Chaos in the Bose-glass phase of a one-dimensional disordered Bose fluid
CChaos in the Bose-glass phase of a one-dimensional disordered Bose fluid
Romain Daviet and Nicolas Dupuis
Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, LPTMC, F-75005 Paris, France (Dated: January 28, 2021)We show that the Bose-glass phase of a one-dimensional disordered Bose fluid exhibits a chaoticbehavior, i.e., an extreme sensitivity to external parameters. Using bosonization, the replica for-malism and the nonperturbative functional renormalization group, we find that the ground stateis unstable to any modification of the disorder configuration (“disorder” chaos) or variation of theLuttinger parameter (“quantum” chaos, analog to the “temperature” chaos in classical disorderedsystems). This result is obtained by considering two copies of the system, with slightly differentdisorder configurations or Luttinger parameters, and showing that inter-copy statistical correlationsare suppressed at length scales larger than an overlap length ξ ov ∼ | (cid:15) | − /α ( | (cid:15) | (cid:28) α can be obtained by computing ξ ov or by studying the instability of the Bose-glassfixed point for the two-copy system when (cid:15) (cid:54) = 0. The renormalized, functional, inter-copy disordercorrelator departs from its fixed-point value – characterized by cuspy singularities – via a chaosboundary layer, in the same way as it approaches the Bose-glass fixed point when (cid:15) = 0 through aquantum boundary layer. Performing a linear analysis of perturbations about the Bose-glass fixedpoint, we find α = 1. CONTENTS
I. Introduction 1II. Model and FRG formalism 2A. Introducing two copies and n replicas 3B. Effective action and FRG 3C. Flow equations 4III. Disorder chaos 5A. Approach to the BG fixed point 51. Quantum boundary layer 52. Linear analysis 6B. Escape from the BG fixed point 61. Chaotic behavior and overlap length 62. Chaos boundary layer 73. Linear analysis 84. Comparison with Duemmer and LeDoussal’s work 10IV. Quantum chaos 11V. Conclusion 11Acknowledgments 12References 12 I. INTRODUCTION
In a Bose fluid with short-range interactions, disordercan induce a quantum phase transition between a su-perfluid phase and a localized phase dubbed Bose glass(BG).
The latter is characterized by a nonzero com-pressibility, a vanishing dc conductivity and the absenceof gap in the optical conductivity. As its name indicates, the BG phase is expected to be analogous to the Fermi-glass phase of interacting fermions in a strong disorderpotential and exhibit some of the characteristic proper-ties of glassy systems. In one dimension, the analogy of the BG phase withother disordered systems exhibiting glassy properties isstrongly supported by the nonperturbative functionalrenormalization group (FRG).
In this approach, onefinds that the BG phase is described by an attractivefixed point analog to the zero-temperature fixed pointcontrolling the low-temperature phase of many classicaldisordered systems. The role of temperature is playedby the Luttinger parameter K ∼ k θ which, as the mo-mentum scale k approaches zero, vanishes with an ex-ponent θ = z − z . Moreover, the renormalized disorder correla-tor assumes a cuspy functional form associated with theexistence of metastable states. At nonzero momentumscale, quantum tunneling between the ground state andthese metastable states leads to a rounding of the cuspsingularity into a quantum boundary layer (QBL). Thelatter controls the low-energy dynamics and is responsi-ble for the ω behavior of the (dissipative) conductivity.Thus the FRG approach reveals some of the glassy prop-erties (pinning, “shocks” or static avalanches) of the BGphase and, to some extent, can be understood withinthe “droplet” picture put forward for the description ofglassy (classical) systems. One of the peculiar features of glassy systems is the ex-treme sensitivity of the ground state with respect to smallchanges in external parameters like the disorder configu-ration or the temperature. In some cases, an infinitesimalperturbation is sufficient to lead to a complete reorgani-zation of the ground state at large length scales. Thissituation is referred to as chaos, e.g. disorder chaosor temperature chaos according to the external parame-ter being considered. Chaos is usually characterized by a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n an overlap length ξ ov beyond which the ground state com-pletely changes as a result of the variation in the externalparameter. The overlap length diverges as ξ ov ∼ | (cid:15) | − /α where α is called the chaos exponent ( | (cid:15) | (cid:28) To our knowledge, the only quantumdisordered system where (disorder) chaos was studied isthe two-dimensional Anderson insulator. In this paper, we study chaos in the BG phase of aone-dimensional Bose fluid. In Sec. II we briefly recallthe FRG formalism used in Refs. 4 and 5 to study the BGphase and generalize it to include two copies of the systemsubjected to slightly different disorder configurations. In particular we introduce the main quantities of in-terest: the running Luttinger parameter K k and theintra- and inter-copy renormalized disorder correlators, δ ,k ( u ) = δ ,k ( u ) and δ ,k ( u ) respectively. Here k isa running momentum scale and u ≡ φ a − φ b stands forthe difference between the fields in two different repli-cas. The flow equations for K k and δ ij,k ( u ) are similarto those obtained for pinned disordered periodic man-ifolds by Duemmer and Le Doussal (DLD) in Refs. 17and 18, with K k playing the role of the temperature.In section III A we first consider the approach to theBG fixed point when the two copies are identical (i.e.,experience the same disorder potential: (cid:15) = 0). In thatcase, the π -periodic functions δ ii,k ( u ) and δ ,k ( u ) ap-proach a fixed-point function δ ∗ ( u ) exhibiting cusps at u = pπ ( p integer). At nonzero momentum scale thecusp singularity at u = pπ is rounded into a QBL. A lin-ear analysis of the perturbations about δ ∗ ( u ) shows thatthe less irrelevant eigenvalue λ = − θ is associated withan eigenfunction which is more and more peaked around u = pπ as the number n max of circular harmonics of δ ij,k ( u ) (used in the numerical solution of the linearizedflow equations) increases, whereas all other eigenfunc-tions remain extended over the whole interval [0 , π ].In Sec. III B we show that the BG fixed point is unsta-ble for any nonzero (cid:15) since in that case δ ii,k ( u ) → δ ∗ ( u )but δ ,k ( u ) → k →
