Spin-Orbit Coupling and Strong Correlations in Ultracold Bose Gases

Spin-orbit coupling and strong correlations in ultracold Bose gases

DISSERTATION

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By William S. Cole, Jr., B.S. Graduate Program in Physics

The Ohio State University 2014

Dissertation Committee: Professor Nandini Trivedi, Advisor

Professor Mohit Randeria

Professor P. Chris Hammel

Professor Richard J. Furnstahl c Copyright by

William S. Cole, Jr.

2014 Abstract

The ability to create artificial gauge fields for neutral atoms adds a powerful new dimension to the idea of using ultracold atomic gases as “quantum simulators” of models that arise in conventional solid state physics. At present, cold atom experiments are able to simulate orbital magnetism and a certain kind of spin-orbit coupling at scales which are quite difficult to achieve in the solid state. This takes us beyond the realm of simulation into questions about new states of matter which might only be possible in cold atom experiments. Indeed, as a byproduct of finding methods to simulate traditional gauge potentials for neutral gases, it has been realized that gauge potentials with no solid state analog can also be created and finely tuned to manipulate the few- and many-body physics of bosons and fermions in remarkable new ways.

Motivated along these lines, in this dissertation I address several issues related to bosons with spin-orbit coupling. After an introduction to synthetic gauge ﬁelds in general I describe

Rashba spin-orbit coupling specifically and how it modifies the usual behavior of bosons in the continuum and in a harmonic trapping potential. Following this, I study the effects of interactions induced by combining spin-orbit coupling with an optical lattice potential. I use a variety of theoretical tools to characterize and classify the exotic phases which emerge.

In the weakly interacting limit, mean-ﬁeld theory suggests Bose-Einstein condensed states with the possibility of magnetically ordered phases arising through interference between macroscopically occupied single particle states. In the deep lattice limit, the ground state is a Mott insulator with magnetic order driven by bosonic superexchange. I address the Mott transition, as well as the nature of superﬂuid states near the Mott transition, and introduce a novel “slave-boson” approach to understand these results. Finally, using some of the

ii insight developed for this general case, I restrict my attention to a more experimentally relevant implementation of SOC, where I can use numerically exact calculations to justify the mean-ﬁeld results.

In the final Chapter, I address a problem which perhaps seems, at first glance, to be unrelated to bosons with Rashba SOC. The problem is identifying the ground states of spinless hardcore bosons on a frustrated honeycomb lattice. The connection between these problems is the delicate question of how bosons condense in a flat band. In that section I provide numerical support to a recently proposed scenario of “statistical transmutation,” where an approximate ground state of the hard core bosons can be obtained from a mean-

ﬁeld approximation in terms of composite fermions.

iii Acknowledgments

First and foremost, I am deeply thankful to my advisor Nandini Trivedi for her guidance along my winding Ph.D. path. Her enthusiasm for physics and her breadth of interests are inspiring. I would also like to give special thanks to Mohit Randeria, who often served a similar role, and together the examples they have set as teachers and mentors I can only hope to emulate.

I am also particularly indebted to Arun Paramekanti and Shizhong Zhang for their collaboration on much of the work presented in this dissertation, and for enlightening discussions over the past few years.

The condensed matter community at Ohio State has been a stimulating, nurturing environment, full of talented post-docs and students from whom I have learned a great deal. Special thanks go to Yen Lee Loh, who helped me to really get my bearings in the folklore of condensed matter theory. Also I must thank Eric Duchon and Weiran Li for being ideal oﬃcemates, along with several other fellow students (Onur, Nganba, Mason,

Tim, Daniel, Jim, John) who have helped to guide my own development as a physicist.

That this thesis was completed at all is a great credit to my wife Alexis. Her unfailing support and encouragement have kept me aﬂoat through every challenge. I have also been blessed with a wonderful, loving family who sparked my initial interest in math and science and continue to fan the ﬂames.

iv Vita

May, 2008 ...... B.S. in Physics, University of Central Florida, Orlando, FL September, 2008 – present ...... Graduate Teaching and Research Assis- tant, and Ohio State Presidential Fellow, the Ohio State University, Columbus, OH

Publications

William S. Cole, Shizhong Zhang, Arun Paramekanti, Nandini Trivedi, Bose Hubbard Models with Synthetic Spin-Orbit Coupling: Mott Insulators, Spin Textures and Superﬂuidity, Phys. Rev. Lett. 109 085302 (2012)

Anamitra Mukherjee, William S. Cole, Nandini Trivedi, Mohit Randeria, Patrick Wood- ward, Theory of strain controlled magnetotransport and stabilization of the ferromagnetic insulating phase in manganite thin ﬁlms, Phys. Rev. Lett. 110 157201 (2013)

O. Nganba Meetei, William S. Cole, Mohit Randeria, Nandini Trivedi, Novel magnetic state in d4 Mott insulators, arXiv:1311.2823

Zhihao Xu, William Cole, Shizhong Zhang, Mott-superﬂuid transition for spin-orbit-coupled bosons in one-dimensional optical lattices, Phys. Rev. A 89 051604(R) (2014)

William S. Cole, Nandini Trivedi, Statistical transmutation of hard core bosons on the frustrated honeycomb lattice, in preparation

Shizhong Zhang, William S. Cole, Arun Paramekanti, Nandini Trivedi, Synthetic gauge ﬁelds in optical lattices, in preparation for Annual Reviews of Cold Atoms and Molecules

Fields of Study

v Major Field: Physics

vi Table of Contents

Page Abstract...... ii Acknowledgments...... iv Vita...... v List of Figures ...... ix List of Tables ...... x

Chapters

1. Introduction, motivations, and outline ...... 1 1.1 Spin-orbit coupling...... 2 1.2 Ultracold bosons ...... 4 1.2.1 The Bose-Hubbard model...... 5 1.3 Synthetic gauge ﬁelds ...... 6 1.3.1 Berry’s connection and the adiabatic principle ...... 8 1.3.2 Raman induced spin-orbit coupling...... 12 1.3.3 “Realistic” Rashba spin-orbit coupling...... 15

2. Exotic Bose matter from nonabelian gauge ﬁelds...... 18 2.1 Bose-Einstein condensation with spin, and with spin-orbit coupling . . . . . 18 2.2 The Rashba hamiltonian...... 21 2.2.1 Cylindrical coordinates ...... 23 2.2.2 The Rashba hamiltonian with a harmonic trapping potential . . . . 26 2.3 Adding interactions...... 27 2.4 An exact two-body ground state ...... 29

3. The Bose-Hubbard model with spin-orbit coupling...... 32 3.1 Spin-orbit coupling in a periodic optical potential...... 34 3.2 Model hamiltonian...... 36 3.3 Weak-coupling approximation...... 36 3.4 Strong-coupling approximation ...... 42 3.5 Mean-field theory of the superfluid-insulator transition...... 51 3.5.1 Description of the mean-field method...... 51 3.5.2 Results ...... 54 3.6 Slave-boson theory...... 56 3.7 Exact numerics in the one-dimensional limit...... 60

vii 4. Hard core bosons on the frustrated honeycomb lattice ...... 63 4.1 Hard core bosons and Jordan-Wigner fermions ...... 63 4.2 Model and methods ...... 67 4.3 Comparing the ground-state energies of hard core bosons and fermions . . . 70 4.4 Monte Carlo results ...... 72 4.4.1 Variational bounds on the ground state energy...... 72 4.4.2 One-body density matrix ...... 72 4.5 Discussion and conclusions...... 73

Bibliography...... 75

Appendices

A. Numerical solution of the Rashba hamiltonian in a harmonic trap . . . 83 A.1 Series expansion in harmonic trap solutions...... 84

B. Eﬀective spin hamiltonian from two-site perturbation theory ...... 90

C. The variational principle and variational Monte Carlo...... 94 C.1 Application of VMC to fermionized hamiltonians...... 97

viii List of Figures

Figure Page

1.1 Mean-ﬁeld phase diagram of the Bose-Hubbard model...... 7 1.2 Schematic implementation of the Raman scheme for spin-orbit coupling . . 13 1.3 Spectrum of the hamiltonian for Raman-induced one-dimensional spin-orbit coupling...... 15

2.1 Energy spectrum of the Rashba hamiltonian...... 24 2.2 Energy spectrum of the Rashba hamiltonian in a harmonic trap ...... 27 2.3 Spatial variation of the eigenstates of the Rashba hamiltonian in a harmonic trap ...... 28

3.1 Energy spectrum of a single particle in a 2D square optical lattice with spin orbit coupling...... 38 3.2 Density of states and lower-band spin wavefunction for a Rashba coupled particle in a 2D square optical lattice...... 39 3.3 Weak-coupling energy expectation values of various ordered condensate states 41 3.4 Spin densities of various ordered condensate states...... 41 3.5 Illustration of frustration for compass model and Dzyaloshinski-Moriya interactions...... 46 3.6 Magnetic phase diagram in the deep Mott limit...... 47 3.7 Magnetic order in Mott insulating states of lattice bosons with spin-orbit coupling...... 48 3.8 Variational comparison of several classical spin states...... 50 3.9 Mean-ﬁeld Mott lobes in the Rashba coupled Bose-Hubbard model . . . . . 55 3.10 Mean-ﬁeld solutions to the Rashba coupled Bose-Hubbard model ...... 56 3.11 DMRG results for the Bose-Hubbard model with abelian spin-orbit coupling 62

4.1 The honeycomb lattice model...... 67 4.2 Comparison of fermion and boson ground state energies...... 71 4.3 Variational Monte Carlo energy and variance for the VMFT state ...... 73 4.4 Condensate fraction in the VMFT state...... 74

ix List of Tables

Table Page

3.1 Virtual state matrix elements for calculating the low-energy eﬀective spin hamiltonian in the Mott insulator...... 43

x Chapter 1 Introduction, motivations, and outline

One of the primary motivations for studying ultracold atomic gases is the notion of quantum simulation. Remarkably rich phenomena, like magnetism and superconductivity, are encoded into the solutions of the familiar non-relativistic Schr¨odingerequation; however, analytic and even numerical estimates for physically meaningful quantities are often available only approximately or else in very special limiting cases. Even minimalistic “eﬀective” models are notoriously hard to solve when there are many interacting degrees of freedom.

On the other hand, with the unprecedented control that experimentalists have achieved over dilute gases of ultracold atoms, interesting model hamiltonians can now be engineered in the laboratory and their properties explored directly without recourse to approximations, across wide parameter regimes [1].

The typical starting point for most many-body theories is an eﬀectively non-interacting

(i.e., mean-ﬁeld) limit where the properties of a single quasiparticle are well understood.

p2 Deviations from the simplest single-particle behavior, H = 2m∗ , however, can have dramatic eﬀects on the many-body physics. The origin of such deviations could be, for example, the presence of a periodic potential, which in turn gives rise to a band spectrum where certain energies are prohibited. In the cold atoms context, a static periodic potential can be created with lasers – the so-called “optical lattice.” In addition, substantial progress has recently been made in driving the occupation of higher energy Wannier states of the individual potential wells of an optical lattice [2,3], simulating the spatially anisotropic

1 p and d orbitals occupied by real electrons in solids. Other typical sources of nontrivial single-body physics are orbital magnetism, which gives rise to the Landau level spectrum, and various ﬂavors of spin-orbit coupling (SOC) which generically give rise to Dirac points and spin-momentum locking within the single-particle states.

While optical lattices are now a standard tool in cold atom experiments, the implementation of orbital magnetism and spin-orbit coupling are both very new and active areas of current research. Both of these phenomena arise in conventional solid-state experiments as a direct consequence of the electric charge carried by the electron and the coupling of that charge to electric and magnetic fields. Since cold atom experiments are performed using neutral atoms, one must find clever ways to artificially endow these atoms with the effects of minimal coupling to a gauge field, but obviously with a substantially different underlying cause. The key insight here will prove to be the use of Berry’s “geometric phase.” From this perspective, it will also turn out to be quite natural to view spin-orbit coupling in an identical framework, arising from the imprinting of spin-carrying particles with a nonabelian geometric phase [4].

In the following chapters, I narrow my focus to exploring the consequences of large spin-orbit coupling for interacting ultracold bosons, with and without an additional optical lattice potential. In the remainder of this introduction, however, I will describe why spin- orbit coupling and Bose gases are interesting, separately, and will also introduce the essential notions that lay the foundation for engineering synthetic gauge ﬁelds for neutral atoms.

Upon combining these ingredients, some of the usual eﬀects of spin-orbit coupling survive

(such as the spin Hall eﬀect), however, the underlying Bose statistics lead to states of matter that are quite foreign to the intuition developed either for solid-state spin-orbit coupled Fermi systems or for the conventional spinless Bose superﬂuids.

1.1 Spin-orbit coupling

Traditionally, spin-orbit coupling is understood as a consequence of special relativity. It appears in the ﬁrst-order (in v/c) relativistic correction to the Schr¨odingerequation for an

2 electron [5], −|e| v Hrel = × E · S (1.1) 2mec c where, intuitively, the spin S of the electron (with charge e = −|e|) will precess as it moves in a static electric ﬁeld E, because under a Lorentz transformation to the particle’s rest

0 v frame, that electric ﬁeld becomes a magnetic ﬁeld B = − c × E, which in turn yields a Zeeman coupling to the spin, ∝ B0 ·S. In the fully relativistic Dirac equation, this coupling between spin and spatial motion is made manifest

2 HDirac = cα · p + βmc (1.2) where (α, β) are a set of 4 × 4 matrices which act in the space of internal states for the electron (two spin projections, particle and antiparticle projections).

Na¨ıvely, one would expect relativistic corrections to be essentially negligible in the low- energy, nonrelativistic limit where condensed matter physics is typically studied. However, there are a variety of situations where the eﬀects of spin-orbit coupling are simply unavoid- able. The familiar L · S coupling of atomic physics arises from the spherically symmetric

Coulomb ﬁeld that binds electrons to positively charged nuclei. The relativistic nature of the atomic “ﬁne-structure” is manifest in the scaling of the level splittings as ∆E ∼ (Zα)4

[5], with α ∼ 1/137 the ﬁne structure constant. However, the presence of the atomic number Z means that these splittings can easily be of the same order as the bandwidth in narrow-band materials that include heavy elements, for example the iridium (ZIr = 77) and osmium (ZOs = 76) oxides. Furthermore, in a bulk crystalline lattice, electrons can experience a nonzero average gradient of the electric potential. This can only happen in crystals that lack inversion symmetry (noncentrosymmetric lattices) and is called Dresselhaus spin-orbit coupling [6,7].

Here, it is the electron’s crystal momentum k, rather than orbital angular momentum, that couples to the spin. Similar to this is the Rashba effect, again related to an explicitly broken inversion symmetry, that in this case occurs at the surface of a crystal or at an interface between two different materials [8]. Rashba proposed this effect in the context of the two-

3 dimensional electron gas in semiconductor quantum wells, although it is also an important ingredient for the electronic structure at the interface between bulk LaAlO3 and SrTiO3 [9]. In recent years, interest in these materials with strong spin-orbit coupling has been on the rise. This has been driven both by the desire to use spin-orbit coupling as a resource in the blooming ﬁeld of spintronics, where it is a crucial ingredient for new circuit devices such as spin transistors [10], as well as an appreciation of the fundamental role that spin-orbit coupling plays in stabilizing new exotic states of matter, for example, the quantum spin

Hall insulators and topological insulators [11].

1.2 Ultracold bosons

Condensed matter physics abounds with examples of macroscopic quantum phenomena.

The usual examples are the dramatic and closely-related phenomena of superconductivity and Bose-Einstein condensation (BEC), but even simple metals can be well understood by starting from an ideal gas of explicitly quantum mechanical fermions. These facts indeed hint at the notion that perhaps the most substantial eﬀects imposed by quantum mechanics on condensed matter systems already appear at the level of exchange statistics, that is, the fundamental constraints satisﬁed by the many-body wavefunction of identical particles when those particles are exchanged.

Of course, the vast majority of matter that we might be interested in studying is made of fermions, whether we are considering the electron gases in metals and semiconductors or the protons and neutrons in nuclei or even the quarks that make up those nucleons.

One might naturally wonder what “boson matter” even means. The distinction between fermions and bosons ﬁrst appears in the two possibilities for how the wavefunction of several indistinguishable particles changes when the coordinates of the particles are permuted. If we were to take two hydrogen atoms and swap their positions then the wavefunction necessarily accumulates a minus sign from swapping the two electrons, as well as a minus sign from swapping the two protons, meaning that in total there is no sign change, as expected for bosons. Each hydrogen atom is made up of two (distinguishable) fermions bound together

4 with an energy of 1 Ry, so unless we probe the atom at that energy scale (or on length scales shorter than the Bohr length), the electron and proton do not have separate identities.

Rather than working in terms of the electron and proton coordinates individually, then, it is much more useful to work with the center of mass coordinate and the speciﬁc set of quantum numbers that the electron possesses as it sits in the Coulomb potential of the proton. This remains the case even for much more complicated atoms.

So now, supposing that we do have some collection of noninteracting atoms, built in such a way that they are composite bosons, how does this collection behave? If their number is conserved, there is a finite temperature phase transition at which the atoms all pile into the lowest energy state available. This insight provides the first steps toward understanding the exotic properties of superfluid 4He, which was discovered in the 1930s and, for many decades, remained the paradigm of a bosonic condensed matter system. It was not, however, until 1995 that Bose-Einstein condensation was observed for the first time in a dilute (that is, essentially noninteracting) atomic gas after an experimental effort spanning decades, by

Cornell and Wieman, and then shortly thereafter reproduced independently by Ketterle, for which the trio shared the 2001 Nobel prize in physics [12, 13]. Since then, this field has blossomed in many directions. The study of bosonic matter is no longer restricted to the superfluidity of 4He, but instead encompasses a wide range of activity, such as systematically studying the effects of interactions on condensation, the role of internal degrees of freedom in “spinor” BECs, the effects of rotation, and the implementation and consequences of synthetic gauge fields such as the spin-orbit coupling I study in this thesis.

1.2.1 The Bose-Hubbard model

Well before the experimental realization of an ideal Bose gas in the cold atoms context, there were experiments where 4He was absorbed into porous media, as well as various studies of disordered “granular” superconductors, where the Cooper pairs could be imagined as approximately behaving like bosons, because their length scale is smaller than that of an individual grain. In either situation, Fisher et al. [14] posited that these experiments could be modeled by a network of localized regions that bosonic degrees of freedom (represented

5 † with creation and annihilation operators which satisfy [bi , bj] = δij) could occupy, weakly coupled together by quantum tunneling. The resulting model, called the “Bose-Hubbard model,” is characterized then by a hamiltonian

X U X H = (V − µ) b†b + b†b b†b − 1 + t b†b (1.3) BHM i i i 2 i i i i ij i j i ij

Where µ is a chemical potential that governs the density, Vi models a local (usually disorder) potential, U is the potential energy penalty that every pair of bosons costs when occupying the same region i, while tij represents the amplitude for a boson in region j to tunnel into i. For a regular lattice network, and vanishing disorder, this is also an ef-

fective description of Josephson junction arrays (those being, in a sense, a special case of

granular superconductors). More relevant to this thesis, however, is that the above model

is a remarkably accurate description of a cold atomic gas in an optically induced periodic

potential, as demonstrated by Jaksch et al. [15].

Looking at this model on its own terms, there are two obvious limits. For U = 0, we have the lattice analog of an ideal Bose gas and, therefore, condensation in the lowest- lying eigenstate of HBHM. A small but nonzero value of U perturbs this result, however

it is insuﬃcient to destroy the condensate [16]. On the other hand, taking all tij = 0,

the eigenstates of HBHM have a ﬁxed, integer number of bosons at each lattice site and clearly do not describe a condensate. One might question if this state is stable against

small, nonzero tij and, indeed, it is. This is the celebrated incompressible “Mott insulator.” In Fig. 1.1, I also plot a zero-temperature mean-ﬁeld phase diagram as a function of the

chemical potential and hopping strength, showing where these phases appear. The mean-

ﬁeld method used here is expanded on in Chapter3.

1.3 Synthetic gauge ﬁelds

As the main thrust of this thesis involves cold atoms interacting with static gauge ﬁelds,

it will be useful to have some understanding of how these gauge ﬁelds are created as a

matter of principle. A charged particle couples to a static magnetic ﬁeld, described by a

6 Figure 1.1: (a) Cartoon representation of bosons in an optical lattice. Each atom can tunnel between neighboring potential wells with an amplitude −t, while two bosons can occupy the same well with an energy penalty U. (b) Mean-field phase diagram of the Bose-Hubbard model. From this contour plot of the condensate order parameter, we observe large domes, the “Mott lobes,” where the condensate vanishes and an incompressible commensurate solid resides. For sufficiently large t, however, these states give way to a superfluid ground state.

vector potential A which satisﬁes B = ∇ × A through the “minimal coupling” substitution

(throughout, I will use natural units so that ~ = c = 1)

π = p − qA (1.4) representing the effect of the field on the canonical momentum π that enters the single particle hamiltonian. However, we are studying neutral atoms. Because q = 0, there is no coupling, and so the effects of orbital magnetism, such as the integer quantum Hall effect for fermions, or the creation of quantized vortices and vortex lattices in superfluids, seem irreparably out of reach. In the following sections, I will provide an elementary introduction to the methods by which the momenta of cold atoms can be modified, through the acquisition of a Berry’s phase or through optical processes that impart momentum to an atom in a way that depends on its internal state, to adopt the above form.

7 1.3.1 Berry’s connection and the adiabatic principle

We will build up the theoretical framework behind synthetic gauge ﬁelds (following somewhat the discussions in [17, 18]) by considering a single atom with some internal degree of freedom (for concreteness, this is usually the hyperﬁne spin state). Now, imagine that we engineer a spatially varying, but time-independent1, coupling of these internal states with a general matrix U(r) , so that the atom is described by the Hamiltonian

1 H = − ∇2 + V (r) + U(r) (1.5) 2M I where the kinetic energy operator is, of course, agnostic to the internal state and V counts the part of the potential energy that is diagonal in the space of internal states. We can diagonalize U in this space, generating a spatially varying set of eigenvectors |χm(r)i (the “dressed states”), and then we can expand the state vector of the atom as

X |Ψ(r, t)i = ψm(r, t)|χm(r)i (1.6) m The time evolution of this vector is governed by the Schr¨odingerequation,

∂ X X 1 i ψ (r, t)|χ (r)i = − ∇2 + V (r) + U(r) ψ (r, t)|χ (r)i (1.7) ∂t m m 2M I m m m m

Acting on this from the left with hχ`| we obtain

∂ 1 X i ψ (r, t) = (V (r) + ) ψ (r, t) − hχ (r)|∇2ψ (r, t)|χ (r)i (1.8) ∂t ` ` ` 2M ` m m m where ` is the eigenvalue of U associated with |χ`i and the last term of the right-hand side

1Time dependence in the coupling is also easily included in this formalism and, indeed, some schemes for generating synthetic gauge ﬁelds rely on it, however, it is omitted here for a smoother presentation of the main ideas.

