Fractional statistics and the butterfly effect

Journal of High Energy Physics, Aug 2016

Fractional statistics and quantum chaos are both phenomena associated with the non-local storage of quantum information. In this article, we point out a connection between the butterfly effect in (1+1)-dimensional rational conformal field theories and fractional statistics in (2+1)-dimensional topologically ordered states. This connection comes from the characterization of the butterfly effect by the out-of-time-order-correlator proposed recently. We show that the late-time behavior of such correlators is determined by universal properties of the rational conformal field theory such as the modular S-matrix and conformal spins. Using the bulk-boundary correspondence between rational conformal field theories and (2+1)-dimensional topologically ordered states, we show that the late time behavior of out-of-time-order-correlators is intrinsically connected with fractional statistics in the topological order. We also propose a quantitative measure of chaos in a rational conformal field theory, which turns out to be determined by the topological entanglement entropy of the corresponding topological order.

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Fractional statistics and the butterfly effect

Received: March Fractional statistics and the butter y e ect Yingfei Gu 0 1 Xiao-Liang Qi 0 1 0 Stanford , CA 94305 , U.S.A 1 Department of Physics, Stanford University Fractional statistics and quantum chaos are both phenomena associated with the non-local storage of quantum information. In this article, we point out a connection between the butter y e ect in (1+1)-dimensional rational conformal eld theories and fractional statistics in (2+1)-dimensional topologically ordered states. This connection comes from the characterization of the butter y e ect by the out-of-time-order-correlator proposed recently. We show that the late-time behavior of such correlators is determined by universal properties of the rational conformal eld theory such as the modular S-matrix and conformal spins. Using the bulk-boundary correspondence between rational conformal eld theories and (2+1)-dimensional topologically ordered states, we show that the late time behavior of out-of-time-order-correlators is intrinsically connected with fractional statistics in the topological order. We also propose a quantitative measure of chaos in a rational conformal eld theory, which turns out to be determined by the topological entanglement entropy of the corresponding topological order. Anyons; Field Theories in Lower Dimensions; Topological Field Theories 1 Introduction 2 Out-of-time-ordered-correlators in rational conformal eld theories 3 4 5 2.1 2.2 2.3 2.4 De nitions and conventions An overview of existing results Rational conformal eld theories Examples The bulk-boundary correspondence Out-of-time-ordered-correlators of random operators Conclusion and discussions A Notations and conventions B The monodromy matrix Mf[a; b] C The residue value r in SU(N)2 WZW models e ect occurs for small perturbation to the initial position x(0) ! x(0) + x(0) if the change j x(t)j / e Ltj x(0)j grows exponentially in time, with variation xx((0t)) can be calculated by the Poisson braket fx(t); p(0)gPB. More generally, to obtain all Lyapunov exponents one should study the Poisson brakets fqi(t); qj (0)gPB with L the Lyapunov exponent. The qi di erent components of coordinates and canonical momenta. The Poisson bracket formula suggests a natural generalization to quantum systems. In Heisenberg picture, quantum chaos can be characterized by the growth of operators of the form i [x(t); p(0)]. More precisely, for a given density matrix of a quantum particle, one should study Tr [x(t); p(0)]2 , which measures the size of the commutator in the state .1 (It should be noted that the square of the commutator should be considered since 1This type of measure of chaos rst appears in the semi-classical treatment of a superconductor by Larkin and Ovchinnikov [1]. { 1 { we are interested in the size of the operator.) Recently, a generalization of such quantities have been studied in many-body systems, where operators x and p are replaced by generic many-body operators. [2{4]. If we take the thermal equilibrium state of the many-body system, the quantum butter y e ect refers to the increase of the thermal expectation value of the \commutator norm square": C(t) := hj[W (t); V (0)]j2i (1.1) where W (t) and V (0) are generic Heisenberg operators at time t and 0. The thermal expectation value h-i is evaluated via the trace: C(t) = Z 1 Tr e H j[W (t); V (0)]j2 . When expanding the \commutator norm square," there are four terms in the function C(t): C(t) = hV y(0)W y(t)W (t)V (0) + V y(0)W y(t)W (t)V (0) | {z \Out-of-time-ordered correlators" } W y(t)V y(0)W (t)V (0) V y(0)W y(t)V (0)W (t)i (1.2) The rst two terms have the time order that appears in the response functions, therefore, are \accessible" [3]. However, the last two terms have a special \out-of-time-order" that is di cult to measure in conventional experiments. In a generic quantum many-body system, we expect the thermal averages of the accessible correlators to approach constant after the thermal time scale ; while the out-of-time-ordered correlators (OTOCs), possesses a nontrivial time dependence: it starts at large value and then decreases. The decrease of OTOCs corresponds to the increase of C(t); therefore, it characterizes the quantum butter y e ect. The OTOC has been extensively discussed in the connection between quantum gravity and quantum chaos, see ref. [2, 3, 5{7]. In those settings, for generic operators W and V , the OTOC stays large until \scrambling time" [8{10] tscr, then it decreases rapidly to 0 and stays at 0 in late time t tscr. Those behaviors are indicated by gravity, and are expected for a strongly chaotic quantum system. On the other hand, also interesting is the measurement of OTOC in \less chaotic" or \non-chaotic" models (e.g. see ref. [ 11, 12 ]) to see how di erent aspects of the model a ect the chaos. In this paper, we measure OTOCs in (1 + 1)-dimensional rational conformal eld theories (RCFTs),2 which are known to be integrable. We focus on the late time regime t tscr and see how the \integrability" stops the late time value from vanishing in RCFTs. Remarkably, our result shows that when time t