QQuantum Mechanics in the Infrared
Dorde Radiˇcevi´c
Stanford Institute for Theoretical Physics and Department of PhysicsStanford UniversityStanford, CA 94305-4060, USA [email protected]
Abstract
This paper presents an algebraic formulation of the renormalization group flow in quantummechanics on flat target spaces. For any interacting quantum mechanical theory, the fixed pointof this flow is a theory of classical probability, not a different effective quantum mechanics. Eachenergy eigenstate of the UV Hamiltonian flows to a probability distribution whose entropy is anatural diagnostic of quantum ergodicity of the original state. These conclusions are supportedby various examples worked out in some detail. a r X i v : . [ h e p - t h ] A ug Introduction
Quantum chaos is an old topic motivated by a simple question: what is the analogue of classical chaosin quantum systems? Equivalently, how are apparent randomness of motion and sensitivity to initialconditions encoded in quantum dynamics at long times? This question has appeared in many guises,e.g. in the study of the spectrum of highly excited states of quantum systems [1–4], in connectionto the localization/thermalization transition in many-body systems (see e.g. [5]), in attempts tounderstand the black hole information paradox [6, 7], and even in relation to reformulating andproving the Riemann hypothesis [8].Ultimately, the study of quantum chaos reduces to understanding the long-time (but not neces-sarily long-distance) behavior of a quantum system. For instance, spectral statistics of a Hamiltoniancan be probed using correlation functions at long times, or, equivalently, by calculating a certainpath integral over long classical trajectories that fill up the available phase space [9–13]. By nowthere exists a wealth of information about the long-time regime of many quantum systems, andmany universal properties that diagnose quantum chaos (or lack thereof) are known. In particular,correlators at the longest time scales available in the system govern spacings between nearby eigen-values of the Hamiltonian, and these spacings are universally captured by simple random matrixtheory in all chaotic systems [3, 14, 15].Given the overwhelming evidence for long-time universality, and more generally given the im-portance of studying quantum chaos in connection to questions in both the condensed matter andthe high energy communities, developing a systematic renormalization group (RG) procedure thattakes us to long times would be of great interest. This cannot be the usual RG in the sense of Wil-son and Kadanoff [16, 17]. For instance, the desired procedure should work in quantum mechanics,i.e. in 0 + 1 dimensions, where there is no analogue of “spin-blocking” available. Another reasonis that this procedure should also work for conformal field theories, which are fixed points underWilsonian RG.In this paper we will formulate long-time RG for quantum mechanics (QM). There are no spatialdimensions here, so it is possible to think of QM as a QFT in 0 + 1 dimensions and apply the usualWilsonian ideas to the path integral in order to orient ourselves; we will later comment on why theyare not enough. Let us consider a QM theory given by (cid:90) [d q ] exp (cid:40) i (cid:90) d t (cid:34)(cid:18) ∂q∂t (cid:19) − V ( q ) (cid:35)(cid:41) . (1)2he field q has mass dimension − / The linear size σ of the target is a dimensionful coupling in the theory. Under Wilsonian RG, σ will flow towardszero, and the theory will become strongly coupled. More precisely, the dimensionless parameterthat sets the strength of interactions is σ /τ , where τ is the time scale of interest. For fixed σ , shorttimes τ are described by coherent, classical dynamics, while long times represent the limit in whichquantum fluctuations are important. Wilsonian RG has very limited use here. QM dynamics is quasiperiodic whenever there areat least two incommensurate frequencies in the problem, so unless the theory is free or otherwisefine-tuned, the usual RG equations will be oscillatory and will not reduce long-time dynamics tosome universal fixed point [18]. Another way to see this problem is to notice that the IR is noteasily expressed as an effective field theory because all powers of q are, na¨ıvely, relevant operators;an explicit regularization is needed to make sense of the IR. This issue is related to the fact thatapproximate conformal symmetry in QM only arises in conjunction with singular potentials andother pathologies [19, 20] (see also [21, 22] for new results on the dual nearly-AdS space).In this paper we avoid path integral