0. Thus the two copiesbecome statistically independent in the large-distancelimit, which corresponds to disorder chaos. From thenumerical solution of the flow equations we find that δ ,k (0) satisfies a scaling form with a characteristiclength ξ ov ∼ | (cid:15) | − /α but the chaos exponent α seemsto converge very slowly with n max . The instability of theBG fixed point occurs via a chaos boundary layer (CBL) reminiscent of the QBL observed in the approachto the BG fixed point. The linear analysis of the per-turbations about the fixed-point solution δ ∗ ( u ) reveals asingle positive eigenvalue λ ≡ α associated with a func-tion which is more and more peaked around u = pπ as n max increases. The convergence of α with n max is ex-tremely slow but in the limit n max → ∞ the solution canbe found analytically (and is essentially given by a Dirac comb) and yields the chaos exponent α = 1. We are thenable to show that the convergence of α with n max is log-arithmic. The agreements and differences between ourresults and those of DLD are discussed in Sec. III B 4.Finally, in Sec. IV, we show that chaos is also obtainedwhen one considers a slight change in the Luttinger pa-rameter. II. MODEL AND FRG FORMALISM
We consider a one-dimensional Bose fluid described bythe Hamiltonian ˆ H . At low energies ˆ H can be approx-imated by the Tomonaga-Luttinger Hamiltonian ˆ H = ˆ dx v π (cid:26) K ( ∂ x ˆ ϕ ) + K ( ∂ x ˆ θ ) (cid:27) , (1)where ˆ θ is the phase of the boson operator ˆ ψ ( x ) = e i ˆ θ ( x ) ˆ ρ ( x ) / and ˆ ϕ is related to the density operator via ˆ ρ ( x ) = ρ − π ∂ x ˆ ϕ ( x ) + 2 ρ cos(2 πρ x − ϕ ( x )) , (2)where ρ is the average density and ρ a nonuniver-sal parameter that depends on microscopic details. ˆ ϕ and ˆ θ satisfy the commutation relations [ˆ θ ( x ) , ∂ y ˆ ϕ ( y )] = iπδ ( x − y ). v denotes the sound-mode velocity and thedimensionless parameter K , which encodes the strengthof boson-boson interactions, is the Luttinger parameter.The ground state of ˆ H is a Luttinger liquid, i.e., a su-perfluid state with superfluid stiffness ρ s = vK/π andcompressibility κ = dρ /dµ = K/πv . The disorder contributes to the Hamiltonian a term ˆ H dis = ˆ dx (cid:26) − π η∂ x ˆ ϕ + ρ [ ξ ∗ e i ˆ ϕ + h . c . ] (cid:27) , (3)where η ( x ) (real) and ξ ( x ) (complex) denote random po-tentials with Fourier components near 0 and ± πρ , re-spectively. η can be eliminated by a shift of ˆ ϕ and is notconsidered in the following. In the functional-integral formalism, after integratingout the field θ , one obtains the Euclidean (imaginary-time) action S [ ϕ ; ξ ] = ˆ X (cid:26) v πK (cid:2) ( ∂ x ϕ ) + v − ( ∂ τ ϕ ) (cid:3) + ρ [ ξ ∗ e iϕ + c . c . ] (cid:27) , (4)where we use the notation X = ( x, τ ), ´ X = ´ β dτ ´ dx and ϕ ( X ) is a bosonic field with τ ∈ [0 , β ]. The modelis regularized by a UV cutoff Λ acting both on mo-menta and frequencies. We shall only consider the zero-temperature limit β = 1 /T → ∞ but β will be kept finiteat intermediate stages of calculations. A. Introducing two copies and n replicas To investigate the chaotic nature of the BG phase, weconsider two copies of the system with slightly differentrealizations of the disorder, ξ ( x ) = ξ ( x ) + (cid:15)ζ ( x ) ,ξ ( x ) = ξ ( x ) − (cid:15)ζ ( x ) , (5)where | (cid:15) | (cid:28)
1. The random potentials ξ and ζ are uncor-related and identically distributed, i.e., assuming Gaus-sian distributions with zero mean, ξ ( x ) = ζ ( x ) = 0 ,ξ ∗ ( x ) ξ ( x (cid:48) ) = ζ ∗ ( x ) ζ ( x (cid:48) ) = Dδ ( x − x (cid:48) ) (6)(all other correlators, e.g. ξ ( x ) ξ ( x (cid:48) ), vanish). We usean overline to denote disorder averaging. Equations (6)imply ξ ∗ i ( x ) ξ j ( x (cid:48) ) = D ij δ ( x − x (cid:48) ) ,D ii = D (1 + (cid:15) ) , D = D = D (1 − (cid:15) ) . (7)The statistical correlations between the two systems arecharacterized by the correlation functions C ij ( X − X (cid:48) ) = (cid:104) ( ϕ i ( X ) − ϕ i ( X (cid:48) ))( ϕ j ( X ) − ϕ j ( X (cid:48) )) (cid:105) . (8)Since the two copies are independent before disorder av-eraging, C ii ( X − X (cid:48) ) = 2[ G c,ii (0) + G d,ii (0) − G c,ii ( X − X (cid:48) ) − G d,ii ( X − X (cid:48) )] ,C ( X − X (cid:48) ) = 2[ G d, (0) − G d, ( X − X (cid:48) )] , (9)where G c,ij ( X − X (cid:48) ) = (cid:104) ϕ i ( X ) ϕ j ( X (cid:48) ) (cid:105) − (cid:104) ϕ i ( X ) (cid:105)(cid:104) ϕ j ( X (cid:48) ) (cid:105) ,G d,ij ( X − X (cid:48) ) = (cid:104) ϕ i ( X ) (cid:105)(cid:104) ϕ j ( X (cid:48) ) (cid:105) − (cid:104) ϕ i ( X ) (cid:105) (cid:104) ϕ j ( X (cid:48) ) (cid:105) (10)are the connected and disconnected propagators, respec-tively. The long-distance part of both C ii ( X − X (cid:48) ) and C ( X − X (cid:48) ) is determined by G d,ij ( X − X (cid:48) ). In the replica formalism, one considers n replicas ofthe system and the disorder-averaged partition function Z [ { J ia } ] = n (cid:89) a =1 2 (cid:89) i =1 Z [ J ia ; ξ i ] , (11)where the 2 n external sources { J ia } act on each replicaindependently and Z [ J ia ; ξ i ] = ˆ D [ ϕ ia ] e − S [ ϕ ia ,ξ i ]+ ´ X J ia ( X ) ϕ ia ( X ) (12) is the partition function of the a th replica of the i th copybefore disorder averaging. Using (7) to perform the dis-order average, one obtains Z [ { J ia } ] = ˆ D [ { ϕ ia } ] e − S [ { ϕ ia } ]+ (cid:80) a,i ´ X J ia ( X ) ϕ ia ( X ) , (13)with the replicated action S [ { ϕ ia } ] = (cid:88) i,a ˆ x,τ v πK (cid:26) ( ∂ x ϕ ia ) + ( ∂ τ ϕ ia ) v (cid:27) − (cid:88) a,b,i,j D ij ˆ x,τ,τ (cid:48) cos[2 ϕ ia ( x, τ ) − ϕ jb ( x, τ (cid:48) )] , (14)where D ij = ρ D ij . B. Effective action and FRG
To implement the nonperturbative FRGapproach, we add to the action (14) the infraredregulator term ∆ S k [ { ϕ ia } ] = 12 (cid:88) i,a,q,ω ϕ ia ( − q, − iω ) R k ( q, iω ) ϕ ia ( q, iω ) , (15)where k is a (running) momentum scale varying fromthe UV scale Λ down to zero and ω ≡ ω n = 2 πn/β ( n integer) is a Matsubara frequency. The cutoff function R k ( q, iω ) is chosen so that fluctuation modes satisfying | q | , | ω | /v k (cid:28) k are suppressed while those with | q | (cid:29) k or | ω | /v k (cid:29) k are left unaffected (the k -dependent sound-mode velocity v k is defined below). In practice we choose R k ( q, iω ) = Z x (cid:18) q + ω v k (cid:19) r (cid:18) q + ω /v k k (cid:19) , (16)where r ( y ) = α/ ( e y −
