Generalized Exact Holographic Mapping with Wavelets
GGeneralized Exact Holographic Mapping with Wavelets
Ching Hua Lee
Institute of High Performance Computing, 138632, Singapore ∗ andDepartment of Physics, National University of Singapore,117542, Singapore † (Dated: October 15, 2018)The idea of renormalization and scale invariance is pervasive across disciplines. It has not onlydrawn numerous surprising connections between physical systems under the guise of holographicduality, but has also inspired the development of wavelet theory now widely used in signal processing.Synergizing on these two developments, we describe in this paper a generalized exact holographicmapping that maps a generic N-dimensional lattice system to a N+1-dimensional holographic dual,with the emergent dimension representing scale. In previous works, this was achieved via theiterations of the simplest of all unitary mappings, the Haar mapping, which fails to preserve theform of most Hamiltonians. By taking advantage of the full generality of biorthogonal wavelets,our new generalized holographic mapping framework is able to preserve the form of a large class oflattice Hamiltonians. By explicitly separating features that are fundamentally associated with thephysical system from those that are basis-specific, we also obtain a clearer understanding of howthe resultant bulk geometry arises. For instance, the number of nonvanishing moments of the highpass wavelet filter is revealed to be proportional to the radius of the dual Anti deSitter (AdS) spacegeometry. We conclude by proposing modifications to the mapping for systems with generic Fermipockets. I. INTRODUCTION
The theme of holographic duality has fascinated ageneration of physicists in both high energy and con-densed matter circles. Also known as the Anti-de-Sitterspace/Conformal Field Theory (AdS/CFT) correspon-dence, it was pioneered by Witten, Maldacena, Klebanovand others in 1998, when an equivalence was made be-tween a D + 1-dimensional quantum field theory a D + 2-dimensional gravitational theory at the partition functionlevel. The canonical example of holographic duality is thecorrespondence between 3+1-dimensional super-Yang-Mills theory and 4+1-dimensional supergravity, with thelarge N (strongly-coupled) limit of the super-Yang-Millstheory being dual to the classical (weakly-coupled) limitof the gravitational theory. At the core of holographicduality is the interpretation of a quantum field theory asa ”hologram” of a dual gravitational system with onehigher dimension, with the extra emergent dimensionrepresenting scale. This provides an avenue to under-standing renormalization group (RG) flow dynamics interms of bulk gravitational dynamics . Inspired bythat, holographic duality has also been used as a toolfor understanding the nature of quantum criticality andhigh temperature superconductivity , for which theexact role of the underlying strong coupling mechanismremains elusive.In face of evidence for the existence of holographic du-ality in various contexts, it will be very desirable to havea microscopic description of holography. This allows for aclear, constructive approach for understanding the dualtheory, when it exists. For this purpose, an approachknown as the Exact Holographic Mapping (EHM) wasproposed by Qi for generic lattice systems. Throughrecursive applications of local unitary transforms, thismapping maps a given “boundary” system onto a “bulk” system with a unitary equivalent Hilbert space, but hav-ing an extra emergent dimension representing scale .Geodesics distances in the bulk system can be deter-mined from the decay behavior of their correlators. Al-though bulk systems obtained in this way via the EHMare not semiclassical bulk geometries corresponding tothe large N limit, in the strict sense of AdS-CFT, theypossess geometries agreeing with expectations from theRyu-Takayanagi formula . Notable examples includethe AdS bulk geometry from a critical boundary fermionat zero temperature, and the BTZ (Ba˜nados, Teitelboim,and Zanelli) black hole geometry at nonzero tempera-ture. As shall be elaborated in this paper, these geo-metric properties arise due to the fundamental scalingbehaviors of the systems under consideration, and holdseven for N = 1 free fermions. Besides defining a bulkgeometry, the EHM procedure is also useful in analyz-ing the RG properties of topological quantities. For in-stance, the holographic decomposition of the Berry cur-vature of a boundary Chern insulator interestingly re-veals a Z topological insulator living in the holographicbulk, thereby providing a holographic interpretation ofthe parity anomaly .Parallel to these developments in holography is thedevelopment of wavelet transforms in computer sci-ence, with applications ranging from image compres-sion to multiscale music texture to financial data anal-ysis. In essence, wavelet transforms are “lossless” RGtransforms probing details of different spatial or tem-poral scales, very analogous to the objective of hologra-phy. As such, there has been a symbiosis of ideas betweenthese two developments; in fact, the EHM is mathemat-ically a Haar wavelet transform acting on the quantummechanical Hilbert space rather than the space of signals.Recently, wavelets bases have also been shown to pro-vide good approximations to certain critical ground a r X i v : . [ c ond - m a t . o t h e r] N ov states in the framework of the multi-scale entanglementrenormalization ansatz (MERA) , a tensor networkapproach pioneered by Vidal et.al. that is closely relatedto the EHM . Described as a quantum circuit, theEHM has proposed implementations with Gaussian en-tangled states in optical networks, circuit QED setups aswell as trapped cold ions .In this work, we shall bring this symbiosis furtherby extending the Exact Holographic Mapping to arbi-trary (discrete) wavelets transforms . This more gen-eral framework allows for a more physically motivated,basis agnostic interpretation of the bulk geometry, sinceit explicitly isolates features associated with the choiceof wavelet basis. Just as importantly, an EHM based ongeneric wavelet bases can preserve the functional formof a much larger class of Hamiltonians, in the spirit ofconventional RG procedures (The existing EHM basedon the Haar wavelet can only preserve linearly disper-ing Hamiltonians). This will be relevant, amongst vari-ous reasons, for the very interesting holographic analysisof topological phases protected by symmetries that alsocreate extra degeneracies in the bandstructure, such astype-II Dirac cones and nodal rings and links .This paper is structured as follows. In Section II,we provide a pedagogical introduction to the construc-tion of wavelet bases in a language familiar to physi-cists, and highlight some properties that play a crucialrole in the describing the emergent geometry of the holo-graphic bulk. Following that, we explain in Section IIIhow Hamiltonians are renormalized under the EHM, andhow to find the appropriate wavelet basis, if it exist, thatkeeps a given Hamiltonian invariant. In Section IV, wederive the dependence of the bulk correlators, mutual in-formation and hence bulk geometry on the wavelet basis,focusing on how it arises from the branch cut topologyof the boundary propagator. Finally, in Section V webriefly discuss generalizations to other configurations ofFermi points, and also anisotropy in the resultant bulkgeometry for multi-dimensional EHM. II. EXACT HOLOGRAPHIC MAPPING (EHM)THROUGH WAVELETSA. Conceptual overview of the EHM
