Phase Fluctuations and Sign Problems Michael Wagman MIT Lattice 2018 East Lansing, Michigan
Lattice QCD and Nuclei Nuclear theory predictions are needed to extract or constrain new physics from intensity frontier experiments Lattice QCD can inform and test EFT power counting and models of heavy nuclei by calculating properties of simple nuclei Increasing the range of nuclei directly accessible to LQCD will increase the reliability of low-energy nuclear theory predictions LUX CUORE DUNE � 2
The Signal-to-Noise Problem LQCD nuclear correlation functions have StN ratios that decrease exponentially with increasing baryon number 1 H 2 H 3 He 4 He E t Parisi, Phys Rept 103 (1984), Lepage, TASI (1989), NPLQCD, PRD 79 (2009), Detmold and Endres, PRD 90 (2014), … � 3
The Sign(al-to-Noise) Problem Average correlators are real. Individual correlators in generic gauge fields are complex G N ( p , t ) = ⟨ C N ( p , t ) ⟩ = ⟨ e R N ( p , t )+ i θ N ( p , t ) ⟩ Complex phase fluctuations give path integrals representing correlators sign problems G N ( p , t ) = ∫ U e − S ( U )+ R N ( p , t ; U )+ i θ N ( p , t ; U ) = 1 N ∑ e R N ( p , t ; U i )+ i θ N ( p , t ; U i ) N i =1 An exponentially decaying average phase always has exponential StN degradation ⟨ e i θ ⟩ 2 ∼ ⟨ e i θ ⟩ ∼ e − M θ t StN ( Re [ e i θ ( t ) ]) = 2 ⟨ e 2 i θ ⟩ − ⟨ e i θ ⟩ 1 2 + 1 � 4
Correlation Function Phases Empirically, correlator magnitudes decay at a rate set by the pion mass, phase factors contribute remaining effective mass M R = − ∂ t ln ⟨ e R ( 0 , t ) ⟩ ∼ 3 M θ = − ∂ t ln ⟨ e i θ ( 0 , t ) ⟩ ∼ M N − 3 2 m π 2 m π M R M θ t t MW and Savage, PRD 96 (2016) � 5
Correlation Function Statistics Generic real, positive correlation functions, as well as early-time nucleons in LQCD, are log-normally distributed Hamber, Marinari, Parisi and Rebbi, Nucl Phys B225 (1983) Guagnelli, Marinari, and Parisi, PLB 240 (1990) Endres, Kaplan, Lee and Nicholson, PRL 107 (2011) Grabowska, Kaplan, and Nicholson, PRD 87 (2012) DeGrand, PRD 86 (2012) Porter and Drut, PRE 93 (2016) Log-normal distributions arise in two-body potential models and products of generic random positive numbers Beane, Detmold, Orginos, Savage, J Phys G42 (2015) Kaplan showed large-time nucleon correlators are better described by heavy-tailed stable distributions Broad, symmetric large-time distributions consistent with moment analysis by Savage � 6
Complex Log-Normal Distributions Products of phase factors have different central limit theorems, approach “wrapped normal” and eventually uniform distributions See e.g. N. I. Fisher, “Statistical Analysis of Circular Data ” (1995) Real part of nucleon correlation functions well-described by marginalization of “complex log-normal distribution” ∞ PDF ( R , θ ) = e − ( R − μ R ) 2 /(2 σ 2 ∑ e − n 2 θ 2 /(2 σ 2 R ) θ ) n = −∞ Re [ C ( t = 7)] Re [ C ( t = 30)] � 7 MW and Savage, PRD 96 (2016) MW, LATTICE 2017
