CConstraints on beta functions in field theories
Han Ma (cid:52) , Sung-Sik Lee (cid:52) , †(cid:52) Perimeter Institute for Theoretical Physics,Waterloo, Ontario N2L 2Y5, Canada,and † Department of Physics & Astronomy,McMaster University,1280 Main St. W., Hamilton,Ontario L8S 4M1, Canada
Abstract
The β -functions describe how couplings run under the renormalization group flow in field theories.In general, all couplings allowed by symmetry and locality are generated under the renormalizationgroup flow, and the exact renormalization group flow takes place in the infinite dimensional spaceof couplings. In this paper, we show that the renormalization group flow is highly constrainedso that the β -functions defined in a measure zero subspace of couplings completely determine the β -functions in the entire space of couplings. We provide a quantum renormalization group-basedalgorithm for reconstructing the full β -functions from the β -functions defined in the subspace. Thegeneral prescription is applied to two simple examples. a r X i v : . [ h e p - t h ] O c t CONTENTS
I. Introduction 3II. Main result: the full β -functions are fixed by β -functions in a subspace 6III. Quantum renormalization group 10A. Action-state correspondence 11B. RG flow as quantum evolution 14C. Reconstruction of the Wilsonian RG from the quantum RG 151. Stable IR fixed point as the bulk ground state 162. Scaling operators as local excitations of the bulk theory 163. Mixing matrix 184. Operator product expansion 195. β -functionals 20IV. 0-dimensional Example 22A. Fixed point 23B. Scaling operators and their OPEs 24C. Full β -functions 24V. D -dimensional example 27A. The RG Hamiltonian 28B. Fixed point 32C. Scaling operators 331. Z odd sector 342. Z even sector 36D. Operator Product Expansion 39VI. Conclusion and discussion 40Acknowledgement 41References 41A. RG Hamiltonian 43B. Solving Non-Hermitian Harmonic oscillator 44C. β -functions for the 0-dimensional model in limiting cases 45D. Solution to the Schrodinger equation for the time-dependent harmonic oscillator 46E. Computation of A s,z , ω s,z , Λ s,z and ∆ s,z A s,z ω s,z s,z s,z J ∗ ,x − x (cid:48) I. INTRODUCTION
One of the greatest advances in modern theoretical physics is the invention of the renor-malization group (RG) [1–12]. The idea is to organize a complicated many-body system interms of length scales of constituent degrees of freedom. Thanks to locality that greatly lim-its the way short-distance modes influence long-distance modes, one can understand coarsegrained properties of the system in terms of effective field theories without considering alldegrees of freedom in the system. This opens the door for systematic understandings ofmany physical phenomena that are otherwise too difficult to study theoretically.The central object in RG is the β -function. It describes how an effective theory graduallychanges as the length scale is increased. The renormalization of the couplings for long-wavelength modes is driven by fluctuations of short-wavelength modes, which creates theRG flow in the space of theories. While the β -function contains the full information on thefate of a system in the infrared limit, it is in practice hard to keep track of the exact RGflow. Even if one starts with a relatively simple theory with a small number of couplingsat a short distance scale, all couplings allowed by symmetry and locality are eventuallygenerated at larger distances. In general, one has to deal with the RG flow in the infinitedimensional space of couplings.Therefore, it is desirable to take advantage of constraints that β -functions satisfy if thereis any. In the free field theories, it is easy to see that not all β -functions are independent. Thescaling dimension of a local operator kinematically determines those of composite operators,and the β -function of the latter is fixed by that of the former to the linear order in thecouplings. It is then natural to ask whether such constraints exist for interacting theoriesand, if so, what the general rules are. There are proposals under special circumstances [13,14]. In this paper, we show that β -functions in all field theories are highly constrained : the β -functions defined in a measure zero subspace of couplings completely fix the β -functionsin the entire space of couplings .To uncover the constraints that β -functions satisfy in general field theories, we use thequantum RG[15, 16]. Quantum RG reformulates the Wilsonian RG by projecting the fullRG flow onto a subspace of couplings. The subspace is spanned by couplings for the so-called single-trace operators. Single-trace operators are basic building blocks of generaloperators in that all operators in a given symmetry sector can be written as compositesof single-trace operators. In large N theories, the set of single-trace operators consists ofthe operators that involve one trace of flavor or color indices[17]. However, the notionof single-trace operators can be defined in any field theory once the fundamental degreesof freedom and the symmetry of the theory are set[16]. Although quantum RG does notinclude composites of the single-trace operators (called multi-trace operators) directly, itexactly takes into account their effects by promoting the single-trace couplings to dynamicalvariables. The pattern of fluctuations of the single-trace couplings precisely captures themulti-trace couplings. As a result, the classical Wilsonian RG flow defined in the full spaceof couplings is replaced with a sum over RG paths defined in the subspace of single-tracecouplings in the quantum RG. The β -functions of the Wilsonian RG is then replaced withan action for dynamical single-trace couplings that determines the weight of fluctuating RGpaths.For a D -dimensional field theory, the theory for the dynamical single-trace couplingstakes the form of a ( D + 1)-dimensional theory, where the dynamical couplings depend onthe D -dimensional space and the RG scale. The theory includes dynamical gravity becausethe coupling functions for the single-trace energy-momentum tensor is nothing but a metricthat is promoted to a dynamical variable in quantum RG[16]. For this reason, quantum RGprovides a natural framework for the AdS/CFT correspondence [18–20] in which the extradimension in the bulk is interpreted as the RG scale [21–28] .In the Wilsonian RG, a field theory is represented as a point in the space of all couplings.In quantum RG, a field theory is represented as a wavefunction defined in the subspace ofthe single-trace couplings. The peak position of the wavefunction indicates the value of thesingle-trace coupling. Around the peak, the second and higher moments of the fluctuatingsingle-trace coupling contain the information on the double-trace and higher-trace couplings.The classical flow of couplings in the Wilsonian RG is replaced with a quantum evolutionof the wavefunction in quantum RG. The bulk theory that governs the quantum RG flowis entirely fixed by the β -functions defined in the subspace of single-trace couplings[15, 16]. In order to construct a background independent gravitational theory for fluctuating couplings in quantumRG, one needs to use a coarse graining scheme[29] that does not introduce a fixed background [15, 30] andsatisfy a consistency condition[31, 32]. For our purpose of demonstrating the existence of constraints of β -functions, however, the issue is not crucial. The quantum RG is an exact reformulation of the WilsonianRG in any coarse graining scheme irrespective of whether the scheme is background independent or not. J J -2 -1 0 1 20.00.51.01.5 FIG. 1. RG flow of a toy model considered in Sec. IV. J ( J ) is the single-trace (double-trace)coupling. The β -functions in the J = 0 subspace, which is denoted as the red horizontal line, fixthe full β -functions in the space of J and J . The full β -functions exhibit rich structures thatinclude one stable and one unstable fixed points away from the single-trace subspace, which are fullyencoded in the β -functions within the subspace of J = 0. Since the wavefunction at an RG scale encodes the full information on all couplings at thescale, the bulk theory fully determines the β -functions of all multi-trace couplings. Thisimplies that the full β -functions is fixed by β -functions defined in the subspace of single-trace couplings. A simple example that illustrates the main result of this paper is shown inFig. 1.In this work, we provide a general algorithm for extracting the full β -functions from the β -functions defined in the subspace of single-trace couplings. To be concrete, the algorithmis explicitly executed for toy models, but the same prescription can be applied to anytheory. The algorithm consists of the following steps. First, we construct the bulk theoryfor quantum RG from the β -functions defined in the space of single-trace couplings. Second,we solve the (functional) Schrodinger equation that evolves an initial state fixed by the UVtheory to IR. Finally, we identify the ground state of the quantum RG Hamiltonian as theIR fixed point of the theory, and states with local excitations as the IR fixed point deformedwith local operators. This allows us to extract the full β -functions from the spectrum ofthe quantum RG Hamiltonian. This dictionary in the algorithm is summarized in Tab. I. TABLE I. Dictionary of correspondence between boundary field theory and bulk theory. boundary field theory bulk theory for quantum RG
RG time (logarithmic length scale) extra bulk dimensionsingle-trace coupling (operator) dynamic field (conjugate momentum)Boltzmann weight in the partition function bulk stateRG transformation radial quantum evolutionstable fixed point ground state of the RG Hamiltonianlocal scaling operators local excitationsscaling dimensions spectrum of the RG Hamiltonian
The rest of the paper is organized as follows. In Sec. II, we outline the main idea withminimal technical details. In Sec. III, we review the formalism of quantum RG. Afteridentifying the space of quantum field theories as a vector space in Sec. III A, we derive thebulk RG Hamiltonian from the β -functions defined in the subspace of single-trace couplingsin Sec. III B. From the spectrum and eigenstates of the RG Hamiltonian that generatesthe quantum RG evolution, we extract the stable fixed point, scaling operators and theirscaling dimensions. The general algorithm is presented in Sec. III C. Next, we apply theseprocedures to a 0-dimensional theory in Sec. IV and a D -dimensional field theory in Sec. V,respectively. II. MAIN RESULT: THE FULL β -FUNCTIONS ARE FIXED BY β -FUNCTIONSIN A SUBSPACE Let us consider a field theory in the D -dimensional Euclidean space with the partitionfunction, Z = (cid:90) D φ e − S [ φ ] , (1)where φ ( x ) represents a set of fundamental fields and S is an action. The RG flow is definedin the space of local theories in a given symmetry sector. To describe the RG flow, one firstneeds to coordinatize the space of theories. For this, we divide S into a reference action S and a deformation, S = S [ φ ] + (cid:88) M (cid:90) d D x J M ( x ) O M ( x ) . (2)Here the reference action S sets the origin in the space of theories. { O M ( x ) } represents thecomplete set of local operators allowed by symmetry, and J M ( x ) is the coupling functionthat deforms the reference theory. One infinitesimal step of coarse graining consists ofintegrating out fast modes of the fundamental fields, and rescaling the space and fields.The one cycle of coarse graining puts the theory into the same form as before except forrenormalized sources, Z = (cid:90) D φ e − S [ φ ] − (cid:80) M (cid:82) d D x (cid:16) J M ( x ) − β M [ J ; x ] dz (cid:17) O M ( x ) , (3)where dz is an infinitesimal parameter and β M [ J ; x ] is the beta function for the local operator O M ( x ). β M [ J ; x ] is a function of x and a functional of coupling functions J N ( x ). Inweakly coupled field theories, one may ignore operators whose couplings remain small inthe perturbative series. In general, one has to keep all couplings. Successive applicationsof the coarse graining give rise to the exact Wilsonian renormalization group (RG) flow inthe infinite dimensional space of couplings[6].Alternatively, the RG flow can be projected to a subspace of couplings at the price ofpromoting the deterministic RG flow to a path integration over RG paths (quantum RG)within the subspace[15, 16]. To see this, we define a quantum state from the action bypromoting the Boltzmann weight to a wave function[30] as | S (cid:105) = (cid:90) D φ e − S [ φ ] | φ (cid:105) , (4)where | φ (cid:105) is the basis state whose inner product is given by (cid:104) φ (cid:48) | φ (cid:105) = (cid:81) x δ [ φ (cid:48) ( x ) − φ ( x )].This correspondence between a D -dimensional action and a D -dimensional quantum stateis not the same as the correspondence between a D -dimensional action and the ground statedefined on a ( D − Z = (cid:104) | S (cid:105) , (5)where | (cid:105) = (cid:82) D φ | φ (cid:105) is a state whose wavefunction is 1. It represents the trivial fixedpoint with zero correlation length. In this picture, one infinitesimal step of coarse grainingis generated by a quantum operator inserted between the overlap Z = (cid:104) | e − dzH | S (cid:105) , (6)where H is the RG Hamiltonian that generates the coarse graining transformation thatsatisfies (cid:104) | H = 0[30]. A concrete example of RG Hamiltonian is discussed in Appendix A.Since | (cid:105) is invariant under the evolution generated by H , the partition function remains thesame under the insertion of the operator. Nonetheless, H generates a non-trivial evolutionof | S (cid:105) once it is applied to the right in Eq. (6) . In more generality, one may choose | S ∗ (cid:105) = (cid:82) D φ e − S ∗ | φ (cid:105) with a different fixed point action S ∗ instead of | (cid:105) [30]. In this case, the partition function is written as Z = (cid:104) S ∗ | S (cid:105) , where S = S − S ∗ is the deformationmeasured with respect to S ∗ . In this case, the coarse graining Hamiltonian that satisfies (cid:104) S ∗ | H (cid:48) = 0 isrelated to H through a similarity transformation, H (cid:48) = e S ∗ He − S ∗ . The one-to-one correspondence between states and actions implies that the resultingstate corresponds to a renormalized action, e − dzH | S (cid:105) = | S + δS (cid:105) , (7)where δS = − dz (cid:88) M (cid:90) dxβ M [ J ; x ] O M ( x ) . (8)Successive applications of the coarse graining transformations give rise to a scale dependentquantum state which corresponds to the scale dependent Wilsonian action, e − zH | S (cid:105) = (cid:90) D φ e − S [ φ ] − (cid:80) M (cid:82) d D xJ M ( x,z ) O M ( x ) | φ (cid:105) , (9)where J M ( x, z ) is the renormalized coupling function that satisfies ∂J M ( x, z ) ∂z = − β M [ J ; x ] (10)with the initial condition J M ( x,
0) = J M ( x ). In Eq. (9), J M ( x, z )’s are classical parametersthat keep track of the exact Wilsonian RG flow. However, Eq. (10) is a rather inefficient wayof keeping track of the evolution of quantum state in that the number of classical parametersone needs to keep is far greater than the number of linearly independent quantum states.Once we realize that the space of theories can be viewed as a vector space, a more naturaldescription of the RG flow is to take advantage of the linear superposition principle. Insteadof labeling a quantum state in terms of classical parameters, a quantum state is expressedas a linear superposition of basis states, which as a set is much smaller than { J M ( x ) } . Acomplete set of basis states can be chosen to be | j (cid:105) = (cid:90) D φ e − S [ φ ] − (cid:80) m (cid:82) d D x j m ( x ) O m ( x ) | φ (cid:105) , (11)where {O m ( x ) } is a subset of symmetry-allowed local operators from which all symmetry-allowed local operators can be written as composites, O M ( x ) = O m ( x ) (cid:104) ∂ µ ..∂ µ l O m ( x ) (cid:105) (cid:104) ∂ µ ..∂ µ l O m ( x ) (cid:105) .. (cid:104) ∂ µ ..∂ µ lk − O m k ( x ) (cid:105) . (12)We call this subset of operators single-trace operators because they are the set of operatorsthat involve one trace in large N matrix models. It is straightforward to see that Eq. (11)forms a complete basis, and Eq. (4) with Eq. (2) can be written as [15, 16] | S (cid:105) = (cid:90) D j Ψ J [ j ] | j (cid:105) , (13)where Ψ J [ j ] is a wavefunction defined in the space of the single-trace sources. In the nextsection, we will provide a prescription to find Ψ J [ j ] from a general action. The measure andthe integration path of the sources in Eq. (13) will be carefully defined when we discussexamples in the following sections. The state for a general theory with multi-trace operatorscan be written as a linear superposition of those whose wavefunctions only include single-trace operators. The multi-trace operators that are not explicitly included in the descriptionendows the single-trace couplings with quantum fluctuations. z z + dz | S | S + δSe − Hdz D j Ψ [ j ] | j D j Ψ [ j ] | je − Hdz β M [ J , J , . . . ; x ] β M [ j, , . . . ; x ] FIG. 2. A field theory S at scale z is represented by a quantum state | S (cid:105) on the upper left corner.The vertical arrows represent an infinitesimal step of coarse graining generated by H . The coarsegraining directly applied to | S (cid:105) , denoted as the vertical arrow on the left, maps | S (cid:105) to | S + δS (cid:105) . S + δS is the renormalized action at scale z + dz , and δS encodes the information on the full β -functions.Alternatively, | S (cid:105) is first written as a linear superposition of basis states whose action only includessingle-trace deformations through the horizontal arrow that points to the right. The result of thecoarse graining applied to that state, denoted as the vertical arrow on the right, only depends onthe β functions defined in the space of single-trace couplings. The resulting coarse grained state onthe lower right corner is finally mapped back to a renormalized action on the lower left corner. Thecommutativity of the diagram implies the full β -functions are fixed by the β -functions defined in thesubspace of single-trace couplings. Because H is a linear operator, the RG flow in Eq. (7) is entirely fixed by how the coarsegraining operator acts on the basis states, e − dzH | S (cid:105) = (cid:90) D j Ψ J [ j ] e − dzH | j (cid:105) . (14)To figure out the resulting state, we simply use the expression in Eq. (9) by turning off allmulti-trace sources. From Eq. (11), we obtain e − dzH | j (cid:105) = (cid:90) D φ e − S [ φ ] − (cid:80) n (cid:82) d D xj n ( x ) O m ( x ) − dz (cid:80) M (cid:82) d D xβ M [ j, ,.. ; x ] O M ( x ) | φ (cid:105) . (15)0Here the beta functions are expressed as β M [ J ; x ] = β M [ j, j [2] , .. ; x ], where j is the single-trace coupling and j [ k ] ’s with k ≥ k -trace operators which arecomposites of k single-trace operators. β M [ j, , .. ; x ] is the beta function defined in the sub-space of single-trace sources. Because {| j (cid:105)} is complete, Eq. (15) can in turn be expressedas a linear superposition of | j (cid:105) as e − dzH | j (cid:105) = (cid:90) D j (cid:48) Φ dz (cid:2) j ; j (cid:48) (cid:3) | j (cid:48) (cid:105) , (16)where Φ dz [ j ; j (cid:48) ] is a propagator of the RG transformation that is determined from β M [ j, , .. ; x ].This allows us to write the resulting state after a coarse graining transformation as e − dzH | S (cid:105) = (cid:90) D j (cid:48) Ψ (cid:48)(cid:48) (cid:2) j (cid:48) (cid:3) | j (cid:48) (cid:105) , (17)where Ψ (cid:48)(cid:48) (cid:2) j (cid:48) (cid:3) = (cid:90) D j Ψ J [ j ] Φ dz (cid:2) j ; j (cid:48) (cid:3) . (18)In the end, the RG transformation leads to the evolution of wave function, Ψ (cid:48)(cid:48) = e −H dz Ψ,where H is an induced coarse graining operator defined by H Ψ[ j (cid:48) ] = − dz (cid:16)(cid:82) D j Ψ [ j ] Φ dz [ j ; j (cid:48) ] − Ψ[ j (cid:48) ] (cid:17) . By equating Eq. (14) with Eq. (7), we obtain e − dzH | S (cid:105) = (cid:90) D φ e − S (cid:20)(cid:90) D j (cid:48) ( x )Ψ (cid:48)(cid:48) (cid:2) j (cid:48) (cid:3) e − (cid:80) n (cid:82) d D xj (cid:48) n ( x ) O m ( x ) (cid:21) | φ (cid:105) = (cid:90) D φe − S − (cid:80) M (cid:82) dx (cid:16) J M ( x ) − dzβ M [ J ; x ] (cid:17) O M ( x ) | φ (cid:105) . (19)This shows that (cid:90) D j (cid:48) D j Ψ J [ j ] Φ dz (cid:2) j ; j (cid:48) (cid:3) e − (cid:80) n (cid:82) d D xj (cid:48) n ( x ) O m ( x ) = e − (cid:80) M (cid:82) dx (cid:16) J M ( x ) − dzβ M [ J ; x ] (cid:17) O M ( x ) . (20)Eq. (20) is the main result of the paper. On the left hand side of Eq. (20), Ψ J [ j ] isfixed by the theory at scale z through Eq. (13), and Φ dz [ j ; j (cid:48) ] is entirely determined from β M [ j, , .. ; x ] through Eqs. (15) and (16). This, in turn, fixes the full beta functions β M [ J ; x ]through Eq. (20). Therefore, the beta functions defined in the subspace of the single-traceoperators completely fix the full beta functions away from the subspace.