8 remains to be evaluated. Suppressing the dependence on r for clarity, we can write

2 2 2 hχ`|∇ ψm|χmi =hχ`| ∇ ψm |χmi + 2 (∇ψm) · |∇χmi + ψm|∇ χmi

where, in the ﬁnal line, I have introduced Berry’s connection A`m = ihχ`|∇χmi. As of now, this is simply a formal deﬁnition that will be expanded on momentarily. First, however, we

can substitute this result into Eq. (1.8) above and ﬁnally obtain

∂ X 1 i ψ (r, t) = (p − A)2 + W + (V (r) + ) δ ψ (r, t) (1.10) ∂t ` 2M I `m `m ` `m m m Already this equation has a formal similarity to what we were trying to achieve: we have expressed something that looks like a Schrödingerequation for the wavefunction of a single spin component in terms of an effective hamiltonian with an apparent vector potential A and a scalar potential, ! 1 X W = h∇χ | · |∇χ i − A · A (1.11) `m 2M ` m `k km k Nonetheless, we look at the right-hand side and notice that this equation actually couples all of ψm(r, t). All we have accomplished so far is a formal rearrangement of terms. The key ingredient that allows us to proceed further is the adiabatic theorem. The internal state behaves as a “fast” degree of freedom compared to the “slow” motion of the atom; thus, if the atom is prepared in one of the local eigenstates of U(r), it will necessarily stay in the corresponding eigenstate as r is slowly varied, so long as the energy separating this state from the other eigenstates is sufficiently large. Heuristically, we demand an energy gap ∆ 1/T , where T is a characteristic timescale for the motion of the atom. This can be further expanded to the statement that the internal state will stay in some manifold of

9 nearly degenerate states, so long as the gap between that manifold and other manifolds of states stays suﬃciently large. We will require the use of this expanded case for non-abelian gauge ﬁelds, but, for a moment, let us consider the consequences of the simpler version for

Eq. (1.10).

Suppose now that the atom is prepared with an internal state |χm(r)i that is well separated energetically from all other states at all r. We then have ψ`6=m(r, 0) = 0 and the adiabatic theorem demands that these coefficients remain zero for all times. Now Eq. (1.10) is no longer a set of coupled differential equations for a set of spinor coefficients, but a single

Schr¨odingerequation for the time evolution of a spinless particle

∂ 1 2 i ψ (r, t) = p − A˜ + W˜ + (V (r) + ) ψ (r, t) (1.12) ∂t m 2M m m

We can then interpret this as an eﬀective vector potential A˜ = Amm and scalar potential

W˜ = Wmm arising from the adiabatic elimination of excited states. We know that a charged particle moving in a vector potential acquires an Aharonov-Bohm phase [19, 20]; our neutral atom acquires an Aharonov-Bohm phase in the same way, and this phase is precisely the geometric phase that it accumulates under adiabatic transport in Berry’s analysis [21].

As an aside, this is a very general outline and in fact many “natural” physical systems can also be well understood through an appreciation of the above framework. In metallic magnets with non-coplanar spin ordering, the adiabatic elimination of the electron spin- state antiparallel to the magnetic texture generates a synthetic orbital magnetism which underlies the so-called topological Hall eﬀect [22]. In the atomic physics context, it was actually appreciated quite shortly after the realization of a BEC that the inhomogeneity of the magnetic trapping potentials inherent to the experiments imposed a nontrivial Berry’s phase for the condensate that could, in turn, be interpreted as coupling to a gauge ﬁeld

[23].

Now if we imagine that, rather than tracking a single isolated spin eigenstate, we follow some set of nearly degenerate states M, then Eq. (1.10) can be projected into that sub- ˜ space and, in turn, describes the coupling to a matrix-valued vector potential A`m. If the component matrices A˜x, A˜y, A˜z are non-commuting, then this is called a non-abelian gauge

10 ﬁeld. ˜ It is crucial at this point to clarify explicitly that the A`m that appears in the projected hamiltonian is the projection of the general matrix A`m into the subspace M,

˜ X A = A`m|`ihm| (1.13) `,m∈M which, however, does not mean that we can take the elements A`m to be zero for states outside this subspace. We will see in just a moment why this is so important.

We are now in a position to ask again what made the adiabatic elimination of states so important. It certainly seems crucial for generating the scalar-valued gauge potential, but could we not interpret any multicomponent problem, by taking M to include all of the internal states, as being coupled to a non-abelian gauge field? The answer is, of course, yes, but now we must recall that a gauge potential can only have physical consequences when the corresponding gauge invariant field strength is non-vanishing. The rule for constructing this field strength, also called the Berry curvature B, is [24]

1 h i B = F ,F = ∂ A˜ − ∂ A˜ − i A˜ , A˜ (1.14) i 2 ijk jk jk j k k j j k or, B = D × A˜ ≡ ∇ − iA˜ × A˜ = ∇ × A˜ − iA˜ × A˜ (1.15)

The A˜ × A˜ term might be unfamiliar; it obviously vanishes for the abelian gauge ﬁelds of electromagnetism. As a consequence, a spatially uniform abelian gauge ﬁeld gives B = 0.

On the other hand, a spatially uniform non-abelian gauge ﬁeld can have a nonzero B, arising solely from this term. We can also simplify the above expression further, as

˜ ∇ × A`m = ih∇χ`| × |∇χmi + ihχ`|(∇ × ∇)|χmi (1.16)

The second term on the right-hand side is zero and, for the ﬁrst term, we can insert a

11 complete set of states to get

˜ X ∇ × A`m = i h∇χ`|χni × hχn|∇χmi n X X = −i hχ`|∇χni × hχn|∇χmi = i A`n × Anm (1.17) n n

where I have made use of the identity h∇χ`|χmi + hχ`|∇χmi = 0 that follows from the

gradient of hχ`|χmi = δ`m. Notice that the basis of M is generally not complete; it is the unprojected A`m that appear in the ﬁnal expression. The matrix elements of the curvature are then

X X X B`m = i A`n × Anm − i A`n × Anm = i A`n × Anm (1.18) n n∈M n/∈M This expression makes it explicit that the adiabatic projection is ultimately required to generate a nonzero curvature, which would be impossible if M spanned the whole space of internal states. Similar algebra can also be used to show that the scalar potential also vanishes unless some states are projected out,

1 X W˜ = A · A (1.19) `m 2M `n nm n/∈M

1.3.2 Raman induced spin-orbit coupling

Several theoretical proposals exist for generating various types of spin-orbit coupling specifically within the framework above. In this section, however, I will describe the implementation designed by the NIST group [25, 26, 27, 18]. Although their method is not an obvious application of the presentation in the previous section, it is presently state-of-the-art for experiments, and so it is worth describing in detail. This approach has also been well described in several recent reviews, for example [28, 29, 30].

Ultimately the experiment rests on the application of a proposal by Higbie and Stamper-

Kurn [31, 32] to generate a momentum-space potential by coupling two internal states through a Raman transition (a two-photon process illustrated in Fig. 1.2) so that a change in the internal state is accompanied by a momentum boost k. Formally, this can be written

12 Figure 1.2: Schematic implementation of the Raman scheme for spin-orbit coupling. (a) Experiment geometry. Two counter propagating lasers alongx ˆ impinge on an optically trapped cloud of cold atoms that are also in a Zeeman ﬁeld alongz ˆ, which deﬁnes the spin quantization axis. (b) Schematic level diagram. The Raman process is a two-photon process, wherein an atom in state |ai absorbs a photon from beam 1 (and acquires a momentum boost k1 and energy ω1) but immediately de-excites through stimulated emission of a photon into beam 2 and falls to state |bi, with a total momentum boost k = k1 − k2. The reverse process occurs as well, |bi → |ai accompanied with a momentum boost of −k. The lasers are detuned from the excited state by ∆, and detuned from the Raman resonance by δ.

as a coupling matrix in the space of two internal states |ai, |bi as

Ω V (r, t) = e−i(k·r−ωt)|aihb| + h.c. (1.20) 2

Consider now the following geometry: two Raman beams, with frequencies ωR and

ωR + ∆ωR are oriented respectively along the +ˆx and −xˆ directions with wavenumbers 2π k = ± λ xˆ. In the NIST experiments on alkali atoms, these two lasers couple the hyperﬁne

process. A bias Zeeman ﬁeld alongy ˆ splits these states by the Zeeman energy ωZ and, thus,

the detuning from the Raman resonance is δ = ∆ωR −ωZ , while the quadratic Zeeman effect splits the |1, 1i state off with a sufficiently large energy that this state can be neglected.

With these ingredients, and using the above coupling, we can write a hamiltonian   2 k + ωZ Ω e−i(2kRx−∆ωRt) H =  2M 2 2  (1.21)  2  Ω i(2kRx−∆ωRt) k ωZ 2 e 2M − 2

13 ∂ P and seek solutions to the Schr¨odingerequation i ∂t φa(r, t) = b Habφb(r, t). First, we can ˜ P eliminate the time and space dependence by a unitary transformation φa = b Uabφb, with

  i exp + [2kRx − ∆ωRt] 0 U =  2  (1.22)  i  0 exp − 2 [2kRx − ∆ωRt] so that     ˜ ˜ ∂ φa ∂ φa i   = U †HU − iU † U   (1.23) ∂t  ˜  ∂t  ˜  φb φb ˜ ˜ or, in other words, the φa, φb are governed by a time-independent, translation invariant hamiltonian

∂ H˜ = U †HU − iU † U ∂t   k2+k2 2 y z + (kx+kR) + ωZ −∆ωR Ω =  2M 2M 2 2  (1.24) 2 2 2  Ω ky+kz (kx−kR) ωZ −∆ωR  2 2M + 2M − 2

Finally, after writing H˜ in terms of Pauli matrices (and recalling that δ = ∆ωR − ωZ ), we can rotate our spin axes {σx, σy, σz} → {σz, σx, σy} to get a familiar expression,

1 1 δ Ω H = k2 + (k + k σ )2 − σ + σ (1.25) s.o. 2M ⊥ 2M x R y 2 y 2 z

This one-dimensional spin-orbit coupling is quite easy to analyze. In the absence of the σz term, the spin eigenstates are just the eigenstates of σy. However, even for nonzero Ω this remains approximately true as long as |kx| Ω, while the eigenstates are essentially σz eigenstates for |kx| Ω. The spectrum has a “double-well” structure (see Fig. 1.3) and is symmetric around kx = 0 for δ = 0. This fairly simple structure nonetheless leads to several exotic consequences, such as an anisotropic critical velocity for superﬂuids [31, 33]

(which can be traced to the explicit breaking of Galilean invariance [34]), a zoo of ordered quantum and thermal phases [25, 35], zitterbewegung of BECs in this Dirac-like spectrum

[36, 27], and a semiclassical spin Hall eﬀect in a 87Rb BEC [26]. In the presence of attractive s-wave interactions, this hamiltonian also becomes reminiscent of one recently engineered in strongly spin-orbit coupled semiconductor nanowires to probe the possibility of Majorana

14 modes [37].

To compare with the “strongly” spin-orbit coupled solid-state systems pointed out earlier in the chapter, there is a natural dimensionless measure of the SOC produced by this scheme.

The spin-orbit coupling leads to an energy scale that is simply the recoil energy of the applied

2 kR Raman beams, ER = 2M . Most of the present experiments that apply this Raman induced 87 SOC to Rb condensates use lasers with wavelength λ ∼ 800 nm, so ER ∼ h × 3.5 kHz. Since these experiments are performed on an optically trapped condensate, it is interesting

to compare with the trapping frequency along the SOC direction, which is typically on the

order ωx ∼ 100−200 Hz. Together, this implies that the SOC energy in present experiments is on the order of 20 − 30 times larger than the trap energy.

Figure 1.3: Spectrum of the hamiltonian for Raman-induced one-dimensional spin-orbit coupling as a function of kx, with ky = kz = 0. The spectrum is plotted for (a) detuning δ = 0 and (b) δ 6= 0. The different contours correspond to different effective Zeeman couplings Ω. When Ω = 0 the two bands cross, giving rise to a one dimensional analog of a Dirac point. The double-well structure is eliminated entirely above a critical coupling Ω > 4ER.

1.3.3 “Realistic” Rashba spin-orbit coupling

Although there are currently no reported experimental implementations of Rashba (or otherwise isotropic) spin-orbit coupling, there appear to be no fundamental barriers to it.

Nonetheless, there are the standard variety of nontrivial technical barriers such as heating,

15 collisional losses, and so on, and thus many proposals have been made to alleviate these issues. Here I review a recent proposal by Campbell et. al [38, 18] for implementing spin-orbit coupling speciﬁcally of the Rashba type, which is particularly promising as it represents a minimal extension of the Raman scheme from the previous section.

The proposal begins with a set of N ≥ 3 internal states, cyclically coupled through

P 1 Raman transitions j Ωj+1,j|j + 1ihj|, with matrix elements Ωj+1,j = 2 Ω exp (ikj · x) (the time dependence is eliminated as in the last section by working in a co-rotating frame).

Cyclic coupling means that “periodic boundary conditions” |N + 1i = |1i are implied. The P momenta kj are chosen such that j kj = 0, so that the atom does not acquire any net momentum during a cycle. Written in second-quantized operators, this coupling leads to a hamiltonian

N Z d2k X k2 Ω h i H = φ† φ + φ† φ + h.c. (1.26) (2π)2 2M j,k j,k 2 j+1,k+kj j,k j=1

which can be simpliﬁed using the transformation φj,k → ϕj,k−Kj , where the vectors Kj are

deﬁned through kj = Kj+1 − Kj. Making this substitution directly yields

N Z d2k X k2 Ω h i H = ϕ† ϕ + ϕ† ϕ + h.c. (1.27) (2π)2 2M j,k−Kj j,k−Kj 2 j+1,k+kj −Kj+1 j,k−Kj j=1 Z 2 N 2 d k X (k + Kj) Ω h i = ϕ† ϕ + ϕ† ϕ + h.c. (1.28) (2π)2 2M j,k j,k 2 j+1,k j,k j=1

which is an integral over a set of N × N matrices

1 2 Ω Ω h 0 (k) = (k + K ) δ 0 + δ 0 + δ 0 (1.29) j,j 2M j j,j 2 j,j +1 2 j+1,j

Now, notice that the oﬀ-diagonal part of hj,j0 has the form of a nearest-neighbor hopping model in the space of internal states. This matrix is diagonalized with the discrete Fourier

transformϕ ˜† = √1 exp 2πi `j ϕ† and leads to the spectrum E = Ω cos 2π` . The kinetic ` N N j ` N energy is not diagonal in this basis, however, as it depends on j through Kj. Cleverly choosing the Kj such that they are the vertices of a regular N−gon,

2πj 2πj K = −k sin xˆ + k cos yˆ (1.30) j L N L N

16 gives a hamiltonian kernel, in the Fourier transformed basis,

2 h`,`0 (k)/ER = E`/ER + k + 1 δ`,`0 + (ikx + ky) δ`,`0+1 + h.c. (1.31)

where the momenta are rescaled by kL = |Kj| and the energy unit is the recoil energy

1 2 ER = 2M kL. This already has an appealingly familiar form, but now let us specialize to 1 the case N = 3. Now there are three dressed states ` = 1, 2, 3 with E1 = E2 = − 2 Ω and

E3 = Ω. If we eliminate the state with ` = 3 (i.e., we assume Ω ER), then we are left 1 with a degenerate two-level subspace – a pseudo-spin 2 – which obeys to zeroth order in

ER/Ω the Rashba hamiltonian

2 H = k I + (σxky − σykx) (1.32)

The remainder of [38] analyzes the higher-order corrections to this and outlines realistic parameters for imposing Rashba spin-orbit coupling on a BEC of 87Rb using the F = 1 and

F = 2 hyperﬁne manifolds.

I have now provided the theoretical framework that underlies artificial gauge fields, and briefly reviewed the experimental implementation of “abelian” (one-dimensional) spin-orbit coupling as well as a realistic proposal which minimally extends the Raman scheme to

Rashba spin-orbit coupling. In the following chapters, I move on to an analysis of what spin-orbit coupling does. In the next chapter, I consider the effects of Rashba SOC on a Bose gas in the continuum and, in the following chapter, in the presence of an optical lattice. In both chapters, the central conclusions reveal novel interaction effects that arise because of the nontrivial properties of spin-orbit coupled single-particle states. The final chapter explores an exotic approach to the question of how bosons condense in a flat-band spectrum (as is the case for bosons with Rashba SOC), but in a quite different model.

17 Chapter 2 Exotic Bose matter from nonabelian gauge fields

2.1 Bose-Einstein condensation with spin, and with spin-orbit coupling

Although multicomponent Bose condensates were previously studied in detail in the context of superfluid helium and unconventional superconductivity, the field of “spinor” BECs was expanded significantly by T.-L. Ho in 1998 [39] in response to experiments being carried out by the MIT group using optical trapping methods [40]. The bosons used for real cold atom experiments are not simply point particles, but have internal magnetic degrees of freedom

(ﬁne and hyperﬁne states) that are neglected in the standard theory of Bose condensation.

In the cold atom context, this is justiﬁed as these internal degrees are “frozen out” by magnetic atom traps. However, in the presence of an optical trap, we must keep some subspace of these internal magnetic states; this in turn aﬀords the condensate the ability to support a vector order parameter, to break ever more exotic symmetries (beyond the traditional U(1) associated with number conservation), and to have more complicated and even fractionalized topological excitations (e.g., half -quantum vortices [41]).

A prescient early study of spin-orbit coupling in spinor Bose gases, motivated by dark state schemes for implementing non-abelian synthetic gauge ﬁelds, was carried out by

Stanescu et al. [42]. They considered a generally anisotropic SOC, which includes both the one-dimensional abelian SOC and the Rashba interaction as limiting cases, and then

18 calculated the critical temperature for Bose condensation as a function of the anisotropy, noting that this Tc formally vanishes in the pure Rashba limit. This is a density of states effect – for the three dimensional Bose gas, the Rashba coupling yields a constant low-energy density of states, reminiscent of a free two-dimensional gas where Tc also vanishes. Not long after, and in fact almost simultaneously with the NIST experiment [25], Ho and Zhang carried out a detailed study of the anisotropic Raman-induced SOC [43]. In addition to the proposal of a “generalized adiabatic” scheme which conceptually bridges the adiabatic and

Raman approaches to implementing artificial gauge fields, their work also predicted precisely the mean-field phases which were measured in [25]. In particular, there are two types of condensates: (1) the bosons condense into a single minimum along kx, or (2) the bosons condense into a superposition state with equal weight in the two minima. In this latter case, interference between the spin wavefunctions at the two minima leads to a spin-modulated or “stripe” state.

It is interesting to compare this with the ground state suggested in [42], which is a fragmented condensate. It consists of a superposition of an N boson condensate in one minimum, and an N boson condensate in the other, thus the name |N00Ni state. This was also the state originally predicted by Higbie and Stamper-Kurn [32] after noting that the

p2 combination of a real-space double well potential and harmonic kinetic energy 2m has an intriguing duality in the Raman scheme, which describes a real-space harmonic conﬁning potential and a double-well in momentum space. Exact diagonalization of two-component

[44] and three-component bosons [45] with Rashba SOC have also suggested fragmented ground states. This sort of macroscopic superposition is, however, known to be very fragile

[46, 47], and has not been observed.

Also shortly after the Stanescu work, papers by Wang et al. [48] and Wu et al. [49] further addressed the question of condensation in the Rashba limit. Even if Tc vanishes, the zero-temperature state could still host a condensate, and under the assumption that a condensate exists, these studies ﬁnd that there are also only two classes of condensate which are very similar to the anisotropic result [43, 50]. For Rashba SOC, however, choosing a single energy minimum to condense in breaks an additional U(1) symmetry from the

19 continuous degeneracy of single-particle states on the “Rashba ring” (see Fig. 2.1), as does choosing two opposite momentum minima to construct the striped condensate. No other solutions (for example, one might imagine an equal superposition of states from the entire ring) are ever found within the mean-ﬁeld approach of these references. Zhou and Cui [51] have further emphasized the instability of the noninteracting condensate, and also that the perturbative treatment of interactions in the presence of a ﬂat band is quite dangerous. They propose that in the presence of interactions, a condensate can exist even where the noninteracting condensate does not, and that the interacting condensate is again fragmented, although their mechanism does not predict fragmentation in the anisotropic limit.

Also in Wu [49] and in following work by other authors [52, 53] the effect of a trapping potential was investigated, and a variety of interesting mean-field condensate states beyond the plane-wave or stripe were identified, as the combination of spin-orbit coupling and a non-uniform spatial potential lead to exotic spatial arrangements of spin density. The single particle problem will be reviewed in Section 2.2.2, and has rather interesting consequences.

For reasons discussed in this Chapter, Wu has called the spectrum “generalized Landau levels” [54].

Perhaps the most exotic suggestion to date, however, is the proposal by Sedrakyan et al. that the ground state in two-dimensions is fundamentally fermionic [55]. Fermion wavefunctions have nodes when two particles approach the same position, as a consequence of the Pauli exclusion principle. Although this allows fermion states to conveniently avoid paying a short-range interaction energy, fermions must also generically occupy high kinetic energy single-particle states, also because of exclusion. This kinetic energy cost vanishes if the low-energy density of states is suﬃciently large to accommodate the fermions, as happens in one spatial dimension. In turn, the ground state of interacting bosons in 1D is

exactly speciﬁed by the absolute value of the wavefunction for non-interacting fermions [56].