goes to 1, OTOCs in (1 + 1)-dimensional RCFTs are intrinsically related to the fraction statistics [14, 15] in (2 + 1)-dimensional topological order. Fractional statistics and chaos are both interesting phenomena associated with nonlocal storage of information in strongly interacting quantum many-body systems. Fractional statistics usually arise in two-dimensional topological ordered states such as fractional quantum Hall states (FQH) [16, 17]. Theoretically, one can model the edge of FQH states using rational conformal eld theories that possess the same algebraic structures/topological order as the corresponding FQH states. We use this type of bulk 2For review, see ref. [13]. { 2 { t W (t) V jxi = W (t)V j i (a) W (t) t V jyi = V W (t)j i (b) and jyi. The operator V and W act on a pure state j i (the grey disk) which is a puri cation of the thermal state. We can imagine W (t) as a small perturbation in late time t : if W = Identity is trivial, then state jxi = jyi; if W is a non-trivial perturbation and the system is \chaotic", we expect the \butter y" W causes a big di erence on states jxi and jyi. Therefore, we can use the inner product hyjxi to quantify the butter y e ect. boundary correspondence3 to build the connections between the butter y e ect on the boundary and the fractional statistics in the bulk. In section 2, we present the CFT computation of OTOC in the content of RCFTs, following the technique developed in ref. [18], and show that the OTOC at t = 1 only depends on the modular S-matrix. Section 2 also includes more examples and discussion of small late time values. Section 3 is devoted to an alternative derivation of the result via bulk-boundary correspondence, which shows the non-trivial \topology" behind the OTOC. We also consider the OTOC between \random operators" in section 4, where we show an unexpected connection between the boundary butter y e ect and the bulk topological entanglement entropy. Section 5 contains conclusions and discussions on real condensed matter experiments. Most of the technical details are placed in the appendices. Out-of-time-ordered-correlators in rational conformal eld theories De nitions and conventions Before going to the detailed discussions, we rst declare some general de nitions for OTOC. More speci cally, we are considering thermal expectation value of OTOC in a quantum system with Hilbert space H: f (t) := hW y(t)V y(0)W (t)V (0)i , V and W act on H. Alternatively, one can interpret f (t) as an inner product between two pure states (see gure 1): f (t) = hyjxi; jxi = W (t)V j i; jyi = V W (t)j i (2.1) where j i is a puri cation of the thermal system at temperature T = 1= , e.g., a thermo eld double state [ 19, 20 ] j i = Z 1=2 Pn e En=2jnijni 2 H H. We require all operators act on one side, say H. 3This is the type of correspondence between (1+1)-dimensional RCFTs and (2+1)-dimensional TQFTs, e.g. WZW/CS correspondence. One should avoid confusing with the AdS/CFT correspondence. { 3 { As a measure of the di erence between jxi and jyi, it is convenient to use the normalized (2.2) HJEP08(216)9 value: fe(t) := hyjxi phxjxihyjyi Previous studies [2, 3, 7, 18] of such correlation function focused on the early time < t < tscr, i.e., between the dissipation time and the scrambling time, and found interesting \Lyapunov behavior"4 for those systems that can be holographically described by Einstein gravity. However, for a generic system that does not have a large separation between dissipation time td and scrambling time tscr, the \Lyapunov behavior" is not wellde ned. Instead, one can focus on the later time regime t tscr, which characterized the residue part of the system that \survived" under the butter y e ect. For later convenience, we denote the inner product hxjxi = hV yW y(t)W (t)V i by g(t). In the regime t , such a four point function generically factorizes to g(t) hV yV i hW yW i , which represents a normalization for operators W and V . The same applies to hyjyi. In the following section, we will study the late time behavior of fe(t) in the context of RCFTs. 2.2 An overview of existing results As a starting point of our discussion, we brie y review the calculation of OTOC in (1 + 1)-d CFT in ref. [18]. To setup the CFT computation, it is essential to use complexi ed time tc = t i , where t stands for the Minkowski time, and for the Euclidean time. The strategy is to begin with a pure Euclidean computation with t = 0 and then analytically continue it to the desired Minkowski time t. Such strategy enables us to manipulate the order of operators by tuning the auxiliary small imaginary part = . After the conformal mapping z = exp (2 w= ) ; z = exp (2 w= ), with w = x tc and w = x + tc,5 the thermal expectation value is mapped to the vacuum expectation value. In the end, it is essential to consider the vacuum expectation value of the four point function which has a general decomposition in terms of conformal blocks [18, 21, 22] 1 1 z122hw z324hv z212hw z234hv p;p X gp;pFp ( ) F p ( ) (2.3) where zij = zi zj , and = zz1123zz3244 ; = zz1123zz2344 are cross ratios, gp;p is the pairing coe cient for holomorphic block p and anti-holomorphic block p. It is important to note that the summation is only over conformal families labeled by fp; pg rather than all the primaries and descendants individually. For the parameter regime t that we are interested in, 4Here Lyapunov behavior refers to the functional dependence of f (t) on t at early time: f (t) 1 e Lt + : : : 1 N where N is some big number in the model, and the exponent L is recognized as Lyapunov exponent. Physically, the Lyapunov behavior characterizes how fast the chaos develops in the system. 