This is illustratedin Fig. 2.
III. QUANTUM RENORMALIZATION GROUP
In this section, we further illustrate the main idea using a simple example where thereis only one type of single-trace operator. It is straightforward to generalize the discussionin this section to cases with multiple single-trace operators.1
A. Action-state correspondence
To be concrete, we consider a partition function given by Z [ J , J , . . . ] = (cid:104) | S J ,J ,... (cid:105) , (21)where | S J ,J ,... (cid:105) = (cid:90) D φ e − S − (cid:82) d D x (cid:16) (cid:80) n J n ( x ) O n [ φ ( x )]+ ... (cid:17) | φ (cid:105) . (22)Here S is the reference action. O ( x ) is a real and local single-trace operator. O n ( x ) with n > O with a positive coupling.We can remove the multi-trace operator in the action by using an identity, f ( O ( y )) = (cid:90) R D j (cid:48) D p (cid:48) e − i (cid:82) d D x j (cid:48) ( x )[ p (cid:48) ( x ) −O ( x )] f ( p (cid:48) ( y ))= (cid:90) I D j D p e − i (cid:82) d D x j ( x )[ p ( x ) − i O ( x )] f ( − ip ( y )) , (23)for any f ( x ). The integration of j (cid:48) ( x ) and p (cid:48) ( x ) are defined along the real axes. In thesecond line, we define j ( x ) = − ij (cid:48) ( x ) and p ( x ) = ip (cid:48) ( x ) so that the integration for j ( x ) and p ( x ) are defined along the imaginary axes . The path of the integration variables is denotedby the subscripts R (real) and I (imaginary). Then, Eq. (22) can be rewritten as | S J ,J ,... (cid:105) = (cid:90) D φ e − S (cid:90) I D j D p e − i (cid:82) d D xj ( x )[ p ( x ) − i O ( x )] e − (cid:82) d D x (cid:16) (cid:80) n J n ( x )[ − ip ( x )] n + ... (cid:17) | φ (cid:105) = (cid:90) I D j D p e − (cid:82) d D x (cid:16) ij ( x ) p ( x )+ (cid:80) n ( − i ) n J n ( x ) p n ( x )+ ... (cid:17) (cid:20)(cid:90) D φ e − S − (cid:82) dxj ( x ) O ( x ) | φ (cid:105) (cid:21) . (24)This implies that the state can be written as a linear superposition of the basis states as | S J ,J ,... (cid:105) = (cid:90) I D j Ψ J ,J ,... [ j ] | j (cid:105) , (25)where | j (cid:105) = (cid:90) D φ | φ (cid:105) T φ,j (26) This Wick’s rotation has the advantage that the source for O is simply represented by j ( x ). T φ,j = e − S − (cid:82) d D xj ( x ) O ( x ) , (27)and Ψ J ,J ,... [ j ] = (cid:90) I D p e − (cid:82) d D x (cid:16) ij ( x ) p ( x )+ (cid:80) n ( − i ) n J n ( x ) p n ( x )+ ... (cid:17) (28)is the wavefunction of the dynamical single-trace source. The integration over p in Eq.(28) is convergent for deformations that are bounded from below. This allows us to writethe original partition function in terms of partition functions that involve only single-tracedeformations, Z [ J , J , . . . ] = (cid:82) I D j Ψ J ,J ,... [ j ] Z [ j, , . . . ] with (cid:104) | j (cid:105) = Z [ j, , . . . ].Let us first consider a simple example with an ultra-local deformation (no derivativeterms) with J n> = 0. In this case, the wave function is Gaussian,Ψ J ,J [ j ] = (cid:90) I D p e − (cid:82) d D x (cid:16) ijp − iJ p − J p (cid:17) = (cid:34)(cid:89) x (cid:114) πJ ( x ) (cid:35) e (cid:82) d D x [ j ( x ) − J x )]24 J x ) . (29)We note that j in Eq. (29) is to be integrated along the imaginary axis, and the wavefunctionis normalizable if J >
0. This shows how both single-trace and multi-trace couplings areencoded in the wavefunction : (cid:104) Ψ J ,J | j ( x ) | Ψ J ,J (cid:105) = J ( x ) , (cid:104) Ψ J ,J | j ( x ) | Ψ J ,J (cid:105) − (cid:104) Ψ J ,J | j ( x ) | Ψ J ,J (cid:105) = − J ( x ) . (30)The expectation value of j ( x ) gives the single-trace coupling, and the second cumulant givesthe double-trace coupling. The second cumulant is negative because j ( x ) fluctuates alongthe imaginary axis.If double-trace operators have a support over a finite region, the action becomes S (cid:48) = S + (cid:90) d D x (cid:16) J ( x ) O ( x ) + (cid:90) d D x (cid:48) J ( x − x (cid:48) ) O ( x ) O ( x (cid:48) ) (cid:17) , (31)where J ( x − x (cid:48) ) is the source for the bi-local double-trace operator. The correspondingwave function is written asΨ J ,J [ j ] = (cid:90) I D p ( x ) e − (cid:82) d D x (cid:16) ij ( x ) p ( x ) − iJ ( x ) p ( x ) − (cid:82) d D x (cid:48) J ( x − x (cid:48) ) p ( x ) p ( x (cid:48) ) (cid:17) = (cid:113) det (cid:2) πJ − ( x − x (cid:48) ) (cid:3) e (cid:82) d D x (cid:82) d D x (cid:48) [ j ( x ) − J ( x )] J − ( x − x (cid:48) )[ j ( x (cid:48) ) − J ( x (cid:48) )] . (32)The non-zero correlation length for j gives (cid:104) j ( x ) j ( x (cid:48) ) (cid:105) − (cid:104) j ( x ) (cid:105)(cid:104) j ( x (cid:48) ) (cid:105) = − J ( x − x (cid:48) ) . (33)3This shows that the correlation between fluctuating single-trace sources encodes the in-formation on how the source for the bi-local double-trace operator decays in space. Ingeneral, all multi-trace couplings can be extracted from higher order cumulants. It is notedthat the single-trace coupling has non-trivial quantum fluctuations only in the presence ofmulti-trace couplings Moreover, Eq. (22) can be written as | S J ,J ,... (cid:105) = (cid:90) D φ W φ [ J , J , . . . ] | φ (cid:105) , (34)where W φ [ J , J , . . . ] = (cid:90) I D j T φ,j Ψ J ,J ,... [ j ] . (35)We denote the vector spaces formed by {W φ } and { Ψ J [ j ] } as W and V , respectively. W is the space of Boltzmann weights within a given symmetry sector with inner product,( W , W (cid:48) ) = (cid:82) D φ W ∗ φ W (cid:48) φ . V is the space of wavefunctions of the single-trace sources withinner product, (Ψ , Ψ (cid:48) ) = (cid:82) D j D j (cid:48) Ψ ∗ [ j ] (cid:104) j | j (cid:48) (cid:105) Ψ (cid:48) [ j (cid:48) ]. Eq. (35) provides a bijective map, T : V → W . Accordingly, for every linear operator ˆ A φ that acts on W , there exists a linearoperator ˆ A j that acts on V such that( ˆ A φ W ) φ [ J , J , . . . ] = (cid:90) I D j ˆ T φ,j ( ˆ A j Ψ) J ,J ,... [ j ] . (36)In order to find the correspondence between ˆ A φ and ˆ A j , we consider the detailed form ofΨ in Eq. (28). ˆ A j , generated by δδj and j , gives the relations δδj ( x ) Ψ J ,... [ j ] = (cid:90) I D p ( − i ) p ( x ) e − (cid:82) d D x (cid:16) ij ( x ) p ( x )+ (cid:80) n ( − i ) n J n ( x ) p n ( x ) (cid:17) ,j ( x )Ψ J ,... [ j ] = (cid:90) I D p (cid:16) − i δδp ( x ) e − (cid:82) d D x (cid:80) n ( − i ) n J n ( x ) p n ( x ) (cid:17) e − i (cid:82) d D xj ( x ) p ( x ) . (37)Since p ( x ) is already identified as i O ( x ) in Eq. (24), j ( x ) and δδj ( x ) acting on V correspondto following operator acting on W . δδj ( x ) ⇔ O ( x ) ,j ( x ) ⇔ − e − S δδ O ( x ) e S . (38) In the context of the holographic renormalization group, this amounts to the fact that integrating outbulk degrees of freedom generates multi-trace operators at a new UV boundary [26, 33]. We can explicitly derive this correspondence as (cid:90) I D j (cid:16) δδj Ψ J ,J ,... [ j ] (cid:17) | j (cid:105) = − (cid:90) I D j Ψ J ,J ,... [ j ] δδj | j (cid:105) = − (cid:90) I D j Ψ J ,J ,... [ j ] (cid:90) D φ δδj e − S − (cid:82) d D xj ( x ) O ( x ) | φ (cid:105) = (cid:90) D φ ( x ) (cid:16) OW φ (cid:17) | φ (cid:105) , (cid:90) I D j (cid:16) j ( x )Ψ J ,J ,... [ j ] (cid:17) | j (cid:105) = (cid:90) I D j (cid:90) D φ Ψ J ,J ,... [ j ] j ( x ) e − S − (cid:82) d D xj ( x ) O ( x ) | φ (cid:105) = (cid:90) D φ (cid:16) − e − S δδ O e S W φ (cid:17) [ J , J , . . . ] | φ (cid:105) . A j (cid:104) j, δδj (cid:105) ⇔ ˆ A φ (cid:2) − e − S δδ O e S , O (cid:3) . B. RG flow as quantum evolution
Identifying the space of theories as a vector space naturally leads to the quantum RG[16].We first consider the field theory in Eq. (21) defined in a finite box with the linear systemsize 1 /λ , where λ corresponds to an IR cutoff energy scale. From Eq. (24), the partitionfunction is equivalent to Z λ [ J , J , . . . ] = (cid:90) I D j (0) Ψ J ,J ,... (cid:104) j (0) (cid:105) Z λ (cid:104) j (0) , , . . . (cid:105) , (39)where Ψ J ,J ,... [ j (0) ] is the wavefunction defined in Eq. (28), and Z λ (cid:2) j (0) , , . . . (cid:3) = (cid:104) | j (0) (cid:105) = (cid:82) D φ e − S − (cid:82) /λ d D xj (0) ( x ) O ( x ) . The subscript λ of Z keeps track of the IR cutoff. Next weperform a coarse graining transformation on Z λ (cid:2) j (0) , , . . . (cid:3) : integrating out high-energymodes of φ which reduces the UV cutoff from Λ to Λ (cid:48) = b Λ with b = e − dz <
1. In general,not only the single-trace source is renormalized but also multi-trace operators are generated.Under rescaling of the field and the space, φ (cid:48) ( x (cid:48) ) = b − ∆ φ φ ( x ) with x (cid:48) = bx , Λ (cid:48) is broughtback to Λ, but the system size decreases as 1 /λ (cid:48) = b/λ . The resulting partition functionbecomes Z λ (cid:104) j (0) , , . . . (cid:105) = (cid:90) D φ e − S [ φ ] e − (cid:82) b/λ d D x (cid:48) (cid:110) j (0) ( x (cid:48) ) O ( x (cid:48) ) − dz (cid:80) n ≥ β n [ j (0) , ,... ] O n ( x (cid:48) ) (cid:111) . (40)The change of the coupling for O n ( x ) with increasing length scale is given by − β n dz . − β dz corresponds to the free energy contributed from the high-energy modes that are integratedout in the infinitesimal coarse graining step. In general, the β -functions depend on all thecouplings J n . However, what enters in Eq. (40) are the beta functions in the subspace ofsingle-trace couplings only. From Eq. (24), the partition function in Eq. (40) can be writtenas Z λ (cid:104) j (0) , , ... (cid:105) = (cid:90) I D j (1) ( x ) D p (1) ( x ) e − (cid:82) b/λ d D x (cid:20) ip (1) (cid:16) j (1) − j (0) (cid:17) − dz (cid:80) n β n dz ( − ip (1) ) n (cid:21) Z b − λ (cid:104) j (1) , , . . . (cid:105) . (41)After M steps of coarse graining, we take the dz → M dz = z ∗ fixed. Thisresults in Z λ [ J , J , . . . ] = (cid:90) I D j ( x, z ) D p ( x, z )Ψ J ,J ,... [ j (0)] e − (cid:82) z ∗ dzL [ j,p,z ] Z e z ∗ λ [ j ( z ∗ ) , , . . . ] , (42) Here we omit an additional subscript λ/b in S to avoid clutter in notation. z is the extra direction in the bulk that labels the logarithmic length scale. The bulkLagrangian is given by L [ j, p, z ] = (cid:90) e − z /λ d D x (cid:16) ip ( x, z ) ∂ z j ( x, z ) − (cid:88) n ≥ β n [ j ( z ) , , .. ; x ] ( − ip ( x, z )) n (cid:17) . (43)In Eq. (42), the RG flow of the D -dimensional field theory is replaced with a ( D + 1)-dimensional path integration of the dynamical single-trace sources[15, 16]. The fluctuationsof the RG paths encode the information of the multi-trace operators. The sum over allpossible RG paths within the subspace of the single-trace sources is weighted with the ( D +1)-dimensional bulk action. Equivalently, the RG flow is described by the quantum evolutionof the wavefunction of the single-trace source. In the Hamiltonian picture, j ( x ) and p ( x ) arecanonical conjugate operators that satisfy the commutation relation [ j ( x ) , p ( x (cid:48) )] = − iδ ( x − x (cid:48) ). The RG evolution can be viewed as an imaginary time evolution, R ( z + dz, z ) = e −H λ [ j,p,z ] dz generated by the bulk RG Hamiltonian, H λ [ j, p, z ] = − (cid:88) n ≥ (cid:90) e − z /λ d D x β n [ j, , .. ; x ] ( − ip ( x )) n , (44)where the superscript λ represents the IR cutoff scale associated with the finite system size. H λ depends on z explicitly through λ as the system size decreases with increasing z . Inthe thermodynamic limit ( λ = 0), H is independent of z . Being an operator that actson wavefunctions defined in the space of single-trace sources, H λ generates the quantumevolution of the state associated with the RG flow. The RG Hamiltonian is fixed by the β -functions within the subspace of the single-trace sources only. C. Reconstruction of the Wilsonian RG from the quantum RG
In this section, we explain how the full β -functions of a field theory can be reconstructedfrom the quantum evolution with the RG Hamiltonian, although the latter is fixed by the β -functions within the subspace of the single-trace couplings. Suppose that there exists aunique IR fixed point in the thermodynamic limit. The fixed point action is invariant underthe RG transformation, and the corresponding quantum state should be an eigenstate of H λ at λ = 0. Furthermore, the stable IR fixed point must correspond to the ground state of H because generic initial state is always projected onto it in the large z limit. More generally,one can consider excited states of H . States of particular interest are eigenstates thatsupport excitations local in space. Those states correspond to the IR fixed point perturbedwith local operators with definite scaling dimensions. These scaling dimensions are givenby the energy differences between the excited states and the ground state. In the rest ofthis section, we establish the correspondence between the ground state (excited states) andthe stable fixed point (the stable fixed point with operator insertions).6
1. Stable IR fixed point as the bulk ground state
We begin with the discussion of the ground state. The ground state of H satisfies H ψ [ j ] = E ψ [ j ] (45)with the lowest eigenvalue. The partition function for the the IR fixed point is given by Z ∗ [ J ∗ , J ∗ , . . . ] = (cid:82) I D j ψ [ j ] (cid:82) D φ e − S − (cid:82) d D xj ( x ) O ( x ) , (46)where O is the single-trace operator. To extract the multi-trace couplings at the fixed point,we use the cumulant expansion (cid:104) e − Ω (cid:105) = e −(cid:104) Ω (cid:105) + ( (cid:104) Ω (cid:105)−(cid:104) Ω (cid:105) )+ ... to rewrite Eq. (46) as Z ∗ [ J ∗ , J ∗ , . . . ] = (cid:90) D φ e − S (cid:104) e − (cid:82) d D xj ( x ) O ( x ) (cid:105) ψ = (cid:90) D φ e − S e − (cid:82) d D x (cid:104) j ( x ) (cid:105) ψ O ( x )+ (cid:82) d D xd D y ( (cid:104) j ( x ) j ( y ) (cid:105) ψ −(cid:104) j ( x ) (cid:105) ψ (cid:104) j ( y ) (cid:105) ψ ) O ( x ) O ( y )+ ... , (47)where (cid:104) F [ j ] (cid:105) ψ ≡ (cid:82) I D j ψ [ j ] F [ j ]. Identifying this as the fixed point action, Z ∗ [ J ∗ , J ∗ , . . . ] = (cid:82) D φ e − S − (cid:82) d D xJ ∗ ( x ) O ( x ) − (cid:82) d D xd D yJ ∗ ( x − y ) O ( x ) O ( y ) − .. , (48)we obtain the couplings at the fixed point, J ∗ ( x ) = (cid:104) j ( x ) (cid:105) ψ J ∗ ( x − y ) = − (cid:16) (cid:104) j ( x ) j ( y ) (cid:105) ψ − (cid:104) j ( x ) (cid:105) ψ (cid:104) j ( y ) (cid:105) ψ (cid:17) . (49)Sources for higher-trace operators at the fixed point are given by the higher cumulants.