The EHM was first introduced in Ref. 16 as a specialtype of tensor network that implements a lossless RG-type procedure through a hierarchy of local mappings.It was then extended to more than one RG dimension inRef. 60, where its various mathematical properties werealso elaborated.We start from a given original ”boundary” system with2 N l sites (Fig. 1). At each iteration, the degrees of free-dom (qubits) on each 2 l ( l ≥
1) adjacent sites are sepa-rated into l small-scale (ultraviolet or UV) and l large-scale (infrared or IR) degrees of freedom (DOFs) via aunitary rotation whose form will be elaborated later. The IR UV UV IR FIG. 1. Left) Illustration of a single EHM iteration with l = 2.The degrees of freedom of 2 l input sites are separated into l UV and l IR sites via a unitary transform. Right) An EHMnetwork with 2 iteration levels. The IR DOFs from each groupof 2 l sites of the input ”boundary” system is fed into thenext iteration, until only l sites remain. The collection of thediscarded UV (red) DOFs, together with the last remainingIR sites (blue), form the ”bulk” system containing the samenumber of DOFs as the original system. l IR DOFs will then be used as the input for the nextiteration, while the UV DOFs will be discarded. Thisprocedure is repeated until we are left with the last setof l sites.Since degrees of freedom at larger scales will undergomore iterations before being discarded, the discardedDOFs from all the iterations collectively form an N + 1-level pyramid-like array arranged hierarchically accord-ing to scale. We shall define these discarded DOFs as the”bulk” system corresponding to the original ”boundarysystem”. Evidently, the bulk system contains the same2 N l DOFs, but are arranged in levels with 2 N − l , 2 N − l ,etc. sites according scale. B. Introduction to wavelets
The abovementioned EHM procedure is mathemati-cally a discrete wavelet transform. Here, we shall pro-vide a pedagogical introduction for its concrete imple-mentation.A 1-dimensional wavelet system consists of a set ofself-similar basis functions defined in exact analogy tothe bulk EHM DOFs. It can be described by a scalingfunction φ ( x ) and mother wavelet function w ( x ) (see Fig.2) pair obeying the recursion relations φ ( x ) = 2 l (cid:88) r =0 c ( r ) φ (2 x − r ) (1) w ( x ) = 2 l (cid:88) r =0 d ( r ) φ (2 x − r ) (2)where d ( r ) and c ( r ) are the high pass and low pass filtervectors, characterized by spatial fluctuations with shorterand longer length scales respectively. Both c and d arelength l + 1 vectors normalized such that (cid:80) r | c ( r ) | = | c | = | d | = 1. In Eq. 1, φ ( x ) is self-similar in the sensethat it is equal to the convolution of c ( r ) and a rescaledversion of itself. The mother wavelet w ( x ), by contrast, is not self-similar, but is the convolution of d ( r ) and φ (2 x ).In the simplest ( l = 1) case of the Haar wavelet used inRefs. 16, 22, and 60, we have c = (1 , / √ d =(1 , − / √ n iterations of Eqs. 1 and 2, one obtains level n wavelets w n,t ( x ) = w (2 n t − x ) (3)possessing characteristic length scales of ∝ n . To studythe properties of w n,t ( x ), it is useful to define the z-transforms C ( z ) = l (cid:88) r =0 c ( r ) z r (4) D ( z ) = l (cid:88) r =0 d ( r ) z r , (5)such that C ( z ) , D ( z ) with z = e ik are the Fourier trans-forms of the low pass and high pass filters respectively.(For the whole of this paper, we shall use the same sym-bol for a function whether its argument is given by k or z = e ik ) The RG properties of the EHM are mostsuccinctly described by the spectral properties of thesefilters. For future reference, we shall denote by C ∗ and D ∗ the polynomial C, D with coefficients (but not theargument z ) conjugated.The possible choices for filters polynomials C ( z ) and D ( z ) are constrained by biorthogonality, that is, by therequirement that φ ( x ) and w ( x ) should be orthogonal totheir translates and among themselves.For instance, the constraint ( φ ( x ) , w ( x + x )) = 0where x ∈ Z stipulates that the low pass and high passfilters project onto orthogonal subspaces. This requiresthat0 = (cid:88) x φ ∗ ( x ) w ( x + x ) ∝ (cid:88) x (cid:88) r,r (cid:48) c ∗ ( r ) d ( r (cid:48) ) φ ∗ (2 x − r ) w (2 x + 2 x − r (cid:48) ) ∝ (cid:88) r,r (cid:48) c ∗ ( r ) d ( r (cid:48) ) δ r,r (cid:48) − x = (cid:88) r c ∗ ( r ) d ( r + 2 x ) (6)which implies that0 = (cid:88) k e ix k C ∗ ( e − ik ) D ( e ik ) = 12 πi (cid:73) C ∗ ( z − ) D ( z ) dzz − x . (7)By the residue theorem, C ∗ ( z − ) D ( z ) must hence haveno term with even power, including the constant term. This can be guaranteed by the alternating-flip construc-tion d ( r ) = ( − r c ( l − r ), i.e. C ( z ) = z l D (cid:18) − z (cid:19) (8)where l is the degree of the polynomials C ( z ) and D ( z ).Also, φ ( x ) and a translated copy of itself φ ( x + x ), x ∈ Z / { } should be orthogonal in order to form a localbasis. This requires that δ x , = (cid:88) x φ ∗ ( x ) φ ( x + x ) ∝ (cid:88) x (cid:88) r,r (cid:48) c ∗ ( r ) c ( r (cid:48) ) φ ∗ (2 x − r ) φ (2 x + 2 x − r (cid:48) ) ∝ (cid:88) r,r (cid:48) c ∗ ( r ) c ( r (cid:48) ) δ r,r (cid:48) − x = (cid:88) r c ∗ ( r ) c ( r + 2 x ) , (9)so C ∗ ( z − ) C ( z ) has a constant term of 1, but no non-constant term with even power. An analogous constraintholds for D ( z ). Since the latter is a high-pass filter, itshould satisfy the additional constraint that it has zeroweight in the long wavelength limit k = 0 (or z = 1). Assuch, D (1) = (cid:80) r d ( r ) e i · r = (cid:80) r d ( r ) = 0 (But see Sect.V A for a reason to break this constraint).All in all, the wavelet basis is completely determinedby the autocorrelation Laurent polynomial P ( z ) = C ∗ ( z − ) C ( z ) = D ( − z − ) D ∗ ( − z )= 1 + (cid:88) j =1 (cid:20) p j − z j − + p ∗ j − z j − (cid:21) (10)whose coefficients p j − take values such that P ( z ) ≥ | z | = 1 on the unit circle, and normalized such that P (1) = 2. The absence of nontrivial even powers of z alsoimplies that P ( z )+ P ( − z ) = 2. C ( z ) and D ( z ), which arerelated by Eq. 7, can be obtained via a factorization of P ( z ).We are now ready to derive specific allowed forms forthe wavelet functions. Since φ ( x ) is a convolution of c ( r )and φ (2 x ) (Eq. 1), its z-transform obeysΦ( z ) = Φ( √ z ) C ( √ z )= Φ( z / ) C ( z / ) C ( z / )= ... = ∞ (cid:89) b =1 C (cid:16) z − b (cid:17) (11)This is the explicit expression for the (z-transform of the)scaling function Φ in terms of its recursive definition.Of course, the infinite product should terminate finitelywhen we are in a discrete system. For that, we can obtainfrom Eqs. 2 and 11 wavelet spectral functions W n ( z )
