Quantum Phase Transitions in Disordered Superconductors and Detection of Modulated Superfluidity in Imbalanced Fermi Gases

Home , Density of states, Quantum Monte Carlo, Quantum rotor model

Quantum phase transitions in disordered superconductors and detection of modulated superfluidity in imbalanced Fermi 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 Mason Swanson, B.S. Graduate Program in Physics

The Ohio State University 2014

Dissertation Committee:

Professor Nandini Trivedi, Advisor

Professor Eric Braaten

Professor Thomas Lemberger

Professor Mohit Randeria c Copyright by

Mason Swanson

2014 Abstract

Ultracold atomic gas experiments have emerged as a new testing ground for ﬁnding elusive, exotic states of matter. One such state that has eluded detection is the Larkin-Ovchinnikov

(LO) phase predicted to exist in a system with unequal populations of up and down fermions.

This phase is characterized by periodic domain walls across which the order parameter changes sign and the excess polarization is localized Despite ﬁfty years of theoretical and experimental work, there has so far been no unambiguous observation of an LO phase. In this thesis, we propose an experiment in which two fermion clouds, prepared with unequal population imbalances, are allowed to expand and interfere. We show that a pattern of staggered fringes in the interference is unequivocal evidence of LO physics.

Finally, we study the superconductor-insulator quantum phase transition. Both superconductivity and localization stand on the shoulders of giants – the BCS theory of superconductivity and the Anderson theory of localization. Yet, when their combined eﬀects are considered, both paradigms break down, even for s-wave superconductors. In this work, we calculate the dynamical quantities that help guide present and future experiments. Specif- ically, we calculate the conductivity σ(ω) and the bosonic (pair) spectral function P (ω)

from quantum Monte Carlo simulations across clean and disorder-driven superconductor-

insulator transitions (SIT). Using these quantities, we identify characteristic energy scales

in both the superconducting and insulating phases that vanish at the transition due to en-

hanced quantum ﬂuctuations, despite the persistence of a robust fermionic gap across the

SIT. The clean and disordered transition are compared throughout, and we ﬁnd that disor-

der leads to enhanced absorption in σ(ω) at low frequencies and a change in the universality

class, although the underlying T = 0 quantum critical point remains in both transitions.

ii Vita

January 27, 1986 ...... Born—Minot ND

May 2009 ...... B.S.—North Dakota State University, Fargo, ND September 2010 - present ...... Graduate Research Fellow—The Ohio State University, Columbus, OH

Publications

Spectral properties and duality across the superconductor-insulator transition, M. Swanson, N. Trivedi, and M. Randeria, in preparation

Observation of the Higgs Mode in Disordered Superconductors Close to a Quantum Phase Transition, D. Sherman, U. S. Pracht, B.Gorshunov, S. Poran, J. Jesudasan, P. Raychaud- huri,5 M. Swanson, A.Auerbach, Nandini Trivedi, M. Scheﬄer, A. Frydman, and M. Dres- sel2, in preparation

Dynamical conductivity across the disorder-tuned superconductor-insulator transition, M. Swanson, Y. L. Loh, N. Trivedi, and M. Randeria, Phys. Rev. X, 4 021007 (2014)

Proposal for interferometric detection of the topological character of modulated superﬂuidity in ultracold Fermi gases, M. Swanson, Y. L. Loh, and N. Trivedi, New Journal of Physics, 14 033036 (2012)

Spin-wave instabilities and non-collinear phases of a frustrated triangular-lattice antiferromagnet, J.T. Haraldsen, M. Swanson, R.S. Fishman, and G. Alvarez, Phys. Rev. Lett. 102 237204 (2009)

Critical anisotropies of a geometrically frustrated triangular lattice antiferromagnet, M. Swanson, J.T. Haraldsen, and R.S. Fishman, Phys. Rev. B 79 184413 (2009)

iii Fields of Study

Major Field: Physics

Studies in theoretical condensed matter physics: Nandini Trivedi

iv Table of Contents

Page Abstract ...... ii Vita ...... iii List of Figures ...... vii List of Tables ...... xiii

Chapters

1 Introduction 1 1.1 Detection of the elusive FFLO state of matter ...... 2 1.2 Superconductor-insulator quantum phase transition ...... 4

2 Interferometric detection of the topological character of modulated su- perﬂuidity in ultracold Fermi gases 6 2.1 Introduction ...... 6 2.2 Experimental Proposal ...... 8 2.3 Interference between coupled tubes ...... 10 2.4 Time-of-Flight Simulation ...... 12 2.5 Discussion ...... 15

3 Superconductor-insulator transition 20 3.1 Introduction ...... 20 3.1.1 Superconductor-insulator transition in 2D thin ﬁlms ...... 21 3.1.2 Nature of the transition ...... 23 3.2 Conductivity across the SIT ...... 26 3.2.1 Bosonic model of SIT and methods ...... 26 3.2.2 Conductivity across the charge-tuned SIT ...... 29 3.3 Spectral function P (ω) across the SIT ...... 31 3.3.1 Bosonic spectral function in the phase model ...... 32 3.3.2 Frequency and momentum resolved spectral-function ...... 33 3.3.3 P (ω) across the clean SIT ...... 33 3.3.4 Compressibility and dispersing sound mode in SC ...... 35 3.4 Dynamical response across the disorder-tuned SIT ...... 37 3.4.1 Implementation of disorder ...... 37 3.4.2 Dynamical quantities σ(ω) and P (ω)...... 38 3.4.3 Quantum critical region ...... 40

v 3.4.4 Universal conductivity ...... 41 3.4.5 Nature of the dirty insulating phase ...... 46 3.5 Experimental comparison ...... 47 3.6 Concluding remarks ...... 50

Appendices

A Quantum to classical mapping of phase models 58 A.1 Quantum classical correspondence ...... 58 A.2 Particle-hole symmetry and the quantum rotor model ...... 60 A.3 Equivalence of 2D JJA to (2+1)D XY model ...... 61 A.3.1 Mapping to XY phase model ...... 63 A.3.2 Mapping to current model ...... 64 A.3.3 Summary and duality of models ...... 66

B Observable quantities and analytic continuation 68 B.1 Kubo formula for the XY model ...... 68 B.2 Current-current correlation functions in electronic systems ...... 71 B.3 Sum rule derivation and comparison to Maximum Entropy results . . . . . 75 B.3.1 Sum rules on σ(ω)...... 75 B.3.2 Sum rules on Im P (ω)...... 76

vi List of Figures

Figure Page

1.1 Schematic of a fully paired superfluid, an FFLO state, exists in a field range hc1 < h < hc2, with excess fermions in domain walls, and a polarized Fermi liquid. The superfluid pairing amplitude changes sign at each domain wall, a feature crucial to the experiment proposed in this thesis...... 3

2.1 (Top) Schematic of a fully paired superfluid, an LO state with excess fermions in domain walls, and a polarized Fermi liquid. (Center) Pairing amplitude ∆ and magnetization m as a function of the Zeeman field h, which is the difference between the chemical potentials of up and down spins. The LO phase exists in a field range hc1 < h < hc2. (Bottom) Real space behavior of ∆(x) and m(x) in each phase...... 7 2.2 Principle of our cold atom interference experiment. Two cigar-shaped condensates are allowed to expand. After a suitable time-of-flight, the shadow of a probe laser in the y-direction gives the interference pattern projected onto the x-z plane. If both clouds are in the uniform SF phase, the interference pattern is similar to the familiar double-slit experiment (left). In contrast, interference between an LO phase and a SF phase gives staggered fringes (right)...... 9 2.3 Proposed experimental geometry. The fermion gas is confined in a harmonic trap. An optical lattice with a large spacing in the z-direction is used to separate the gas into two independent quasi-2D layers. A second optical lattice in the y-direction cuts each layer into a series of weakly coupled tubes – the optimal geometry for LO physics. The trap and lattices are turned off abruptly, allowing the two layers to expand and interfere with one another, generating fringes as in Fig. 2.2...... 11

vii 2.4 Interference patterns for three different configurations: fully paired SF state (top left), locked LO state (top right), and unlocked LO state (bottom). In each case we consider a 2 5 array of coupled tubes as shown in Fig. 2.3, × using a fully paired SF as a reference phase in the bottom layer. The pairing amplitude ∆(x) in each of the five top layer tubes is shown to the left of the interference pattern. The locked LO states were taken to be ∆(x) = −x2/2σ2 e sn(x/λLO k), for k 1 with period λLO, where sn(x k) is the Jacobi | . | elliptic function. In the unlocked case we added a random displacement of −x2/2σ2 the domain walls ∆(x) = e sn((x + δ)/λLO k) where δ [ λLO, λLO]. | ∈ − This interference pattern contains signatures of the LO phase even when the domain wall locations fluctuate between tubes. In the bottom right, we show fixed-z cuts of the interference pattern for the fully locked LO state (top) and unlocked LO state (bottom). For the locked domain walls, the original domain wall spacing λLO is the same as the peak-to-peak distance of the horizontal interference fringes (since there is no expansion in the x- direction). While the visibility is reduced in the unlocked case, from the peak-to-peak distance we can still identify the domain wall spacing. . . . . 13 2.5 Interference patterns I ∆(x, z) 2 obtained from time-of-flight simulations. ∝ | | The initial configuration (left) consists of a fully paired superfluid in the upper tube and an LO state in the lower tube. The domain walls characterizing the LO state can be seen in the figure. After a suitable intermediate time-of-flight expansion, the interference pattern develops with no significant expansion in the longitudinal (x) direction (center) consistent with the analysis above, and staggered interference fringes are clearly visible. For longer times, expansion along the x-axis occurs, but LO signatures remain (right). 14 2.6 Results of BdG simulations on a 50 7 lattice in a harmonic potential Vr. × The parameters for the system are U = 2.5t and µavg = 0.1t, with har- | 2| 2 monic potential V (x, y) = 4t (2x/Lx) + (2y/Ly) , where the origin is at the center of the lattice. (Left) Pairing amplitude ∆ for intertube coupling strengths t⊥/t = 0.1 and 0.3. At low temperatures, an LO state is seen in the central three tubes of the trap for T < 0.12t. Thermal domain wall fluctuations similar to those studied in Fig. 2.4 can be seen in the simulations with small intertube coupling t⊥/t = 0.1. With increasing intertube coupling, the LO domain wall phases become locked between the different tubes. (Right) Pairing amplitude ∆ and magnetization m in the central tube for t⊥/t = 0.3 at T = 0 (a) and T = 0.15t (b). Because of the harmonic confinement, the excess magnetization can reside in the wings of the trap for both the LO and BCS states, but for the LO state (a), there are also peaks in m at the LO domain walls...... 16 2.7 lllustration of imaginary time worldlines of LO domain walls and corresponding magnetizations with (right) and without (left) quantum fluctuations. Pluses and minuses correspond to the sign of the pairing amplitude ∆(x)in each region. In the bottom panel, the quantum fluctuations enforce a spatial profile for the magnetization that is less sharp, but which still clearly maintains the features of domain walls...... 18

viii 3.1 The first experiment demonstrating the phenomenon of superconductivity, discovered by Kammerlingh Onnes in 1911. When cooled below a critical temperature Tc = 4.2K, mercury displayed a complete lack of electrical resistance...... 22 3.2 (Left) Measured resistance per square versus temperature for a series of Bis- muth films varying in thickness from 9 A˚ (most disordered) to 15 A˚ (least disordered). The inset shows isothermal cuts of the same data; the crossing indicates the critical disorder strength. (Right) Scaling collapse of the −1/νz same data when plotted in the scaling variable t d dc , where t = T . | − | The inset shows the best fit of ∂R/∂d d=d used to determine the value of | c νz 1.2. The scaling demonstrates the underlying quantum critical point in ≈ the disorder tuned SIT. All figures taken from [1]...... 24 3.3 (a) Generic description of quantum phase transition phase diagram for the SIT. The insulating and superconducting phases each have characteristic energy scales that vanish at the quantum critical point. Between these regions, a quantum critical region is present near the critical point. (b) Schematic phase diagram for the disordered Josephson junction model. A transition in the disorder-free (“clean”) model can be tuned by changing the ratio of charging energy to Josephson energy (Ec/EJ ). Alternatively, at any intermediate Ec/EJ , a disordered transition can be tuned by removing a fraction of Josephson sites (p). Both transitions are considered below...... 25 3.4 Results for fermionic models of the SIT. (a) Bogoliubov-de-Genne 0 (BdG) calculation of superfluid stiffness Ds and pairing gap Egap as a function of disorder V and two different interaction strengths U in the attractive 0 Fermi-Hubbard model. While BdG cannot fully capture the SIT (Ds > 0 for all disorder strengths), it is clear that the pairing gap remains of the same order of magnitude even when the superfluid stiffness is largely suppressed, evidence of a localization driven transition. (b) Similar quantities calculated using determinental quantum Monte Carlo (DQMC), which fully captures the SIT. The insulating state is characterized by an energy gap ωpair in the pair spectral function that vanishes at the SIT. (c) DQMC results for the density of states in the SC state (top) and insulating state (bottom) showing clear evidence of a hard gap across the transition...... 26 3.5 The emergent inhomogeneity of the local pairing amplitude ∆(r) in a disordered superconductor in the left panel and the robustness of the single- particle gap [2, 3, 4] across the SIT suggests an effective low-energy description in terms of a disordered quantum XY model shown on the right. The quantum phase transition occurs when long range phase coherence is lost between weakly connected “superconducting islands” tuned by the ratio Ec/EJ of charging energy to Josephson coupling as well as by disorder, modeled by removing a fraction p of the Josephson bonds...... 27

ix 3.6 (left) Free energy of superconducting state showing the gapless Nambu- Goldstone mode (green) and the gapped Higgs mode (red). (right) Energy scales, in units of EJ , as a function of the control parameter Ec/EJ in the clean system. From the SC side, the superfluid stiffness ρs and the Higgs “mass” ωHiggs, and from the insulating side, the optical conductivity threshold ωσ and the boson energy scales ωB and ωeB, vanish at the transition creating a fan-shaped region where quantum critical fluctuations dominate. 30 3.7 Real (top) and imaginary (bottom) portions of the conductivity σ(ω) across the clean superconductor-insulator transition. Superconductor (a) and (d): Re σ(ω) contains a δ-function contribution proportional to the superfluid density ρs and an absorption threshold from the Higgs amplitude mode. ωRe σ(ω) shows deviations from a constant value (also proportional to ρs) from the charging fluctuations. As the SIT, (b) and (e) is approached, there is a transfer of spectral weight from ρs to low frequencies. Insulator (c) and (f): The insulating state is characterized by a lack of superfluid density and an absorption gap ωσ equal to twice the gap in the spectral function ωb. At the SIT, there is a crossover from inductive (ωRe σ(ω) > 0) to capacitive (ωRe σ(ω) < 0) behavior at low frequencies...... 32 3.8 Properties of the Spectral function Im Pq(ω) in the superconducting phase (left), at the SIT (middle), and in the insulating phase (right). The top two rows show the full spectral function Pq(ω), the third row shows the momentum distributions nq, and the bottom row shows the density of states. 34 p 3.9 Compressibility κ, superfluid density ρs, and speed of sound c = ρs/κ across the disorder-free SIT. The compressibility is vanishing at the SIT, consistent with the gapped insulating phase, and the speed of sound goes to a constant value...... 36 3.10 Full spectral functions Im Pq(ω) for different Ec/EJ values in the superconducting phase. The black line indicates the phonon mode ω = c q , where p | | c = ρs/κ is shown in Fig. 3.9. The speed of sound increases up to the SIT, where a gap in Im Pq(ω) opens. The speed of sound is a quantity derived entirely from (two separate) imaginary-time correlation functions, and its agreement with the analytically continued function Im Pq(ω) provides further evidence of the validity of the maximum entropy method...... 37 3.11 (a-f): Re σ(ω) across the (a,c,e) clean (p=0) and (b,d,f) disorder-tuned (fixed Ec/EJ ) superconductor-insulator transitions. (g): Boson spectral function Im P (ω)/ω for a clean superconducting (blue) and insulating (red) state. The energy scales shown in Fig. 3.6 are indicated in (a-g). All quantities are at fixed temperature T/EJ = 0.156, and fixed system size, 256 256 × for the clean case and 64 64 for the disordered. In the disordered system, × the spectral functions are marked by a significant increase in low frequency weight, obscuring the gap scales of the clean system. (h): Schematic phase diagram showing how the SIT can be crossed by either increasing Ec/EJ or by tuning the disorder p...... 39

x 3.12 Dynamical response functions across the disorder-tuned SIT. The critical disorder pc = 0.337 is marked as a dashed line; T/EJ = 0.156, Ec/EJ = 3.0 and L = 64. (a) In the conductivity Re σ(ω) the superfluid response is evident as a zero-frequency delta function of strength ρs. Deep in the insulator there is a gap in Re σ(ω) that grows with disorder. (b) ωIm σ(ω) shows a crossover from “inductive” (ωIm σ(ω) = ρs > 0) to “capacitative” (ωIm σ(ω) < 0) behavior at small ω across the transition. (c) The boson spectral function Im P (ω)/ω, which has a peak centered about zero frequency in the superconductor, develops a characteristic scale ωeB in the insulator that grows with disorder...... 40 3.13 (a,b) Superfluid stiffness ρs (green), bosonic scale ωeB (red) in the insulator, and low-frequency conductivity σ∗ (blue), defined in the text, as functions of disorder p at two different temperatures shown in panel (c). The quantities 2 are in units of EJ and σQ = 4e /h, respectively. The quantum critical region is shaded gray in all three panels. (c) Phase diagram with Tc determined by ∗ zν vanishing of ρs and T by the vanishing of ωB. The lines are fits to p pc e | − | with pc 0.337 and zν 0.96...... 41 ≈ ≈ 3.14 (a) Comparison of two methods for obtaining the low-frequency conductivity ∗ near the SIT at T/EJ = 0.156, with σ from the integrated spectral weight ∗ in Eq. (3.22), and σΛ from the current correlator Λxx at imaginary time τ = β/2 (see text). (b) Plot of σ∗(T ; p) as a function of the disorder p at various temperatures. The various curves cross at the critical disorder ∗ ∗ strength pc at which σ is T -independent with the critical value σ 0.5σQ. ∗ ≈ (c) Scaling collapse of the σ (T ; p) data with pc = 0.337 and zν = 0.96, consistent with Fig. 3.13...... 42 3.15 (a) Illustration of low frequency integrated weight σ∗ defined in Eq. 3.21. σ∗ is a universal quantity and a reliable estimate of the dc conductivity that can be extracted from different types of numerical or experimental data. (b) Il- lustration of low frequency estimate of σ∗ directly from correlation functions. Here Kβ/2 = 1/2csch(βω/2) denotes the Kernel of the integral relation between Λxx(τ = β/2) and Λxx(ω)...... 43 3.16 Compressibility κ, superfluid density ρs, and spectral function gap ωeB across the disorder-driven SIT. These results are for T/EJ = 0.156 and show a finite compressibility tail into the insulating phase. Whether or not this tail scales to zero in the insulating phase with lowering temperature has not been explored in our model...... 46 3.17 Tunneling density of states G/GN and optical conductivity σ1 of disordered NbN thin films. (a) shows a weakly disordered film where the gap scale extracted by a BCS fit (blue line) to the tunneling density of states correctly predicts the absorption gap in the conductivity. In (b), a more strongly disordered film, the best fit to the tunneling data does not predict the absorption gap. The green line shows the best BCS fit to the conductivity data, and predicts a reduced gap, min[σ1] attributed to phase fluctuations. (c) shows the discrepancy between gap scales predicted by these two experimental tech- clean niques; Tec = Tc/Tc ...... 48

xi 3.18 Comparison of the experimental subgap conductivity of NbN films to the theoretical predictions made in this work. The experimental subgap conductivity exp BCS is obtained by subtracting off the best BCS fit to the data: σH = σ σ . 1 − The experimental and theoretical data show qualitatively similar features. 49

A.1 Diagram of θ configuration (represented as spins) with discretized imaginary time...... 62 A.2 Configurations of the superfluid and insulator phases expressed in the current representation (top) and in the phase representation (bottom). The ordered phase of the phase model is the superfluid phase, and the ordered phase of the current model is the insulating phase...... 67

B.1 Sum rules for quantities calculated using Maximum Entropy analytic continuation. (a) and (b) show the conductivity sum rule for the clean (p = 0) and disorder tuned (Ec/EJ ) transition. (c) and (d) show the sum rules given by Eqs. B.60 and B.61 for the boson spectral function P (ω)for the disordered tuned transition. In all cases, the results shown are for T/EJ = 0.156, but hold at all temperatures considered in this work...... 77

xii List of Tables

Table Page

3.1 Table of experimental critical exponents for various superconductor-insulator transitions...... 23 3.2 Table of numerical estimates of the dc conductivity for the disorder-free 2D Josephson-junction array...... 45

A.1 Summary of classical-to-quantum mapping of Josephson-junction array. . . 66

xiii Chapter 1 Introduction

The problem of understanding the behavior of many-particle systems is one that is central to nearly every area of physics. In strongly-correlated systems, the many-body ground state that emerges can have exotic and unexpected properties. One of the most striking examples of this was the discovery of superconductivity in 1911, where at a critical temperature, a new phase of matter emerges, characterized by having zero electrical resistance. A theory explaining this phenomenon eluded physicists for almost 50 years before Bardeen, Cooper, and Schrieﬀer explained it as resulting from the condensation of paired-electrons. A plethora of new novel states have been discovered, including High-Tc superconducting materials, the quantum and fractional quantum Hall eﬀects, and, recently, topological insulators.