2. Scaling operators as local excitations of the bulk theory
Next, let us study excited states of H . We start by considering states with local exci-tations. The n -th excited state that supports a local excitation at x satisfies e −H z ψ n,x = e −E n z ψ n,e − z x , (50)where E n is the n -th eigenvalue. For general x , this is not a genuine eigenvalue equationbecause the dilatation generator in the RG Hamiltonian preserves only one point in spacewhich is chosen to be x = 0 here. A local operator inserted at x is transported to e − z x underthe RG flow. Only the states that support local excitations at x = 0 can remain invariantunder the RG evolution. { ψ n, } forms the complete basis of states that support local7excitations at x = 0. Generic excited states can be obtained by applying local operators,denoted as ˆ A n,x,j , to the ground state as ψ n,x [ j ] = ( ˆ A n,x,j ψ ) [ j ] . (51)ˆ A n,x,j , which consists of j ( x ) and δδj ( x ) , creates local excitations at x , where ˆ A ,x,j = . Fromthe correspondence in Eq. (36), ˆ A n,x,j is dual to an operator that acts on the Boltzmannweight, (cid:90) I D j ( ˆ A n,x,j ψ ) [ j ] (cid:90) D φ e − S − (cid:82) d D xj ( x ) O ( x ) = (cid:90) D φ ( ˆ A n,x,φ W ) φ [ J ∗ , J ∗ , . . . ] , (52)where ˆ A n,x,φ consists of O ( x ) and δδ O ( x ) . ˆ A n,x,j ( ˆ A n,x,φ ) is the representation of the operatorin space V ( W ). Henceforth, we use ˆ A n,x to denote the operator itself when there is noneed to specify its representation.We now verify that excited states indeed correspond to the fixed point perturbed withlocal operators that have definite scaling dimensions. For this, we consider the IR fixedpoint theory with a small perturbation added at the origin, e (cid:15) n, ˆ A n, | S J ∗ ,J ∗ ,... (cid:105) = (cid:90) D φ ( e (cid:15) n, ˆ A n, ,φ W ) φ [ J ∗ , J ∗ , . . . ] | φ (cid:105) , (53)where (cid:15) n, is an infinitesimal parameter. The RG Hamiltonian generates an evolution ofthis state as e − Hdz (cid:104) e (cid:15) n, ˆ A n, | S J ∗ ,J ∗ ,... (cid:105) (cid:105) = (cid:90) I D j e −H dz (cid:16) ψ [ j ] + (cid:15) n, ψ n, [ j ] (cid:17) (cid:90) D φ e − S − (cid:82) d D xj ( x ) O ( x ) | φ (cid:105) = e −E dz e (cid:15) n, e − ( E n −E dz ˆ A n, | S J ∗ ,J ∗ ,... (cid:105) (54)to the linear order in (cid:15) n, . This implies that the infinitesimal source evolves as d(cid:15) n, dz = − ( E n − E ) (cid:15) n, (55)under the RG flow, and ˆ A n, ,φ is a scaling operators with the scaling dimensions∆ n = E n − E . (56)The spectrum of H encodes the information of all scaling operators and their scalingdimensions. ˆ A n,x ’s create local excitations with definite scaling dimensions, and are calledscaling operators. On the other hand, O M ( x )’s represent general multi-trace operators interms of which the UV action is written, and are called UV operators. In general, thescaling operators can be written as linear superpositions of UV operators :ˆ A n,x,φ = (cid:88) M g n,M O M ( x ) . (57)8The inverse of Eq. (57) can be written as O M ( x ) = (cid:80) n ( g − ) M,n ˆ A n,x,φ .From the local scaling operators, one can generate translationally invariant eigenstatesby turning on the sources uniformly in space. For example, ψ n [ j ] = (cid:90) d D x ψ n,x [ j ] (58)is an eigenstate with energy ∆ n − D . The shift in the energy by − D is from the spatialdilatation included in H , e −H dz ψ n [ j ] = e − ∆ n dz (cid:90) d D x ψ n,e − dz x [ j ] = e − (∆ n − D ) dz (cid:90) d D ˜ x ψ n, ˜ x [ j ] , (59)where x = e dz ˜ x is used. If ∆ n > D for all n , the fixed point is stable as all deformations withspatially uniform sources are irrelevant. Local excitations with energy gap less than (equalto) D correspond to relevant (marginal) operators. Throughout this paper, we assume thatlocal excitations of the RG Hamiltonian have a non-zero gap (∆ n >
3. Mixing matrix
According to Eq. (57), g n,M encodes the relation between scaling operators, ˆ A n,x,φ andUV operators, O M ( x ). By writing (cid:88) n,x (cid:15) n,x ˆ A n,x,φ = (cid:88) M,x (cid:15)
UVM ( x ) O M ( x ) (60)and using Eq. (57), we obtain the relation between the sources for the scaling operatorsand the UV operators, (cid:15) UVM ( x ) = (cid:88) n (cid:15) n,x g n,M , (cid:15) n,x = (cid:88) M (cid:15) UVM ( x )( g − ) M,n . (61)To the linear order in (cid:15) , the β -functions of these UV couplings are d(cid:15) UVM ( x ) dz = − (cid:88) n ( E n − E ) (cid:15) n,x g n,M = − (cid:88) n,M (cid:48) (cid:15) UVM (cid:48) ( x )( g − ) M (cid:48) n g n,M ( E n − E ) . (62)This gives the mixing matrix M M (cid:48) M = − (cid:88) n ( g − ) M (cid:48) n g n,M ( E n − E ) . (63)9It is noted that what appears on the right hand side of Eq. (63) is determined from thespectrum of H which is fixed by the beta functions defined in the subspace of the single-trace couplings. Eq. (63) shows that this small set of data completely fixes the full mixingmatrix that involves all multi-trace operators. This shows that the β -functions for generalmulti-trace operators are fixed by those for the single-trace operator to the linear orderin the deformation. In the following two subsections, we show that this holds beyond thelinear order.
4. Operator product expansion ψ n,n , ( x/ , − x/ T e H z x e −H z x T ˆ A n , − x/ ,φ × ˆ A n,x/ ,φ ψ n,n , ( a/ , − a/ c n , n , l ψ l, ˆ A l, ,φ C n , n , l ( | x | / a ) ˆ A l, ,φ FIG. 3. The procedure of extracting OPE between two local operators. Two local operatorsˆ A n (cid:48) , − x/ ,φ and ˆ A n (cid:48) ,x/ ,φ shown in the upper left corner undergo a series of transformations followingthe arrows in the clockwise direction : 1) the operators inserted to the IR fixed point corresponds tothe ground state with two local excitations through the action-state correspondence (right arrow T );2) the state is evolved with the RG Hamiltonian for z x = ln | x | a (down arrow e −H z ); 3) the resultingstate supports local excitation near the origin which is then expressed as a linear superposition ofthe eigenstates ψ l, with local excitations at the origin weighted with c n,n (cid:48) ,l ; 4) the resulting statecorresponds to the fixed point with local operator insertions at the origin (left arrow T ); 5) thetheory is evolved backward in RG time by − z x (up arrow e H z ). The identification of the finaloperator with the initial product of two operators gives the desired operator product expansion. The operator product expansion (OPE) between general multi-trace operators is alsofully encoded in the spectrum of H . Suppose we insert two local scaling operators ˆ A n,x/ and ˆ A n (cid:48) , − x/ at x/ − x/
2, respectively. The wavefunction for the resulting theory is ψ n,n (cid:48) , ( x/ , − x/ [ j ] = ( ˆ A n,x/ ,j ˆ A n (cid:48) , − x/ ,j ψ ) [ j ]. Under the evolution generated by H , theseparation between the two operators decreases exponentially in z . For z (cid:28) ln | x | a , where a = Λ − is the short distance cutoff length scale, the two local excitations remain well-separated in space, and evolve independently, e −H z ( ˆ A n,x/ ,j ˆ A n (cid:48) , − x/ ,j ψ ) [ j ] ≈ e − (∆ n +∆ n (cid:48) + E ) z ( ˆ A n,xe − z / ,j ˆ A n (cid:48) , − xe − z / ,j ψ ) [ j ] . (64)0This follows from the facts that 1) ˆ A n (cid:48) , − x/ and ˆ A n,x/ create local excitations with energies∆ n and ∆ n (cid:48) above the ground state; 2) H is local at length scales larger than a , and twooperators evolve independently with the total energy given by E n,n (cid:48) = E + (∆ n + ∆ n (cid:48) ) inthe limit that their separation is large. As two excitations approach, they interact, and thestate evolves into a more complicated state. Nonetheless, the state obtained after evolvingthe initial state for z x = ln | x | a can be written as a linear superposition of eigenstates of theRG Hamiltonian with local excitations located at the origin, e −H z x ( ˆ A n,x/ ,j ˆ A n (cid:48) , − x/ ,j ψ ) [ j ] = e − (∆ n +∆ n (cid:48) + E ) z x (cid:88) l F n,n (cid:48) ,l ( x )( ˆ A l, ,j ψ ) [ j ] , (65)where F n,n (cid:48) ,l ( x ) is a function that captures the effect of interaction. It is a regularizationdependent function which can be computed from the RG Hamiltonian. There is no inter-action between two operators when the separation is much larger than a . This follows fromthe fact that one only integrates out modes whose wavelengths are order of a in each coarsegraining step. As a result, F n,n (cid:48) ,l ( x ) exponentially approaches a constant in the large | x | limit, lim | x |→∞ | F n,n (cid:48) ,l ( x ) − c n,n (cid:48) ,l | ∼ e −| x | /a , where c n,n (cid:48) ,l ≡ F n,n (cid:48) ,l ( ∞ ). Now the state inEq. (65) is evolved backward in RG time z x , which results in( ˆ A n,x/ ,j ˆ A n (cid:48) , − x/ ,j ψ ) [ j ] = (cid:88) l F n,n (cid:48) ,l ( x ) e (∆ l − ∆ n − ∆ n (cid:48) ) z x ( ˆ A l, ,j ψ ) [ j ] . (66)From this, we obtain the OPE of scaling operators :ˆ A n,x/ ,φ × ˆ A n (cid:48) , − x/ ,φ = (cid:88) l C n,n (cid:48) ,l ( x ) ˆ A l, ,φ , (67)where C n,n (cid:48) ,l ( x ) = F n,n (cid:48) ,l ( x ) | x/a | ∆ n +∆ n (cid:48) − ∆ l . (68)In the large | x | limit, this reduces to the standard form, C n,n (cid:48) ,l ( x ) ∼ c n,n (cid:48) ,l | x | ∆ n +∆ n (cid:48)− ∆ l . Theprocedure used to extract the OPE is summarized in the Fig. 3. β -functionals Now we are ready to derive the full beta functions. For generality, we consider the case inwhich the sources are position dependent, and derive the beta functionals of the WilsonianRG from the quantum RG. We consider a UV theory with general deformations added tothe fixed point theory S ∗ parametrized by J ∗ n,x , S = S ∗ + (cid:90) d D x (cid:15) n,x ˆ A n,x,φ , (69)1where (cid:15) n,x = J n,x − J ∗ n,x , and repeated indices are summed over.The state that corresponds to this deformed theory is given by | Ψ (cid:105) = | ψ (cid:105) − (cid:90) d D x (cid:18) (cid:15) n,x − (cid:90) d D x (cid:48) (cid:15) l,x + x (cid:48) (cid:15) m,x − x (cid:48) C lmn ( x (cid:48) ) (cid:19) | ψ n,x (cid:105) + O ( (cid:15) ) (70)to the second order in (cid:15) n , where we use ˆ A l,x + x (cid:48) / ,φ × ˆ A m,x − x (cid:48) / ,φ = (cid:80) n C lmn ( x (cid:48) ) ˆ A n,x,φ obtained in Eq. (67). Under the evolution generated by the RG Hamiltonian, the stateevolves to | Ψ( z ) (cid:105) = e −E dz (cid:20) | ψ (cid:105) − e − ∆ n dz (cid:90) d D x (cid:18) (cid:15) n,x − (cid:90) d D x (cid:48) (cid:15) l,x + x (cid:48) (cid:15) m,x − x (cid:48) C lmn ( x (cid:48) ) (cid:19) | ψ n,xe − dz (cid:105) (cid:21) + O ( (cid:15) )= e −E dz (cid:20) | ψ (cid:105) − e − ∆ n dz (cid:90) d D ˜ xe Ddz (cid:18) (cid:15) n,e dz ˜ x − (cid:90) d D ˜ x (cid:48) e Ddz (cid:15) l,e dz (˜ x + ˜ x (cid:48) ) (cid:15) m,e dz (˜ x − ˜ x (cid:48) ) C lmn ( e dz ˜ x (cid:48) ) (cid:19) | ψ n, ˜ x (cid:105) (cid:21) + O ( (cid:15) ) , (71)where the renormalized spatial coordinate is defined as x = ˜ xe dz in the second line. Thefinal state can be written in the form of Eq. (70) provided (cid:15) n,x is replaced with renormalizedsource, (cid:15) (cid:48) n, ˜ x = e − ˜∆ n dz (cid:15) n,e dz ˜ x + 12 (cid:90) d D ˜ x (cid:48) (cid:15) l,e dz (˜ x + ˜ x (cid:48) ) (cid:15) m,e dz (˜ x − ˜ x (cid:48) ) (cid:104) e − ( ˜∆ l + ˜∆ m ) dz C lmn (˜ x (cid:48) ) − e ( D − ˜∆ n ) dz C lmn ( e dz ˜ x (cid:48) ) (cid:105) + O ( (cid:15) ) , (72)where ˜∆ n = ∆ n − D . | Ψ( z ) (cid:105) corresponds to the action, S ( z ) = S ∗ + (cid:90) d D x (cid:15) (cid:48) n,x ˆ A n,x,φ , (73)where the renormalized source is given by (cid:15) (cid:48) n,x = (cid:15) n,x + dz (cid:110) − ˜∆ n (cid:15) n,x + x ∂∂x (cid:15) n,x + 12 (cid:90) d D x (cid:48) (cid:15) l,x + x (cid:48) (cid:15) m,x − x (cid:48) G lmn ( x (cid:48) ) (cid:111) + O ( (cid:15) )(74)to the linear order in dz with G lmn ( y ) = − C lmn ( y ) y ∂∂y ln F lmn ( y ) . (75)The term quadratic in (cid:15) in Eq. (74) describes the fusion of two operators into one. Since G lmn ( y ) decays exponentially in the large y limit, operators whose separation is smallerthan a mainly contribute to the fusion process. This shows that the β -functionals of J n,x are local, ddz J n,x = − (∆ n − D )( J n,x − J ∗ n,x ) + x ∂∂x ( J n,x − J ∗ n,x )+ 12 (cid:90) d D x (cid:48) ( J l,x + x (cid:48) − J ∗ l,x + x (cid:48) )( J m,x − x (cid:48) − J ∗ m,x + x (cid:48) ) G lmn ( x (cid:48) ) + O (cid:16) ( J − J ∗ ) (cid:17) . (76)2This also confirms that couplings are irrelevant (relevant) if ∆ n > D (∆ n < D ). It is notedthat these are β functions for sources of the scaling operators. The β functions for thesources of the UV operators are obtained from Eq. (76) through a linear transformation inEq. (61).In this section, we show that the β -functions defined in the subspace of the single-tracecouplings encode the entire β -functions of the theory. In the following sections, we applythe prescription to two models explicitly. IV. -DIMENSIONAL EXAMPLE In this section, we consider a toy model of a 0-dimensional theory. For simplicity, weassume that there is only one single-trace operator, and that only single-trace and double-trace operators are generated under the coarse graining when the reference theory S isdeformed by the single-trace operator. We assume that the reference theory is invariantunder a Z symmetry, and the single-trace operator is odd under the symmetry. Thesymmetry constrains the form of the β -functions. We assume that the β -functions in thesubspace of the single-trace deformation take the following form β ( j, , .. ) = f − wj ,β ( j, , .. ) = aj,β ( j, , .. ) = − b,β n> ( j, , .. ) = 0 . (77)Here j stands for the source for the single-trace operator. β describes the flow of theidentity operator. β ( β ) is the beta function for the single (double)-trace coupling. a , b , w and f are non-zero real parameters. Under the Z transformation, the single-trace couplingtransforms as j → − j . This guarantees that β and β are even in j , and β is odd in j .Eq. (77) describes how couplings are renormalized when S is deformed by the single-traceoperator. In particular, b (cid:54) = 0 implies that the double-trace operator is generated and theRG flow leaves the subspace of single-trace deformation even if the UV theory has onlysingle-trace deformation. From Eq. (77), it is unclear where the double-trace coupling j eventually flows under the RG flow. Depending on the sign of β ( j, j ) at large j , thetheory may or may not flow to a scale invariant fixed point in the IR. Remarkably, β ( j, j )at general values of j is already encoded in Eq. (77), which determines the fate of the RGflow in the space of all couplings.Following the formalism in Sec. III B, we obtain the bulk RG Hamiltonian in Eq. (44).It is convenient to shift the RG Hamiltonian to remove a constant piece,˜ H = H + (cid:16) f − a (cid:17) = b (cid:20) ˜ p + ia b ˜ j (cid:21) + (cid:20) a − bw b (cid:21) ˜ j , (78)3where ˜ j = ij and ˜ p = − ip fluctuate along the real axis . They satisfy commutation relation (cid:2) ˜ p, ˜ j (cid:3) = i . This is a TP-symmetric non-Hermitian quadratic Hamiltonian[34]. As is shownin Appendix B, the spectrum of this RG Hamiltonian is given by E n = ( n + 12 ) √ η, (79)where η = a − bw . Unlike the Hermitian cases, the left and right eigenstates take differentforms, ψ Rn (˜ j ) = 1 √ n n ! ( ε π ) / e − ξ R ˜ j H n (cid:20)(cid:114) ε j (cid:21) ,ψ Ln (˜ j ) = 1 √ n n ! ( ε π ) / e − ξ L ˜ j H n (cid:20)(cid:114) ε j (cid:21) , (80)where ξ R,L = b ( √ η ± a ), ε = b √ η and H n ( x ) is the Hermite polynomial : H ( x ) = 1, H ( x ) = 2 x , H ( x ) = 4 x − . . . .The spectrum of the Hamiltonian is determined by the parameters in the β -functions, a , b and w . First, the eigenvalues of the Hamiltonian are real for η >
0. Second, theeigenstates are square-integrable for ξ R,L >
0. These conditions are satisfied for b > w < a . In the following, we first focus on this parameter region that supportsa real spectrum with normalizable eigenstates. At the end of this section, we will see thatviolation of these conditions is associated with a loss of stable fixed point in the IR. A. Fixed point
A generic initial state evolves to the right ground state of ˜ H in the large z limit, ψ R (˜ j ) = ( ε π ) / e − ξ R ˜ j . (81)As discussed in Sec. III C, the right ground state corresponds to the stable fixed point ofthe theory. To extract the fixed point action, we write the right ground state as | (cid:105) = (cid:90) D φ (cid:90) R D ˜ j ψ R (˜ j ) e − S + i ˜ j O | φ (cid:105) = ( ε π ) / (cid:114) πξ R (cid:90) D φ e − S − ξR O | φ (cid:105) . (82)The logarithm of its wavefunction in the φ basis gives the fixed point action, S ∗ = S + J ∗ O + J ∗ O (83)with J ∗ = 0 and J ∗ = ξ R . We emphasize that the stable fixed point exists away from thesubspace of the single-trace coupling, yet the position of the fixed point is fully determinedfrom the beta functions defined in the subspace. The additional shift a/ B. Scaling operators and their OPEs
Now we turn our attention to excited states of ˜ H . Each right eigenstate corresponds tothe fixed point theory with an operator insertion. The n -th excited state is | n (cid:105) = (cid:90) D φ (cid:90) R D ˜ j ψ Rn (˜ j ) e − S + i ˜ j O | φ (cid:105) = (cid:90) D φ √ n n ! ( ε π ) / (cid:114) πξ R e − S H n (cid:20) − i (cid:114) ε ∂∂ O (cid:21) e − ξR O | φ (cid:105) . (84)The excited states can be reached by applying ‘raising’ operators to the ground state, | n (cid:105) = ˆ A n | (cid:105) , (85)where ˆ A n is the operator that maps the ground state to the n -th excited states that has scal-ing dimension √ ηn . In general, the n -th scaling operator is given by a linear superpositionof all k -trace operators with k = n, n − , n − , .. . For n = 0 , , , ,
4, we obtainˆ A = , ˆ A = i √ ε ξ R O , ˆ A = 1 √ ε ξ R − − √ ε ξ R O , ˆ A = − i √ ε √ ξ R O + i √ ε √ ξ R (cid:16) ε ξ R − (cid:17) O , ˆ A = ε √ ξ R O + √ ε √ ξ R (cid:16) − ε ξ R (cid:17) O + √ √ − ε ξ R ) . (86)It is straightforward to identify all scaling operators in this way.The OPE coefficient can be computed accordingly. For instance, two ˆ A fuse toˆ A × ˆ A = (1 − ε ξ R ) ˆ A + √ A , (87)and the associated OPE coefficients are given by C = (1 − ε ξ R ), C = √
2. Similarly, allOPE coefficients can be extracted from the eigenstates of the RG Hamiltonian. In Tab. II,we list all OPE for ˆ A n × ˆ A m up to n, m = 2. C. Full β -functions Based on Sec. III C 5, we can immediately write down the full β -functions of the theory.In 0 dimension, there is no x dependence of the couplings and OPE coefficients. By setting5 TABLE II. Operator product expansion of ˆ A n × ˆ A m for n, m = 0 , , m n A ˆ A ˆ A A (1 − ε ξ R ) ˆ A + √ A √ − ε ξ R ) ˆ A + √ A A √ − ε ξ R ) ˆ A + √ A (1 − ε ξ R ) ˆ A + 2 √ − ε ξ R ) ˆ A + √ A D = 0 and C lmn ( x ) = c lmn in Eq. (72), we readily obtain the β -function, ddz J n = − ∆ n ( J n − J ∗ n )+ 12 (∆ n − ∆ l − ∆ m ) c lmn ( J l − J ∗ l )( J m − J ∗ m ) + O (cid:16) ( J − J ∗ ) (cid:17) , (88)From Eq. (79) that implies ∆ n = √ ηn and Table II, we obtain the beta functions for thecouplings of ˆ A and ˆ A , ddz J = −√ η (cid:34) ε √ ξ R − √ ξ R (cid:35) J − (cid:112) η (1 − ε ξ R ) J J + O (cid:16) ( J − J ∗ ) (cid:17) , (89) ddz J = √ η ξ R + ε √ η √ ξ R − √ η √ ξ R − √ η (cid:34) ε √ ξ R − √ ξ R (cid:35) J − (cid:112) η (1 − ε ξ R ) J + O (cid:16) ( J − J ∗ ) (cid:17) , (90)where ddz J n = − β n . Although β n> = 0 in the subspace of single-trace coupling, they arein general non-zero away from the subspace. It is straightforward to compute β n for any n order by order in ( J − J ∗ ). This shows that the full beta functions are indeed encoded inthe RG Hamiltonian that is fixed by the beta functions defined in the space of single-tracecouplings.The beta functions for multi-trace operators allow us to explore the RG flow away fromthe subspace of the single-trace coupling. Eqs. (89) and (90) computed to the quadraticorder in δJ = ( J − J ∗ ) can be trusted near J = J ∗ . To describe the RG flow far away fromthe stable fixed point, one needs to take into account terms that are higher order in δJ andhigher trace couplings. Here we focus on the flow in the space of J and J near J = J ∗ . Tothe quadratic order in δJ , J n with n > J = J ∗ . The RG flow in the space of J and J is shown in Fig. 1 for a = − . bb = 0 . b = 0 . ( b ) ( J ∗ ) ± b = 0 . w ( a ) ( J ∗ ) ± -0.5 -0.4 -0.3 -0.2 -0.1 1.01.52.02.50.5 J J ( c ) -2 -1 0 1 2-0.4-0.20.00.20.4 J ( c ) -2 -1 0 1 2-0.5 0.00.51.01.5 ( c ) J -2 -1 0 1 2-0.4-0.20.00.20.4 0.2 0.4 0.6 0.80.40.60.81.01.21.41.6 b ( J ∗ ) ± b ( b ) ( J ∗ ) ± b ( b ) ( J ∗ ) ± FIG. 4. The double-trace coupling J at the stable (blue line) and unstable (orange line) fixed pointas a function of w at b = 0 . b at w = − . ), w = − . ), w = − . ), w = − .
01 (b ). Three RG flow diagrams at w = − . b = 0 . ), b = 0 .
001 (c ), b = 0 . ), b = 0 . a = − . b = 0 . w = − .
5. We find two fixed points at( J ∗ , J ∗ ) − = (cid:18) , ξ R (cid:19) , (91)( J ∗ , J ∗ ) + = (cid:32) , ξ R + √ ξ R ( ε − ξ R ) (cid:33) , (92)where 0 < ( J ∗ ) − < ( J ∗ ) + because ξ R > ε > ξ R . In either the small ξ R or large ε limit, the second fixed point is close to the stable fixed point, and the terms that arehigher order in δJ in Eqs. (89) and (90) are negligible near these two fixed points. Thefirst fixed point in Eq. (91) is the stable fixed point identified from the ground state of theRG Hamiltonian in Eq. (83). Both δJ and δJ are irrelevant whose scaling dimensionsare −√ η and − √ η , respectively. The second fixed point in Eq. (92) is an unstable fixedpoint. Both δJ and δJ are relevant with scaling dimensions, √ η and 2 √ η , respectively.In Fig. 4(a) and Fig. 4(b ∼ ), we plot the value of J at the two fixed points as b and w are varied, respectively. For b > w <
0, the spectrum of the RG Hamiltonian is realand the eigenstates are normalizable. In this case, ξ R and ε − ξ R are both positive andfinite such that the two fixed points remain separated, as shown in Fig. 4(c ∼ ). The RGflow changes qualitatively if b or w approaches 0. In Appendix C, we examine the RG flowin the w → − and b → + limits in more details.7 V. D -DIMENSIONAL EXAMPLE In this section, we extend the discussion in the previous section to a D -dimensional fieldtheory. For simplicity, we continue to assume that there is only one single-trace operator,and that multi-trace operators higher than double-trace operator are not generated whenthe reference action is deformed only by the single-trace operator. We also assume that thereference theory is invariant under the spatial translation, the rotation and the inversionsymmetry, and has an internal Z symmetry under which the single-trace operator is odd.The symmetry largely fixes the form of the β -functionals in the subspace of the single-trace couplings order by order in the coupling. To be concrete, we consider the following β -functionals in the subspace of single-trace couplings, β [ j, , .. ; x ] = f − g [ ∂ x j ( x )] − wj ( x ) ,β [ j, , .. ; x ] = aj ( x ) − x∂ x j ( x ) ,β [ j, , .. ; x ] = − b,β k ≥ [ j, , .. ; x ] = 0 . (93)Here β k [ j (1) , j (2) , j (3) , .. ; x ] represents the β -functional for the k -trace operator at (cid:0) j (1) , j (2) , j (3) , .. (cid:1) ,where j ( m ) is the m -trace coupling. ( ∂ x j ) ≡ (cid:80) Dµ =1 ( ∂ µ j ) , and x∂ x j ≡ (cid:80) Dµ =1 x µ ∂ µ j . f, g, w, a, b are constants that represent the contributions to the beta functions generatedfrom integrating out short distance modes and rescaling the fundamental fields at everyRG step. The last term in β dilates the space because the coordinate in the ( l + 1)-th RGstep is related to that in the previous step through x ( l +1) = x ( l ) e − dz . The rescaling makessure that the UV cutoff remains invariant under the RG flow, and the same coarse grainingcan be applied at all steps. On the other hand, the rescaling of space reduces the size ofthe system in real space by e − dz at every RG step.Eq. (93) fixes the bulk theory in Eq. (43), which in turn determines the fate of the fieldtheory in the low-energy limit. The wavefunction that fully determines the renormalizedaction at scale z is given by the path integration of the single-trace source and its conjugatevariable, Ψ[ j, z ] = (cid:82) I D j ( x, z (cid:48) ) D p ( x, z (cid:48) ) Ψ J ,J ,... [ j (0)] e − (cid:82) z dz (cid:48) L [ j,p,z (cid:48) ] (cid:12)(cid:12)(cid:12) j ( z )= j . (94)While the bulk Lagrangian is quadratic in the present case, it depends on x explicitlybecause of the dilatation term in β of Eq. (93). This gives rise to a mixing betweendifferent Fourier modes in the momentum space . The mixing makes it hard to computethe path integration directly. To bypass this problem, we follow the three steps describedbelow. The mixing arises because the momentum in the ( l + 1)-th RG step is related to the momentum in theprevious step as k ( l +1) = k ( l ) e dz .
