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Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Phase Manipuation of Fermionic Cold Atoms in Mixed Dimensions Defense of the Dissertation Kyle Irwin Department of Physics University of California, Riverside September 25th, 2014
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Cold Atoms in Condensed Matter Cold atoms can be made to exist in specially engineered lattice configurations by overlapping laser light Techniques gained popularity in the study of Bose-Einstein condensates Recent experiments have shifted focus to fermioc atoms trapped in optical lattices Feshbach resonances can be used to tune the interactions between atoms Magnetic trapping, traditional and laser cooling Engineering lasers to couple to different species dipole moment
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results The Ising Model H Z(K, h) = {S} eK R,i SR SR+i +h R SR = e−βF Lattice of spins SR coupled to local field H Nearest neighbor interactions J Parameters coupling constants K = −βJ, h = βH Can define an order parameter M = 1/N R SR As T → Tc (K → Kc), F develops a singularity
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Block Spins H L L Kadanoff (’67) proposed a “blocking scheme” where spins over blocks of size Ld are replaced with a block spin Hypothesized free energy F is invariant with respect to new coupling constants: K → KL, h → hL Order parameter transforms as well: M → µ
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Scaling and Fixed Points For the free energy to remain invariant, physical quantities must transform as F(K, h) = L−d F(KL, hL) ξ(K, h) = Lξ(KL, hL) After many blockings the coupling constants tend to fixed points KL → K∗ , hL → h∗ and once reached this signals the Divergence of the correlation length ξ(K∗ , h∗ ) = ξ(K∗ , h∗ )/l
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Phase Transitions and Wilson’s “Flow Equations” As the system reaches criticality: Phase transition signaled by order parameter acquiring non-zero value Fluctuations of the order parameter grow as signaled by diverging susceptibility Long range physics remain after taking partial trace of partition function with RG procedure The Beta Function Wilson showed (’71) infintesimal renormal- ization steps lead to a differential equation for the flow of couplings: the beta function
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Constructing the Partiton Function Generic interacting hamiltonian: ˆH = ij i|ˆT|j c† i cj + 1 2 ijkl ij|ˆU|lk c† i c† j ck cl Thermodynamics determined by the many-body path integral partition function: Z = e−βΩ = Tre−β(ˆH−µˆN) Trace taken with coherent states. β → β 0 dτ, c† i → ¯φi (τ), Z → D ¯φφ e−S(¯φi (τ)φi (τ)) The Action S ¯φi (τ)φi (τ) = i β 0 dτ ¯φi (τ)∂τ φi (τ) + ˆH ¯φi (τ)φi (τ) − µˆN ¯φi (τ)φi (τ)
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Hubbard Fermions Hubbard fermions are a tight binding simplification of the generic Hamiltonian with in the lattice basis, i.e. i ≡ xσ t U x kinetic term non-zero only for nearest neighbor hopping with strength −t generic interaction reduces to on-site repulsion/attraction with strength U Transform from lattice to momentum basis, and from τ to Matsubara frequeny ωn: φσ(xτ) = 1√ Nβ kω ei(kx−ωτ) φσ(kω) ωn = (2n + 1)π/β due to feature of fermionic coherent states