For the two-dimensional Rashba spectrum, we also have a diverging low-energy density of

states. Sedrakyan et al. suggest that a similar mechanism occurs, and at suﬃcienttly low

densities the ground state of Rashba bosons is not a condensate but rather a composite

fermi sea coupled to a Chern-Simons ﬁeld which implements the change in statistics.

20 In this chapter we first detail the rather fascinating physics of the Rashba hamiltonian on its own, but particularly with an eye to cold bosons. In this introduction I have described several recent stabs at the physics of interacting bosons with a Rashba spectrum, with the intention of pointing out that there are a variety of approaches leading to a variety of potentially inconsistent but nevertheless intellectually stimulating results. Thus, after describing the general model, I present some preliminary progress in understanding interacting Rashba bosons beyond mean-field theory. Unless otherwise specified, I will usually be focusing on two spatial dimensions (the third being locked out by a confining potential larger than any other energy scale in the hamiltonian). This is mostly for ease of visualization, as well as the expectation that quantum effects become more pronounced in lower dimensions.

2.2 The Rashba hamiltonian

Substantial experimental and theoretical eﬀort has gone into studying spin-orbit coupled

BECs with anisotropic SOC. In particular, most experiments are done in the limit described above, where only one component of the momentum is coupled to spin. These eﬀorts have been exciting in their own right, but a “holy grail” for experiments is to eventually achieve the isotropic Rashba coupling. As noted Chapter1, the Rashba hamiltonian was originally proposed in the context of 2D electron gases trapped at asymmetric semiconductor interfaces, but its simple structure also shows up in a variety of modern contexts, for example in describing the surface states of topological and Chern insulators. It is precisely this universality – the Rashba hamiltonian is essentially the Dirac hamiltonian, but it shows up for non-relativistic electrons – that has driven such intense interest in adding Rashba spin-orbit coupling to the list of quantum phenomena that can be simulated with cold atoms.

In the continuum, the Rashba hamiltonian has a simple form,

p2 H = + λ (σ p − σ p ) (2.1) R 2M I x y y x with σ denoting the Pauli matrices and I the 2 × 2 identity matrix in spin space. Recalling

21 the discussion in Chapter1, it is also sometimes convenient to write this at the hamiltonian of a free particle coupled to a spatially-uniform but nonabelian gauge potential,

(p − A)2 1 H = − A2, A = Mλ (σ xˆ − σ yˆ) (2.2) R 2M 2M y x

It is helpful to begin our analysis by identifying some of the symmetries of this hamiltonian.

• It is translationally invariant; the eigenstates are plane-waves with a momentum-

dependent spin wavefunction.

• It is invariant under combined spin and real-space rotation, and we can equally well

ﬁnd a set of simultaneous eigenstates of HR and total (L + S) angular momentum.

• Spatial inversion symmetry I (under which p → −p) is explicitly broken, but a

symmetry called parity is preserved; this is represented by the operator P = σzI.

This operator still takes p → −p but simultaneously takes σx → −σx, σy → −σy.

• Denoting complex conjugation as C, the hamiltonian is symmetric under time reversal,

provided by the anti-unitary operator T = iσyC. This implies that energy eigenstates come in “Kramer’s pairs,” |ψi and T |ψi.

Because of the translational symmetry we can expect solutions of the form ψk(r) =

√1 eik·rχ , where Ω is the volume of the system and χ is an as-yet undetermined 2- 2Ω k k component spin wavefunction with no spatial dependence. We can evaluate

2 1 k ik·r 1 ik·r HRψk(r) = √ I + λ (σxky − σykx) e χk = √ e χk (2.3) 2Ω 2M 2Ω so that the eigenvalues and eigenvectors are determined by the matrix equation   1 2 2M k λ(ky + ikx)   χ±,k = ±(k)χ±,k (2.4)  1 2  λ(ky − ikx) 2M k

22 which has solutions   k2 1 (k) = ± λ|k|, χ =   , φ = arctan (k /k ) (2.5) ± 2M ±,k   k y x ∓ieiφk

It is also useful to give a name to the inverse-length scale set by the spin-orbit coupling strength through ks.o. = Mλ. This is the quantity that was given by the wavevector kR in the Raman scheme. In terms of this, the spectrum can also be written ±(k) =

1 2 1 2 2M (|k|±ks.o) − 2M ks.o.. From the above we can also now see one of the general consequences of spin-orbit coupling quite clearly: the spin eigenfunctions χ±,k each represent a point on the Bloch sphere, where they correspond to spin projections s = (sin θ cos φ, sin θ sin φ, cos θ)

π with θ = π/2 and φ = φk ± 2 . Thus the spin and momentum are locked together, with the spin oriented perpendicular to the momentum. This is discussed further in the following chapter, where I address how this spectrum is modiﬁed in the presence of an additional periodic lattice potential.

2.2.1 Cylindrical coordinates

As mentioned above, the rotational symmetry of the hamiltonian ensures that Jz = Lz + Sz is conserved, and deﬁnes a good quantum number for its eigenstates,

1 J ψ (r) = m + ψ (r) (2.6) z m 2 m

This strongly constrains the angular dependence of possible eigenfunctions, and we can see immediately that they must be expressed by the spinor   eimφ fm↑(r) ψm(r) = √   (2.7) 2π  iφ  fm↓(r)e

To deﬁne the functions fmσ(r) uniquely, we can act on this state with HR and solve.

First, however, we must write HR in cylindrical coordinates, in its explicit matrix form,   1 2 −iφ ∂ i ∂ − 2M ∇ λe ∂r − r ∂φ HR =   (2.8)  iφ ∂ i ∂ 1 2  −λe ∂r + r ∂φ − 2M ∇

23 Figure 2.1: Energy spectrum E±(k) of the isotropic Rashba hamiltonian. Beneath is a contour plot of the lower band energy, with the “Rashba ring” of degenerate ground states highlighted, along with corresponding spin wavefunctions along the ring. The spectrum is perhaps most easily visualized as the solid of rotation generated by two intersecting parabola (black lines).

±iφ ∂ i ∂ where we have made use of the identity ∂x ±i∂y = e ∂r ± r ∂φ and understand that the Laplacian should be expressed in cylindrical coordinates. After making this transformation, it is easily veriﬁed that the eigenstates are   imφ im Jm(kr)e k2 ψ±,m(r) = √   , ±,m = ± λk (2.9) 2Ω  i(m+1)φ  2m ±Jm+1(kr)e

As an aside, we could have also obtained this result recalling that plane waves were already energy eigenstates, and expanding the plane wave in functions appropriate to a

24 cylindrically symmetric problem, through the identity

X m im(φr−φk) exp (ik · r) = i Jm(kr)e (2.10) m which in turn yields   1 X 1 √ m im(φr−φk)   ψ±,k(r) = i Jm(kr)e   (2.11) 2Ω m ∓ieiφk   imφr 1 X Jm(kr)e √ m −imφk   = i e   (2.12) 2Ω i(m+1)φr m ±Jm+1(kr)e and then ﬁnally we recover Eq. (2.9) by calculating

Z 2π dφ k imφk ψ±,|k|,m(r) = e ψ±,k(r) (2.13) 0 2π

Returning to the thread of the discussion, looking at the single particle spectrum in terms of angular momentum eigenstates has yielded something quite remarkable. If we consider

Landau levels from the symmetric gauge point of view, the macroscopic degeneracy of each eigenvalue can be tied to the fact that the energy does not actually depend at all on angular momentum. We have precisely the same situation here – the energy has no dependence on the angular momentum eigenvalue m. On the other hand, an important feature of the

Landau level spectrum is the cyclotron frequency gap which separates the levels from one another. Here we ﬁnd that the energy varies continuously with the radial momentum |k|, and so no such gaps exist.

It turns out however that a harmonic conﬁning potential does introduce analogous gaps, but at the expense of also introducing a weak dependence of the energy on the angular momentum. Wu has called this spectrum “generalized Landau levels,” and I will explain this in the following section. Furthermore, while the translation invariant problem is fundamentally interesting, actual experiments are most commonly done in a harmonic trap that breaks that translation invariance anyway. In this case the present description in cylindrical coordinates is a more useful starting point.

25 2.2.2 The Rashba hamiltonian with a harmonic trapping potential

Having understood the spectrum of HR in some detail, it will also be useful to understand the eﬀect of a harmonic conﬁnement, which is typically present in cold atom experiments, by including a contribution to the hamiltonian

1 H = Mω2 r2 (2.14) trap 2 T

This term explicitly breaks the translation invariance, so the eigenstates of HR + Htrap will no longer be speciﬁed by k. It does not appear that an exact analytic solution is known, and many authors have looked at the spectrum of this model by numerically calculating the expansion of the eigenstates in the well-known exact eigenstates of Htrap [52, 53, 44]. I do so as well, but the bookkeeping involved in constructing these solutions is relegated to

AppendixA. In the remainder of this section I focus instead on describing the properties of these solutions.

In Fig. 2.2 I plot the energy spectrum as a function of the angular momentum quantum number m as well as a radial quantum number n that is produced by the combined SOC and trapping potential. Looking at the spectrum itself, the intriguing features that emerge with large SOC are: (1) Well separated groups of states, distinguished by a radial index n, and split apart by an energy gap ∼ ωT . (2) Very little change in the energy as the angular momentum index m is increased, up to some large value of m, which can be arbitrarily increased by increasing γ.

The way this spectrum emerges from the usual trap states (at γ = 0) is quite diﬀerent from the true Landau levels that emerge by rotating the trap. Under rotation, in the rotating frame, the hamiltonian becomes Htrap − ΩLz. The eigenstates of this are still trap eigenstates, but large Lz states are reduced in energy, and for Ω ' ωT , the lowest energy manifold is highly degenerate, containing the largest Lz state from every excited trap level. Just looking at the spectrum, SOC appears to do something similar, except the low-energy states are not trap eigenstates at all, but superpositions of many trap states which conspire to produce this band ﬂattening.

26 I also plot the spin-resolved radial part for some of these wavefunctions in Fig. 2.3, which can be compared with the Bessel functions that appeared in Eq. (2.9). A seemingly crude approximation to the n = 0 states (the lowest generalized Landau level) is to take the states from the Rashba ring (i.e., demand k = ks.o.) and account for the trapping potential with a gaussian factor multiplying those wavefunctions.   imφ 2 J (γr/`)e − 1 ( r ) m φm,n=0(r) ≈ e 2 `   (2.15)  i(m+1)φ  Jm+1(γr/`)e

In fact looking at the Figure, this approximation does an extremely admirable job until we approach large m, where the character of the state is dominated by the harmonic conﬁnement and the Rashba-only solutions are no longer suﬃcient.

2 Figure 2.2: Energy spectrum εnm + γ /2 of the isotropic Rashba hamiltonian in a harmonic trap, in units of the trap energy ωT , for several values of the dimensionless SOC strength γ = √Mλ . The spectrum is plotted against the angular momentum index m, and each MωT curve corresponds to a set of states with identical radial index n. There are also states with m < 0, however the spectrum is symmetric, so we only plot the m ≥ 0 half. For large spin-orbit coupling, states with larger |m| come down in energy, while an approximately nωT energy penalty is associated with increasing values of the radial index.

2.3 Adding interactions

As pointed out by the authors of [48], the diﬀerent pseudospin components for our spinor bosons are, in some current experimental implementations of SOC, superpositions of atomic hyperﬁne states; because of this, the interactions between the particles could take on quite a

27 Figure 2.3: Spin-resolved n = 0 eigenstates at several characteristic m values, all at γ = 10. For small m the states are weighted near the center of the trap and dominated by the Rashba contribution. The dashed lines (largely indistinguishable from the colored, solid curves) are from Eq. (2.15). For suﬃciently large m, the approximation breaks down.

complicated form. Nonetheless, this is not a universal feature of all synthetic SOC schemes, so we will consider here (and in the following chapter) a standard representation of the usual short-ranged interaction hamiltonian,

Z X gσσ0 H = d2r ψ† (r)ψ† (r)ψ (r)ψ (r) (2.16) int 2 σ σ0 σ0 σ σσ0 It is often useful to expand the ﬁeld operators in terms of the single-particle eigenstates

† P ∗ † described above for cylindrical coordinates, i.e., ψσ(r) = m ϕm(rσ)am. The mode index m includes the angular momentum and helicity, as well as the appropriate radial quantum number. The hamiltonian then becomes

X † X † † HR + Hint = mamam + Vm1m2m3m4 am4 am3 am2 am1 (2.17) m m1m2m3m4 where Z X gσσ0 V = d2r ϕ∗ (rσ)ϕ∗ (rσ)ϕ (rσ)ϕ (rσ) (2.18) m1m2m3m4 2 m4 m3 m2 m1 σσ0 I note here that the total angular momentum conservation that has proven useful for understanding the Rashba hamiltonian to this point still holds when interactions are turned on. The Rashba hamiltonian with this interaction, in a harmonic trap, and for parameters where it is appropriate to work in the lowest generalized Landau level, has previously been studied at the level of mean-ﬁeld theory (the operators am are replaced by complex numbers

which represent their average value hami, and then the energy is minimized with respect to these values subject to some constraints) as well as exact diagonalization for few particles

28 (N = 2 and 8) in the weakly interacting limit. I now extend these prior results by building new intuitions from an exact calculation.

2.4 An exact two-body ground state

To build some initial intuition about the effect interactions will have in the many-body ground state, we can first attempt to construct a two-body ground state. To minimize the kinetic energy, both bosons must occupy a state (or linear combination of states) coming from the ring of single-particle minima, and should also occupy states with opposite momentum, so that the center of mass momentum vanishes. If we can take such a state, and, due to the short range nature of the interaction hamiltonian, find a corresponding wavefunction

Ψ(r1, r2) such that Ψ(r1 = r2) vanishes, then we will have found not only an eigenstate

but the ground state of HR + Hint. We begin by occupying opposite linear momentum states on the degenerate ring of the

lower helicity band, k and −k with |k| = ks.o.. We can characterize these instead by their angle along the ring (deﬁning a† ≡ a† ) and then make an arbitrary linear θ −,ks.o.(cos θ,sin θ) combination Z π dθ † † |Ψi = Aθaθaθ+π|0i (2.19) 0 π which yields the spin-dependent wavefunction

Z π dθ † † Ψ(r1σ1, r2σ2) = hr1σ1, r2σ2|Ψi = Aθh0|ψσ1 (r1)ψσ2 (r2)aθaθ+π|0i (2.20) 0 π

This is easily evaluated by ﬁnding the commutator

Z † X 2 ik·r † h † i ik·r a−,k = d re χ−,k(σ)ψσ(r) ⇒ ψσ(r), a−,k = e χ−,k(σ) (2.21) σ which leads to the result

Z π dθ ik·r1−ik·r2 Ψ(r1σ1, r2σ2) = Aθ e χθ(σ1)χθ+π(σ2)+ 0 π −ik·r1+ik·r2 e χθ+π(σ1)χθ(σ2) (2.22)

Now recall that the goal was to ﬁnd a wavefunction that vanishes as r1 = r2, independent

29 of spin orientation. We need to ﬁnd

Z π dθ Ψ(r1 = r2; σ1, σ2) = Aθ [χθ(σ1)χθ+π(σ2) + χθ+π(σ1)χθ(σ2)] (2.23) 0 π

and ﬁnally, upon explicitly evaluating this with the spinors in Eq. (2.5), we obtain

Z π dθ Ψ↑↑(r1 = r2) = Aθ × 2 0 π Z π dθ i2θ Ψ↓↓(r1 = r2) = Aθ × 2e 0 π

Ψ↑↓(r1 = r2) = Ψ↓↑(r1 = r2) = 0 (2.24)

Regardless of Aθ, the spin-mixed two-body wavefunctions are identically zero because of interference from the spinor part of the single-particle states. To satisfy the constraint that Aθ = Aθ+π as well as making the other two integrals vanish, it is easily seen that the

in2θ choice Aθ = e does the job for any integer value of n except n = −1, 0. What is the lesson of this two-body calculation? For the mean-ﬁeld Gross-Pitaevski solutions, the lowest-energy states host an N-boson condensate in either a linear combination of opposite-momentum states on the ring, or just a single state along the ring. No more complicated solutions are found, but these solutions also do not eliminate the interaction energy. For exactly two particles, by making use of all of the states along the ring, with

the particular phase relationship demanded by Aθ, the interaction energy can be made to vanish.

in2θ The requirement that Aθ = e is also quite intriguing. Once a particular value of n is chosen, this state appears to break time reversal symmetry. It is therefore useful to

calculate the total angular momentum carried by this state. To do so, however, it is much

more convenient to transform to the angular momentum basis that I used in the cylindrically

† P −imθ † symmetric case. The transformation is carried out by substituting aθ = m e am.I obtain Z π dθ X X |Ψi = e−imπe−i(m+`−2n)θa†a† |0i = (−1)`a†a† (2.25) π ` m ` 2n−` 0 `,m ` which allows the pair of bosons to carry any odd integer total angular momentum projection

1 1 Jz = (` + 2 ) + (2n − ` + 2 ) = 2n + 1, except the previous constraint on n requires Jz 6= ±1. 30 This also means, however that Jz = 0 is blocked; this state must break time reversal symmetry to avoid paying the interaction energy.

At this point, it is not obvious how to generalize this result to more particles, but it does indicate that the behavior of interacting bosons with spin-orbit coupling holds many surprises. Rather than pursue this further in this context, I now proceed to the next chapter, where I describe an interacting lattice system of spinor bosons.

31 Chapter 3 The Bose-Hubbard model with spin-orbit coupling

In the previous chapter, I described how isotropic spin-orbit coupling enables rather unconventional physics. This is especially true for spin-orbit coupled Bose gases, as no obvious solid-state analog exists. The appearance of a “ﬂat” single-particle energy spectrum makes the fate of Bose condensation in particular a delicate question, and the interactions between bosons in the low-energy states can become very complicated. In this chapter I will consider the eﬀects of a particularly popular route to generating tunable interactions between the atoms. The idea here is to introduce a periodic potential for the atoms to move in – an optical lattice [15, 57].

The optical lattice approach has the additional beneﬁt of stabilizing a commensurate solid (the Mott insulator) for suﬃciently large interaction strength, which can support or potentially compete with other interesting correlated states. For bosons in particular, introducing an optical lattice leads to the Bose-Hubbard model [14] described in Chapter1, with parameters t describing the tunneling rate between neighboring potential wells of the lattice and U the energy cost associated to each pair of particles occupying the same well. This model predicts a BEC to Mott insulator transition, and this has now been clearly observed in experiment [58]. Starting with the work of [59], the study of spinor bosons was extended to the optical lattice. Magnetic order deep in the Mott limit [60], as well as the Mott transition [61], were explored extensively. On a square lattice, in the absence of frustrated tunneling terms, there is no sign problem, and detailed quantum Monte

32 Carlo simulations have been performed [62]. These studies yield magnetically ordered Mott phases, elucidate the role that this magnetism plays on the Mott transition, and uncover rather exotic superfluid phases, such as the “counter-flow” superfluid [63, 64]. A goal of this chapter therefore is not merely to supplement the ideas presented in the previous chapter with a lattice potential, but also to explore how standard results for spinor bosons on the lattice are modified by introducing a spin-orbit coupling.

There is in fact a quickly growing literature on ideas for experimental realization of the models envisioned in this chapter, and I describe some of the more well-studied routes.

The most obvious one is to simultaneously apply the optical potentials that give rise both to spin-orbit coupling and the lattice [65], and then to construct the spin-orbit coupled

Wannier orbitals and tunneling matrix elements between them as carried out by [66, 67].

We will describe the physics of this in the following section. A less obvious route, but one which may also have some advantages in terms of lower loss rates and less heating, makes explicit use of more complicated lattice potentials and light-assisted tunneling processes, generalizing the scheme of [68] (which proposes an experimental configuration to simulate the Hofstadter problem) to non-abelian gauge coupling. More discussion of this approach can be found in, e.g., [69, 70]. Although it is a very different kind of spin orbit coupling, there have been related proposals that use similar principles to generate an abelian but spin-dependent gauge field in two-dimensions [71]. Finally, it has also been proposed to produce an optical lattice and, through careful control of the polarization of the optical lattice lasers, have the optical lattice also provide a spatially varying (with the same period of the lattice) spin-dependent potential [72].

Regardless of how it is ultimately realized in an experiment, in this chapter I investigate the general properties of the Rashba spin-orbit coupled two-species Bose-Hubbard model: i.e., the quantum phases it supports at zero temperature, and the transitions between them.

While many of the results presented here were originally demonstrated in [73], a number of complementary investigations have been carried out by other authors [74, 66, 67, 75, 76, 77] and I draw comparisons with their work when appropriate.

33 3.1 Spin-orbit coupling in a periodic optical potential

The microscopic degrees of freedom that will ultimately be described by our spin-orbit

† coupled Bose-Hubbard model are not the boson ﬁeld operators ψσ(r) we have been using † up to here, but instead a set of lattice bosons biα, associated with a corresponding set of Wannier basis functions localized near each potential well (labeled by i), and a two-

component pseudospin that need not be identical to the original (e.g. hyperﬁne) degree

† of freedom. In this section I will demonstrate heuristically how the biα are obtained. To begin, I write down the contributions to the free particle hamiltonian coming from the

optical lattice and the spin-orbit coupling individually.

For the remainder of this chapter I only consider an ideal square lattice, although gen-

eralization to more complicated lattices is straightforward. The optical lattice introduces a

potential [15, 57]:

2 2 Vo.l.(x, y) = V0 sin (ko.l.x) + sin (ko.l.y) (3.1)

2π π with ko.l. = λ = a representing the wave vector of the applied laser which in turn leads to a lattice spacing a. This also sets a natural energy scale for the problem, the recoil energy

1 2 of the lattice Eo.l. = 2M ko.l.. We also consider a spin-orbit coupling which, for convenience, takes the Rashba form HR described in the previous chapter,

1 H = − ∇2 − iλ (σ ∂ − σ ∂ ) (3.2) R 2M x y y x so that the full single particle hamiltonian is given by

Z X 2 † H = d r ψσ(r)hσσ0 ψσ0 (r) (3.3) σσ0 with the matrix

hσσ0 = Vo.l.(r)δσσ0 + (HR)σσ0 (3.4)

In general, the lattice periodicity implies that the eigenfunctions of Eq. (3.4) will be

Bloch states. We can adopt our desired local picture by Fourier transforming these Bloch states which generates a set of well-localized Wannier states. Then, the tunneling amplitudes

34 can be calculated either from the matrix elements of H between these Wannier states or

ﬁtting the tight-binding bands to the spectrum determined by a band-structure calculation

[66, 67], with both approaches requiring somewhat detailed numerics. On the other hand, all

of the necessary physical insight can be determined from a straightforward approximation

which generalizes a similar and well-accepted approximation found in the original work of

[15].