81. We introduce new variables in the bulk, j ( x, z ) = − i ˜ j ( xe z , z ) e D z , p ( x, z ) = i ˜ p ( xe z , z ) e D z . (95)Besides the rescaling of spatial coordinate that undoes the dilatation, the fields arealso multiplied with a factor e D z to compensate the z -dependent volume of the space. ± i is multiplied so that ˜ j and ˜ p fluctuate along the real axis. In the new variables,the dilatation effect disappears and Fourier modes with different momenta do notmix as will be shown later.2. The path integration in Eq. (94) is performed in ˜ j and ˜ p . This is done in theHamiltonian picture.3. The scale transformation is reinstated by expressing the z -dependent state in termsof j (cid:48) ( x, z ) = ˜ j ( xe z , z ) e D z , p (cid:48) ( x, z ) = ˜ p ( xe z , z ) e D z . (96)In the following sections, we implement these steps to identify the IR fixed point and thespectrum of scaling operators at the fixed point. A. The RG Hamiltonian
In terms of the variables introduced in Eq. (95), the bulk Lagrangian is written as L [ j, p, z ] = (cid:90) /λ d D X (cid:16) i ˜ p∂ z ˜ j + i D p ˜ j − (cid:88) n ≥ ˜ β n [˜ j ; X ]˜ p n e ( n − D z (cid:17) , (97)where ˜ β [˜ j ; X ] = β [ j ; x ] − ie D z X∂ X ˜ j, ˜ β n [˜ j ; X ] = β n [ j ; x ] for n (cid:54) = 1 . (98)with X = xe z . ˜ j ( X ) and ˜ p ( X ) obey the canonical commutation relation, (cid:2) ˜ j ( X ) , ˜ p ( X (cid:48) ) (cid:3) = − iδ ( X − X (cid:48) ) . (99)The RG Hamiltonian density is given by H (cid:2) ˜ j, ˜ p, z (cid:3) = i D p ˜ j − (cid:88) n ≥ ˜ β n [˜ j ; X ]˜ p n e ( n − D z . (100)9By shifting the Hamiltonian by a constant, we write the Hamiltonian density as˜ H (cid:2) ˜ j, ˜ p, z (cid:3) = H (cid:2) ˜ j, ˜ p, z (cid:3) + e − Dz f −
12 ( a − D δ (0) (101)= − ge z (cid:2) ∂ X ˜ j ( X ) (cid:3) − w ˜ j ( X ) + i
12 ( a + D (cid:2) ˜ j ( X )˜ p ( X ) + ˜ p ( X )˜ j ( X ) (cid:3) + b ˜ p ( X ) . As expected, the dilatation in Eq. (93) cancels with that in Eq. (98). Instead, the RGHamiltonian acquires explicit z dependence.In the Fourier basis,˜ j ( X ) = 1 √ V (cid:88) K e iK · X ˜ j K , ˜ p ( X ) = 1 √ V (cid:88) K e iK · X ˜ p K , (102)where V = λ − D is the volume of the system, the RG Hamiltonian can be written as˜ H ( z ) = (cid:88) K ˜ h K , (103)where ˜ h K = b (cid:110) (cid:2) ˜ p K + iζ ˜ j K (cid:3) (cid:2) ˜ p − K + iζ ˜ j − K (cid:3) + Ω K,z ˜ j K ˜ j − K (cid:111) (104)with Ω K,z = σ + αe z K . (105)Here, ζ = b ( a + D ), σ = b ( a + D ) − wb and α = − g/b . Henceforth, we set b = 1 / ζ = a + D , σ = ζ − w and α = − g . ˜ j K and ˜ p − K are canonical conjugatevariables that satisfy (cid:2) ˜ j K , ˜ p K (cid:48) (cid:3) = − iδ K, − K (cid:48) . While ˜ j K =0 and ˜ p K =0 are real, ˜ j K (cid:54) =0 and ˜ p K (cid:54) =0 are complex with ˜ j K = ˜ j ∗− K and ˜ p K = ˜ p ∗− K . The Hamiltonian can be decomposed into asum of time-dependent harmonic oscillators,˜ H ( z ) = ˜ h + (cid:48) (cid:88) K> (˜ h ( R ; K ) + ˜ h ( I ; K ) ) , (106)where (cid:80) (cid:48) K> runs over the half of non-zero momenta with K identified with − K , and˜ h = b (cid:110) (cid:2) ˜ p + iζ ˜ j (cid:3) + Ω ,z ˜ j (cid:111) , ˜ h ( R ; K> = b (cid:110) (cid:2) ˜ p ( R ; K ) + iζ ˜ j ( R ; K ) (cid:3) + Ω K,z (˜ j ( R ; K ) ) (cid:111) , ˜ h ( I ; K> = b (cid:110) (cid:2) ˜ p ( I ; K ) + iζ ˜ j ( I ; K ) (cid:3) + Ω K,z (˜ j ( I ; K ) ) (cid:111) (107)with ˜ j ( R ( I ); K ) = √ j K and ˜ p ( R ( I ); K ) = √ p K that satisfy the commutationrelation (cid:2) ˜ j ( S ; K ) , ˜ p ( S (cid:48) ; K (cid:48) ) (cid:3) = − iδ K, − K (cid:48) δ SS (cid:48) with S, S (cid:48) = I, R .0The RG flow is described by the imaginary time Schrodinger equation,˜ H ( z )Ψ (cid:2) ˜ j, z (cid:3) = − ∂∂z Ψ (cid:2) ˜ j, z (cid:3) . (108)The three parameters ζ , σ and α fully determine the solution Ψ (cid:2) ˜ j, z (cid:3) . The problem of theharmonic oscillator with time-dependent frequency has been studied extensively in Refs.[35–37], which is reviewed in Appendix D. We consider a UV theory obtained by addingthe single-trace and double-trace couplings to the reference theory S in a translationallyinvariant way. In this case, the initial wavefunction is a Gaussian product state in the K -space. Because the Hamiltonian is non-interacting, Ψ (cid:2) ˜ j, z (cid:3) remains Gaussian at all z .The solution is written asΨ (cid:2) ˜ j, z (cid:3) = Ψ (cid:2) ˜ j , z (cid:3) (cid:48) (cid:89) K> (cid:110) Ψ ( R ; K ) (cid:2) ˜ j ( R ; K ) , z (cid:3) Ψ ( I ; K ) (cid:2) ˜ j ( I ; K ) , z (cid:3) (cid:111) , (109)where (cid:81) (cid:48) K> runs over the half of the non-zero momenta. The wavefunction for each modesatisfies ˜ h s Ψ s (cid:2) ˜ j s , z (cid:3) = − ∂∂z Ψ s (cid:2) ˜ j s , z (cid:3) , where the subscript s stands for 0, ( R ; K ) or ( I ; K ).The initial state can be written asΨ s (cid:2) ˜ j s , (cid:3) = (cid:88) m c m Ψ m,s (cid:2) ˜ j s , (cid:3) , (110)where { Ψ m,s (cid:2) ˜ j s , (cid:3) } represents the eigenstates of the Hamiltonian ˜ h s at z = 0 and { c m } isa set of z -independent coefficients. Under the RG flow, the state evolves toΨ s (cid:2) ˜ j s , z (cid:3) = (cid:88) m c m Ψ m,s (cid:2) ˜ j s , z (cid:3) , (111)where Ψ m,s (cid:2) ˜ j s , z (cid:3) = 1 π / √ m m ! e − ∆ s,z exp (cid:20) − s,z ˜ j s (cid:21) × exp (cid:20) ω s,z A s,z ˜ j s (cid:21) H m (cid:34) − A s,z (cid:112) Ω s, δδ ˜ j s (cid:35) exp (cid:20) − ω s,z A s,z ˜ j s (cid:21) . (112)Here ω s,z = (cid:20)(cid:82) z dz (cid:48) A s,z (cid:48) + s, (cid:21) − . A s,z is a function that satisfies ¨ A s,z − A s,z Ω s,z = 0 with A s, = 1 and ˙ A s, = 0 ( ˙ A ≡ ∂ z A ). e − ∆ s,z = ω s,z A s,z √ Ω s, . s,z = ζ + ˙ A s,z A s,z + ω s,z A s,z . At z = 0, s,z is reduced to Ω s, + ζ , and e − ∆ s,z becomes (cid:112) Ω s, .1Finally, the z -dependent state is written in terms of the variables in Eq. (96),Ψ (cid:48) [ j (cid:48) , z ] = Ψ (cid:2) j (cid:48) , z (cid:3) (cid:48) (cid:89) K> Ψ ( R ; K ) (cid:104) j (cid:48) ( R ; Ke z ) , z (cid:105) Ψ ( I ; K ) (cid:104) j (cid:48) ( I ; Ke z ) , z (cid:105) = Ψ (cid:2) j (cid:48) , z (cid:3) (cid:48) (cid:89) k> Ψ ( R ; ke − z ) (cid:104) j (cid:48) ( R ; k ) , z (cid:105) Ψ ( I ; ke − z ) (cid:104) j (cid:48) ( I ; k ) , z (cid:105) . (113)Here we use ˜ j K = j (cid:48) k for the Fourier modes, where x = e − z X and k = e z K . For a finitesystem size, k and K are discrete, K = 2 πL ( n , n , .., n D ) , k = 2 πe z L ( n , n , .., n D ) , (114)where L = V /D is the linear system size and n i ’s are integers.Our next goal is to extract the fixed point of the full Wilsonian RG and local scalingoperators with their scaling dimensions from the scale dependent state obtained from thequantum RG. As discussed in the previous sections, the asymptotic ground state thatemerges in the large z limit corresponds to the stable fixed point, and eigenstates withlocal excitations and eigenvalues give scaling operators and scaling dimensions, respectively.However, it is not easy to extract the asymptotic state in the large z limit because the RGHamiltonian is z -dependent. Even if one prepares an initial state to be an eigenstate of theinstantaneous RG Hamiltonian at z = 0, the state does not remain the same under the RGevolution as is shown in Eq. (112). Therefore, we use the following strategy. Given thatthe RG Hamiltonian is invariant under the Z symmetry, we consider a generic initial statein each of the Z even sector and the Z odd sector. Under the quantum RG flow, thoseinitial states evolve within each sector aslim z →∞ | Ψ + ( z ) (cid:105) = (cid:88) n e −E + n z | n ; + (cid:105) , lim z →∞ | Ψ − ( z ) (cid:105) = (cid:88) n e −E − n z | n ; −(cid:105) , (115)where | n, ±(cid:105) corresponds to the eigenstates of the RG Hamiltonian that emerges in thelarge z limit in each parity sector, and E ± n is the corresponding eigenvalue. From this,we identify the eigenstate with the lowest eigenvalue in the even sector as the groundstate that represents the stable IR fixed point. The excited states in each parity sectorcorrespond to the states obtained by deforming the ground state with scaling operatorswith the corresponding Z parity and scaling dimension, E ± n − E +0 . This follows from˜ j K = (cid:90) /λ d D X √ V e − iKX ˜ j ( X, z ) = (cid:90) /λ d D X √ V e − iKX e − D z j (cid:48) ( Xe − z , z )= (cid:90) e − z /λ d D x √ V e − Dz e − ikx j (cid:48) ( x, z ) = j (cid:48) k . B. Fixed point
In this section, we identify the IR fixed point of the theory from quantum RG. As aninitial state, we choose the ground state of the instantaneous RG Hamiltonian at z = 0,which has the translational invariance and even Z parity,Ψ (cid:2) ˜ j, (cid:3) = Ψ ,K =0 (cid:2) ˜ j , (cid:3) (cid:48) (cid:89) K> Ψ , ( R ; K ) [˜ j ( R ; K ) , , ( I ; K ) [˜ j ( I ; K ) , . (116)In the large z limit, the z -dependent wavefunction for each s -mode becomesΨ ,s (cid:2) ˜ j s , z (cid:3) = π − / e − ∆ s,z exp (cid:20) − s,z ˜ j s (cid:21) . (117)The asymptotic many-body wavefunction is written asΨ (cid:2) ˜ j, z (cid:3) = N (cid:48) ( z ) exp (cid:34) − (cid:88) K ˜ j K ˜ j − K K,z (cid:35) , (118)where N (cid:48) ( z ) = (cid:104)(cid:81) K π − / e − ∆ K,z (cid:105) and Λ
K,z are expressed in terms of α , ζ and σ asΛ K,z = [ G Ke z ( α, σ ) + ζ ] − , (119)where G k ( α, σ ) = 12 √ α | k | I σ [ √ α | k | ] ( I − √ σ (cid:2) √ α | k | (cid:3) + I √ σ (cid:2) √ α | k | (cid:3) ) (120)in the large z limit with fixed k = Ke z (see Appendix E for the details). This shows thatΛ K,z converges to a z -independent function when viewed as a function of k . Physically, thisis due to the fact that the scale invariance becomes manifest if one zooms in toward the K = 0 point progressively as z increases. The overall normalization of the wavefunctiondecreases with increasing z due to the damping associated with the imaginary time evolution . To show that the wavefunction approaches a scale invariant asymptotic in the large z limit, we need to go back to the scaled variable in Eq. (96). The wavefunction for j (cid:48) k , p (cid:48) k iswritten as Ψ (cid:48) [ j (cid:48) , z ] = N (cid:48) ( z ) exp (cid:16) − (cid:88) k
12 ˜Λ k j (cid:48) k j (cid:48)− k (cid:17) , (121) The normalization factor N (cid:48) ( z ) is determined by e − ∆ K,z ≈ σ / A Ke z ( α, σ ) e −√ σz in the large z limit, where A k is a function of k = Ke z (Eq. (E3) in Appendix E), A k ( α, σ ) = − − √ σ πα − √ σ sin( π √ σ ) I √ σ [ √ α | k | ]Γ( −√ σ ) | k | −√ σ k = Λ ke − z ,z . In the large z limit for a fixed k , ˜Λ k takes the following forms (seeAppendix E), ˜Λ k = (cid:40) √ σ + ζ ) for | k | (cid:28) , [ √ α | k | + ζ ] for | k | (cid:29) . (122)This confirms that in the large z limit Ψ (cid:48) [ j (cid:48) , z ] evolves to a z -independent state up to the z -dependent normalization factor.Similar to what we studied in Sec. IV A, the state in the large z limit encodes theinformation on the IR fixed point. Defining J ∗ ,k ≡ ˜Λ k , we rewrite the asymptotic state inthe large z limit aslim z →∞ | Ψ( z ) (cid:105) = N (cid:48) ( z ) (cid:90) D φ e − S (cid:90) R D j (cid:48) e − (cid:16) (cid:80) k ( J ∗ ,k ) − j (cid:48) k j (cid:48)− k − i (cid:80) k j (cid:48) k O − k (cid:17) | φ (cid:105) = N ( z ) (cid:90) D φ e − S ∗ | φ (cid:105) , (123)where N ( z ) = N (cid:48) ( z ) det [2 πJ ∗ ] / and the fixed point action S ∗ is given by S ∗ = S + 12 (cid:88) k J ∗ ,k O k O − k = S + 12 (cid:90) d D xd D x (cid:48) J ∗ ,x − x (cid:48) O x O x (cid:48) , (124)where J ∗ ,x − x (cid:48) = 1 V e − Dz (cid:88) k J ∗ ,k e − ik ( x − x (cid:48) ) = (cid:90) d D k (2 π ) D ˜Λ k e − ik ( x (cid:48) − x ) . (125)Here we use O k = √ V e − Dz (cid:82) d D x O x e − ikx .As is shown in Fig. 5 (see Appendix F for the details), lim z →∞ J ,x − x (cid:48) converges to auniversal profile in the thermodynamic limit. J ∗ ,x − x (cid:48) is peaked at x − x (cid:48) = 0 with a finitewidth that is order of the short distance cutoff. It decays exponentially at large | x (cid:48) − x | .We emphasize that the IR fixed point that exists away from the subspace of the single-tracecouplings has been extracted solely from the β -functions that are defined in the subspace. C. Scaling operators
In this subsection, we extract scaling operators from excited states of the RG Hamilto-nian.4 -4 -2 2 40.10.20.30.4 J ∗ ,x − x x − x ( a ) -5 50.20.40.60.8 ( b ) x − xJ ∗ ,x − x FIG. 5. (a) J ∗ ,x − x (cid:48) as a function of x − x (cid:48) at σ = 2 .
01 and ζ = − . D = 1 at z = 22 (orange), z = 23 (green), z = 24 (blue). For the computation, the lattice regularization is used with the totalnumber of sites L/a = e . (b) J ∗ ,x − x (cid:48) at σ = 0 .
01 and ζ = 0 . z for each color as in (a). α = 1. Z odd sector In the Z odd sector, we consider an initial state in which one of the Fourier modes isexcited. Suppose that the mode with momentum P is in the first excited state with respectto the RG Hamiltonian at z = 0, where the momentum is measured in the coordinate systemdefined at z = 0. In the large z limit, the state evolves to (see Appendix G for derivation) | Ψ ,P ( z ) (cid:105) = ( i √ N ( z ) (cid:90) D φ (cid:16) σ / A P e z ˜Λ P e z e −√ σz O P e z (cid:17) e − S ∗ | φ (cid:105) . (126) N ( z ) is the normalization of the ground state defined in Eq. (123). Compared to the groundstate, the weight of the first excited state with a definite momentum decays as e −√ σz in thelarge z limit. The state that supports an excitation at P (cid:54) = 0 at z = 0 can not be invariantunder the RG flow because a non-zero P is pushed toward larger momenta in the large z limit due to the rescaling. Namely, a source that is added periodically in space at UV flowsto a periodic source with a shorter wavelength at larger z when measured in the rescaledcoordinate system.The excited state with P = 0 is an exception. In the presence of a uniform source,the excited state flows to a scale invariant state in the large z limit. Using O p = √ V e − Dz (cid:82) d D x e − ipx O x for the Fourier transformation at z , where p = P e z and x = Xe − z ,we rewrite Eq. (126) for P = 0 as | Ψ , ( z ) (cid:105) = e − ( √ σ − D/ z i √ N ( z ) σ / ˜Λ √ V A (cid:90) D φ (cid:18)(cid:90) d D x O x (cid:19) e − S ∗ | φ (cid:105) . (127)5The first excited state with the uniform source flows to (cid:82) D φ (cid:0)(cid:82) d D x O x (cid:1) e − S ∗ | φ (cid:105) in thelarge z limit with the z -dependent amplitude, e − ( √ σ − D/ z relative to the ground state.This implies that the spatially uniform deformation of the Z odd single-trace operator isrelevant (irrelevant) if √ σ < D/ √ σ > D/ -15 -10 -5 5 10 150.050.100.150.200.250.30 -15 -10 -5 5 10 150.050.100.150.20 FIG. 6. J (1 , x at z = 22 (orange), z = 23 (green), z = 24 (blue) with L/a = e in D = 1. We setparameters to be α = 1 and (a) σ = 2 .
01 and ζ = − .
1; (b) σ = 0 .