50 100 150 200 250 300 350 x (cid:45) (cid:45) w (cid:72) x (cid:76) a (cid:61)
50 100 150 200 250 300 350 x (cid:45) (cid:45) (cid:45) w (cid:72) x (cid:76) a (cid:61)
50 100 150 200 250 300 350 x (cid:45) w (cid:72) x (cid:76) a (cid:61)
50 100 150 200 250 300 350 x (cid:45) w (cid:72) x (cid:76) a (cid:61)
50 100 150 200 250 300 350 x (cid:45) w (cid:72) x (cid:76) a (cid:61)
50 100 150 200 250 300 350 x (cid:45) w (cid:72) x (cid:76) a (cid:61)
50 100 150 200 250 300 350 x (cid:45) (cid:45) w (cid:72) x (cid:76) a (cid:61)
50 100 150 200 250 300 350 x (cid:45) (cid:45) (cid:45) w (cid:72) x (cid:76) a (cid:61) FIG. 2. Illustration of the real-space wavelets w n,t =0 ( x ) (Eq. 12) for various instances with length of unit cell 2 l = 4. Thesewavelets are based on the ansatz C ( z ) = (1 − a ∗ )(1 − az )+(1+ a )( a ∗ z + z ) √ | a | ) , where a >
0. It is a more general ansatz than Eq. 15,admitting complex a , and can be shown to be consistent with having an odd P ( z ) − | z | = 1, and which also P ( z ) + P ( − z ) = 2. Physically, w n,t =0 ( x ) represents the ”orbital shape” of the UV wavelet basis: In the a = 1 Haar case forinstance, the basis contains two rectangular regions of opposite signs, representing an antisymmetric (short-wavelength) degreeof freedom. At other a , these basis wavefunctions become either more rounded or jagged. The beauty of wavelet bases is thatbasis wavefunctions can possess very detailed internal structures, such that only selected features will be ”zoomed-in” acrosswavelet levels. corresponding to the wavelets w n ( x ) at scale level n : W n ( z ) = 1 √ π W (cid:16) z n (cid:17) = 1 √ π D (cid:16) √ z n (cid:17) Φ (cid:16) √ z n (cid:17) = 1 √ π D (cid:16) z n − (cid:17) n − (cid:89) b =0 C (cid:16) z b (cid:17) (12)where the additional normalization factor of √ π is in-troduced for future notational consistency. Hence theconstruction of a (1-dimensional) wavelet basis involvesthese three basic steps:1. Choosing a polynomial P ( z ) = P ( e ik ) with desiredspectral properties (Eq. 10).2. Factorization of P ( z ) into C ( z ) and D ( z ).3. Construction of wavelet spectral functions W n ( z )via Eq. 12.As a simplest illustration, the Haar wavelet is character-ized by P ( z ) = 1+ z + z − , which factorizes to C ( z ) = z √ , D ( z ) = − z √ . From Eq. 12, the Haar wavelet spec-tral functions are thus given by W n ( z ) = √ − n (1 − z b − ) (cid:81) n − b =0 (cid:16) z b (cid:17) = 2 − n/ (cid:16) − z n − (cid:17) − z . This is illus-trated by the α = 1 case shown in Fig. 3.Finally, one can compute the spectral weight | W n ( z ) | of each wavelet level directly through the autocorrelationfunction: | W n ( z ) | = W ∗ n ( z − ) W n ( z )= 12 π P (cid:16) − z − n − (cid:17) n − (cid:89) b =0 P (cid:16) z b (cid:17) (13) C. Implementation of wavelets in the EHM
The Exact Holographic Mapping is most easily un-derstood in terms of its wavelets in momentum space.Writing the second quantized operators of the original(boundary) system as a † k = √ N l (cid:80) x e ikx a † x , the EHMis just a unitary transform to the basis of (bulk) statescreated by b † nx = (cid:88) k W ∗ n ( e − ik ) e − i n kx a † k (14)where n ≥ x = 1 , , ..., N − n l denotes the position within level n . Hence the original2 N l DOFs a † x | (cid:105) are re-distributed into a pyramid with2 N − n l sites (DOFs) b † nx | (cid:105) at level n (Fig. 1). Note that k refers to the momentum defined within each level: On the n th level with 2 N − n l sites, k = πj N − n l where j ∈ Z . ThatEq. 14 represents a unitary transformation of the Hilbertspace can be seen from the biorthogonality of W n ( z ) z n x ,which is proven in Appendix A. D. Wavelet properties relevant to holography
We have seen that a wavelet basis naturally providesa way to decompose information into a hierarchy of ba-sis vectors at various scales. Furthermore, these waveletbases are local and thus suitable candidates for describ-ing physical degrees of freedom in real space. This shouldbe contrasted with Fourier transforming into the momen-tum space basis, where each momentum mode is periodicand not compactly supported.Below, we highlight a few properties of the waveletbasis that play a key role in the EHM. Of most signifi-cance is the smoothness of the IR filter C ( z ) in the longwavelength limit z = 1 (or k = 0). This smoothness ischaracterized by an integer κ , which is the order of thefirst nonzero derivative (number of vanishing moments)of C ( z ) at z = 1, i.e. C ( κ ) (1) (cid:54) = 0 but C ( κ (cid:48) ) (1) = 0 for κ (cid:48) < κ . Equivalently, P ( z ) has 2 κ − (cid:45) (cid:45) (cid:144) Π (cid:200) W (cid:72) k (cid:76)(cid:200) (cid:45) (cid:45) (cid:144) Π (cid:200) W (cid:72) k (cid:76)(cid:200) (cid:45) (cid:45) (cid:144) Π (cid:200) W (cid:72) k (cid:76)(cid:200) (cid:45) (cid:45) (cid:144) Π (cid:200) W (cid:72) k (cid:76)(cid:200) FIG. 3. The spectral weights | W n ( e ik ) | for levels n = 1 , , , l = 2 wavelets described by Eq. 15 for α = 1 . , , . , , − . α = 1 case (Red)corresponds to the simplest Haar wavelet with no z de-pendence. The next lowest Daubechies wavelet is given by α = 1 .