There also exist fascinating examples of phase transitions between these quantum states of matter. These quantum phase transitions occur in many-body systems at zero temperature, where the tuning of a parameter in the Hamiltonian results in the dramatic reorganization of the system from one ground state to another. This field incorporates ideas from condensed matter and high energy physics, where powerful theoretical tools have been developed to understand the many-body problem. Simple toy models can be constructed that show quantum phase transitions, but many experimental realizations – the ferromagnet-quantum paramagnet transition in LiHoF4 [5], the superfluid-Mott insulator phase transition in optical lattice experiments with ultracold atoms [6], and the superconductor-insulator transition in disordered thin films [7], to name a few – are avail- able, too.

1 In this thesis we will present work on both quantum phase transitions and elusive quantum states of matter. The work that we have done is theoretical and computational, but we have made strong eﬀorts to keep experiments in mind, making speciﬁc connections and proposals for experiments whenever possible. While traditionally, the states of matter we are considering were found only in solid state systems, advances in atomic physics, notably the realization of Bose-Einstein condensation and the development of optical lattices, has enabled many of these ideas to be tested in atomic physics labs. We will keep both scenarios in mind.

The organization of this thesis has two main thrusts: (i) We propose an experiment to detect the proposed Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in imbalanced Fermi systems. Although proposed to exist in systems ranging from exotic superconductors to neutron stars, this subtle state of matter proposed over 50 years ago has yet to be observed.

In this work, we uncover distinct signatures of an FFLO phase in the interference pattern between two expanding ultracold atomic gases. (ii) We use quantum Monte Carlo simulations combined with analytic continuation techniques to study the evolution of dynamical quantities across the superconductor-insulator quantum phase transition in disordered superconducting systems. The model that we use captures the superconducting phase ﬂuc- tuations that drive the transition and the quantities we calculate, the optical conductivity and pair spectral function make experimentally testable predictions. We make comparisons to preliminary experiments motivated by our theoretical ﬁndings. Additionally, our work addresses several questions about the underlying universality class of various models of the superconductor-insulator transition (SIT). Finally, we include several extensive Appendices detailing the numerical and analytical tools employed throughout.

1.1 Detection of the elusive FFLO state of matter

Given a system of fermions with attractive interactions in a Zeeman field, what is the ground state of the system? The attractive interaction favors pairing of the spins while the field favors a polarized Fermi liquid state. Though the simplest theories predict a first-order

2 polarized Papairediring  FFLO state superfluid Fermi liquid

Polarization m

hc1 h =(µ µ )/2 hc2 ↑ − ↓ Pairing  Figure 1.1: Schematic of a fully paired superfluid, an FFLO state, exists in a field range h < h < h , with excess fermions in domain walls, and a polarized Fermi liquid. The c1 m c2 superfluid pairing amplitude changes sign at each domain wall, a feature crucial to the experiment proposed in this thesis. x x x x Paired SF Strong LO Weak LO Polarized FL superfluid-FermiPolariza liquidtion m transition, this competition can lead to far more subtle physics, the FFLO state shown inh Fig.c1 2.1. Inh this=(µ state,µ the)/2 excess spinshc2 are localized in domain ↑ − ↓ walls separating regions of paired superfluid. Despite being theoretically proposed over 50  years ago and sought after in systems ranging from exotic superconductors [8, 9] to neutron m stars [10], there has so far never been any unambiguous evidence of the FFLO phase. x x x x In Chapter 2, we outline an ultracold atomic gas experiment that would definitely show Paired SF Strong LO Weak LO Polarized FL the existence of the FFLO state. Since the first observation of Bose-Einstein condensation

(BEC) in 1995, ultracold atomic systems have become a powerful playground for exploring exotic quantum systems, due to the unprecedented control and tunability they provide. The experiment that we propose is an interference experiment between two isolated mixtures of two different species of fermions. When the confining potentials are turned off the atomic gases expand in plane-wave states. The far-field absorption of the clouds effectively gives the momentum distribution of the atomic clouds in the trap. This method was used to verify that the first BEC experiments contained a macroscopic number of particles in the zero momentum state. When two or more atomic clouds expand into one another before being imaged, an interference pattern is created, and signatures in this interference pattern can tell you information about the state of the system prior to turning off the traps. This method has been used successfully to image thermal vortices in 2D atomic Bose systems [11].

3 In our experimental proposal, we show how the domain walls predicted for the FFLO state could be imaged as a distinctive “tire tread” pattern in the interference between two isolated mixtures of diﬀerent species of fermions. We discuss the consequences of various experimental geometries, and the robustness of this experiment to thermal and quantum

ﬂuctuations.

1.2 Superconductor-insulator quantum phase transition

Both superconductivity and localization stand on the shoulders of giants – the BCS theory of superconductivity and the Anderson theory of localization – and their interplay has proven to be a rich and intriguing problem. This is especially true in two dimensions.

While Anderson showed [12] that superconductivity is quite robust against localization in three dimensions and exists in a variety of systems, it is more subtle in two dimensions where both superconductivity and localization are marginal. While it has been shown that superconductivity does exist in two dimensions, it can be lost by many parameters including temperature, disorder, gate voltage, and magnetic ﬁeld. The system can enter a metallic state under these tunings, but more often than not, a superconductor-insulator phase transition occurs. This is an example of a quantum phase transition of a strongly- correlated sermonic system.

The first studies of the superconductor-insulator phase transition tuned by disorder, which we will focus on in this thesis, were dc transport measurements in thin films, where the disorder is tuned by the inverse thickness of the films. While superconductivity is usually lost by the breaking of paired electrons, it has been shown [32, 33, 34] in model fermionic Hamiltonians with attraction between electrons and disorder arising from random potentials, that the single-particle density of states continues to show a hard gap across the disorder-driven quantum phase transition and that pairs continue to survive into the insulating state. The superconducting transition temperature Tc, however, does decrease with increasing disorder and vanishes at a critical disorder signaling a superconductor- insulator transition (SIT). The picture of the transition that emerges then is a bosonic one,

4 where phase ﬂuctuations between localized Cooper-pairs in superconducting regions are responsible for the loss of superconductivity. These theoretical predictions are supported by scanning tunneling spectroscopy experiments [35, 36, 37, 38] and by magnetoresistance oscillations [30] in disordered thin ﬁlms.

In Chapter 3, we focus speciﬁcally on the phase ﬂuctuations that drive the SIT and their imprint on dynamical quantities as the transition is tuned. We use quantum Monte

Carlo simulations combined with maximum entropy techniques to analytically continue imaginary-time correlation functions to calculate real frequency quantities. Both the optical conductivity σ(ω) and the bosonic (pair) spectral function Pq(ω) are investigated in detail across SITs tuned by increased charging ﬂuctuations (“clean” transition) and disorder

(“dirty transition”). Both of these quantities are experimentally accessible, and we focus

on elucidating properties that are currently being measured and ones that can be measured

in the future. In addition to experimental considerations, we are able to derive quantities

out of these dynamical quantities that help shed light on outstanding theoretical questions

regarding the transitions, such as the universality class of the SIT and the nature of the

disordered insulating phase.

Finally, many of the numerical methods and analytic tools are derived in detail in the

Appendices of this thesis. The quantum Monte Carlo techniques that we use to study the

SIT problem employ a mapping between the 2D quantum rotor model and the (2+1)D

classical XY model. The quantum-to-classical mapping is well known, but it is remarkable

how much exciting physics can be extracted from such a well-known classical model. Ad-

ditionally, much of our work relies on using the maximum entropy method to perform a

delicate analytic continuation procedure from the imaginary time correlation function data

to real frequency functions. This is a notoriously diﬃcult procedure, and often the results

are only trusted in certain cases. With this in mind, we have been very careful throughout

to perform as many internal consistency checks on the maximum entropy results that we

have. These results are also presented in the appendix, and we feel that they provide strong

evidence for the conﬁdence in the numerical calculations.

5 Chapter 2 Interferometric detection of the topological character of modulated superfluidity in ultracold Fermi gases

2.1 Introduction

As a fascinating example of self-organized quantum matter, FFLO states have long been sought after in exotic superconductors [8, 9] and in ultracold atomic gases [13], and they may even occur in neutron stars [10]. In superconductors, modulated superﬂuidity results from the interplay between superconducting pairs and an applied parallel magnetic ﬁeld.

The inter-electron attraction favors a superfluid (SF) state consisting of pairs of up- and down-spin electrons, whereas the field favors a polarized Fermi liquid (FL) state with a lower Zeeman energy. The simplest theories predict a first-order SF-FL transition[14, 15].

However, the competition between pairing and polarization can produce far more subtle physics – an intermediate Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [16, 17, 18, 19,

20, 21], henceforth referred to as LO1 as depicted in Fig. 1. As the ﬁeld is increased beyond hc1 it forces excess fermions into the superﬂuid in the form of domain walls where the

1Multiple studies have found that the condensation energy in a LO state is signiﬁcantly (at least 16 times) greater than that of an FF state.

6 Pairing !

Polarization m

hc1 h =(µ µ )/2 hc2 ↑ − ↓

! m

x x x x Paired SF Strong LO Weak LO Polarized FL

Figure 2.1: (Top) Schematic of a fully paired superfluid, an LO state with excess fermions in domain walls, and a polarized Fermi liquid. (Center) Pairing amplitude ∆ and magnetization m as a function of the Zeeman field h, which is the difference between the chemical potentials of up and down spins. The LO phase exists in a field range hc1 < h < hc2. (Bottom) Real space behavior of ∆(x) and m(x) in each phase.

order parameter changes sign between positive and negative values. (This is analogous to how a perpendicular field forces vortices into a superconducting film.) The wavelength of these modulations decreases with increasing field and ultimately the system gives way to a polarized Fermi liquid.

Cold atom experiments provide a highly tunable testing ground for ﬁnding LO states, and there have been several proposals to do so through modulations of the polarization in real space [21], peaks in the pair momentum distribution at the modulation wavevector [22,

23], shadow features in the single-particle momentum distribution [21], and Andreev bound states in the density of states [21]. A recent experiment on an array of one-dimensional tubes found density proﬁles in agreement with Bethe ansatz calculations [24], which predict power-law LO correlations at zero temperature. However, direct evidence of the sign changes

7 of the order parameter – the deﬁning feature of an LO state – is still lacking.

In this chapter, we propose an interference experiment in which two isolated superfluids expand into each other, as illustrated in Fig. 2.2. The ideal situation for detecting the LO phase is shown in the lower panel of this figure, where one layer is a uniform fully-paired superfluid, which serves as a reference phase, and the other layer is a modulated (LO) superfluid. The resulting interference pattern is directly sensitive to real-space modulations of the order parameter, and should provide an unequivocal signature of the elusive LO phase.

2.2 Experimental Proposal

LO states have been predicted to exist in various situations [21, 25, 26]. The most likely of these to be realized in the near future is an array of tubes with a small intertube coupling [27,

28, 22]. This quasi-one-dimensional geometry provides good Fermi surface nesting at the LO wavevector qLO = kF ↑ kF ↓, together with Josephson coupling between the order parameter − in adjacent tubes which is necessary to stabilize true long-range order.

Therefore, it seems that the idea can be best realized in the following way. Two species of fermions (referred to as and ) are loaded into an optical trap. The cloud is separated ↑ ↓ into two independent quasi-2D “pancake” layers using an optical lattice with a wide spacing in the vertical direction z (created with two laser beams intersecting at a shallow angle [11]).

In order to generate an LO phase it is necessary for the two layers to have diﬀerent relative

population imbalances. This may happen by chance due to natural number ﬂuctuations,

or alternatively, one can induce hyperﬁne transitions using Raman transitions, where the

two lasers are detuned at the appropriate RF frequency. In order to address each layer

seperately, it would be necessary to make the beam diameters smaller than the interlayer

distance. In this paper, we do the analysis of the interference patterns assuming that one

of the layers is balanced while the other contains a population imbalance, but this is not

necessary. As long as there are diﬀerent relative population imbalances between the two layers, LO signatures will be observable. An additional optical lattice is further used to

8 1 1 ! LO / SF SF / SF LO / SF ! -1 -1 -1 z z

y y 1 x x

Figure 2.2: Principle of our cold atom interference experiment. Two cigar-shaped condensates are allowed to expand. After a suitable time-of-ﬂight, the shadow of a probe laser in the y-direction gives the interference pattern projected onto the x-z plane. If both clouds are in the uniform SF phase, the interference pattern is similar to the familiar double- slit experiment (left). In contrast, interference between an LO phase and a SF phase gives staggered fringes (right).

create a 2D array of weakly-coupled tubes, conducive to the formation of an LO state The

resulting geometry is shown in Fig. 2.3.

Once the gases are allowed to equilibrate for suﬃcient time, the interactions in the

system are quickly ramped to the BEC side of the Feshbach resonance [29] to “freeze” or

“project” the pair wavefunction into a boson wavefunction, so that from this point onwards

the pairs move as independent bosons (instead of disintegrating into fermions). Then, the

conﬁning potentials are abruptly turned oﬀ. As the clouds expand into one another, they

interfere and form a 3D matter wave interference pattern. The projection of this interference

pattern onto the x-z plane can be measured by absorption of a resonant probe beam along

the y direction. Any LO phase modulation features will be captured in these projected

interference patterns. It may be desirable to image the minority ( ) atoms, which gives the ↓ ﬁnal distribution of the tightly bound pairs without contamination from unpaired majority

( ) atoms. ↑ An earlier paper [30] has discussed the interferometric detection of the FFLO phase

through correlation functions [31] obtained by averaging over many interference snapshots.

We emphasize that our proposal considers the information in the individual snapshots them-

9 selves. Most notably the “tire-tread” staggered interference fringes corresponding to the sign change of the pairing amplitude across the domain walls. The information in these individual snapshots contains distinct information from that in the correlation functions.

For example, if the domain wall spacing is not uniform in multiple snapshots, the LO information could be lost in the averaging necessary to obtain correlation function data.

2.3 Interference between coupled tubes

We now discuss analytically the salient features of the interference patterns of a 2D array of coupled tubes. We begin by considering two layers, each containing N coupled tubes, with separation d in the z-direction. After ramping up the interaction to produce a molecular

BEC (of fermion pairs), the wavefunction is

N 2 2 2 2 −(z−d/2) /2σz X T −(y−an) /2σy ∆(x, y, z) = e ∆n (x)e n=1 N 2 2 2 2 −(z+d/2) /2σz X B −(y−an) /2σy + e ∆n (x)e (2.1) n=1 where a is the separation between in-plane tubes and σy and σz are the Gaussian conﬁne-

th T ments in the respective directions. The wavefunction in the n tube is denoted by ∆n (x) in B the top layer and by ∆n (x) in the bottom layer. When the trap and lattices are switched off, the clouds expand predominantly in the tightly confined directions (y and z). After a suitable time-of-flight t, the wavefunction is effectively Fourier-transformed in the y and z direc-

tions. That is, the ﬁnal wavefunction ∆(x, y, z, t) is approximately proportional to the ini-

tial momentum distribution in the y and z directions, i.e., ∆(x, y, z, t) ∆(x, ky, kz, t = 0) ≈ where y = tky/m and z = tkz/m:

−z2σ2/2 −y2σ2/2 ∆(x, ky, kz) = σyσze z e y (2.2) " N N # X X eikzd/2 ∆T (x)eikyan + e−ikzd/2 ∆B(x)eikyan . × n n n=1 n=1

2 The 3D density of the cloud, after expansion, is given by I(x, ky, kz) ∆(x, ky, kz) . ∝ | | The imaging process measures the integrated density along the y direction, I(x, kz) ∝

10 z a

y x d

Imaging beam

Figure 2.3: Proposed experimental geometry. The fermion gas is conﬁned in a harmonic trap. An optical lattice with a large spacing in the z-direction is used to separate the gas into two independent quasi-2D layers. A second optical lattice in the y-direction cuts each layer into a series of weakly coupled tubes – the optimal geometry for LO physics. The trap and lattices are turned oﬀ abruptly, allowing the two layers to expand and interfere with one another, generating fringes as in Fig. 2.2.

R 2 dky ∆(x, ky, kz) : | | N N −σ2z2 X X −(n−m)2a2/4σ2 I(x, kz) e z e y (2.3) ∝ n=1 m=1 T T B B T B ∆ (x)∆ (x) + ∆ (x)∆ (x) + 2∆ (x)∆ (x) cos kzd . × n m n m n m

We can consider the behavior of the above interference formula in its two limits: widely

separated tubes (a/σy ) and overlapping tubes (a/σy 0). In the overlapping limit, → ∞ → the total interference pattern reduces to that of two isolated 2D layers. In the limit of

widely separated tubes, similar to the proposed experiment, the total interference pattern

reduces to the sum of the interference patterns from adjacent tubes in the two layers.

Realistic parameters are interlayer spacing d = 3µm, layer thickness σz 200nm, and ≈ optical lattice spacing 532nm in the y direction [24, 11]. This lattice should be shallow

enough to allow suﬃcient intertube hopping to lock the position of domain walls between

adjacent tubes. Our analysis is valid when the domain wall spacing is much larger than σz, which is true for small imbalances.

The above analysis involves just two layers, which are necessary and suﬃcient for gener-

ating interference. Introducing more layers would complicate the analysis, and reduce the

11 visibility of the interference pattern2; nevertheless, there would still be observable effects of p the order of 1/ Nlayers. Figure 2.4 illustrates the projected interference pattern, described by Eq. 2.3, for three configurations of the upper layer: a uniform SF, an LO phase with domain walls locked between tubes, and an LO phase with domain walls fluctuating between tubes. In each case we assume that the lower layer has been prepared in a uniform SF state. The form of the

LO state that we use is the Jacobi elliptic function sn(x k), which is a good approximation | of the pair wavefunction in the limit of quasi-1D tubes with small intertube coupling [32].

Each wavefunction is multiplied by Gaussian envelopes in the x, y, and z directions to

mimic the eﬀect of the trapping potential. The staggered fringe pattern in the lower two

panels is a clear signature of oscillations of the relative phase between the SF and LO layers,

in contrast to the straight interference fringes in the top panel.

2.4 Time-of-Flight Simulation

To better understand how the interference pattern evolves once the trapping potentials have

been switched oﬀ, we have simulated the time-of-ﬂight evolution of our experimental setup.

To do this, we numerically evolve the free-particle Schr¨odingerequation

∂ i~ 2 ∆(x, y, z, t) = ∆(x, y, z, t) (2.4) ∂t 2m ∇

subject to the initial pairing amplitude when the trap is abruptly turned oﬀ (t = 0).

We have analyzed the evolution of the interference pattern for a two-tube geometry

similar to what is shown in the right panel of Fig. 2.2, where one of the tubes is in a fully

paired SF state and the adjacent tube is in an LO state. The LO state we used is the same

as that used in the analysis of Fig. 2.4 above. This provides the opportunity to test the

validity of the approximation that we need to only consider expansion in the two transverse

directions, and that a suitable time-of-ﬂight can be found in experiments that achieves this

“far ﬁeld” limit. The results of our simulation are shown in Fig. 2.5. To set the time and

2 In fact, a three-layer geometry has the advantage that the central layer and outer layers naturally acquire diﬀerent population imbalances when loaded. We have checked that this geometry still produces visible fringes. 12 Top layer pairing Integrated interference Top layer pairing Integrated interference

amplitude: t = 0 pattern: t = ttof amplitude: t = 0 pattern: t = ttof

Fully paired SF Phase locked LO

y z y z x -1 ! 1 x x -1 ! 1 x

Top layer pairing Integrated interference amplitude: t = 0 pattern: t = t tof 20

15 !LO Phase unlocked LO 10 5 0 20 15 10 5 y z 0 !10 !5 0 5 10 x -1 ! 1 x x

Figure 2.4: Interference patterns for three different configurations: fully paired SF state (top left), locked LO state (top right), and unlocked LO state (bottom). In each case we consider a 2 5 array of coupled tubes as shown in Fig. 2.3, using a fully paired SF as a × reference phase in the bottom layer. The pairing amplitude ∆(x) in each of the five top layer tubes is shown to the left of the interference pattern. The locked LO states were taken −x2/2σ2 to be ∆(x) = e sn(x/λLO k), for k 1 with period λLO, where sn(x k) is the Jacobi | . | elliptic function. In the unlocked case we added a random displacement of the domain −x2/2σ2 walls ∆(x) = e sn((x + δ)/λLO k) where δ [ λLO, λLO]. This interference pattern | ∈ − contains signatures of the LO phase even when the domain wall locations fluctuate between tubes. In the bottom right, we show fixed-z cuts of the interference pattern for the fully locked LO state (top) and unlocked LO state (bottom). For the locked domain walls, the original domain wall spacing λLO is the same as the peak-to-peak distance of the horizontal interference fringes (since there is no expansion in the x-direction). While the visibility is reduced in the unlocked case, from the peak-to-peak distance we can still identify the domain wall spacing.

