theorem 11. (Main Result) Let |ψi ∈ H be a simultaneous eigenstate of G (i.e. a state belonging to a one-dimensional G-invariant subspace), where G is a group of ﬁnite order n.

Then, for any given δ ∈ [0, 1] and every pair of disjoint intervals I1 and I2 separated by a distance of at least one lattice site, at least one of the following holds:

1. ∃g ∈ G − {1} s.t. |ψi is δ/Q1(g) weakly-ordered on {I1,I2} with respect to g, where Q1(g) =#Conj[G]−#Conj[G/hC(g)i]

2. |ψi is (1 − δ)/Q2 weakly-disordered on {J1,J2}, where J1 and J2 are the intervals complementary to I1 and I2 (as shown in Figure 2 below) and where Q2 : G−{1} → Z+ is deﬁned by Q2(g) = #Conj[Z(g)].

Figure 2: Intervals I1, I2, J1, and J2 from Theorem 11 partitioning the spin chain.

Proof. Just like Michael Levin in his proof of Lemma 1 from Ref. [4], we use the Fuchs-van de Graaf inequality [2]. A consequence of this inequality is that for any two states |ψi and |ψ0i and any subset X ⊂ {1, 2, ..., L}, we have 1 max |hψ| A |ψi − hψ0| A |ψ0i | + max |hψ| U |ψ0i |≥ 1, (24) supp(A)⊂X 2 supp(U)⊂Xc where operators A satisfy kAk ≤ 1 and operators U are unitary. Suppose that |ψi is a 0 simultaneous eigenstate of G, |ψ i = gI2 |ψi for some g ∈ G − {1}, and X = I1 ∪ I2. Substituting these into Eq. (24) yields

1 ∗ max |hψ| (A − gI2 AgI2 ) |ψi | + max |hψ| UgI2 |ψi |≥ 1. (25) supp(A)⊂I1∪I2 2 supp(U)⊂J1∪J2 It is clear from Eq. (25) that for any given δ ∈ [0, 1], either

12 (i) ∃g ∈ G − {1} s.t. the ﬁrst term of Eq. (25) is greater than or equal to δ.

(ii) ∀g ∈ G − {1}, the second term of Eq. (25) is greater than or equal to 1 − δ.

We claim that in case (i), |ψi is δ/Q1(g) weakly-ordered on {I1,I2} with respect to g, while in case (ii), it is (1 − δ)/Q2 weakly-disordered on {J1,J2}. Case (i): Given g ∈ G − {1} s.t. the first term of Eq. (25) is no less than δ, we define ∗ F := AM − gI2 AM gI2 , where AM is a choice of operator A that gives the maximum in the first term of Eq. (25). In addition, let ! dim ρ X X B := π (R ⊗ R∗) ◦ π (R ⊗ R∗ )(F/2) = a∗ χ (b)b∗ F b a, (26) ρ ρ0 ρ I2 I2 2n2 ρ I2 I2 a∈G b∈G where ρ0 is the trivial irreducible representation of G and ρ is any of the irreducible representations of G. Clearly, when we take the sum of all double projections Bρ of F/2, we get back the projection of F/2 onto the trivial isotypical of R ⊗ R∗: X 1 X B = π (R ⊗ R∗)(F/2) = a∗F a, (27) ρ ρ0 2n ρ∈D a∈G where again D is the set of all isomorphism classes of irreducible representations of G.

Consider a projection Bρ with ρ such that ρ(g) = 1. The right cosets of hgi form a partition of G, so we can rewrite Eq. (26) as   dim ρ X ∗ X X ∗ Bρ = a  χρ(b)bI F bI  a (28) 2n2 2 2 a∈G K∈hgi\G b∈K

Of course, since representations are homomorphisms, hgi ⊂ ker(ρ), which implies that ρ

and its character χρ are constant over every coset of hgi. Furthermore, we can choose a representative element b(K) for every coset K ∈ hgi\G. Thus, we can rewrite Eq. (28) as

 o(g)−1   dim ρ X ∗ X ∗ X ∗ v v Bρ = a  χρ(b(K))b(K)I  gI F gI  b(K)I  a, (29) 2n2 2 2 2 2 a∈G K∈hgi\G v=0

where o(g) is the order of element g. Notice that

o(g)−1 o(g)−1 X ∗ v v X ∗ v ∗ v gI2 F gI2 = gI2 (AM − gI2 AM gI2 )gI2 = 0, (30) v=0 v=0 because the middle expression is a telescoping sum. Hence, by Eqs. (29) and (30), we have

that ∀ρ s.t. ρ(g) = 1, operator Bρ = 0. Thus, if we let N be the set of all isomorphism classes of irreducible representations of G s.t. ρ(g) 6= 1, then X X Bρ = Bρ. (31) ρ∈D ρ∈N