1) with α a constant of order unity. Z x is defined below.The partition function Z k [ { J ia } ] = ˆ D [ { ϕ ia } ] exp (cid:110) − S [ { ϕ ia } ] − ∆ S k [ { ϕ ia } ] + (cid:88) i,a ˆ X J ia ϕ ia (cid:111) (17)thus becomes k dependent. The expectation value of thefield reads φ ia ( X ) = δ ln Z k [ { J jf } ] δJ ia ( X ) = (cid:104) ϕ ia ( X ) (cid:105) (18)(to avoid confusion in the indices we denote by { J jf } the2 n external sources).The scale-dependent effective actionΓ k [ { φ ia } ] = − ln Z k [ { J ia } ] + (cid:88) i,a ˆ X J ia φ ia − ∆ S k [ { φ ia } ](19)is defined as a modified Legendre transform which in-cludes the subtraction of ∆ S k [ { φ ia } ]. Assuming thatfor k = Λ the fluctuations are completely frozen by theterm ∆ S Λ , Γ Λ [ { φ ia } ] = S [ { φ ia } ]. On the other handthe effective action of the original model (14) is givenby Γ k =0 since R k =0 vanishes. The nonperturbative FRGapproach aims at determining Γ k =0 from Γ Λ using Wet-terich’s equation ∂ t Γ k [ { φ ia } ] = 12 Tr (cid:110) ∂ t R k (cid:0) Γ (2) k [ { φ ia } ] + R k (cid:1) − (cid:111) , (20)where Γ (2) k is the second functional derivative of Γ k and t = ln( k/ Λ) a (negative) RG “time”. The trace in (20)involves a sum over momenta and frequencies as well ascopy and replica indices.To solve (approximately) the flow equation (20) weconsider the following ansatz for the effective action Γ k [ { φ ia } ] = (cid:88) a Γ ,k [ φ a ] − (cid:88) a,b Γ ,k [ φ a , φ b ] , (21)where φ a = { φ a , φ a } andΓ ,k [ φ a ] = (cid:88) i ˆ X Z x (cid:26) ( ∂ x φ ia ) + ( ∂ τ φ ia ) v k (cid:27) , Γ ,k [ φ a , φ b ] = (cid:88) i,j ˆ x,τ,τ (cid:48) V ij,k (cid:0) φ ia ( x, τ ) − φ jb ( x, τ (cid:48) ) (cid:1) , (22)with initial conditions Z x = v/πK , v Λ = v and V Λ ,ij ( u ) = 2 D ij cos(2 u ). The form of Γ ,k and Γ ,k is strongly constrained by the statistical tilt symme-try (STS) due to the invariance of the disorder part ofthe action (14) in the time-independent shift ϕ ia ( X ) → ϕ (cid:48) ia ( X ) = ϕ ia ( X ) + w ( x ) with w ( x ) an arbitrary functionof x . The STS yieldsΓ k [ { φ (cid:48) ia } ] = Γ k [ { φ ia } ] + nβZ x ˆ x ( ∂ x w ) + Z x (cid:88) i,a ˆ X ( ∂ x w )( ∂ x φ ia ) . (23)This implies that Z x remains equal to its initial valueand no other space derivative terms are allowed; for in-stance the term ( ∂ x φ a )( ∂ x φ a ) is not possible. The term( ∂ τ φ a )( ∂ τ φ a ) is a priori not excluded by the STS but isnot generated by the flow equation. Since the two copiesare equivalent ( D = D ), the velocity v k is copy inde-pendent. In addition to v k one may define a k -dependentLuttinger parameter by Z x = v k /πK k . The STS alsoensures that the two-replica potential V ij,k ( φ ia , φ jb ) is afunction of φ ia − φ jb only.Thus the main quantities of interest are K k , v k andthe two-replica potential V ij,k ( u ). It is convenient to in-troduce the dimensionless function δ ij,k ( u ) = − K v V (cid:48)(cid:48) ij,k ( u ) k . (24) For a single copy, the BG fixed point is characterized by avanishing of K k and v k : K k , v k ∼ k θ for k →
0. The van-ishing of K k implies that quantum fluctuations are sup-pressed at low energies and therefore a pinning of the field ϕ ( x, τ ) by the random potential. On the other hand the π -periodic function δ ∗ ( u ) = lim k → δ k ( u ) exhibits cuspsat u = pπ ( p integer). This cuspy nonanalytic form is re-lated to the existence of metastable states. At nonzeromomentum scale, quantum tunneling between the groundstate and these metastable states leads to a rounding ofthe nonanalyticity into a QBL. The latter is responsiblefor the vanishing of the optical conductivity σ ( ω ) ∼ ω in the low-frequency limit. With the ansatz (21,22), the disconnected propagatorof the two-copy system is given by G d,ij,k ( q, iω ) = βδ ω, v k K δ ij,k (0)[ Z x q + R k ( q, . (25) Stricto sensu this expression is valid only for | q | (cid:28) k sincethe ansatz (21,22) is based on a derivative expansion.However, we expect q to act as an infrared cutoff in theflow so that G d,ij,k =0 ( q,
0) can be approximately obtainedby setting k ∼ | q | , i.e., G d,ij,k =0 ( q, iω ) ∼ βδ ω, π δ ij, | q | (0) | q | . (26)Since the intracopy correlation function C ii ( X ) is notmodified by the inter-copy statistical correlations, wehave C ii ( X ) (cid:39) πδ ∗ (0) ln | x | . The system exhibits chaosif lim x →∞ C ( X ) = 0 for any (cid:15) >
0, which requireslim k → δ ,k (0) = 0: The two copies are then statisticallyindependent at long distances regardless of the (nonzero)difference in the random potentials ξ ( x ) and ξ ( x ). Inthe following we shall therefore consider the flow of F k = 1 + (cid:15) − (cid:15) δ ,k (0) [ δ ,k (0) + δ ,k (0)] , (27)with initial condition F Λ = 1. C. Flow equations
A detailed derivation of the flow equations for a singlecopy can be found in Ref. 5. The generalization to twocopies is straightforward and gives ∂ t δ ii,k ( u ) = − δ ii,k ( u ) − l K k δ (cid:48)(cid:48) ii,k ( u )+ π ¯ l { δ (cid:48)(cid:48) ii,k ( u )[ δ ii,k ( u ) − δ ii,k (0)] + δ (cid:48) ii,k ( u ) } , (28) ∂ t δ ,k ( u ) = − δ ,k ( u ) − l K k δ (cid:48)(cid:48) ,k ( u )+ π ¯ l { δ (cid:48)(cid:48) ,k ( u )[ δ ,k ( u ) − δ ii,k (0)] + δ (cid:48) ,k ( u ) } , (29)and ∂ t K k = θ k K k , ∂ t ( K k /v k ) = 0 , (30)where θ k = z k − π δ (cid:48)(cid:48) ii,k (0) ¯ m τ (31)with z k the running dynamical critical exponent. Thethresholds functions l , ¯ l and ¯ m τ are defined in Ref. 5.Note that the π periodicity of δ ij,k ( u ), as well as theproperty ´ π du δ ij,k ( u ) = 0, are maintained by the flowequations. III. DISORDER CHAOS
We consider only the case where the parameters ofthe microscopic action (i.e., the initial conditions of theRG flow) are such that the system is in the BG phase.The flow equations are integrated numerically using thefourth-order Runge-Kutta method with adaptative stepsize. The functions δ ij,k ( u ) = n max (cid:88) n =1 δ ij,n,k cos(2 nu ) (32)are expanded in circular harmonics with n max in therange [100 − δ ij, ,k necessary vanishessince ´ π du δ ij,k ( u ) = 0. A. Approach to the BG fixed point
As expected, the RG equations for δ ii,k ( u ), K k and v k are identical to the one-copy case. The function δ ii,k ( u )approaches the π -periodic fixed-point solution δ ∗ ( u ) = 12 π ¯ l (cid:20)(cid:16) u − π (cid:17) − π (cid:21) , u ∈ [0 , π ] , (33)which exhibits cusps at u = pπ ( p integer). The BGfixed point is a “critical”, scale-invariant, fixed point asfar as the disorder correlator δ ii,k ( u ) is concerned, i.e., inthe zero-frequency sector. The finite localization length,which characterizes the BG phase, appears only in thenonzero-frequency sector of the theory, a feature whichis related to the nonanalytic structure of the propagatorat zero frequency.