methods and present a very natural way to describe amonotonic RG flow in the Hamiltonian formalism. The idea is very much in the spirit of WilsonianRG: at each step, we reduce the algebra of observables to an appropriately chosen subalgebra. (UsualKadanoff decimation can be expressed in this language.) As we will argue in detail in Section 2,at short times this algebraic coarse-graining naturally corresponds to changing the cutoff used todiscretize time in the definition of the QM path integral. The true benefit of our framework, however,is that it allows us to rather transparently understand what happens in the deep IR, i.e. when thetime step becomes finite, or even when it is of the order of the recurrence time. The upshot is thatgoing to longer time scales (or smaller algebras) does not take us to a different, effective QM; rather,each decimation turns some degrees of freedom into classical random variables, and flowing to thedeep IR turns a QM theory into a theory of classical probability whose event space corresponds tothe remaining Abelian algebra of observables. This is decoherence. In order to meaningfully talk about chaos, the phase space has to have finite volume. This is why we only workwith compact target spaces in this paper. Note that we set (cid:126) = 1 as in any other QFT. Literature on quantum chaos often studies the semiclassical limit (cid:126) → (cid:126) = 1 and using otherparameters — in this case σ /τ — as dimensionless couplings that govern quantum fluctuations.It is also important to contrast the two kinds of classical regimes one can talk about. At σ /τ → ∞ , the pathintegral is dominated by trajectories around a unique classical saddle point; the kinetic term controls most of thedynamics and there is typically ballistic transport with very little diffusion. At σ /τ →
0, the path integral receivescontributions from all possible long classical trajectories/saddle points. The conventional lore is that this regimereflects the statistics of energy eigenstates, as these are the only states with trivial time evolution — it is this propertythat allows them to probe long times without getting averaged to death. Looking at the Hamiltonian − (cid:126) µ ∂ ∂q + V ( q ),one sees that studying the spectrum at (cid:126) → ∼ / (cid:126) ) have finite energy, therebyallowing us to study them in a controlled way. This way the classical limit gives us access to the long-time dynamics. no effective, UV-independent, quantum description of interacting IR dynamics in 0 + 1dimensions. The only exception is a trivial one: time-independent states in a given theory will havewell-defined IR behavior. This means that there is no universal quantum-mechanical fixed pointin any model, and the only sensible fixed points in QM are time-independent classical probabilitydistributions that come from decimating energy eigenstates in the UV. For models whose targetspaces are Abelian groups, we will put these claims on solid footing by the end of Section 2.In Sections 3 and 4 we study algebraic coarse-graining of energy eigenstates and its connectionto diagnosing chaos. We restrict ourselves to flat target spaces with various potentials. In each casewe decimate the algebra of observables, and we follow how eigenstates of the Hamiltonian decohereand become mixed following a very specific pattern governed by the center of the subalgebra.Along the way we note that the von Neumann entropy of the resulting mixed states acts both as aZamolodchikov c -function and as a measure of quantum ergodicity or “eigenstate thermalization,”i.e. as a measure of how randomly distributed the eigenstates are compared to those of a freeHamiltonian that is naturally defined on the target group.In the Conclusion, we emphasize the lessons learned and offer more comments on related topics.In particular, we point out other systems of potential interest that this procedure extends to. As abonus, we also clarify the nature of the Hilbert space of a single Majorana fermion. We will exclusively work with QM theories whose target G is a d -torus or, when regulated, a discretegroup of the form ( Z N ) d . The generalization of our results to other (nonabelian) compact groupsis straightforward but often tedious, and we will briefly review how it is done at the end of thisSection.The operator algebra inherits the group structure of the target space in the following way. Let g i be the d generators of G . There are then 2 d generators of the algebra. These are the d positionoperators, U i , and the d shift or momentum operators, L i . The Hilbert space appropriate to thegiven target group G is spanned by a set of vectors {| g n · · · g n d d (cid:105)} labeled by elements g = (cid:81) i g n i i ∈ G with n i = 0 , . . . , N −