Near the minima of the lattice potential, the atoms essentially experience harmonic

2 2 conﬁnement: Vo.l. ∝ (x − xi) + (y − yi) . For a suﬃciently deep lattice, I can reliably approximate the lowest-energy Wannier states at each site with the ground states for a −1/4 spin-orbit coupled particle in a single harmonic well with trap length ` = 1 V0 . ko.l. Eo.l. In the previous chapter I have already described the solution to this problem. In the present

approximation we then have a two component degree of freedom (a Kramer’s doublet) at

1 each lattice site, given by the total angular momentum Jz = ± 2 states:   2 J (γr/`) − 1 r 0 ∗ 2 ( ` )   φ+ 1 (r) ≈ e , φ− 1 (r) = iσyφ+ 1 (r) (3.5) 2  iφ  2 2 J1(γr/`)e where γ = Mλ` is the dimensionless spin-orbit coupling strength familiar from the previous chapter. It is worth emphasizing that this deep lattice approximation corresponds to a much narrower trap (i.e., a higher energy scale) than the physical traps. As such, it will be typical here for γ ∼ ks.o. ∼ 1. This is also a very important conceptual point. For too large ko.l. γ, these two states become nearly degenerate with higher total-angular-momentum states, and the notion of a two component pseudospin for each site must be abandoned entirely.

Coming back to the central point, the character of the localized boson operators is not given by the bare spin, but rather by the above wavefunctions that account for the competition between the spin-orbit and lattice scales.

Z † 2 † biα = d r φασ(r − Ri)ψσ(r) (3.6)

Because these wavefunctions are spatially anisotropic in their phase – the relative eiθ be-

tween the up and down spin components – the hopping amplitudes for processes that ﬂip a

35 pseudospin will share this spatial anisotropy identically.

Lastly, it is worth recalling that in current experiments using the Raman scheme, an additional Zeeman term appears in the single-particle hamiltonian, proportional to the Rabi frequency of the Raman beam that implements the SOC [25]. In the lattice context, the consequences of such a term have been considered by [75], and including such a term in the above analysis is straightforward as well. Nonetheless, as we are interested in singling out the eﬀects of spin-orbit coupling speciﬁcally, we consider here only time-reversal symmetric models.

3.2 Model hamiltonian

The previous section motivates how our microscopic model can result from a speciﬁc experimental implementation. However, we can write it in a rather aesthetically appealing form that turns out to be quite robust regardless of the speciﬁc details:

X 1 X H = H + H = R (x, x0)b† b + h.c. + U b† b† b b (3.7) K U αβ xα x0β 2 αβ xα xβ xβ xα hxx0i,αβ x,αβ where R(x, x0) = −teiA·(rx0 −rx) implements minimal coupling (the “Peierls’ substitution”) to a non-abelian gauge ﬁeld A = (θxσy, θyσx, 0). Indeed, this appears to be exactly what we would naively have written down as the lattice discretization of the continuum model considered in the previous chapter. Nonetheless,

† appreciating the character of the local states that the biα operators describe will be crucial to interpreting experiments.

3.3 Weak-coupling approximation

It will prove useful to ﬁrst understand what this model describes in the U = 0 limit.

HK remains lattice-translation invariant, so after replacing the boson operators with their

36 Fourier transforms, b† = √1 P eik·rx b† , one is left with a block diagonal hamiltonian, xα N k kα   X bk↑ H = −2t † † h   (3.8) K bk↑ bk↓ k   k bk↓

with

hk = (cos kx cos θx + cos ky cos θy)I + (sin ky sin θy)σx + (sin kx sin θx)σy (3.9)

± Using well-known properties of the Pauli matrices, the energy spectrum Ek and spin ± eigenstates χk (the eigenvalues and eigenvectors of hk) are easily evaluated as: q ± 2 2 Ek = −2t (cos kx cos θx + cos ky cos θy) ± (sin ky sin θy) + (sin kx sin θx) (3.10)

  ϑ cos k sin θ sin k χ± =  2  , ϑ = π, ϕ = arctan y y (3.11) k   k k iϕk ϑk sin θx sin kx ∓ie sin 2 which have an appealing formal similarity to the results absent the lattice.

To get a better sense of these results, we can observe that at each point in momentum space, the operator hk = hk ·σ deﬁnes a vector in the xy plane through the coeﬃcients of σx and σy. This is formally a momentum space Zeeman coupling, and so the spin eigenstates in the lower (higher) energy band are parallel (anti-parallel) to this vector. That is, the two bands are distinguished by their so-called helicity quantum number, with ± referring to the two eigenvalues of hk · σ/|hk|. As long as θx, θy 6= 0, interesting characteristic features of this energy spectrum are the four degenerate energy minima in the lower band and four

Dirac points at the time-reversal-invariant momenta. Finally, we can write boson operators in the helicity basis, which will be useful as the low-energy physics should be dominated by † P ± † the lower-helicity band. These operators are bk± = α χk α bkα. The energy spectrum is − plotted in Fig. 3.1, while the resulting density of states and the spin projection of χk are shown in Fig. 3.2.

Now we will restrict our attention to the Rashba limit, where θx = −θy ≡ θ. In this case, √ the minima appear at km = (±k0, ±k0) where tan k0 = (tan θ)/ 2, as shown in Fig. 3.1. The four-fold degeneracy of the single-particle ground state allows us to write down a large

37 Figure 3.1: Energy spectrum E(k) of a single particle in a 2D square optical lattice with isotropic spin orbit coupling, θ = π/3. Below it is a contour plot of the lower band energy, showing low energy states at ﬁnite momenta; in particular the four minima at (±k0, ±k0) are highlighted, along with the corresponding spin wavefunction. Also notable are the Dirac cones at k = (0, 0), (0, π), (π, 0), (π, π).

class of degenerate many-body ground states for the non-interacting bosons,

X cN ,N ,N ,N N1 N2 N3 N4 |Ψi = √ 1 2 3 4 b† b† b† b† |0i (3.12) k1− k2− k3− k4− N1!N2!N3!N4! N1+N2+N3+N4=N

(N+3)! with N!·3! coeﬃcients which are arbitrary up to a normalization constraint. However, it is expected that when weak interactions are taken into account, a unique ground state will be

38 Figure 3.2: (Left) Band-resolved density of states for the spectrum plotted in Fig. 3.1. The low-energy singularity is reminiscent of the continuum DoS, while the higher energy singularity comes from the “saddles” at the Brillouin zone edge. The Dirac points are buried, since the bands overlap, however the characteristic linear behavior can be seen when the contributions to the DoS are viewed separately. (Right) The spin expectation − value s = (hσxi, hσyi, hσzi) in the negative-helicity-band wavefunction χ (k). This is again reminiscent of the continuum Rashba eigenstates, however the circulation of the spin wavefunction is counter-clockwise for contours that enclose k = (0, 0) or (π, π) (in the usual counter-clockwise sense), and clockwise for those that surround k = (0, π) or (π, 0). These two species of Dirac points are therefore topologically distinct from one another, with winding number ±1, respectively.

selected. To explore this, we ﬁrst assume that the bosons condense into one single particle

state (fragmented condensates can be ruled out at this level of approximation on very

general grounds [47]) with the generic wavefunction ϕ(r) = P4 c eikm·rχ− , where the m=1 m km cm are a set of normalized complex variational parameters. The optimal set of cm minimizes the interaction energy EU [{cm}] ≡ hΦ|HU |Φi, and fully characterizes the properties of the

condensate. It is convenient to rewrite HU in terms of the eigenstate operators of HK , and then the interaction energy only receives contributions from terms where all 4 operators correspond to the 4 minima. This process yields an expression

X X 0 − ∗ − ∗ − − EU ∝ Uαβ(cp+q−kχp+q−k,α) (ckχk,β) (cpχp,β)(cqχq,α). (3.13) αβ kpq

The primed sum indicating that we only consider terms where all momentum indices cor-

respond to the energy minima.

From a numerical minimization of EU , I ﬁnd – similar to studies in the absence of an

39 optical lattice [48] – that either a single minimum is occupied (leading to a “plane wave” condensate) or two opposite momenta are equally occupied (leading to a uniform density but spin-polarization-striped condensate). Which state is chosen depends on the deviation from a totally spin-isotropic interaction Uαβ = U, with the striped condensate being energetically favorable for U↑↑ = U↓↓ ≡ U < U↑↓. The conceptual explanation of this result is very straightforward, and it is useful to ﬁrst write down the wavefunctions which are macroscopically occupied. We have the plane wave,

  1 1 √   ΨPW(r) = exp (ik1 · r)   (3.14) 2 eiπ/4 the stripe,   1 cos(k1 · r)   Ψstripe(r) = [ΨPW,k1 (r) + ΨPW,k3 (r)] = (3.15) 2  i3π/4  e sin(k1 · r) and we may also consider, although it fails to appear as a ground state in this approach, a state where an equal combination of all four minima is occupied,   4 1 X 1 ΨSkyrmion(r) = √ exp (ikm · r)   (3.16)  i(2m−1)π/4  2 2 m=1 e

The variational energies of each of these solutions are shown in Fig. 3.3 as a function of the interaction anisotropy parameter g = U↑↓/U↑↑. The spatial distribution of spin density that these states describe are illustrated in Fig. 3.4.

The spin striped and plane wave solutions are the only two possible which have a spatially uniform number density, which is favored by the spin-isotropic part of the interaction. The striped phase additionally has a sort of “phase separation” into regions where the two spin densities minimize their spatial overlap. This is favored when the interspecies interaction

U↑↓ is dominant. The Skyrmion state describes a local minimum in then energy, but never a global minimum. It does not have a uniform number density, and is less eﬃcient than the stripe phase at minimizing the spatial overlap of the two spin components. We will see shortly however that a similar state can nonetheless be stabilized at stronger interactions. It 40 Figure 3.3: Weak-coupling energy expectation values for various ordered condensate states as a function of the anisotropy parameter g. As described in the text, the plane wave solution is optimal for g < 1, while the striped condensate has lower energy for g > 1. I include the Skyrmion solution although it always has a higher energy than the other two. Interactions beyond the type considered here (for example, dipolar interactions) could in principle stabilize similar Skyrmion density-wave states.

Figure 3.4: Spin densities of the various ordered condensates: (left) single plane wave, (middle) stripe, (right) Skyrmion. The z projection of the spin density is indicated by color, while the x and y projections are indicated by the white arrows. The plane wave and stripe solutions have uniform total number density, while the Skyrmion has a density wave, with peaks in the dark blue regions and vanishing density in the interstitial regions where all spin components are zero.

is interesting to note that other authors have observed that density-modulated condensates, including quasicrystals, can be stabilized by long-ranged dipolar interactions, even in the weak-coupling approximation [78, 79].

Finally, some comments are in order about the reliability of these results. In the contin-

41 uum – where the single particle spectrum has a continuous degeneracy – the weak coupling solutions are unstable (with respect to temperature; recall the vanishing Tc) and are therefore suspect. As pointed out in the previous chapter, a number of authors have therefore attempted to produce non-perturbative solutions for the ground state of the Rashba Bose gas [55, 51, 44]. In the lattice potential, the degeneracy is reduced to a fourfold discrete degeneracy, which in turn stabilizes the mean-ﬁeld solutions at extremely weak coupling, however for interaction strengths U on the order of the barrier between the minima, these results could be suspect as well. This makes the intermediate coupling regime on the lattice quite interesting, and the nonperturbative approaches of those references could ultimately be relevant in this context as well.

3.4 Strong-coupling approximation

We now turn our focus to the opposite limit, sufficiently large Uαβ (for all αβ) that number fluctuations are effectively blocked, leaving only virtual hopping processes to reduce the ground-state degeneracy. An effective magnetic hamiltonian governing these residual spin

ﬂuctuations can be derived for the Mott insulator by ordinary second-order perturbation theory. The starting point is to consider a restriction to two lattice sites, and to decompose the hamiltonian as H = H0 + H1 where H0 includes only the onsite (i.e., the interaction) terms and

X † H1 = b1αhαβb2β + h.c. (3.17) αβ

On these two sites, the set of ground states of H0 is spanned by

G = {| ↑1, ↑2i, | ↑1, ↓2i, | ↓1, ↑2i, | ↓1, ↓2i} (3.18)

which all have eigenvalue −2µ. The calculation of perturbative shifts to the eigenvalues and

eigenstates can be combined into an eﬀective hamiltonian in this basis [80], whose matrix

elements between states |s1i, |s2i ∈ G are

X hs1|H1|γihγ|H1|s2i (Heff )s1s2 = − 1 (3.19) γ Eγ − 2 (Es1 + Es2 )

42 where |γi are the six available excited states which can be obtained by acting on the four ground states with H1. The algebra that goes into constructing the spin model is somewhat tedious, but is partially included here for completeness. To start, we can calculate

X † † ∗ † † H1|µ1, ν2i = b1αhαβb2β + b2βhαβb1α b1µb2ν|0i (3.20) αβ

X † † † ∗ † = b1αhαβb1µδβ,ν + b2βhαβb2νδα,µ |0i (3.21) αβ

X † † X ∗ † † = hανb1αb1µ|0i + hµβb2βb2ν|0i (3.22) α β

While substituting in the four ground-state spin orientations yields

√ √ ∗ ∗ H1| ↑1, ↑2i = 2h↑↑|(↑↑)1, 02i + h↓↑|(↑↓)1, 02i + 2h↑↑|01, (↑↑)2i + h↑↓|01, (↑↓)2i (3.23) √ √ ∗ ∗ H1| ↑1, ↓2i = 2h↑↓|(↑↑)1, 02i + h↓↓|(↑↓)1, 02i + 2h↑↓|01, (↓↓)2i + h↑↑|01, (↑↓)2i (3.24) √ √ ∗ ∗ H1| ↓1, ↑2i = 2h↓↑|(↓↓)1, 02i + h↑↑|(↑↓)1, 02i + 2h↓↑|01, (↑↑)2i + h↓↓|01, (↑↓)2i (3.25) √ √ ∗ ∗ H1| ↓1, ↓2i = 2h↓↓|(↓↓)1, 02i + h↑↓|(↑↓)1, 02i + 2h↓↓|01, (↓↓)2i + h↓↑|01, (↑↓)2i (3.26) √ The factors of 2 arise from normalization, e.g., |(↑↑) , 0 i = √1 b† b† |0i. At this point 1 2 2 1↑ 1↑ it is convenient to rewrite this as a table of the matrix elements which go into Eq. (3.19),

provided in Table 3.1.

hγ|H |si | ↑ , ↑ i | ↑ , ↓ i | ↓ , ↑ i | ↓ , ↓ i E − 1 (E + E ) 1 √1 2 √1 2 1 2 1 2 γ 2 s1 s2 h(↑↑)1, 02| 2h↑↑ 2h↑↓ 0 0 U↑↑ h(↑↓)1, 02| h↓↑ h↓↓ √h↑↑ √h↑↓ U↑↓ h(↓↓)1, 02| √ 0 0 √2h↓↑ 2h↓↓ U↓↓ ∗ ∗ h01, (↑↑)2| 2h↑↑ 0 2h↓↑ 0 U↑↑ h0 , (↑↓) | h∗ h∗ h∗ h∗ U 1 2 ↑↓ √ ↑↑ ↓↓ √ ↓↑ ↑↓ ∗ ∗ h01, (↓↓)2| 0 2h↑↓ 0 2h↓↓ U↓↓ Table 3.1: Virtual state matrix elements which enter into Eq. (3.19) for calculating the low- energy eﬀective spin hamiltonian deep in the Mott insulating limit. The rightmost column gives the energy gap to the corresponding virtual excitation.

Now, at this point I could simply insert the matrix elements in Table 3.1 into the 43 expression Eq. 3.19 in the full generality presented so far. This is quite cumbersome, and the resulting expression is not particularly illuminating. However, now that we are armed with the tools to calculate Heff as a 4 × 4 matrix given some input tunneling matrix hαβ, should we so desire, let us pursue an alternative representation with a more physical character.

Each term in the eﬀective hamiltonian can be replaced by a boson operator expression, since Heff can be expanded in projection operators

0 0 † † |σ1σ2ihτ1τ2| = b1σb2σ0 b2τ 0 b1τ (3.27) and the right hand side has the same matrix elements as the left hand side in the space of single-occupancy states. Additionally, we can rewrite this as

† † † † b1σb2σ0 b2τ 0 b1τ = (b1σb1τ )(b2σ0 b2τ 0 ) (3.28) which represent local spin-density operators [81]

1 1 b† b = + Sz, b† b = − Sz, b† b = S+, b† b = S− (3.29) i↑ i↑ 2 i i↓ i↓ 2 i i↑ i↓ i i↓ i↑ i

The remaining algebra, which simply entails writing all of these terms out explicitly and reorganizing them, is relegated to AppendixB. Explicitly setting   cos θ sin θ +ˆx   h = Rαβ(i, i + x) = −t   (3.30) − sin θ cos θ   cos θ i sin θ +ˆy   h = Rαβ(i, i + y) = −t   (3.31) i sin θ cos θ appropriate for Rashba SOC, and

U↑↑ = U↓↓ = U, U↑↓ = U↓↑ = gU (3.32) we eventually arrive at a more intuitive looking model, which is clearly a description of the

44 residual low-energy spin dynamics deep in the Mott insulator:   X X a ab b Hmag = JSi · Si+µ + Dµ · (Si × Si+µ) + Si Γµ Si+µ (3.33) i,µ=x,y a,b=x,y,z

4t2 The natural energy scale here is given by J = gU . In terms of this scale, we can write the exchange constant J = −J cos 2θ which accompanies the spin-isotropic Heisenberg interaction; this is clearly ferromagnetic in the limit of vanishing spin-orbit coupling, as it must be.

The remaining terms account for anisotropies in spin space that are generated either by the explicitly spin-anisotropic interactions of the bosons (g 6= 1) or by the explicit coupling of spin to orbital motion (θ 6= 0). The vectors Dx = −yˆ(J g sin 2θ) and Dy = −xˆ (J g sin 2θ) arise purely from SOC, and characterize the antisymmetric Dzyaloshinski-Moriya (DM) interaction [82, 83]. The SOC also generates symmetric anisotropic interactions of a “compass

xx yy zz zz model” type, Γy = Γx = −J (1 − cos 2θ), while an out-of-plane anisotropy Γx = Γy =

−2J (g − 1) arises from the original spin-anisotropy of HU (all other components of the

ab tensor Γµ are zero).

We now have a rigorously derived model for how the Jz = ±1/2 pseudospin degrees of freedom at each site interact with one another. To develop some insight into how this model behaves, we additionally make a mean-field approximation, by promoting the quantum mechanical spin operators into classical vector degrees of freedom. In other words, we replace quantum spins by their expectation values in spin coherent states, Sˆ i → hSi|Sˆ i|Sii = Si. This gives a classical model for interacting spins, however finding general solutions for frustrated classical spin models is still a nontrivial problem. This frustration is illustrated in Fig. 3.5. For the DM interaction, this frustration can also be appreciated more formally on the following grounds. To minimize the energy of a bond along the x direction, the DM vector demands that the spins be related by a rotation about the y-axis. Similarly the spins along a y direction bond are related by a rotation about x. Now, suppose a spin is placed on lattice site i. We can find a unique spin at site i +x ˆ to minimize that bond’s energy, and the same for i +y ˆ. However, because of the non-commutativity of rotation operators

(RxRySi 6= RyRxSi), we cannot write down a unique spin at i +x ˆ +y ˆ that minimizes the

45 Figure 3.5: Illustration of frustration for compass model and Dzyaloshinski-Moriya in- y y teractions. In (a) and (b), spin configurations which minimize the ferromagnetic Si Si+ˆx x x interactions and Si Si+ˆy interactions are shown respectively. A combination of these configurations where the spins all point in the same in-plane direction also gives the same constrained minimum of the energy, but the point is that there is no spin configuration which simultaneously fully satisfies every bond. In (c) we see a similar effect, explained in detail in the text, where if the bonds (i, i +x ˆ) and (i, i +y ˆ) are satisfied, there is no spin state at site i +x ˆ +y ˆ that allows the remaining bonds to be fully satisfied.

energy on all bonds.

Given the general diﬃculty of ﬁnding ground states of frustrated spin models, we use

classical Monte Carlo annealing to determine the ground state phase diagram shown in

Fig. 3.6. Some notes are in order about the implementation of our Monte Carlo routine.

The most important issue is that we consider ﬁnite lattices (typically 36 × 36 sites) with

periodic boundary conditions. However, we will ﬁnd that much of the phase diagram in-

cludes spiral phases which will be generically incommensurate with the underlying lattice.

The boundary conditions on the ﬁnite size simulation then typically “lock in” a commen-

surate conﬁguration, even if that would not be the true thermodynamic ground state. This

makes the analysis somewhat diﬃcult, as, for example, a ferromagnetic solution is indistin-

guishable from a spiral with a pitch much larger than the simulation cell. Our numerics

could be improved greatly by using twisted boundary conditions which would accommo-

date incommensurate order, however the scope of our investigation is of a more qualitative

nature.

Upon ﬁnding stable spin conﬁgurations, we produce the classical phase diagram through

a variational comparison of the energies of these conﬁgurations over the parameter space.

46 Figure 3.6: Classical magnetic phases for the hamiltonian in Eq. (3.33), determined by Monte Carlo annealing on a 36 × 36 site square lattice. Although the spiral states are generally incommensurate, the shaded green area in the Spiral-2 region corresponds to a likely commensurate state. In this region the spiral unit cell contains 4 sites, with the spin winding by π/2 along each bond in the spiral direction. This pattern maximizes the cross- product of neighboring spins, and is therefore quite favorable in the region θ ∼ π/4, where the isotropic ferromagnetic interaction vanishes.

The general features of the classical phase diagram can therefore be largely understood by visualizing the various stable spin conﬁgurations, shown in Fig. 3.7, and which we now describe in some detail.