1. Quantum boundary layer
For any nonzero momentum scale k , the cusp singular-ity at u = pπ is rounded into a boundary layer. In thevicinity of the BG fixed point, K k →
0, the solution canbe written in the form δ ii,k ( u ) = δ ii,k (0) + K k f (cid:18) uK k (cid:19) (34)near u = 0 and for an arbitrary value of the ratio u/K k .The k -independent even function satisfies f (0) = f (cid:48) (0) =0 and f (cid:48)(cid:48) (0) <
0. From (28) we obtain ∂ t δ ii,k (0) (cid:39) − δ ii,k (0) − l f (cid:48)(cid:48) + π ¯ l ( f (cid:48)(cid:48) f + f (cid:48) ) (35) using K k → ∂ t K k = θ k K k →
0. The right-handside must be independent of x = u/K k and equal to − δ ii,k (0) − l f (cid:48)(cid:48) (0) since f (0) = f (cid:48) (0) = 0, i.e., − l f (cid:48)(cid:48) + π ¯ l ( f (cid:48)(cid:48) f + f (cid:48) ) = − l f (cid:48)(cid:48) (0) . (36)This yields f ( x ) = l π ¯ l (cid:40) − (cid:20) − π ¯ l l f (cid:48)(cid:48) (0) x (cid:21) / (cid:41) . (37)Since the solution (34) must approach the fixed-point so-lution δ ∗ ( u ) when K k →
0, we deduce f (cid:48)(cid:48) (0) = − π/ l ¯ l .From (35) and (36) we also obtain ∂ t δ ii (0) = − δ k,ii (0) − l f (cid:48)(cid:48) (0) , (38)i.e., δ ii,k (0) = Ce − t − l f (cid:48)(cid:48) (0) . (39)Since δ ,k (0) approaches a finite value as t → −∞ , theconstant C must necessarily vanish. From the flow equa-tion (28) it is easy to see that the relevant eigenvalue 3 isassociated with a constant solution δ k ( u ) = const, whichis not allowed as it violates the condition ´ π du δ ii,k ( u ) =0. We therefore obtain δ ii,k (0) = − l f (cid:48)(cid:48) (0) = π l = δ ∗ (0) , (40)which is the expected result in the limit k →
0. For k >
0, we expect δ ii,k (0) = δ ∗ (0) − l π ¯ l K k (41)to leading order in K k , where the prefactor of K k isdetermined by requiring that δ ii,k ( u ) − δ ∗ ( u ) vanishesto order K k when | u | /K k (cid:29)
1. The QBL definedby (34,37,41) is in very good agreement with the numer-ical solution of the flow equations (see the discussion ofthe QBL and CBL in Sec. III B 2 and Fig. 5).The preceding results imply the deviation from the BGfixed point g k ( u ) = δ ii,k ( u ) − δ ∗ ( u ) (cid:39) (cid:26) − l π ¯ l K k + | u | l if | u | (cid:28) K k , | u | (cid:29) K k . (42)We conclude that when K k → k →
0) the function g k ( u ), which characterizes the approach to the BG fixedpoint via the QBL, tends to ∼ K k X ( K ) u, where X ( K ) u, isthe π -periodic Kronecker comb. In the next section weshall see how this result can be reproduced from a linearanalysis of the perturbations about the fixed point.
2. Linear analysis
Let us consider a small perturbation about the BGfixed point ( K ∗ = 0 , δ ∗ ( u )), δ ii,k ( u ) = δ ∗ ( u ) + g ( u ) e − λt ,K k = K ∗ + Ke − λt , (43)where g ( u ) and K (not to be confused with the bare value K Λ of the Luttinger parameter) are k independent. Tofirst order in g and K , we obtain the flow equations λg ( u ) = 3 g ( u ) + Kl δ ∗(cid:48)(cid:48) ( u ) − π ¯ l { g (cid:48)(cid:48) ( u )[ δ ∗ ( u ) − δ ∗ (0)]+ δ ∗(cid:48)(cid:48) ( u )[ g ( u ) − g (0)] + 2 δ ∗(cid:48) ( u ) g (cid:48) ( u ) } (44)and λK = − θK, (45)with θ = lim k → θ k . We now expand the functions g ( u )and δ ∗ ( u ) in circular harmonics as in (32) with δ ∗ n = 12 π ¯ l n . (46)Note that this expression implies that δ ∗(cid:48)(cid:48) ( u ) = 1 π ¯ l [1 − π X ( u )] , (47)where X ( u ) the π -periodic Dirac comb, and differsfrom the naive result δ ∗(cid:48)(cid:48) ( u ) = 1 /π ¯ l (which violates thecondition ´ π du δ ∗(cid:48)(cid:48) ( u ) = 0) obtained from (33). We thusrewrite Eqs. (44) and (45) as λg n = n max (cid:88) n (cid:48) =1 A n,n (cid:48) g n (cid:48) − l π ¯ l K,λK = − θK, (48)with A n,n (cid:48) = (cid:18) − π n (cid:19) δ ( K ) n,n (cid:48) − n (cid:34) n + n (cid:48) ) + 1 − δ ( K ) n,n (cid:48) ( n − n (cid:48) ) (cid:35) , (49)where δ ( K ) n,n (cid:48) denotes the Kronecker delta.The solution with eigenvalue λ = − θ , correspondingto a vanishing of the Luttinger parameter K k ∼ e θt , canbe found analytically in the limit n max → ∞ . Setting K = 1 and g n = − l /π ¯ l n max , and using n max (cid:88) n (cid:48) =1 ( A n,n (cid:48) + 2) = 2 + O (cid:18) n max (cid:19) , (50)one easily sees that the first equation in (48) is satisfiedto leading order in 1 /n max . Thus, for n max → ∞ , weobtain the linear perturbation K = 1 , g ( u ) = − l π ¯ l X ( K ) u, . (51) . . . . . . u/π − . . . . g ( u ) λ = − θλ ’ − λ ’ − λ ’ − Figure 1. Solutions g ( u ) = (cid:80) n max n =1 g n cos(2 nu ) deduced fromthe numerical solution of Eqs. (48) with n max = 400 and θ = 0 .
5, and corresponding to the largest eigenvalues: λ = − θ = − . λ (cid:39) − . / − . / − . g (0) = 1. The result g ( u ) ∝ X ( K ) u, agrees with the QBL analysis ofSec. III A 1.This result is confirmed by the numerical solution ofthe linear system (48). As n max increases, we find a so-lution g ( u ), associated with an eigenvalue λ (cid:39) − θ , andwhich becomes more and more localized about u = pπ ( p integer) with the ratio g ( pπ ) /K taking a nonzero limit,in agreement with (51). There are also n max eigenvectorswith K = 0 and a function g ( u ) which typically extendsover the whole interval [0 , π ] (see Fig. 1). The largesteigenvalues converge to {− , − , − , − , · · · } when n max → ∞ . The convergence is fast and, for the largesteigenvalues, already obtained with a two-digit precisionfor n max = 400. Although θ is not precisely known, itsatisfies θ < − θ is therefore the largest eigenvalue andcontrols the approach to the BG fixed point.If we set K ≡ K Λ = 0 in the flow equation (28) thecusp in δ ii,k ( u ) arises for k > This finite-scale singu-larity is not accounted for in the linear analysis. Indeed,if we set K = 0 in (44) and (45), we find that the func-tion g ( u ) e λt goes smoothly to zero and the fixed point δ ∗ ( u ) is recovered only for k = 0. Thus it seems that theboundary layer induced by a nonzero K k is a necessarycondition for the linear analysis to be valid. B. Escape from the BG fixed point
1. Chaotic behavior and overlap length
When (cid:15) = 0 the two copies are identical, δ ,k ( u ) = δ ii,k ( u ), and δ ,k ( u ) approaches δ ∗ ( u ) when k →
0. Forsmall but nonzero (cid:15) , implying δ , Λ ( u ) (cid:54) = δ ii, Λ (u), δ k, ( u )is first attracted to the BG fixed-point solution δ ∗ ( u ) but ln(Λ /k ) . . . . l n ( δ ∗ ( ) − δ , k ( )) δ ,k (0) δ ,k (0) δ ∗ (0) ln(Λ /k ) − . − . − . − . − . l n ( δ ∗ ( ) − δ , k ( )) θt + cst − λ max t + cst Figure 2. (top) δ ,k (0) = δ ,k (0) and δ ,k (0) vs ln(Λ /k ) = − t as obtained from the numerical solution of the flow equa-tions with n max = 800 and (cid:15) (cid:39) × − . When (cid:15) (cid:54) = 0, δ ,k (0) first approaches the BG fixed-point value δ ∗ (0) butis eventually suppressed, lim k → δ ,k = 0, thus showing thatthe two copies become statistically uncorrelated at long dis-tances. (bottom) The ln-ln plot shows that the approach of δ ,k (0) to its fixed point value δ ∗ (0) is controlled by the ex-ponent − θ and the escape by λ max (cid:39) .