13 t = 0.0 ms t = 2.96 ms t = 8.87 ms

20 ) µm

( 0 z

-20

-20 0 20 -20 0 20 -20 0 20 x (µm) x (µm) x (µm)

Figure 2.5: Interference patterns I ∆(x, z) 2 obtained from time-of-flight simulations. ∝ | | The initial configuration (left) consists of a fully paired superfluid in the upper tube and an LO state in the lower tube. The domain walls characterizing the LO state can be seen in the figure. After a suitable intermediate time-of-flight expansion, the interference pattern develops with no significant expansion in the longitudinal (x) direction (center) consistent with the analysis above, and staggered interference fringes are clearly visible. For longer times, expansion along the x-axis occurs, but LO signatures remain (right).

length scales, we have set the length of our tubes to be approximately 50 µm, and used

the mass of 6Li, typical values for such an experiment [24]. The spacing of the domain

walls is about 10 µm which is also the spacing of the bright and dark interference signals in

the horizontal direction. This length scale is accessible in similar time-of-ﬂight experiments

that have a resolution of 3µm. In the central panel of Fig. 2.5, we see that there is a ≈ suitable intermediate time-of-ﬂight where expansion in the longitudinal (x) direction can

be neglected, and the interference patterns shown in Fig. 2.4 are valid. The expansion

time of this snapshot is t mdR⊥/~, where d is the separation of the tubes and R⊥ is the ≈ transverse radius of the tube, can be associated with the “far ﬁeld” limit of expansion in

the transverse (z) direction [33].

Our simulations also allow us to comment on the eﬀects of expansion along the x-axis

in our experimental proposal. In the ﬁnal frame of Fig. 2.5, we show the interference

pattern after long time expansion. Despite the fact that the analysis of Section 2.3 for

the interference pattern breaks down in this regime, the LO interference signatures remain 14 visible.

Our time-of-ﬂight simulation contains two isolated tubes for computational simplicity.

In the experiment proposed above, we have considered an array of coupled tubes in order to quench the fluctuations. The time-of-flight calculation can be extended to an array of tubes, similar to the proposed experiment. In this geometry, the expansion in the y-direction would approach the far-field limit at times ty maR⊥/~. Since a d/6, ty < tz = mdR⊥/~ which ≈ ≈ implies that when the far-field limit is satisfied in the z-direction, the far-field limit is also satisfied in the y-direction, and the analysis discussed in Section 2.3 is valid.

2.5 Discussion

Even with intertube coupling to stabilize quasi-long-range-order, we expect thermal and quantum ﬂuctuations to have an impact on the interference patterns in our proposed experiment.

Thermal fluctuations: The interference experiment that we have proposed essentially takes “snapshots” of the wavefunction after the time-of-flight. For this reason, the thermal phase fluctuations are not integrated over, and can be considered as domain wall fluctuations between tubes. However, as shown in the lowest panel of Fig. 2.4, these fluctuations are not severe enough to destroy the occurrence of staggered interference fringes, the signature of an LO phase. This is further detailed in the bottom right panel of the same figure, where we show a cross section of the interference pattern.

A remarkable feature of cold atom experiments is that control parameters (interaction, lattice depth, and trap depth) can be turned oﬀ very quickly, much faster than the typical timescale of domain wall movement. This allows us to take a snapshot of the wavefunction

(resolved in real time), in a way not possible in condensed matter experiments. Thus, even above the critical temperature where thermal ﬂuctuations destroy long-range order, it may still be possible to detect LO physics in the form of “temporary” domain walls!

In order to study the eﬀect of thermal ﬂuctuations quantitatively, we have performed

15 t!!t ! 0.1 t!!t ! 0.3

T!t ! 0. 0.2 (a) ! 0.1 m T!t ! 0.03 0.0 !0.1 T!t ! 0.06 !0.2 10 20 30 40 0.2 (b) T!t ! 0.09 0.1 0.0 T!t ! 0.12 !0.1 !0.2 ! ! 0.15 T t 10 20 30 40 x y y x -0.2 ! 0.2 x

Figure 2.6: Results of BdG simulations on a 50 7 lattice in a harmonic potential Vr. The × parameters for the system are U = 2.5t and µavg = 0.1t, with harmonic potential V (x, y) = 2 2 | | 4t (2x/Lx) + (2y/Ly) , where the origin is at the center of the lattice. (Left) Pairing amplitude ∆ for intertube coupling strengths t⊥/t = 0.1 and 0.3. At low temperatures, an LO state is seen in the central three tubes of the trap for T < 0.12t. Thermal domain wall fluctuations similar to those studied in Fig. 2.4 can be seen in the simulations with small intertube coupling t⊥/t = 0.1. With increasing intertube coupling, the LO domain wall phases become locked between the different tubes. (Right) Pairing amplitude ∆ and magnetization m in the central tube for t⊥/t = 0.3 at T = 0 (a) and T = 0.15t (b). Because of the harmonic confinement, the excess magnetization can reside in the wings of the trap for both the LO and BCS states, but for the LO state (a), there are also peaks in m at the LO domain walls.

16 Bogoliubov-de Gennes (BdG) simulations of the 2D attractive Hubbard Hamiltonian

X † † X H = trr0 (c cr0σ + c 0 crσ) (µσ Vr)nrσ (2.5) − rσ r σ − − hr,r0i,σ r,σ X 1 1 U (nr↑ )(nr↓ ) (2.6) − | | r − 2 − 2 with hopping trr0 = t for nearest-neighbor bonds along the length of the tube (in the x direction), trr0 = t⊥ for nearest-neighbor bonds between tubes, fermion creation and † † annihilation operators crσ and crσ, number operators nrσ = crσcrσ, chemical potentials

µσ = µ σh for the two spin species, Zeeman ﬁeld h, local Hubbard attraction U , and − | | conﬁning harmonic potential Vr. The results of these simulations on a system of 7 tubes with 50 sites in each tube are

shown in Fig. 2.6. The harmonic potential we consider mimics the experimental proposal.

The behavior of the BdG pairing amplitude ∆(r) cr↑cr↓ in the central three tubes as a ∝ h i function of temperature for intertube coupling strengths t⊥ = 0.1t and 0.3t is shown. For

small intertube coupling t⊥/t = 0.1, we see domain wall ﬂuctuations between the tubes whose eﬀect we have included in our interference pattern analysis in Fig. 2.4. As the

intertube coupling is increased at low temperatures, the phases of the domain walls in the

diﬀerent tubes “lock” together. Our BdG results show that the LO state survives up to

a temperature of TLO 0.12t. We expect phase ﬂuctuations to suppress TLO and to also ≈ depend on the intertube coupling: the larger the coupling, the smaller the suppression.

However, as we have emphasized above, in spite of these ﬂuctuations, the interferometric

∗ signature would still be visible up to T TLO (which is a signiﬁcant fraction of the ∼ temperature for the onset of pairing TBCS 0.2t for these parameters). ≈ Quantum ﬂuctuations: At low temperatures, the excess fermions residing in the

domain walls quantum tunnel from one position to another. This can be viewed as the

diﬀusion of domain walls in imaginary time τ; measurements are necessarily averaged over

τ. For isolated tubes, quantum ﬂuctuations of the domain walls prevent long-range LO

order, and there is only quasi-long-range order at zero temperature; hence, the interference

pattern will be washed out. This is why we recommend using suﬃcient coupling between

17 Figure 2.7: lllustration of imaginary time worldlines of LO domain walls and corresponding magnetizations with (right) and without (left) quantum fluctuations. Pluses and minuses correspond to the sign of the pairing amplitude ∆(x)in each region. In the bottom panel, the quantum fluctuations enforce a spatial profile for the magnetization that is less sharp, but which still clearly maintains the features of domain walls.

the tubes to lock their phases so that the pairing amplitude modulations remain even after averaging over quantum ﬂuctuations (see Fig. 2.7).

Stabilization of ﬂuctuations with intertube coupling: It is known that in one- dimensional systems at ﬁnite temperatures, there is no long-range-order. Indeed, interference experiments between two quasi-1D systems have demonstrated the exponential decay of correlations in the system [34]. Here again, we emphasize that our proposal suggests that experimentalists look at the interference between planes of coupled 1D tubes, which

has the eﬀect of stabilizing the LO order. Although this order at ﬁnite temperatures is only

algebraic at best (since the planes are still 2D), the system is eﬀectively long range ordered

if the correlation length is larger than the system size. Similar experiments on 2D Bose

systems [11] have found “zipper” interference patterns which were attributed to unbound

vortex/anti-vortex pairs. These vortex/anti-vortex ﬂuctuations are only observed near the

Kosterlitz-Thouless transition. In contrast, domain walls are an integral feature of the LO

ground state and should exist in the whole temperature range 0 < T < Tc. In our experimental geometry it is easy to distinguish between the interference signatures

of the vortices and LO domain wall defects. Since vortices are point defects in 2D, they

will produce interference signatures regardless of the direction of the imaging beam. In

18 particular, if the imaging beam were positioned along the axis of the tubes (the x direction),

LO signatures would disappear and vortex signatures, if they were to occur, would be visible.

Regardless, we do not expect vortices like those seen in the 2D Bose gas experiments to occur because the coupling between tubes is small.

Conclusions: Interferometric techniques have proven to be powerful methods to detect the internal structure of the pairs in cuprates [35], ruthenates, pnictides, and other unconventional superconductors [36]. Our proposal diﬀers from those experiments in that it measures the sign changes of the order parameter as a function of the center of mass of the pairs. Such a measurement would be the ﬁrst to directly image the real-space modulation predicted for the LO phase and provide unequivocal evidence of LO physics.

19 Chapter 3 Superconductor-insulator transition

3.1 Introduction

The interplay of superconductivity and localization has proven to be a rich and intriguing problem, especially in two dimensions [7, 37, 38, 39, 40, 41, 42]. Both paradigms stand on the shoulders of giants – the BCS theory of superconductivity and the Anderson theory of localization. Yet, when the combined eﬀects of superconductivity and disorder are considered, both paradigms break down, even for s-wave superconductors.

While superconductivity is usually lost by the breaking of paired electrons, it has been shown [2, 3, 4] in model fermionic Hamiltonians with attraction between electrons and disorder arising from random potentials that the single-particle density of states continues to show a hard gap across the disorder-driven quantum phase transition and that pairs continue to survive into the insulating state. The superconducting transition temperature Tc, however, does decrease with increasing disorder and vanishes at a critical disorder signaling a superconductor-insulator transition (SIT). The picture of the transition that emerges is a bosonic one, where phase ﬂuctuations between localized Cooper-pairs in superconducting regions are responsible for the loss of superconductivity. These theoretical predictions are supported by scanning tunneling spectroscopy experiments [43, 44, 45, 46] and by magne-

20 toresistance oscillations [41] in disordered thin ﬁlms.

In this chapter, we focus speciﬁcally on the phase ﬂuctuations that drive the SIT and their imprint on dynamical quantities as the transition is tuned. We use quantum Monte

(“dirty transition”). Both of these quantities are experimentally accessible, and we focus

on elucidating properties that can be tested in the coming future. In addition to experi-

mental considerations, we are able to derive quantities from these dynamical quantities that

help shed light on outstanding theoretical questions regarding the transitions, such as the

universality class of the SIT and the nature of the disordered insulating phase.

3.1.1 Superconductor-insulator transition in 2D thin ﬁlms

First discovered, quite unexpectedly, by Kamerlingh Onnes in 1911, superconductivity is

characterized by the total absence of electrical resistance. In conventional superconductors,

this phenomena is quite fragile and only exists up to a few degrees Kelvin. The mechanism

behind superconductivity eluded physicists until 1957, when Bardeen, Cooper, and Schreif-

fer explained superconductivity by pairing of electrons (Cooper pairs) mediated by phonon

interactions. While Anderson showed [12] that superconductivity is quite robust against

localization in three dimensions and exists in a variety of systems, it is more subtle in two

dimensions where both superconductivity and localization are marginal. While it has been

shown that superconductivity does exist in two dimensions, it can be lost by tuning many

parameters, including temperature, disorder, gate voltage, and magnetic ﬁeld. The system

can enter a metallic state under these tunings, but more often than not, a superconductor-

insulator phase transition occurs. This is an example of a quantum phase transition of

a strongly-correlated fermionic system. The ﬁrst experiments studying the disorder-tuned

SIT in 2D were dc transport measurements on disordered superconducting ﬁlms [47], where

the disorder is enhanced by reducing the thickness of the ﬁlms. Essentially the sheet re-

21 Figure 3.1: The ﬁrst experiment demonstrating the phenomenon of superconductivity, discovered by Kammerlingh Onnes in 1911. When cooled below a critical temperature Tc = 4.2K, mercury displayed a complete lack of electrical resistance.

sistance in the normal state at a temperature slightly above Tc is found to increase as the thickness is decreased.

These experiments were quick to characterize the transition as an example of a quantum phase transition. Classical and quantum phase transitions exhibit divergent correlation lengths near the transition, and scaling analysis of measurable quantities can lend insight into the nature of the underlying universality class of the transition. The correlation length

ν divergence is characterized by the critical exponent ν, ξ T Tc . Temporal ﬂuctuations ∼ | − | are governed by the dynamical exponent z such that the relaxation time is τ ξz in the ∼ critical ﬂuctuation region. As discussed in Appendix A, dynamics and thermodynamics are

intertwined in quantum mechanical systems at ﬁnite temperatures, and a quantum system

in d dimensions behave like a classical system in d + z dimensions, where the temporal dimension contains information about the quantum and thermal ﬂuctuations of the phase transition.

Fisher [48] presented a scaling theory of the SIT that make a prediction for the scaling

22 Material tuning parameter critical exponents Amorphous Bi [1] thickness νz = 1.2 0.2 ± thickness in applied perpendicular field νz = 1.4 0.2 ± perpendicular field νz = 0.7 0.2 ± Amorphous MoGe [49] perpendicular field νz = 0.7 0.2 ± YBCO [50] electrostatic gate voltage νz = 2.2 LSCO [50] electrostatic gate voltage νz = 1.5

Table 3.1: Table of experimental critical exponents for various superconductor-insulator transitions.

form of the resistance near the transition

−1/νz 1/ν(z+1) R = RcF (δT , δE ), (3.1) where Rc is the resistance at the critical point, and δ is the tuning parameter for the

quantum phase transition. For example, δ = d dc /dc for the disorder tuned transition or | − | δ = H Hc /Hc for a magnetic ﬁeld driven transition. Experimentally, a set of resistance | − |

curves R = R(d, T ) can be plotted in the scaling variables, and with the correct choice of critical exponents ν and z, the curves should collapse on to a single function. An example

of this is shown in Fig. 3.2 for the disorder-tuned transition in Bismuth ﬁlms. In this case,

it was determined that νz 1.2 0.2. In Table 3.1, we provide an sample of experimental ≈ ± critical exponents found throughout the SIT literature. For reference, two likely candidates for the universality of the SIT are the (2 + 1)D XY model (νz = 0.67) and 2D percolation

(νz = 4/3) are found to be inadequate. The span of values in this table testiﬁes to the amount of physical variety in the superconductor-insulator transition.

3.1.2 Nature of the transition

Because the superconductor-insulator transition involves both the physics of paired electrons

(BCS theory) and disorder-induced localization (Anderson localization), a natural question to ask is how these two mechanisms conspire to induce the quantum phase transition. Is the mechanism primarily fermionic, resulting from disorder-induced breaking of Cooper-pairs?

Or is it primarily bosonic, resulting from the localization of paired states? One of the earliest

23 Figure 3.2: (Left) Measured resistance per square versus temperature for a series of Bismuth films varying in thickness from 9 A˚ (most disordered) to 15 A˚ (least disordered). The inset shows isothermal cuts of the same data; the crossing indicates the critical disorder strength. (Right) Scaling collapse of the same data when plotted in the scaling variable −1/νz t d dc , where t = T . The inset shows the best fit of ∂R/∂d d=d used to determine | − | | c the value of νz 1.2. The scaling demonstrates the underlying quantum critical point in ≈ the disorder tuned SIT. All figures taken from [1].

theoretical investigations of this was by Ghosal, Randeria, and Trivedi [51, 3] who studied the Fermi Hubbard model with onsite disorder by means of Bogolioubov-de-Gennes (BdG) mean-field-theory. While this mean-field-theory only includes amplitude fluctuations of the order parameter and neglects phase fluctuations, it provided early and important insight into the problem.

Fig. 3.4 shows the main result of this early BdG work. While BdG cannot capture the

SIT (there is some finite superfluid stiffness at all disorder strengths), this work provides striking insights into the SIT. First, the single-particle gap remains finite and of the same order of magnitude as disorder in increased, strong evidence for the localization of superconducting pairs across the SIT. Further, this theoretical work allows local quantities to be calculated, such as the local pairing amplitude, see Fig. 3.4. What can be seen even in the earliest work is emergent inhomogeneity; the regions of superconducting order occur over a length scale given by the correlation length, while the disorder is determined by homogeneous randomness. The picture that emerges is that phase fluctuations between regions of

24 Figure 3.3: (a) Generic description of quantum phase transition phase diagram for the SIT. The insulating and superconducting phases each have characteristic energy scales that vanish at the quantum critical point. Between these regions, a quantum critical region is present near the critical point. (b) Schematic phase diagram for the disordered Josephson junction model. A transition in the disorder-free (“clean”) model can be tuned by changing the ratio of charging energy to Josephson energy (Ec/EJ ). Alternatively, at any intermediate Ec/EJ , a disordered transition can be tuned by removing a fraction of Josephson sites (p). Both transitions are considered below.

localized superconducting pairs induce the phase transition.

More recently, quantum Monte Carlo methods have been employed to look at the same model. While more computationally intensive, determinantal quantum Monte Carlo

(DQMC) captures both the amplitude and the phase ﬂuctuations that drive the phase transition, and consequently, unlike BdG, DQMC does capture the SIT. For additional insoght, the imaginary time correlation functions that are calculated within Monte Carlo methods can be analytically continued to real frequencies to calculate dynamical quantities such as spectral functions or densities-of-states. Bouadim, Loh, Randeria, and Trivedi [4] tracked the evolution of the density-of-states across the SIT and veriﬁed that the hard-gap in the single particle spectrum indeed persists into the insulating phase.

The emergent inhomogeneity of local superconducting quantities and the persistence of a hard gap in the insulating state have been seen in thin films across the SIT in STM experiments [43, 44, 45, 46], in excellent agreement with the numerical work. Other experimental evidence that the SIT is bosonic in nature has been provided by magnetoresistance oscillations in nano-honeycomb patterned films [41]. In the work presented here, we will focus on these systems and use a bosonic model of the SIT to see how phase fluctuations 25 Figure 3.4: Results for fermionic models of the SIT. (a) Bogoliubov-de-Genne (BdG) 0 calculation of superfluid stiffness Ds and pairing gap Egap as a function of disorder V and two different interaction strengths U in the attractive Fermi-Hubbard model. While BdG 0 cannot fully capture the SIT (Ds > 0 for all disorder strengths), it is clear that the pairing gap remains of the same order of magnitude even when the superfluid stiffness is largely suppressed, evidence of a localization driven transition. (b) Similar quantities calculated using determinental quantum Monte Carlo (DQMC), which fully captures the SIT. The insulating state is characterized by an energy gap ωpair in the pair spectral function that vanishes at the SIT. (c) DQMC results for the density of states in the SC state (top) and insulating state (bottom) showing clear evidence of a hard gap across the transition.

contribute to experimentally measurable quantities across this quantum phase transition.

While this assumption necessarily ignores the single-particle physics of the underlying problem, it allows us to perform calculations on much larger systems, which is necessary for the speciﬁc dynamical quantities that we want to calculate.