1. (For continuous G , we will use e iθ i e i with θ i ∈ [0 , π ) instead.) The position4perators are chosen such that each of these basis states is an eigenstate of all the U i ’s; the spectrumof U i is then given by a unitary representation of U (1) or Z N . The shift operators in the discrete caseare chosen so that L i | g (cid:105) = | g i g (cid:105) ; in the continuous case, L i = exp (cid:110) i d θ i ∂∂θ i (cid:111) . Since G is Abelian,all shift generators commute with each other, and the only nontrivial commutation relation is U i L i = g i L i U i . (2)For simplicity, we now specialize to d = 1. Consider first the continuous case: a particle on acircle. Let us focus on propagation over a very short time interval d t (relative to any other timescale in the problem). The length d t can be thought of as a regulator in the path integral, whichis formally defined as a product of ∼ t ordinary integrals. If the kinetic term has two derivatives,any two trajectories that differ by less than √ d t at all times will have the same action and willtherefore constructively interfere in the path integral. Higher derivative terms in the action willmerely change the power of d t without changing the qualitative picture.We can thus conclude that a temporal cutoff induces a target space cutoff, in the sense that aQM path integral whose time is measured in steps of d t → t sums overthe target space with step √ d t . Conversely, a target space cutoff d x induces a natural time stepd x , the time it takes to diffuse across length d x ; any trajectories that started within length d x ofeach other will be mixed together after time d x . However, if d x ∼ G = Z , then thisheuristic makes no sense, as d t is no longer infinitesimal.Now we see in more detail how Wilsonian RG leads to trouble. As we begin the flow in a theorywith target U (1) = lim N →∞ Z N , the natural time step is d t ∼ /N →
0. As d t is increased, N willdecrease, and at some point it will stop being the largest quantity in the problem. In fact, aftersufficiently many na¨ıve Wilsonian decimations, N will become finite, and at that point the naturaltime step is finite and we have no right to view the effective theory as being given by a path integral.This means that we need a better framework for understanding what happens after many deci-mations. The connection between d t and d x (or N ) provides a natural way forward in a Hamiltonianframework: one flows to the IR by coarse-graining the target space without any explicit referenceto time. We will henceforth refer to the starting target space as the UV, and to the final targetspace as the IR. In order to flow from the UV while maintaining all the IR correlations, we followthe same strategy that proved useful in understanding entanglement in general QFTs [23–25]: weconstruct a sequence of operator algebras, each contained in the previous one, and we choose thatthe position generator in each new algebra can distinguish fewer different positions. For instance, if U was a position generator in the UV Z N theory, U will be a position generator appropriate for a5 N/ theory, U will be appropriate for a Z N/ theory, and so on. States — represented by densitymatrices that belong to algebras — are then naturally projected along the flow in such a way as toensure that expectations of all IR operators are preserved.In the next subsection we will present this construction in detail, but here we highlight how itdiffers from Wilsonian RG as we typically understand it:1. We flow to longer time scales, but not necessarily to smaller energies. However, the long-timedynamics encodes information about gaps between nearby eigenvalues in the Hamiltonian,while dynamics at shorter times encodes coarser properties of the spectrum. In this (admit-tedly vague) sense we are flowing towards smaller energy scales.2. States in the IR will typically be mixed, even if we started from a pure state in the UV. InQFT we work with the pure state sector in the IR, but here we will not be able to do thisbecause the UV and IR sectors are strongly coupled. The coupling between the UV and IR ina QFT is governed by the small width of the momentum shell being integrated out; here wehave no such parameter. Thus, evolution of a pure IR state will almost always make it mixed.3. In a similar vein, time evolution in the IR is not governed by an effective Hamiltonian builtout of IR operators. The mixing of states is described by a classical probability