6. (These figures areobtained for K = 0 .
1, other figures use K = 0 . is eventually suppressed as shown in Fig. 2:lim k → δ ,k ( u ) = 0 . (52)Linearizing the equation ∂ t δ ,k , we find δ ,k ( u ) ∼ k − π / cos(2 u ) for k → G d, ,k =0 ( q, ∼ | q | − π / , so that C ( X ) ∼ | x | − π / (54)decays with the exponent π / − (cid:39) . ξ ov ≡ ξ ov ( (cid:15) ) associated with the instability of the BG fixedpoint when (cid:15) (cid:54) = 0 and signaling the loss of statistical cor-relations between the two copies at large length scales. − − − − − (cid:15) ξ o v Λ ξ ov ∝ | (cid:15) | − / . Figure 3. ξ ov vs (cid:15) in a log-log plot as obtained from thecriterion F k =1 /ξ ov = 0 .
1. The blue line shows a linear fitcorresponding to the power-law behavior (55) with α (cid:39) . We use the criterion F k =1 /ξ ov = γ where γ (cid:28) F k is defined by (27). ξ ov divergesfor (cid:15) → ξ ov ∝ | (cid:15) | − /α , (55)where α is the chaos exponent. If we plot F k as a functionof kξ ov for various values of (cid:15) , we observe a data collapsethus showing that F k satisfies the one-parameter scalingform F k = F ( kξ ov ) , (56)where F ( x ) is a universal scaling function, as expectedfor a scale-invariant fixed point with a single relevantdirection (Fig. 4).On can also obtain the chaos exponent directly fromthe flow equations. For (cid:15) →
0, when δ ,k (0) is near itsfixed point value δ ∗ (0), one has δ ,k (0) (cid:39) δ ∗ (0) + Ae θt + Be − λ max t , (57)as shown in Fig. 2. The leading irrelevant eigenvalue − θ ,as discussed in Sec. III A 2, controls the approach to theBG fixed point. The relevant eigenvalue λ max controlsthe departure from the fixed point at very long RG time | t | . λ max also determines the divergence of the overlaplength when (cid:15) → ξ ov ∼ | δ , Λ (0) − δ ii, Λ (0) | /λ max ∼ | (cid:15) | /λ max , (58)so that α = λ max /
2. The estimate of the chaos exponentobtained from (57) is in very good agreement with thecalculation of ξ ov using the criterion F k =1 /ξ ov = γ . Theresults are shown in Table I for various values of n max .
2. Chaos boundary layer
The boundary layer analysis of Sec. III A 1 can be gen-eralized to the case where (cid:15) (cid:54) = 0. Since the flow equation
Table I. Chaos exponent α vs number n max of harmonics used in the numerics obtained from ξ ov , δ ,k (0) − δ ∗ (0) or the linearanalysis [Eqs. (66)]. n max
100 200 300 400 500 600 1000 10000 20000 30000 40000 50000 100000from ξ ov δ ,k (0) − δ ∗ (0) 0.800 0.811 0.817 0.820 0.823from linear analysis 0.717 0.748 0.764 0.773 0.781 0.786 0.800 0.847 0.857 0.862 0.865 0.868 0.876 of δ ii,k ( u ) and K k are independent of (cid:15) , the intracopydisorder correlator δ ii,k ( u ) is still given by (34) when | u | , K k (cid:28)
1. The equation for δ ,k ( u ) can be writtenas ∂ t δ ,k ( u ) = − δ ,k ( u ) − l (cid:18) K k + π ¯ l l ˆ K k (cid:19) δ (cid:48)(cid:48) ,k ( u )+ π ¯ l { δ (cid:48)(cid:48) ,k ( u )[ δ ,k ( u ) − δ ,k (0)] + δ (cid:48) ,k ( u ) } , (59)where ˆ K k = δ ii,k (0) − δ k, (0) . (60) ln(Λ /k ) . . . . . . F k (cid:15) = 10 − (cid:15) = 10 − (cid:15) = 10 − (cid:15) = 10 − (cid:15) = 10 − − − −
10 0 10 − ln( kξ ov ) . . . . . . F k (cid:15) = 10 − (cid:15) = 10 − (cid:15) = 10 − (cid:15) = 10 − (cid:15) = 10 − Figure 4. (top) F k vs k for various values of (cid:15) . Statistical cor-relations between the two copies are lost when 1 /k is largerthan the overlap ξ ov length defined by F k =1 /ξ ov = 0 .
1. (bot-tom) F k vs kξ ov showing the data collapse expected from thescaling form (56). We assume that the system is near the BG fixed point,so that both K k , ˆ K k and their derivatives ∂ t K k , ∂ t ˆ K k aresmall. Near u = 0, but for an arbitrary ratio u/K tot ,k ,the solution of (59,60) can be written in the form δ ,k ( u ) = δ ,k (0) + K tot ,k f (cid:18) uK tot ,k (cid:19) , (61)where K tot ,k = K k + π ¯ l l ˆ K k . (62)Using the fact that K tot ,k and ∂ t K tot ,k are small, Eq. (59)implies that f ( x ) satisfies (36) and is therefore givenby (37) with f (cid:48)(cid:48) (0) = − π/ l ¯ l . This is expected sincewhen (cid:15) = 0, one has K tot ,k = K k and δ ,k ( u ) = δ ii,k ( u )is given by (34). Similarly to (41) we find δ ,k (0) = δ ∗ (0) − l π ¯ l K tot ,k . (63)When δ ,k ( u ) approaches the fixed point, i.e., when K tot ,k (cid:39) K k , the decreasing width of the boundary layeris controlled by quantum fluctuations as discussed inSec. III A 1 ( δ ,k ( u ) (cid:39) δ ii,k ( u ) in that case). At lat-ter RG times | t | , when K tot ,k (cid:39) ( π ¯ l /l ) ˆ K k , the widthof the boundary layer increases as a result of the loss ofstatistical correlations between the two copies due to thechaotic behavior of the system. Equations (61,63) are invery good agreement with the numerical solution of theflow equations as shown in Fig. 5.