3.2 Conductivity across the SIT

3.2.1 Bosonic model of SIT and methods

As described above, the robustness of the single-particle gap across the SIT suggests that the low-energy physics near the SIT can be described by an eﬀective “bosonic” Hamiltonian, the disordered quantum XY model, where the relevant degrees of freedom are the phases of the local superconducting order parameter. We map the 2D quantum XY Hamiltonian to an anisotropic classical (2+1)D XY model [52, 53, 54] and simulate the model using

Monte Carlo methods. We focus on the behavior of two dynamical quantities of funda-

26 NATURENATURE PHYSICS PHYSICSDOI: 10.1038/NPHYS2037DOI: 10.1038/NPHYS2037NATURE PHYSICS DOI: 10.1038/NPHYS2037 ARTICLESARTICLES ARTICLES abab3 ab3 0.43 c0.4 c1.50.4 c1.5 1.5

0 0 Δ0

¬3 ¬3 0¬3 0 0 0 0

Figure 5FigureEmergent 5 Emergent granularity. granularity.a, Disordera, Disorder realizationFigure realization 5V(EmergentR) onV a( 36R) granularity. on36 a 36lattice36a at, lattice DisorderV 3 att. bV realization, Local3t. b pairing, LocalV(R amplitude) pairing on a 36 amplitude136op( latticeR) from1op at(R aV) BdG from3t calculation. b a, BdG Local calculation pairing at amplitude at 1op(R) from a BdG calculation at | | | ⇥ ⇥ = = ⇥ = U 1.5tU, T 10,.5t and, T n0, and0.875.n Note0.875. the Note emergent theU emergent ‘granular’1.5t, T ‘granular’ structure0, and Figuren structure where0.875. the 3.5: where Note pairing the Thethe emergent amplitude pairing emergent amplitude ‘granular’ ‘self-organizes’ inhomogeneity ‘self-organizes’ structure into where superconducting into the superconductingof pairing the amplitude localislands pairingon islands ‘self-organizes’ the on amplitude the into superconducting ∆(r) in a disorderedislands on the | |= | |== = = = | |= = = scale ofscale the coherence of the coherence length, even length, though even the though ‘homogeneous’scale the of ‘homogeneous’ the coherence disordersuperconductor length, disorder potential even potential in thougha varies in thea inonvaries ‘homogeneous’ the the scale on left the of panelscale a lattice disorder of a andspacing. lattice potential the spacing.c, Local robustness in a energycvaries, Local on gap energy the of!dos scale the gap(R) of! single-particledos a lattice(R) spacing. c, gap Local [2,energy 3, gap 4] across!dos(R) from BdG,from defined BdG, definedas the smallest as the smallest energy at energy whichfrom at the which BdG, local defined the DOS local is as non-zero DOS the smallest is non-zero (N(R energy,!) (>N(0R at.,004).! which) > 0 Note. the004). localthat Note this DOS that gap is non-zerothis is finite gap everywhereis (N finite(R,! everywhere) > and0.004). that and Notethe that that the this gap is finite everywhere and that the smallestsmallest gaps occur gaps on occur the SC on islands the SC defined islands definedbysmallest the largest by gaps the pairing largest occur onamplitude. pairingthe the SIT SC amplitude. islands suggests defined by an the effective largest pairing low-energy amplitude. description in terms of a disordered quantum XY model shown on the right. The quantum phase transition occurs when long range phase 1op(R)1opc(R )cR cgeneratedR cR generated in the presence in the1 presenceop of( largeR) of disorder,cR largecR disorder,generatedcoherence as we asComparison in we the is presenceComparison lost between with of large experiments. with disorder, weakly experiments. as connectedWe we describeComparisonWe describe “superconducting the connection with the connectionexperiments. islands”We describe tuned by the the connection ratio now explain.= h # ="ih # "i = h # "i between our predictions and experiments on the disorder-tuned now explain. now explain. E /E of chargingbetween energy our predictions to Josephson and experiments couplingbetween onour as the well predictions disorder-tuned as by anddisorder, experiments modeled on the by disorder-tuned removing We showWe in show Fig. in5 that Fig. even5 that for even ‘homogeneous’ for ‘homogeneous’We show disorder, in Fig. disorder, 5thatc thatJ even thatSIT in for systemsSIT ‘homogeneous’ in systems such as such indium disorder, as indium oxide, that titanium oxide,SIT titanium nitride,in systems nitride, and such niobium and as indium niobium oxide, titanium nitride, and niobium is, an uncorrelatedis, an uncorrelated random random potential potentialV (is,R) an (Fig.V ( uncorrelatedR 5a),) (Fig. the 5a), pairinga random the fraction pairingnitride potentialp ofnitride films, theV ( forR films, Josephson) which (Fig. for 5a), our which theory the bonds. our pairing seemstheory to seemsnitride be the to mostfilms, be the appropriate. for most which appropriate. our theory seems to be the most appropriate. amplitudeamplitude1op(R)1 exhibitsop(R) exhibits an emergent an emergent ‘granular’amplitude ‘granular’ structure1op( structureR) (shown exhibits (shown anFirst, emergent letFirst, us ‘granular’ discuss let us discuss the structure insulating the insulating (shown side of side theFirst, SIT. of let the The us SIT. discuss existence The the existence of insulating of side of the SIT. The existence of in Fig. 5b).in Fig. The 5b). system The system self-organizes self-organizes into superconductingin into Fig. 5b). superconducting The system islands, self-organizes islands,a gap ina gaptheinto insulator in superconducting the insulator implies implies activated islands, activated transport,a gap transport, in the consistent insulator consistent with implies with activated transport, consistent with 5 5 5 on the scaleon the of scale the coherence of the coherence length, length,with finiteon with the1 finiteop scale(R),1 of interspersedop the(R), coherence interspersedearly length, measurementsearly with measurementsfinite 1op on(R amorphous), interspersed on amorphous InOx filmsearly InOx. measurements Furthermore,films . Furthermore, there on amorphous there InOx films . Furthermore, there 8 8 8 with insulatingwith insulating regions regions where where1op(R)1 isopwith( negligible.R) insulating is negligible. The regions spatialmental The wherespatialis significance, evidence1opis(R evidence) for is negligible. pairs the for onconductivity pairs the The on insulating spatial the insulatingσ side(isω) evidence of and side the the oftransition for the boson pairs transition onin (“pair”) thein insulating spectral side of function the transitionP (ω)in variationsvariations of spectral of spectral features features (asymmetry (asymmetryvariations and coherence and of spectral coherence peaks) features peaks)specially (asymmetryspecially patterned patternedand amorphous coherence amorphous bismuth peaks) bismuth films.specially films. patterned amorphous bismuth films. in thisin inhomogeneous this inhomogeneous state were state already werein already this discussed inhomogeneous discussed aboveobtained above in state in byRecent were analytic alreadyRecent scanning continuation scanning discussed tunnelling tunnelling above microscpy from in microscpy imaginary (STM)Recent (STM) experiments scanning time experiments using tunnelling are the are maximum microscpy (STM) entropy experiments method are connectionconnection with Fig. with 4. Fig. 4. connection with Fig. 4. directlydirectly relevant relevant to our to predictions our predictions on thedirectly on superconducting the relevant superconducting to our side predictions side on the superconducting side 18 18 18 The closeThe connection close connection between between inhomogeneity inhomogeneityThe close and energy connection andsupplemented energy gaps between gapsof the inhomogeneitySIT.of by the Experiments SIT.sum Experiments rules. and on energy homogeneously Focusing on gaps homogeneously onof disordered the a bosonic SIT. disordered Experiments TiN model films TiN on filmsallows homogeneously us to determine disordered TiNexactly films is madeis clear made in clear Fig. in5b,c, Fig. which 5b,c, demonstrates which demonstratesis made two clear striking two in Fig. striking facts. 5b,c, facts. whichhave shown demonstrateshave shown that, whereas twothat, striking whereasTc goes facts.Tc togoes zerohave to at zero shown the atSIT, that,the the SIT, whereas STM the STMTc goes to zero at the SIT, the STM ! ! We seeWe that see (1) that there (1) is there an energy is an energy gap inWe gapthe see LDOS in thatthe atLDOS (1) every therewhat at site, every is an physics site,gap energydosgap gaparisesremains indos the outremains finite, LDOS of inthe finite, at agreement every phase in agreement site, fluctuations withgap Fig. with! 1.dos Furthermore, Fig.remains driving 1. Furthermore, finite, the the in SIT, agreement the and haswith theFig. 1. added Furthermore, benefit the and (2)and small (2) gaps small!dos gaps(R)! indos the(R) LDOS in the are LDOSand spatially (2) are small spatially correlated gaps correlated!dos with(R) in withgap the inLDOSgap the are LDOSin the spatially LDOSshows correlated markedshows marked withinhomogeneity, inhomogeneity,gap in the which LDOS supportswhich shows supports marked inhomogeneity, which supports large 1largeop(R)1 SCop( islands.R) SC islands. large 1op(R) SC islands. our pictureour picture of emergent of emergent granularity granularity (seeour Figs (see picture 4 and Figs 5). 4 of and Afteremergent 5). our After granularity our (see Figs 4 and 5). After our A simpleA simpleway to way understand to understand these results theseA resultsis simple to use is thewayto use pairing- tothat the understand pairing- it allowspaper these waspaper us results written,to was perform is written, to we use became simulations the we pairing- became aware ofaware thatpaper new areof was experiments new much written, experiments larger thatwe became than that those aware achievedof new experiments in DQMC that of-exact-eigenstatesof-exact-eigenstates approach approach generalized generalizedof-exact-eigenstates to highly to highly disordered disorderedapproachcorroborate generalizedcorroborate our to predictions. our highly predictions. disordered STM experiments STMcorroborate experiments on InO our onx (ref. predictions.InO 31),x (ref. 31), STM experiments on InOx (ref. 31), systemssystems15. In the15. limit In the of limit weak of attraction, weak attraction, pairingsystems pairing leads15. In to theleads a gap limit tosimulations in a theof gap weak inTiN the attraction, (ref. ofTiN the 32), pairing (ref. attractive and 32), leads NbN and to a Fermifilms NbN gap in33 films thehave Hubbard33 TiN allhave found (ref. model. all 32),found a pseudogap and a pseudogap NbN films33 have all found a pseudogap low-energylow-energy DOS in DOS the underlying in the underlying Andersonlow-energy Anderson insulator insulator DOS and inleads the and to underlying leadspersisting to Andersonpersisting up to insulator upmany to timesmany and leadsT timesc. In to T particular,.persisting In particular, they up observeto they many observe a times aT . In particular, they observe a The quantum XY model is equivalentc to a Josephson-junctionc array, with the Hamilto- the islandsthe islands with non-zero with non-zero1op and1 aop smalland athe energy small islands energygap. with On gap. the non-zero On other the1 otheropmarkedand amarked small suppression energy suppression gap. of On the theof low-energy otherthe low-energymarked DOS together DOS suppression together with of a with the alow-energy DOS together with a hand, thehand, insulating the insulating sea corresponds sea corresponds to thehand, higher-energy to the the higher-energy insulating strongly sea strongly correspondsdestructiondestruction to the of coherence higher-energy of coherence peaks strongly above peaksT abovec,destruction in completeTc, in complete of agreement coherence agreement peaks above Tc, in complete agreement localizedlocalized states in states the system. in the system. localized states in thenian system.with ourwith predictions. our predictions. with our predictions. From thisFrom perspective this perspective one can one see can that see the that gap the! gap, observed! , observedWe hopeWe that hope future that! future STM experiments STM experiments will study will in study detail in detail From thisdos perspectivedos one can see that the gap dos, observedEc X 2WeX hope that future STM experiments will study in detail in thein spatially the spatially average average DOS, initially DOS, initially decreasesin the decreases spatially with increasing with average increasing DOS,the initially anticorrelationthe anticorrelation decreases that withHˆ weJ that increasing= predict we predict betweenntheˆ between anticorrelation the heightJij thecos height of ( thatθˆ thei of weθˆj the) predict between the height (3.2) of the 2 i − − disorderdisorder owing owing to a reduction to a reduction in the in DOSdisorder the near DOS owing the near chemical to the a reductionchemicalcoherence incoherence the peaks DOS (associated peaks near (associated the with chemical large withi pairing largecoherence pairing amplitude)hiji peaks amplitude) (associatedand the and thewith large pairing amplitude) and the potentialpotential in our in model. our model. (In a real (In amaterial, realpotential material, the in coupling ourthe couplingmodel. will (Insmall will a real energysmall material, energy gaps in thegaps thecoupling local in the DOS. local will The DOS. obvioussmall The energy obvious quantum gaps quantum critical in the criticallocal DOS. The obvious quantum critical 29 29 29 also decreasealso decreasewith disorder.)with disorder.) However, However,also at decrease high at disorder, highwith disorder,where the disorder.) thescaling the number However,scaling between between operator atTc highandT disorder,⇢c sn ˆ(0)andi at at⇢s the(0)site the atSIT,i thescalingis well canonically SIT, studied between well studied inT ratherc conjugateand in⇢ rathers(0) at to the the SIT, phase well studied operator in ratherθî. 2 2 34 34 2 34 gap growsgap (consistentgrows (consistent with Fig. with 1) Fig. like 1)!gapdos like grows!Udos/(2 (consistent⇠Uloc),/(2 where⇠loc), with wheredifferent Fig. 1)different like systems!dos systems,U also/(2 remains,⇠loc also), where remains to be testeddifferent to be experimentally tested systems experimentally, also in remains in to be tested experimentally in ⇠ is the⇠ single-particleis the single-particle localization localization length⇠ 15 length⇡. This| |15⇡ is. This| due| isto due the tos the-waves superconducting-wave superconducting15⇡ | | films. films. loc loc loc is the single-particleHere localizationEc is the length charging. This energy. is due to The the Josephsons-wave superconducting couplings films. are Jij = EJ with probability enhancedenhanced effective effective attraction attraction between between fermionsenhanced fermions confined effective confined to attraction a to a between fermions confined to a smallersmaller localization localization volume volume⇠ 2 . ⇠ 2 . smaller localization volume ⇠Conclusion2 . Conclusion Conclusion loc loc (1 p) andloc Jij = 0 with probability p. The clean system (p = 0) is a coherent superconductor The phaseThe stiffness phase stiffness (or superfluid (or superfluid density)The density)⇢s phase(T ⇢0), stiffnesss(T on0),− the (or on superfluidIn the conclusion,In conclusion, density) we have⇢s( weT obtained have0), obtained on detailed the detailedIn insights conclusion, insights and predictions we and have predictions obtained detailed insights and predictions = = = other hand,other decreaseshand, decreases monotonically monotonically withother disorder with hand, disorder decreases as thewhen as SC monotonicallytheEfor SCdominates observablefor observable with properties over disorder propertiesE of,as the with the highly of the SCphases disordered highlyfor aligned observable disordered superconducting across properties superconducting all of the the junctions.highly disordered However, superconducting large islandsislands become become smaller smaller and the and Josephson the Josephson couplingislands coupling become between smaller between islands and islands theandJ Josephson insulatingand insulating coupling states instates between 2Dc films,in islands 2D and films, ofand and the insulating oftransition the transition states between in between 2D films, and of the transition between becomesbecomes weaker. weaker. Thus, even Thus, if even one ifstarts onebecomes with starts a withweaker. weak-coupling a weak-coupling Thus, eventhese if one states.these starts states. Although with Although a weak-coupling we focused we focused on sthese-wave on states.s-wave SC films, Although SC it films, has we it hasfocused on s-wave SC films, it has Ec/EJ favors a well-defined number eigenstate, leads to strong phase fluctuations, and BCS superconductorBCS superconductor with ! withdos !⇢doss, disorderBCS⇢s, superconductor disorder will necessarily will necessarily with not! escapednot⇢ , escaped disorder our attention our will attention necessarily that aspects thatnot aspects of escapedour of results our our results bear attention a bear that a aspects of our results bear a ⌧ ⌧ dos ⌧ s drive itdrive into it the into!dos the !⇢doss regime.⇢s regime. Eventually,drive Eventually, it quantuminto the quantum! phasedos phase⇢sstrikingregime.striking resemblance Eventually, resemblance to quantum the completelyto the phase completely different—andstriking different—and resemblance much less muchto the less completely different—and much less drives the system into an insulating state. Thus Ec/EJ can be used to tune across the SIT fluctuationsfluctuations destroy destroy long-range long-range order at orderfluctuationsT at0,T leading destroy0, leading to long-range an tounderstood—problem an orderunderstood—problem at T 0, of leading the of pseudogap to the an pseudogapunderstood—problem in the ind-wave the d high--waveT ofc high- theT pseudogap in the d-wave high-T = c c insulatorinsulator with low-energy with low-energy excitations excitations thatinsulator are that pairs are with= localized pairs low-energy localized on excitationssuperconductors. on superconductors. that are Features= pairs Features localized such as such the on loss assuperconductors. the of loss low-energy of low-energy spectral Features spectral such as the loss of low-energy spectral SC islands.SC islands. SC islands. in the cleanweight system.weight persisting persisting A across quantum acrossthermal phase thermal or quantum transitionweight or quantum phase persisting can transitions, phase also across transitions, be thermal induced or by quantum increasing phase disorder transitions, The low-The⇢s low-regime⇢s regime on the on SC the side SC of side theThe SIT of thelow- leads SIT⇢s to leadsregime a finite- to on a finite- theeven SC as sideeven coherence of as the coherence SIT peaks leads are peaks to destroyed, a finite- are destroyed, mayeven well as may coherence be well common be peaks common to are todestroyed, may well be common to temperaturetemperature transition transition driven bydriven thermal by thermal phasetemperature fluctuations phase transition fluctuations30 with driven30 withall by systems thermalall systems where phase thewhere fluctuations small the superfluid small30 with superfluid stiffnessall systems stiffness drives where thedrives loss the the ofsmall loss superfluid of stiffness drives the loss of Tc ⇢sT(0). The⇢ (0). large The energy large energy gap then gap leads thenT to leads a⇢ marked(0). to The a marked deviation large energy deviationphase gap then coherence.phase leads coherence. to The a marked pseudogap The deviationpseudogap in underdoped inphase underdoped27 coherence. cuprates cuprates is The driven pseudogap is driven in underdoped cuprates is driven ⇠ c s c s from conventionalfrom⇠ conventional BCS theory, BCS theory, with a with pairingfrom⇠ a pairingconventional pseudogap pseudogap in BCS the theory, inby the the withby proximity thea pairing proximity to pseudogap the toMott the insulator in Mott the insulator andby the further and proximity further complicated to complicated the Mott insulator and further complicated < < < < the temperaturethe temperature range T rangec TTc !dosT. This!thedos pseudogap. temperatureThis pseudogap exists range even existsTc < evenbyT < competing!bydos. competingThis order pseudogap parameters, order exists parameters, with even disorder withby competing disorder probably probably order playing parameters, playing a a with disorder probably playing a in the weak-couplingin the weak-coupling regime,⇠ regime, provided⇠ ⇠ provided⇠ onein isthe close one weak-coupling is enough close enough to the regime, to⇠secondary the provided⇠ secondary role, one unlike is role, close the unlike enough disorder-induced the to disorder-induced the secondary pseudogap pseudogap role, near unlike the near SIT the the disorder-induced SIT pseudogap near the SIT SIT so thatSIT so⇢s that!dos⇢s . !dos. SIT so that ⇢ ! . discusseddiscussed in this paper. in this paper. discussed in this paper. ⌧ ⌧ s ⌧ dos

NATURENATURE PHYSICS PHYSICSADVANCEADVANCE ONLINE PUBLICATION ONLINE PUBLICATIONNATUREwww.nature.com/naturephysics PHYSICSwww.nature.com/naturephysicsADVANCE ONLINE PUBLICATION www.nature.com/naturephysics 5 5 5 | | | | | | p (bond dilution) for ﬁxed Ec/EJ . (Fig. 3.11(h)). Thus Eq. 3.2 is a simple yet non-trivial model that describes a disorder-tuned SIT with a dynamical exponent z = 1. Disorder is

introduced into the quantum model by breaking bonds (“Josephson couplings”) on a 2D

square lattice with a probability p. We compare the results of the disorder-driven SIT with

the clean system [52, 55], where the SIT is tuned by Ec/EJ , the charging energy relative to the Josephson coupling.

Our results are obtained from calculations of the superfluid stiffness ρs, the complex conductivity σ(ω), and the boson spectral function Im P (ω). We estimate the superfluid stiffness ρs using

ρs/π = Λxx(qx 0, qy =0, iωn =0) Λxx(qx =0, qy 0, iωn =0), (3.3) → − → which is the difference of the longitudinal and transverse pieces of the current-current correlation function Λxx. Here jx(r, τ) sin [θ(r +x, ˆ τ) θ(r, τ)] is the current and ωn = 2πnT ∼ − are Matsubara frequencies. A detailed derivation of the different limits of the response functions and how quantities like the superfluid density relate to them can be found in

Appendix B.

We use the Kubo formula for the complex conductivity

kx Λxx(ω) σ(ω) = h− i − (3.4) ω + i0+ where kx is the average kinetic energy associated with bonds in the x-direction. The h− i real-frequency current-current correlation function is calculated from the imaginary-time

QMC by inverting

Z −∞ dω e−ωτ Λxx(q = 0, τ) = Im Λxx(q = 0, ω). (3.5) π 1 e−βω ∞ −

(From here on, we will use Λxx(τ) or Λxx(ω) to implicitly denote the q = 0 component of this function.) This analytic continuation procedure is delicate, and we accomplish it using the maximum entropy method (MEM). We have checked our results extensively using sum rules and compared the MEM results with direct estimates in imaginary time, as described in detail below. Details of the method and the checks are provided in Appendix B.