13 Combining Eqs. (27) and (31) yields X 1 X B = a∗F a, (32) ρ 2n ρ∈N a∈G Recall that |ψi is a simultaneous eigenstate of G. Since R is a unitary representation, we also know that ∀a ∈ G, the eigenvalue of |ψi with respect to operator a lies on the unit circle in C. Therefore,

1 X 1 X n 1 hψ| a∗F a |ψi = hψ| a∗F a |ψi = |hψ| F |ψi |= |hψ| F |ψi |. (33) 2n 2n 2n 2 a∈G a∈G Combining Eqs. (32) and (33) yields

1 1 X ∗ hψ| Bρ |ψi = |hψ| F |ψi |= max |hψ| (A − gI2 AgI2 ) |ψi | ≥ δ. (34) 2 supp(A)⊂I1∪I2 2 ρ∈N

Hence, ∃ρ˜ ∈ N s.t |hψ| Bρ˜ |ψi |≥ δ/card(N). It is easy to check that Bρ˜ satisﬁes condition 1 ∗ met by operator B in Deﬁnition 4 of weak order. By Lemma 3, projections πρ0 (R ⊗ R ) and ∗ πρ˜(RI2 ⊗ RI2 ) commute, so

∗ ∗ ∗ ∗ Bρ = πρ0 (R ⊗ R ) ◦ πρ˜(RI2 ⊗ RI2 )(F/2) = πρ˜(RI2 ⊗ RI2 ) ◦ πρ0 (R ⊗ R )(F/2). (35)

This means Bρ˜ satisfies conditions 2 and 3 of Definition 4 as well. Condition 5 that kBρ˜k ≤ 1 holds too, because by the triangle inequality, kAM k ≤ 1 implies that kF/2k ≤ 1, and since Bρ˜ ≤ 1 is a double projection of F/2, its operator norm must be no greater than that of F/2. To complete the proof of our claim for case (i), all we have left is to find card(N). An irreducible ρ of G is not in N if and only if ker ρ contains the smallest normal subgroup containing g, because ker ρ is a normal subgroup of G. As discussed at the very beginning of this section, hC(g)i is the smallest normal subgroup containing g, so ρ∈ / N if and only if hC(g)i ⊂ ker ρ. Therefore, the number of irreducibles of G not contained by N is equal to the number of irreducible representations of G/hC(g)i, which in turn equals to #Conj[G/hC(g)i]. Since we also have that the total number of irreducible representations of G is #Conj[G], we deduce that

card(N) = #Conj[G] − #Conj[G/hC(g)i] = Q1(g),

so |hψ| Bρ˜ |ψi |≥ δ/Q1(g). In all, we found an operator Bρ˜ that satisfies the conditions for B in Definition 4 with δ replaced by δ/Q1(g), so |ψi qualifies as a δ/Q1(g) weakly-ordered state on I1,I2. Case (ii): ∀g ∈ G − {1}, we define for all irreducible representations ρ of Z(g) the following double projection:

(g) ∗ ∗ (g) Cρ := πρ0 ((R ⊗ R )|Z(g)) ◦ πρ((RJ2 ⊗ RJ2 )|Z(g))(UM )   dim ρ X ∗ X ∗ (g) = a  χρ(b)bJ U bJ  a, (36) |Z(g)|2 2 M 2 a∈Z(g) b∈Z(g)

14 (g) where UM is a choice of unitary operator U that gives the maximum in the second term of Eq. (25). Let D(g) be the set of all isomorphism classes of irreducible representations of Z(g). By the same reasoning as for Eq. (27), we have

X 1 X (g) C(g) = a∗U a, (37) ρ |Z(g)| M ρ∈D(g) a∈Z(g)

Since every element of Z(g) commutes with gI2 , we know that

X (g) (g) hψ| Cρ gI2 |ψi = |hψ| UM gI2 |ψi |= max |hψ| UgI2 |ψi |≥ 1 − δ. (39) supp(U)⊂J1∪J2 ρ∈D(g)

(g) (g) Therefore, there exists an irreducible representationρ ˜ s.t |hψ| Cρ˜ gI2 |ψi |≥ (1−δ)/card(D ) (g) = (1 − δ)/#Conj[Z(g)]. By analogy with Case (i), one can easily check that Cρ˜ satisﬁes the other properties (properties 1, 2, 3, and 5) in Deﬁnition 8, so |ψi is a (1 − δ)/Q2 weakly- disordered state on J1,J2.