01 and ζ = 0 . The other type of eigenstates that are invariant under the RG evolution is the ones thatsupport excitations localized in space. In order to find local scaling operators associatedwith states with local excitations, we consider an initial state in which the single-traceoperator is inserted at X . For z >
0, the state becomes | Ψ ,X ( z ) (cid:105) = 1 √ V (cid:88) P e iP X | Ψ ,P ( z ) (cid:105) . (128)In the large z limit, the state evolves to lim z →∞ | Ψ ,X ( z ) (cid:105) = ( i √ N ( z ) e − ( √ σ + D/ z (cid:90) D φ ˆ A ( Xe − z ) e − S ∗ | φ (cid:105) , (129)where ˆ A ( x ) = (cid:90) d D x (cid:48) J (1 , x − x (cid:48) O x (cid:48) (130) lim z →∞ | Ψ ,X ( z ) (cid:105) = ( i √ N ( z ) √ V (cid:90) D φ (cid:16) (cid:88) P σ / A Pe z ˜Λ Pe z e −√ σz O Pe z e iPX (cid:17) e − S ∗ | φ (cid:105) = ( i √ N ( z ) e −√ σz √ V (cid:90) D φ (cid:16) (cid:88) p σ / A p ˜Λ p O p e ip ( Xe − z ) (cid:17) e − S ∗ | φ (cid:105) = ( i √ N ( z ) e −√ σz V e − Dz/ (cid:90) D φ (cid:16) (cid:90) d D x (cid:48) (cid:88) p σ / A p ˜Λ p e ip ( e − z X − x (cid:48) ) O x (cid:48) (cid:17) e − S ∗ | φ (cid:105) = ( i √ N ( z ) e − ( √ σ + D/ z (cid:90) D φ ˆ A ( Xe − z ) e − S ∗ | φ (cid:105) , where we used V e − Dz (cid:80) p = (cid:82) d D p (2 π ) D at z . J (1 , x = (cid:90) d D p (2 π ) D σ / A p ˜Λ p e ipx . (131)ˆ A ( x ) inserts a single-trace operator around x with distribution given by J (1 , x − x (cid:48) . Henceforth,we use J ( n,m ) to denote the contribution of the m -trace operator to the n -th scaling operator.The local operator inserted at X at the UV boundary evolves to a distribution of localoperators centered at Xe − z at z >
0. The shift of the central position is due to the rescalingof the space. In the large z limit, the local operator evolves to ˆ A (0), which we identifyas the local scaling operator inserted at the origin. The broadening of the distribution in J (1 , x is due to the correlation in the fluctuations of the single-trace coupling at the fixedpoint. Irrespective of the initial profile of the local operator at the UV, it converges to theuniversal profile J (1 , x at large z in the thermodynamic limit. In Fig. 6, we numerically plot J (1 , x ( z ) which converges to a z -independent profile in the large z limit. Compared to theground state, the overall weight of the first excited state with the local excitation decays as e − ( √ σ + D/ z in the large z limit. This implies that the operator O x has scaling dimension∆ = √ σ + D/
2. This is consistent with the fact that the operator is relevant (irrelevant)if √ σ < D/ √ σ > D/ Z symmetry, the Z odd operator is not allowed. To see if the low-energy fixed point is stable in the presence of the Z symmetry, we need to consider localscaling operators in the even parity sector. The operator with the smallest scaling dimensionin the even sector is the identity operator. In the following section, we obtain the next lowestscaling operator in the even sector. Z even sector Excited states in the Z even sector should include even number of excited modes. Letus consider an initial state with two excited modes labelled by momenta P and P (cid:48) . Underthe RG evolution, the state in general evolves into a linear superposition of the ground state(for P + P (cid:48) = 0) and excited states. Since we are interested in the excited state above theground state, we discard the slowest decaying state (the ground state). The state with thenext slowest decaying amplitude in the large z limit is given by (see Appendix H) | Ψ ,P,P (cid:48) ( z ) (cid:105) = −√ N ( z ) e − √ σz (cid:90) D φ (cid:16) σ / A P e z A P (cid:48) e z ˜Λ P e z ˜Λ P (cid:48) e z O P e z O P (cid:48) e z − δ P + P (cid:48) , (cid:34) σ / A P e z ˜Λ P e z + 12 √ σ √ αP + σ W P e z (cid:35) (cid:17) e − S ∗ | φ (cid:105) , (132)where N ( z ) encodes the rate at which the ground state decays and e − √ σz is the additionaldecay for the next slowest decaying state. Again, the state with non-zero momenta can7not be invariant under the RG evolution due to the rescaling that shifts momenta to largervalues with increasing z . To find a local scaling operator, we consider the state that evolvesfrom an initial state that supports local excitations at positions X and X (cid:48) , | Ψ ,X,X (cid:48) ( z ) (cid:105) = V (cid:80) P,P (cid:48) e iP X + iP (cid:48) X (cid:48) | Ψ ,P,P (cid:48) ( z ) (cid:105) . (133)In the large z limit, the state evolves to | Ψ ,X,X (cid:48) ( z ) (cid:105) = −√ N ( z ) e − (2 √ σ + D ) z (cid:90) D φ ˆ A ( Xe − z , X (cid:48) e − z ) e − S ∗ | φ (cid:105) , (134)where ˆ A ( x, x (cid:48) ) = (cid:90) d D yd D y (cid:48) J (2 , x − y,x (cid:48) − y (cid:48) O y O y (cid:48) − J (2 , x − x (cid:48) (135)with J (2 , x,x (cid:48) = (cid:90) d D pd D p (cid:48) (2 π ) D σ / A p A p (cid:48) ˜Λ p ˜Λ p (cid:48) e ipx + ip (cid:48) x (cid:48) ,J (2 , x = (cid:90) d D p (2 π ) D e ipx (cid:34) σ / A p ˜Λ p + 12 W p (cid:35) . (136) | Ψ ,X,X (cid:48) ( z ) (cid:105) = 1 V (cid:88) P,P (cid:48) e iPX + iP (cid:48) X (cid:48) | Ψ ,P,P (cid:48) ( z ) (cid:105) = −√ N ( z ) e − √ σz (cid:90) D φ (cid:16) V (cid:88) p,p (cid:48) e ip ( Xe − z )+ ip (cid:48) ( X (cid:48) e − z ) J (2 , p,p (cid:48) O p O p (cid:48) − V (cid:88) p e ip ( X − X (cid:48) ) e − z J (2 , p (cid:17) e − S ∗ | φ (cid:105) = −√ N ( z ) e − (2 √ σ + D ) z (cid:90) D φ (cid:16) (cid:90) d D yd D y (cid:48) V e − Dz (cid:88) p,p (cid:48) e ip ( Xe − z − y )+ ip (cid:48) ( X (cid:48) e − z − y (cid:48) ) J (2 , p,p (cid:48) O y O y (cid:48) (cid:17) e − S ∗ | φ (cid:105) + √ N ( z ) e − (2 √ σ + D ) z (cid:90) D φ (cid:16) V e − Dz (cid:88) p e ip ( X − X (cid:48) ) e − z J (2 , p (cid:17) e − S ∗ | φ (cid:105) , where p = P e z and J (2 , p,p (cid:48) = σ / A p A p (cid:48) ˜Λ p ˜Λ p (cid:48) , J (2 , p = σ / A p ˜Λ p + √ σ (cid:112) αp e − z + σ W p ≈ σ / A p ˜Λ p + 12 W p . in the large z limit. FIG. 7. The profile of J (2 , x,x (cid:48) as a function of x and x (cid:48) at z = 25 for a system with L/a = e in D = 1. We use α = 1, ζ = 0 . σ = 0 .
01 for the plot. ˆ A ( x, y ) is a composite operator that supports two single-trace operators centered atposition x and y respectively. In the large z limit, the initial state flows to the stateobtained by applying ˆ A (0 ,
0) to the ground state. Therefore, we identify ˆ A (0 ,
0) as thelowest scaling operator above the identity operator in the even sector. It is noted thatˆ A (0 ,
0) is a linear superposition of a double-trace operator and an identity operator. Thisis because the double-trace operator and the identity operator mix under the RG flow. J (2 , x,x (cid:48) , that describes the distribution of the two single-trace operators, can be written as J (2 , x,x (cid:48) = J (1 , x J (1 , x (cid:48) , and its profile in the real space is determined by that of J (1 , x . InFig. 7, J (2 , x,x (cid:48) is shown as a function of x and x (cid:48) at a fixed z . It has a peak at the originand decays exponentially away from the peak with the width that is comparable to theshort distance cutoff scale. According to Eq. (134), the local deformation induced by thisscaling operator decays with rate 2 √ σ + D relative to the ground state. Thus, its scalingdimension of the local operator is 2 √ σ + D which is twice of the single-trace operator. Itis an irrelevant operator due to √ σ > n (cid:0) √ σ + D (cid:1) for n = 1 , , , .. . See Appendix I for the details. All operators in the evensector are irrelevant for √ σ >
0. This shows that the fixed point in Eq. (123) is stable inthe presence of the Z symmetry. The fact that the scaling dimensions are additive is afeature of the generalized free theory for which the bulk RG Hamiltonian is quadratic. Forgeneral theories whose RG Hamiltonian is not quadratic, this is no longer the case. It will beof great interest to consider large N theories whose RG Hamiltonian includes interactionsthat are suppressed by 1 /N , and compute 1 /N corrections to the scaling dimensions fromthe quantum RG.9 D. Operator Product Expansion
Now we consider the OPE between two parity-odd operators with the lowest scalingdimension. For this, we insert ˆ A at X and − X in the stable fixed point theory. Thedeformed theory corresponds to the initial state with two local excitations, | Ψ × X, − X (0) (cid:105) = −√ N (0) (cid:90) D φ ˆ A ( X ) ˆ A ( − X ) e − S ∗ | φ (cid:105) . (137)Then, following the steps explained in Sec. III C 4, we evolve the state with the RG Hamil-tonian for z = ln | Xa | to obtain | Ψ × X, − X ( z ) (cid:105) = −√ N ( z ) e − (2 √ σ + D ) z (cid:90) D φ ˆ A ( a ) ˆ A ( − a ) e − S ∗ | φ (cid:105) = −√ N ( z ) e − (2 √ σ + D ) z (cid:90) D φ (cid:90) d D yd D y (cid:48) J (1 , a − y J (1 , − a − y (cid:48) O y O y (cid:48) e − S ∗ | φ (cid:105) , (138)where a is the short distance cutoff length scale. In the second equality, we use the expressionfor ˆ A in Eq. (130). By using Eq. (135) for ˆ A , we rewrite Eq. (138) as | Ψ × X, − X ( z ) (cid:105) = −√ N ( z ) e − (2 √ σ + D ) z (cid:90) D φ (cid:104) J (2 , a + ˆ A ( a, − a ) (cid:105) e − S ∗ | φ (cid:105) . (139)We expand ˆ A ( a, − a ) in a to obtain | Ψ × X, − X ( z ) (cid:105) = −√ N ( z ) e − (2 √ σ + D ) z (cid:90) D φ (cid:104) J (2 , a + ˆ A (0 ,
0) + ... (cid:105) e − S ∗ | φ (cid:105) , (140)where the ellipsis includes derivative terms such as a∂ x ˆ A ( x, y ) | x = y =0 and − a∂ y ˆ A ( x, y ) | x = y =0 .Finally, the backward evolution for RG time − z restores the initial state, | Ψ × X, − X (0) (cid:105) = −√ N (0) (cid:90) D φ (cid:104) e − (2 √ σ + D ) z J (2 , a + ˆ A (0 ,
0) + ... (cid:105) e − S ∗ | φ (cid:105) . (141)Comparing this with Eq. (137), we obtain the OPE for two ˆ A operators inserting at X and − X as ˆ A ( X ) × ˆ A ( − X ) = J (2 , a (cid:12)(cid:12)(cid:12) aX (cid:12)(cid:12)(cid:12) ˆ A + ˆ A (0 ,
0) + ..., (142)where ˆ A = and ∆ = √ σ + D/
2. Eq. (142) shows the channels in which two single-traceoperators fuse into a local double-trace operator with spin 0 and the identity operator. Theellipsis includes double-trace operators with larger spins and descendants. Following theprocedure explained in Sec. III C 5, one can compute the β -functions for general multi-tracecouplings. F n,n (cid:48) ,l ( x ) introduced in Eq. (65) is independent of x because the RG Hamiltonian is quadratic in thiscase. VI. CONCLUSION AND DISCUSSION
In this paper, we show that the full β -function of the exact Wilsonian RG is completelyfixed by the β -function defined in the subspace of single-trace couplings. We establish thisgeneral constraints on β -functions using the quantum RG, which is an exact reformulationof the Wilsonian RG. In quantum RG, the conventional RG flow in the space of couplings isreplaced with a quantum evolution of a wavefunction defined in the subspace of single-tracecouplings, where fluctuations of the dynamical single-trace couplings encode the informationabout all multi-trace couplings. Since the quantum evolution of the RG flow is completelydetermined from the β -functions defined in the subspace of single-trace couplings, the fullWilsonian β -functions can be extracted from the β -functions defined in the subspace.We explain the algorithm for reconstructing the full Wilsonian RG using toy examples.The procedure consists of two steps. First, we construct the RG Hamiltonian that gener-ates the quantum RG flow from the β -functions defined in the subspace of the single-tracecouplings. Second, we establish the correspondence between the ground state of the RGHamiltonian with the stable IR fixed point. Similarly, excited states with local excitationsare mapped to the IR fixed point deformed with corresponding local operators. The ener-gies of the excited states determine the scaling dimensions of the local operators. From thecompleteness of the eigenstates of the RG Hamiltonian, one can also extract the OPE co-efficients among general operators and reconstruct the full β -functions. The toy models weuse in this paper are generalized free theories that give rise to quadratic RG theories. Nev-ertheless, the same prescription can be applied to general theories whose RG Hamiltonianis not quadratic. In large N theories, interactions in the bulk are suppressed by 1 /N , andquadratic bulk theories are a good starting point to include 1 /N corrections perturbativelyin the bulk.We conclude with a few open questions. First, the β -functions in the subspace of single-trace couplings include the information on all fixed points that exist away from the subspaceas is discussed in Sec. IV. As multiple fixed points collide with a parameter of the theorytuned, the stable fixed point can disappear in the space of real couplings. It will be ofinterest to understand how the loss of conformality or an appearance of non-unitary fixedpoints [38–41] manifests itself in the bulk. Second, it would be interesting to consider casesin which the bulk RG Hamiltonian supports multiple ground states. One can consider afew scenarios in which degenerate ground states for the RG Hamiltonian arise. Degenerateground states can be related to each other through symmetry, in which case the emergenceof degenerate ground state is a sign of a spontaneous symmetry breaking. A degeneracy canalso arise due to a suppression of tunneling between topologically distinct RG paths[42].Finally, an exactly marginal deformation gives rise to degenerate ground states,1 ACKNOWLEDGEMENT
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In this appendix, we show the explicit form of an RG Hamiltonian that generates theexact RG flow for the φ - theory[6, 30]. Let us consider a D -dimensional scalar field theorywhose Euclidean action is written as S = S + S . (A1)Here S is a quadratic action, S = 12 (cid:90) d D k G − ( k ) φ k φ − k , (A2)where k is momentum, and G − ( k ) = e k k is a regularized kinetic term that suppressesfluctuations at momenta larger than UV cutoff Λ. S is a deformation that includes inter-actions and higher derivative terms. The standard exact RG flow equation can be obtainedby lowering the UV cut-off as Λ → Λ e − dz followed by a rescaling of field and momentum, φ k → e D +22 dz φ e dz k [6]. The correction to the effective action generated from this coarsegraining is obtained by applying an RG Hamiltonian to the wavefunction e − S as e − Hdz e − S = e − S − δS , (A3)where δS is the correction to the effective action, andˆ H = e − S (cid:90) d D k (cid:34) ˜ G ( k )2 π k π − k − i (cid:18) D + 22 φ k + k∂ k φ k (cid:19) π − k + C (cid:35) e S (A4)is the RG Hamiltonian. Here π k = − i δδφ − k is the conjugate momentum of φ k . ˜ G ( k ) = ∂G Λ ( k ) ∂ ln Λ is the propagator of the high-energy modes that are integrated out in the coarse4graining scheme. C = − (cid:82) d D kδ D (0) (cid:104) ˜ G G − + 1 (cid:105) is a constant. One can check that theRG Hamiltonian leaves the trivial state invariant, (cid:104) | H = 0, and the partition function isinvariant under the RG evolution, (cid:104) | S (cid:105) = (cid:104) | e − dzH | S (cid:105) . (A5) Appendix B: Solving Non-Hermitian Harmonic oscillator