28 3.2.2 Conductivity across the charge-tuned SIT

Previous conductivity studies of the SIT in the disorder-free Josephson-junction array (JJA) model have focused on properties only at the critical point. Because we have carefully used analytic continuation methods, we are able to trace the evolution of the conductivity across the entire quantum phase transition. We will describe the features of σ(ω) in each of the phases and how the various energy scales that can be extracted go soft as the quantum critical point is approached.

Superconductor: Our results for the conductivity are shown in Fig. 3.7. The SC state is characterized primarily by a non-zero superfluid stiffness ρs, which appears in Re σ(ω) as a δ-function. Beyond this, Re σ(ω) shows finite spectral weight above a threshold. Note

that in the bosonic model, the cost of making electron-hole excitations is essentially inﬁnite

(i.e., much larger than all scales of interest), and we emphasize that this gap is not coming

from pair-breaking, which is not included in our model. The situation here is indeed more

interesting. Phase ﬂuctuations of the order parameter, Ψ = A exp(iθ), lead to a current

j Im Ψ∗ Ψ A 2 θ. (3.6) ∼ ∇ ∼ | | ∇

This then leads to the absorption threshold [56, 57] for creating a massive amplitude excitation (Higgs mode) and a massless phase excitation (phonon), see Fig. 3.6. Hence, we identify the threshold in Re σ(ω) with the Higgs scale ωHiggs. We emphasize that even though the microscopic model (3.2) has only phase degrees of freedom, its long-wavelength behavior upon coarse-graining contains both amplitude (Higgs) and phase ﬂuctuations (phonons and vortices). In addition, one can show that Re σ(ω) has a ω5 tail at low energies arising from

three-phonon absorption in a clean SC. The large power-law suppression, together with a

very small numerical prefactor [58], however, makes this spectral weight too small to be

visible in our numerical results for Re σ(ω).

As Ec/EJ is tuned to reach the SIT in the clean system, ρs decreases and vanishes at the transition; see Fig. 3.6. We also ﬁnd that the Higgs scale goes soft upon approaching

29 4 Ρs H´10L 3 2ΩB æ

æ 2 ìæà æ æ Ω ìæà æ Higgs Ω 1 æ à Σ æ ìæ æ æ ìæà 2ΩB 0 æ 3.5 4.0 4.5 5.0

EcEJ

Figure 3.6: (left) Free energy of superconducting state showing the gapless Nambu- Goldstone mode (green) and the gapped Higgs mode (red). (right) Energy scales, in units of EJ , as a function of the control parameter Ec/EJ in the clean system. From the SC side, the superfluid stiffness ρs and the Higgs “mass” ωHiggs, and from the insulating side, the optical conductivity threshold ωσ and the boson energy scales ωB and ωeB, vanish at the transition creating a fan-shaped region where quantum critical fluctuations dominate.

the quantum critical point, as expected. We have also used ρs to test the sum rule for the MEM-derived optical conductivity, which is an important veriﬁcation of the analytic

continuation procedure. The total spectral weight is given by

Z ∞ dω Re σ(ω) = π kx /2, (3.7) 0 h− i

R ∞ + where kx is the kinetic energy. We find that + dω Re σ(ω) (note the lower limit of 0 ) h− i 0 calculated from the MEM result differs from kx by an amount that is exactly accounted h− i for by the delta function ρsδ(ω). A complete discussion of the sum rule properties and verifications can be found in Appendix A.

While the delta function in Re σ(ω) cannot be directly detected in dynamical experi-

ments, its Kramers-Kronig transform in the reactive response Im σ(ω) = ρs/ω can indeed be measured. In the SC, the ﬁnite low-frequency absorption in Re σ(ω) (due to the single-

phonon processes discussed above) causes ωIm σ(ω) to deviate from a constant, as is evident

in Fig. 3.12. Our results are qualitatively similar to what has been seen in recent experi-

ments, which, however, have focused on ﬁnite-temperature transitions in weakly disordered

30 samples [59].

Insulator: The clean insulator shows a hard gap in Re σ(ω) with an absorption threshold that we denote by ωσ; see Fig. 3.11(e). The nature of this gap will become clear in our discussion of the boson spectral function Pq(ω) below, which has an energy gap ωB associated with the cost of inserting a boson (or superconducting pair) into the insulating state.

The simplest process contributing to the conductivity is described diagrammatically as the convolution of two boson Greens functions leading to ωσ = 2ωB as seen in Fig. 3.6. We also see that both of these energy scales go soft as the SIT is approached from the insulating

side.

The reactive response Im σ(ω) in the insulating phase shows general qualitative features

that can be experimentally veriﬁed. The crossover from inductive behavior in the supercon-

ductor to capacitive behavior in the insulator is clearly identiﬁable by the negative portion

of ωIm σ(ω) seen in Fig. 3.7. Additionally, the insulator is characterized by a diamagnetic

susceptibility whose coefficient is given by Im σ(ω) = ω/ at small ω. We find that − this result is consistent with our findings, although quantitative extraction of is problem-

atic because the maximum entropy method is less reliable at frequencies within the lowest

Matsubara frequency 2π/T . The appropriate way to study this would be to begin with a

current model dual to the phase model studied here, discussed in detain in Appendix A.

3.3 Spectral function P (ω) across the SIT

At any quantum phase transition, the disordered phase is expected to have an energy gap

that goes soft at the quantum critical point. Given that the single-particle gap remains

ﬁnite across the transition, what is the energy scale that vanishes on the insulating side of

the SIT? It was proposed by Bouadim, Loh, Trivedi, and Randeria [4] in their DQMC work

that it is energy scale two-particle (pair) excitations.

In this work, we have begun under the assumption that the single particle gap is much

larger than any other energy scales, and we consequently only consider bosonic excitations.

The notion of a “pair” spectral function in our model is then simply the bosonic Greens

31 Figure 3.7: Real (top) and imaginary (bottom) portions of the conductivity σ(ω) across the clean superconductor-insulator transition. Superconductor (a) and (d): Re σ(ω) contains a δ-function contribution proportional to the superfluid density ρs and an absorption threshold from the Higgs amplitude mode. ωRe σ(ω) shows deviations from a constant value (also proportional to ρs) from the charging fluctuations. As the SIT, (b) and (e) is approached, there is a transfer of spectral weight from ρs to low frequencies. Insulator (c) and (f): The insulating state is characterized by a lack of superfluid density and an absorption gap ωσ equal to twice the gap in the spectral function ωb. At the SIT, there is a crossover from inductive (ωRe σ(ω) > 0) to capacitive (ωRe σ(ω) < 0) behavior at low frequencies.

function. Exactly as with the conductivity, we calculate this function in imaginary time

and use maximum entropy to analytically continue it to real frequencies.

3.3.1 Bosonic spectral function in the phase model

The bosonic Greens function, in imaginary time, is given by P (r, r0; τ) = a†(r, τ)a(r0, 0) . h i We can write the particle creation operator in terms of an amplitude and phase a†(r, τ) = p n(r, τ) exp[iθ(r, τ)], but we are ignoring the on-site amplitude ﬂuctuations here and therefore

P (r, r0; τ) = exp θ(r, τ) θ(r0, 0) , (3.8) h − i which is just a spin-spin correlation function in the classical phase model. The spectral function, which is the imaginary part of the real-frequency Greens function comes for the

32 analytic continuation of the imaginary-time quantity

Z ∞ dω e−τω P (q, τ) = Im P (q, ω), (3.9) π 1 e−βω −∞ − exactly as before.

3.3.2 Frequency and momentum resolved spectral-function

While the idea of the pair spectral function was originally introduced to provide an energy scale that in ﬁnite in the insulating state, the full spectral function Im P (q, ω) contains the full information about the excitations in the system, and we can derive many useful quantities from it. For example, the momentum distribution nq is given by

Z ∞ 1 nq = P (q, τ = 0) = dω Im P (q, ω), (3.10) 1 e−βω −∞ − R ∞ which in the T 0 limit, reduces to the more familiar expression nq = dωIm P (q, ω). → 0 This expression provides an energy scale for the SC state (a ﬁnite condensate fraction

nq=0 > 0) and another check on the analytic continuation. Similarly, we can consider the sum over momentum of Im P (ω), which serves the purpose

of a bosonic “density-of-states:”

X Im P (ω) = Im P (q, ω) = Im P (r = 0, ω). (3.11) q

The notation may be slightly confusing here, but in this work, Im P (ω) will be used to

denote the local quantity Im P (r = 0, ω), equal to the sum over the momentum components

of the full spectral function Im P (q, ω). This quantity is the simplest component of the

spectral function that shows a ﬁnite gap in the insulating state.

3.3.3 P (ω) across the clean SIT

Fig. 3.8 shows the results for the spectral function across the clean SIT in the Josephson

junction array model. The full spectral function shows a dispersing mode in both the SC and

the insulator, with the insulating mode being gapped. The gapless mode in the SC state,

which we attribute to phonon modes, appears to have linear behavior at low frequencies, 33 Figure 3.8: Properties of the Spectral function Im Pq(ω) in the superconducting phase (left), at the SIT (middle), and in the insulating phase (right). The top two rows show the full spectral function Pq(ω), the third row shows the momentum distributions nq, and the bottom row shows the density of states.

which you would expect for such excitations. We will discuss this in more detail in the section on compressibility below.

34 Also shown in Fig. 3.8 is the momentum distribution nq calculated in two ways, directly from the imaginary-time data, nq = P (q, τ = 0), and as the integral over the analytically- continued data Z ∞ −βω −1 nq = dω(1 e ) Im P (q, ω). (3.12) −∞ − These two methods are indistinguishable from one another, an important check on the maximum entropy method. The characteristics of the momentum distribution are also clear. In the SC state, the presence of a condensate results in a jump in the momentum distribution at q = 0. The size of this jump is reduced to zero upon approach to the transition and is absent in the insulating phase. P The density-of-states Im P (ω) = q Im P (q, ω) also contains information about the excitations of the system. As can be seen in Fig. 3.8, the SC shows a large peak at low frequencies. Again, this peak is due to the presence of the condensate, and its height diminishes on approaching the critical point. The oddness of Im P (ω) makes this a curious feature, as it doesn’t show up as a true δ-function in Im P (ω) as might be expected. This is further complicated by the analytic continuation procedure, which is less reliable as low frequencies. The insulator is marked by a gap to excitations, as seen in the full spectral function.

3.3.4 Compressibility and dispersing sound mode in SC

Phonon modes are important characteristics of the superﬂuid state. These modes are expected to disperse with a sound velocity ω c q for small momenta, where the sound ≈ | | velocity is related to the compressibility κ and the superﬂuid density ρs:

p c = ρs/κ. (3.13)

(There is in principle a mass factor in this expression, too, which in our calculations has been set equal to one.) We have already discussed the superﬂuid density as the appropriate long-wavelength limit of the transverse current-current correlation function, and is plotted in Fig 3.6. Similarly, the compressibility is the long-wavelength limit of the density-density

35 p Figure 3.9: Compressibility κ, superﬂuid density ρs, and speed of sound c = ρs/κ across the disorder-free SIT. The compressibility is vanishing at the SIT, consistent with the gapped insulating phase, and the speed of sound goes to a constant value.

correlation function. Previously, we showed that the current is given by a spatial derivative of the superconducting phase j(x, τ) sin(δxθ(x, τ)); similarly, the number density at site ∼ i is given by a temporal derivative of the phase ni(τ) = i∂τ θi(τ). The compressibility is − given by

2 κ = lim Sq(iωn = 0) n , (3.14) q→0 − h i where Sq(iωn) is the the Fourier transform of the density-density correlation function

1 X Z β Sq(iωn) = dτ n(r, τ)n(0, 0) , (3.15) Nβ r 0 h i and n is the average density. h i The compressibility and speed of sound across the clean (Ec/EJ ) transition are shown in Fig 3.9. The compressibility vanishes as the SIT is crossed, which you would expect for the

incompressible insulating phase. The speed of sound, though, remains ﬁnite, indicating a

compensation between the vanishing superﬂuid density and the divergent inverse compress-

ibility. This is consistent with previous studies of the superﬂuid-Mott insulator transition

in the Bose-Hubbard Model [60]. In this work, the speed of sound was ﬁnite only when

the superﬂuid-to-Mott Insulator transition was tuned through the tip of the Mott lobe,

precisely where the Bose-Hubbard Model and the Josephson junction array Hamiltonian

are equivalent.

36 Figure 3.10: Full spectral functions Im Pq(ω) for diﬀerent Ec/EJ values in the supercon- p ducting phase. The black line indicates the phonon mode ω = c q , where c = ρs/κ is | | shown in Fig. 3.9. The speed of sound increases up to the SIT, where a gap in Im Pq(ω) opens. The speed of sound is a quantity derived entirely from (two separate) imaginary-time correlation functions, and its agreement with the analytically continued function Im Pq(ω) provides further evidence of the validity of the maximum entropy method.

The comparison between the speed of sound calculated from the imaginary-time cor-

relation functions and the analytically-continued spectral function Im Pq(ω) is shown in

Fig. 3.10 for various Ec/EJ values in the SC state. The two calculations show remarkable agreement and provide strong evidence for the maximum entropy methods we have

employed. The disagreement as higher frequencies can be understood as distortion to the

dispersion coming from the underlying lattice.

3.4 Dynamical response across the disorder-tuned SIT

3.4.1 Implementation of disorder

The disorder in our system is implemented by choosing the Josephson couplings Jij from a binary distribution   0 with probability p Jij = (3.16)  EJ with probability 1 p −

37 see Fig. 3.5. For each disorder value, we perform simulations over 100-200 disorder conﬁg- urations to get self-averaged results.

Because we only have disorder in the physical 2D quantum system, this translates into correlated disorder in the imaginary-time direction. This distinction is important in considerations of the Harris criteria for disorder-driven quantum phase transitions. The Harris criteria states that disorder will remain irrelevant to the universality of a phase transition if

dν > 2, (3.17)

where d is the dimension and ν is the correlation length exponent. Even though the quantum

problem in 2D has a classical interpretation in (2+1)D, the dimensions that enters the Harris

criteria must be the dimensionality of the disorder. In our case, then d = 2 and ν = 0.67

(for the 3D XY model) and dν = 1.34 < 2, the Harris criteria is violated, and we expect the presence of disorder to change the universality class between the clean and disorder-driven transitions. We will see below that this is indeed the case, and we’ll be able to provide an estimate on the critical exponents of the disorder-driven transition.

3.4.2 Dynamical quantities σ(ω) and P (ω)

Superconductor: The disordered SC results differ in several ways from those of the clean system. First, the superfluid stiffness ρs is reduced by disorder, vanishing at the SIT upon tuning the transition by disorder p. An important difference is the absence of a discernible Higgs threshold in Re σ(ω) for the disordered SC; see Fig. 3.11(b). Qualitatively we can understand this by the fact that once disorder breaks momentum conservation even single-phonon absorption is permitted and one no longer needs a multi-phonon process for absorption. The effect of long-range Coulomb interactions, which change the phonon dispersion ( q) to that of a 2D plasmon ( √q), is an important open problem. ∼ ∼ insulator: In contrast to the hard gap of the clean system, the dirty insulator exhibits absorption down to arbitrarily low frequencies (see Fig. 3.11(f)), which is, at least in part, due to rare regions. This then raises the question: what is the characteristic energy scale

38 Clean Hp = 0L Disordered HEcEJ = 3.0L

0.08 HaL EcEJ = 3.22 0.08 HbL p = 0.025 L 0.06 SC 0.06 Ω H

Σ 0.04 0.04 Ρs Ω Re 0.02 Higgs 0.02 0.00 0.00 0 2 4 6 8 0 2 4 6 8 0.8 HcL EcEJ = 4.17 0.8 HdL p = 0.290 L

Ω 0.6 SIT 0.6 H

Σ 0.4 0.4

Re 0.2 0.2 0.0 0.0 0.40 2 4 6 8 0.40 2 4 6 8 HeL EcEJ = 4.76 HfL p = 0.425 L 0.3 ΩEJ 0.3 ΩEJ

Ω INS H

Σ 0.2 0.2

Re 0.1 ΩΣ 0.1 0.0 0.0 0 2 4 6 8 0 2 4 6 8 ΩEJ ΩEJ

2.0 HgL E E = 4.08 5 HhL c J INS

Ω E E = 4.55 4 c J J L 1.5 E Ω 3 H c P 1.0 ΩB ΩB E 2

Im SC 0.5 1 0.0 0 0 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 ΩEJ p

Figure 3.11: (a-f): Re σ(ω) across the (a,c,e) clean (p=0) and (b,d,f) disorder-tuned (fixed Ec/EJ ) superconductor-insulator transitions. (g): Boson spectral function Im P (ω)/ω for a clean superconducting (blue) and insulating (red) state. The energy scales shown in Fig. 3.6 are indicated in (a-g). All quantities are at fixed temperature T/EJ = 0.156, and fixed system size, 256 256 for the clean case and 64 64 for the disordered. In the disordered × × system, the spectral functions are marked by a significant increase in low frequency weight, obscuring the gap scales of the clean system. (h): Schematic phase diagram showing how the SIT can be crossed by either increasing Ec/EJ or by tuning the disorder p.

that goes soft as one approaches the SIT from the insulating side? We ﬁnd that this scale is

the location of the low-energy peak at ωeB in the boson spectral function Im P (ω)/ω, whose evolution with disorder is most readily seen in the “slingshot-like” plot in Fig. 3.12(c). The

corresponding changes in Re σ(ω) are shown in Fig. 3.12(a). We also note that there is a

marked change in Im σ(ω) across the SIT. We see from Fig. 3.12(b) that it changes sign

at low frequencies from an inductive (Im σ(ω) > 0) to a capacitive (Im σ(ω) < 0) response

going through the disorder-tuned SIT.

39 Figure 3.12: Dynamical response functions across the disorder-tuned SIT. The critical disorder pc = 0.337 is marked as a dashed line; T/EJ = 0.156, Ec/EJ = 3.0 and L = 64. (a) In the conductivity Re σ(ω) the superﬂuid response is evident as a zero-frequency delta function of strength ρs. Deep in the insulator there is a gap in Re σ(ω) that grows with disorder. (b) ωIm σ(ω) shows a crossover from “inductive” (ωIm σ(ω) = ρs > 0) to “capacitative” (ωIm σ(ω) < 0) behavior at small ω across the transition. (c) The boson spectral function Im P (ω)/ω, which has a peak centered about zero frequency in the superconductor, develops a characteristic scale ωeB in the insulator that grows with disorder.

3.4.3 Quantum critical region

We have already discussed the various scales that go soft on approaching the SIT from either side. The results for the clean system, with SIT tuned by Ec/EJ , are summarized in

Fig. 3.6. At the low temperature (T/EJ = 0.156) where these results have been calculated, there is no discernible critical fluctuation region. This is not the case for the disorder- tuned transition. The finite temperature QMC data, taken at face value, suggest a finite separation between the disorder value at which ρs goes to zero from the SC side and the disorder value at which characteristic boson scale ωeB vanishes from the insulating side; see Fig. 3.13 (a,b).

There is some speculation about whether a “bose metal” phase intervenes between the superconducting and insulation phases of the SIT for a ﬁnite range of disorder strengths, but we have deﬁnitively ruled out this possibility for our model by performing careful scaling analysis of our data. We have determined that this intermediate region is not a Bose metal separating the SC and insulator, but rather the quantum critical region. As shown in

Fig. 3.13 (c), the SC transition temperature Tc, at which ρs vanishes, and the crossover

40 1.0 æ æ HaL æ æ æ 0.8 æ * æ 0.6 Σ æ æ æ æ Ω 0.4 Ρs æ B æ æ 0.2 æ æ æ æ æ æ æ æ æ æ æ æ 0.0 æ æ 0.1 0.2 0.3 0.4 0.5æ 1.0 H L æ b æ æ 0.8 æ Σ* æ æ 0.6 æ ΩB 0.4 Ρs æ æ æ 0.2 æ æ æ æ 0.0æ æ æ æ 0.1 0.2 0.3 0.4 0.5 p 0.35 0.30 HcL æ Quantum Critical * 0.25 T æ T C Region 0.20 æ à T HaL 0.15 æ à 0.10 æ à HbL æ à 0.05

0.00 æ 0.1 0.2 0.3 0.4 0.5 p

Figure 3.13: (a,b) Superfluid stiffness ρs (green), bosonic scale ωeB (red) in the insulator, and low-frequency conductivity σ∗ (blue), defined in the text, as functions of disorder p at two different temperatures shown in panel (c). The quantities are in units of EJ and 2 σQ = 4e /h, respectively. The quantum critical region is shaded gray in all three panels. ∗ (c) Phase diagram with Tc determined by vanishing of ρs and T by the vanishing of ωeB. zν The lines are fits to p pc with pc 0.337 and zν 0.96. | − | ≈ ≈

∗ scale T , at which ωeB vanishes, deﬁne this fan-shaped critical region. (We have used z = 1 in scaling the system size as we go down in temperature in Fig. 3.13 (c); see Appendix A.)