5We use a di erent sign convention comparing to ref. [18] for later convenience. { 4 { the cross ratio is always small: ; 1. Therefore, we can formally expand Fp( ) according to the powers of : 1 n=0 Fp( ) = X Fp;n hp+n; if NapaNbpb 6= 0 (2.4) termediate channel p. Fp = 0 if NapaNbb either a a or b where hp denotes the dimension of the primary p (and hp + n for descendants, n 1). fFp;ng are coe cients depending on the details of operator W and V . a; b label the conformal families of W; V , correspondingly. (For simplicity we can assume W; V are primaries. We sometimes make the family label explicit by denoting W = W [a] and V = V [b].) The fusion multiplicity Napa counts the dimension of operator product algebra from a; a to inp = 0, i.e., if p is absent in the fusion channels of The behavior of OTOC is determined by the dependence of conformal blocks on cross ratio : for arbitrary four points on complex plane, it is convenient to use a conformal map to move three of them to standard positions: 0; 1 and +1, and leave over with a free parameter . Therefore, the conformal blocks fFp( )g will have three singularities in general: = 0; 1 and 1 (see gure 2a). The same is true for F ( ). When implementing the analytic continuation to Minkowski time t, the path (t) might go around a singular point and leave from the principal sheet to the second sheet. One can check (e.g. see ref. [18]) that among conformal blocks in the OTOC f (t), only the holomorphic winds a topologically nontrivial loop,6 as is shown in gure 2a. In contrast, the winding of or is trivial in the in-timeordered correlator g(t). This is the key di erence that leads to a nontrivial ratio fe(t). 2.3 Rational conformal eld theories For general CFTs, the number of conformal families can be in nite and there is not much general information we can tell. One can instead restrict to certain subclass of CFTs. For example in ref. [18], the authors analyzed the behavior of the Virasoro identity block of a holographic CFT [23{25] and deduced a result about the butter y e ect that is consistent with the holographic analysis.7 In this paper, we will choose to work in another rich subclass of CFTs | diagonal rational conformal eld theories, which have a well-controlled algebraic structure on conformal blocks (for a review of RCFTs, see ref. [13] and reference therein). In this subclass, the four point functions in Euclidean time have a simpler nite sum presentation: 6Technically speaking, to make the statement more precise, one need to place W y and W at x 7To avoid confusion, we clarify that the previous studies [2, 3, 18] were mainly interested in the Lyapunov exponent and scrambling time, which are in the \early time" regime, while the current work will focus on the \late time" regime instead. 1 1 N X z122hw z324hv z212hw z234hv i=1 Fi ( ) F i ( ) (2.5) , s.t. { 5 { 0 (t) 1 1 (a) Monodromy (analytic). M f a j a b a b b := a j b N i=1 = P Mfij a i (b) Monodromy (algebraic). a b b clockwisely and induces a linear transformation on the space of conformal blocks Vaabb ' Vbaab, which de nes the the monodromy matrix. (b) The diagrammatic representation of the monodromy matrix Mf = Mf[a; b]. For xed a; b, Mf[a; b] acts by braiding the two lines a; b. See appendix A for detailed conventions of the diagrammatics. (t) 2 C f0; 1; +1g winds around z = 1 where holomorphic conformal blocks fFig form a nite dimensional vector space: Vaabb. This linear space is parametrized by the cross ratio , and the same applies to the anti8 holomorphic F i and . 9 In this setting, we are able to discuss the e ect of analytic continuation to the second sheet more concretely: according to the general principle of RCFTs [27], a full winding around a singularity induces a linear transformation Mf[a; b] in space Vaabb, known as the monodromy [21]. See gure 2 for its de nition in terms of diagrams. With this de nition, one can compute the matrix element of Mf in terms of the general algebraic data of RCFT, known as the F-matrix and the R-matrix. This computation is well-known in the literature, and we include a summary of it in the appendix B. The monodromy matrix is determined by universal data that only depends on the algebraic structure of RCFTs, independent of the particular state chosen in the family and the details of locations of the operators. Now with Mfij de ned in gure 2, we can rewrite the formula of normalized OTOC more explicitly: fe(t) = i;j=1 N P N i=1 Mfij Fi( )F j ( ) P Fi( )F i( ) (2.6) 1 1 the normalization cancels the unimportant common prefactor z122hw z324hv z212hw z234hv . Nevertheless, the actual functional form of fe(t) is still messy because of the appearance of descendants when one explicitly expand the conformal blocks as power series of . However, the result is simpli ed in the late time t regime, with ; exp( 2 t= ). 8Here we use an index i instead of the intermediate channel label p due to the subtlety that a xed channel p might have dimension NapaNbpb > 1. Later we will ignore this subtlety and use indices i; j to label the intermediate channels as well. Alternatively, one can treat channel labels as labeling a subspace instead of a vector. 9Alternatively, one can formulate conformal blocks as vector bundle over the moduli space [26]. The monodromy is de ned as that of the vector bundle. Such a vector bundle is also equipped with a ber-wise metric for computing physical correlation functions. In the diagonal theory here, the metric is ij. { 6 { First of all, the residue value at t ! 1 is free of complicated coe cients (i.e., those Fi;ns): r[a; b] = lim fe(t) = t!1 jF1(0)j2 Mf[a; b]11jF1(0)j2 = Mf[a; b]11 (2.7) only depends on universal data: (1; 1) component of monodromy matrix Mf[a; b]. Interestingly, the (1; 1) element of monodromy matrix can be expressed solely by the modular S-matrix (see appendix B for a more detailed derivation): where D = 1=s11 is the total quantum dimension, and da = s1a=s11 are the quantum dimensions of individual conformal families. (As a side remark, Mf[a; b]11 is known as monodromy scalar in the literature of anyon interferometry [28, 29], which we will discuss more in later part of the article.) From the equation (2.8), one can immediately see that Abelian theories10 have jrj = 1. In general, one can prove a physically intuitive inequality jrj 1 by noticing jsabj = 1 D c X dcNacb ac b D c 1 X dcNacb = dadb D Therefore, jrj = Djsabj=dadb Nacb is nonzero for only one c, the inequality picks equal sign. 1. When there is only one fusion channel in a b, i.e. when From this somewhat \trivial" inequality, we see an interesting indication: \scrambling",11 which refers to the suppression of r here, only comes from non-Abelian anyons. Physically, the scrambling occurs due to nontrivial interference between di erent fusion channels, which describes the spreading of quantum information more and more non-locally among di erent conformal families. For Abelian anyons, the long-time value of OTOC is only di erent from the short-time value by a phase. Although such a phase is still a topological feature, it does not lead to scrambling since there