Conventional ferro- and antiferromagnetic phases In the absence of spin orbit coupling,

it is well known that the two-species Bose-Hubbard model has a ferromagnetic ground state

in the Mott limit. This is contrary to the Fermi-Hubbard model, where antiferromagnetism

emerges because the energy-lowering virtual hopping processes are Pauli blocked between

two sites with the same spin orientation. Bose statistics, contrarily, leads to an enhancement

factor that favors aligned spin orientations, and therefore ferromagnetism. Now, if g > 1, we

have an additional out-of-plane anisotropy to the interaction which favors order along the z-

axis (both for ferromagnetic interaction without SOC, and the antiferromagnetic interaction

that emerges near θ = π/2), while g < 1 favors order in the xy plane, giving rise to the xy

ferromagnet. These anisotropies preserve the states against weak spin-orbit coupling, and

so it is only for the special isotropic line g = 1 that the presence of a ferromagnetic solution

47 Figure 3.7: Classical magnetic states in the Mott insulating limit of the spin-orbit coupled Bose-Hubbard model. (a) Spiral-1 state; coplanar spin-orientation rotating along (11). (b) Spiral-2 state; coplanar spin-orientation rotating along (10). (c) Vortex crystal; 2 × 2 unit cell with π/2 rotation along each bond. (d) Skyrmion crystal; the 3×3 unit cell is highlighted with a gray box. The central spin points in the positive z direction, while the remaining spins tumble outward toward −z.

should be viewed cautiously.

Vortex crystal As θ approaches π/2, the out-of-plane interaction is antiferromagnetic,

yielding the z-AFM phase, but for g < 1, the dominant contribution to the hamiltonian

comes from an in-plane compass model interaction. This interaction is frustrated, and we

ﬁnd that it is best satisﬁed by a state we call the “vortex crystal,” where the spin rotates

by π/2 in the xy plane along every bond around a square plaquette. This name is rather

48 suggestive; as we will see below in the mean-ﬁeld analysis, this order persists into an exotic superﬂuid order where the order-parameter phase possesses the same winding. Intuitively, this would correspond to an alternating pattern of phase vortices and anti-vortices at the centers of the plaquettes.

Spiral order For g > 1 and intermediate values of the spin-orbit coupling, we recover a magnetic phase reminiscent of the “stripe” condensate described in the previous section.

Here we have an incommensurate spin spiral along the (11) or (11)¯ direction of the lattice.

At weak coupling, the pitch of the stripe condensate was determined solely by the location of the energy minima km. At strong coupling, it is determined by the ratio of the DM interaction to the spin-isotropic interaction.

For g < 1, coplanar spiral order is also found, but the spiral vector is along the (10) or (01) direction. This kind of order does the most to compromise between the DM term

(tumbling the spins in one direction so that the cross product between spins in that direction does not vanish) while also satisfying the large compass interaction (by aligning spins along the other direction)

Skyrmion crystal In a small parameter regime, we find that the energy is minimized by a superposition of stripes in the (10) and (01) directions. This superposition leads to a magnetic texture that again is reminiscent of the Skyrmion condensate described above. In this case it has a unit cell of 3×3 lattice sites. The central spin in the unit cell points in either the positive or negative z direction, while the remaining spins tumble outward toward the opposite z direction. For g < 1, the extra planar anisotropy prevents these off-center spins from having a significant z-component, however. As the only non-coplanar arrangement of P spins, this state also carries a non-zero spin chirality i Si · (Si+ˆx × Si+ˆy). In this lattice discretization of the spin chirality, there is no need for the result to be quantized, however it is useful to note that this definition is inspired by the continuum formulation, wherein this spin chirality is topologically quantized, and simply counts the number of Skyrmions present in the texture [84].

To identify the phase boundaries, it is helpful to compare the Monte Carlo annealing results with a variational comparison of the energies of these states. An example of this

49 analysis is shown in Fig. 3.8. Plotted are the calculated ground-state energies at low temperature from several Monte Carlo runs, along with the variational energy of states that are likewise determined by the Monte Carlo analysis at a certain parameter point, but then evaluated over the parameter space. I use these level crossings to locate the phase boundaries between magnetic states. From this plot we can also observe that the conventional ferromagnetic and antiferromagnetic states persist as metastable solutions in the Monte

Carlo annealing outside of the phase-space where they represent the true minimum.

Figure 3.8: Variational comparison of several classical spin states at g = 0, as detailed in the main text. The red symbols are the results of several independent Monte Carlo simulations, while the various continuous curves correspond to the variational energies of corresponding spin conﬁgurations, without any of the defects that typically are trapped in the Monte Carlo ground state. The higher energy of the Monte Carlo simulations occur both from thermal defects as well as trapped topological defects (such as domain walls) that are quite diﬃcult to eliminate in the local-update scheme I have adopted.

Closing this section, we note that the spin-rotation symmetry is explicitly broken by the hamiltonian for most of the (g, θ) parameter space, so it is expected that most of these

50 classical spin conﬁgurations are stable against quantum ﬂuctuations due to the absence of Goldstone modes. On the other hand, it is also interesting that many of these magnetic states additionally spontaneously break the rotational and translational symmetries of the square lattice. Recent experiments on a trapped gas with Raman induced SOC have demonstrated multiple phase transitions as a function of increasing temperature: striped condensate → single-minimum condensate → thermal Bose gas [35]. Likewise, multiple transitions might be observed as these spatial symmetries are restored deep in the Mott limit, en route to the thermally disordered paramagnet.

3.5 Mean-ﬁeld theory of the superﬂuid-insulator transition

The weak interaction and strong interaction limits of the spin-orbit coupled Bose Hubbard model give already rather exotic results, but there are appealing similarities between the two. The weak-coupling condensates have the ability to break spin symmetry and develop magnetic order, as do the Mott insulating states. However there is a rather dramatic transition between these limits (namely, the superﬂuid-insulator transition) that we have yet to describe. However the limits are suggestive, and several questions arise quite naturally.

(1) Does the magnetic order in the insulator persist across the Mott transition? (2) If so, how does this impact the nature of the transition, compared to the traditional Bose-

Hubbard model? (3) Just above the Mott transition, can there exist spin-ordered superﬂuid phases which have no weak-coupling analog? The remainder of this chapter aims to address these issues.

3.5.1 Description of the mean-ﬁeld method

As a first approximation to locating the Mott transition and examining how it is affected by spin-orbit coupling, I generalize a mean-field approach introduced by Sheshadri et al. [85] which provides a good qualitative description of the Mott transition for lattice bosons. We begin with an exact substitution, writing the boson creation and annihilation operators in terms of their expectation value plus the fluctuations around it, biα = ϕiα + δbiα, where

51 ϕiα ≡ hbiαi, and the operators δbiα satisfy the same boson commutation algebra as the original operators. The mean-ﬁeld approximation consists of neglecting terms that are second-order in the ﬂuctuations:

† ∗ † ∗ biαbjβ ≈ ϕiαbjβ + biαϕjβ − ϕiαϕjβ (3.34)

Applying this procedure only to the kinetic energy operator HK decouples the sites from one another, leaving the full hamiltonian as a direct sum over lattice sites, on each of which

the interaction term remains to be diagonalized exactly. Formally, we have

MF X X h ∗ † ∗ i HK = Rαβ(i, i + µ) ϕiαbi+µ,β + biαϕi+µ,β − ϕiαϕi+µ,β + h.c. (3.35) i,αβ µ=ˆx,yˆ

For notational convenience, we can introduce a c-number ﬁeld at each lattice site,

X ∗ hiα = Rαβ(i, i + µ)ϕi+µ,β + Rβα(i − µ, i)ϕi−µ,β (3.36) µ,β

which accounts for the coupling to neighbor sites. Using this notation, we can simplify the

kinetic energy further,

MF X ∗ † ∗ HK = hiαbiα + hiαbiα − hiαϕiα (3.37) i,α

where the simplifying power of our mean ﬁeld assumption is now much more apparent.

Introducing now a chemical potential that governs the total particle number, the full mean

ﬁeld hamiltonian is given by

MF X † H = HK + HU − µ biαbiα (3.38) iα Solving this mean-ﬁeld hamiltonian is substantially easier than solving the original prob-

lem, since each “site-hamiltonian” can be diagonalized independently. However each of these

site-hamiltonians also depends parametrically through hi on the value of the ϕ-ﬁeld on all of the neighboring sites, which in turn must be self-consistently evaluated from the solution

to the single-site hamiltonian for those sites.

For the spinless, translation invariant Bose-Hubbard model (or in the spinful case with-

52 out spin-orbit coupling), a homogeneous approximation ϕiα = ϕα ∀i is typically justified. The assumption of a homogeneous state clearly simplifies the problem spectacularly, but it is apparent from the mean-field hamiltonian with spin-orbit coupling present that inhomo- geneous states can have substantially lower energy in certain parameter regimes.

It is conceptually useful to point out that the mean-field approach thus described can be equivalently cast as a variational approach, where we choose a trial density matrix ρvar = exp(−βHMF ) and minimize the variational free energy with respect to field configurations

{ϕiα}. At zero temperature, this density matrix is speciﬁed by a Gutzwiller variational wave function [16]   Y X |Ψi =  ci,n↑,n↓ (ϕi↑, ϕi↓)|n↑, n↓ii (3.39) i n↑,n↓=0 where |n↑, n↓ii is the joint number state of up- and down-spin bosons on lattice site i. Either approach provides us with a variational upper-bound to the ground state energy, which is a function of the 2Ns complex variational parameters:

X (i) Evar [{ϕ}] = 0 [ϕi↑, ϕi↓] (3.40) i

(i) with 0 the ground state energy of the site-hamiltonian at site i. The goal is ultimately to minimize this energy subject to the self-consistency constraints. In [73] this was carried out through iteration to a ﬁxed point, but other methods could be pursued for parameter regions where convergence proves diﬃcult.

Before moving on to the results, a few more technical details should be emphasized.

To work in a finite Hilbert space, it is necessary to impose a cutoff ncut on the number of bosons on a lattice site. For high accuracy results we would like to choose ncut very large, but this comes at the cost of computational efficiency. A more practical route is to

ﬁnd the smallest value of ncut, such that increasing it further doesn’t impact the results beyond some threshold accuracy. There are a number of measures by which to make this determination, for example convergence of the energy. This is perhaps the most obvious route (and the one employed in [85]); the idea is to increase ncut until the ground state

53 energy no longer changes. However, the order parameter need not be as strongly inﬂuenced by the low-occupation states as the energy. Thus we can also use the convergence of the order parameter as a metric, since this is often more interesting to us than the ground state energy anyway. Finally, another possible metric is to continue increasing ncut until the amplitude for the state with ncut bosons is below a cutoﬀ threshold. We have employed these checks to ensure the accuracy of our calculations.

3.5.2 Results

First I characterize the transition by locating the “Mott lobes” in the t/U − µ/U plane for various values of spin-orbit parameter and anisotropy. In the limit t/U = 0, we can consider a single well, which will contain exactly one atom of any spin orientation for

0 < µ < min(U, gU). This same well will contain two atoms for min(U, gU) < µ < min(2U, 2gU, U + gU), and so on. There is a nontrivial Mott insulating state if this persists to nonzero t/U. In Fig 3.9 I present three representative phase diagrams, all for fixed g = 1.2, but with θ = 0, π/4, π/2. From these we observe a substantial change in the location of the Mott transition as a function of θ. The conceptual interpretation of this result is simply that spin-orbit coupling frustrates the kinetic energy and reduces the bandwidth, therefore a larger bare hopping amplitude t is required at larger SOC parameter to stabilize a superfluid state. However, it is also the case that the character of the superfluid state just above the transition is dramatically different for different values of θ. In fact, by considering the local magnetic moments in the superfluid state

∗ mx = φxασαβφxβ (3.41) we find that the superfluid solutions at specific parameter points shown in Fig. 3.10 neatly match the magnetic ordering of the underlying Mott phase. It is also of interest to determine if strong correlations and spin-orbit coupling might conspire to create spin or number currents in the various superfluid ground states, analogous to the ground state currents generated by orbital magnetism [86].

Because the gauge is “ﬁxed” – or, in other words, there are potentials that mimic

54 Figure 3.9: Mott lobes in the (t, µ) plane at SOC parameters θ = 0, π/4, π/2, for anisotropies (a) g = 1.2 and (b) g = 0.8. The signiﬁcant features are a large enhancement, with increasing θ, of the bare t/U required to reach the superﬂuid phase. For g < 1, the n = 1 Mott state at t = 0 is only stable up to µ = gU, since the n = 2 Mott state with opposite- species bosons on each lattice site is also available at that energy.

the appearance of a gauge potential, but the notion of gauge invariance holds no special signiﬁcance – the conventional number and spin currents are observable [87, 88]. For the lattice problem, currents live on the bonds between sites,

a † a jx,µ = −i hbx,ασαβbx+µ,βi − c.c. (3.42) where a = 0, 1, 2, 3, and σ0 is the identity matrix. These are observable precisely because

we are not considering a dynamical SU(2) gauge theory. However it is also quite appealing

to continue with the analogy and look at the gauge invariant currents [88]

˜a † a µ jx,µ = −i hbx,α σαβ − iAαβ bx+µ,βi − c.c. (3.43)

In Fig. 3.10 I show the structure that emerges in the mean-ﬁeld ground-state solution

{ϕxα} through these currents and spin densities at a few representative parameter points. In all cases these are at t/U = 0.07, near the Mott transition. Although the anti-ferromagnetic

superﬂuid found at g = 1.5 and θ = π/2 has an analogous weak-coupling condensate – the

“striped” state previously described – the solution at g = 0.5 has the same basic structure

as the VX magnetic insulator, for which no weak coupling analog exists. In both of these

states, there are microscopic circulating density currents.

55 Figure 3.10: Mean-field solutions {ϕxα} for various parameter points (top row g = 1.5, bottom row g = 0.5, from left to right θ = 0, π/4, π/2). In this figure I characterize these z states according to the z-projection of their on-site spin-density mx, as all states have uniform number density, as well as the number density currents that live on the bonds, ˜0 jx,µ. These states have similar magnetic structure to the underlying Mott insulating states, except that the region of Stripe-2 phase space in the Mott insulator is overtaken by the xy- ferromagnetic superfluid. These states are calculated on a 12 × 12 site cluster with periodic boundary conditions, although only a smaller representative region is shown.

Before closing this section it is useful to note that, while numerically demanding, the above approach is not a particularly sophisticated mean field theory. It is incapable of describing Mott states that are not featureless trivial product states. It is therefore quite promising that a recent study of the present model by He et al. [89] finds similar magnetically ordered Mott insulators and superfluids using a bosonic extension of dynamical mean-field theory (BDMFT).

3.6 Slave-boson theory

If we believe the apparent separation between magnetic ordering and the superﬂuid-insulator transition that is suggested by the Gutzwiller theory and the strong-coupling Mott limit,

56 it is natural to try to unify these limiting cases in a theory where explicit spin-“charge” separation can be exploited. For this purpose, inspired by slave-particle approaches in the study of correlated electron problems [90], we formulate a new kind of slave-boson decomposition of the physical boson operators by writing

† 1 † † bxα = q axfxα (3.44) † axax

† † where both ax and fxα are boson operators. The “charge” degree of freedom is represented † † † by ax, while fxα carries the spin degree of freedom. The condensation of ax produces † superﬂuidity, while condensation of the fxα is responsible for any magnetic ordering. We † will use nx,(a) = axax for the number of charges on site x. We note that there is a local U(1) gauge redundancy remaining in the slave boson formation,

† † † † ax → ax exp(iϕx), fxα → fxα exp(−iϕx) (3.45)

that leaves the physical boson operator unchanged.

† A straightforward calculation demonstrates that bxα and its conjugate satisfy all of † † the necessary commutation relations so long as both ax and fxα also satisfy the bosonic commutation relations.

The physical Hilbert space is given by specifying the integer number of b-bosons with (b) (b) spin-up and spin-down, |n↑ , n↓ i, at each lattice site. This space is enlarged in the slave (a) (f) (f) particle formulation to |n ; n↑ , n↓ i. The projection back into the physical Hilbert space is implemented by the constraint that there must always be an equal number of a-charge

degrees of freedom and f-spin degrees of freedom on each lattice site. Thus the physical

Hilbert space can be written as

(f) (f) (f) (f) |n↑ + n↓ ia ⊗ |n↑ , n↓ if (3.46)

or in terms of operators, we can demand

† X † axax = fxαfxα, ∀x (3.47) α

57 Following the standard procedure for slave-particle theories, we may construct a slave boson mean-ﬁeld theory by replacing the exact local constraint by its average; that is, we require that

† X † haxaxi = h fxαfxαi, ∀x (3.48) α

which is implemented by introducing a Lagrange multiplier λx at each site. When Eq. (3.44) is substituted into the Hamiltonian Eq. (3.7), we ﬁnd

X 1 † 1 † 0 H = −t √ axax0 √ fxαRαβ(x, x )fx0β + h.c. (3.49) nax nax0 x,x0 U X + n2 + n2 + 2gn n (3.50) 2 x,(f↑) x,(f↓) x,(f↑) x,(f↓) x with the understanding that the constraint in Eq. (3.48) should be satisﬁed.

Now, to formulate a tractable mean field theory, we posit the following: (1) there is a superfluid-Mott transition for the chargons and (2) magnetic ordering persists across this transition. We can then treat the spinon field classically, and allow any structure to emerge from variational energy minimization. Furthermore, if we consider only the vicinity of the superfluid-Mott insulator transition (where charge fluctuations are heavily suppressed), it is reasonable to replace nax by its average value of one particle per site. This means that P † † † ∗ ∗ αhfxαfxαi = 1 and so we parametrize the spinon field by zxzx = 1, where zx = (zx↑, zx↓). After these simplifications, we obtain two coupled mean-field Hamiltonians:

X ∗ 0 † U X † † H = −t z R 0 (x, x )z 0 a a 0 + H.c. + a a a a (3.51) a xα αα xα x x 2 x x x x x,x0 x

X ∗ 0 X 2 2 Hf = −t (zxαRαα0 (x, x )zxα0 Axx0 + c.c.) + U(g − 1) |zx↑| |zx↓| (3.52) x,x0 x

† where we have defined Axx0 = haxax0 i, the average value of the bond operators associated with hopping of the chargons. The Hamiltonian Hf is purely classical and couples to the chargons only through the c-number Axx0 . As for the chargon part, the effective hopping matrix element for charge is modified to become

˜ ∗ 0 tx,x0 = zxαRαβ(x, x )zx0β. (3.53)

58 A fully self-consistent minimization would proceed as follows: first we propose initial values for zx, then solve for the chargon Hamiltonian Ha, obtain the value of Axx0 and then minimize Hf to find the corresponding updated zx. In the following, I instead explore some consequences for specific limiting cases of Eq. (3.53), where some simple intuition can be given to the results.

z-FM : This corresponds to taking zx↑ = 1 and zx↓ = 0 for all x. The effective Hamil- tonian Ha then takes the same form as a conventional spinless Bose-Hubbard model, but with an effective hopping t˜xx0 = t cos θ. The only effect of changing θ is to reduce the bandwidth, and the critical hopping (t/U)c of the superfluid-Mott transition is multiplied 1 by cos θ compared to the transition in the convention Bose-Hubbard model. xy-FM : Similarly, in the xy ferromagnetic phase, the spins align in the xy-plane with an √ √ angle ϕ from thex ˆ-axis – that is, zx↑ = 1/ 2 and zx↓ = 1/ 2 exp(iϕ). The kinetic energy part of the chargon hamiltonian becomes

(K) X ˜ † ˜ † Ha = − (txaxax+ˆx + tyaxax+ˆy + H.c.) (3.54) x with t˜x = t(cos θ − i sin θ sin ϕ) and t˜y = t(cos θ + i sin θ cos ϕ). Although the eﬀective hopping amplitudes are complex, a hopping path around a plaquette contains no ﬂux.

Nonetheless, the magnitude of the hopping diﬀers alongx ˆ andy ˆ, except for the special values ϕ = (2n + 1)π/4. The chargon band structure contains a unique minimum at a nonzero k for θ 6= 0, and condensation in this state corresponds naturally to the weak- coupling plane-wave condensate.

z-AFM : here the unit cell is doubled and we have zx↑ = 1, zx↓ = 0 on the A sublattice, while on the B sub-lattice we take zx↑ = 0, zx↓ = 1. Since hopping occurs exclusively between A and B sub-lattices, we obtain ( ) (K) X † † X † † Ha = −t sin θ (−axax+ˆx + iaxax+ˆy + H.c.) − (axax+ˆx + iaxax+ˆy + H.c.) x∈A x∈B (3.55)

Taking the product of the hopping amplitudes around a plaquette yields a value −t4 sin4 θ, the negative sign implying that each plaquette encloses a π ﬂux for for the chargons. As

59 a result, there are symmetry breaking plaquette currents for the chargons, consistent with those that appeared in the more conventional mean-ﬁeld approach. Therefore, in addition to the usual U(1) symmetry breaking at the Mott-to-superﬂuid transition, we have also an

Ising order associated with the orientation of the plaquette currents. √ Vortex Crystal: In this state, the spin conﬁguration is characterized by zx↑ = 1/ 2 and √ zx↓ = 1/ 2 exp(iϕx), where the phase angle now depends on the lattice coordinate. There are four sites in the unit cell, with φA = π/4, φB = 3π/4, φC = 5π/4, and φD = 7π/4. In this arrangement, the amplitude associated with a hop from an A site to a B site within

the unit cell is given by

1 t˜ = teiπ/4 cos θ + e−iπ/2 sin θ (3.56) BA 2

It is also quickly veriﬁed that, within the unit cell, t˜BA = t˜CB = t˜DC = t˜AD. The overall phase factor therefore ensures that for a counterclockwise traversal of a plaquette, the

4 1 2 product of amplitudes is −t 4 (1 + sin θ) . This again corresponds to a π ﬂux through each unit cell for the chargons.

3.7 Exact numerics in the one-dimensional limit

The discussion throughout this chapter has focused on the isotropic Rashba spin-orbit coupling, while the current experiments have been performed in a regime where the spin- orbit coupling operates in only one direction (e.g., Hs.o. ∼ pxσy). The eﬀects of this sort of spin-orbit coupling in two spatial dimensions at the mean-ﬁeld level are quite easy to guess from the discussion thus far in this chapter. While exact numerical studies of the

Rashba spin-orbit coupled Bose-Hubbard model remain out of reach, it is possible to ﬁnd

(numerically) exact results for the experimentally relevant unidirectional SOC in a one- dimensional lattice model, by using the Density Matrix Renormalization Group (DMRG) technique. The results presented in this section are discussed in detail [91], and so here I intend instead only to point out two features which speak to the issues highlighted in the present Chapter.