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Hubbard Action in the Thermodynamic Limit at Zero Temperature Take the action to the thermodynamic (N → ∞) and zero-temperature (β → ∞) limits. 1 N k → dk 2π , 1 β ω → dω 2π S = S0 + SI S0 = kω σ G−1 0 (kω)¯φσ(kω)φσ(kω) SI = 1 2 {kω} σ Γ(43; 12)¯φσ(4)¯φ−σ(3)φ−σ(2)φσ(1) G0(kω) = 1/(−iω + (k) − µ) is the Green function Γ(43; 12) = U¯δ(k1 +k2 −k3 −k4)δ(ω1 +ω2 −ω3 −ω4) is the coupling fucntion
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Hubbard Dispersion at Zero-Temperature −π −π 2 0 π 2 π ka −2t −t 0 t 2t ε(k) Λ−Λ−ΛΛ 1D Hubbard Dispersion (k) = −2tcos(k) At half-filling (µ = 0) the fermi surface consists of two points located at ±π/2 Introduce cut-off ±Λ around fermi points restricting low-energy dynamics Linearize dispersion within cut-off. Fields live on “left” or “right” branches
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Scattering Processed at Half-Filling Restricting momenta to the cut-off reduces scattering events to four unique possibilites at half filling categorized by momentum transfer between particles 1 and 4 known as the g-ology. −π 2 π 2 −π 2 π 2 g2 g1 1 2 3 4 1 2 3 4 −σ σ −σ σ −π 2 π 2 −π 2 π 2 g3 g4 1 2 3 4 1 2 3 4 −σ σ −σ σ dotted-line left branch, solid-line right branch g1 back-scattering, g2/g4 forward-scattering Umklapp g3 processes only possible at half-filling
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Low Energy Effective Action Divide fields into fast and slow modes φ(kω) = Θ(| k | − Λ)η(kω) + Θ(Λ − | k |)φ(kω) π 2 dΛ −Λ Λ φ(kω) η(kω) Separate the partition function Z = DφeS0(φ) DηeS0(η) eSI (φη) = Zη DφeS0(φ) eSI (φη) η = Dφe S0(φ)+SI (φ)+ln eSI (φη) η +lnZη
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results One-Loop Corrections 4 32 1 UΛ−dΛ 4 32 1 UΛ 4 32 1 UΛ UΛ 3 42 1 UΛ UΛ 4 32 1 UΛ UΛ UΛ σ σ Corrections to the one-body terms can be lumped in with the chemical potential Evaluate ln eSI (φη) η to one loop using the linked cluster theorem Only marginal terms (g-ology) grow under renormalization of the interactions
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Flow Equations for the g-ology Cutoff parameterized as Λ(l) = Λ0e−l and reduces to the fermi energy in differential reductions of Λ(l + dl) = Λ(l)e−dl Action transforms as SΛ0 → SΛ0e−dl → SΛ0e−2dl → . . . Flow Equations d dl g1 = −g1 2 d dl (2g2 − g1) = g3 2 d dl g3 = g3(2g2 − g1) d dl g4 = 0 Initial conditions to flow equations come from constant term int Taylor expansion of SI around the fermi points
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Source Fields and the Susceptibility Correlations can be calculated by adding a set of source fields coupled to an order parameter. S(φ, hδ) = S(φ) + qδ(h∗ δ (q)∆δ(q) + h.c.) The susceptibility χ = −β ∂Ω ∂h∗ δ (q)∂hδ(q ) = − ¯∆δ(q)∆δ(q ) Table of order parameters Phase Order Parameter 1D Coupling CDWπ α ¯φα(k)φα(k + π) g2 − 2g1 g3 SDWπ i α,β ¯φα(k)σi αβφ(k + π) g2 ± g3 SS0 1√ 2 α αφα(k)φα(−k) −g1 − g2 ST0 α 1√ 2 φα(k)φ−α(−k) + φα(k)φα(−k) g1 − g2
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Flow of the Susceptibility Ul l + dl l l hδ hδ hδ hδ hδ hδ hδ hδ hδ l + dl l Ul l S(¯φ, φ, h)Λ(l) = S(¯φ, φΛ(l)) + δq (zδ(l, q)∆δ(q)h∗ δ (q) + h.c.) − χδ(l, q)h∗ δ (q)hδ(q) + . . . d dl lnz = 1 2 gδ d dl lnχδ = gδ
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Mixed Hubbard System 1D UAtA tB tB UB UI UI 1D-2D Mixed Hubbard System Energetics 1D interacting Hubbard system A 2D interacting Hubbard gas system B density-density interaction on 1D line at energy cost UI