25 (Black), and has the special property that of having2 κ − P ( w ) at k = 0. Consequently, with contributions away from the peaksmost strongly suppressed, it has the strongest peaks amongall the other α . As α decreases, the IR DOFs become lesseffectively suppressed, leading to higher secondary peaks. The significance of κ is illustrated in Fig. 3, where thespectral weights | W n ( e ik ) | for levels n = 1 , , P ( z ) of the form P ( z ) = 1 + 1 + α (cid:0) z + z − (cid:1) + 1 − α (cid:0) z + z − (cid:1) (15)One readily checks that P (1) = 2, P ( z ) + P ( − z ) = 2and P ( z ) ≥ − < α < . For the special caseof α = , P ( z ) factorizes to (cid:16) z + z (cid:17) (cid:16) − z + z − (cid:17) =(1 + cos k ) (cid:0) − cos k (cid:1) , i.e. P (1) = P (cid:48) (1) = P (2) (1) = P (3) (1) = 0, implying that κ = 2. This case is rep-resented by the black curve in Fig. 3, which possessesa spectral weight that is strongly suppressed at k = 0even for the first level n = 1. This strong suppressionis further magnified in subsequent levels, with the corre-sponding D (cid:16) z n − (cid:17) factor giving rise to the sharpest IRpeaks compared to the other cases with fewer vanishingmoments, i.e. κ = 1.In general, wavelet mappings with higher κ are moreeffective at suppressing DOFs away from the limiting IRpoint, and thus have more pronounced spectral peaks at k = ± π n at the n th level. In essence, wavelet map-pings represent a trade-off between locality and sharpnessof scale resolution: A sharp momentum cutoff requiresnon-local (power-law decaying) real space components,while the most local mapping (the Haar wavelet) lead torounded spectral peaks. With a given length 2 l for themother wavelet, the maximal κ and hence best possiblespectral resolution is realized by the Daubechies’ waveletfamily with P ( z ) (Fig. 4) given by P Daub ( z )= 2 (cid:18) z + z − (cid:19) l l − (cid:88) j =0 (cid:18) l + j − j (cid:19) (cid:18) − z + z − (cid:19) j / l + j (16)which reproduces the abovementioned α = waveletwhen l = 2, and the Haar wavelet when l = 1. That P Daub ( z ) has κ = l , the maximum possible value fora given l , can be seen by expressing it in terms of y = − z + z − , which yields P (cid:48) Daub ( y ) ∝ y l − (1 − y ) l − . (cid:45) (cid:45) (cid:45) (cid:144) Π (cid:200) W n (cid:72) k (cid:76)(cid:200) (cid:72) Κ(cid:61) (cid:76) (cid:45) (cid:45) (cid:45) (cid:144) Π (cid:200) W n (cid:72) k (cid:76)(cid:200) (cid:72) Κ(cid:61) (cid:76) (cid:45) (cid:45) (cid:45) (cid:144) Π (cid:200) W n (cid:72) k (cid:76)(cid:200) (cid:72) Κ(cid:61) (cid:76)
FIG. 4. Spectral weight of levels n = 1 , , , κ = 1 ,
10 and 100. We seeextremely smoothness at k = 0 in the large κ limit, since thewavelet filter has a zero with κ − III. RENORMALIZATION OF HAMILTONIANSUNDER THE EHM
Regarded as a lossless renormalization group (RG) pro-cedure, the Exact Holographic Mapping should ideallypreserve the form of the Hamiltonian under renormaliza-tion. Below we shall discuss when this is possible, andhow can the renormalization scale parameter be deter-mined. This will greatly generalizes the scope of previ-ous literature , where the special choice of the Haarwavelet basis preserves the form of Dirac-type Hamilto-nians sin kσ + ( m + 1 − cos k ) σ only.Let h n be the input of the n th EHM iteration of theoriginal Hamiltonian h . From Eq. 11, h n +1 is related to h n via a multiplication with the wavelet spectral weight | C | . Writing h ( w ) as h ( k/
2) (with a slight misuse ofnotation), such that w = e ik/ , we have2 h n +1 ( w )= h n ( w ) C ( w ) C ∗ (cid:0) w − (cid:1) + h n ( − w ) C ( − w ) C ∗ (cid:0) − w − (cid:1) = (cid:88) ± h n ( ± w ) P ( ± w ) (17)The two copies of momenta w = e ik/ and − w = e i ( k + π ) / in the summation arise due to a folding of theBrillouin zone, since level n has twice as many sites aslevel n + 1. Hence, we have h n +1 ( w ) given by the av-erage of h ( ± w ) weighted by the wavelet autocorrelationfunction from both ± w .To find conditions on the wavelet that leaves theHamiltonian invariant, we set h n and h n +1 in Eq. 17to have the same functional form h : λh ( w ) = 12 (cid:88) ± h ( ± w ) P ( ± w )= h even ( w ) + [ P ( w ) − h odd ( w ) (18)where λ is the (constant) scale factor for each RG step,and h even/odd ( w ) = ( h ( w ) ± h ( − w )). In other words,given a Hamiltonian h ( w ) = h even ( k ) + h odd ( k ), thewavelet that fixes it must have the autocorrelation func-tion P ( w ) = C ∗ ( z − ) C ( z ) = 1 + λh ( w ) − h even ( w ) h odd ( w ) (19)Here are a few caveats about Eq. 19:1. The RG scale factor can only take nontrivial valuesof λ (cid:54) = 1 if the Hamiltonian is gapless (critical) inthe long-wavelength limit k = 0 ( w = 1). Thisfollows immediately by setting w = 1 and notingthat P (1) = 2.2. Given h ( w ) of degree d , the degree of P ( w ) (or C ( w )) is fixed by comparing the leading powers ofEq. 18 to be l = 2 (cid:98) d (cid:99) + 13. There may not exist a wavelet that leaves the formof a given h ( w ) invariant. Existence of the formeris contingent on the RHS of Eq. 19 being factoriz-able into an odd Laurent polynomial P ( w ) with oddpowers − l to l , such that it is real for | w | = 1, andthat P ( w ) + P ( − w ) = 2 (i.e. of the form Eq. 10).Further discussion is given in Appendix B; refer tothe next subsection for specific examples of invari-ant Hamiltonians and their associated wavelets.In the critical case h (1) = 0, λ is determined by theconstraint P (1) = 2. Eq. 19 gives λ = lim w → h even ( w ) + [ P ( w ) − h odd ( w ) h ( w )= lim w → h (cid:48) even ( w ) + [ P ( w ) − h (cid:48) odd ( w ) + P (cid:48) ( w ) h odd ( w )2 wh (cid:48) ( w )= 12 + lim w → [ P ( w ) − h (cid:48) odd ( w ) + P (cid:48) ( w ) h odd ( w )2 wh (cid:48) ( w ) (20)If h (cid:48) (1) (cid:54) = 0, the limit on the last line is easily taken and λ = , unless P (cid:48) (1) and h odd (1) are both nonzero. Thiscan occur only if P ( w ) do not have real coefficients, and h ( w ) is neither odd nor even. Letting γ be the order ofthe first nonzero derivative of the Hamiltonian h ( w ) at w = 1, we have λ | γ =1 = 12 (cid:18) P (cid:48) (1) h odd (1) h (cid:48) (1) (cid:19) (21)Frequently, the Hamiltonian is not linearly dispersiveat w = 1, and to evaluate λ we will need to invokeL’Hˆopital’s rule a total of γ number of times. For γ = 2,we get λ | γ =2 = 14 (cid:18) P (cid:48)(cid:48) (1) h odd (1) + 2 P (cid:48) (1) h (cid:48) odd (1) h (cid:48)(cid:48) (1) (cid:19) (22)and, in general, λ | γ = 12 γ (cid:32) (cid:80) γj =1 P ( j ) (1) h (3 − j ) odd (1) h ( γ ) (1) (cid:33) = 12 γ (cid:32) − (cid:80) γj =1 P ( j ) (1) h (3 − j ) even (1) h ( γ ) (1) (cid:33) (23)As such, an EHM iteration rescales the Hamiltonian bya factor of for each vanishing order of h ( w = 1) if it iseither fully even or odd. Otherwise, λ will be more com-plicated, depending on the derivatives of the resultantwavelet autocorrelation P ( w ). A. Renormalization examples
1. Simplest case: Haar wavelet
With the Haar wavelet basis, C ( w ) = w √ and P ( w ) = C ( w ) C ∗ ( w − ) = 1 + w + w − . It is easy to verify that thetwo linearly independent solutions to Eq. 18 are h ( w ) = w − w − i ⇒ h ( k ) = sin k and h ( w ) = − w − w − ⇒ h ( k ) = − cos k , both with the RG rescaling λ = , consistentwith Eqs. 21 and 22 respectively.