60 For the Raman-induced abelian spin-orbit coupling described in Chapter1, there are, as I mentioned, two possible condensate states at weak coupling. Either a single energy minimum or a linear combination of the two energy minima is occupied, and the preference between these can be tuned by the spin-anisotropy of the interactions between atoms. The strong coupling eﬀective magnetic model can be read oﬀ from Eq. (3.33), and after a rotation of the spin axes by π/2 aroundx ˆ (done for notational convenience in [91]), we have

4t2 X H = − cos(2θ)SxSx + (2g − 1) cos(2θ)SySy + SzSz mag gU i i+1 i i+1 i i+1 i x y y x +g sin(2θ) Si Si+1 − Si Si+1 (3.57)

For g > 1 this model is dominated either by the Ising-like interactions of the y-component of the spins or by the Dzyaloshinskii-Moriya contribution depending on the value of θ. For g < 1, the model is dominated by Ising-like interactions between the z-component of the spins for all θ 6= 0, π/2. DMRG calculations allow us to obtain the magnetically ordered,

Mott-insulating ground states of the one-dimensional analog of Eq. (3.7), which are indeed well-described by Eq. (3.57). Similar to the mean-ﬁeld results, there are magnetic insulators which are analogous to the magnetically ordered condensates, and also magnetic insulators which do not have a weak-coupling analog.

DMRG also enables us to go beyond the mean-field description of the Mott state, and locate the transitions exactly in the t − µ plane. In Fig. 3.11 I show the n = 1 “Mott lobe,” where the magnetic phases described previously are found. In the inset, I show that as t/U is increased there are two phase transitions: (1) A continuous Mott transition from a y − FM insulator to a superfluid, described by the appearance of a superfluid density n0; and (2) a first-order transition to the xy-chiral superfluid, which is smoothly connected to the weak-coupling condensate. In between these two transitions is a y − FM superfluid phase, which retains its magnetic order across the Mott transition.

61 Figure 3.11: Figures adapted from [91]. Left, the Mott lobes in the t − µ plane at various values of the SOC parameter θ. The reentrant structure is a well-known feature of the Bose- Mott transition in one spatial dimension. In the inset, two order parameters are plotted as a function of t/U. First, the superﬂuid order parameter n0 becomes nonzero at the Mott transition. At a larger t/U, the “chiral” order parameter (see [91]) that describes the weak-coupling striped condensate discontinuously jumps to a nonzero value. On the right, the magnetic phase diagram deep in the n = 1 Mott lobe is shown in the g − θ plane.

62 Chapter 4 Hard core bosons on the frustrated honeycomb lattice

The present chapter might seem superﬁcially unrelated to the previous two, however they are all tied together by the deep and to a large degree open question of how bosons condense

(or possibly fail to condense!) when the lowest-lying single-particle energy eigenvalues are part of a highly degenerate manifold [92], as we saw with the “Rashba ring.” I consider here a model of hard core bosons on the two-dimensional honeycomb lattice with frustrated hopping amplitudes. The honeycomb lattice has two sites in its unit cell, and this sublattice degree of freedom plays the role of a pseudospin. This pseudospin in turn is quite naturally tied to the orbital motion of the particles, providing a straightforward conceptual analog of the Rashba spin-orbit coupling I have considered up to this point. More importantly, this model has a single particle spectrum with a continuous ring of lowest-energy states. In the concluding section, I tie the lessons of this chapter back to the problem of the isotropic

Rashba-Bose gas.

4.1 Hard core bosons and Jordan-Wigner fermions

Hard core bosons on a lattice (without any other internal degrees of freedom) are de-

† 0 ﬁned by their commutation relations [bx, bx0 ] = 0 for x 6= x and on-site anti-commutation † {bx, bx} = 1. These “mixed” commutation relations make it rather diﬃcult to study otherwise free hardcore bosons by analogy to either free bosons or fermions. Since the on-site anti-commutation gives us part of the usual fermion algebra for free, one might be inclined

63 to ask if a transformation exists where hard core bosons can be represented by fermions. For one spatial dimension, a constructive answer to this was provided by Jordan and Wigner already in 1928. Writing

Px−1 † Px−1 † † 0 c 0 c 0 † 0 c 0 c 0 bx = (−1) x =1 x x cx, bx = (−1) x =1 x x cx (4.1)

(where c† creates a conventional fermion) leads to the oﬀ-site commutation relations (taking,

without loss of generality, x0 > x)

0 0 † Px−1 c† c Px −1 c† c † Px −1 c† c † Px−1 c† c x00=1 x00 x00 x00=1 x00 x00 x00=1 x00 x00 x00=1 x00 x00 [bx, bx0 ] = (−1) cx(−1) cx0 − (−1) cx0 (−1) cx 0 0 Px −1 c† c † † Px −1 c† c x00=x x00 x00 x00=x x00 x00 = cx(−1) cx0 − cx0 (−1) cx 0 0 Px −1 c† c Px −1 c† c x00=x+1 x00 x00 † † x00=x+1 x00 x00 = −(−1) cx0 cx − cx0 cx(−1) 0 Px −1 c† c x00=x+1 x00 x00 † = −(−1) {cx, cx0 }

= 0 (4.2)

† † For the on-site algebra, {bx, bx} = {cx, cx} = 1, which is easily found by inspection from the previous equation if we had taken x = x0 in the second line. The interpretation of this

is that we could imagine that hardcore bosons are in fact actually fermions, but the price

we pay is that these fermions interact nonlocally with one another through the “string”

of negative signs that must accompany the fermion operators to restore the original boson

statistics. The remarkable thing that occurs in some one-dimensional models is that these

“string” factors cancel identically, so that the spectrum of, say, otherwise noninteracting

hard core bosons with nearest-neighbor hopping is identical to that of free fermions.

In a certain sense, the Jordan-Wigner transformation formalizes the notion that ex-

change statistics in one spatial dimension are not “fundamental.” The distinctions between

hardcore bosons and fermions produce certain constraints for formulating some speciﬁc cal-

culation, but we can switch between these representations at will. It has only more recently

been appreciated that the exchange statistics of a two-dimensional system of identical par-

ticles are likewise not fundamental in this same sense; any desired statistics can be obtained

by coupling the original particles to a Chern-Simons gauge ﬁeld. Fradkin explored the lat-

64 tice formulation of this sort of “statistical transmutation” as a natural generalization of the

Jordan-Wigner transformation to two spatial dimensions [93].

Ultimately Fradkin’s generalized Jordan-Wigner transformation turns a two-dimensional model of hard core bosons into a model of spinless fermions with hopping phases that, as in the original Jordan-Wigner transformation, depend on the locations of all of the other fermions (this is reviewed in Section 4.2). Although this transformed hamiltonian does not appear any simpler to solve (unlike in 1d, there are no cancellations of the phases), it does suggest a non-obvious mean-ﬁeld approximation in terms of the fermions, which in principle might have some unique properties that are missed in traditional mean-ﬁeld approaches for the original hard core bosons.

Some early numerical work attempted to quantify the accuracy of this so-called “vector mean-ﬁeld theory” (VMFT) [94, 95, 96], but focused only on the regime where condensation of the bosons is unfrustrated. The goal of the present work is to extend some of this analysis to a case where Bose condensation is frustrated by the appearance of a “Moat band” [97]

– a spectrum where the lowest-lying single particle states host some continuous degeneracy which leads to a divergent density of states, just as we saw occur for the continuum Rashba hamiltonian.

In addition to this very general motivation, we are also interested in a speciﬁc current controversy. Several groups have recently considered the possibility of a Bose metal phase at half-ﬁlling in the model considered in this Chapter (see Fig. 4.1). This is neither a condensate nor a commensurate solid, but a gapless quantum-disordered liquid of bosons.

As the ratio t2/t1 of the next-nearest and nearest neighbor tunneling amplitudes is increased, three distinct regions are encountered in existing numerical studies: (I) a ﬁnite momentum condensate at the unique single particle minimum; (II) an intermediate regime, where a

Bose metal state was initially identiﬁed in exact diagonalization calculations [98]; and (III) another condensate at a diﬀerent, but still unique minimum.

After the initial work from Varney et al., the intermediate phase has been variously understood as this Bose metal [99], a gapped but non-condensed liquid [100], a gas of composite fermions[97], and a charge density wave (CDW) solid [101]. Out of all of these

65 approaches, the most numerically rigorous (in view of extrapolating to the thermodynamic limit) is DMRG. Although that work has plausibly ruled out the Bose metal state, the CDW order (since conﬁrmed in other numerical investigations [102, 103]) poses a serious mystery since the underlying hamiltonian has no terms which would stabilize such behavior. In this chapter I will demonstrate that these correlations are readily obtained by starting from an underlying composite fermion state.

However there is a signiﬁcant barrier to comparing VMFT results with the detailed numerical calculations in the literature. There is no variational principle governing the energies determined by the VMFT, and so any direct comparison with other results in terms of energetics is simply not meaningful. The variational principle can be restored, but the cost is evaluating the energy of free fermion VMFT states with respect to the complicated non-local fermion hamiltonian that results from the generalized Jordan-Wigner transformation. This is precisely what I do in this Chapter.

As a preview the content of this Chapter, the following points are illustrated through several numerical calculations:

(1) Recent work by Nie et al. [104, 105] has emphasized a so-called “natural inequality for comparing the ground state energies of fermions and hard core bosons subject to the same hamiltonian. Following their analysis, we determine a lower bound on the ground- state energy of the present model, and show that this bound is higher than the energy of the corresponding Fermi gas precisely in the range of t2/t1 where various authors have suggested an exotic ground state.

(2) The composite fermion mean-field state introduced by Sedrakyan et al. [97] and described in detail in this work is sufficiently interesting that its properties should be rigorously compared with other approaches. Since the mean-field energy in their approximation is not a variational upper bound on the ground state energy of the original hamiltonian of hard core bosons, we use Monte Carlo calculations both to set a variational upper bound on the true ground state energy via this mean-field state.

(3) With the same Monte Carlo method, we also calculate the one-body density matrix

(OBDM). Some care must be taken here; although the mean-ﬁeld state has the character of

66 a Fermi gas, when calculating the OBDM one must also include the non-local operators that implement the statistical transmutation. As a matter of principle these non-local operators can restore oﬀ-diagonal long-range or algebraic order (this is similar to the situation one has with the one-dimensional hard core Bose gas, for which the Jordan-Wigner transformation yields the exact ground state).

4.2 Model and methods

I am considering here a model of hard core bosons, i.e., otherwise commuting boson opera-

† † tors bx subject to a local constraint bxbx = 0 or 1, which is enforced in the construction of the Hilbert-Fock space. The hamiltonian can be thought of as a Bose-Hubbard model with

U → ∞, and so all that remains are the hopping terms

X † HB = txx0 bxbx0 (4.3) xx0

0 with txx0 = t1 or t2 when xx is a pair of nearest-neighbor or next-neighbor lattice sites, respectively. The lattice and available hopping processes are illustrated in Fig. 4.1.

Figure 4.1: (a) The honeycomb lattice, with nearest neighbor bonds (solid lines) and next- neighbor bonds (dashed lines) indicated. Hopping occurs along nearest neighbor bonds with amplitude t1 and along next-neighbor bonds with amplitude t2. There are two inequivalent sites in the unit cell. (b) Contour plot of the energy spectrum E(kx, ky) for the lower band, plotted in the ﬁrst Brillouin zone. Similar to the Rashba problem, the lowest energy states are degenerate and appear along a ring of constant |k|= 6 0.

67 Before applying it to the present model, I quickly review Fradkin’s generalized Jordan-

Wigner transformation (which I will refer to hereafter as the FJW transformation). Again, the idea is essentially to replicate the underlying premise of the Jordan-Wigner transformation that a boson can be represented by a fermion times a “string” that depends on the other fermion coordinates, and serves to restore the oﬀ-site boson commutation relations. At the same time, the on-site hard core constraint is exactly handled by the fermion statistics. The transformation is given by [93, 95, 96]   X bx = Wxcx,Wx = exp i Gxx0 nx0  (4.4) x06=x

† † where cx annihilates a fermion on site x, nx = bxbx = cxcx is the number operator, and

Gxx0 can be any two-point function that satisﬁes Gxx0 − Gx0x = π mod 2π. I will follow 0 x2−x2 most authors in taking the simple choice Gxx0 = arctan 0 . x1−x1

The next step is to rewrite HB as an operator in the Hilbert-Fock space for the fermions † cx. This can be done by direct substitution. After evaluating  

† X −iΦxx0 WxWx0 = exp −i Gx,x00 − Gx0,x00 nx00 − i [Gxx0 nx0 − Gx0xnx] ≡ e (4.5) x006=x,x0 the hamiltonian takes on a deceptively simply appearance:

X † −iΦxx0 HFJW = txx0 cxe cx0 (4.6) xx0 which describes fermions hopping on the original honeycomb lattice, but now interacting with an emergent U(1) gauge ﬁeld which depends on the coordinates of all of the other particles in the system.

Although solving HFJW does not appear any easier than solving HB, my goal here is more modest. I instead study the properties of states which are constructed as the single

Slater determinant ground state of a noninteracting “vector mean-ﬁeld” hamiltonian with the form

X † HVMF = txx0 cx exp (−iAxx0 ) cx0 (4.7) xx0

68 which now represents completely free spinless fermions hopping in a static gauge field, which to a first approximation can be obtained by substituting nx = hnxi = ρ in HFJW . This is the construction and the vector mean-field theory considered by Sedrakyan et al. [97].

Before proceeding further, I would like to highlight an ambiguity that is often overlooked in this approach: HFJW only acts between Fock states with nx0 = 1 and nx = 0. This means that, for the exact hamiltonian, nothing is lost by taking

X Φxx0 = Gx,x00 − Gx0,x00 nx00 (4.8) x006=x,x0

The mean-field gauge configuration Axx0 , however, differs by ρ (Gxx0 − Gx0x) depending on whether or not this identity is used. Whether or not this term should be included is left to be determined by the variational minimization.

Also prior to moving on, it is useful to point out some prior work on the validity of vector mean-field theory [94, 95, 96]. Both Canright and Matsui performed exact diagonalization of HF with a few fermions on small, unfrustrated square lattices. They considered the overlaps between VMFT states determined by this construction and the exact ground states at various system sizes, with promising results. Gros et al. pointed to the difference between the VMFT energy and a proper variational energy, however they were also restricted in their scope to an unfrustrated square lattice. In the absence of frustration, it is easily proved that one can take any wavefunction for the bosons and then find a lower energy one by taking the absolute value. The energy of this state is lower than the VMFT energy, and is also a variational upper bound, and thus the VMFT energy is also a looser variational upper bound. However the logic that leads to this breaks down in the presence of frustration, and it becomes necessary to actually calculate the “true” variational energy to make meaningful comparisons with other numerical studies.

69 4.3 Comparing the ground-state energies of hard core bosons and fermions

The intuition developed by Sedrakyan et al. [55, 97] starts from the observation that a

Fermi liquid will generically have a lower energy than a weakly interacting Bose condensate, at suﬃciently low densities, in situations where the low-energy density of states is divergent. This is the situation for hard core bosons in one spatial dimension, where it is well known that the ground state wavefunction is simply the absolute value of the ground state wavefunction for the non-interacting Fermi gas (i.e., a Slater determinant) [56, 106].

This argument is physically appealing but heuristic, however, as I show in this section, this intuitive argument can be made rigorous and quantitative.

The comparison between the ground state energies for free fermions and hard core bosons otherwise obeying the same hamiltonian (EF and EB, respectively) was recently explored in detail by Nie et al. [104, 105] with a signiﬁcant emphasis on the instances where the

“natural inequality” EB ≤ EF is violated. Following their procedure, I will now prove that there is a region of t2/t1 in this honeycomb model where a rigorous lower bound on EB is necessarily greater than the energy of corresponding free fermions. This is indeed precisely the region where the nature of the ground state is mysterious. It is worth emphasizing that this argument makes no reference to the existence or lack of a condensate; rather it proves that the lowest energy of an arbitrarily exotic many-body state of hard core bosons, in the thermodynamic limit, must be higher than that of free fermions.

The essence of the argument is a theorem [107] that, upon decomposing the hamiltonian P in terms of clusters H = i hi (with overlapping and therefore generically non-commuting P hi), the ground state energy is bounded from below: E0 ≥ i ei0, where ei0 are the lowest- lying eigenvalues of the hi. For our purposes, it is suﬃcient to show that there is some clustering for which the free fermion ground state is lower in energy. Our choice of cluster is a single hexagonal plaquette, illustrated in Fig. 4.2.

70 Explicitly, we write

X X † H = hP , hP = txx0 bxbx0 (4.9) P x,x0∈P with txx0 = t1/2 for nearest-neighbor bonds (the factor of 1/2 comes because each nearest- neighbor bond is a member of two overlapping clusters) and txx0 = t2 for next-neighbor bonds.

Figure 4.2: On the left is the plaquette tiling used to set the lower bound on the boson ground state energy. All nearest-neighbor bonds are shared between overlapping plaquettes, while the next-neighbor bonds are all captured within a single plaquette. On the right I plot the diﬀerence (E˜B − EF )/N versus t2, both in units of t1. When this diﬀerence is positive, the fermion ground state energy is lower. This is the region where the noninteracting fermion band structure has “moats” (see [97]) and where the boson ground state is potentially not a condensate.

Exact diagonalization of HB on this plaquette at µ = 0 then gives a lowest-lying energy

ei0(t2/t1) that, considering there are N/2 plaquettes in a honeycomb lattice of N sites, can

be used to bound the lowest-eigenvalue of HB from below,

E E˜ e (t /t ) B ≥ B = 0 2 1 (4.10) N N 2 which is readily calculated. Now we can ask what the fermion ground-state energy is. This

71 in turn is easily evaluated from the non-interacting energy spectrum, √ " !# t2 kx 3ky E±(kx, ky)/t1 = 2 cos kx + 4 cos cos t1 2 2 v √ u ! u kx 3ky ± t3 + 2 cos k + 4 cos cos (4.11) x 2 2

We can follow Nie [104] and calculate the energy of the Fermi gas at µ = 0 although this does not correspond to half-filling, or we can set µ such that we are at half-filling but now we are comparing the systems at different chemical potential. It turns out that this choice actually does not have any qualitative effect. In Fig. 4.2 I plot the energy difference E˜B −EF (choosing to set µ = 0). Where it is positive, the fermion ground state has a lower energy than a lower bound on the boson ground state energy, and this turns out to be precisely the region where the studies mentioned in the introduction propose that there is no condensate.

4.4 Monte Carlo results

We can now discuss the results of variational Monte Carlo calculations. The details of the algorithm can be found in AppendixC.

4.4.1 Variational bounds on the ground state energy

First I discuss the true variational energy, which is not hψ|HVMF |ψi, with |ψi the ground state of HVMF . Instead, we must take this state and evaluate hψ|HFJW |ψi. Since HFJW by construction has the same spectrum as HB, this is the upper bound we seek. In Fig. 4.3 I present results both for this variational energy bound as well as the associated variance.

4.4.2 One-body density matrix

There is some subtlety to calculating the one-body density matrix. Although we are working

72 Figure 4.3: (a) Variational energy versus linear dimension of the system for the bare VMFT state described in the text. Without any tuning parameters, I estimate that the inﬁnite size ground state energy per site is ∼ −0.2554, which is within 15% of the best estimate from large-scale DMRG simulations [101]. (b) The variance in the energy becomes extremely small as we scale up to larger system sizes, indicating proximity to an eigenstate of HFJW .

out though that as a matter of principle free fermion density matrices cannot exhibit oﬀ- diagonal long-range order (ODLRO), but here our estimate for the density matrix of the hardcore bosons is not just a free fermion density matrix but a complicated, non-local, many-body expectation value. This is similar to what occurs in one dimension [106], and so the restoration of ODLRO cannot be ruled out a priori.

I test for the presence of a condensate by evaluating ρxx0 , ﬁnding its largest eigenvalue, associated with the condensate fraction fc, and seeing how that eigenvalue scales with the system size. This is shown in Fig. 4.4, where it is clearly vanishing with increasing system size. This trial wavefunction does not describe a condensate.

4.5 Discussion and conclusions

In this Chapter, I have rigorously verified the intuition that suggests a fermionized ground state for hard core bosons on the frustrated honeycomb lattice in the parameter regime t2/t1 ∼ 0.2 − 0.3. I then took the fermion mean-field trial state and calculated its variational energy using the Monte Carlo method, as the mean-field energy by itself does not satisfy a variational principle. The true variational energy of this bare mean-field state is not particularly competitive with exact results and other much more complicated varia-

73 Figure 4.4: Condensate fraction fc versus the linear dimension of the system for the bare VMFT state described in the text. As the system size is increased, fc falls oﬀ rapidly.

tional states (for example the value Evar/N = -0.2815 found in [100]), but it is also not unreasonably high considering the absence of any variational parameters. Early attempts

at minimizing the energy with a simple short-ranged Jastrow factor show a substantial

reduction in energy, making this state much more competitive.

More broadly, in this chapter I have attempted to chip away at one small aspect of

the problem of Bose condensation in ﬂat bands, and the potentially exotic physics that

can emerge from the frustration of condensation. Although the model of this chapter was

particularly well suited to numerical investigation, it bears substantial similarity to the

Rashba model of the previous chapters, and although the ﬁeld is still in the earliest stages

of creating and manipulating spin-orbit coupled condensates, extreme spin-orbit coupling

is a very fertile ground for reﬁning our understanding of Bose condensation, superﬂuidity,

and even quantum magnetism and topological matter.

74 Bibliography

[1] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger. Many-body physics with ultracold gases. Rev. Mod. Phys., 80:885–964, Jul 2008.

[2] G. Wirth, M. Olschl¨ager,and¨ A. Hemmerich. Evidence for orbital superﬂuidity in the P-band of a bipartite optical square lattice. Nature Physics, 7:147–153, February 2011.