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Mixed Dimensional Action Z = DAeSA DBeSB+SI SA = kω G−1 A (kω)¯φσ(kω)φσ(kω) + UA 2Nβ 1+2−3=4 ¯φσ(4)¯φ−σ(3)φ−σ(2)φσ(1) Ignore interactions in the 2D system SB = pΩ G−1 B (pΩ)¯Φσ(pΩ)Φσ(pΩ) SI = UI N2β px =k+px −k σσ ¯φσ(kω)φσ(k ω )¯Φσ (pΩ)Φσ (p Ω ) Green Functions GA = 1 −iω+ A−µA A(k) = −2tAcos(k) GB = 1 −iΩ+ B−µB B(p) = −2tb(cos(px ) + cos(py ))
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Effective 1D Action 4 32 1 UI UI UI Integrate out B-fermions for an effective action composed solely of A-fermions. Z = DAe−Seff Seff = SA + ln e−SI B − lnZB One-loop effective interaction Ueff = 2UI 2 N3β pqΩ GB(p; Ω)GB(px + k41; Ω + ω41) = 2UI 2 N3 pq n[ B(px +k41,q]−n[ B(px ,py )] iω41+ B(px +k41,q)− B(px ,py )
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Density Controlled Interaction -0.637 -0.574 -0.512 -0.45 µB =0.0 -0.598 -0.543 -0.488 -0.433 µB =0.1 -0.567 -0.516 -0.465 -0.414 µB =0.2 -0.541 -0.494 -0.447 -0.399 µB =0.3 -0.518 -0.474 -0.431 -0.387 µB =0.4 -0.498 -0.457 -0.416 -0.375 µB =0.5 -0.478 -0.44 -0.402 -0.365 µB =0.6 -0.459 -0.424 -0.389 -0.355 µB =0.7 -0.442 -0.409 -0.377 -0.345 µB =0.8 -0.424 -0.395 -0.365 -0.335 µB =0.9 -0.407 -0.38 -0.353 -0.326 µB =1.0 -0.39 -0.366 -0.342 -0.317 µB =1.1 -0.374 -0.352 -0.33 -0.308 µB =1.2 -0.357 -0.338 -0.319 -0.3 µB =1.3 -0.34 -0.324 -0.308 -0.292 µB =1.4 -0.323 -0.31 -0.297 -0.284 µB =1.5 -0.305 -0.295 -0.285 -0.275 µB =1.6 -0.287 -0.281 -0.274 -0.268 µB =1.7 -0.268 -0.263 -0.258 -0.253 µB =1.8 -0.254 -0.249 -0.243 -0.238 µB =1.9 -0.246 -0.237 -0.228 -0.219 µB =2.0 -0.238 -0.223 -0.208 -0.192 µB =2.1 -0.231 -0.211 -0.192 -0.172 µB =2.2 -0.222 -0.2 -0.177 -0.155 µB =2.3 -0.214 -0.189 -0.165 -0.14 µB =2.4 -0.206 -0.179 -0.153 -0.126 µB =2.5 -0.198 -0.17 -0.141 -0.113 µB =2.6 -0.189 -0.16 -0.131 -0.102 µB =2.7 -0.18 -0.151 -0.121 -0.091 µB =2.8 -0.171 -0.141 -0.111 -0.081 µB =2.9 -0.162 -0.132 -0.102 -0.071 µB =3.0 -0.153 -0.123 -0.093 -0.062 µB =3.1 -0.143 -0.113 -0.084 -0.054 µB =3.2 -0.132 -0.104 -0.075 -0.046 µB =3.3 -0.122 -0.094 -0.066 -0.039 µB =3.4 π −π 2 0 π 2 π -0.11 -0.084 -0.058 -0.031 µB =3.5 π −π 2 0 π 2 π -0.097 -0.073 -0.049 -0.025 µB =3.6 π −π 2 0 π 2 π -0.083 -0.061 -0.04 -0.018 µB =3.7 π −π 2 0 π 2 π -0.066 -0.048 -0.03 -0.012 µB =3.8 π −π 2 0 π 2 π -0.043 -0.031 -0.018 -0.006 µB =3.9 A-fermion interaction mediated through B-fermion scattering events Coupling function prior to RG process: UA + Ueff (4 − 1) As µB is varied, shape of Ueff changes Relative values of Ueff (0) and Ueff (π) change as will be important for RG intial conditions
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Effective Interaction in Lattice Space -0.53 -0.34 -0.15 0.04 μB =0.0 -0.52 -0.33 -0.15 0.04 μB =0.1 -0.49 -0.32 -0.14 0.04 μB =0.2 -0.48 -0.31 -0.13 0.04 μB =0.3 -0.46 -0.29 -0.13 0.03 μB =0.4 -0.44 -0.28 -0.13 0.03 μB =0.5 -0.42 -0.27 -0.12 0.03 μB =0.6 -0.41 -0.26 -0.12 0.03 μB =0.7 -0.39 -0.25 -0.11 0.03 μB =0.8 -0.38 -0.24 -0.11 0.02 μB =0.9 -0.36 -0.24 -0.11 0.02 μB =1.0 -0.35 -0.23 -0.1 0.02 μB =1.1 -0.34 -0.22 -0.1 0.02 μB =1.2 -0.32 -0.21 -0.1 0.01 μB =1.3 -0.31 -0.2 -0.1 0.01 μB =1.4 -0.3 -0.2 -0.09 0.01 μB =1.5 -0.28 -0.19 -0.09 0.0 μB =1.6 -0.27 -0.18 -0.09 0.0 μB =1.7 -0.26 -0.17 -0.09 0.0 μB =1.8 -0.25 -0.16 -0.08 0.0 μB =1.9 -0.24 -0.16 -0.08 0.0 μB =2.0 -0.22 -0.15 -0.07 0.0 μB =2.1 -0.21 -0.14 -0.07 0.0 μB =2.2 -0.2 -0.13 -0.06 0.0 μB =2.3 -0.19 -0.13 -0.06 0.01 μB =2.4 -0.18 -0.12 -0.06 0.01 μB =2.5 -0.17 -0.11 -0.05 0.01 μB =2.6 -0.16 -0.1 -0.05 0.01 μB =2.7 -0.15 -0.1 -0.04 0.01 μB =2.8 -0.14 -0.09 -0.04 0.01 μB =2.9 -0.12 -0.08 -0.04 0.0 μB =3.0 -0.11 -0.07 -0.04 0.0 μB =3.1 -0.1 -0.07 -0.03 0.0 μB =3.2 -0.09 -0.06 -0.03 0.0 μB =3.3 -0.08 -0.05 -0.02 0.0 μB =3.4 0 a 2a 3a 4a -0.07 -0.04 -0.02 0.0 μB =3.5 0 a 2a 3a 4a -0.06 -0.04 -0.02 0.0 μB =3.6 0 a 2a 3a 4a -0.04 -0.03 -0.01 0.0 μB =3.7 0 a 2a 3a 4a -0.03 -0.02 -0.01 0.0 μB =3.8 0 a 2a 3a 4a -0.02 -0.01 -0.01 -0 μB =3.9 On site interaction is always attractive As µB is varied nearest and further neighbor attraction or repulsion varies
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Initial Conditions to the RG To determine phases, solve the flow equations for the g-ology couplings and susceptibility simultaneously subject to the initial conditions: Initial conditions gl=0 1 = gl=0 3 = UA + Ueff (k41 = π) gl=0 2 = gl=0 4 = UA + Ueff (k41 = 0)
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results The Phase Diagram The phase diagram is parameterized by |Ueff (π)| and |Ueff (0)| Fixing the chemical potential fixes the form of Ueff |Ueff (k)| at 0 and π determine which couplings will grow Fluctuations for certain order parameters grow giving rise to diverging susceptibilities TLL TLL FL C D W /SS UA UA |Ueff (π)| |Ueff (0)| SDW CDW SS ST (SS)
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Spin Density Wave Phase 1.80 1.82 1.84 1.86 1.88 1.90 1.92 1.94 RG Step l 50 0 50 100 150 200 250 g-ology U0 =0.6UA Uπ =0.2UA g1 g2 g3 g4 0.0 0.5 1.0 1.5 RG Step l 2 0 2 4 6 8 10 12 |gδ| st sdwb ss cdws cdwb sdws Forward and Umklapp scatterings diverge accompanied by a divergence in the SDW susceptibility
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Tomnaga-Luttinger Liquid Phase 0 200 400 600 800 RG Step l 0.1 0.0 0.1 0.2 0.3 0.4 0.5 g-ology U0 =0.6UA Uπ =1.0UA g1 g2 g3 g4 g1, g3 are null. Forward scattering remains constant and can be bosonized yielding the Tomonaga–Luttinger Model
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Charge Density Wave Phase 2.15 2.20 2.25 2.30 RG Step l 200 150 100 50 0 50 100 g-ology U0 =0.6UA Uπ =1.2UA g1 g2 g3 g4 0.0 0.5 1.0 1.5 2.0 RG Step l 2 0 2 4 6 8 10 12 |gδ| cdwb sdws ss st sdwb cdws Forward and Umklapp scatterings diverge accompanied by a divergence in the CDW susceptibility
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Charge Density Wave/Spin Singlet Phase 4.5 4.6 4.7 4.8 4.9 RG Step l 100 80 60 40 20 0 20 g-ology U0 =1.2UA Uπ =1.2UA g1 g2 g3 g4 0 1 2 3 4 RG Step l 2 0 2 4 6 8 10 12 |gδ| sdwb st sdws cdwb ss cdws Back and Forward scattering diverge to negative values accompanied by simultaneously diverging SS/CDW susceptibilities
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Spin Singlet Phase 4.80 4.85 4.90 4.95 RG Step l 100 80 60 40 20 0 20 g-ology U0 =1.6UA Uπ =1.2UA g1 g2 g3 g4 0 1 2 3 4 RG Step l 2 0 2 4 6 8 10 12 14 |gδ| st sdwb cdwb sdws cdws ss Back and Forward scattering continue to diverge, but in a way that enhances the SS susceptibility
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Fermi Liquid Phase 0 50 100 150 200 RG Step l 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 g-ology U0 =1.0UA Uπ =0.5UA g1 g2 g3 g4 Forward scattering terms remain null, back and Umklapp scatterings quickly renormalize to zero.
Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Triplet (Singlet) Phase Back and Umklapp scattering flow to zero. Forward scatterings flow to constant negative values. Since no couplings diverge, phase is determined by the slowest decaying correlation function Correlations decay ∝ 1/xgδ ST decays the slowest. SS next slowest. 0 50 100 150 200 RG Step l 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 g-ology U0 =1.5UA Uπ =0.5UA g1 g2 g3 g4

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final_talk

  • 1. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Phase Manipuation of Fermionic Cold Atoms in Mixed Dimensions Defense of the Dissertation Kyle Irwin Department of Physics University of California, Riverside September 25th, 2014
  • 2. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
  • 3. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Cold Atoms in Condensed Matter Cold atoms can be made to exist in specially engineered lattice configurations by overlapping laser light Techniques gained popularity in the study of Bose-Einstein condensates Recent experiments have shifted focus to fermioc atoms trapped in optical lattices Feshbach resonances can be used to tune the interactions between atoms Magnetic trapping, traditional and laser cooling Engineering lasers to couple to different species dipole moment
  • 4. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
  • 5. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results The Ising Model H Z(K, h) = {S} eK R,i SR SR+i +h R SR = e−βF Lattice of spins SR coupled to local field H Nearest neighbor interactions J Parameters coupling constants K = −βJ, h = βH Can define an order parameter M = 1/N R SR As T → Tc (K → Kc), F develops a singularity
  • 6. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Block Spins H L L Kadanoff (’67) proposed a “blocking scheme” where spins over blocks of size Ld are replaced with a block spin Hypothesized free energy F is invariant with respect to new coupling constants: K → KL, h → hL Order parameter transforms as well: M → µ
  • 7. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Scaling and Fixed Points For the free energy to remain invariant, physical quantities must transform as F(K, h) = L−d F(KL, hL) ξ(K, h) = Lξ(KL, hL) After many blockings the coupling constants tend to fixed points KL → K∗ , hL → h∗ and once reached this signals the Divergence of the correlation length ξ(K∗ , h∗ ) = ξ(K∗ , h∗ )/l
  • 8. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Phase Transitions and Wilson’s “Flow Equations” As the system reaches criticality: Phase transition signaled by order parameter acquiring non-zero value Fluctuations of the order parameter grow as signaled by diverging susceptibility Long range physics remain after taking partial trace of partition function with RG procedure The Beta Function Wilson showed (’71) infintesimal renormal- ization steps lead to a differential equation for the flow of couplings: the beta function
  • 9. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
  • 10. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Constructing the Partiton Function Generic interacting hamiltonian: ˆH = ij i|ˆT|j c† i cj + 1 2 ijkl ij|ˆU|lk c† i c† j ck cl Thermodynamics determined by the many-body path integral partition function: Z = e−βΩ = Tre−β(ˆH−µˆN) Trace taken with coherent states. β → β 0 dτ, c† i → ¯φi (τ), Z → D ¯φφ e−S(¯φi (τ)φi (τ)) The Action S ¯φi (τ)φi (τ) = i β 0 dτ ¯φi (τ)∂τ φi (τ) + ˆH ¯φi (τ)φi (τ) − µˆN ¯φi (τ)φi (τ)
  • 11. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Hubbard Fermions Hubbard fermions are a tight binding simplification of the generic Hamiltonian with in the lattice basis, i.e. i ≡ xσ t U x kinetic term non-zero only for nearest neighbor hopping with strength −t generic interaction reduces to on-site repulsion/attraction with strength U Transform from lattice to momentum basis, and from τ to Matsubara frequeny ωn: φσ(xτ) = 1√ Nβ kω ei(kx−ωτ) φσ(kω) ωn = (2n + 1)π/β due to feature of fermionic coherent states