2. Odd Hamilonians
For generic Hamiltonians odd in w , Eq. 19 nicely sim-plifies to P ( w ) − λ h ( w ) h ( w ) = 12 γ h ( w ) h ( w ) (24)This equation can always be satisfied by h ( w ) = (cid:89) j (cid:18) w a j − w − a j i (cid:19) b j (25)i.e. h ( k ) = (cid:81) j sin b j ( a j k ) for (cid:80) j a j b j ∈ odd. Fromthe familiar relation sin 2 x x = cos x , we see that P ( w )is a valid wavelet autocorrelation polynomial given by P ( w ) = 1 + (cid:81) j (cid:16) w aj + w − aj (cid:17) b j . Two interesting specialcases are elaborated below.Hamiltonians of the form h ( k/
2) = sin ak = w a − w − a i , a odd, are invariant under the IR filter C ( w ) = w a √ or P ( w ) = 1 + w a + w − a , with λ = . We need a to be oddas P ( w ) can never have even nontrivial even powers.The above results are applicable to Hamiltonians evenin k too, as long as they are odd in w = e ik/ , i.e. Hamil-tonians of the form h ( k ) = cos ak − cos bk ∼ b − a (cid:18) k + a + b k (cid:19) (26)since h ( k/
2) = w a + w − a − w b + w − b , a, b odd, are invari-ant under P ( w ) = C ( w ) C ∗ ( w − ) = 1 + w a + w − a + w b + w − b ,with λ = . To find C ( w ), note that P ( w ) can alwaysbe factorized into C ( w ) and C ∗ ( w − ) because it is sym-metric in w and w − , and its roots hence comes in pairsof w and w − . This factorization admits no general an-alytic solution, but for simple cases like a = 3 , b = 1, wecan (with a bit of effort) find the nice solution C ( w ) = − i ( w + w )+ w . This defines the wavelet basis for which h ( k ) = cos 3 k − cos k remains invariant.With odd Hamiltonians, one can directly check fromthe form of h ( w ) if the corresponding wavelet is of κ = 1.Such bases are characterized by a nonvanishing P (cid:48)(cid:48) (1),which can be obtained via direct differentiation of Eq.24: P (cid:48)(cid:48) (1) = 3 h (cid:48)(cid:48) (1) + 2 h (cid:48)(cid:48)(cid:48) (1)2 h (cid:48) (1) − (cid:18) h (cid:48)(cid:48) (1) h (cid:48) (1) (cid:19) (27)Evidently, some fine-tuning is needed to necessitate awavelet with κ > P (cid:48)(cid:48) (1) = P (cid:48)(cid:48)(cid:48) (1) = 0). IV. WAVELET DEPENDENCE OF BULKGEOMETRY
One of the most attractive features of the Exact Holo-graphic Mapping is that it reproduces, for various im-portant cases, bulk geometries in agreement the Ryu-Takayanagi (RT) formula . Specifically, it yields for anynumber of dimensions the AdS space for critical systemsat zero temperature, and BTZ/Lifshitz black holes forcritical linear/nonlinear dispersing systems at nonzerotemperature .The RT formula proposes that the the entanglemententropy of a boundary region is proportional to the areaof its corresponding minimal surface in the bulk. Inspiredby this information theoretic definition of area, theEHM framework proposed that geodesic distancesin the EHM bulk are determined by mutual information,i.e. the upper bound of the correlation functions betweentwo endpoints. This is a paradigm shift from the usualconceptual relationship between correlation and distance: Conventionally, we think of the correlator decay behavioras a function of separation distance but now, we invertthis relationship by defining the distance based on theextent of correlator decay.In this section, we shall focus on the the dependenceof the bulk geometry on the wavelet basis, which is anaspect not studied in Ref. 60. A. Definition of the bulk geometry
Consider two points 1 and 2 in the bulk system withcoordinates ( (cid:126)x , n , t ) and ( (cid:126)x , n , t ), where (cid:126)x is the siteindex within a level, n the level index and t the time.These two points are separated by a spatial coordinateinterval of ∆ (cid:126)x = (2 n x − n x , n − n ) sites and tem-poral coordinate interval of ∆ t . Recall that each levelin the bulk contains ∝ − n DOFs with spectral weight | W n ( k ) | , such that we approach the low energy limit inthe limit of large n .With the EHM, we define the physical distance d between these two points in the bulk by d = − d I ∼ − d log C (28)where I is the mutual information between points 1and 2 and C is the two-point bulk correlation functionbetween them. The length scale d can be interpreted asthe inverse mass scale of the massive field associated with C living in the curved bulk geometry. The asymptoticequality on the RHS was shown in Ref. 60, that I be-haves asymptotically like 8 C . Eq. 28 also applies fortemporal intervals if we perform a Wick rotation to imag-inary time τ = it , so that temporal oscillations becomeexponential decay. With that, we have C (∆ (cid:126)x, τ ) = (cid:104) T b n x ( τ ) b † n x (0) (cid:105) = (cid:88) k W ∗ n ( e − ik ) W n ( e ik ) e − ik (2 n x − n x ) G k ( τ )= (cid:73) | z | =1 dzz W ∗ n ( z − ) W n ( z ) z n x − n x G z ( τ )(29)in terms of the boundary correlation function G k ( τ ) ( G z and G k are used interchangeably, depending on the ar-gument used) given by G k ( τ ) = e τh ( k ) I + e βh ( k ) (30)for the Hamiltonian h ( k ), with β the inverse temperature.Near a gapless point z = e ik = 1, the energy manifolds(eigenenergy bands of h ( z )) generically exhibit branchpoints . As we see later, the power-law decay of C shall depend crucially on the existence of these complexsingularities. In a typical case without accidental de-genaracy, the band crossing involves two bands and G k possesses a square-root branch cut u ∼ √ z = e ik/ or u ∼ √ z − = e − ik/ . To see this explicitly, consider thecanonical two-band Dirac model h ( k ) = sin kσ + ( m +1 − cos k ) σ , σ , the Pauli matrices, with eigenenergies E k = E z = (cid:113) m + 1) − ( m + 1)( z + z ) and gap m .In matrix form, h ( z ) = (cid:18) i ( z − (1 + m )) − i ( z − (1 + m )) 0 (cid:19) (31)with the correlator G z given by (cid:18) cosh( τ E z ) I + h ( z ) E z sinh( τ E z ) (cid:19) (cid:18) I − h ( z ) E z tanh βE z (cid:19) (32)Crucial to the analytic structure of this matrix is the”flattened hamiltonian” h ( z ) E z = (cid:113) m +1 z (cid:114) z − m +1 z − ( m +1) (cid:113) zm +1 (cid:114) z − ( m +1) z − m +1 → m =0 (cid:18) √ z √ z (cid:19) (33)Its branch cut topology crucially affects the bulk corre-lator because it dictates the deformation of the contourin Eq. 29. In the gapped case with nonzero m , h ( z ) E z has4 branch points (0 , ∞ , m + 1 , m +1 ), two within and twooutside the unit circle. Hence C can be evaluated with-out deforming the unit circle, giving rise to results dependent on the position of the singularities introducedby either mass or temperature scale, but independent ofthe wavelet basis.In the gapless ( m = 0) case which we shall focus on,the only branch cut extends from z = 0 to z = ∞ ,which is unavoidable. In the following, we shall evaluatethe bulk correlator and hence bulk geodesic distances bydeforming the unit circle to a keyhole-like contour, fromwhich the dependence of the correlator decay behavioron the branch cut becomes apparent. We shall considerthe general case where the unitary transforms (and hencefilters C j and D j ) at each iteration j are not necessarilythe same. B. Geodesic distances and bulk geometry for acritical 1D free fermion
1. Intra-level direction