[3]M. Olschl¨ager,T.¨ Kock, G. Wirth, A. Ewerbeck, C. Morais Smith, and A. Hemmerich. Interaction-induced chiral px ±ipy superﬂuid order of bosons in an optical lattice. New Journal of Physics, 15(8):083041, August 2013.

[4] Frank Wilczek and A. Zee. Appearance of gauge structure in simple dynamical systems. Phys. Rev. Lett., 52:2111–2114, Jun 1984.

[5] Gordon Baym. Lectures on quantum mechanics. Westview Press, Boulder, Colo. u.a., 2010.

[6] G. Dresselhaus. Spin-orbit coupling eﬀects in zinc blende structures. Phys. Rev., 100:580–586, Oct 1955.

[7] Roland Winkler. Spin-orbit coupling eﬀects in two-dimensional electron and hole systems. Springer, Berlin; New York, 2003.

[8] Y. A. Bychkov and E.´ I. Rashba. Properties of a 2D electron gas with lifted spectral degeneracy. Soviet Journal of Experimental and Theoretical Physics Letters, 39:78, January 1984.

[9] S. Banerjee, O. Erten, and M. Randeria. Ferromagnetic exchange, spin-orbit coupling and spiral magnetism at the LaAlO3/SrTiO3 interface. Nature Physics, 9:626–630, October 2013.

[10] Igor Zuti´c,Jaroslavˇ Fabian, and S. Das Sarma. Spintronics: Fundamentals and ap- plications. Rev. Mod. Phys., 76:323–410, Apr 2004.

[11] M. Z. Hasan and C. L. Kane. Colloquium. Rev. Mod. Phys., 82:3045–3067, Nov 2010.

75 [12] E. A. Cornell and C. E. Wieman. Nobel lecture: Bose-einstein condensation in a dilute gas, the ﬁrst 70 years and some recent experiments. Rev. Mod. Phys., 74:875–893, Aug 2002.

[13] Wolfgang Ketterle. Nobel lecture: When atoms behave as waves: Bose-einstein condensation and the atom laser. Rev. Mod. Phys., 74:1131–1151, Nov 2002.

[14] Matthew P. A. Fisher, Peter B. Weichman, G. Grinstein, and Daniel S. Fisher. Boson localization and the superﬂuid-insulator transition. Phys. Rev. B, 40:546–570, Jul 1989.

[15] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller. Cold bosonic atoms in optical lattices. Phys. Rev. Lett., 81:3108–3111, Oct 1998.

[16] Daniel S. Rokhsar and B. G. Kotliar. Gutzwiller projection for bosons. Phys. Rev. B, 44:10328–10332, Nov 1991.

[17] Jean Dalibard, Fabrice Gerbier, Gediminas Juzeli¯unas,and Patrik Ohberg.¨ Collo- quium. Rev. Mod. Phys., 83:1523–1543, Nov 2011.

[18] N. Goldman, G. Juzeliunas, P. Ohberg, and I. B. Spielman. Light-induced gauge ﬁelds for ultracold atoms. ArXiv e-prints, August 2013.

[19] Y. Aharonov and D. Bohm. Signiﬁcance of electromagnetic potentials in the quantum theory. Phys. Rev., 115:485–491, Aug 1959.

[20] Akira Tonomura, Nobuyuki Osakabe, Tsuyoshi Matsuda, Takeshi Kawasaki, Junji Endo, Shinichiro Yano, and Hiroji Yamada. Evidence for aharonov-bohm eﬀect with magnetic ﬁeld completely shielded from electron wave. Phys. Rev. Lett., 56:792–795, Feb 1986.

[21] M. V. Berry. Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 392(1802):45– 57, 1984.

[22] P. Bruno, V. K. Dugaev, and M. Taillefumier. Topological hall eﬀect and berry phase in magnetic nanostructures. Phys. Rev. Lett., 93:096806, Aug 2004.

[23] Tin-Lun Ho and V. B. Shenoy. Local spin-gauge symmetry of the bose-einstein condensates in atomic gases. Phys. Rev. Lett., 77:2595–2599, Sep 1996.

[24] B Estienne, S M Haaker, and K Schoutens. Particles in non-abelian gauge potentials: Landau problem and insertion of non-abelian ﬂux. New Journal of Physics, 13(4):045012, 2011.

[25] Y.-J. Lin, K. Jim´enez-Garc´ıa,and I. B. Spielman. Spin-orbit-coupled Bose-Einstein condensates. Nature, 471:83–86, March 2011.

[26] M. C. Beeler, R. A. Williams, K. Jim´enez-Garc´ıa,L. J. Leblanc, A. R. Perry, and I. B. Spielman. The spin Hall eﬀect in a quantum gas. Nature, 498:201–204, June 2013.

76 [27] L J LeBlanc, M C Beeler, K Jim´enez-Garc´ıa,A R Perry, S Sugawa, R A Williams, and I B Spielman. Direct observation of zitterbewegung in a boseeinstein condensate. New Journal of Physics, 15(7):073011, 2013.

[28] H. Zhai. Spin-Orbit Coupled Quantum Gases. International Journal of Modern Physics B, 26:30001, 2012.

[29] V. Galitski and I. B. Spielman. Spin-orbit coupling in quantum gases. Nature, 494:49– 54, February 2013.

[30] H. Zhai. Degenerate Quantum Gases with Spin-Orbit Coupling. ArXiv e-prints, March 2014.

[31] J. Higbie and D. M. Stamper-Kurn. Periodically dressed bose-einstein condensate: A superﬂuid with an anisotropic and variable critical velocity. Phys. Rev. Lett., 88:090401, Feb 2002.

[32] J. Higbie and D. M. Stamper-Kurn. Generating macroscopic-quantum-superposition states in momentum and internal-state space from bose-einstein condensates with repulsive interactions. Phys. Rev. A, 69:053605, May 2004.

[33] D M Stamper-Kurn. Anisotropic dissipation of superﬂuid ﬂow in a periodically dressed bose-einstein condensate. New Journal of Physics, 5(1):50, 2003.

[34] Qizhong Zhu, Chuanwei Zhang, and Biao Wu. Exotic superﬂuidity in spin-orbit coupled bose-einstein condensates. EPL (Europhysics Letters), 100(5):50003, 2012.

[35] S.-C. Ji, J.-Y. Zhang, L. Zhang, Z.-D. Du, W. Zheng, Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan. Experimental determination of the ﬁnite-temperature phase diagram of a spin-orbit coupled Bose gas. Nature Physics, 10:314–320, April 2014.

[36] Chunlei Qu, Chris Hamner, Ming Gong, Chuanwei Zhang, and Peter Engels. Obser- vation of zitterbewegung in a spin-orbit-coupled bose-einstein condensate. Phys. Rev. A, 88:021604, Aug 2013.

[37] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven. Signatures of majorana fermions in hybrid superconductor- semiconductor nanowire devices. Science, 336(6084):1003–1007, 2012.

[38] D. L. Campbell, G. Juzeli¯unas,and I. B. Spielman. Realistic rashba and dresselhaus spin-orbit coupling for neutral atoms. Phys. Rev. A, 84:025602, Aug 2011.

[39] Tin-Lun Ho. Spinor bose condensates in optical traps. Phys. Rev. Lett., 81:742–745, Jul 1998.

[40] D. M. Stamper-Kurn, M. R. Andrews, A. P. Chikkatur, S. Inouye, H.-J. Miesner, J. Stenger, and W. Ketterle. Optical conﬁnement of a bose-einstein condensate. Phys. Rev. Lett., 80:2027–2030, Mar 1998.

[41] Chao-Ming Jian and Hui Zhai. Paired superﬂuidity and fractionalized vortices in systems of spin-orbit coupled bosons. Phys. Rev. B, 84:060508, Aug 2011.

77 [42] Tudor D. Stanescu, Brandon Anderson, and Victor Galitski. Spin-orbit coupled bose- einstein condensates. Phys. Rev. A, 78:023616, Aug 2008.

[43] Tin-Lun Ho and Shizhong Zhang. Bose-einstein condensates with spin-orbit interaction. Phys. Rev. Lett., 107:150403, Oct 2011.

[44] B. Ramachandhran, Hui Hu, and Han Pu. Emergence of topological and strongly correlated ground states in trapped rashba spin-orbit-coupled bose gases. Phys. Rev. A, 87:033627, Mar 2013.

[45] S.-W. Song, Y.-C. Zhang, H. Zhao, X. Wang, and W.-M. Liu. Fragmentation of Spin-orbit Coupled Spinor Bose-Einstein Condensates. ArXiv e-prints, May 2014.

[46] Tin-Lun Ho and Sung Kit Yip. Fragmented and single condensate ground states of spin-1 bose gas. Phys. Rev. Lett., 84:4031–4034, May 2000.

[47] Anthony J Leggett. Quantum liquids : Bose condensation and Cooper pairing in condensed-matter systems. Oxford University Press, Oxford, 2011.

[48] Chunji Wang, Chao Gao, Chao-Ming Jian, and Hui Zhai. Spin-orbit coupled spinor bose-einstein condensates. Phys. Rev. Lett., 105:160403, Oct 2010.

[49] Wu Cong-Jun, Ian Mondragon-Shem, and Zhou Xiang-Fa. Unconventional bose-einstein condensations from spin-orbit coupling. Chinese Physics Letters, 28(9):097102, 2011.

[50] S.-K. Yip. Bose-einstein condensation in the presence of artiﬁcial spin-orbit interaction. Phys. Rev. A, 83:043616, Apr 2011.

[51] Qi Zhou and Xiaoling Cui. Fate of a bose-einstein condensate in the presence of spin-orbit coupling. Phys. Rev. Lett., 110:140407, Apr 2013.

[52] Subhasis Sinha, Rejish Nath, and Luis Santos. Trapped two-dimensional condensates with synthetic spin-orbit coupling. Phys. Rev. Lett., 107:270401, Dec 2011.

[53] Hui Hu, B. Ramachandhran, Han Pu, and Xia-Ji Liu. Spin-orbit coupled weakly interacting bose-einstein condensates in harmonic traps. Phys. Rev. Lett., 108:010402, Jan 2012.

[54] Xiangfa Zhou, Yi Li, Zi Cai, and Congjun Wu. Unconventional states of bosons with the synthetic spin-orbit coupling. Journal of Physics B: Atomic, Molecular and Optical Physics, 46(13):134001, 2013.

[55] Tigran A. Sedrakyan, Alex Kamenev, and Leonid I. Glazman. Composite fermion state of spin-orbit-coupled bosons. Phys. Rev. A, 86:063639, Dec 2012.

[56] M. Girardeau. Relationship between systems of impenetrable bosons and fermions in one dimension. Journal of Mathematical Physics, 1(6):516–523, 1960.

[57] E. Duchon, Y. L. Loh, and N. Trivedi. Optical Lattice Emulators: Bose and Fermi Hubbard Models. ArXiv e-prints, November 2013.

78 [58] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, and I. Bloch. Quantum phase transition from a superﬂuid to a Mott insulator in a gas of ultracold atoms. Nature, 415:39–44, January 2002.

[59] Eugene Demler and Fei Zhou. Spinor bosonic atoms in optical lattices: Symmetry breaking and fractionalization. Phys. Rev. Lett., 88:163001, Apr 2002.

[60] L.-M. Duan, E. Demler, and M. D. Lukin. Controlling spin exchange interactions of ultracold atoms in optical lattices. Phys. Rev. Lett., 91:090402, Aug 2003.

[61] Ehud Altman, Walter Hofstetter, Eugene Demler, and Mikhail D Lukin. Phase diagram of two-component bosons on an optical lattice. New Journal of Physics, 5(1):113, 2003.

[62] A. Isacsson, Min-Chul Cha, K. Sengupta, and S. M. Girvin. Superﬂuid-insulator transitions of two-species bosons in an optical lattice. Phys. Rev. B, 72:184507, Nov 2005.

[63] A. B. Kuklov and B. V. Svistunov. Counterﬂow superﬂuidity of two-species ultracold atoms in a commensurate optical lattice. Phys. Rev. Lett., 90:100401, Mar 2003.

[64] Anatoly Kuklov, Nikolay Prokof’ev, and Boris Svistunov. Superﬂuid-superﬂuid phase transitions in a two-component bose-einstein condensate. Phys. Rev. Lett., 92:030403, Jan 2004.

[65] C. Hamner, Y. Zhang, M. A. Khamehchi, M. J. Davis, and P. Engels. Spin-orbit coupled Bose-Einstein condensates in a one-dimensional optical lattice. ArXiv e- prints, May 2014.

[66] J. Radi´c,A. Di Ciolo, K. Sun, and V. Galitski. Exotic quantum spin models in spin-orbit-coupled mott insulators. Phys. Rev. Lett., 109:085303, Aug 2012.

[67] Zi Cai, Xiangfa Zhou, and Congjun Wu. Magnetic phases of bosons with synthetic spin-orbit coupling in optical lattices. Phys. Rev. A, 85:061605, Jun 2012.

[68] D Jaksch and P Zoller. Creation of effective magnetic fields in optical lattices: the hofstadter butterfly for cold neutral atoms. New Journal of Physics, 5(1):56, 2003.

[69] K. Osterloh, M. Baig, L. Santos, P. Zoller, and M. Lewenstein. Cold atoms in non- abelian gauge potentials: From the hofstadter ”moth” to lattice gauge theory. Phys. Rev. Lett., 95:010403, Jun 2005.

[70] N Goldman, F Gerbier, and M Lewenstein. Realizing non-abelian gauge potentials in optical square lattices: an application to atomic chern insulators. Journal of Physics B: Atomic, Molecular and Optical Physics, 46(13):134010, 2013.

[71] Colin J. Kennedy, Georgios A. Siviloglou, Hirokazu Miyake, William Cody Burton, and Wolfgang Ketterle. Spin-orbit coupling and quantum spin hall eﬀect for neutral atoms without spin ﬂips. Phys. Rev. Lett., 111:225301, Nov 2013.

79 [72] Artem M. Dudarev, Roberto B. Diener, Iacopo Carusotto, and Qian Niu. Spin-orbit coupling and berry phase with ultracold atoms in 2d optical lattices. Phys. Rev. Lett., 92:153005, Apr 2004.

[73] William S. Cole, Shizhong Zhang, Arun Paramekanti, and Nandini Trivedi. Bose- hubbard models with synthetic spin-orbit coupling: Mott insulators, spin textures, and superﬂuidity. Phys. Rev. Lett., 109:085302, Aug 2012.

[74] T. Graß, K. Saha, K. Sengupta, and M. Lewenstein. Quantum phase transition of ultracold bosons in the presence of a non-abelian synthetic gauge ﬁeld. Phys. Rev. A, 84:053632, Nov 2011.

[75] Saptarshi Mandal, Kush Saha, and K. Sengupta. Superﬂuid-insulator transition of two-species bosons with spin-orbit coupling. Phys. Rev. B, 86:155101, Oct 2012.

[76] M. Gong, Y. Qian, V. W. Scarola, and C. Zhang. Dzyaloshinskii-Moriya Interaction and Spiral Order in Spin-orbit Coupled Optical Lattices. ArXiv e-prints, May 2012.

[77] Y. Qian, M. Gong, V. W. Scarola, and C. Zhang. Spin-Orbit Driven Transitions Between Mott Insulators and Finite Momentum Superﬂuids of Bosons in Optical Lattices. ArXiv e-prints, December 2013.

[78] Ryan M. Wilson, Brandon M. Anderson, and Charles W. Clark. Meron ground state of rashba spin-orbit-coupled dipolar bosons. Phys. Rev. Lett., 111:185303, Oct 2013.

[79] Sarang Gopalakrishnan, Ivar Martin, and Eugene A. Demler. Quantum quasicrystals of spin-orbit-coupled dipolar bosons. Phys. Rev. Lett., 111:185304, Oct 2013.

[80] Subir Sachdev. Quantum phase transitions. Cambridge University Press, Cambridge, 2011.

[81] Assa Auerbach. Interacting electrons and quantum magnetism. Springer, New York, 1998.

[82] I. Dzyaloshinsky. A thermodynamic theory of “weak” ferromagnetism of antiferro- magnetics. Journal of Physics and Chemistry of Solids, 4(4):241 – 255, 1958.

[83] TˆoruMoriya. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev., 120:91–98, Oct 1960.

[84] Michael Stone and Paul Goldbart. Mathematics for physics : a guided tour for graduate students. Cambridge University Press, Cambridge, 2010.

[85] K. Sheshadri, H. R. Krishnamurthy, R. Pandit, and T. V. Ramakrishnan. Superﬂuid and insulating phases in an interacting-boson model: Mean-ﬁeld theory and the rpa. EPL (Europhysics Letters), 22(4):257, 1993.

[86] Matthew Killi and Arun Paramekanti. Use of quantum quenches to probe the equilib- rium current patterns of ultracold atoms in an optical lattice. Phys. Rev. A, 85:061606, Jun 2012.

80 [87] B. Berche, E. Medina, and A. López. Spin superfluidity and spin-orbit gauge symmetry fixing. EPL (Europhysics Letters), 97(6):67007, 2012.

[88] P. Nikoli´c.Vortices and vortex states in Rashba spin-orbit-coupled condensates. ArXiv e-prints, June 2014.

[89] L. He, A. Ji, and W. Hofstetter. Bose-Bose Mixtures with Synthetic Spin-Orbit Coupling in Optical Lattices. ArXiv e-prints, April 2014.

[90] Patrick A. Lee, Naoto Nagaosa, and Xiao-Gang Wen. Doping a mott insulator: Physics of high-temperature superconductivity. Rev. Mod. Phys., 78:17–85, Jan 2006.

[91] Zhihao Xu, William S. Cole, and Shizhong Zhang. Mott-superﬂuid transition for spin- orbit-coupled bosons in one-dimensional optical lattices. Phys. Rev. A, 89:051604, May 2014.

[92] Sebastian D. Huber and Ehud Altman. Bose condensation in ﬂat bands. Phys. Rev. B, 82:184502, Nov 2010.

[93] Eduardo Fradkin. Jordan-wigner transformation for quantum-spin systems in two dimensions and fractional statistics. Phys. Rev. Lett., 63:322–325, Jul 1989.

[94] G. S. Canright, S. M. Girvin, and A. Brass. Statistics and ﬂux in two dimensions. Phys. Rev. Lett., 63:2291–2294, Nov 1989.

[95] Claudius Gros, S. M. Girvin, G. S. Canright, and M. D. Johnson. Variational theorem for vector-mean-ﬁeld theories of statistical transmutation. Phys. Rev. B, 43:5883– 5907, Mar 1991.

[96] Tetsuo Matsui, Kazuhiko Sakakibara, and Hiroshi Takano. Fermions in magnetic ﬁeld versus hard-core bosons in two dimensions. Physics Letters A, 194:413 – 418, 1994.

[97] Tigran A. Sedrakyan, Leonid I. Glazman, and Alex Kamenev. Absence of bose condensation on lattices with moat bands. Phys. Rev. B, 89:201112, May 2014.

[98] Christopher N. Varney, Kai Sun, Victor Galitski, and Marcos Rigol. Kaleidoscope of exotic quantum phases in a frustrated xy model. Phys. Rev. Lett., 107:077201, Aug 2011.

[99] C. N. Varney, K. Sun, V. Galitski, and M. Rigol. Quantum phases of hard-core bosons in a frustrated honeycomb lattice. New Journal of Physics, 14(11):115028, November 2012.

[100] Juan Carrasquilla, Andrea Di Ciolo, Federico Becca, Victor Galitski, and Marcos Rigol. Nature of the phases in the frustrated xy model on the honeycomb lattice. Phys. Rev. B, 88:241109, Dec 2013.

[101] Zhenyue Zhu, David A. Huse, and Steven R. White. Unexpected z-direction ising antiferromagnetic order in a frustrated spin-1/2 j1-j2 xy model on the honeycomb lattice. Phys. Rev. Lett., 111:257201, Dec 2013.

81 [102] J. Oitmaa and R. R. P. Singh. Phase diagram of the frustrated quantum-xy model on the honeycomb lattice studied by series expansions: Evidence for proximity to a bicritical point. Phys. Rev. B, 89:104423, Mar 2014.

1 [103] R. F. Bishop, P. H. Y. Li, and C. E. Campbell. Frustrated spin- 2 J1˘J2 isotropic xy model on the honeycomb lattice. Phys. Rev. B, 89:214413, Jun 2014.

[104] Wenxing Nie, Hosho Katsura, and Masaki Oshikawa. Ground-state energies of spinless free fermions and hard-core bosons. Phys. Rev. Lett., 111:100402, Sep 2013.

[105] W. Nie, H. Katsura, and M. Oshikawa. Particle Statistics, Frustration, and Ground- State Energy. ArXiv e-prints, January 2014.

[106] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol. One dimensional bosons: From condensed matter systems to ultracold gases. Rev. Mod. Phys., 83:1405– 1466, Dec 2011.

[107] Rolf Tarrach and Roser Valent. Exact lower bounds to the ground-state energy of spin systems: The two-dimensional s=1/2 antiferromagnetic heisenberg model. Phys. Rev. B, 41:9611–9613, May 1990.

[108] Ramamurti Shankar. Principles of quantum mechanics. Plenum Press, New York, 1994.

82 Appendix A Numerical solution of the Rashba hamiltonian in a harmonic trap

In this appendix, I provide the details that enter into calculating the single-particle eigenvalues and eigenstates of the Rashba hamiltonian in the presence of an additional harmonic conﬁnement. Our starting point is the hamiltonian introduced in Chapter2, which we will consider in two spatial dimensions.

p2 1 H = + λ (σ × p) · e + Mω2 r2 (A.1) 2M z 2 T

In that chapter I also pointed out that the combined spin and real-space rotational symmetry implies that eigenstates should take the form of a spinor   eimϕ φm↑(r) ψm(r) = √   (A.2) 2π  iϕ  φm↓(r)e

I will also make use of the fact that time-reversal symmetry implies that all of the single-

particle states are doubly degenerate. In our case, this degeneracy means each of the

negative total angular momentum states (m < 0) can be constructed explicitly from its

time-reversed (m ≥ 0) partner. In everything that proceeds we therefore take a positive

semideﬁnite m. The Kramers partners can be constructed afterwards.