Heavy-Tailed Phase Velocity Nucleon phase empirically well-described by wrapped-normal distribution Phase and log-magnitude time derivatives approach time independent, heavy-tailed wrapped stable distributions at late times � 8
Dynamical Source Construction Generalized pencil-of-functions (GPoF): an interpolating operator that has been time evolved is still a good interpolating operator G N ( p , t , τ src ) = 1 e i p ⋅ x Γ αβ ⟨ N α ( x , t ) e H τ src N β (0) e − H τ src ⟩ = G N ( p , t + τ src ) V ∑ x Aubin and Orginos (2010) Generalized GPoF (GGPoF): an interpolating operator time evolved with a modified Hamiltonian is still a good interpolating operator ( p , t , τ src ) = 1 e i p ⋅ x Γ αβ ⟨ e i θ N ( p ,0) − i θ N ( p , − τ src ) N α ( x , t ) N β ( 0 , − τ src ) ⟩ V ∑ G ( θ N ) t N x t + τ src Phase fluctuations during source construction can be removed by adding phase reweighting to the time evolution operator used StN [ G ( θ N ) ( p , t , τ src ) ] ∼ e − ( E ( p ) − 3 StN [ G N ( p , t , τ src ) ] ∼ e − ( E ( p ) − 3 2 m π ) t 2 m π )( t + τ src ) N � 9 MW and Savage (2017)
Phase Reweighted GGPoF Noise independent of after variance excited-state region τ src M θ ρ ( t, τ src ) Correct ground-state energies empirically reproduced* ** τ src = 0 t, t + τ src t τ src = 0 t, � 10 t + τ src t, τ src = 30
Phase Reweighted GGPoF Noise independent of after variance excited-state region τ src M θ ρ ( t, τ src ) Correct ground-state energies empirically reproduced* ** τ src = 0 t, t + τ src *Can this be proven? Phase reweighting factors are non-local in time, spoiling t standard spectral representation τ src = 0 t, **Except in the isovector channel… 0 ++ � 11 t + τ src t, τ src = 30
Auxiliary Charged Static Fermions Auxiliary fields representing static quarks and an Abelian gauge field in the zero-coupling limit can be freely added to path integrals 4 e 2 V μν ( x ) V μν ( x ) δ ( ∂ μ V μ − ∂ 2 f ( x ) ) −∑ x H ( x ) [ e iV 4( x ) H ( x + ̂ 4) − H ( x ) ] + 1 e → 0 ∫ V H H e 1 = lim = ∫ H H e −∑ x H ( x ) [ e if ( x + ̂ 4) − H ( x ) ] μ ) − if ( x ) H ( x + ̂ Static fermion two-point function given by auxiliary field Wilson line, depends on auxiliary function gauge-fixing function G H ( t , f ) = ⟨ H s ( x , t ) H s ′ � (0) ⟩ = δ x , y δ s , s ′ � e if ( 0 , t ) − if ( 0 ,0) The spectrum of auxiliary-charge zero states is independent of the auxiliary field gauge-fixing function � 12
GGPoF with Auxiliary Fields Spectral representation for correlators with hadrons and auxiliary fermions depends on gauge-fixing function (only) between auxiliary source/sink N ( p , t , τ src ) = 1 ( f ) e i p ⋅ x Γ αβ ⟨ N α ( x , t ) H 1 (0) H 1 ( 0 , − τ src ) N β ( 0 , − τ src ) ⟩ V ∑ G ( f ) x ⟨ 0 | N α | 𝔬 ( p ) ⟩ e − E 𝔬 ( p ) t ⟨ 𝔬 ( p ) | H 1 | 𝔩 ( f ) ⟩ e − E ( f ) 𝔩 τ src ⟨ 𝔩 ( f ) | H 1 N β | 0 ⟩ = Γ αβ ∑ 𝔬 , 𝔩 = ∑ 𝔬 ( p , τ src ) e − E 𝔬 ( p ) t Z 𝔬 ( p ) Z ( f ) 𝔬 GGPoF nucleon two-point function reproduced by choosing a gluon-field dependent auxiliary gauge-fixing function f ( t ) = θ N ( t , U ) = arg C N ( t , U ) e i p ⋅ x Γ αβ ⟨ e i θ N ( p ,0) − i θ N ( p , − τ src ) N α ( x , t ) N β ( 0 , − τ src ) ⟩ = ∑ ( p , t , τ src ) = 1 Z 𝔬 ( p ) Z ( θ N ) ( p , τ src ) e − E 𝔬 ( p ) t V ∑ G ( θ N ) 𝔬 N 𝔬 x � 13