6 Discussion of Future Work To precisely figure out how to generalize Theorems 1 and 2 in Ref. [4], I am analyzing how Levin’s Lemmas 2 and 3 can be reformulated. Currently, I suspect that generalizing these lemmas when G is Abelian might be straightforward, although I still have verify more carefully. One aspect that I paid particular attention to is that Levin relies on a famous result by Hastings [3] to prove Lemma 3. While Levin adapted this result for the lowest-energy even eigenstate of an Ising-symmetric local Hamiltonian on the spin-chain, I think that the result can also be adapted for lowest-energy G-invariant eigenstate of an onsite G-invariant local Hamiltonian, where G is any finite group. In order to generalize Levin’s adaptation of the Hastings result, I think all it takes is to replace the projection onto the even subspace by the projection onto the trivial isotypical component of representation R of G. Theorems 1 and 2 from [4] rely on rigorous definitions of order and disorder parameters that are specific to Ising symmetric spin chains. Levin provided these definitions in his paper. Based on how we defined weak order and disorder, I find it most intuitive to generalize Levin’s definitions of order and disorder parameters as follows:

Deﬁnition 12. Let g ∈ G−{1}. A collection of operators {Oi : i ∈ X} with X ⊂ {1, 2, ..., L} is called a (δ, `, g) order parameter for a state |ψi if and only if

∗ 1. Oi belongs to an isotypical component R ⊗ R corresponding to a nontrivial irreducible representation ρ s.t. ρ(g) 6= 1.

15 2. Oi is supported on [i − `, i + `].

∗ 3. |hψ| Oi Oj |ψi |≥ δ for all i, j ∈ X with |i − j|≥ 2`.

4. kOik ≤ 1.

The set X is called the domain of deﬁnition of the order parameter. S (g) Deﬁnition 13. A collection of operators g∈G−{1}{Oi : i ∈ X} with X ⊂ {1, 2, ..., L} is called a (δ, `) disorder parameter for a state |ψi if and only if

(g) ∗ 1. Oi belongs to some isotypical component of (R ⊗ R )|Z(g).

(g) 2. Oi is supported on [i − `, i + `].

(g)∗ (g) 3. |hψ| Oi Oj g[i+1,j] |ψi |≥ δ for all i, j ∈ X with |i − j|≥ 2`.

(g) 4. Oi ≤ 1. The set X is called the domain of deﬁnition of the disorder parameter.

Relying on these deﬁnitions, I hope to generalize Theorem 1 into a claim of the following form: in a circular spin chain, for any translationally invariant and onsite G-invariant Hamiltonian such that its lowest-energy G-invariant eigenstate |Ωi is non-degenerate, |Ωi must either have

1. a (δ, `, g) order parameter for some g ∈ G − {1} or

2. a (δ, `) disorder parameter deﬁned on the whole spin chain (i.e. X = {1, 2, ..., L}) with δ equal to some ﬁxed positive number less than 1 and ` somehow bounded from above. The upper bound on ` might again depend on the dimension d of each site’s vector space Hi and the energy gap within the trivial isotypical component of R. Additionally, I hope to achieve a similar generalization of Theorem 2.

7 Acknowledgments Special thanks to Professors Daniel S. Freed and Lukasz Fidkowski for their outstanding co-supervision of my work. This thesis was also made possible by my participation in the Summer 2019 Physics REU at the University of Washington, so I would like to acknowledge the National Science Foundation, as well as the faculty and staﬀ from the Physics Department at the University of Washington. I thank the program directors Dr. Gray Rybka and Dr. Subhadeep Gupta and the program administrators Linda Vilett and Cheryl McDaniel for coordinating that REU and making it possible in the ﬁrst place. Furthermore, special thanks to Dr. Fidkowski’s students Tyler Ellison, Ryan Lanzetta, Joseph Merritt, and Sujeet Shukla for their outstanding mentorship while I was at the REU.

16 References [1] A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge University Press, Cambridge, 2010).

[2] C. Fuchs and J. van de Graaf, Cryptographic distinguishability measures for quantum- mechanical states, IEEE Trans. Inform. Theory 45, p. 1216 (1999).

[3] M. B. Hastings, An area law for one-dimensional quantum systems, J. Stat. Mech. Theory Exp. 2007, p. 8024 (2007).

[4] M. Levin, Constraints on order and disorder parameters in quantum spin chains, arXiv:1903.09028.