  • 12. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Hubbard Action in the Thermodynamic Limit at Zero Temperature Take the action to the thermodynamic (N → ∞) and zero-temperature (β → ∞) limits. 1 N k → dk 2π , 1 β ω → dω 2π S = S0 + SI S0 = kω σ G−1 0 (kω)¯φσ(kω)φσ(kω) SI = 1 2 {kω} σ Γ(43; 12)¯φσ(4)¯φ−σ(3)φ−σ(2)φσ(1) G0(kω) = 1/(−iω + (k) − µ) is the Green function Γ(43; 12) = U¯δ(k1 +k2 −k3 −k4)δ(ω1 +ω2 −ω3 −ω4) is the coupling fucntion
  • 13. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Hubbard Dispersion at Zero-Temperature −π −π 2 0 π 2 π ka −2t −t 0 t 2t ε(k) Λ−Λ−ΛΛ 1D Hubbard Dispersion (k) = −2tcos(k) At half-filling (µ = 0) the fermi surface consists of two points located at ±π/2 Introduce cut-off ±Λ around fermi points restricting low-energy dynamics Linearize dispersion within cut-off. Fields live on “left” or “right” branches
  • 14. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Scattering Processed at Half-Filling Restricting momenta to the cut-off reduces scattering events to four unique possibilites at half filling categorized by momentum transfer between particles 1 and 4 known as the g-ology. −π 2 π 2 −π 2 π 2 g2 g1 1 2 3 4 1 2 3 4 −σ σ −σ σ −π 2 π 2 −π 2 π 2 g3 g4 1 2 3 4 1 2 3 4 −σ σ −σ σ dotted-line left branch, solid-line right branch g1 back-scattering, g2/g4 forward-scattering Umklapp g3 processes only possible at half-filling
  • 15. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Low Energy Effective Action Divide fields into fast and slow modes φ(kω) = Θ(| k | − Λ)η(kω) + Θ(Λ − | k |)φ(kω) π 2 dΛ −Λ Λ φ(kω) η(kω) Separate the partition function Z = DφeS0(φ) DηeS0(η) eSI (φη) = Zη DφeS0(φ) eSI (φη) η = Dφe S0(φ)+SI (φ)+ln eSI (φη) η +lnZη
  • 16. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
  • 17. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results One-Loop Corrections 4 32 1 UΛ−dΛ 4 32 1 UΛ 4 32 1 UΛ UΛ 3 42 1 UΛ UΛ 4 32 1 UΛ UΛ UΛ σ σ Corrections to the one-body terms can be lumped in with the chemical potential Evaluate ln eSI (φη) η to one loop using the linked cluster theorem Only marginal terms (g-ology) grow under renormalization of the interactions
  • 18. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Flow Equations for the g-ology Cutoff parameterized as Λ(l) = Λ0e−l and reduces to the fermi energy in differential reductions of Λ(l + dl) = Λ(l)e−dl Action transforms as SΛ0 → SΛ0e−dl → SΛ0e−2dl → . . . Flow Equations d dl g1 = −g1 2 d dl (2g2 − g1) = g3 2 d dl g3 = g3(2g2 − g1) d dl g4 = 0 Initial conditions to flow equations come from constant term int Taylor expansion of SI around the fermi points
  • 19. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Source Fields and the Susceptibility Correlations can be calculated by adding a set of source fields coupled to an order parameter. S(φ, hδ) = S(φ) + qδ(h∗ δ (q)∆δ(q) + h.c.) The susceptibility χ = −β ∂Ω ∂h∗ δ (q)∂hδ(q ) = − ¯∆δ(q)∆δ(q ) Table of order parameters Phase Order Parameter 1D Coupling CDWπ α ¯φα(k)φα(k + π) g2 − 2g1 g3 SDWπ i α,β ¯φα(k)σi αβφ(k + π) g2 ± g3 SS0 1√ 2 α αφα(k)φα(−k) −g1 − g2 ST0 α 1√ 2 φα(k)φ−α(−k) + φα(k)φα(−k) g1 − g2
  • 20. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Flow of the Susceptibility Ul l + dl l l hδ hδ hδ hδ hδ hδ hδ hδ hδ l + dl l Ul l S(¯φ, φ, h)Λ(l) = S(¯φ, φΛ(l)) + δq (zδ(l, q)∆δ(q)h∗ δ (q) + h.c.) − χδ(l, q)h∗ δ (q)hδ(q) + . . . d dl lnz = 1 2 gδ d dl lnχδ = gδ
  • 21. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
  • 22. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Mixed Hubbard System 1D UAtA tB tB UB UI UI 1D-2D Mixed Hubbard System Energetics 1D interacting Hubbard system A 2D interacting Hubbard gas system B density-density interaction on 1D line at energy cost UI