∗ Both Tc and T extrapolate to zero at the same critical disorder pc 0.337 (for the chosen ≈ zν value of Ec/EJ ) with the scaling p pc where z = 1 and ν = 0.96 0.06. This change | − | ± in universality is expected for a system that, like ours, violates the Harris criteria. The critical exponents for the disordered system have few comparisons in the literature, which we explore further in the next section.

3.4.4 Universal conductivity

Finally, we turn to the important question of the dc conductivity at the SIT [52, 53, 54, 55,

61]. At the quantum critical point, the dc conductivity has been measured to be very near 41 1.0 æ 1.0ææ à * 0.8 HaL æ Σ à Σ* TEJ = 0.313 è L ò HbL ò 1 0.6 æ æà ì 0.250

- à

Q 0.8 æà

Σ ò 0.4 æà 0.208 ì æà æà à ì æ 0.2 à æ æà 0.156 ò à æ æà æà æ à à ô æà æ æà æ æ æà æà à æà æ 0.104 ô 0.0àæ 0.6 òô ì

0.0 0.1 0.2 0.3 0.4 0.5 Q ô æàìò 0.078 è

Σ à

p ô * æ

Σ à ìò HcL ìò 0.4 æ 0.8 ìò ô æì à à æ ì æ ææô ò Q 0.6 ìôò à æ àìæòà Σ ì æ æà à * ìò 0.4 æ 0.2 ô ò ì

Σ ô à æìæ à òòà æ æ ò ì ì àæ ô 0.2 ô òì à ò æ ô ô 0.0 0.0 æ æ 0.0 0.1 0.2 0.3 0.4 0.5 0.30 0.35 0.40 0.45 -1zv Èp-pcÈT p

Figure 3.14: (a) Comparison of two methods for obtaining the low-frequency conductivity ∗ near the SIT at T/EJ = 0.156, with σ from the integrated spectral weight in Eq. (3.22), ∗ and σΛ from the current correlator Λxx at imaginary time τ = β/2 (see text). (b) Plot of σ∗(T ; p) as a function of the disorder p at various temperatures. The various curves cross ∗ at the critical disorder strength pc at which σ is T -independent with the critical value ∗ ∗ σ 0.5σQ. (c) Scaling collapse of the σ (T ; p) data with pc = 0.337 and zν = 0.96, ≈ consistent with Fig. 3.13.

2 the quantum of conductance σQ = (2e) /h. This quantity is a universal number, characteristic of the universality class underlying the transition. It’s extraction is complicated by the

diﬃculty of extracting this quantity reliably from numerical data. Damle and Sachdev [62]

showed that at the critical point the conductivity has a universal form

σ(ω) = σQΦ(ω/T ), (3.18)

where Φ is the universal scaling function; that is, σ(ω) only depends on the scaling variable

ω/T at the critical point. This allows us to deﬁne two distinct zero-frequency limits of the

42 (a) (b)

Figure 3.15: (a) Illustration of low frequency integrated weight σ∗ deﬁned in Eq. 3.21. σ∗ is a universal quantity and a reliable estimate of the dc conductivity that can be extracted from diﬀerent types of numerical or experimental data. (b) Illustration of low frequency ∗ estimate of σ directly from correlation functions. Here Kβ/2 = 1/2csch(βω/2) denotes the Kernel of the integral relation between Λxx(τ = β/2) and Λxx(ω).

conductivity

σ(0) = σ(ω 0,T = 0) (3.19) → σ( ) = σ(ω = 0,T 0). (3.20) ∞ →

The dc limit accessed in experiments is equivalent to σ( ), which requires ω 0 ﬁrst and ∞ → then T 0. This limit is unfortunately not possible when analytically continuing Matsubara → data [62], because we are always limited frequency values greater than the lowest Matsubara frequency ω1 = 2πT . Regardless, we have devised a way to extract a reliable universal quantity from analytically continued QMC data by exploiting quantum critical scaling and sum rules. This is accomplished through the following observations: (i) the MEM results satisfy the conductivity sum rule, which integrates over all frequencies, and (ii) these results are reliable for high frequencies ω > 2π. Taking the diﬀerence of integrated spectral weights, we can reliably estimate Z 2πT σ∗ = (2πT )−1 dωRe σ(ω, T ; p). (3.21) 0+ −1/zν We may now use the universal scaling form [62] Re σ(ω, T ; p) = σQΦ ω/T ; p pc T , | − |

43 2 with σQ = 4e /h, to obtain

Z 2π ∗ σQ −1/zν σ (T ; p) = dx Φ(x; p pc T ). (3.22) 2π 0+ | − | where x = ω/T . Thus σ∗ is a T -independent universal constant at the quantum critical point p = pc and closely related to the low frequency conductivity measured in experiments. The low-frequency conductivity can also be estimated directly from the current-current correlation function either in imaginary-time or as a function of the Matsubara frequencies.

In imaginary-time, the low frequency conductivity is related to [63].

2 ∗ β σ = Λxx(τ = β/2) (3.23) Λ π

To see this, ﬁrst note that

Z ∞ dω e−βω/2 Λxx(τ = β/2) = Λxx(ω) π 1 e−βω −∞ − Z ∞ dω 1 = csch(βω/2) Λxx(ω) (3.24) −∞ π 2

The csch(βω/2 kernel is singular at the origin and decays to zero with a width proportional

∗ to the temperature T = 1/β. Assuming that Λxx(ω) = σΛσQω over this frequency range, ∗ the integral can be evaluated to give the abo‘ve expression for σΛ. This is illustrated in Fig. 3.22. The σ∗ estimates obtained by the two methods show excellent agreement

(Fig. 3.14(a)) and provide a non-trivial check on the analytic continuation. Alternatively, the low frequency conductivity can be related to the current-current correlation function expressed in Matsubara frequencies. The relationship between Λxx(τ) and Im Λxx(ω) can be rewritten in terms of Λxx(iωn) as

Z ∞ 2 dω ωn D(iωn) kx Λxx(iωn) = ρs + 2 2 Im Λxx(ω). (3.25) ≡ h− i − −∞ π ω + ωn

∗ The same approximation for the conductivity at the critical point, that σ(ω) σ σQ, ≈ Λ ∗ determines that D(iωn) σ σQωn at small ωn. While this method has been used as an ≈ Λ ∗ estimate of σΛ in other works, we ﬁnd that it is an unreliable method for our results. In Fig. 3.14 (b) we plot σ∗(T ; p) as a function of p for various temperatures. In the

44 ∗ Method σ /σQ σ(0)/σQ σ( )/σQ ∞ QMC + MEM (this work) 0.4 - - QMC + Pade Approximates [55] - 0.45 0.05 - ± Holography [64] 0.45 0.475 0.36 Holography [65] 0.443 0.45 0.36

Table 3.2: Table of numerical estimates of the dc conductivity for the disorder-free 2D Josephson-junction array.

superconductor (p < pc) the conductivity increases with decreasing T , while the opposite trend is observed in the insulator (p > pc). Precisely at the SIT p = pc, we ﬁnd a T - independent crossing point which also allows us to estimate the critical σ∗. Another way

∗ −1/zν to scale the data is to plot σ (T ; p) as a function of the scaling variable p pc T . We | − | ﬁnd data collapse for pc = 0.337 and zν = 0.96 (consistent with Fig. 3.13) with a critical

∗ value of σ 0.5σQ. ≈ We can do the same for the disorder-free model, which is in the (2+1)D XY universality class. The value we find here is σ∗ 0.4, which is consistent with the recent result of ≈ Smakov and Sorensen [55], where they found σ(0) = 0.45 0.05 using Padéapproximates ± to analytically continue MC data at the critical point. These results differ from the previous estimate [52] of σ(0) = 0.285 0.02 obtained by extrapolation of the current-current ± correlation function for ωn 0. → More recently, groups [64, 65] have exploited the AdS/CFT correspondence to use holographic continuation to perform analytic continuation at the critical point. A distinct benefit of this method is that they are able to access the σ( ) limit of the function. Both ∞ groups find σ( ) = 0.356. This procedure comes with certain caveats though. First, the ∞ discrepancy between the two values is attributed [65] to the extreme precision (up to five significant figures) of location of the critical point. Second, there is currently no procedure known to apply AdS/CFT procedures to disordered problems.

For the disorder-tuned transition, we have obtained

σ∗ 0.50 (3.26) ≈ ν = 0.96 0.06 (3.27) ± 45 Figure 3.16: Compressibility κ, superﬂuid density ρs, and spectral function gap ωeB across the disorder-driven SIT. These results are for T/EJ = 0.156 and show a ﬁnite compressibility tail into the insulating phase. Whether or not this tail scales to zero in the insulating phase with lowering temperature has not been explored in our model.

where z = 1 by deﬁnition for the model. There have been only a few results on disordered transitions that can meaningfully be compared to our work. A Monte Carlo study of the

(2+1)D XY model with onsite charging energy disorder [54] found z = 1.07 0.03, ν 1, ± ≈ and σ(0) = 0.27 0.04 obtained by extrapolation of Λxx for ωn 0. We expect that using ± → analytic continuation could modify this estimate. Studies of the disordered quantum rotor model using strong disorder renormalization group theory [66] have found ν = 1.09 0.04, ± although they have not looked at the universal conductivity.

3.4.5 Nature of the dirty insulating phase

We now turn to the question of the nature of the dirty insulating phase. In the literature, an insulating phase of bosons with disorder has been referred to as a compressible Bose glass phase away from particle-hole symmetry [67] or an incompressible Mott glass phase with particle-hole symmetry [66]. We work with a particle-hole symmetric system, and while we see evidence of a gap-like scale in the insulator, we also ﬁnd a low-frequency tail in the absorption, presumably arising from rare regions. In this respect, we expect our results to be more akin to a Bose glass phase.

46 We can also directly calculate the compressibility across the disorder-tuned transition in the same way that we calculated it across the clean transition. Indeed, we find that the compressibility remains finite into the insulating phase; see Fig. 3.16. This result would seem to provide further evidence that the insulating phase is a compressible Bose glass phase, despite the fact that our model has particle-hole symmetry. It should be noted though, that these results are for a relatively high temperature system (T/EJ = 0.156), where there is a significant quantum critical region. We cannot rule out the possibility that

the finite compressibility tail into the insulating phase is not a finite temperature effect

without seeing the eﬀect of lowering the temperature. Still though, the compressibility

results are a promising direction in addressing this question.

3.5 Experimental comparison

Optical conductivity measurements on disordered thin ﬁlms have been carried out primarily

at frequencies well within the superconducting gap (0–20 GHz) [68, 69, 70, 71, 72, 59].

These low-frequency experiments are able to capture deviations from the expected behavior

2 of Im σ(ω) = ρse /mω that cannot be accounted for by pair-breaking mechanisms. This has been a primary motivation of the theoretical work presented here, but the frequency

range (GHz) of these spectroscopic experiments is insuﬃcient to capture all of the physics

of our models.

More recently, a combination of THz measurements of the conductivity that extend up to the gap scale have been performed in conjunction with tunneling measurements on NbN

films. The combination of techniques allows us to directly look at the role of phase fluctuations across the SIT. Fig. 3.17 summarizes this experimental work. Panels (a) and (b) show the tunneling density of states G/GN and the real part of the optical conductivity σ1 for two different disordered samples of NbN. The strength of the disorder is characterized

by the Tc of the ﬁlms. In each, a BCS ﬁt is made to the density of states and a tunneling

gap ∆t is extracted. Using this gap, Mattis-Bardeen theory can be used to predict the conductivity if only pair-breaking is present in the system. For the weakly disorder system,

47 Figure 3.17: Tunneling density of states G/GN and optical conductivity σ1 of disordered NbN thin films. (a) shows a weakly disordered film where the gap scale extracted by a BCS fit (blue line) to the tunneling density of states correctly predicts the absorption gap in the conductivity. In (b), a more strongly disordered film, the best fit to the tunneling data does not predict the absorption gap. The green line shows the best BCS fit to the conductivity data, and predicts a reduced gap, min[σ1] attributed to phase fluctuations. (c) shows the discrepancy between gap scales predicted by these two experimental techniques; clean Tec = Tc/Tc .

Fig. 3.17(a), the BCS fit correctly predicts the absorption spectrum. But at higher disorder strengths, Fig. 3.17(b), this fit is clearly incorrect. Even the best fit using Mattis-Bardeen theory cannot fit the data, clear evidence that phase fluctuations are important near the

SIT. Fig. 3.17(c) shows the evolution of these two energy scales as you approach the SIT. As expected, the single particle gap remains ﬁnite and has the same order of magnitude as the transition is approached. In contrast, the gap scale from the conductivity measurements, denoted min[σ1], shown in blue, collapses to zero near the SIT. The inset of Fig. 3.17(c) shows these quantities plotted in variables scaled by the critical temperature, which more clearly shows the crossover from physics governed by quasiparticles and by phase ﬂuctuations.

A more direct comparison to the theoretical work we have done can be made by consid-

48 Figure 3.18: Comparison of the experimental subgap conductivity of NbN films to the theoretical predictions made in this work. The experimental subgap conductivity is obtained exp BCS by subtracting off the best BCS fit to the data: σH = σ σ . The experimental and 1 − theoretical data show qualitatively similar features.

ering just the subgap conductivity

exp BCS σH (ω) = σ (ω) σ (ω), (3.28) 1 − that is, the diﬀerence between the experimentally measured conductivity and the conductivity associated with the quasiparticle physics, obtained from the BCS ﬁt to the tunneling spectrum. This quantity is the most closely related to the conductivity we calculate in our bosonic model. Fig. 3.18 shows the comparison between the experimental subgap conductivity and our phase model results. Both results show qualitatively very similar behavior.

As the transition is approached, there is an increase in both the total subgap weight and the weight at very low frequencies. While a quantitative comparison is still difficult, this comparison provides evidence that the absorption in disordered thin films near the SIT is coming from a combination of collective phase modes. As mentioned in our discussion about, the presence of disorder wipes out a discernible Higgs mass scale, which is also not identifiable in the experimental data.

49 3.6 Concluding remarks

We have presented calculations of the complex dynamical conductivity σ(ω) and the boson

spectral function P (ω) across the SIT driven by increasing the charging energy Ec/EJ as well as by increasing disorder p. By comparison of the clean and disordered problems, we see the eﬀect of disorder on the Higgs scale ωHiggs in the superconductor and on the threshold ωσ in the insulator, in generating low frequency weight in absorption in both superconducting and insulating phases, and in expanding the region over which critical

fluctuations are observable. In the literature, an insulating phase of bosons with disorder has been referred to as a compressible Bose glass phase away from particle-hole symmetry [67] or an incompressible Mott glass phase with particle-hole symmetry [66]. We work with a particle-hole symmetric system, and while we see evidence of a gap-like scale in the insulator, we also find a low-frequency tail in the absorption, presumably arising from rare regions. In this respect our insulator seems more akin to a Bose glass. It is important to emphasize that the effects we have calculated have required going beyond mean field theories, even those that included emergent granularity due to the microscopic disorder, by focussing on the role of fluctuations of the order parameter. We have calculated the effect of these fluctuations, both amplitude and phase, on experimentally accessible observables using QMC methods coupled with maximum entropy methods, constrained by sum rules. Recently the AdS-CFT holographic mapping has been used to obtain the dynamical conductivity at the disorder- free bosonic quantum critical point [64, 65]. Our focus here has been on the evolution of the dynamical quantities in both the phases, superconducting and insulating, and across the disorder-driven SIT, for which the holographic formalism has not yet been developed.

Our calculations have laid the foundation for key signatures in dynamical response functions across quantum phase transitions. Many of which are currently being pursued in experiments, with promising preliminary results. Though we have focused on the disorder- driven s-wave SIT in thin ﬁlms, the ideas are equally relevant for a diverse set of problems, including: (i) unconventional superconductors like the high Tc cuprates that have a quantum critical point tuned by doping, (ii) SIT at oxide interfaces like LaAlO3/SrTiO3, (iii) SIT

50 in the next generation of weakly coupled layered materials like dichalcogenide monolayers, and (iv) bosons in optical lattices with speckle disorder.

51 Bibliography

[1] N. Markovi´c,C. Christiansen, and A. M. Goldman. Thickness˘magnetic ﬁeld phase diagram at the superconductor-insulator transition in 2d. Phys. Rev. Lett., 81:5217– 5220, Dec 1998.

[2] Amit Ghosal, Mohit Randeria, and Nandini Trivedi. Role of spatial amplitude ﬂuc- tuations in highly disordered s-Wave superconductors. Phys. Rev. Lett., 81(18):3940, November 1998.

[3] Amit Ghosal, Mohit Randeria, and Nandini Trivedi. Inhomogeneous pairing in highly disordered s-wave superconductors. Phys. Rev. B, 65(1):014501, Nov 2001.

[4] Karim Bouadim, Yen Lee Loh, Mohit Randeria, and Nandini Trivedi. Single- and two- particle energy gaps across the disorder-driven superconductor-insulator transition. Nat Phys, 7:884–889, July 2011.

[5] D. Bitko, T. F. Rosenbaum, and G. Aeppli. Quantum critical behavior for a model magnet. Phys. Rev. Lett., 77:940–943, Jul 1996.

[6] Markus Greiner, Olaf Mandel, Tilman Esslinger, Theodor W. Hansch, and Immanuel Bloch. Quantum phase transition from a superﬂuid to a mott insulator in a gas of ultracold atoms. Nature, 415(6867):39–44, January 2002.

[7] A. F. Hebard and M. A. Paalanen. Magnetic-ﬁeld-tuned superconductor-insulator transition in two-dimensional ﬁlms. Phys. Rev. Lett., 65(7):927–930, Aug 1990.

[8] Jan Zaanen, Sudip Chakravarty, T. Senthil, Philip Anderson, Patrick Lee, J¨org Schmalian, Masatoshi Imada, David Pines, Mohit Randeria, Chandra Varma, Matthias Vojta, and Maurice Rice. Towards a complete theory of high tc. Nature Physics, 2(3):138–143, March 2006.

[9] G. R. Stewart. Heavy-fermion systems. Rev. Mod. Phys., 56(4):755–787, Oct 1984.

[10] Mark Alford, Jeﬀrey A. Bowers, and Krishna Rajagopal. Crystalline color superconductivity. Phys. Rev. D, 63(7):074016, Mar 2001.

[11] Z. Hadzibabic, P. Kruger, , M. Cheneau, B. Battelier, and J. Dalibard. Berezinski- ikosterlitzthouless crossover in a trapped atomic gas. Nature, 411:1118, June 2006. 52 [12] P. W. Anderson. Absence of diﬀusion in certain random lattices. Phys. Rev., 109(5):1492–1505, Mar 1958.

[13] Leo Radzihovsky and Daniel E Sheehy. Imbalanced feshbach-resonant fermi gases. Rep. Prog. Phys., 73(7):076501, 2010.

[14] B. S. Chandrasekhar. A note on the maximum critical ﬁeld of high-ﬁeld superconductors. Appl. Phys. Lett., 1(1):7–8, 1962.

[15] A. M. Clogston. Upper limit for the critical ﬁeld in hard superconductors. Phys. Rev. Lett., 9(6):266–267, Sep 1962.

[16] P. Fulde and R. A. Ferrell. Phys. Rev., 135:A550, 1964.

[17] A. I. Larkin and Y. N. Ovchinnikov. Zh. Eksp. Teor. Fiz., 47:1136, 1964. [Sov. Phys. JETP 20, 762 (1965)].

[18] Kazushige Machida and Hiizu Nakanishi. Superconductivity under a ferromagnetic molecular ﬁeld. Phys. Rev. B, 30(1):122–133, Jul 1984.

[19] H. Burkhardt and D. Rainer. Fulde-ferrell-larkin-ovchinnikov state in layered superconductors. Ann. Physik, 3:181, 1994.

[20] Nobukatsu Yoshida and S.-K. Yip. Larkin-ovchinnikov state in resonant fermi gas. Phys. Rev. A, 75(6):063601, 2007.

[21] Yen Lee Loh and Nandini Trivedi. Detecting the elusive larkin-ovchinnikov modulated superﬂuid phases for imbalanced fermi gases in optical lattices. Phys. Rev. Lett., 104(16):165302, Apr 2010.

[22] M. Rizzi, Marco Polini, M. A. Cazalilla, M. R. Bakhtiari, M. P. Tosi, and Rosario Fazio. Fulde-ferrell-larkin-ovchinnikov pairing in one-dimensional optical lattices. Phys. Rev. B, 77(24):245105, Jun 2008.