The RG Hamiltonian considered in the paper takes the following form, H = 12 m π x + 12 mω ( x + iγ π x ) , (B1)where x ( π x ) corresponds to p ( j ) in the RG Hamiltonian in the main text. The conjugatevariables satisfy the commutation relation [ x , π x ] = i . The RG Hamiltonian is invariantunder the P and T symmetries, P x P = − x , P π x P = − π x , P i P = i, T x T = x , T π x T = − π x , T i T = − i. (B2)Under the similarity transformation S = e γ π x , the non-Hermitian RG Hamiltonian istransformed to an Hermitian RG Hamiltonian as H ≡ SHS − = (cid:20) m π x + 12 mω x (cid:21) . (B3)In terms of the n -th eigenstate | ψ n (cid:105) of H , the right and left eigenstates of H are given by | ψ Rn (cid:105) = S − | ψ n (cid:105) , (cid:104) ψ Ln | = (cid:104) ψ n | S (B4)with eigenvalue ( n + ) ω . Their wavefunctions in the π x basis are given by ψ Rn ( π x ) = (cid:104) π x | ψ Rn (cid:105) = N R e − ( mω + γ ) π x H n ( π x √ mω ) ,ψ Ln ( π x ) = (cid:104) π x | ψ Ln (cid:105) = N L e − ( mω − γ ) π x H n ( π x √ mω ) , (B5)where N R ( L ) are the normalization constants. The right (left) eigenstates are normalizableif γ > − mω ( γ < mω ).From the standard ladder operators of the Hermitian Hamiltonian, a † = (cid:114) mω x − imω π x ) , a = (cid:114) mω x + imω π x ) , (B6)5the raising and lowering operators for the non-Hermitian Hamiltonian can be obtained viathe similarity transformation, a = S − a † S = (cid:114) mω (cid:20) x + i ( γ − mω ) π x (cid:21) ,a = S − aS = (cid:114) mω (cid:20) x + i ( γ + 1 mω ) π x (cid:21) . (B7)The non-Hermitian RG Hamiltonian can be written in terms of a and a as H = ωa a + ω . (B8) Appendix C: β -functions for the -dimensional model in limiting cases In this appendix, we analyze the RG flow of the 0-dimensional example in various limits. • w → − limit with a < b > w → − limit, ξ R approaches 0 + . In this limit, the two fixed points are pushedto the region with large J as is shown in Fig. 4(a). As ξ R vanishes in the small w limit, the eigenstates of the RG Hamiltonian become non-normalizable. • w → − limit with a > b > ξ R = a b and ε = ab are finite and positive, and the right wave function isnormalizable. Because ε − ξ R = 0, the β -functions become ddz J = −√ ηJ + O (cid:16) ( J − J ∗ ) (cid:17) ,ddz J = √ η ξ R − √ ηJ + O (cid:16) ( J − J ∗ ) (cid:17) . The unstable fixed point is pushed to positive infinity, and only the stable fixed pointlocated at ( J ∗ , J ∗ ) − = (cid:16) , ξ R (cid:17) survives. • b → + limit with w < a < b limit, ξ R = b ( | a | + a − bw | a | ) and ε − ξ R = b ( | a | − a − bw | a | ). For a < ξ R = w a and ε − ξ R ≈ − a b → ∞ + . The β -functions in Eq. (89) and Eq. (90)become ddz J = −√ η √ ε − ξ R ) ξ R (cid:34) √ ξ R + ( ε − ξ R )4( ε − ξ R ) ξ R − J (cid:35) J + O (cid:16) ( J − J ∗ ) (cid:17) ,ddz J = (cid:112) η ε − ξ R ξ R (cid:16) J − ξ R (cid:17) − √ η (cid:16) J − ξ R (cid:17) + O (cid:16) ( J − J ∗ ) (cid:17) . ε − ξ R approaches infinity in the small b limit, the second term in ddz J is neg-ligible. Thus, the two fixed points are getting closer to each other until they collideat ( J ∗ , J ∗ ) ± = (cid:16) , ξ R (cid:17) . This is shown in Fig. 4(c , ). Before collision, the stable(unstable) fixed point has two irrelevant (relevant) directions. As the fixed pointscollide, both directions become marginal because the negative and positive scalingdimensions can meet only at 0. It turns out that the perturbations are marginallyirrelevant from one side and marginally relevant from the other side. If b becomes neg-ative, the normalizable ground state disappears, which suggests a loss of stable fixedpoint in the real space of the couplings [38–41]. It will be of interest to understandconstraints on the range of conformality from unitarity. • b → + limit with w < a > a > ξ R = a b → ∞ + and ε − ξ R = − w a is finite. In the b → + limit, theunstable fixed point moves towards the positive infinity, while the stable fixed pointmoves to the origin. The β -functions in this limit are given by ddz J ≈ −√ ηJ + O (cid:16) ( J − J ∗ ) (cid:17) ,ddz J ≈ − √ ηJ + O (cid:16) ( J − J ∗ ) (cid:17) , where η ≈ a . Appendix D: Solution to the Schrodinger equation for the time-dependent harmonicoscillator
We first review the evolution of a time-dependent harmonic oscillator with Hamiltonian H = 12 x + 12 Ω z π x , (D1)where Ω z is a function of imaginary time z . An initial state can be written as a superpositionof the eigenstates of the instantaneous Hamiltonian at z = 0 as Ψ [ π x ,
0] = (cid:80) m c m Ψ m [ π x , m [ π x ,
0] = 1 √ n n ! (cid:16) Ω π (cid:17) / H n (cid:104)(cid:112) Ω π x (cid:105) exp (cid:20) −
12 Ω π x (cid:21) . (D2)The time dependent state satisfies the Schrodinger equation, − ∂ ∂ π x Ψ[ π x , z ] + 12 Ω z π x Ψ[ π x , z ] = − ∂ Ψ [ π x , z ] ∂z . (D3)7We introduce a new variable, ξ = π x A z and write the time-dependent solution asΨ [ π x , z ] = C z exp (cid:2) − A z K z ξ (cid:3) ˜Ψ [ ξ, θ ] , (D4)where K z , C z and θ z are function of z which are related to A z through K z = ˙ A z A z , ˙ C z = − C z K z and ˙ θ ( z ) = 1 /A z with ˙ A z = dA z /dz . ˜Ψ [ ξ, θ ] satisfies12 A z (cid:104) ¨ A z − A z Ω z (cid:105) ξ ˜Ψ [ ξ, θ ] − δ ˜Ψ [ ξ, θ ] δθ + 12 δ δξ ˜Ψ [ ξ, θ ] = 0 (D5)We choose A z so that ¨ A z − A z Ω z = 0, and ˜Ψ satisfies − δ ˜Ψ [ ξ, θ ] δθ + 12 δ δξ ˜Ψ [ ξ, θ ] = 0 . (D6)This is the Schrodinger equation of a particle in the free space, which has a one-parameterfamily of solutions, ˜Ψ (cid:96) [ ξ, θ ] = exp (cid:20) − (cid:96) θ ( z ) (cid:21) exp [ − i(cid:96)ξ ( z )] , (D7)where (cid:96) is a real parameter that corresponds to the momentum conjugae to ξ in the particleanalogy. The general solution is given by a linear superposition of Ψ (cid:96) asΨ [ π x , z ] = 1 √ A z exp (cid:16) − ˙ A z A z π x (cid:17) (cid:90) ∞−∞ d(cid:96) (cid:110) φ [ (cid:96) ] exp (cid:20) − (cid:96) (cid:90) z dz (cid:48) A z (cid:48) (cid:21) exp (cid:20) − i (cid:96) π x A z (cid:21) (cid:111) , (D8)where φ [ (cid:96) ] is the weight for the mode labelled by (cid:96) , and θ ( z ) = (cid:82) z dz (cid:48) A z (cid:48) is used. At z = 0,the initial conditions ˙ A = 0 and A = 1 lead toΨ [ π x ,
0] = (cid:90) d(cid:96) (cid:110) φ [ (cid:96) ] exp( − i(cid:96) π x ) (cid:111) . (D9)Thus, φ [ (cid:96) ] is the Fourier transformation of Ψ [ π x , φ [ (cid:96) ] = 12 π (cid:90) Ψ (cid:2) π (cid:48) x , (cid:3) exp( i(cid:96) π (cid:48) x ) d π (cid:48) x (D10)The n -th eigenstate of the H at z = 0 has the Fourier components given by φ n [ (cid:96) ] = 12 π √ n n ! (cid:16) Ω π (cid:17) / (cid:90) H n (cid:104)(cid:112) Ω π x (cid:105) exp (cid:16) −
12 Ω π x (cid:17) exp [ i(cid:96) π x ] d π x = √ π √ n n (cid:16) π Ω (cid:17) / H n (cid:16) − i (cid:112) Ω δδ(cid:96) (cid:17) exp (cid:20) − (cid:96) (cid:21) . (D11)8Because the Hermite polynomial is complete, φ n [ (cid:96) ] can be decomposed as a linear super-position of ˜ φ n (cid:48) [ (cid:96) ] = √ π √ n (cid:48) n (cid:48) ! (cid:16) π Ω (cid:17) / H n (cid:48) (cid:20) i (cid:96) √ Ω (cid:21) exp (cid:20) − (cid:96) (cid:21) . (D12)Inserting Eq. (D12) into Eq. (D8), we obtain a solution,Ψ n (cid:48) [ π x , z ] = √ π π √ n (cid:48) n (cid:48) ! (cid:16) π Ω (cid:17) / (cid:114) ω z A z exp (cid:16) − ˙ A z A z π x (cid:17) H n (cid:48) (cid:20) − A z √ Ω δδ π x (cid:21) exp (cid:20) − ω z A z π x (cid:21) , (D13)where we use ω z = (cid:20)(cid:82) z dz (cid:48) A z (cid:48) + (cid:21) − as in the main context. First three states are given byΨ [ π x , z ] = 1 √ π (cid:16) π Ω (cid:17) / (cid:114) ω z A z exp (cid:34) − (cid:16) ˙ A z A z + ω z A z (cid:17) π x (cid:35) , Ψ [ π x , z ] = 1 √ π (cid:16) π Ω (cid:17) / (cid:114) ω z A z exp (cid:34) − (cid:16) ˙ A z A z + ω z A z (cid:17) π x (cid:35) (cid:20) ω z A z √ Ω π x (cid:21) , Ψ [ π x , z ] = 12 √ π (cid:16) π Ω (cid:17) / (cid:114) ω z A z exp (cid:34) − (cid:16) ˙ A z A z + ω z A z (cid:17) π x (cid:35) (cid:34) (cid:18) ω z A z √ Ω π x (cid:19) − − ω z Ω (cid:35) . (D14)Now we consider the non-Hermitian RG Hamiltonian which is of our interest : H (cid:48) = 12 ( x + iγ π x ) + 12 Ω z π x . (D15)This is related to Eq. (D1) through the similarity transformation, H (cid:48) = e − γ π x He γ π x .Accordingly, its solution is related to Eq. (D13) through Ψ (cid:48) n = e − γ π x Ψ n ,Ψ (cid:48) n [ π x , z ] = √ π π √ n n ! (cid:16) π Ω (cid:17) / (cid:114) ω z A z exp (cid:34) − (cid:16) γ + ˙ A z A z (cid:17) π x (cid:35) H n (cid:20) − A z √ Ω δδ π x (cid:21) exp (cid:20) − ω z A z π x (cid:21) , (D16)where we use ω z = (cid:20)(cid:82) z dz (cid:48) A z (cid:48) + (cid:21) − as in the main context. Appendix E: Computation of A s,z , ω s,z , Λ s,z and ∆ s,z In this appendix, we provide the expressions for A s,z , ω s,z , Λ s,z and ∆ s,z that appear inthe solution for the RG Hamiltonian in Sec. V. Since the expressions of A s,z , ω s,z , Λ s,z and∆ s,z are the same for s = ( R ; K ) and s = ( I ; K ), we will just denote them as A K,z , ω K,z Λ K,z and ∆
K,z in this appendix. For b = 1 /
2, we express the solution for the Schrodingerequation Eq. (108) in terms of α = − g , ζ = a + D and σ = ζ − w .9 A s,z We start with A s,z that satisfies ¨ A s,z − A s,z Ω s,z = 0 with the initial conditions ˙ A , = 0and A , = 1. For K = 0, A ,z = cosh( √ σz ) is the solution. For general K , A K,z is givenby A K,z = π √ α π √ σ ) | K | (cid:110) I −√ σ (cid:2) √ α | K | e z (cid:3) (cid:16) I − √ σ (cid:2) √ α | K | (cid:3) + I √ σ (cid:2) √ α | K | (cid:3) (cid:17) − I √ σ (cid:2) √ α | K | e z (cid:3) (cid:16) I − −√ σ (cid:2) √ α | K | (cid:3) + I −√ σ (cid:2) √ α | K | (cid:3) (cid:17)(cid:111) . (E1)In the large z limit with fixed k ≡ Ke z , A K,z can be written as˜ A k,z ≡ lim z →∞ A K,z (cid:12)(cid:12)(cid:12) k = A k ( α, σ ) e √ σz , (E2)where A k ( α, σ ) ≈ − − √ σ πα − √ σ sin( π √ σ ) I √ σ [ √ α | k | ]Γ( −√ σ ) | k | −√ σ . (E3)This has the following limiting behaviour in k = Ke z . a. | K | e z (cid:28) : For √ α | K | e z (cid:28) (cid:112) − √ σ < (cid:112) √ σ , one can approximate A K,z to A K,z ∼ A K, e −√ σz + A K, e √ σz , (E4)where A K, = π √ α −√ σ −√ σ sin( π √ σ ) 1Γ(1 − √ σ ) | K | −√ σ (cid:16) I − √ σ (cid:2) √ α | K | (cid:3) + I √ σ (cid:2) √ α | K | (cid:3) (cid:17) , A K, = − π √ α √ σ √ σ sin( π √ σ ) 1Γ(1 + √ σ ) | K | √ σ (cid:16) I − −√ σ (cid:2) √ α | K | (cid:3) + I −√ σ (cid:2) √ α | K | (cid:3) (cid:17) . (E5)In the large z limit with fixed k , Eq. (E5) becomes˜ A k, ≡ lim z →∞ A K, (cid:12)(cid:12)(cid:12) k = 12 + αk e − z √ σ (1 + √ σ ) + O (cid:16) ( ke − z ) (cid:17) , ˜ A k, ≡ lim z →∞ A K, (cid:12)(cid:12)(cid:12) k = 12 + αk e − z √ σ ( √ σ −
1) + O (cid:16) ( ke − z ) (cid:17) . (E6) b. | K | e z (cid:29) : For | K | e z (cid:29) A K,z can be expanded in powers of 1 / | K | e z , A K,z = A K, e √ α | K | e z e z/ (cid:20) −
18 (4 σ − √ α | K | e z + O (cid:18) Ke z ) (cid:19)(cid:21) , (E7)0where A K, = π / α / g K √ π √ σ ) | K | / (E8)and g K = I − √ σ (cid:2) √ α | K | (cid:3) + I √ σ (cid:2) √ α | K | (cid:3) − I − −√ σ (cid:2) √ α | K | (cid:3) − I −√ σ (cid:2) √ α | K | (cid:3) . (E9)In the large z limit with fixed k , we obtain˜ A k, ≡ lim z →∞ A K, (cid:12)(cid:12)(cid:12) k = V k − −√ σ e ( + √ σ ) z + O (cid:16) ( ke − z ) −√ σ (cid:17) , (E10)where V = − π / α − / − √ σ −√ σ + sin( π √ σ ) 1Γ( −√ σ ) . (E11) ω s,z Here, we show that˜ ω k,z ≡ lim z →∞ ω K,z (cid:12)(cid:12)(cid:12) k ≈ √ σ (cid:104) − W k e − √ σz (cid:105) − , (E12)where W k ≈ (cid:110) k (cid:28) k √ σ (cid:104) − √ σ V √ α ( e − √ α − e − √ αk ) (cid:105) k (cid:29) z limit with fixed k = Ke z . a. | K | e z (cid:28) : Using Eq. (E4), we obtain ω K,z = (cid:34) K, + (cid:90) z dz (cid:48) A K,z (cid:48) (cid:35) − ≈ (cid:34) K, + e √ σz − √ σ ( A K, + A K, )( A K, + A K, e √ σz ) (cid:35) − . (E14)According to Eq. (E6), we find˜ ω k,z ≡ lim z →∞ ω K,z (cid:12)(cid:12)(cid:12) k ≈ √ σ (cid:104) − e − √ σz (cid:105) − , (E15)where we used lim z →∞ Ω K, | k = √ σ b. | K | e z (cid:29) : According to Eq. (E7), we have ω K,z ≈ (cid:104) K, + K − √ σ − √ σ ( A K, + A K, )( A K, + A K, K − √ σ ) + 1 A K, (cid:16) − e − √ α | K | e z √ α | K | + e − √ α √ α | K | (cid:17)(cid:105) − . (E16)In the large z limit, based on Eq. (E10) we find˜ ω k,z ≡ lim z →∞ ω K,z (cid:12)(cid:12)(cid:12) k ≈ √ σ (cid:20) − k √ σ (cid:16) − √ σ V √ α ( e − √ α − e − √ αk ) (cid:17) e − √ σz (cid:21) − , (E17)for k (cid:29) Λ s,z z z z Λ K,z K Λ K,z z Λ K,z ( a ) ( b ) ( c )( d ) ( e ) ( f ) Λ K,z
K d Λ K,z /dzd Λ K,z /dz
FIG. 8. Λ
K,z plotted as a functions of K or z with α = 1 in D = 1. For (a), (b), (c), we choose ζ = − . σ = 2 .
01. For (d), (e), (f), we choose ζ = 0 . σ = 0 .
01. (a) Λ
K,z vs z at K = 0 . .
005 (orange), 0 .
01 (green), 0 .
05 (red) and 0 . K,z vs K at z = 2 (blue), 3(orange), 4 (green), 5 (red). (c) ddz Λ K,z vs z for K = 0 . K = 0 .
005 and K = 0 .
01. For eachvalue of K , the minimum occurs at z = 7 . z = 6 . z = 5 . Ke z takes the same value, 2 . K,z vs z at K = 0 .
001 (blue), 0 .
005 (orange), 0 . .
05 (red) and 0 . K,z vs K at z = 2 (blue), 3 (orange), 4 (green), 5 (red).(f) ddz Λ K,z vs z for K = 0 . K = 0 .
005 and K = 0 .
01. For each value of K , the minimum islocated at z = 6 . z = 4 . z = 4 . Ke z takes the same value,0 . k ˜Λ k k ( a ) ( b ) ˜Λ k FIG. 9. Λ
K,z = ˜Λ k vs k for various values of z between 0 and 20 in D = 1. The arrows pointtowards the direction of increasing z . (a) α = 1, ζ = − . σ = 2 .
01 (b) α = 1, ζ = 0 . σ = 0 .
01. The curves converge to a universal one in the large z limit. In this section, we compute Λ s,z defined by1Λ s,z = (cid:34) ζ + ˙ A s,z A s,z + ω s,z A s,z (cid:35) , (E18)where ω s,z = (cid:20)(cid:82) z dz (cid:48) A s,z (cid:48) + s, (cid:21) − . In Fig. 8, we show the profile of Λ K,z for two sets ofparameters. Λ
K,z smoothly interpolates the two limiting behaviours of the Ke z (cid:29) Ke z (cid:28) z limit with fixed k = Ke z , Λ K,z approaches auniversal function as is shown in Fig. 9. Now, let us find the analytic expression for˜Λ k = lim z →∞ Λ K,z (cid:12)(cid:12)(cid:12) k , (E19)where the limit is taken with k = Ke z fixed.In the large z limit, as we shown in Eq. (E2) and Eq. (E12), ˜ ω k,z approaches √ σ ,and ˜ ω k,z (cid:28) ˜ A k,z . So the dominant contribution to Λ K,z in Eq. (E18) is from (cid:104) ˙ A K,z A K,z + ζ (cid:105) − .Therefore, Λ K,z approaches a universal form as a function of k = | K | e z ,˜Λ k = [ G k ( α, σ ) + ζ ] − , (E20)as the large z limit is taken with fixed k , where G k ( α, σ ) = ˙ A K,z A K,z = 12 √ α | k | I σ [ √ α | k | ] ( I − √ σ (cid:2) √ α | k | (cid:3) + I √ σ (cid:2) √ α | k | (cid:3) ) . (E21) G k ( α, σ ) becomes G k ( α, σ ) ≈ √ σ (E22)3for k (cid:28)
1, and G k ( α, σ ) ≈ √ α | k | (E23)for k (cid:29)
1. In order for the wavefunction to be normalizable, the width of the Gaussianwavefunction in Eq. (112) should be finite. This requires Λ
K,z > K and z . This,in turn, implies that Ω K, > − ζ for all K , equivalently √ σ > − ζ . ∆ s,z ∆ K,z K z ∆ K,z z ∆ K,z ∆ K,z K z ∆ K,z z ( a ) ( b ) ( c )( d ) ( e ) ( f ) ∆ K,z
FIG. 10. (a) and (d): ∆
K,z plotted as a function of K at z = 1 (blue), z = 2 (orange), z = 3(green) and z = 4 (red) for (a) ζ = 0 . σ = 0 .