is no nontrivial unitary transformation in the vector space generated. This observation is closely related to the correspondence between scrambling and fractional statistics, which we will elaborate in next section. Now we return to more general components of matrix Mf, which appears in the form of Mfij hi+n hj+m in the numerator of equation (2.6). n; m 2 Z 0 come from possible descendants. As a function of t, such terms decay exponentially as Mfij exp( 2 (hi + hj + m + n)t= ). Except the (1; 1) element we mentioned before, it is generally unpractical to separate a prefactor from an exponential decaying function. However, we can still make non-trivial prediction on the \spectroscopy" of function fe(t) by knowing whether certain Mfij is non-zero. The exponents from the denominator: 10We borrow the terminology from anyon theory [30, 31] that non-Abelian anyons a; b refer to those conformal families having multiple fusion channels e.g. a b = c + d + : : :, and Abelian theories refer to a theory without non-Abelian anyons. 11Scrambling is de ned as the process where a non-entangled state evolves into an almost maximally entangled state, or correspondingly, the process of a simple local operator evolves into a complicated operator that is supported on almost the whole system. For details see ref. [2, 8{10]. { 7 { (2.8) 2 = 1 + , = , sjk = p2N 1 exp 2jNk 2 i , r[Vj ; Vk] = exp 2jNk 2 i . PiN=1 Fi( )F i( ) is always in form of agonal pairing. However, the numerator: PN i;n = 2hi + n; n 2 Z 0 as a consequence of di i;j=1 Mfij Fi( )F j ( ) contains mixed exponents i;j;n = hi + hj + n, which enriches the spectrum. In particular, the slowest decaying rate comes from pairing between identity block with the smallest scaling dimension hz (assuming hz < 1), if the corresponding matrix element Mf1z (or Mfz1) is non-vanishing. Such elements of the monodromy matrix are related to the \generalized" modular S-matrix sz;ab for one-punctured-torus [30, 32] as Mf1z = Ddsazd;abb . More details are presented in appendix B. 2.4 Examples To make the abstract discussion more transparent, in this section we present three examples: (1) Ising model, see table 1; (2) Compacti ed boson, see table 2; and (3) SU(2) WZW model at level k, see table 3. The Ising model was considered as an example of non-chaotic theory in ref. [18]. Our result on late time residue value is consistent with ref. [18]. Method here has the advantage that it only relies on a small amount of universal data rather than the explicit functional form of conformal blocks or four point functions. The compacti ed boson is an example of Abelian theory. The residue value for this model is a pure phase jrj = 1 since there is no non-Abelian anyon. { 8 { SU(2) WZW model at level k Primaries: V0; V1; : : : Vk; . according to equation (2.7) and (2.8). HJEP08(216)9 SU (2)25 SU (2)100 SU (2)50 SU (2)200 In the example of WZW models, residue value r is strongly oscillating in i and j, so that some channels are more scrambled than others. To understand the overall behavior of the magnitude of r, which describes the strength of \scrambling", we show a probability distribution plot of jr [Vi; Vj ]j for the SU(2)k WZW model at level k = 25; 50; 100; 200 in gure 3. One can see from the distribution that the theory is more chaotic in larger k, where most of the jrj is close to zero. In appendix C, we present the statistics of jrj for a related family of RCFT's, the SU(k) WZW models at level 2. These theories are related to SU(2)k by the level-rank duality [22], but they have a large central charge c = 2(k2 k+21) . We prove there in appendix C that the statistics of jrj in SU(k)2 is identical to that of SU(2)k, which makes them interesting examples of RCFT's with stronger chaos in the large central charge limit. { 9 { j i 2 Ha Ha (a) Fixed time slide. t a t left boundary bulk right boundary (b) Space-time picture. xed time t , the CFT state at two Ha; (b) The spacetime picture for a pair of anyons a; a created and passed through the boundary at xed time t . The state was in identity sector before t : (t < t ) 2 H1 H1, and shifted to sector Ha Ha after time t . 3 The bulk-boundary correspondence As can be seen from the discussion in the previous section, the language of anyons (more precisely, the language of unitary modular tensor category (UMC)) is useful in discussing RCFTs, since they share the same algebraic structure. In this section, we will propose a physical setup to demonstrate that the correspondence betweeen scrambling in RCFTs and anyons in (2 + 1)-dimensions is not a formal connection, but has a intrinsic physical reason, coming from the bulk-boundary correspondence of (2 + 1)-d TOS. Chiral part of RCFT (i.e. the holomorphic sectors, or the anti-holomorphic sectors) can be realized as the edge theory of (2+1)-d chiral topological order [ 13, 35 ], such as fractional quantum Hall (FQH) states [36, 37]. In this bulk-boundary correspondence, a non-chiral RCFT can be viewed as the low energy e ective theory of a strip of (2 + 1)-d chiral topological order, as is shown in gure 4. In such a strip, the edge states on the two boundaries are described by the holomorphic and anti-holomorphic sectors of a RCFT, and the bulk topological order is described by the corresponding UMC. There is a one-to-one correspondence between the labels of primary elds in RCFT and the labels of anyon types in the bulk. When the bulk has no anyon (or has anyons that fuse to identity, i.e., have zero total anyonic charge), the topological sector of the two boundaries must be conjugate of each other, so that the low energy Hilbert space of this system is the same as that of a diagonal RCFT: Htotal = Ha Ha M a (3.1) One can also consider the physical process in the (2+1)-d strip corresponding to acting operators in the RCFT in the spacetime picture, see gure 4b. The insertion of a primary operator of family (a; a) at (x; t ) on the boundary corresponds to creating a pair of anyons (a; a) at an earlier time and pass them through the boundary at spacetime point (x; t ). To compute two point functions, one need to get them back at a later time and annihilate them in the bulk. For four point functions, the procedure is similar. However, we will show in details that there is a nontrivial linking structure in the bulk picture for OTOC. In our two-side WLy V y L L by shifting along the light cones (see equation (3.2). setting, one can split each physical operator into left-moving and right-moving parts, acting on the left and right boundaries of the strip: O(x; t) = OL(x; t)OR(x; t). The advantage of such a decomposition is that we are allowed to move the chiral operators along light cones freely: OL(x; t) = OL(x c; t c) and OR(x; t) = OR(x + c; t c) where c is an arbitrary constant. Therefore, we can use this freedom to map the OTOC to a time-ordered four-point function: f (t) = hW (t)yV yW (t)V i = hWLy (2c; t + 2c)VLy(t + c; t + c)WL(0; t)VL(0; 0) WRy ( 2c; t + 2c)VRy( t c; t + c)WR(0; t)VR(0; 0)i (3.2) with c > 0. After this shift, we can compute the OTOC in time ordered way, as shown in gure 5. From the viewpoint of bulk-boundary correspondence ( gure 6a), the insertions of operators V [b]; V y[b] on two boundaries corresponds to creating a pair of anyons labeled by b, b and passing through the boundary at insertion points of VL; VR, then pull the anyons back at the insertion points of VLy; VRy. Similar discussion applies to operators W [a]; W y[a]. Remarkably, this con guration of operators on the boundaries induce a nontrivial linking of anyon worldlines in the bulk, as is shown in gure 6a. One can further compute the linked world lines in the anyon theory, which gives Dsab. The computation is under the assumption that bulk energy gap of topological quantum matter is much larger than the temperature 1= such that we can ignore the actual dynamics of the bulk particles and only focus on the topological content. Alternatively, one can interpret the bulk-boundary correspondence here by starting with a pair of world-lines for anyons a and b, see gure 6b, and then imagine to move the boundaries inwards. The boundaries will nally intersect with the world lines, which induce insertions of operators on the boundaries. If we start with the linked con guration as shown in gure 6b, we are able to arrange the crossing points such that they correspond to the OTOC in equation (3.2). HJEP08(216)9 (a) (b) HJEP08(216)9 for exterior) are for anyon label b, and red lines are for anyon label a. The insertion point at the boundary is determined by equation (3.2) and gure 5. (b) The interpolation between a completely bulk anyon braiding procedure and a boundary OTOC. Starting with a pair of linked anyon worldlines inside the bulk, one can move both boundaries inwards to intersect with the world-lines. Such world lines will induce insertion of operators in the families a; b. By proper arrangement of initial con gurations of the linked world lines, one obtains the OTOC in sub gure (a). t + c t t WLy WL VL x the insertion of operators V [b]; W [a]:: on two boundaries of a strip of topological quantum matter, we shifts the operator along light cone to reach the position in equation (3.3). (b) shows the corresponding anyon world-lines, red for label a and black for label b, such world-line con guration doesn't link, therefore can be deform to two independent loops. (Notice that WL=R and WLy=R share the same location in gures.) As a comparison, one can run the same procedure for ordinary ordered correlation function g(t), which can be split and shifted to time ordered as well: g(t) = hV yW y(t)W (t)V i = hVLy(t + c; t + c)WLy (0; t)WL(0; t)VL(0; 0) R V y( t c; t + c)WRy (0; t)WR(0; t)VR(0; 0)i (3.3) The bulk picture for g(t) was shown in gure 7b, where we can clearly see that the world lines are unlinked, which therefore, correspond to two independent loops with labels a and b. Such loops together have the amplitude dadb according to theory of anyons. In summary, we have provided a physical interpretation to the correspondence between late-time universal behavior of OTOC in RCFT and fractional statistics of anyons in (2 + 1)-d TOS. The intrinsic reason of this correspondence is the fact that these two seemingly unrelated phenomena | the butter y e ect and fractional statistics | are both consequences of nonlocality in unitary time evolutions. The butter y e ect comes from the propagation of quantum information in the Hilbert space from simple local operators to more and more non-local operators (for related discussion see ref. [ 38 ]) which makes it more and more di cult to reveal the information locally. Similarly, fractional statistics is only possible because anyons are intrinsically non-local. The braiding of anyons, especially that of non-Abelian anyons, lead to a nontrivial and non-local unitary transformation on the Hilbert space, which can be viewed as a special example of scrambling. 4 Out-of-time-ordered-correlators of random operators In the previous sections, we have discussed how OTOC of a pair of operators, each in a pre xed conformal family, depends on the algebraic content of the operators. Indeed, by the bulk-boundary correspondence, we show that OTOC in RCFT is related to the braiding of anyons in (2 + 1)-d. In this section, we would like to investigate how OTOC can be used as a diagnostic of RCFTs. For that purpose, we would like to consider OTOC of a generic pair of operators, rather than selecting particular operators by hand. A natural choice is to consider OTOC between two randomly chosen operators. The key question we need to address is what is a natural random ensemble of operators in an RCFT. To specify the random ensemble, we need a proper probability distribution for conformal families. Since the Hilbert space of RCFT is a direct sum of that of each conformal family (equation (3.1)), the probability of a random vector in the Hilbert space to be in a given conformal family is proportional to the Hilbert space dimension of that family, i.e., pa / dim(Ha Ha). Naively this is not meaningful, since the Hilbert space dimension of each family is in nite. Nevertheless, the ratio of the Hilbert space dimensions of di erent sectors is well-de ned and is determined by the quantum dimension: dim Ha = dim H1 a(0) 1(0) = s11 sa1 = da where a( ) = Tra e2 i (L0 c=24) is the character for sector a. a(0) is the high temperature limit of the character.12 Since only the ratio between di erent sectors matter for the de nition of random ensemble, we can de ne a regularized probability pa = d2=D2. When a we consider a random operator, it is chosen to be in sector (a; a) with probability pa. It is interesting to note that this probability distribution is also fusion invariant, i.e., According to ref. [ 40 ], pab!c = dcNacb=dadb. Therefore, Pa;b papbpab!c = pc, where pab!c is the probability of fusing a and b to total charge c. X papbpab!c = a;b X d2adb2 dcNacb = ! 