  • 23. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Mixed Dimensional Action Z = DAeSA DBeSB+SI SA = kω G−1 A (kω)¯φσ(kω)φσ(kω) + UA 2Nβ 1+2−3=4 ¯φσ(4)¯φ−σ(3)φ−σ(2)φσ(1) Ignore interactions in the 2D system SB = pΩ G−1 B (pΩ)¯Φσ(pΩ)Φσ(pΩ) SI = UI N2β px =k+px −k σσ ¯φσ(kω)φσ(k ω )¯Φσ (pΩ)Φσ (p Ω ) Green Functions GA = 1 −iω+ A−µA A(k) = −2tAcos(k) GB = 1 −iΩ+ B−µB B(p) = −2tb(cos(px ) + cos(py ))
  • 24. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
  • 25. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Effective 1D Action 4 32 1 UI UI UI Integrate out B-fermions for an effective action composed solely of A-fermions. Z = DAe−Seff Seff = SA + ln e−SI B − lnZB One-loop effective interaction Ueff = 2UI 2 N3β pqΩ GB(p; Ω)GB(px + k41; Ω + ω41) = 2UI 2 N3 pq n[ B(px +k41,q]−n[ B(px ,py )] iω41+ B(px +k41,q)− B(px ,py )
  • 26. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Density Controlled Interaction -0.637 -0.574 -0.512 -0.45 µB =0.0 -0.598 -0.543 -0.488 -0.433 µB =0.1 -0.567 -0.516 -0.465 -0.414 µB =0.2 -0.541 -0.494 -0.447 -0.399 µB =0.3 -0.518 -0.474 -0.431 -0.387 µB =0.4 -0.498 -0.457 -0.416 -0.375 µB =0.5 -0.478 -0.44 -0.402 -0.365 µB =0.6 -0.459 -0.424 -0.389 -0.355 µB =0.7 -0.442 -0.409 -0.377 -0.345 µB =0.8 -0.424 -0.395 -0.365 -0.335 µB =0.9 -0.407 -0.38 -0.353 -0.326 µB =1.0 -0.39 -0.366 -0.342 -0.317 µB =1.1 -0.374 -0.352 -0.33 -0.308 µB =1.2 -0.357 -0.338 -0.319 -0.3 µB =1.3 -0.34 -0.324 -0.308 -0.292 µB =1.4 -0.323 -0.31 -0.297 -0.284 µB =1.5 -0.305 -0.295 -0.285 -0.275 µB =1.6 -0.287 -0.281 -0.274 -0.268 µB =1.7 -0.268 -0.263 -0.258 -0.253 µB =1.8 -0.254 -0.249 -0.243 -0.238 µB =1.9 -0.246 -0.237 -0.228 -0.219 µB =2.0 -0.238 -0.223 -0.208 -0.192 µB =2.1 -0.231 -0.211 -0.192 -0.172 µB =2.2 -0.222 -0.2 -0.177 -0.155 µB =2.3 -0.214 -0.189 -0.165 -0.14 µB =2.4 -0.206 -0.179 -0.153 -0.126 µB =2.5 -0.198 -0.17 -0.141 -0.113 µB =2.6 -0.189 -0.16 -0.131 -0.102 µB =2.7 -0.18 -0.151 -0.121 -0.091 µB =2.8 -0.171 -0.141 -0.111 -0.081 µB =2.9 -0.162 -0.132 -0.102 -0.071 µB =3.0 -0.153 -0.123 -0.093 -0.062 µB =3.1 -0.143 -0.113 -0.084 -0.054 µB =3.2 -0.132 -0.104 -0.075 -0.046 µB =3.3 -0.122 -0.094 -0.066 -0.039 µB =3.4 π −π 2 0 π 2 π -0.11 -0.084 -0.058 -0.031 µB =3.5 π −π 2 0 π 2 π -0.097 -0.073 -0.049 -0.025 µB =3.6 π −π 2 0 π 2 π -0.083 -0.061 -0.04 -0.018 µB =3.7 π −π 2 0 π 2 π -0.066 -0.048 -0.03 -0.012 µB =3.8 π −π 2 0 π 2 π -0.043 -0.031 -0.018 -0.006 µB =3.9 A-fermion interaction mediated through B-fermion scattering events Coupling function prior to RG process: UA + Ueff (4 − 1) As µB is varied, shape of Ueff changes Relative values of Ueff (0) and Ueff (π) change as will be important for RG intial conditions
  • 27. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Effective Interaction in Lattice Space -0.53 -0.34 -0.15 0.04 μB =0.0 -0.52 -0.33 -0.15 0.04 μB =0.1 -0.49 -0.32 -0.14 0.04 μB =0.2 -0.48 -0.31 -0.13 0.04 μB =0.3 -0.46 -0.29 -0.13 0.03 μB =0.4 -0.44 -0.28 -0.13 0.03 μB =0.5 -0.42 -0.27 -0.12 0.03 μB =0.6 -0.41 -0.26 -0.12 0.03 μB =0.7 -0.39 -0.25 -0.11 0.03 μB =0.8 -0.38 -0.24 -0.11 0.02 μB =0.9 -0.36 -0.24 -0.11 0.02 μB =1.0 -0.35 -0.23 -0.1 0.02 μB =1.1 -0.34 -0.22 -0.1 0.02 μB =1.2 -0.32 -0.21 -0.1 0.01 μB =1.3 -0.31 -0.2 -0.1 0.01 μB =1.4 -0.3 -0.2 -0.09 0.01 μB =1.5 -0.28 -0.19 -0.09 0.0 μB =1.6 -0.27 -0.18 -0.09 0.0 μB =1.7 -0.26 -0.17 -0.09 0.0 μB =1.8 -0.25 -0.16 -0.08 0.0 μB =1.9 -0.24 -0.16 -0.08 0.0 μB =2.0 -0.22 -0.15 -0.07 0.0 μB =2.1 -0.21 -0.14 -0.07 0.0 μB =2.2 -0.2 -0.13 -0.06 0.0 μB =2.3 -0.19 -0.13 -0.06 0.01 μB =2.4 -0.18 -0.12 -0.06 0.01 μB =2.5 -0.17 -0.11 -0.05 0.01 μB =2.6 -0.16 -0.1 -0.05 0.01 μB =2.7 -0.15 -0.1 -0.04 0.01 μB =2.8 -0.14 -0.09 -0.04 0.01 μB =2.9 -0.12 -0.08 -0.04 0.0 μB =3.0 -0.11 -0.07 -0.04 0.0 μB =3.1 -0.1 -0.07 -0.03 0.0 μB =3.2 -0.09 -0.06 -0.03 0.0 μB =3.3 -0.08 -0.05 -0.02 0.0 μB =3.4 0 a 2a 3a 4a -0.07 -0.04 -0.02 0.0 μB =3.5 0 a 2a 3a 4a -0.06 -0.04 -0.02 0.0 μB =3.6 0 a 2a 3a 4a -0.04 -0.03 -0.01 0.0 μB =3.7 0 a 2a 3a 4a -0.03 -0.02 -0.01 0.0 μB =3.8 0 a 2a 3a 4a -0.02 -0.01 -0.01 -0 μB =3.9 On site interaction is always attractive As µB is varied nearest and further neighbor attraction or repulsion varies