[23] Michele Casula, D. M. Ceperley, and Erich J. Mueller. Quantum monte carlo study of one-dimensional trapped fermions with attractive contact interactions. Phys. Rev. A, 78(3):033607, 2008.

[24] Yean An Liao, Ann Sophie C. Rittner, Tobias Paprotta, Wenhui Li, Guthrie B. Par- tridge, Randall G. Hulet, Stefan K. Baur, and Erich J. Mueller. Spin-imbalance in a one-dimensional fermi gas. Nature, 467(7315):567–569, 2010.

[25] Aurel Bulgac and Michael McNeil Forbes. Unitary fermi supersolid: The larkin- ovchinnikov phase. Phys. Rev. Lett., 101(21):215301, Nov 2008.

[26] Zi Cai, Yupeng Wang, and Congjun Wu. Stable fulde-ferrell-larkin-ovchinnikov pairing states in two-dimensional and three-dimensional optical lattices. Phys. Rev. A, 83(6):063621, Jun 2011.

[27] Meera M. Parish, Stefan K. Baur, Erich J. Mueller, and David A. Huse. Quasi-one- dimensional polarized fermi superﬂuids. Phys. Rev. Lett., 99(25):250403, 2007.

53 [28] Erhai Zhao and W. Vincent Liu. Theory of quasi-one-dimensional imbalanced fermi gases. Phys. Rev. A, 78(6):063605, 2008.

[29] C. A. Regal, M. Greiner, and D. S. Jin. Observation of resonance condensation of fermionic atom pairs. Phys. Rev. Lett., 92(4):040403, Jan 2004.

[30] V. Gritsev, E. Demler, and A. Polkovnikov. Interferometric probe of paired states. Phys. Rev. A, 78:063624, Dec 2008.

[31] Anatoli Polkovnikov, Ehud Altman, and Eugene Demler. Interference between independent ﬂuctuating condensates. Proceedings of the National Academy of Sciences, 103(16):6125–6129, 2006.

[32] R. M. Lutchyn, M. Dzero, and V. M. Yakovenko. Spectroscopy of the soliton lattice formation in quasi-one-dimensional fermionic superﬂuids with population imbalance. ArXiv e-prints, October 2010.

[33] F. Gerbier, S. Trotzky, S. F¨olling,U. Schnorrberger, J. D. Thompson, A. Widera, I. Bloch, L. Pollet, M. Troyer, B. Capogrosso-Sansone, N. V. Prokof’ev, and B. V. Svistunov. Expansion of a quantum gas released from an optical lattice. Phys. Rev. Lett., 101:155303, Oct 2008.

[34] T. Betz, S. Manz, R. B¨ucker, T. Berrada, Ch. Koller, G. Kazakov, I. E. Mazets, H.-P. Stimming, A. Perrin, T. Schumm, and J. Schmiedmayer. Two-point phase correlations of a one-dimensional bosonic josephson junction. Phys. Rev. Lett., 106:020407, Jan 2011.

[35] D. A. Wollman, D. J. Van Harlingen, W. C. Lee, D. M. Ginsberg, and A. J. Leggett. Experimental determination of the superconducting pairing state in ybco from the phase coherence of ybco-pb dc squids. Phys. Rev. Lett., 71(13):2134–2137, Sep 1993.

[36] J. D. Strand, D. J. Van Harlingen, J. B. Kycia, and W. P. Halperin. Evidence for complex superconducting order parameter symmetry in the low-temperature phase of upt3 from josephson interferometry. Phys. Rev. Lett., 103(19):197002, Nov 2009.

[37] D. Shahar and Z. Ovadyahu. Superconductivity near the mobility edge. Phys. Rev. B, 46(17):10917–10922, Nov 1992.

[38] P.W. Adams. Field-induced spin mixing in ultra-thin superconducting al and be ﬁlms in high parallel magnetic ﬁelds. Phys. Rev. Lett., 92(6):067003, Feb 2004.

[39] G. Sambandamurthy, L. W. Engel, A. Johansson, and D. Shahar. Superconductivity- related insulating behavior. Phys. Rev. Lett., 92:107005, Mar 2004.

[40] M. A. Steiner, G. Boebinger, and A. Kapitulnik. Possible ﬁeld-tuned superconductor- insulator transition in high-tc superconductors: Implications for pairing at high magnetic ﬁelds. Phys. Rev. Lett., 94(10):107008, Mar 2005.

[41] M. D. Stewart, Aijun Yin, J. M. Xu, and James M. Valles. Superconducting pair correlations in an amorphous insulating nanohoneycomb ﬁlm. Science, 318:1273, nov 2007. 54 [42] V. F. Gantmakher and V. T. Dolgopolov. Superconductor-insulator quantum phase transition. Physics-Uspekhi, 53:3, 2010.

[43] Benjamin Sacépé, Claude Chapelier, Tatyana I. Baturina, Valerii M. Vinokur, Mikhail R. Baklanov, and Marc Sanquer. Pseudogap in a thin film of a conventional superconductor. Nature Communications, 1:140, December 2010.

[44] Benjamin Sacépé,Thomas Dubouchet, Claude Chapelier, Marc Sanquer, Maoz Ovadia, Dan Shahar, Mikhail Feigel’man, and Lev Ioffe. Localization of preformed Cooper pairs in disordered superconductors. Nat. Phys., 7(3):239–244, March 2011.

[45] Mintu Mondal, Anand Kamlapure, Madhavi Chand, Garima Saraswat, Sanjeev Kumar, John Jesudasan, L. Benfatto, Vikram Tripathi, and Pratap Raychaudhuri. Phase ﬂuctuations in a strongly disordered s-wave nbn superconductor close to the metal- insulator transition. Phys. Rev. Lett., 106(4):047001, Jan 2011.

[46] D. Sherman, G. Kopnov, D. Shahar, and A. Frydman. Measurement of a superconducting energy gap in a homogeneously amorphous insulator. Phys. Rev. Lett., 108:177006, Apr 2012.

[47] D. B. Haviland, Y. Liu, and A. M. Goldman. Onset of superconductivity in the two- dimensional limit. Phys. Rev. Lett., 62(18):2180, May 1989.

[48] Matthew P. A. Fisher, G. Grinstein, and S. M. Girvin. Presence of quantum diﬀusion in two dimensions: Universal resistance at the superconductor-insulator transition. Phys. Rev. Lett., 64(5):587, 1990.

[49] Ali Yazdani and Aharon Kapitulnik. Superconducting-insulating transition in two- dimensional a-moge thin ﬁlms. Phys. Rev. Lett., 74:3037–3040, Apr 1995.

[50] Xiang Leng, Javier Garcia-Barriocanal, Shameek Bose, Yeonbae Lee, and A. M. Gold- man. Electrostatic control of the evolution from a superconducting phase to an insulating phase in ultrathin yba2cu3o7−x ﬁlms. Phys. Rev. Lett., 107:027001, Jul 2011. [51] Amit Ghosal, Mohit Randeria, and Nandini Trivedi. Spatial inhomogeneities in disordered d-wave superconductors. Phys. Rev. B, 63(2):020505, December 2000.

[52] Min-Chul Cha, Matthew P. A. Fisher, S. M. Girvin, Mats Wallin, and A. Peter Young. Universal conductivity of two-dimensional ﬁlms at the superconductor-insulator transition. Phys. Rev. B, 44:6883–6902, Oct 1991.

[53] Erik S. Sørensen, Mats Wallin, S. M. Girvin, and A. Peter Young. Universal conductivity of dirty bosons at the superconductor-insulator transition. Phys. Rev. Lett., 69:828–831, Aug 1992.

[54] Min-Chul Cha and S. M. Girvin. Universal conductivity in the boson hubbard model in a magnetic ﬁeld. Phys. Rev. B, 49:9794–9801, Apr 1994.

[55] Jurij Smakovˇ and Erik Sørensen. Universal scaling of the conductivity at the superﬂuid- insulator phase transition. Phys. Rev. Lett., 95:180603, Oct 2005.

55 [56] Netanel H. Lindner and Assa Auerbach. Conductivity of hard core bosons: A paradigm of a bad metal. Phys. Rev. B, 81:054512, Feb 2010.

[57] Snir Gazit, Daniel Podolsky, and Assa Auerbach. Fate of the higgs mode near quantum criticality. Phys. Rev. Lett., 110:140401, Apr 2013.

[58] Daniel Podolsky, Assa Auerbach, and Daniel P. Arovas. Visibility of the amplitude (higgs) mode in condensed matter. Phys. Rev. B, 84:174522, Nov 2011.

[59] M. Mondal, A. Kamlapure, S. C. Ganguli, J. Jesudasan, V. Bagwe, L. Benfatto, and P. Raychaudhuri. Enhancement of the finite-frequency superfluid response in the pseudogap regime of strongly disordered superconducting films. Sci. Rep., 3:1357, 2013.

[60] C. Menotti and N. Trivedi. Spectral weight redistribution in strongly correlated bosons in optical lattices. Phys. Rev. B, 77:235120, Jun 2008.

[61] Fei Lin, Erik S. Sørensen, and D. M. Ceperley. Superﬂuid-insulator transition in the disordered two-dimensional bose-hubbard model. Phys. Rev. B, 84:094507, Sep 2011.

[62] Kedar Damle and Subir Sachdev. Nonzero-temperature transport near quantum critical points. Phys. Rev. B, 56:8714–8733, Oct 1997.

[63] Nandini Trivedi, Richard T. Scalettar, and Mohit Randeria. Superconductor-insulator transition in a disordered electronic system. Phys. Rev. B, 54(6):R3756, 1996.

[64] W. Witczak-Krempa, E. Sorensen, and S. Sachdev. The dynamics of quantum criticality: Quantum Monte Carlo and holography. ArXiv e-prints 1309.2941, September 2013.

[65] K. Chen, L. Liu, Y. Deng, L. Pollet, and N. Prokof’ev. Universal Conductivity in a Two-dimensional Superﬂuid-to-Insulator Quantum Critical System. ArXiv e-prints 1309.5635, September 2013.

[66] S. Iyer, D. Pekker, and G. Refael. Mott glass to superﬂuid transition for random bosons in two dimensions. Phys. Rev. B, 85:094202, Mar 2012.

[67] 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.

[68] R. W. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gr¨uner. Fluctuations, dissipation, and nonuniversal superﬂuid jumps in two- dimensional superconductors. Phys. Rev. B, 75(9):094506, March 2007.

[69] R. Crane, N. P. Armitage, A. Johansson, G. Sambandamurthy, D. Shahar, and G. Gr¨uner. Survival of superconducting correlations across the two-dimensional superconductor-insulator transition: A ﬁnite-frequency study. Phys. Rev. B, 75(18):184530, May 2007.

[70] Wei Liu, Minsoo Kim, G. Sambandamurthy, and N. P. Armitage. Dynamical study of phase ﬂuctuations and their critical slowing down in amorphous superconducting ﬁlms. Phys. Rev. B, 84:024511, Jul 2011.

56 [71] Wei Liu, LiDong Pan, Jiajia Wen, Minsoo Kim, G. Sambandamurthy, and N. P. Ar- mitage. Microwave spectroscopy evidence of superconducting pairing in the magnetic- ﬁeld-induced metallic state of inox ﬁlms at zero temperature. Phys. Rev. Lett., 111:067003, Aug 2013.

[72] G. Lemari´e,A. Kamlapure, D. Bucheli, L. Benfatto, J. Lorenzana, G. Seibold, S. C. Ganguli, P. Raychaudhuri, and C. Castellani. Universal scaling of the order-parameter distribution in strongly disordered superconductors. Phys. Rev. B, 87:184509, May 2013.

[73] Peter B. Weichman and Ranjan Mukhopadhyay. Particle-hole symmetry and the dirty boson problem. Phys. Rev. B, 77:214516, Jun 2008.

[74] Hannes Meier and Mats Wallin. Quantum critical dynamics simulation of dirty boson systems. Phys. Rev. Lett., 108:055701, Jan 2012.

[75] Fawaz Hrahsheh and Thomas Vojta. Disordered bosons in one dimension: From weak- to strong-randomness criticality. Phys. Rev. Lett., 109:265303, Dec 2012.

57 Appendix A Quantum to classical mapping of phase models

In the following appendix, we will outline the correspondence between quantum mechanical path integrals and classical statistical mechanics, which provides many insights into the nature of quantum phase transitions. A concrete example of this is the mapping of the quantum rotor or Josephson Junction array Hamiltonian into a classical XY model, which we demonstrate explicitly.

In the following, as in the remainder of this thesis, we have set ~ = 1 and have reinstated it when describing the actual values of calculated quantities.

A.1 Quantum classical correspondence

The thermodynamics of a quantum mechanical system with Hamiltonian H depends on the calculation of the quantum partition function

X Z = Tr e−βH = n e−βH n (A.1) n h | | i where β = 1/T is the inverse temperature and the free energy is given by F = (1/β) log Z. − In the second expression for Z, the trace as been written as a sum over a complete set of states n . From the partition function Z, all thermodynamic quantities of interest can be | i calculated. For example any observable can be calculated from O 1 = Tr e−βH . (A.2) hOi Z O 58 Looking more closely at the density matrix e−βH , it has the same form as the quantum mechanical evolution operator e−iT H that generates time evolution by an amount = T iβ in imaginary time. The idea here is that thermodynamics of a quantum system can − be thought of as the transition amplitude over an interval of the length of the inverse temperature. This observation becomes a powerful computational tool when expressed in

Feynman’s path integral description of quantum mechanics.

As in the path integral formulation of quantum mechanics, consider dividing the imaginary time interval up into N equally spaced steps ∆τ = β/N. The number of time steps

N must be chosen small enough that ∆τ is small on the scale of all relevant time/energy scales in the problem. Then, we can write

N e−βH = e−∆τH . (A.3)

If we substitute this into the expression for the partition function and insert a complete set of states m between every term in the product, we have | i

X −∆τH −∆τH −∆τH Z = n e m1 m1 e m2 m2 e m3 (A.4) h | | ih | | ih | | i × n,m1,m2...mN −∆τH −∆τH m3 e m4 m4 mN mN e n . ×h | | ih | · · · | ih | | i

Written in this way, the partition function looks formally like the partition function of a classical system expressed in terms of transfer matrices! The only difference is that now imaginary time has become an added dimension (in addition to space) that the configurations must be summed over. While this additional dimension is finite for a finite temperature system (with a length equal to β), in the limit T 0, a d-dimensional quantum system → truly looks like a d+1 dimensional classical system. This equivalence will be made concrete when we consider the mapping of the 2D Josephson junction array Hamiltonian into a (2

+ 1)D XY model. An important observation to make here is that in many cases, once the mapping has been performed, the universality of the higher dimensional classical system has been well studied, and can immediately be applied to the quantum system.

59 A.2 Particle-hole symmetry and the quantum rotor model

We analyze the (2+1)D quantum XY model given by Eq. 3.2, which is a generalization of the full quantum rotor Hamiltonian

Ec X 2 X X HˆJ = nˆ Jij cos (θˆi θˆj) (µ Vi) ni 2 i − − − − i hiji i 2 Ec X d X = Jij cos (θˆi θˆj) (A.5) − 2 dθ2 − − i i hiji X d + i (µ Vi) − dθ i i since ni = id/dθi. The partition function can be expressed as the coherent-state path − integral Z = R D[θ]e−S with action [52]

Z β n 1 X 2 µ Vi S = dτ (∂τ θi) i − ∂τ θi E − E /2 0 c i c X o Jij(1 cos[(θˆi θˆj)]) . (A.6) − − − hiji

For a slowly varying phase, this becomes

Z β n 1 X 2 µ Vi Jij 2o S = dτ (∂τ θi) i − ∂τ θi (∂rθi) . (A.7) E − E /2 − 2 0 c i c

For the pure system (Vi = V ), if (µ V )/(Ec/2) is an integer, then the middle term − R β does not contribute to the free energy because ∂τ θi = 2π integer. In this special case 0 × of particle-hole symmetry, the dynamical exponent is z = 1. The presence of bond disorder

respects this particle-hole symmetry, and the value of z is unchanged [73].

Away from this particle-hole symmetry point, the ﬁrst derivative term remains in the

action, and for the pure system, z = 2. Even at the particle-hole symmetry point, disorder

in the potential term Vi is expected to change the value of z. In recent Monte Carlo

simulations [74], z = 1.83 0.05 was found. We, however, work with a model in which Vi is ± not random. The bond disorder Jij that we consider respects particle-hole symmetry and the dynamical exponent is expected to remain z = 1 as argued in [73]. The good scaling

collapse shown in Figs. 3.13 and 3.14 shows that our data is consistent with z = 1. We also

60 note that a recent Monte Carlo study of the (1+1)D JJA also concluded that z = 1 in the

presence of bond disorder [75].

A.3 Equivalence of 2D JJA to (2+1)D XY model

Now we will explicitly demonstrate the ideas discussed above by showing how to map a 2D

quantum system, the Josephson junction model, to the (2+1)D XY model. While all of

the computations presented in this thesis have been done in the XY (phase) representation,

many previous studies exist approaching the problem form the dual (current) representation,

the Villain model. With this in mind, we will also show how this model arises out of the

quantum-to-classical mapping.

The Hamiltonian for the 2D Josephson-Junction array (JJA) is

Ec X 2 X HJ = nˆ EJ cos (θî θˆj) T + V, (A.8) 2 i − − ≡ i hi,ji where Ec and EJ are the charging and Josephson energies, respectively. The number and phase are conjugate variables, and therefore we have [θî, nî] = i andn î = i∂/∂θî. Using − this, the Hamiltonian can be rewritten in terms of only the phase variable

2 Ec X ∂ X HJ = − EJ cos (θˆi θˆj). (A.9) 2 2 ˆ − − i ∂ θi hi,ji

This is often referred to as the quantum rotor model. The partition function is given by

ZJ = Tr exp βHJ . Following the prescription above, we will discretize imaginary time − into M time-slices such that β = M∆τ:

ZJ = Tr exp ( βHJ ) (A.10) − M = lim Tr exp ( ∆τHJ ) (A.11) M→∞ − M−1 Z Y Dθ θ(τj+i) exp ( ∆τHJ ) θ(τj) , (A.12) ≈ h{ }| − |{ }i j=0 where θ(τj) refers to the 2D θ conﬁguration of the imaginary time-slice τj (see Fig. A.1). { } The path integral is over all phase conﬁgurations.

61 θr(τj +1)

Figure A.1: Diagram of θ con- θ (τ ) Δτ = β / M r j ﬁguration (represented as spins) θr’(τj) with discretized imaginary time. τ y x

In order to proceed, we want to separate the potential and kinetic terms of the Hamil- tonian. To do so, we must take ∆τ suﬃciently small (by choosing M suﬃciently large) so that

e−∆τHJ e−∆τT e−∆τV . (A.13) ≈ · This is the so-call Trotter decomposition. The partition function is now

Z M−1 Y −∆τT −∆τV ZJ Dθ θ(τj+1) e e θ(τj) . (A.14) ≈ h{ }| · |{ }i j=0

The phase conﬁguration θ(τ)j is an eigenstate of the potential term { }

−∆τV n X o e θ(τj) = exp ∆τ EJ cos [θr(τj) θr0 (τj)] θ(τj) , (A.15) |{ }i · − |{ }i hr,r0i so we can separate it from the inner product

Z M−1 Y n X o −∆τT ZJ Dθ exp ∆τ EJ cos [θr(τj) θr0 (τj)] θ(τj+1) e θ(τj) . ≈ · − · h{ }| |{ }i j=0 hr,r0i (A.16)

−∆τT Now we must deal with the Kinetic term. Let Tj θ(τj+1) e θ(τj) be the ≡ h{ }| |{ }i amplitude going from time-slide j to j + 1. We can write this as the product over each

individual site ( 2 ) Y ∆τEc ∂ Tj = θr(τj+1) exp θr(τj) . (A.17) ˆ2 r h | − 2 ∂θr | i

To evaluate this, note that the eigenstates of ∂/∂θˆr are Jr(τj) , where | i

∂ iJr(τj )θr(τj ) Jr(τj) = Jr(τj) Jr(τj) and θr(τj) Jr(τj) = e , (A.18) ∂θˆr | i | i h | i

so we insert a complete set of states at each site r, τj to obtain (the summation runs over { } 62 all such conﬁgurations)

( 2 ) Y X ∆τEc ∂ Tj = θr(τj+1) exp Jr(τj) Jr(τj) θr(τj) (A.19) h | − 2 ∂θˆ2 | i · h | i r Jr(τj ) r Y X ∆τEc 2 = exp J (τj) θr(τj+1) Jr(τj) Jr(τj) θr(τj) (A.20) − 2 r h | i · h | i r Jr(τj ) Y X ∆τEc 2 = exp J (τj) exp iJr(τj)[θr(τj) θr(τj+1)] . (A.21) − 2 r · { − } r Jr(τj )

If we insert this back into the partition function we obtain

M−1 Z Y n X o ZJ Dθ exp ∆τ EJ cos [θr(τj) θr0 (τj)] (A.22) ≈ · − × j=0 hr,r0i Y X ∆τEc 2 exp J (τj) exp iJr(τj)[θr(τj) θr(τj+1)] × − 2 r · { − } r Jr(τj )

This somewhat complicated expression leaves us with two options: (i) we can perform the

summation of the integers Jr(τj) and obtain a phase-only model or (ii) we can integrate

out the phase variables θr(τj). While there are computational considerations in choosing which option to pursue, the distinction is more than just formal. The phase model is a

natural description of a superconducting system, which the dual model naturally describes

that insulating (phase-disordered) state.