0i, b( 1= ) is dominated by the vacuum sector, so that a(0) = sa1 1(i1) [39]. The fusion invariance further justi es the probability pa = d2a=D2 as the correct random ensemble. If we consider a random anyon gas in the bulk, and draw two large adjacient regions A and B, we expect the anyon type of each region (de ned by the fusion of all anyons in that region) to be random, while the same should apply to region A [ B. This is the bulk interpretation why a random distribution pa that emerges from ergodic motion of anyons should be fusion invariant. With this probability distribution, we can compute the random average of the residue value r[a; b]: hri := X papbr[a; b] = a;b a;b D X d2d2 a b Dsab = In the second last step, we used the unitarity of S-matrix in the summation. Interestingly, the nal result only depends on the total quantum dimension D, which also appeared as the characterization of topological entanglement entropy = log D, as was proposed in ref. [ 40, 41 ]. Therefore we have related a measure of the butter y e ect, the random operator OTOC, with a measure of the topological order, the topological entropy:13 hri = e 2 (4.4) We would like to provide some further analysis to this formula. Firstly, we discussed earlier that for xed channels, scrambling only occurs for non-Abelian channels, since for Abelian channels a; b, jr[a; b]j = 1. In contrast, for the random operator case, the average hri < 1 even for an Abelian theory. This is because even in an Abelian theory, nontrivial phase interference can occur between di erent conformal families for an operator that is a superposition of di erent families. Physically, even an Abelian fractional statistics requires fractionalization, which in the (2 + 1)-d language means that even an Abelian anyon is a collective excitation of the system which cannot be created by a local operator. Due to such intrinsic nonlocality in the dynamics of the system, the time evolution of a generic operator looks chaotic, although that of a special operator in a single conformal family does not. In this sense, an Abelian theory is an intermediate case between free (boson or fermion) systems and more chaotic (non-Abelian) RCFTs. Secondly, the total quantum dimension is, roughly speaking, the size of operator content in an RCFT. equation (4.4) means that an RCFT with more elds is on average more chaotic. For an RCFT with N conformal families, D 2 = PN 2 a=1 da N , such that hri Moreover, the equal sign can only be reached if and only if the theory is Abelian. In other words, for the same number of conformal families (or anyon types in the (2 + 1)-d language), non-Abelian theories are more chaotic than Abelian theories. For example, in the SU(2)k WZW model, the number of family is N = k + 1 and the total quantum dimension is D = sin( =(k+2)) k3=2 in large k limit. Therefore, hri k 3 in SU(2)k, while in an Abelian theory with the same N = k + 1 one would have hri = (k + 1) 1. This is 13A similar formula appears in ref. [ 38 ], where the averaged OTOC is related to the second Renyi entropy of a certain region in the doubled state that represents the time evolution operator. It is possible that these two formula are related, although the relation is not clear to us yet. FQH a t1 b t2 for left and right \bridges" respectively. The fractional quantum Hall state occupies the region between the two edges (curves with arrow), with the shaded regions depleted. We denote the anyonic charge on the edge and the central island by a and b, respectively. conceptually consistent with our observation in section 2.4 that most channels in SU(2)k are strongly chaotic in the large k limit. 5 Conclusion and discussions In this article, we studied OTOC in the context of RCFTs, and relate its behavior to the universal algebraic data of RCFT, such as the monodromy matrix and the modular S-matrix. Through the bulk-boundary correspondence of (2 + 1)-d TOS, we pointed out a connection between the OTOC in RCFTs and the fractional statistics in the corresponding TOS. We have shown that the OTOC in an RCFT can be mapped to a time-ordered four-point function which corresponds to a physical process of anyon braiding. In other words, our results point out that the two consequences of \emergent nonlocality" in (2+1)-d TOS | chaos on the boundary and fractionalization in the bulk | always accompany each other. Furthermore, our result shows that for a xed channel (meaning xed conformal families in the boundary, or xed anyon types in the bulk), scrambling only occurs for non-Abelian anyons, as a consequence of nontrivial interference between di erent fusion channels. When we consider a pair of random operators rather than operators in a xed conformal family, the average value of OTOC is determined by the total quantum dimension, so that the \average degree of chaos" in an RCFT is directly related to the topological entanglement entropy in the bulk. Besides providing a physical interpretation of the relation between chaos and topological order, the bulk-boundary correspondence we discussed also suggests a potential approach of measuring the OTOC experimentally. In equation (2.7), we show that the residue value r[a; b] of OTOC is determined by the (1; 1) element of monodromy matrix Mf[a; b]11. Interestingly, the same quantity played an essential role in the interferometry of anyons [28, 29], which has been studied extensively in fractional quantum Hall states, both theoretically and experimentally. [42{48] To be more precise, consider a typical interferometer of FQH state with two point contacts [42], as is shown in gure 8. The physical measurable quantity in this setup is the two terminal conductance (with current owing from left to right) jt2j2 + 2 Re(t1t2ei ab Mf[a; b]11). The phase factor exp(i ab) includes contributions from the xx / jt1j2 + Aharonov-Bohm phase and other dynamical phase factors. Therefore we see that the residual value of normalized OTOC r[a; b] = Mf[a; b]11 plays an essential role in the conductance oscillation. A stronger butter y e ect corresponds to a smaller conductance oscillation. In the extreme case when Mf[a; b]11 = 0, no interference can be observed in the conductance. For example such an absence of conductance oscillation has been