83 A.1 Series expansion in harmonic trap solutions

p2 1 2 2 Absent spin orbit coupling, the eigenfunctions of the hamiltonian H0 = 2M + 2 MωT r obviously retain the form in Eq. (A.2). It remains to solve for the radial dependence. Two

things ﬁrst. We can write the Laplacian in polar coordinates, and also express all lengths

−1/2 in a natural unit, the trap length ` = (MωT ) . Writing ξ = r/` we obtain

2 2 2 2 1 2 ωT 1 ∂ ∂ 1 ∂ ωT 1 ∂ ∂ 1 ∂ − ∇ = − + 2 + 2 2 = − + 2 + 2 2 (A.3) 2M 2(MωT ) r ∂r ∂r r ∂ϕ 2 ξ ∂ξ ∂ξ ξ ∂ϕ

Using the angular dependence we’ve already determined, we’re now left with an equation for the radial function. Let’s consider the up-spin component,

1 1 ∂ ∂2 m2 1 H φ (r) = ω − + − + ξ2 φ (r) = εφ (r) (A.4) 0 m↑ T 2 ξ ∂ξ ∂ξ2 ξ2 2 m↑ m↑

The solutions to this equation are well-known, as this is a textbook problem [108], and

are given by (for any integer n)

(n) (n) φm↑(r) = Rn,m(ξ) and φm↓(r) = Rn,m+1(ξ) (A.5)

where s 1 n! m − 1 ξ2 m 2 R (ξ) = 2 ξ e 2 L (ξ ) (A.6) n,m ` (n + m)! n

H0Rn,m(ξ) = ωT (2n + m + 1)Rn,m(ξ) (A.7)

m where Ln (x) is the generalized Laguerre polynomial, which we have implemented using Mathematica’s LaguerreL[n,m,x]. For future convenience, I deﬁne the units and combi-

1 q n! natoric factor into a coeﬃcient Cn,m = ` 2 (n+m)! . I reiterate that m was the angular momentum quantum number, and n is the usual radial quantum number for the isotropic oscillator.

The next step of course is to restore the spin orbit coupling. The hamiltonian has a

84 matrix form (with the obvious n dependence suppressed for notational convenience)     1 2 1 2 imϕ − 2 ∇ξ + 2 ξ λ(∂x − i∂y) φm↑(r)e H = ωT      1 2 1 2   i(m+1)ϕ  −λ(∂x + i∂y) − 2 ∇ξ + 2 ξ φm↓(r)e   imϕ φm↑(r)e = εm   (A.8)  i(m+1)ϕ  φm↓(r)e where again we can handle all of the angular dependence ﬁrst. We use ∂x ± i∂y =

e±iφ h ∂ i ∂ i ` ∂ξ ± ξ ∂ϕ , and recall the dimensionless spin-orbit coupling strength γ = (Mλ`). We reduce this then to       (m) m+1 H0 γωT ∂ξ + ξ φm↑(r) φm↑(r)     = εm   (A.9)  m (m+1)      −γωT ∂ξ − ξ H0 φm↓(r) φm↓(r)

The superscript on H0 is simply a reminder of which m lives in its diﬀerential operator deﬁnition.

Now, the harmonic oscillator states constructed above will no longer be eigenstates –

but they do provide a complete basis in which to expand any arbitrary state, and the fact

that they are harmonic oscillator eigenstates will still be very useful in what follows. In any

case, we can write

X X φm↑(r) = Am,nRn,m(ξ), φm↓(r) = Bm,nRn,m+1(ξ) (A.10) n n after which the Schr¨odingerequation becomes the pair of coupled equations

X m + 1 ω (2n + m + 1)A R + γB ∂ + R − ε˜ A R = 0 T m,n n,m m,n ξ ξ n,m+1 m m,n n,m n (A.11)

X m ω (2n + m + 2)B R − γA ∂ − R − ε˜ B R = 0 T m,n n,m+1 m,n ξ ξ n,m m m,n n,m+1 n (A.12)

Typically this is the point where one would use the orthogonality of the functions Rn,m, integrating these equations against a “selection” function that would leave us with a set of

85 linear equations that the coefficients {A, B} must satisfy. However the presence of deriva- tives make it clear that the effect of SOC is to couple together many harmonic oscillator states with different radial quantum numbers n. Thus this “bare” n is no longer a good

quantum number. We need to solve the above equations to ﬁnd eigenstates that carry a new

radial quantum number which will be conserved. Solving these equations is not a trivial

task and writing out the bookkeeping involved is the purpose of the present Appendix.

m First we need the orthogonality relationship for Ln ,

Z ∞ m −x m m (m + n)! dx x e Ln (x)Ll (x) = δnl (A.13) 0 n! which immediately implies

Z ∞ 1 2 d(ξ ) Rn,m(ξ)Rl,m(ξ) = δnl (A.14) 2 0

In fact, we could take this and apply it to Eq. (A.12) straight away; we get " # X Z m (2l + m + 2)B − γ A ξdξ R ∂ − R =ε ˜ B (A.15) m,l m,n l,m+1 ξ ξ n,m m m,l n but the evaluation of the integral in the brackets is no easy task. Lacking any particularly elegant way to proceed, I do as follows.

To use the orthogonality, I need the recurrence relationships to raise and lower m-indices.

I also need to work out the action of the derivative operator. Also, I begin with this example because, by coincidence, it is signiﬁcantly simpler than the corresponding integral arising from Eq. (A.11). First I note that

m m+1 − 1 ξ2 m+1 2 m 2 − ∂ − R = C ξ e 2 2L (ξ ) + L (ξ ) (A.16) ξ ξ n,m n,m n−1 n

m+1 − 1 ξ2 m+1 2 m+1 2 = Cn,mξ e 2 Ln−1 (ξ ) + Ln (ξ ) (A.17) Cn,m Cn,m = Rn−1,m+1 + Rn,m+1 (A.18) Cn−1,m+1 Cn,m+1 √ √ = nRn−1,m+1 + n + m + 1Rn,m+1 (A.19)

86 where I have used the recursion identity

m 2 m+1 2 m+1 2 Ln (ξ ) = Ln (ξ ) − Ln−1 (ξ ) (A.20)

Now we can insert this back into the integral, and straightforwardly apply orthogonality.

The ﬁnal result is

√ √ (2l + m + 2)Bm,l + γ( l + 1)Am,l+1 + γ( l + m + 1)Am,l =ε ˜mBm,l (A.21)

Next we have to pursue the evaluation of the integral that shows up when using the orthogonality relation on Eq. (A.11). This is somewhat messier. The recursion identities we will need are

m + 1 n + 1 ξLm+2(ξ2) = − ξ Lm+1(ξ2) − Lm (ξ2) (A.22) n−1 ξ n ξ n+1

2 m+1 2 m 2 m 2 ξ Ln (ξ ) = (n + m + 1)Ln (ξ ) − (n + 1)Ln+1(ξ ) (A.23)

Let us dig in

m + 1 2(m + 1) m+1 − 1 ξ2 m+2 ∂ξ − Rn,m+1 = Rn,m+1 − ξRn,m+1 − 2Cn,m+1ξ e 2 ξLn−1 (A.24) ξ ξ | {z } ≡X The last term on the right-hand side evaluates as

m+1 − 1 ξ2 m + 1 m+1 2 n + 1 m 2 X = 2C ξ e 2 − ξ L (ξ ) − L (ξ ) (A.25) n,m+1 ξ n ξ n+1

2(m + 1) Cn,m+1 = Rn,m+1 − 2ξRn,m+1 − 2 (n + 1)Rn+1,m (A.26) ξ Cn+1,m so that m + 1 √ ∂ − R = ξR + 2 n + 1R (A.27) ξ ξ n,m+1 n,m+1 n+1,m

87 Next we have to “lower” the ﬁrst term on the RHS, using a recursion identity.

m − 1 ξ2 2 m+1 ξRn,m+1 = Cn,m+1ξ e 2 ξ Ln (A.28) m − 1 ξ2 Cn,m+1 m Cn,m+1 m = ξ e 2 (n + m + 1) Cn,mLn − (n + 1) Cn+1,mLn+1 (A.29) Cn,m Cn+1,m √ √ = n + m + 1Rn,m − n + 1Rn+1,m (A.30)

So that ﬁnally putting it all together we get

m + 1 √ √ ∂ − R = n + m + 1R + n + 1R (A.31) ξ ξ n,m+1 n,m n+1,m which we can then put into Eq. (A.11) and again apply the orthogonality relationship.

Finally, after all of this eﬀort, Eqs. (A.11) and (A.12) become a set of coupled linear equations for the coeﬃcients.

√ √ (2l + m + 1)Am,l + γ lBm,l−1 + l + m + 1Bm,l =ε ˜mAm,l (A.32)

√ √ (2l + m + 2)Bm,l + γ l + 1Am,l+1 + l + m + 1Am,l =ε ˜mBm,l (A.33)

T This is now straightforward to handle. For each ﬁxed m, we can express (Al,Bl) as a column vector, write out the explicit matrix of coeﬃcients, and diagonalize it. In a compact notation,       ↑ T H0 Λ Am Am     = εn,m   (A.34)  ↓      Λ H0 Bm Bm with

↑,(m) ↓,(m) (H0 )ll0 = ωT (2l + m + 1)δll0 , (H0 )ll0 = ωT (2l + m + 2)δll0 (A.35) and √ √ (m) (Λ )ll0 = γωT l + 1δl+1,l0 + γωT l + m + 1δll0 (A.36)

To actually implement this on a finite computer, we need to pick some truncation l ∈ [0, lmax]. The required value depends on γ, with higher values obviously necessitating a higher cutoff. We typically set the cutoff by hand, raising it until Am,lmax and Bm,lmax are 88 zero.

Solving this gives us the spectrum εn,m (n being a new quantum number that characterizes which energy-ordered eigenstate of Eq. (A.34) is under consideration) and expansion coeﬃcients {Am,l,Bm,l}. Together with Eqs. (A.2) and Eqs. (A.10) this is a complete and numerically exact speciﬁcation of the single-particle solutions.

89 Appendix B Effective spin hamiltonian from two-site perturbation theory

In Chapter3 I pointed out that the matrix elements of Heff can be written as

0 0 † † |σ1σ2ihτ1τ2| = (b1σb1τ )(b2σ0 b2τ 0 ) (B.1)

which can then be expressed in local spin operators through the transformations

1 1 b† b = + Sz, b† b = − Sz, b† b = S+, b† b = S− (B.2) i↑ i↑ 2 i i↓ i↓ 2 i i↑ i↓ i i↓ i↑ i

In this Appendix, we show the explicit expansion of Heff and rearrangement of terms that give the more familiar magnetic model presented in Chapter3. First the expansion, P which is simply writing the sum Heff = αβ(Heff )αβ|αihβ| out explicitly. We will simplify the notation by using Vαβ ≡ (Heff )αβ

† † † † Heff = V11b1↑b1↑b2↑b2↑ + V22b1↑b1↑b2↓b2↓+

† † † † V33b1↓b1↓b2↑b2↑ + V44b1↓b1↓b2↓b2↓+ † † † † V12b1↑b1↑b2↑b2↓ + V13b1↑b1↓b2↑b2↑+

† † † † V14b1↑b1↓b2↑b2↓ + V23b1↑b1↓b2↓b2↑+ † † † † V24b1↑b1↓b2↓b2↓ + V34b1↓b1↓b2↑b2↓ + h.c. (B.3)

90 Now we insert the substitution of spin operators

1 1 1 1 H = V + Sz + Sz + V + Sz − Sz + eff 11 2 1 2 2 22 2 1 2 2 1 1 1 1 V − Sz + Sz + V − Sz − Sz + 33 2 1 2 2 44 2 1 2 2 1 1 V + Sz S+ + V S+ + Sz + V S+S++ 12 2 1 2 13 1 2 2 14 1 2 1 1 V S+S− + V S+ − Sz + V − Sz S+ + h.c. (B.4) 23 1 2 24 1 2 2 34 2 1 2

Next we begin the process of arranging these terms

1 H = (V + V + V + V ) + (V − V − V + V ) SzSz+ eff 4 11 22 33 44 11 22 33 44 1 2 1 1 (V + V − V − V ) Sz + (V − V + V − V ) Sz+ 2 11 22 33 44 1 2 11 22 33 44 2 1 (V + V )S+ + (V + V )S+ + (V + V )∗S− + (V + V )∗S− + 2 13 24 1 12 34 2 13 24 1 12 34 2 z + + z ∗ z − ∗ − z (V12 − V34)S1 S2 + (V13 − V24)S1 S2 + (V12 − V34) S1 S2 + (V13 − V24) S1 S2 +

+ + + − ∗ − − ∗ − + V14S1 S2 + V23S1 S2 + V14S1 S2 + V23S1 S2 (B.5)

The spin raising and lowering operators are convenient for many purposes, but here we will revert back to the cartesian components through S± = Sx ± iSy, and, throwing out the overall constant term, we can write

z z Heff = (V11 − V22 − V33 + V44) S1 S2 + 1 1 (V + V − V − V ) Sz + (V − V + V − V ) Sz+ 2 11 22 33 44 1 2 11 22 33 44 2 x x Re (V13 + V24) S1 + Re (V12 + V34) S2 −

y y Im (V13 + V24) S1 − Im (V12 + V34) S2 +

z x x z 2 Re (V12 − V34) S1 S2 + 2 Re (V13 − V24) S1 S2 −

z y y z 2 Im (V12 + V34) S1 S2 − 2 Im (V13 − V24) S1 S2 +

x x y y 2 Re (V23 + V14) S1 S2 + 2 Re (V23 − V14) S1 S2 −

y x x y 2 Im (V14 + V23) S1 S2 − 2 Im (V14 − V23) S1 S2 (B.6)

Looking at the above expression, we recognize that it can be written in the much more

91 compact form

X a b Heff = S1 JabS2 + b1 · S1 + b2 · S2 (B.7) ab with

1 b = Re (V + V )x ˆ − Im (V + V )y ˆ + (V + V − V − V )z ˆ (B.8) 1 13 24 13 24 2 11 22 33 44 1 b = Re (V + V )x ˆ − Im (V + V )y ˆ + (V − V + V − V )z ˆ (B.9) 2 12 34 12 34 2 11 22 33 44 and the exchange tensor given by   2 Re (V23 + V14) −2 Im (V14 − V23) 2 Re (V13 − V24)     J =  −2 Im (V + V ) 2 Re (V − V ) −2 Im (V − V )  (B.10)  14 23 23 14 13 24    2 Re (V12 − V34) −2 Im (V12 + V34)(V11 − V22 − V33 + V44)

The last step than we can take with all of this hard won generality is to decompose J

T T into its symmetric JS = (J + J )/2 and antisymmetric JA = (J − J )/2 parts, the latter of which is entirely responsible for the Dzyaloshinski-Moriya interaction in Eq. (3.33).

Having built the above framework, it is now simply bookkeeping (carried out in Math- ematica) to insert a particular model, calculate the matrix elements (Heff )αβ, and ﬁnally generate the exchange matrix and b vectors. Thus, inserting     cos θ sin θ cos θ i sin θ +ˆx   +ˆy   h = −t   , h = −t   , (B.11) − sin θ cos θ i sin θ cos θ as well as U↑↑ = U↓↓ = U, U↑↓ = U↓↑ = gU, we can quickly obtain that the matrix elements conspire such that b1 = b2 = 0 along either bond. This should be expected as the underlying model was time-reversal symmetric, so we could have thrown these terms out by hand. The exchange tensors are, respectively,

  − cos(2θ) 0 g sin(2θ) 2   +ˆx 4t   J =  0 −1 0  (B.12) gU     −g sin(2θ) 0 (1 − 2g) cos(2θ)

92   −1 0 0 2   +ˆy 4t   J =  0 − cos(2θ) −g sin(2θ)  (B.13) gU     0 g sin(2θ) (1 − 2g) cos(2θ) which match precisely the expressions of Eq. (3.33) and the paragraph which follows it.

93 Appendix C The variational principle and variational Monte Carlo

The variational principle in quantum mechanics ensures that we can place an upper bound on the ground-state energy of any hamiltonian, by evaluating its expectation value with respect to any vector in its Hilbert space, i.e.,

1 E ≤ E = hΨ|H|Ψi,A = hΨ|Ψi (C.1) 0 var A

which is easily proved by from the eigenstate expansion of H. This is a powerful tool, since

we often cannot solve H exactly, but might have a good guess for its ground state, which

we would like to check, and compare against other approaches. This guess can be further

reﬁned by minimizing Evar with respect to any parameters that |Ψi might depend on. For a generic many-body problem, however, the diﬃculty that arises is that there might not be any way to analytically calculate hΨ|H|Ψi. This is where Monte Carlo techniques

enter the picture. By expanding the trial state |Ψi in some convenient basis, a natural

1 2 probability distribution emerges, Pσ = A |hσ|Ψi| . Although this basis will usually be too large to sum over explicitly, Metropolis Monte Carlo is a well-known tool for estimating

Evar by stochastically sampling Pσ. Let us make these ideas more explicit. Consider an expansion in real-space conﬁgurations of N identical particles, R = {r1, r2,..., rN },

X |Ψi = ψ(r1, r2,..., rN )|Ri (C.2) R

94 where, to remain in the correct Hilbert space (which is a condition on the validity of the variational principle), the wavefunction ψ(r1, r2,..., rN ) must have the correct exchange symmetry (i.e., symmetric for bosons, antisymmetric for fermions).

We can write the variational bound on the energy as an expectation value with respect

to a probability distribution as follows,

1 1 X E = hΨ|H|Ψi = ψ∗(R)ψ(R0)hR|H|R0i (C.3) var A A R,R0 2 X |ψ(R)| X ψ(R0) = hR|H|R0i (C.4) A ψ(R) R | {z } R0 ≡P (R) | {z } ≡Ω(R)

As noted before, evaluating the explicit sum over R is typically impossible. However we can estimate Evar if we can perform the following steps:

1. Generate a set of M conﬁgurations, C = {R1, R2,..., RM }, where any particular R appears a number of times proportional to its relative probability.

2. For all R ∈ C, evaluate Ω(R).

This yields our estimate 1 X E ' Ω(R) (C.5) var M R∈C Any other expectation value can of course be calculated likewise, by simply replacing H

with the appropriate operator.

Now, the question is, how does one perform the ﬁrst step, ensuring a sequence of con-

ﬁgurations that resemble P (R)? This is precisely the problem solved by the Metropolis

algorithm. Starting from a conﬁguration R1, we make a “move” to a trial conﬁguration R2,

and “accept” this move with some transition probability TR2←R1 . The key to this approach is that the transition probabilities must satisfy the detailed balance condition

TR2←R1 P (R1) = TR1←R2 P (R2) (C.6)

This condition can be met with a simple choice; take TR2←R1 = min (P (R2)/P (R1), 1). It is quickly verified by substitution that this choice satisfies Eq. (C.6). More complicated solu- 95 tions exist, which might be advantageous in cases where this choice yields poor performance, but we find the above sufficient for our purposes.

The above considerations still need to be combined into an algorithm that a computer is capable of carrying out. A sketch of such an algorithm is as follows:

1. Generate an initial conﬁguration Ri entirely at random.

2. Propose a new trial conﬁguration Rt

3. Choose a random number r uniformly from [0, 1], and calculate the transition proba-

bility p = min (P (Rt)/P (Ri), 1).

4. If r < p, accept the trial conﬁguration by taking Ri+1 = Rt. Otherwise, reject the

trial conﬁguration and take Ri+1 = Ri. Either way, increment i and return to step 2 until i = N + M.

5. Throw away the ﬁrst N conﬁgurations as “thermalization” steps. We started from a

completely random point in conﬁguration space, and thereby artiﬁcially increased its

representation in the conﬁguration sequence. Now use the remaining M conﬁgurations

to calculate estimates for observable quantities.

Before moving on, another issue worth consideration is ergodicity. The above approach only gives a reliable estimate if every possible configuration can be reached from any other configuration with a finite number of “moves.” Although this is usually easy to satisfy in principle, in practice we only keep a finite set of configurations, and therefore only get a limited number of moves. It is therefore important to be judicious in the construction of trial moves. If we generate a new configuration R2 through a tiny deviation from R1, then we will typically find TR2←R1 ∼ 1 (using the simple choice above), and will therefore accept most moves. However we will also move rather slowly through the configuration space, generate highly correlated samples, and fail, from a practical standpoint, to meet the spirit of ergodicity. At the other extreme, we could generate the new configuration completely at random. We will very likely construct a series of uncorrelated states, but

96 if these are low probability states, they will not get accepted. Thus we again stay at R1, heavily oversampling that conﬁguration.

C.1 Application of VMC to fermionized hamiltonians

The Jordan-Wigner transformation in Chapter4 yields a hamiltonian HF with the exact same spectrum as HB, which means it satisﬁes the variational principle for the exact (hardcore boson) ground state energy, but must be evaluated with respect to fermion wavefunctions. In this section we outline some of the technical details related to the speciﬁc calculations in that Chapter.

To perform the VMC calculations, we need to know how HF acts on position eigenstates,

  X X † † HF |Ri = txx0 exp i [Gx−x00 − Gx0−x00 ] fx00 fx00  fxfx0 |Ri (C.7) xx0 x006=x,x0 Consider first just a single term from the hamiltonian; it vanishes unless |Ri has a fermion on site x0 and is empty on site x. For a configuration that satisfies this,   xx0 0 0 X 0 0 HF |R, rj = x i = |R , rj = xi × txx exp i [Gx−ri − Gx −ri ] (C.8) i6=j

The result of this is that we need to evaluate

N X X ψ(r1, r2,..., rN |rj ← rj + δ) Ω(R) = × ψ(r1, r2,..., rN ) j=1 δ   X h i t(rj +δ),rj exp i G(rj +δ)−rk − Grj −rk  (C.9) k6=j

The most computationally intensive task in the algorithm is the evaluation of the wave-

function ψ(r1, r2,..., rN ). For the mean-ﬁeld fermion wavefunctions we consider, this has

3 the form of a Slater determinant, which requires O Np operations (Np being the number of particles in the simulation). However when we look at the above formula, we notice

that what we actually need is the ratio of two Slater determinants which diﬀer by a sin-

2 gle column. Somewhat surprisingly, this is a quantity which can be calculated in O Np

97 operations, at the expense of carrying around the inverse of the Slater matrix. Equally surprisingly, updating the inverse of the Slater matrix after changing a single column can

2 also be accomplished in O Np operations. These procedures have the names “matrix determinant lemma” and “Sherman-Morrison-Woodbury formula”, respectively.