To explicitly demonstrate how the bulk geometry de-pend on the choice of wavelet basis, we turn to the sim-plest case of critical 1D free fermion described by a DiracHamiltonian. We stress that this choice of Hamiltonianis made purely due to its analytic tractability; indeed, anEHM generalized to arbitrary wavelet bases will be ableto retain the forms of a far larger class of Hamiltonians(Sect. III). We first study the zero-temperature bulk correlator C due to a displacement of x sites in the intra-leveldirection, so that level indices n = n = n are equaland τ = 0, β → ∞ . Physically, this correlator is betweendegrees of freedom at the same scale and time.A nonzero matrix element u of C is given by u = − (cid:73) | z | =1 dzz W ∗ n ( z − ) W n ( z ) √ z z n x = 12 (cid:90) W ∗ n ( z − ) W n ( z ) z n x ( √ z − √ e πi z ) dzz = (cid:90) W ∗ n ( z − ) W n ( z ) z n x √ z dz = (cid:90) ( W ∗ n ( z − ) W n ( z ) z m n ) z n ( x − m ) − / dz = (cid:90) Q ( z ) z X dz = Q (1) X + 1 − X + 1 (cid:90) Q (cid:48) ( z ) z X +1 dz = (cid:88) j =0 Q ( j ) (1) X !( X + j )! ( − j ∼ Q (2 κ ) (1) X κ +1 ∼ Q (2 κ ) (1)2 n (2 κ +1) x κ +1 (34)In line 4, m is the degree of each factor C or D in W n ( z ) = π D n ( z n − ) (cid:81) n − j =1 C j ( z j − ), introduced suchthat Q ( z ) = W ∗ n ( z − ) W n ( z ) z m n does not have negativepowers of z . In line 5, X = 2 n ( x − m ) − / x , so that the j correc-tions in the third last line can be dropped. The final ex-pression involves κ , the first nonzero derivative of W n ( z )at z = 1 (see Sect. II D). The integer κ , which charac-terizes the wavelet moment at the IR (long-wavelength)point z = 1, shall be a key quantity in determining howthe EHM affects the correlators and hence bulk geometry.Let’s now evaluate Q (2 κ ) (1) by an explicit expansionabout z = 1: Q (2 κ ) (1) (cid:15) κ (2 κ )!= Q (1 − (cid:15) )= (1 − (cid:15) ) n m W ∗ n ((1 − (cid:15) ) − ) W n (1 − (cid:15) ) ≈ W ∗ n (1 + (cid:15) ) W n (1 − (cid:15) )= 12 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:89) j =1 C j (cid:16) (1 − (cid:15) ) j − (cid:17) D n (cid:16) (1 − (cid:15) ) n − (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:89) j =1 C j (1) D n (cid:0) − n − (cid:15) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≈ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:89) j =1 C j (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) D ( κ ) n (1) 2 κ ( n − (cid:15) κ κ ! (cid:12)(cid:12)(cid:12)(cid:12) ≈ π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:89) j =1 C j (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) D ( κ ) n (1) (cid:12)(cid:12)(cid:12) κ ( n − (cid:15) κ κ ! (35)The C factors are evaluated at z = 1 with no need forTaylor expansion because they are IR filters, which arenot supposed to have vanishing values at z = 1. Com-paring coefficients, we see that Q (2 κ ) (1) = 12 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:89) j =1 C j (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) D ( κ ) n (1) (cid:12)(cid:12)(cid:12) κ ( n − (cid:18) κκ (cid:19) (36) D ( κ ) n (1) is the first nonzero derivative of the UV filter D n at the IR point k = 0 or z = 1. Combining Eq. 36 withEqs. 28 and 34, we obtain I ∼ u = (cid:32)(cid:18) κκ (cid:19) | D ( κ ) n (1) | κ π (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:89) j =1 C j (1) √ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x κ +2 = (cid:18) κκ (cid:19) | D ( κ ) n (1) | κ +1 π n − (cid:89) j =1 (cid:18) − | C j ( − | (cid:19) x κ +2 , (37)which coincides with results from Ref. 18 for bosonic systems. All in all we have (plotted in Fig. 5) d ( x ) ∼ d (2 κ + 1) log | x | + const. (38)Explicitly, we see that the mutual information I decayswith x with an exponent of 4 κ + 2, i.e. that the choice ofwavelet basis affects the coefficient 4 κ + 2 of the logarith-mic term, but does not modify its qualitative asymptoticbehavior. Physically, a larger κ leads to faster decay ofmutual information because the additional smoothnessof the UV filter D n at k = 0 extinguishes more DOFs. Notably, there will be no dependence on n , the level in-dex, only if C j ( −
1) = C j ( k = π ) = 0 for all levels j .In other words, each IR filter C j will lead to a suppres-sion of I xy unless C j ( −
1) = 0, i.e. is a perfect IR filtertaking zero value at the UV point k = π . To put thesignificance of this observation in context, consider thefitting of the geodesic distance d ∼ d log C with thatof Anti de-Sitter (AdS) space (Appendix I of Ref. 60): d AdS ( x ) ∼ R log | x | R (39)where R is the AdS radius. If we want to fit d ( x ) of I to d Ads ( x ), which do not depend on the radial coordinate,we will need each iteration of the EHM to discard all ofthe largest scale DOFs, which are at k = π . This can onlyhappen if C j ( −
1) = 0 for all levels j . Merely having all C j ’s equal is not sufficient for ensuring that the geodesicdistance is independent of the scale n .From now, we assume perfect IR filters that have zerosupport at k = π . Comparing Eqs. 38 and 39, we obtain Rd = κ + 12 (40)and R = 12 √ | D ( κ ) n (1) | (cid:18) κκ (cid:19) π κ +1 (41)We see that R depends only on κ and | D ( κ ) n (1) | . In thesimplest case of the Haar wavelet basis, D n ( z ) = − z , so κ = 1 and | D (1) n (1) | = √ . Eq. 40 and 41 then coincideswith numerical results from Ref. 16. x d FIG. 5. The geodesic distances d determined from the bulkcorrelators via Eq. 28. Plotted are the curves for the Diracmodel with the κ = 1 , , κ , i.e. 2 κ + 1 = 3 , , d exhibits power-law decay after just a few sitesprovide a posteriori justification of the approximations in Eq.34.
2. Inter-level (radial) direction
We now consider the case with zero intra-level displace-ment ( x = 0) and temporal displacement ( τ = 0), sothat the interval lies in the ”adial” direction from level 1to level n . This is an interval between different lengthsscales at the same spacetime coordinates. A nontrivialmatrix element u of the bulk correlator takes the form u = − (cid:73) | z | =1 dzz W ∗ ( z − ) W n ( z ) √ z I n = u = − (cid:73) | z | =1 dzz W ∗ ( z − ) D n ( z n − ) n − (cid:89) j =1 C j ( z j − ) √ z (43)and J n − = − (cid:73) | z | =1 dzz W ∗ ( z − ) n − (cid:89) j =1 C j ( z j − ) √ z (44)which is the unprojected correlator in the ( n − th level.For 2 n (cid:29) I n and J n − are approximately related by I n = − (cid:73) | z | =1 dz √ z W ∗ ( z − ) n − (cid:89) j =1 C j ( z j − )( D n (0) + O ( z n − )) ∼ D n (0) J n − (45)since the truncated contributions from monomials of z n − integrate to small quantities n − +const. that canbe discarded for 2 n (cid:29)
1. Hence I n is dominated by theterm containing D n (0), the constant term in D n ( z ). Notethat | D n (0) | = | d (0) | < (cid:80) j | d ( j ) | = 1. Similarly,we can also show that J n ∼ C n (0) J n − . Hence asymp-totically, | u | ∼ D n (0) n − (cid:89) j =1 C j (0) ∝ C (0) n − , (46)the last expression holding when the IR filters C j are allthe same. Hence the radial geodesic distance goes like d (1 , n ) = − d log | u | ∼ ( n −
1) log 1 C (0) + small const.(47)Comparing this with the radial AdS distance d AdS (1 , n ) ∼ R ( n −
1) log 2 , (48)we obtain Rd = − | C (0) | log 2 (49) so that Rd = 1 in the Haar case with C ( z ) = z √ . C (0) is the same-site coefficient in the real-space re-cursion relation of the IR wavelet filter. As such, a small C (0) represents a large ’spreading’ of the EHM tree net-work, and should cause the mutual information to decayfaster as we travel down the different hierarchical levels( n ) of the the tree.