considered in the lling fraction = 5=2 state as an evidence of non-Abelian statistics [44, 45, 47]. According to our results, the absence of conductance oscillation can also be viewed as a measure of complete scrambling in the corresponding conformal families of the RCFT describing the boundary. In the end, let us make some more speculations about possible generalizations of our results. Chaos is suppressed by emergent conservation laws which constrains the dynamics of the theory. Therefore it is natural to guess that similar results on OTOC can be obtained in non-critical one-dimensional systems with emergent conservation laws, such as integrable models. The integrability of (1 + 1)-d integrable models is described by the Yang-Baxter equation, which has a similar algebraic structure as that in RCFTs and topological order. Therefore, it is tempting to guess that OTOC in one dimensional integrable models might also capture universal algebraic properties of the model, which we leave for future study. Acknowledgments We would like to acknowledge helpful discussions with Meng Cheng, Tian Lan and Daniel A. Roberts. This work is supported by the National Science Foundation through the grant No. DMR-1151786. Upon nishing this work, we become aware of the parallel work of Pawel Caputa, Tokiro Numasawa and Alvaro Veliz-Osorio [49]. We would like to thank Tokiro Numasawa for informing us of their work before posting their paper. A Notations and conventions In this appendix, we will introduce necessary backgrounds for the notations we used in main text, especially the diagrams. Both subjects of RCFT and anyons have be extensively studied and properly summarized in literature, and we will follow the presentation of lecture note by G. Moore and N. Seiberg [13] and the reference therein for RCFTs, and ref. [30] by A. Kitaev for anyon theories. The purpose of this section is to review the diagrammatic conventions for anyon theories, and explain why we are allowed to use them to describe RCFTs. Let us start with general RCFTs. Such theories have simple analytic properties in physical correlation functions: h : : : i X gij Fi( )F j ( ) (A.1) Where M is a nite number, counting the dimension of the space of conformal blocks. gij is the coe cient indicating the paring between holomorphic and anti-holomorphic blocks. Holomorphic conformal blocks form a vector space parametrized by moduli (same to anti-holomorphic blocks, by ). The moduli describes the shape of a two dimensional manifold, together with the locations of elds inserted. An alternative geometrical M i;j=1 2 Vcab c (a) Splitting. y 2 Vacb a b (b) Fusion. formulation due to D. Friedan and S. Shenker [26] describes conformal blocks as a vector bundle over moduli space, and the vector bundle is equipped with a ber-wise metric gij for physical correlation functions. To build a connection to the algebraic theory of anyons, it is essential to nd the building blocks on both sides. In RCFT side, such object is the conformal blocks associated to a 3-punctured sphere, or intuitively, the chiral half of three-point functions.14 More explicitly, we assume the three punctures were created by insertions of holomorphic elds of family a; b; c, and we denote the space of conformal blocks associated to such geometry by Vabc. The dimension of this space is known as the fusion multiplicity Nabc = dim Vabc. Such formalism also contains the notion of dual or anti-particles, a; b; c. Then we can lift multiplicity can be lifted or lowered: e.g., Ncab = dim Vcab = dim Vabc = Nabc. and lower the indices in convention: e.g., Vabc ' Vcab. Accordingly, the indices of fusion In the algebraic theory of anyons, the parallel notion is the fusion and splitting spaces between \superselection sectors" a, b and c (anyon labels). More concretely, vectors in the splitting space Vcab represent the di erent ways of splitting c into a and b, or equivalent classes of local operators that operate the splitting. Analogously, vectors in the fusion space Vacb represent the di erent ways of fusing a and b into c. If 2 Vcab, then its dagger y 2 Vacb, see gure 9. Nacb = dim Vcab counts the dimension of splitting/fusion space. On both sides, such objects can be used to build more complicated spaces: in RCFTs, they are conformal blocks of multiple insertions of operators; in anyons, they are fusion/splitting space of multiple anyons. They also have a set of identical consistent conditions to satisfy, of which the crucial one for the fusion theory is the pentagon equation. Furthermore, both sides have a notion of braiding: in RCFTs the conformal blocks naturally have a dependence of complex coordinates and the braiding of operators is de ned. In anyons, braiding of a and b is an element in Vabba: Rab 2 Vabba : Rab := b a a b (A.2) With the braiding we have more consistent conditions to satisfy, known as the hexagon equations. 14More abstractly and probably more precisely, one can use the notion of chiral vertex operators [32]. In the end the RCFTs naturally requires modular invariance as a physical constraint, which thus leads to the algebraic structure of modular tensor category (MTC). In this paper, we further restrict ourselves to diagonal theories, in which one can choose a proper basis such that the \metric" gij = ij . More explicitly, the physical correlation function has a simpler expression in the diagonal basis: M X i=1 h : : : i = Fi( )F i( ) (A.3) Under this condition, we end up with unitary modular tensor category (UMC) [27]. On HJEP08(216)9 the anyon side, after introducing the braiding, the theory is already physically sensible and known as unitary braided fusion category (UBFC). With extra non-degeneracy condition on braiding, we end up with the same algebraic theory: unitary modular tensor category (UMC). We will restrict ourselves in this paper to discuss those theories described by UMC. Unfortunately, the literature on RCFTs and anyons have di erent conventions of drawing diagrams and denoting matrices. We will choose the conventions of anyon theories. More explicitly, we follow the \arrowless" conventions in ref. [30], i.e., the vertical lines should read from bottom to top, and arrows will only be marked on horizontal lines when necessary. For example, in such conventions, the conformal block in equation (2.4) is drawn as a vector in Vaabb ' Vbaab: a(0) a( ) b(1) p b(1) := Fp( ) hp ; we only write