  • 28. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Outline 1 Cold Atoms 2 RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 3 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results
  • 29. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Initial Conditions to the RG To determine phases, solve the flow equations for the g-ology couplings and susceptibility simultaneously subject to the initial conditions: Initial conditions gl=0 1 = gl=0 3 = UA + Ueff (k41 = π) gl=0 2 = gl=0 4 = UA + Ueff (k41 = 0)
  • 30. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results The Phase Diagram The phase diagram is parameterized by |Ueff (π)| and |Ueff (0)| Fixing the chemical potential fixes the form of Ueff |Ueff (k)| at 0 and π determine which couplings will grow Fluctuations for certain order parameters grow giving rise to diverging susceptibilities TLL TLL FL C D W /SS UA UA |Ueff (π)| |Ueff (0)| SDW CDW SS ST (SS)
  • 31. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Spin Density Wave Phase 1.80 1.82 1.84 1.86 1.88 1.90 1.92 1.94 RG Step l 50 0 50 100 150 200 250 g-ology U0 =0.6UA Uπ =0.2UA g1 g2 g3 g4 0.0 0.5 1.0 1.5 RG Step l 2 0 2 4 6 8 10 12 |gδ| st sdwb ss cdws cdwb sdws Forward and Umklapp scatterings diverge accompanied by a divergence in the SDW susceptibility
  • 32. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Tomnaga-Luttinger Liquid Phase 0 200 400 600 800 RG Step l 0.1 0.0 0.1 0.2 0.3 0.4 0.5 g-ology U0 =0.6UA Uπ =1.0UA g1 g2 g3 g4 g1, g3 are null. Forward scattering remains constant and can be bosonized yielding the Tomonaga–Luttinger Model
  • 33. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Charge Density Wave Phase 2.15 2.20 2.25 2.30 RG Step l 200 150 100 50 0 50 100 g-ology U0 =0.6UA Uπ =1.2UA g1 g2 g3 g4 0.0 0.5 1.0 1.5 2.0 RG Step l 2 0 2 4 6 8 10 12 |gδ| cdwb sdws ss st sdwb cdws Forward and Umklapp scatterings diverge accompanied by a divergence in the CDW susceptibility
  • 34. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Charge Density Wave/Spin Singlet Phase 4.5 4.6 4.7 4.8 4.9 RG Step l 100 80 60 40 20 0 20 g-ology U0 =1.2UA Uπ =1.2UA g1 g2 g3 g4 0 1 2 3 4 RG Step l 2 0 2 4 6 8 10 12 |gδ| sdwb st sdws cdwb ss cdws Back and Forward scattering diverge to negative values accompanied by simultaneously diverging SS/CDW susceptibilities
  • 35. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Spin Singlet Phase 4.80 4.85 4.90 4.95 RG Step l 100 80 60 40 20 0 20 g-ology U0 =1.6UA Uπ =1.2UA g1 g2 g3 g4 0 1 2 3 4 RG Step l 2 0 2 4 6 8 10 12 14 |gδ| st sdwb cdwb sdws cdws ss Back and Forward scattering continue to diverge, but in a way that enhances the SS susceptibility
  • 36. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Fermi Liquid Phase 0 50 100 150 200 RG Step l 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 g-ology U0 =1.0UA Uπ =0.5UA g1 g2 g3 g4 Forward scattering terms remain null, back and Umklapp scatterings quickly renormalize to zero.
  • 37. Dissertation Defense Kyle Irwin Cold Atoms RG for 1D Interacting Fermions at Half Filling Historical First Steps RG for 1D Interacting Fermions at Half Filling The Flow Equations 1D-2D Mixed Fermi System The Model The Mediated Interaction RG Results Triplet (Singlet) Phase Back and Umklapp scattering flow to zero. Forward scatterings flow to constant negative values. Since no couplings diverge, phase is determined by the slowest decaying correlation function Correlations decay ∝ 1/xgδ ST decays the slowest. SS next slowest. 0 50 100 150 200 RG Step l 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 g-ology U0 =1.5UA Uπ =0.5UA g1 g2 g3 g4