3. Imaginary time direction
We now focus on the case with ∆ (cid:126)x = 0, but imaginarytime interval τ >
0. From Eq. 32, the leading contribu-tion to the correlator is C ( τ ) = 12 (cid:90) π − π dq | W n ( e iq ) | e − τE q (50)The simplying caveat is that we only have to care aboutthe extreme IR (small q ) contribution to this integral.This is because e − E q τ = e − v F | q | τ decays rapidly for mod-erately large τ . Hence we only need to know the IR be-havior of W n ( z ) = D n ( z n − ) (cid:81) n − j =1 C j ( z j − ), which isgiven by Eq. 35: | W n ( e i ∆ q ) | ≈ | W n (1 − i ∆ q ) | ≈ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n − (cid:89) j =1 C j (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) D ( κ ) n (1) (cid:12)(cid:12)(cid:12) κ ( n − (∆ q ) κ πκ ! = 12 π (cid:12)(cid:12)(cid:12) D ( κ ) n (1) (cid:12)(cid:12)(cid:12) (2 κ +1)( n − (∆ q ) κ κ ! (51)where κ is the order of the first nonzero derivative of D n ( z ), as before. Evaluating Eq. 50 in terms of theincomplete Gamma function (cid:82) q γ e − qτ dq ∼ γ ! τ γ +1 , we ob-tain C ( τ ) ∼ π (cid:12)(cid:12)(cid:12) D ( κ ) n (1) (cid:12)(cid:12)(cid:12) (2 κ +1)( n − τ κ +1 (cid:18) κκ (cid:19) (52)Comparing d ( τ ) = − d log C ( τ ) with the imaginarytime geodesic distance of Euclidean AdS space d AdS ( τ ) = 2 R (cid:16) log τR − n log 2 (cid:17) , (53)we obtain Rd = κ + 12 (54)which agrees exactly with the intra-level result (Eq. 40).The corresponding AdS radius R is also given by Eq. 41.The equivalence of the fitting parameters to AdS spacein the intra-level and imaginary time directions is notsurprising, since there is a global rotation symmetry thatrelates space and imaginary time.1 V. FURTHER GENERALIZATIONSA. “Zooming in” onto arbitary Fermi points
The EHM is essentially a “lossless” RG procedure pro-ducing a series of bulk layers n that represent the orig-inal system viewed from various energy scales. Mathe-matically, that is accomplished by “zooming in” succes-sively closer to the low energy regions of the system. Ina fermionic system, the lowest energy regions are Fermipoints in the case of semimetals, or Fermi surfaces in thecase of metals. It is imperative that we are not just ableto probe the long-wavelength k = 0 limit, but also ableto probe the low energy limit of a given system. Sincethe EHM should fundamentally be a low energy probe,the resultant bulk geometry should not be qualitativelyaffected by that positions of the Fermi points. That thisis true will be evident from the results of this section,where we show that all that is required is a modificationof the wavelet basis.So far, the EHM described involve iterations thatsuccessively “zoom in” onto the long wavelength limit k = 0 (or z = e ik = 1). This is appropriate if thephysical system has a Fermi point at k = 0. How-ever, most real systems like Graphene or specially de-sign metamaterials possess interesting and pos-sibly topologically nontrivial critical points (valleys,line nodes etc.) elsewhere in the Brillouin zone.If the critical point is simply shifted to k (cid:54) = 0, we cantrivially modify the EHM via C ( z ) → C ( ze − ik ) , D ( z ) → D ( ze − ik ) (55)so that its spectral properties are simply translated by k .This modification introduces complex coefficients in thereal-space wavelet functions, which is perfectly permissi-ble for a wavelet mapping acting in quantum mechanicalHilbert space.More interestingly, we can also “split” the spec-tral peaks such that the EHM “zooms in” onto morethan one momentum point. This is achieved by in-terchanging the sequence of UV and IR filters in thetower of C and D filters used in constructing W n ( z )in Eq. 12: Instead of the original definition W n ( z ) = C ( z ) C ( z ) ...C ( z n − ) D ( z n − ), we shall define W n ( z ) = B + n (cid:16) z n − (cid:17) n − (cid:89) j B − j (cid:16) z j − (cid:17) (56)where each B ± n ( z ) can be either C ( z ) or D ( z ). We definea vector (cid:126)v such that v j = 1 if C ( z ) was used at level j ,and v j = − D ( z ) was used. Hence the usual definitionof W n ( z ) will correspond to (cid:126)v = (1 , , , ..., , − C ( z ) D ( z ) D ( z ) C ( z ) D ( z ), for instance, will corre-spond to (cid:126)v = (1 , − , − , , − C and D filters areillustrated in Fig. 6. At each level, the IR filter C (cid:0) z n (cid:1) has vanishing spectral weight when z n = −
1, i.e. k = π n − (2 j + 1), j ∈ Z . If the tower of filters take the form C ( z ) C ( z ) C ( z ) ... , i.e. consists of all IR filters C , k = 0eventually survives as the only peak. In this sense, W n ( z )zooms in onto k = 0. (cid:45) (cid:45) (cid:45) k (cid:72) z (cid:76) , D (cid:72) z (cid:76) (cid:45) (cid:45) (cid:45) k (cid:61) (cid:72) (cid:45) (cid:76) : (cid:200) W (cid:72) x (cid:76)(cid:200) (cid:45) (cid:45) (cid:45) k (cid:61) (cid:72) (cid:45) (cid:45) (cid:76) : (cid:200) W (cid:72) x (cid:76)(cid:200) (cid:45) (cid:45) (cid:45) k (cid:61) (cid:72) (cid:45) (cid:45) (cid:76) : (cid:200) W (cid:72) x (cid:76)(cid:200) FIG. 6. Top Left) Illustration of the profiles of C (black) and D (purple dashed) for the Haar wavelet. The IR filter C (cid:16) z n (cid:17) strengthens the (IR) contribution closer to the existing foldedIR points at π n j , j ∈ Z . The UV filter D (cid:16) z n (cid:17) attempts to“split” these contributions by favoring contributions halfwaybetween the IR points. Top Right) | W ( z ) | as defined in theusual case of CCCCCD . Each C filter strengthens the peakaround k = 0, while the last D filter splits it to support thecontributions just around k = 0. Bottom Left) A given D filter at the second level splits the peaks to ± π/
2. BottomRight) A D filter at the third level splits the peaks to ± π/ κ > Now, suppose that C (cid:0) z m (cid:1) is replaced by D (cid:0) z m (cid:1) ,which suppresses k = π m − j , j ∈ Z . This includes the k = 0 point, which will thus no longer be “zoomed in”onto. But at the same time, the points k = π m − (2 j + 1), j ∈ Z will be allowed to survive. Due to the finiteenvelope of C and D as shown in Fig. 6, the spec-tral weights of these new peaks depends on the dis-tance from the previous IR point. Consider the example (cid:126)v = (1 , − , , , , − m = 2, C ( z ) is replaced by D ( z ). This replaces the “default” IR peak of k = 0 bythe “new” IR peaks ± π . At the next m = 6, C ( z ) isreplaced by D ( z ). Of all the k-points k = π (2 j + 1), j ∈ Z , that thus do not have to vanish, the dominant onesare those closest to the incumbent IR points ± π . In gen-eral, when C is replaced by D at z m , z m , ..., z mr , thedominant pair of IR points eventually zoomed in ontowill be ± π m − ± π m − ± ... ± π mr − , where the j th ± sign ( j >
2) is chosen such that the point to be zoomedin is closer to the “old” IR point ± π m − ± ... ± π mj − − .To zoom in onto arbitrary Fermi points, one can com-bine translations and splittings of the IR points at variouslevels via Eqs. 55 and 56, as well as utilize specific forms2of C j and D j to achieve the desired spectral peaks. B. EHM in higher dimensions - basis anisotropy