A.3.1 Mapping to XY phase model

Starting from Eq. A.23, our goal is to sum over the Jr(τj), leaving only terms involving the

phase variables θr(τj). To do so, ﬁrst use the Poisson summation formula to write

∞ ∞ r X 2 X 2π 2 S(θ) = e−αJ /2 eiJθ = e−(θ−2πm) /2α. (A.23) α J=−∞ m=−∞

This periodic summation of Gaussians is the Villain approximation of the function

∞ r X 2π 2 S(θ) = e−α(θ−2πm) /2 e−αeα cos θ (A.24) m=−∞ α ≈

63 in the limit of large α (here α = 1/Ec∆τ is controlled by the small imaginary time step). Hence

Y 1 Tj = exp cos [θr(τj) θr(τj+1)] (A.25) r Ec∆τ − ( ) 1 X = exp cos [θr(τj) θr(τj+1)] (A.26) Ec∆τ r − ignoring the constant prefactor from the Villain approximation, which is completely irrele-

vant for the partition function. Inserting this into our expression for the partition function,

Eq. A.16, we have

M−1 Z Y n 1 X ZJ Dθ exp cos [θr(τj) θr(τj+1)] + (A.27) ≈ E ∆τ − j=0 c r X o + ∆τEJ cos [θr(τj) θr0 (τj)] − hr,r0i Z n 1 X = Dθ exp cos [θr(τj) θr(τj+1)] + E ∆τ − c r, j X o + ∆τEJ cos [θr(τj) θr0 (τj)] − hr,r0i, j Z −βXY HXY = Dθ e = ZXY

where HXY is the Hamiltonian of an anisotropic XY model

X X HXY = Kτ cos [θr(τj) θr(τj+1)] K0 cos [θr(τj) θr0 (τj)] (A.28) − − − − r, j hr,r0i, j

where r and r0 are sites in the x-y plane and j indexes the anisotropic direction (associated

with imaginary time τ in the quantum problem).

A.3.2 Mapping to current model

Now we will integrate out the phase variables θr(τj), leaving a partition function expressed

only in the integers Jr(τj). To do so, ﬁrst observe that the Fourier series for exp(α cos θ) is

∞ α cos(θ) X iJθ e = IJ (α)e , (A.29) J=−∞

64 where IJ (α) is the modiﬁed Bessel function of the ﬁrst kind. Using this, the potential term in Eq. A.23 can be written as

n X o exp ∆τ EJ cos [θr(τj) θr0 (τj)] = (A.30) · − hr,r0i Y X = IJ (∆τEJ ) exp iJrr0 (τj)[θr(τj) θr0 (τj)] 0 { − } hr,r i Jrr0 (τj )

We can simplify this further by observing that the argument ∆τEJ will necessarily be

small by the choice of an appropriate time step. In this limit, the Bessel function IJ (α) ∝ exp ( 1/2 log(2/α)J 2) and − n X o exp ∆τ EJ cos [θr(τj) θr0 (τj)] (A.31) · − ∝ hr,r0i Y X 2 exp ( 1/2 log(2/∆τEJ )J ) exp iJrr0 (τj)[θr(τj) θr0 (τj)] ∝ 0 − { − } hr,r i Jrr0 (τj )

And the partition function, Eq. A.23 is

M−1 Z 2 Y Y X −Ke0J 0 (τj ) ZJ Dθ e rr exp iJrr0 (τj)[θr(τj) θr0 (τj)] (A.32) ≈ 0 { − } j=0 hr,r i Jrr0 (τj ) 2 Y X −Keτ J (τj ) e r exp iJr(τj)[θr(τj) θr(τj+1)] , × { − } r Jr(τj )

where

Ke0 = 1/2 log (2/∆τEj) (A.33)

Keτ = ∆τEc/2.

We have now both the potential and kinetic terms in the same form, but we still want to

integrate out the phase variables. This is achieved simply but noting that

Z 2π dθ iJθ e = δJ,0. (A.34) 0 2π

The only question then, is when is the integration non-zero? Consider a site in d dimensions.

This site has 2d neighbors: d “ﬂowing in” and d “ﬂowing out.” In order for the δ-function

to be non-zero the sum of the “currents” J at this site must be equal to zero. In that sense

65 Model Charging energy Josephson Energy

2D Josephson junction array Ec/2 EJ (2+1)D Anisotropic XY model Kτ = Lτ /βEc K0 = βEJ /Lτ (2+1)D Anisotropic Villain model Keτ = βEc/2Lτ Ke0 = log(2Lτ /βEJ )

Table A.1: Summary of classical-to-quantum mapping of Josephson-junction array.

the configurations that we have to consider are only those that obey a Kirchhoff-law like sum rule. The partition function expressed only in terms of the current variables is   X 0  X 2 X 2  ZJ exp Ke0Jrr0 (τj) Keτ Jr (τj) , (A.35) ≈ − 0 − [Jr(τj )]  hr,r i r  where the prime on the summation indicates the Kirchhoff constraint. This classical model is often referred to as the Villain model or the Link-current model. Numerical methods to study it are well known, and it has been used extensively to look at superconductor-insulator problems.

A.3.3 Summary and duality of models

Summarizing these results, we have shown that the 2D quantum XY model can be mapped to a (2+1) D classical model expressed in a phase variable (the XY model) or in an integer current variable (the Villain model). The relationship between the parameters of the original

Hamiltonian and the classical models is shown in Table A.1

The duality of the two models is shown in Fig. A.2. While each model exhibits an order-disorder transition, they have opposite ordered and disordered phases.

66 Figure A.2: Configurations of the superfluid and insulator phases expressed in the current representation (top) and in the phase representation (bottom). The ordered phase of the phase model is the superfluid phase, and the ordered phase of the current model is the insulating phase.

67 Appendix B Observable quantities and analytic continuation

B.1 Kubo formula for the XY model

In the presence of a vector potential A(r), the XY Hamiltonian is modiﬁed:

X HXY = Kb cos (θb Ab), (B.1) − − b R where the sum is over all bonds b and Ab = dr A(r) is the value of the vector potential b · along that bond. From linear response theory, the current in the x-direction induced by the

x-component of the vector potential is

jx(q, ω) = Υxx(q, ω)Ax(q, ω), (B.2) h i

+ where Υxx(q, ω) is the analytic continuation (iωn w + i0 ) of the response function →

Υxx(q, iωn) = kx Λxx(q, iωn) (B.3) h− i −

and the current-current correlation function Λxx(q, iωn) is given above. The full derivation of Eq. B.3 is below.

The optical conductivity is related to the q = 0 values of Υxx(q, ω)

Υxx(q = 0, ω) σ(ω) = , (B.4) iω

68 from which it follows that the real part of σ(ω) is given by

Λ00 (ω) σ0(ω) = Dδ(ω) + xx (B.5) ω where 0 D π kx Λ (q = 0, ω 0) (B.6) ≡ h− i − xx →

is the Drude weight. The full set of limits of Λxx(q, ω) are

0 = kx Λxx(qx 0, qy = 0, iωn = 0) (B.7) h− i − →

Ds/π = kx Λxx(qx = 0, qy 0, iωn = 0) (B.8) h− i − →

D/π = kx Λxx(qx = 0, qy = 0, iωn 0), (B.9) h− i − →

where Ds is the superﬂuid weight. All of these limits are derived below.

Derivation of Kubo formula We will now derive the full electromagnetic response to

linear order. First consider the free energy functional

F [A] = ln Z[A] (B.10) − where the partition function Z[A] is represented as a path integral over all phase conﬁgu-

rations ( ) Z X Z[A] = d[θ] exp Kb cos (θb Ab) . (B.11) − b

The current jb is the term that couples linearly to Ab, that is

δF [A] jb[A] = (B.12) δAb 1 δZ[Ab] = − (B.13) Z[A] δAb ( ) 1 Z δ X = − d[θ] exp Kb cos (θb Ab) (B.14) Z[A] δAb − b ( ) 1 Z X = d[θ] Kb sin (θb Ab) exp Kb cos (θb Ab) (B.15) Z[A] − − b = Kb sin(θb Ab) . (B.16) h − i

69 The full electromagnetic response tensor Υbb0 is the induced current on a bond b due to an applied potential on the bond b0

δjb Υbb0 [A] = (B.17) δAb0 δ2F = (B.18) δAbδAb0 δ 1 δZ[A] = − (B.19) δAb0 Z[A] δAb 1 δ2Z[A] 1 δZ[A] δZ[A] = − + 2 . (B.20) Z[A] δAbδAb0 Z [A] δAb δAb0

Where ( ) δ2Z[A] Z δ X = d[θ] Kb sin (θb Ab) exp Kb cos (θb Ab) (B.21) δAbδAb0 δAb0 − − b Z h 2 0 0 i = d[θ] δb,b0 K cos (θb Ab) Kb sin (θb Ab)Kb0 sin (θ A ) (B.22) b − − − b − b × ( ) X exp Kb cos (θb Ab) . × − b And therefore

0 0 Υbb0 [A] = Kb cos (θb Ab) δb,b0 Kb sin (θb Ab)Kb0 sin (θ A ) + (B.23) h − i − h − b − b i 0 0 + Kb sin (θb Ab) Kb0 sin (θ A ) h − ih b − b i

Since we are studying linear response, we now set Ab = 0. In this limit, the terms

Kb sin (θb) vanish, and we have h i

0 Υbb0 = Kb cos (θb) δb,b0 Kb sin (θb)Kb0 sin (θ ) . (B.24) h i − h b i

We are interested in the response in one of the in-plane directions, sayx ˆ, to the vector

potential component in the same direction, so we will only consider bonds b, b0 in thex ˆ

direction. Further, the system is translationally invariant, so we can set one of the bonds

to be the origin without loss of generality. The response function is now (expressing the

70 coordinates as r = (x, y), and τ as above)

Υxx(r, τ) = K0 cos [θ(r +x, ˆ τ) θ(r, τ)] δ(r, τ) j(r, τ)j(0) (B.25) h − i − h i

= kx δ(r, τ) Λxx(r, τ). (B.26) h− i −

Where kx is the average kinetic energy associated with the x-bonds (which has no h− i (r, τ))dependence and Λxx(r, τ) is the current-current correlation function, where the current is

jx(r, τ) = K0 sin [θ(r +x, ˆ τ) θ(r, τ)]. (B.27) − We are more interested in the wavevector and frequency dependent correlation function given by

Υxx(q, iωn) = kx Λxx(q, τ), (B.28) h− i − where

2 X −i2πq·r/L −iωnτ Λxx(q, τ) = Λxx(q, τ)e e . (B.29) r,τ

B.2 Current-current correlation functions in electronic systems

We will now show that similar considerations in electronic systems can be used as a di- agnostic to determine whether a system is insulating, metallic, or superconducting. We will show that these characteristics of the system follow from taking the appropriate limits qi 0 and ω 0 of the current-current correlation function. → → Consider a lattice electron system with kinetic energy of the form

X † † K = t c cj,σ + c ci,σ , (B.30) − i,σ j,σ hi,ji,σ where the summation is over nearest-neighbor sites, t is the hopping from site to site coming † from the one-electron overlap, and ci,σ creates an electron of spin σ at site i. Throughout, we will use the convention that ~ = c = 1. We will calculate the current response of this

71 system to an applied vector potential in the x direction Ax(l, t). Under such a perturbation, the hopping element along an x bond gains a phase factor

† ieAx(l,t) † c clσ e c cl,σ l+xσ → l+x,σ 2 2 † 1 + ieAx(l, t) e A (l, t) c cl,σ ≈ − x l+x,σ

† −ieAx(l,t) † c clσ e c cl+x,σ l+x,σ → l,σ 2 2 † 1 ieAx(l, t) e A (l, t) c cl+x,σ. (B.31) ≈ − − x l,σ

Therefore, to order A2, the kinetic energy with the perturbation included has three terms:

2 X † † e 2 † † KA = K t ieAx(l, t) c cl,σ c cl+x,σ A (l, t) c cl,σ + c cl+x,σ . − l+x,σ − l,σ − 2 x l+x,σ l,σ l,σ (B.32)

The paramagnetic current-density in the x direction is given by

P X † † jx (l, t) = it cl+x,σcl,σ cl,σcl+x,σ . (B.33) σ −

The second additional term in KA we recognize as the kinetic energy density along just the x bonds: X † † kx(l, t) = t cl+x,σcl,σ + cl,σcl+x,σ . (B.34) − σ

Using these we can rewrite KA as

2 X P e 2 KA = K ej (l, t)A(l, t) + kx(l, t)A (l, t) . (B.35) x 2 l

The total induced current density jx(l, t) along the x direction is given by δKA/δAx(l, t): − 0 2 0 X P δA(l , t) e 0 δA(l , t) jx(l, t) = ej (l, t) + Ax(l , t) x δA(l, t) 2 δA(l, t) l0 2 P e 0 = ej (l, t) + Ax(l , t) , (B.36) x 2 | {z } | {z } paramagnetic diamagnetic

0 since the variation δA(l , t)/δA(l, t) = δl,l0 .

72 If the vector potential has the form

n iq·l−iωto Ax(l, t) = Re Ax(q, ω)e , (B.37) then the current induced will have the same form

n iq·l−iωto jx(l, t) = Re jx(q, ω) e . (B.38) h i h i

We can calculate jx(q, ω) using linear response theory. Since the diamagnetic term is h i already of (A2), we only need to calculate the paramagnetic current response to the O paramagnetic term in KA. Denoting the paramagnetic current-current response function as Λxx(q, ω), we have 2 jx(q, ω) = e [ kx + Λxx(q, ω)] , (B.39) h i h i where Λxx(q, ω) is the analytic continuation (ωm = m2π/β ω + iη) of → Z β 1 iωmτ P P Λxx(q, iωm) = dτ e jx (q, τ), jx ( q, 0) (B.40) N 0 h − i

P (note that there is a factor of i in the deﬁnition of jx (l, t) above). To connect this to the theory of superconductivity, recall the London equation describing

the Meissner eﬀect. For a static (ω = 0), transverse (q A(q, ω) = 0), slowly varying vector · potential, the current response is

ns j(q, ω) = e2 A(q, ω), (B.41) h i − m where ns is the superﬂuid density. Since this is a transverse function, we can write it in the long wavelength limit as qiqj ji(q) = f(q) δi,j Aj(q). (B.42) h i − q2

From Eq. B.41, we see that for a superconductor

ns Ds f(q 0) = e2 , (B.43) → − m ≡ − π

73 where Ds is deﬁned as the superﬂuid “weight.” Combining Eq. B.39 and B.42

2 qx f(q) Λxx(q, ω = 0) = kx 1 , (B.44) −h i − − q2 e2

and therefore Ds Λxx(qx = 0, qy 0, ω = 0) = kx (B.45) → −h i − πe2 and

Λxx(qx 0, qy = 0, ω = 0) = kx . (B.46) → −h i

The superfluid weight Ds comes out as the difference between the transverse and longitudinal components of the current-current correlation function in exactly the same way as the superfluid density ρs came out in the previous section. We can also consider the limit q = 0, ω 0. Recall that the optical conductivity is → given by

j(q = 0, ω) i = σij(ω)Ej = iωσij(ω)Aj, (B.47) h i and therefore

2 kx + Λxx(q = 0, ω) σxx(ω) = e h i . (B.48) iω In order to properly evaluate this expression, we shift the pole at ω = 0 by letting ω ω+iη → for inﬁnitesimal η, consistent with the analytic continuation of Λxx described above. Now recall that 1 1 lim = iπδ(ω). (B.49) η→0 ω + iη P ω −

From this, we see that there is a delta function contribution to Re σxx at ω + iη = 0, this { } we identify as the Drude weight D:

2 lim Re σxx = πe kx + Re Λxx(q = 0, ω 0) = D. (B.50) ω+iη→0 { } − h i { → }

The Drude weight, which is a measure of the ratio of charge carriers to their mass, distin- guishes between metals (D > 0) and insulators (D = 0).

74 To summarize, we have found that

Ds Λxx(qx = 0, qy 0, ω = 0) = kx → −h i − πe2

Λxx(qx 0, qy = 0, ω = 0) = kx → −h i D Re Λxx(q = 0, ω 0) = kx , { → } −h i − πe2

which can be used to characterize

Insulator: D = 0 and Ds = 0

Metal: D > 0 and Ds = 0

Superconductor: D > 0 and Ds > 0

B.3 Sum rule derivation and comparison to Maximum En- tropy results

Sum rules provide an important consistency check on all of our results. Here, we provide derivations of the important sum rules for the quantities we have calculated and show their

B.3.1 Sum rules on σ(ω)

The conductivity sum rule is given by

Z ∞ Iσ = dω Re σ(ω) = π kx . (B.51) −∞ h− i

To see this, we will express Re σ(ω) in terms of Λxx(ω) and use the properties we derived above to express the sum rule. First consider

Z ∞ Z ∞ n o dω Re σ(ω) = dω Dδ(ω) + σreg(ω) (B.52) −∞ −∞ Z ∞ Λ00 (ω) = D + dω xx (B.53) −∞ ω

75 Now, we can rewrite this as the limit

Z ∞ Z ∞ Im Λxx(ω) dω Re σ(ω) = D + lim dω (B.54) ω0→0 ω ω0 −∞ −∞ − = D + π lim Re Λxx(ω), (B.55) ω0→0

where we have used the Kramers-Kronig relation to relate the real and imaginary parts of

Λxx in the last line. This limit is given by Eq. B.50 above, and we have

Z ∞ dω Re σ(ω) = D + π( kx D/π) (B.56) −∞ h− i −

= π kx . (B.57) h− i

This includes the spectral weight contained in the delta-function response proportional to

the superfluid stiffness ρs. The regular part of the spectrum (which we obtain from analytic continuation) satisfies Z ∞ 2 dω Re σ(ω) = π kx ρs. (B.58) 0+ h− i − We emphasize that this sum rule is not built into the MEM routine, and provides an independent verification of the procedure. The sum rule is shown in Fig. B.1 for both the clean and the disorder driven transitions.

B.3.2 Sum rules on Im P (ω)

The boson spectral function is related to the boson Greens function via

Z ∞ dω e−ωτ P (τ) = Im P (ω) (B.59) π 1 e−βω −∞ − which we invert using the MEM in exactly the same way as for σ(ω).

The boson spectral functions obeys the following two sum rules

Z ∞ (1) dω 1 I = Im P (ω) = P (τ = 0) = 1, (B.60) P π 1 e−βω −∞ − which follows trivially from Eq. B.59, and

Z ∞ Z β (2) dω Im P (ω) (3) IP = = dτP (τ) = IP . (B.61) −∞ π ω 0

76 àà ææ à à æ æà 0.25 æà 0.25 HbL X-k \ - Ρ HaL æ à x s à à æ æ æ à æà æ à æà I 0.20 à æ 0.20 à Σ æ à æà æ à æ æà æ à à à æà æ æ æ à æà 0.15 à 0.15 æ æà æ æà æà æà æà æà æà æà æà æà 0.10 0.10 æà æà æà X-k \ - Ρ æà x s æà æà æà à æà æà æ æà 0.05æà 0.05à IΣ æ 0.00 0.00 3.5 4.0 4.5 5.0 0.0 0.2 0.4 0.6 0.8 EcEJ p

0.5æà æà æà æà æà 1.2 æà æà æà HdL 0.4 HcL æà æà æ æ æ æ æ æ æ æ æ æ æà 1.0 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æà æà 0.3 æà æà æà æà æà æà æà 0.8 æà æà æà æà æà æà æà 0.2 æà æà æà æà æà H2L æà IP 0.6 H1L 0.1 H3L IP IP 0.0 0.4 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 p p

Figure B.1: Sum rules for quantities calculated using Maximum Entropy analytic continuation. (a) and (b) show the conductivity sum rule for the clean (p = 0) and disorder tuned (Ec/EJ ) transition. (c) and (d) show the sum rules given by Eqs. B.60 and B.61 for the boson spectral function P (ω)for the disordered tuned transition. In all cases, the results shown are for T/EJ = 0.156, but hold at all temperatures considered in this work.

This second sum rule can be seen by integrating both sides Eq. B.59 over τ from 0 to

β. Note that, in the limit of T 0, Eq. B.60 reduces to a sum rule on Im P (ω) itself → R ∞ 0 dωIm P (ω) = 1. Results for the sum rules are shown in Fig. B.1.