01, and (d) ζ = − . σ = 2 .
01. (b) and (e): ∆
K,z plotted as a function of z at K = 0 .
001 (blue), K = 0 .
01 (orange), K = 0 .
05 (green) and K = 0 . ζ = 0 . σ = 0 .
01, and (e) ζ = − . σ = 2 .
01. (c) and (f) are (b) and (e) shown in thelogarithmic scale. In all plots, we set D = 1 and α = 1. According to Eq. (E2) and Eq. (E12), we have expressed Ω K, , A K,z and ω K,z in termsof k and z . In the large z limit, we have e − ∆ K,z = ω K,z A K,z (cid:112) Ω K, ≈ σ / A Ke z ( α, σ ) e −√ σz . (E24)This analytical expression is consistent with the numerical plot of ∆ K,z shown in Fig. 10.As a function of k , ∆ K,z at different z behave in the same way except for a vertical shift, asis shown in Fig. 11. This agrees with our analytical expression, ∆ K,z = − log 2 − log σ +log A k + √ σz . Under the RG transformation from length scale z to z + dz followed by therescaling of K to Ke − z , ∆ K,z transforms to ∆
K,z + √ σdz .4 ∆ K,z k ( a ) ( b ) ∆ K,z k FIG. 11. ∆
K,z as a function of k = Ke z at different values of z ranging from z = 22 (bottom) to z = 60 (top) for (a) ζ = 0 . σ = 0 .
01, and (b) ζ = − . σ = 2 . D = 1 and α = 1 are used forboth plots. Appendix F: Numerical calculation of J ∗ ,x − x (cid:48) It is hard to obtain a full expression for J ∗ ,x − x (cid:48) in a closed form. In this appendix, wecompute it numerically for D = 1. This requires UV and IR regularizations. Consider aone-dimensional lattice with N sites. Before doing the scale transformation in Eq. (96), ifthe lattice spacing is a , a function f X = xe z ,z at z = 0 can be expressed as f X, = 1 N N (cid:88) m =1 f πNa m e i πNa mX = a (cid:90) Λ ∼ πa dK π f K, e iKX . (F1)For z (cid:54) = 0, the lattice spacing increases to ae z and the number of sites decreases to N ( z ) = N e − z . Then Eq. (F1) becomes f xe z ,z = 1 N ( z ) N ( z ) (cid:88) m =1 f πN ( z ) aez m e i πN ( z ) a mx = a (cid:90) Λ e − z e z dK π f K,z e iKX = a (cid:90) Λ dk π f ke − z ,z e ikx , (F2)where k = Ke z and x = e − z X . If f ke − z ,z = ˜ f k is independent of z for a fixed k , ˜ f x = f xe z ,z is scale invariant, i.e. z -independent. Since Λ K,z = ˜Λ k , the profile of J ,x − x (cid:48) would beinvariant under RG transformation.Now, let us numerically compute J ∗ ,x − x (cid:48) , which is expressed as J ∗ ,x − x (cid:48) = 1 N ( z ) N ( z ) (cid:88) m =1 ˜Λ πmN ( z ) cos (cid:20) πmN ( z ) ( x − x (cid:48) ) (cid:21) (F3)5with a = 1. The profile is shown in Fig. 5. For large enough system size N , the coupling inthe real space reaches a z -independent profile at large z provided N e − z (cid:29)
1. This profileis universal because it does not depend on Ω K, at UV. We note that there are regions ofnegative coupling at large | x − x (cid:48) | . We attribute this phenomenon as a finite size effect. InFig. 12, as the system size N increases, the coupling becomes more positive. -4 -2 2 40.10.20.30.4 x − xJ ∗ ,x − x ( a ) -5 50.20.40.60.8 J ∗ ,x − x x − x ( b ) FIG. 12. (a) J ∗ ,x − x (cid:48) in D = 1 plotted as a function of x − x (cid:48) for σ = 2 .
01 and ζ = − . z = 26at N = e (red), N = e (green), N = e (blue), N = e (black), N = e (orange). (b) Theprofile at σ = 0 .
01 and ζ = 0 .
1. Curves with a same color are the ones with a same N . α is set to 1. Appendix G: Wavefunction with one excited mode in the D-dimensional example
Suppose that the mode s is in its first excited state, where s can be either P = 0, ( R ; P )or ( I ; P ). The wavefunction for mode s is given byΨ ,s (cid:2) ˜ j s , z (cid:3) = π − / e − ∆ s,z (cid:16) √ e − ∆ s,z ˜ j s (cid:17) exp (cid:20) − s,z ˜ j s (cid:21) , (G1)where H ( x ) = 2 x is used. The excited state corresponds to the following state in therescaled variables in the large z limit, | Ψ , ( z ) (cid:105) = N ( z ) (cid:90) D φ e − S (cid:16) i √ σ / A e −√ σz ˜Λ O (cid:17) e − S ∗ | φ (cid:105) , | Ψ , ( R ; P ) ( z ) (cid:105) = N ( z ) (cid:90) D φ (cid:16) i σ / A P e z e −√ σz ˜Λ P e z ( O P e z + O − P e z ) (cid:17) e − S ∗ | φ (cid:105) , | Ψ , ( I ; P ) ( z ) (cid:105) = N ( z ) (cid:90) D φ (cid:16) σ / A P e z e −√ σz ˜Λ P e z ( O P e z − O − P e z ) (cid:17) e − S ∗ | φ (cid:105) , (G2)where we used e − ∆ K,z ≈ σ / A Kez ( α,σ ) e −√ σz and k = e z K . N ( z ) is the z -dependent normal-ization of the ground state in Eq. (123). P labels the initial momentum of the excitedmode. It is scaled to be P e z as z increases. For P (cid:54) = 0, we can construct the excited state6with a momentum ± P by making linear superpositions of | Ψ , ( R ; P ) ( z ) (cid:105) and | Ψ , ( I ; P ) ( z ) (cid:105) : | Ψ , ± P ( z ) (cid:105) = √ (cid:16) | Ψ , ( R : P ) ( z ) (cid:105) ± i | Ψ , ( I : P ) ( z ) (cid:105) (cid:17) . This leads to Eq. (126). Appendix H: Possible wavefunctions with two excited modes in the D-dimensionalexample
In order to derive Eq. (132), we first list the wave functions for two excited modes asΨ , , (cid:2) ˜ j , z (cid:3) = π − / e − ∆ ,z √ (cid:20) e − ∆ ,z ˜ j ) − − ω ,z Ω , (cid:21) exp (cid:20) − ,z ˜ j (cid:21) (H1)= 1 √ , (cid:2) ˜ j , z (cid:3) × Ψ , (cid:2) ˜ j , z (cid:3) − √ ω ,z Ω , )Ψ , (cid:2) ˜ j , z (cid:3) , Ψ , ( S ; P ) , ( S ; P ) (cid:2) ˜ j S ; P , z (cid:3) = π − / e − ∆ P,z √ (cid:20) e − ∆ P,z ˜ j S ; P ) − − ω P,z Ω P, (cid:21) exp (cid:20) − P,z ˜ j S ; P (cid:21) = 1 √ , ( S ; P ) (cid:2) ˜ j S ; P , z (cid:3) × Ψ , ( S ; P ) (cid:2) ˜ j S ; P , z (cid:3) − √ ω P,z Ω P, )Ψ , ( S ; P ) (cid:2) ˜ j S ; P , z (cid:3) , Ψ , ( S ; P ) , ( S (cid:48) ; P (cid:48) ) (cid:54) =( S ; P ) = Ψ ,S ; P (cid:2) ˜ j S ; P , z (cid:3) × Ψ ,S (cid:48) ; P (cid:48) (cid:2) ˜ j S (cid:48) ; P (cid:48) , z (cid:3) . Here S and S (cid:48) can be R or I . Using Eq. (E12) and Eq. (E24), we rewrite the excited statesin terms of rescaled variables p in the large z limit as | ¯Ψ , , ( z ) (cid:105) = N ( z ) √ (cid:90) D φ (cid:16) − σ / A (cid:104) ˜Λ O (cid:105) (cid:17) e − S ∗ | φ (cid:105) + 1 √ (cid:16) σ / A e − √ σz ˜Λ − − (cid:104) − W e − √ σz (cid:105) (cid:17) | Ψ ( z ) (cid:105) , | Ψ , ( R ; P ) , ( R ; P ) ( z ) (cid:105) = N ( z ) √ (cid:90) D φ (cid:16) − σ / A P e z ˜Λ P e z ( O P e z + O − P e z ) (cid:17) e − S ∗ | φ (cid:105) + 1 √ (cid:16) σ / A P e z e − √ σz ˜Λ P e z − − √ σ √ αP + σ (cid:104) − W P e z e − √ σz (cid:105) (cid:17) | Ψ ( z ) (cid:105) , | Ψ , ( I ; P ) , ( I ; P ) ( z ) (cid:105) = N ( z ) √ (cid:90) D φ (cid:16) σ / A P e z ˜Λ P e z ( O P e z − O − P e z ) (cid:17) e − S ∗ | φ (cid:105) + 1 √ (cid:16) σ / A P e z e − √ σz ˜Λ P e z − − √ σ √ αP + σ (cid:104) − W P e z e − √ σz (cid:105) (cid:17) | Ψ ( z ) (cid:105) , | Ψ , ( R ; P ) , ( I ; P ) ( z ) (cid:105) = N ( z ) (cid:90) D φ (cid:16) i σ / A P e z ˜Λ P e z (cid:2) O P e z − O − P e z (cid:3) (cid:17) e − S ∗ | φ (cid:105) , (H2)7 | Ψ , , ( R ; P ) ( z ) (cid:105) = N ( z ) (cid:90) D φ (cid:16) − √ σ / A P e z A ˜Λ ˜Λ P e z O [ O − P e z + O P e z ] (cid:17) e − S ∗ | φ (cid:105) , | Ψ , , ( I ; P ) ( z ) (cid:105) = N ( z ) (cid:90) D φ (cid:16) i √ σ / A P e z A ˜Λ ˜Λ P e z O [ O P e z − O − P e z ] (cid:17) e − S ∗ | φ (cid:105) , | Ψ , ( R ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) = N ( z ) (cid:90) D φ (cid:16) − σ / A P e z A P (cid:48) e z ˜Λ P (cid:48) e z ˜Λ P e z [ O P e z + O − P e z ] [ O P (cid:48) e z + O − P (cid:48) e z ] (cid:17) e − S ∗ | φ (cid:105) , | Ψ , ( R ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) = N ( z ) (cid:90) D φ (cid:16) iσ / A P e z A P (cid:48) e z ˜Λ P (cid:48) e z ˜Λ P e z [ O P e z + O − P e z ] [ O P (cid:48) e z − O − P (cid:48) e z ] (cid:17) e − S ∗ | φ (cid:105) , | Ψ , ( I ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) = N ( z ) (cid:90) D φ (cid:16) iσ / A P e z A P (cid:48) e z ˜Λ P (cid:48) e z ˜Λ P e z [ O P e z − O − P e z ] [ O P (cid:48) e z + O − P (cid:48) e z ] (cid:17) e − S ∗ | φ (cid:105) , | Ψ , ( I ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) = N ( z ) (cid:90) D φ (cid:16) σ / A P e z A P (cid:48) e z ˜Λ P (cid:48) e z ˜Λ P e z [ O P e z − O − P e z ] [ O P (cid:48) e z − O − P (cid:48) e z ] (cid:17) e − S ∗ | φ (cid:105) , (H3)where N ( z ) = N ( z ) e − √ σz . For P (cid:54) = P (cid:48) , we can superpose the wavefunctions above toobtain | Ψ ,P,P (cid:48) ( z ) (cid:105) = √ (cid:16) | Ψ , ( R ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) + i | Ψ , ( R ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) − | Ψ , ( I ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) + i | Ψ , ( I ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) (cid:17) , | Ψ ,P, − P (cid:48) ( z ) (cid:105) = √ (cid:16) | Ψ , ( R ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) − i | Ψ , ( R ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) + | Ψ , ( I ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) + i | Ψ , ( I ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) (cid:17) , | Ψ , − P,P (cid:48) ( z ) (cid:105) = √ (cid:16) | Ψ , ( R ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) + i | Ψ , ( R ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) + | Ψ , ( I ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) − i | Ψ , ( I ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) (cid:17) , | Ψ , − P, − P (cid:48) ( z ) (cid:105) = √ (cid:16) | Ψ , ( R ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) − i | Ψ , ( R ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) − | Ψ , ( I ; P ) , ( I ; P (cid:48) ) ( z ) (cid:105) − i | Ψ , ( I ; P ) , ( R ; P (cid:48) ) ( z ) (cid:105) (cid:17) , | Ψ , ,P ( z ) (cid:105) = 12 ( | Ψ , , ( R ; P ) ( z ) (cid:105) + i | Ψ , , ( I ; P ) ( z ) (cid:105) ) , | Ψ , , − P ( z ) (cid:105) = 12 ( | Ψ , , ( R ; P ) ( z ) (cid:105) − i | Ψ , , ( I ; P ) ( z ) (cid:105) ) . (H4)For non-zero P , we have | Ψ ,P,P ( z ) (cid:105) = 12 (cid:16) | Ψ , ( R ; P ) , ( R ; P ) ( z ) (cid:105) − | Ψ , ( I ; P ) , ( I ; P ) ( z ) (cid:105) + √ i | Ψ , ( R ; P ) , ( I ; P ) ( z ) (cid:105) (cid:17) , | Ψ , − P, − P ( z ) (cid:105) = 12 (cid:16) | Ψ , ( R ; P ) , ( R ; P ) ( z ) (cid:105) − | Ψ , ( I ; P ) , ( I ; P ) ( z ) (cid:105) − √ i | Ψ , ( R ; P ) , ( I ; P ) ( z ) (cid:105) (cid:17) , | Ψ ,P, − P ( z ) (cid:105) = 12 (cid:16) | Ψ , ( R ; P ) , ( R ; P ) ( z ) (cid:105) + | Ψ , ( I ; P ) , ( I ; P ) ( z ) (cid:105) (cid:17) + 1 √ (cid:20) √ σ √ αP + σ (cid:21) | Ψ ( z ) (cid:105) . (H5)Finally, together with | Ψ , , ( z ) (cid:105) = | ¯Ψ , , ( z ) (cid:105) + √ | Ψ ( z ) (cid:105) , (H6)8 | Ψ ,P,P (cid:48) ( z ) (cid:105) for any P and P (cid:48) can be written in the general form given in Eq. (132). Appendix I: Other scaling operators
In this section, we consider general excited states. The wavefunction for n excited modesis | Ψ n, { P } ( z ) (cid:105) = (cid:34) ( − i ) n √ n √ n ! (cid:35) N ( z ) (cid:90) D φ (cid:16) σ n/ (cid:81) ni ˜Λ P i e z n (cid:81) ni A P i e z e − n √ σz (cid:34) n (cid:89) i O P i e z (cid:35) + . . . (cid:17) e S ∗ | φ (cid:105) , (I1)where . . . includes terms with less number of O P e z operators. This wave function leads tostate of the system as | Ψ n, { X } ( z ) (cid:105) = 1 V n/ (cid:88) { P } e i (cid:80) ni P i X i | Ψ n, { P } ( z ) (cid:105) = (cid:34) ( − i ) n √ n √ n ! (cid:35) N ( z ) e − ( n √ σ + nD ) z (cid:90) D φ ˆ A n ( { x ≡ Xe − z } ) e − S ∗ | φ (cid:105) + . . . , (I2)where the scaling operator is defined asˆ A n ( { x } ) = [ n/ (cid:88) m =0 (cid:90) (cid:34) n − m (cid:89) i d D y i (cid:35) J ( n,n − m ) { x − y } (cid:34) n − m (cid:89) i O y i (cid:35) (I3)with J ( n,n − m ) { x − y } = (cid:90) (cid:34) n − m (cid:89) i d D p i (2 π ) D (cid:35) J ( n,n − m ) { p } e i (cid:80) n − mi p i ( x i − y i ) e i (cid:80) ni = n − m +1 p i x i δ ( n (cid:88) i = n − m +1 p i ) . (I4)Here J ( n,n − m ) represents the weight for ( n − m )-trace operators to the n -th scaling oper-ator. For example, the contribution from the n -trace operator is given by J ( n,n ) { p } = σ n/ (cid:81) ni ˜Λ p i n (cid:81) ni A p i . (I5)The local operator ˆ A n has scaling dimension n (cid:0) √ σ + D (cid:1)(cid:1)