Weak Order and Disorder of Quantum States Invariant Under Onsite Action by an Arbitrary Finite Group on a Spin Chain by 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 pa- rameters 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 state- ment 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 2 f1; 2; :::; Lg, 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 observ- able 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 ⊂ f1; 2; :::; Lg 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) j2 =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 i2X 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) ⊂ fi; i + 1g and kHi;i+1k ≤ 1: The operators Hi;i+1 correspond to interactions between neighboring spins.