the leading term here for the late time t The monodromy matrix Mf[a; b] In this section, we review the algebraic expression of Mf[a; b] in terms of F-matrices and R-matrices, and also its diagrammatic expression. Relevant discussions can be found in ref. [50] Operator Mf is a linear map: Vbaab ! Vbaab. M f j := j = (RbaRab) 1 j (B.1) a a b a a b b b b Use bases transformation (F-matrix), we can express (RbaRab) 1 operator in terms of R and F -matrices. (RbaRab) 1 j = (RbaRab) 1 X hFbaabi a a b b = X hF aabi b = X hF aabi b k k;i jk jk k RkbaRkab RkbaRkab jk 1 a a b b k a a b b k 1 hF aabiy b ki a a b i b In terms of matrix elements: axioms in both RCFTs [27] and anyons [30]. We should clarify again to avoid confusion that Mf[a; b]ij is in general a matrix itself. The space of the conformal block with intermediate channel label \i" (or j; k) in the equation is in general NaiaNbib dimensional, which could be greater than 1. Therefore, labels i; j; k should be read as labels for space in general. However, the \monodromy" scalar Mf[a; b]11 (denoted as Mab in anyon interferometry literature) is indeed a scalar: Na1aNb1b = 1 by It is also convenient to rewrite matrix RkbaRab 1 in terms of the topological spin a, k which is related to conventional spin sa by a = exp(2 isa) when the latter is de ned. The diagrammatic expression of this equation is easier to memorize. We use the inner product to single out the matrix element Mf[a; b]ij = r , where denominator Tfr iyMf j Tfr iy i Tfr jy j (normalization): Tfr( iy i) = a b = dadb = Tfr( jy j ) iy i (B.2) (B.3) (B.4) (B.5) a a b b 1y C C C C A a a b j b 3 7 7 5 We add back the arrows for horizontal lines to avoid confusion. Therefore, the nal diagrammatic representation for element Mf[a; b]ij is HJEP08(216)9 (B.6) (B.7) (B.8) (B.9) (C.1) In particular, the (1; 1) element has a nice expression in terms of the modular S-matrix: where Lz = L More generally, if one introduce the \generalized" S-matrix [27, 30] Sz 2 Aut (Lz), b Vbbz is the space associated to torus with one puncture of label z, the (1z) or (z1) element also has simpler algebraic expression in terms of the modular data: Mf[a; b]1z = 1 dadb a b z = Dsz;ab ; dadb Mf[a; b]z1 = 1 dadb a b z = Dsz;ab dadb a factor of p normalization: Pb sz;absz;bc = dz ac.) If we take z = 1, sz;ab = sz;ab = sab = sab goes back to the familiar S-matrix. (To avoid confusion, we remark here that sz;ab we de ned via diagram is di erent from ref. [30] by dz due to the di erent conventions for diagram normalization. Our sz;ab has C The residue value r in SU(N)2 WZW models In this section, we study the residue value r in SU(N) WZW models at level 2, which are related to the SU(2) level N models we study in the main text by the level-rank duality. The explicit formula for modular S-matrix of general SU(N )k is complicated. However, due to the simplicity of SU(2)N , it is possible to have a simple formula for SU(N )2, which is given by a level-rank duality on S-matrix (for example, see ref. [22]): s ; = N r k e2 ij jj j=Nks t; t where reduced Young diagram ; 2 SU(N )k, and their transpose t; t 2 SU(k)N , (e.g., see gure 10 as a demonstration). For k = 2, diagram can be conveniently parametrized reduce (b) (a) SU(2)N . (We assume N > 3 here). We can label in SU(N )2 and (b) its transpose t in by x = 3, y = 1 and reduced t by x y = 2. by two integers: N 1 x y 0, which count the number of boxes for column 1 and 2. Its transpose is not generally reduced in SU(2)N , but reduce to a t 2 SU(2)N , which can be parametrized by one integer x y 2 [0; N 1] \ Z. And j j counts the total number of also useful to mention the total number of families in SU(N )k : (kk!+(NN boxes in reduced diagram: j j = x + y. The identity sector corresponds to x = y = 0. It is We can derive the modular S-matrix of SU(N )2 by the explicit formula of the S-matrix of SU(2)N : s(x;y);(x0;y0) = 2 pN (N + 2) e2 i (x+y)(x0+y0) 2N sin (x y + 1)(x0 y0 + 1) N + 2 Here integers x; y; x0; y0 satisfy N 1 x y 0 and N 1 x0 y0 0. An immediate observation is that when N those in SU(2)N : N (N + 1)=2 on a smaller set of real numbers js t; t j in SU(2)N . 1, the number of families in SU(N )2 is much greater than N + 1. Therefore, the norm js ; j = q N2 js t; t j depends We can also derive the jr[ ; ]j of SU(N )2 from the formula C.1: jr[ ; ]j = js t; t jjs11j = jr[ t; t] j js1; t jjs1; t j which is identical to those in the dual theory. In the following we will show explicitly that not only the spectrum of jrj are identical for the dual theories SU(N )2 and SU(2)N , but also the distributions of jrj are identical. jr[i; j]j for two labels i; j 2 SU(2)N , i.e. i; j = 0; 1; 2; : : : ; N has the following expression: jr[i; j]j = sin N+2 sin (i+1)(j+1) N+2 sin (i+1) N+2 sin (j+1) N+2 First of all, it can be directly veri ed that jr[i; j]j = jr[i; N j]j, 8i; j, since sin (i + 1)(j + 1) N + 2 (j + 1) N + 2 = sin 1 (i + 1) = sin (N j + 1)(i + 1) N + 2 Next, we count how many labels in SU(N )2 are mapped to j and N j (we assume j 6= N=2 for now, and comment later) in SU(2)N . In general, j in SU(2)N corresponds to all pairs of integers (x; y) satisfying x y = j; N 1 x y 0. There are (C.2) (C.3) (C.4) (C.5) in total N j pairs. So together with those corresponds to label N j, there are N channels in SU(N )2 that correspond to the pair of channels fj; N jg in SU(2)N . In other words, for generic two labels i; j, there are four identical normed residue values jr[i; j]j = jr[N i; j]j = jr[i; N j]j = jr[N i; N identical residual values in SU(N )2. Therefore each jr[i; j]j in SU(2)N has N 2=4 duplicates j]j in SU(2)N , and there are N 2 in SU(N)2. For the case when j = N SU(N )2, so that the counting also holds. j = N=2 2 Z, there are N j = N=2 labels in The above counting argument is su cient to prove that the probability distributions for jrj in SU(2)N and SU(N )2 are identical. We should comment here that we have only discussed the norm of r. r has a strongly uctuating phase, which can distinguish between di erent 's that have same reduced transpose t 2 SU(2)N . This is consistent with the fact that the two dual theories have di erent average OTOC hri since they have di erent quantum dimensions. 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Yingfei Gu, Xiao-Liang Qi. Fractional statistics and the butterfly effect, Journal of High Energy Physics, 2016, 129, DOI: 10.1007/JHEP08(2016)129