3. Linear analysis
We now consider a linear analysis of the perturbationsabout the BG fixed point in the case (cid:15) (cid:54) = 0. Writing δ ii,k ( u ) = δ ∗ ( u ) + g ii ( u ) e − λt ,δ ,k ( u ) = δ ∗ ( u ) + g ( u ) e − λt ,K k = K ∗ + Ke − λt (64)(note that ˆ K k , introduced in the preceding section, is notan independent variable), we obtain λg ( u ) = 3 g ( u ) + Kl δ ∗(cid:48)(cid:48) ( u ) − π ¯ l { g (cid:48)(cid:48) ( u )[ δ ∗ ( u ) − δ ∗ (0)]+ δ ∗(cid:48)(cid:48) ( u ) g ( u ) + 2 δ ∗(cid:48) ( u ) g (cid:48) ( u ) } (65) . . . . . u . . . δ k , ( u ) δ k , ( u ) δ ∗ ( u ) Figure 5. δ ,k ( u ) near u = 0 as obtained from the numer-ical solution of the flow equations (lines) and the analyticexpression (61) (symbols). δ ,k =Λ e − . shows the QBL thatforms during the approach to the BG fixed point whereas δ ,k =Λ e − . shows to the CBL due to the escape of thefixed point. When using expression (61), the value of K tot ,k is obtained from the numerical solution of the flow equations. to first order in g while g ii and K satisfy (44,45).Expanding both g ij ( u ) and δ ∗ ( u ) in circular harmonicsyields the linear system λg ii,n = n max (cid:88) n (cid:48) =1 A n,n (cid:48) g ii,n (cid:48) − l π ¯ l K,λg ,n = ∞ (cid:88) n (cid:48) =1 B n,n (cid:48) g ,n (cid:48) − l π ¯ l K,λK = − θK, (66)where A n,n (cid:48) is defined in (49) and B n,n (cid:48) = A n,n (cid:48) + 2 . (67)Using ∞ (cid:88) n (cid:48) =1 B n,n (cid:48) = 2 , (68)which follows from (50), we see that K = 0 and g ,n =const, i.e., g ( u ) = π X ( u ) − , (69)is solution with eigenvalue λ = 2 and satisfies ´ π du g ( u ) = 0. It qualitatively reproduces the resultobtained from the boundary layer analysis in Sec. III B 2although the latter gives a Kronecker comb and not aDirac comb (see the discussion in Sec. III A 2). We do notexpect the difference between the Kronecker and Diraccombs to bear a particular physical meaning. In bothcases the singular function g ( u ) originates from theboundary layer near u = pπ , be it a QBL or a CBL[Eq. (61)]. Similarly we find that g ( u ) defined by (69), together with K = π ¯ l l ( θ + 2) , (70)is solution with eigenvalue λ = − θ .Let us now look for the other eigenfunctions, with λ (cid:54) =2 , − θ and therefore K = 0, in the form g ( u ) = π X ( u ) − h ( u ) , (71)where h ( u ) is assumed to be free of Dirac peaks but itsderivative may be discontinuous at u = pπ . From theequations satisfied by g ( u ) and π X ( u ) −
1, we easilyobtained( λ − π X ( u ) −
1] + λh ( u ) = 3 h ( u ) − π ¯ l { δ ∗(cid:48)(cid:48) ( u ) h ( u ) h (cid:48)(cid:48) ( u )[ δ ∗ ( u ) − δ ∗ (0)] + 2 δ ∗(cid:48) ( u ) h (cid:48) ( u ) } . (72)Collecting all terms involving Dirac peaks, we obtain( λ − π X ( u ) = π X ( u ) h ( u )= π X ( u ) h (0) . (73)This equation is satisfied if h (0) = λ − . (74)The terms free of Dirac peaks lead to the equation0 = (4 − λ )( h −
1) + u ( π − u ) h (cid:48)(cid:48) + 2( π − u ) h (cid:48) (75)for u ∈ [0 , π ]. Setting h = 1+ f and introducing x = u/π ,we finally obtain0 = (4 − λ ) f + x (1 − x ) f (cid:48)(cid:48) + 2(1 − x ) f (cid:48) , (76)where the function f ( x ) must satisfy ˆ dx f ( x ) = − . (77)Equation (76) was studied by DLD. The solutions thatare symmetric about x = 1 / ,
1] (thiscondition follows from h ( u ) being even and π periodic)can be expressed in terms of hypergeometric functions.The condition (77) of integrability selects a discrete setof values of λ , for which the hypergeometric function be-comes a polynomial function of finite order, f ( x ) = m (cid:88) m =0 c m x m . (78)For f ( x ) to be solution of (76), we must require c m +1 = c m m ( m + 3) − λ ( m + 1)( m + 2) , (79)whereas c is determined from (77). Imposing c m +1 = 0then gives λ = 2 − m ( m + 3)2 . (80)0 . . . . . . u/π − . − . . . . g ( u ) λ → λ → − λ → − λ → − Figure 6. Functions g ( u ), corresponding to the four largesteigenvalues, obtained from the numerical solution of (66) with n max = 1000 (solid lines). Away from the points u = 0 and u = π , these functions are well approximated by the functions f ( u ) solutions of (76) associated with the eigenvalues 2 / − / − / −
25, with an appropriate normalization (dashed lines).
For m = 0 we obtain λ = 2 and f ( x ) = −
1, i.e., h ( u ) =0, which reproduces the solution (69). The choice m =1, and more generally m odd, must be discarded sincethe corresponding solutions do not satisfy f (0) = f (1).For m = 2, one finds λ = − f ( x ) = − − x + 5 x ) . (81)The condition (74), λ = 2 + h (0) = 3 + f (0), is satisfied.The next solution ( m = 4) corresponds to λ = −
12 and f ( x ) = − − x + 56 x − x + 42 x ) , (82)and satisfies (74). All other solutions can be obtainedsimilarly and are associated with eigenvalues that aremore and more irrelevant as m increases. The negativeeigenvalue spectrum {− , − , − , − , − , − , · · · } is the same as that obtained numerically for the approachto the BG fixed point.In Fig. 6 we show the solution g ( u ) obtained from anumerical solution of (66) with a finite number n max ofcircular harmonics. We find that there is a single posi-tive eigenvalue (in agreement with the analytic results),associated with a function g ( u ) which, as n max increases,is more and more strongly peaked near u = pπ , withhowever a nonzero value away from these two points toensure that ´ π du g ( u ) = 0. This behavior is in quali-tative agreement with (69). The functions g ( u ) associ-ated with negative eigenvalues are also strongly peakednear u = 0 and u = π , and their behavior away fromthese two points is well approximated by the function f ( u ) = g ( u ) − π X ( u ) found analytically above (Fig. 6).The convergence with n max of the eigenvalues to thespectrum (80) is however extremely slow. Even fora relatively large value n max = 100 000 we find that λ max (cid:39) .
75 is still far from its expected converged value . . . . . − n max . . . . . . λ m a x n max3 . . . . / ( a − λ m a x ) Figure 7. Largest eigenvalue λ max vs n max obtained fromthe numerical solution of (66). The continuous (orange) lineshows the logarithmic convergence (83) of λ max with n max . λ max = 2. Our results agree with a logarithmic conver-gence, λ max ( n max ) (cid:39) a − bc + ln( n max ) (83)with a (cid:39) . b (cid:39) .
70 and c (cid:39) .