While we have so far focused on 1-dimensional EHM,all the results so far can be directly generalized to ahigher dimensional EHM relating a d + 1-dim bound-ary system to a d + 2-dim bulk. This generalizationcan be simply accomplished by taking direct productsof the wavelet filters in various dimensions, as describedat length in Section V of Ref. 60. With possibly different κ parameters κ , ..., κ d for the wavelet basis in each di-rection, one may naively think that we will arrive at bulkgeodesic distances given by d ,j ( x j ) ∼ d (2 κ j + 1) log x j with anisotropic AdS radii R j = d (cid:0) κ + (cid:1) . This is ac-tually not true. To understand why, note that each factorof | W n ( e ik ) | near k = 0 acts as a derivative on the (orig-inal) boundary correlator G x ∝ (cid:90) G k e ikx ∼ | (cid:126)x | = 1 (cid:112) x + ... + x d , (57)so that for a wavelet filter in the direction j , (cid:82) | W n ( e ik j ) | e i n (cid:126)k · (cid:126)x G k dk j ∼ ∂ κ j j | (cid:126)x | ∼ θ − | (cid:126)x | wherecos θ is the d th component ratio of (cid:126)x .In general, the bulk correlator C ( (cid:126)x ) will be domi-nated by terms involving the lowest κ j in almost all di-rections, not just in the j th direction. To see why, con-sider the d = 2 case κ = 1 and κ = 2. The two leadingcontributions to C ( (cid:126)x ) are proportional to ∂ | (cid:126)x | = 3 cos θ − | (cid:126)x | (58)and ∂ | (cid:126)x | = 3(3 cos θ −
24 cos θ sin θ + 8 sin θ ) | (cid:126)x | (59)In the asymptotic limit of large | (cid:126)x | , the decay exponentis always θ − d min( κ , ..., κ d ) + d / VI. CONCLUSION
Motivated by the desire for a holographic mappingthat preserves the form of a wide class of Hamiltonians, we generalized the Exact Holographic Mapping to con-sist of the most general unitary transformation based onbiorthogonal wavelets. Compared to the original EHMbased on the Haar wavelet, our generalized EHM canpreserve Hamiltonians with various exotic band touch-ings, and not just those of linear Dirac type. The preciserelationship between the Hamiltonian and the waveletmapping that preserves it is summarized in Eq. 19, whichcan also be shown to determine the renormalization scalefactor λ .We also derived the dependence of the bulk geometryon the wavelet basis, and showed that the latter onlyaffects quantities arising from branch cuts in the prop-agator. These include the correlator decay exponent ofa critical system at zero temperature and hence its dualAdS radius, but not the spatial event horizon of the dualgeometry due to mass or temperature scale. Of primarysignificance is the integer κ , which is the order of thefirst nonzero derivative of the IR wavelet filter C ( z ) inthe long-wavelength limit. It is κ , and not the length 2 l of the mother wavelet, that controls the bulk geometry.The generality of the wavelet EHM formulation alsoenables us to “zoom in” onto Fermi points away fromthe long wavelength limit. This can be accomplished, forinstance, by reversing the roles of the UV and IR filters atcertain scale levels n . Finally, we discussed the implica-tions of having higher dimensional EHM with anisotropicbases.We also took this opportunity to provide a pedagogi-cal introduction to the construction of wavelets, a topicintimately related to renormalization but rarely coveredin detail in the physics literature. ACKNOWLEDGMENTS
CH thanks Xiao-Liang Qi, Guifre Vidal and YingfeiGu for helpful discussions.
Appendix A: Biorthogonality of wavelets
Consider the generic definition with the roles of the C and D wavelet filters possibly interchanged (Sect. V A).Eq. 12 is generalized to W n ( z ) = B + n ( z n − ) n − (cid:89) j B − j ( z j − ) (A1)where B ± is the z-transform of b ± . It is possible to provethe orthogonality of the W n ’s from Eq. 56. Suppose m > n :3( w n , w m ) ∝ (cid:73) | z | =1 dzz W ∗ n ( z − ) W m ( z )= (cid:73) | z | =1 dzz B + ∗ ( z − n − ) B + ( z m − ) n − (cid:89) a =0 B −∗ ( z − a ) m − (cid:89) b =0 B − ( z b )= (cid:73) | z | =1 dzz (cid:104) B + ∗ ( z − n − ) B − ( z n − ) (cid:105) n − (cid:89) a =0 (cid:104) B −∗ ( z − a ) B − ( z a ) (cid:105) (cid:16) O ( z n ) (cid:17) = (cid:73) | z | =1 dzz (nonconstant)= 0 (A2)The first term in line 3, which is equal to C ∗ ( z − n − ) D − ( z n − ) or D ∗ ( z − n − ) C − ( z n − ), has noconstant term by Eq. 6, and has a smallest power of ± n − r , r an odd positive integer. This power can-not be canceled by any combination of terms in theproduct in the second term, since each term is equal to C ∗ ( z − a ) C − ( z a ) or D ∗ ( z − a ) D − ( z a ) and has no evenpower of z ± a . Explicitly, each postive power term in theproduct has the form z (cid:80) n − j =0 j (2 m j +1) = z (cid:80) n − j =1 m j − j +2 n − − where m j is either a non-negative integer or − , thelatter corresponding to the case when z j is not used.The exponent is thus odd and unable to cancel the powerin z − n − r . This holds for the negative power terms too.The remaining terms from W m are either constant orhave degree exceeding ± n − , and so cannot form a con-stant term. Hence the integral is zero by the residuetheorem.If w n or w m were to be displaced from each other by adistance x , there will be an addition factor of z ± m x or z ± n x in the integral. However, it is clear from the aboveargument that such a term also cannot be combined withan other term to produce a constant term. Hence thedisplaced wavelet bases are also orthogonal, as requiredearlier on. Note that this above proof does not require C and D to be the same for each level j , but only that they mustall satisfy the conditions mentioned in Sect. II. Appendix B: Discussion on finding RG-invariantHamiltonians
Here we give a matrix approach to solving for the P ( z )of the appropriate wavelet transform that leaves a givenHamiltonian h ( z ) invariant.General real Laurent polynomials h ( z ) and P ( z ) for z = e ik , k ∈ R can be written as h ( z ) = l (cid:88) j =0 a j z j + c.c. = h even ( z ) + h odd ( z ) (B1) P ( z ) = 1 + l (cid:88) j odd p j z j + c.c. (B2)For h ( z ) to be invariant under the wavelet transform de-scribed by P ( z ) Eq. 18, λh ( z ) = h even ( z ) + ( P ( z ) − h odd ( z )must be satisfied. By equating the coefficients of non-negative powers of z on both sides (there are only evenpowers, of course), we obtain the relation a l ... a l − a l ... a l − a l − a l ... a ∗ l − a ∗ l − a ∗ l − ... a l a ∗ l a ∗ l − a ∗ l − ... a l − a l p l p l − p l − ... p ∗ l − p ∗ l = λ a l a l − a l − ... a a − a a (B3)where l is the (odd) degree of P ( z ), which is also themaximum possible degree of h ( z ). In matrix equationform, Eq. B3 becomes A(cid:126)p = λ(cid:126)a − (cid:126)a e , where A is the lower triangular Toeplitz matrix comprising the coefficients of h odd ( z ) and their complex conjugates, (cid:126)p and (cid:126)a the vectorsof coefficients of P ( z ) and h odd ( z ) respectively, and (cid:126)a e the4vector of h even ( z ).Fortuitously, the lower triangular matrix A can be in-verted easily. Writing A = a l ( I + N ) where N is aNilpotent Toeplitz matrix, we easily find that A − =( I − N + N − ... + N l − ) /a l . Upon a bit more algebra,we find that p l p l − p l − ... p ∗ l − p ∗ l = b ... b b ... b l − b l − ... b b l − b l − ... b b b l b l − ... b b b λa l λa l − ... λa − a λa − a λa − a (B4) where b j , j, , ..., l are the coefficients of y j in the expan-sion of 1 a l + a l − y + a l − y + ... + a ∗ l y l = 1 y l/ h odd ( y − / )(B5)Since P ( z ) + P ( − z ) = 2, we can fix λ by requiring that (cid:80) lj p l − j = 1. A solution of Eq. 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