Meson GGPoF Results Identical construction for generic hadrons, e.g. isovector mesons e i p ⋅ x ⟨ e i θ Γ (0) − i θ Γ ( τ src ) [ ¯ u Γ d ]( 0 , − τ src ) ⟩ = ∑ ( p , t , τ src ) = 1 𝔬 ( p ) Z Γ ,( θ N ) V ∑ G ( θ Γ ) Z Γ ( p , τ src ) e − E 𝔬 ( p ) t d Γ u ]( x , t )[ ¯ 𝔬 Γ 𝔬 x Possible for to be non-zero in cases where 𝔬 ( p ) Z Γ ,( θ N ) Z Γ Z Γ 𝔬 ( p ) Z Γ 𝔬 ( p ) = 0 ( p , τ src ) 𝔬 Isovector mesons: ud equivalent to U (1) u − d ud → e i θ ¯ ¯ background field: breaks conservation of total isospin! Baryons and nuclei: qqq → e i θ qqq equivalent to U (1) B PRELIMINARY background field: preserves all symmetries of interest � 14
Scalar Signal-to-Noise Problems Is exponential StN degradation of complex correlators inevitable? Toy model: free (or interacting) complex scalar field theory in (0+1)D L − 1 ∑ ( φ *( t + 1) − φ *( t ))( φ ( t + 1) − φ ( t )) − M 2 | φ 2 | S = t =0 G Q ,2 P = ⟨ φ ( t ) Q | φ ( t ) | 2 P φ *( t ) Q | φ (0) | 2 P ⟩ ∼ e − E Q ,2 P t Scalar correlators have exponential StN degradation set by total charge contained in spacetime volume StN [ G Q ,2 P ] ∼ e − E Q ,0 t ∼ e − M | Q | t � 15
Scalar Sign(al-to-Noise) Problems Scalar field phase gives correlation function path integrals a sign problem, responsible for exponential StN problem G 1,0 = ⟨ e ℛ ( t )+ i Θ ( t ) ⟩ = ∫ φ * φ e − S + ℛ ( t )+ i Θ ( t ) Distribution of phase fluctuations approximately wrapped normal I 0 ( κ ( t )) ] t ′ � =1 [ ≈ 1 t I | n | ( κ ( t )) PDF ( Θ ) = 1 2 π ∑ e − in Θ e − tn 2 /(2 ⟨ κ ⟩ ) 2 π ∑ ∏ e − in Θ n ∈ℤ n ∈ℤ No magnitude fluctuations, small phase fluctuations κ ( t ) = 2 | φ ( t ) || φ ( t − 1) | � 16
Phase Unwrapping Wrapped normal approximation has exponential StN problem StN [ ⟨ cos Θ⟩ ] ∼ e − t /(2 κ ) What if we “unwrap” the phase? ˜ t ∑ Θ ( t ) = Θ ( t ′ � ) − Θ ( t ′ � − 1) + 2 πν ( t ′ � ) Θ t ′ � =1 Average phase can be reconstructed from unwrapped phase cumulants ⟨ cos Θ⟩ = ⟨ cos ˜ n =1 κ n ( ˜ Θ ⟩ = e ∑ ∞ Θ )/ n ! Unwrapped cumulants avoid ˜ exponential StN problem Θ StN [ e − ˜ 2 κ Θ 2 /2 ] ∼ t t � 17 Detmold, Kanwar, MW (2018)
Phase Unwrapping Systematics Large phase jumps in regions of small magnitude lead to ambiguities in phase unwrapping Different definitions lead to large numerical discrepancies for all points after a large phase jump Heavy-tailed phase jump distributions appear in 1D scalar field correlators as well as LQCD baryons — Are large phase jumps a generic feature of LQFT? � 18
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