77, as shown in Fig. 7.The various values of the chaos exponent obtained nu-merically from either ξ ov , δ ,k (0) − δ ∗ (0) or the lin-ear analysis, are shown in Table I. Our analytic result λ max = 2 implies a chaos exponent α = λ max / λ max with n max implies that itis necessary to probe the system at very long length scalesto observe the value α = 1 of the chaos exponent. Anyfinite length L indeed introduces an effective upper cutoff n max ∼ /L on the number of circular harmonics in theFourier series expansion of the function δ ,k ( u ). In par-ticular this means that the critical behavior ξ ov ∼ | (cid:15) | − /α will be observed only if ξ ov is sufficiently large (i.e., | (cid:15) | sufficiently small). Thus, to determine the chaos expo-nent from the numerical solution of the flow equations,one would need both a very small value of (cid:15) and an ex-tremely large n max , which cannot be realized in practice.
4. Comparison with Duemmer and Le Doussal’s work
Equations (28,29) and (65) are identical to those ob-tained by DLD in their study of periodic elastic manifoldspinned by disorder (with the temperature playing the roleof the Luttinger parameter). But our analysis of the lin-earized equation (65) differs in a crucial way: DLD use δ ∗(cid:48)(cid:48) ( u ) = 1 /π ¯ l which contradicts (47) and is not correctsince it violates the condition ´ π du δ ∗(cid:48)(cid:48) ( u ) = 0. As a re-sult, they obtain equation (76) for the function g insteadof f = g − π X . Since none of the solutions of this equa-tion have a vanishing integral over the interval [0 ,
1] theyconclude that the linear analysis of perturbations aboutthe fixed point fails.1To circumvent this difficulty DLD consider a two-dimensional system where the temperature is marginal, θ = 0, and therefore does not flow under RG. When (cid:15) = 0one obtains a line of fixed points indexed by T >
0. Thefunction δ ∗ ( u ) is analytic and exhibit a thermal bound-ary layer (TBL) of width T instead of the cusp, whichmakes the linear analysis about the fixed point free of thedifficulties that arise when the temperature flows towardzero. DLD find that outside the TBL (i.e. for | u | (cid:38) T )the eigenfunction g ( u ) corresponding to the largest eigen-value λ must be chosen among the solutions f ( u ) of (76),whereas g ( u ) = 16 T λ − (cid:104) (cid:0) u T (cid:1) (cid:105) (84)for | u | (cid:46) T . The eigenvalue λ (cid:39) − /T ) (85)converges logarithmically toward 2 when T →
0. Weconclude that the T → θ = 0 agree with the conclusions ofSec. III B 3 obtained in the case where the temperature T (i.e., the Luttinger parameter in our notations) flowsto zero ( θ > T → g ( u ) inside the TBL [Eq. (84)] is given by thesingular function X ( u ). The temperature dependence ofthe eigenvalue in (85) is similar to the dependence of λ with respect to n max in (83). These logarithmic correc-tions are due to the finite length scale introduced by thefinite temperature in the marginal case ( θ = 0) or thefinite number of circular harmonics in our study ( θ > θ > δ ,k ( u ) − δ ii,k ( u ), occurs with eigen-value λ = 2 if u is outside the CBL and 2 + θ if u is insidethe CBL, which gives the chaos exponent α = 2 / (2 + θ ).This latter result disagrees with our conclusions. IV. QUANTUM CHAOS
In this section we consider two copies of the systemsubjected to the same disorder potential but with differ-ent Luttinger parameters, K = K + (cid:15),K = K − (cid:15). (86)The replicated action is then given by S [ { ϕ ia } ] = (cid:88) i,a ˆ x,τ v πK i (cid:26) ( ∂ x ϕ ia ) + ( ∂ τ ϕ ia ) v (cid:27) − (cid:88) a,b,i,j D ˆ x,τ,τ (cid:48) cos[2 ϕ ia ( x, τ ) − ϕ jb ( x, τ (cid:48) )] . (87) In order to implement the FRG, we choose a cutoff func-tion which depends on the copy index, R i,k ( q, iω ) = Z ix (cid:32) q + ω v i,k (cid:33) r (cid:32) q + ω /v i,k k (cid:33) , (88)where Z ix = v/πK i and v i,k is the renormalized velocityof the i th copy.The ansatz for the effective action Γ k [ { φ ia } ] is givenby (22) with Z x and v k replaced by Z ix and v i,k . Theflow equations for K i,k , v i,k and δ ij,k ( u ) = − K i K j v k V (cid:48)(cid:48) ij,k ( u ) (89)are given by ∂ t K i,k = θ i,k K i,k , ∂ t ( K i,k /v i,k ) = 0 ,θ i,k = π δ (cid:48)(cid:48) ii,k (0) ¯ m τ . (90)and ∂ t δ ii,k ( u ) = − δ ii,k ( u ) − l K i,k δ (cid:48)(cid:48) ii,k ( u )+ π ¯ l { δ (cid:48)(cid:48) ii,k ( u )[ δ ii,k ( u ) − δ ii,k (0)] + δ (cid:48) ii,k ( u ) } , (91) ∂ t δ ,k ( u ) = − δ ,k ( u ) − l K ,k + K ,k δ (cid:48)(cid:48) ,k ( u )+ π ¯ l { δ (cid:48)(cid:48) ,k ( u )[ δ ,k ( u ) − δ ,k (0)] + δ (cid:48) ,k ( u ) }− π ¯ l ˆ K k δ (cid:48)(cid:48) ( u ) , (92)where ˆ K k = δ ,k (0) + δ ,k (0)2 − δ ,k (0) . (93)These equations are similar to those discussed in Secs. IIand III. Although the parameter ˆ K Λ vanishes, since δ ij, Λ ( u ) ≡ δ Λ ( u ) is independent of the copy indices i, j ,it takes a nonzero value as soon as k < Λ. This can beseen by noticing that Eqs. (91,92) imply ∂ t ˆ K k (cid:12)(cid:12) k =Λ = 14 ( K − K ) l δ (4)Λ (0) . (94)Therefore all conclusions reached in Sec. III remain validas can be explicitly verified by solving numerically theflow equations. The chaotic behavior now originates inthe difference in the quantum fluctuations of the twocopies of the system. Although they are suppressed inthe long-distance limit, K i,k → k →
0, they selectdifferent ground states in the two copies. This “quan-tum” chaos is analog to the “temperature” chaos in clas-sical disordered system, as shown by the analogy betweenEqs. (91,92,94) and the equations derived in Ref. 17.
V. CONCLUSION
We have investigated the chaotic behavior of the BGphase of a one-dimensional disordered Bose fluid. By2solving numerically the nonperturbative FRG equations,we find that two copies of the system with slightly dif-ferent disorder configurations become statistically uncor-related at large distances. The chaos exponent α canbe obtained from the overlap length ξ ov ∼ | (cid:15) | − /α orthe growth of δ ,k (0) − δ ∗ (0) ∼ e − αt at long RG time | t | , but the convergence with the number n max of cir-cular harmonics used for the disorder correlators δ ij,k ( u )turns out to be logarithmic and therefore extremely slow.From the linear analysis of perturbations about the BGfixed point, we are however able to show analytically that α = 1.Although the chaos exponent is related to the rele-vant RG eigenvalue λ max = 2 α of the linearized flownear the BG fixed point, as for a standard critical point,the peculiar nature of the fixed point makes the situa-tion somewhat unusual. The fixed-point disorder corre-lator δ ∗ ( u ) exhibits cusps at u = pπ and δ ,k ( u ) ap-proaches and departs from its nonanalytic fixed-pointform via a QBL and a CBL, respectively. This hasstrong consequences for the linear analysis of the per-turbations about the fixed point. The eigenfunctions g ,k ( u ) ≡ δ ,k ( u ) − δ ∗ ( u ) = π X ( u ) + f ( u ), solutions of the linearized flow equations, are singular at u = pπ ( f ( u ) is a regular function). Although this could callinto question the linear analysis, the agreement with theresults obtained from the numerical analysis of the flowequations, where the function δ ,k ( u ) remains analyticat all scales k ≥
0, strongly supports its validity, evenif this agreement is obtained for a finite number n max ofcircular harmonics for which the chaos exponent signifi-cantly differs from its converged value.The chaotic behavior of the BG phase can also be in-duced by a modification of quantum fluctuations due toa slight variation of the Luttinger parameter.Finally we note that all these conclusions also apply tothe Mott-glass phase of a disordered Bose fluid inducedby long-range interactions. ACKNOWLEDGMENTS
We thank Gilles Tarjus for a critical reading of themanuscript. T. Giamarchi and H. J. Schulz, “Localization and interac-tion in one-dimensional quantum fluids,” Europhys. Lett. , 1287 (1987). T. Giamarchi and H. J. Schulz, “Anderson localization andinteractions in one-dimensional metals,” Phys. Rev. B ,325 (1988). M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S.Fisher, “Boson localization and the superfluid-insulatortransition,” Phys. Rev. B , 546 (1989). Nicolas Dupuis, “Glassy properties of the Bose-glass phaseof a one-dimensional disordered Bose fluid,” Phys. Rev. E , 030102(R) (2019). Nicolas Dupuis and Romain Daviet, “Bose-glass phase of aone-dimensional disordered bose fluid: Metastable states,quantum tunneling, and droplets,” Phys. Rev. E ,042139 (2020). Nicolas Dupuis, “Is there a mott-glass phase in a one-dimensional disordered quantum fluid with linearly con-fining interactions?” Europhys. Lett. , 56002 (2020). Romain Daviet and Nicolas Dupuis, “Mott-glass phase ofa one-dimensional quantum fluid with long-range interac-tions,” Phys. Rev. Lett. , 235301 (2020). L. Balents, J.-P. Bouchaud, and M. M´ezard, “The LargeScale Energy Landscape of Randomly Pinned Objects,” J.Phys. I , 1007 (1996). D. S. Fisher and D. A. Huse, “Equilibrium behavior of thespin-glass ordered phase,” Phys. Rev. B , 386 (1988). Susan R. McKay, A. Nihat Berker, and Scott Kirk-patrick, “Spin-glass behavior in frustrated ising modelswith chaotic renormalization-group trajectories,” Phys.Rev. Lett. , 767–770 (1982). A. J. Bray and M. A. Moore, “Chaotic Nature of the Spin-Glass Phase,” Phys. Rev. Lett. , 57–60 (1987). D. S. Fisher and D. A. Huse, “Nonequilibrium dynamics of spin glasses,” Phys. Rev. B , 373 (1988). D. S. Fisher and D. A. Huse, “Directed paths in a randompotential,” Phys. Rev. B , 10728 (1991). Yonathan Shapir, “Response of manifolds pinned byquenched impurities to uniform and random perturba-tions,” Phys. Rev. Lett. , 1473–1476 (1991). I. Kondor and A. Vegso, “Sensitivity of spin-glass order totemperature changes,” J, Phys. A: Math. Gen. , L641(1993). Jens Kisker and Heiko Rieger, “Application of a minimum-cost flow algorithm to the three-dimensional gauge-glassmodel with screening,” Phys. Rev. B , R8873–R8876(1998). P. Le Doussal, “Chaos and Residual Correlations in PinnedDisordered Systems,” Phys. Rev. Lett. , 235702 (2006). O. Duemmer and P. Le Doussal, “Chaos in the thermalregime for pinned manifolds via functional RG,” (2007),arXiv:0709.1378 [cond-mat.dis-nn]. G. Lemari´e, “Glassy Properties of Anderson Localization:Pinning, Avalanches, and Chaos,” Phys. Rev. Lett. ,030401 (2019). In the single-copy case, the nonperturbative FRG approachhas also been used to study the random-field Ising model,see Refs. 37–41. T. Giamarchi,
Quantum physics in one dimension (OxfordUniversity Press, Oxford, 2004). F. D. M. Haldane, “Effective Harmonic-Fluid Approachto Low-Energy Properties of One-Dimensional QuantumFluids,” Phys. Rev. Lett. , 1840 (1981). M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, andM. Rigol, “One dimensional bosons: From condensed mat-ter systems to ultracold gases,” Rev. Mod. Phys. , 1405(2011). Juergen Berges, Nikolaos Tetradis, and Christof Wet- terich, “Non-perturbative renormalization flow in quantumfield theory and statistical physics,” Phys. Rep. , 223–386 (2002), arXiv:hep-ph/0005122. P. Kopietz, L. Bartosch, and F. Sch¨utz,
Introduction tothe Functional Renormalization Group (Springer, Berlin,2010). B. Delamotte, “An Introduction to the NonperturbativeRenormalization Group,” in
Renormalization Group andEffective Field Theory Approaches to Many-Body Systems ,Lecture Notes in Physics, Vol. 852, edited by A. Schwenkand J. Polonyi (Springer Berlin Heidelberg, 2012) pp. 49–132. N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M.Pawlowski, M. Tissier, and N. Wschebor, “The nonper-turbative functional renormalization group and its appli-cations,” Physics Reports , arXiv:2006.04853 (2021). C. Wetterich, “Exact evolution equation for the effectivepotential,” Phys. Lett. B , 90 (1993). Ulrich Ellwanger, “Flow equations for n point functionsand bound states,” Z. Phys. C , 503 (1994). T. R. Morris, “The exact renormalization group and ap-proximate solutions,” Int. J. Mod. Phys. A , 2411 (1994). The boundary layer, of width ∼ K k , implies that δ k (0) − δ ∗ (0) ∼ K k to leading order in the limit K k → The π -periodic Kronecker and Dirac combs are defined by X ( K ) u, = (cid:80) ∞ n = −∞ δ ( K ) u,nπ and X ( u ) = (cid:80) ∞ n = −∞ δ ( u − nπ ),respectively ( δ ( K ) u, denotes the Kronecker delta). Near the fixed point, ∂ t K k (cid:39) θK k and ∂ t ˆ K k (cid:39) − α ˆ K k . Note that even if h (cid:48)(cid:48) ( u ) contains Dirac peaks, h (cid:48)(cid:48) ( u )[ δ ∗ ( u ) − δ ∗ (0)] is a regular function since δ ∗ ( u ) − δ ∗ (0) vanishes for u = pπ . Equation (76) corresponds to Eq. (22) of Ref. 18 with (cid:15) = 3and a = 2 λ . When comparing with DLD’s work, we have set the di-mensionality to one (i.e., (cid:15) = 3 in DLD’s notations) in theresults of Ref. 18. G. Tarjus and M. Tissier, “Nonperturbative functionalrenormalization group for random field models and relateddisordered systems. I. Effective average action formalism,”Phys. Rev. B , 024203 (2008). M. Tissier and G. Tarjus, “Nonperturbative functionalrenormalization group for random field models and relateddisordered systems. II. Results for the random field O ( N )model,” Phys. Rev. B , 024204 (2008). M. Tissier and G. Tarjus, “Nonperturbative functionalrenormalization group for random field models and relateddisordered systems. III. Superfield formalism and ground-state dominance,” Phys. Rev. B , 104202 (2012). M. Tissier and G. Tarjus, “Nonperturbative functionalrenormalization group for random field models and relateddisordered systems. IV. Supersymmetry and its sponta-neous breaking,” Phys. Rev. B , 104203 (2012). Gilles Tarjus and Matthieu Tissier, “Random-field Isingand O( N ) models: theoretical description through thefunctional renormalization group,” Eur. Phys. J. B93