distributionwhose evolution is set by the time-dependent expectation values of the UV operators in theHamiltonian.4. The Kadanoff decimation in QFT can also be represented as a reduction in algebras, as we arethere removing position and shift operators associated to modes with high spatial momentum.In QM we have no sense of spatial momentum, and instead we are removing position operatorswhile leaving the shift operators intact. Unlike in QFT, our IR algebras will always have anontrivial center. The superselection sectors labeled by the center eigenvalues are the sectorsthat are being mixed; center operators are the classical variables. After many decimations,we will be left only with the center. This is now a wholly classical regime, with each densitymatrix diagonal in the center eigenbasis.5. There exists scheme dependence. For instance, coarse-graining shift operators instead of theposition operators leads to a different IR with different center operators. For theories with aquadratic kinetic term and a position-dependent potential, the scheme in this paper is natural.6. We will be particularly interested in the IR fixed point in which we have no more positionoperators to remove (the “deep IR”). In principle, we can then continue coarse-graining the6hift operator algebra. This leads to a trivial algebra that contains just the identity, which isthe ultimate fixed point of our RG. However, in this paper we focus on the fixed point of thescheme in which just the position generators are coarse-grained.7. The dimension of the Hilbert space never changes in our setup, but all density matrices becomeblock-diagonal in the center eigenbasis. The labels of these sectors are classical variables, andso one can say that the effective amount of quantum variables in the theory decreases aftereach coarse-graining. Let us now describe in some detail the proposed target space coarse-graining in a Z N theory. Forsimplicity, we take N to be a power of two. The starting algebra of observables A consists of allpossible products of position and shift generators U and L , with U N = L N = . In position basis, U = diag( g n ) where g = e πi/N and n = 0 , . . . , N −
1. We define a decimation to be the map A r (cid:55)→ A r +1 , where A r +1 is generated by the same shift operator as A r and by the square of theposition generator of A r . Since we are starting from A , this means that A r is generated by U r and L . The center of A r is the Abelian group generated by L r ≡ L N/ r , as can be easily checkedby using the commutation relation (2).Whenever we represent elements of A r as matrices on the N -dimensional Hilbert space (whichis the same for every r ), we will always work in a basis that diagonalizes L r . All elements of A r will be block-diagonal in this basis. We will refer to these blocks as superselection sectors, and wewill index them with k = 0 , . . . , r − L r in sector k is g kr ≡ g Nk/ r .Any density matrix ρ is an element of the algebra of observables. It can be represented as ρ = (cid:88) O∈A ρ O O . (3)In our setup, it can be shown [28] that ρ O = 1 N (cid:10) O − (cid:11) . (4)Decimation also maps ρ r (cid:55)→ ρ r +1 such that Tr( ρ r O ) = Tr( ρ r +1 O ) for all O ∈ A r +1 ⊂ A r . Thisdecimation is simple in the eigenbasis of L : one just restricts ρ r to the block-diagonal form with2 r +1 blocks, and sets all the other entries to zero. Only eigenstates of L r +1 remain pure after thisdecimation. 7t is instructive to see how certain states map. Let us take as an example the Z theory.After one decimation, the position eigenstates | (cid:105) = | g (cid:105) and | g (cid:105) both become uniform mixes of √ (cid:0) | (cid:105) + | g (cid:105) (cid:1) and √ (cid:0) | (cid:105) − | g (cid:105) (cid:1) . On the other hand, the L eigenstates √ (cid:0) | (cid:105) ± | g (cid:105) (cid:1) remainpure. The same thing happens for the pair of states | g (cid:105) and | g (cid:105) . In general, any linear combinationof pure nondegenerate eigenstates of L r will become mixed, i.e. ρ = (cid:88) k, l α k α ∗ l | ψ k (cid:105)(cid:104) ψ l | (cid:55)→ ρ r = (cid:77) k | α k | | ψ k (cid:105)(cid:104) ψ k | . (5)for any set {| ψ k (cid:105)} of nondegenerate eigenstates of L r .Going back to the Z example, we may thus view the decimated system as a coupled set of twoHilbert spaces (sectors) of dimension two, or of two Z theories. Each sector is spanned by stateswith the same L eigenvalue; e.g. one sector is spanned by √ (cid:0) | (cid:105) + | g (cid:105) (cid:1) and √ (cid:0) | g (cid:105) + | g (cid:105) (cid:1) . Eachsector can thus be viewed as a Z theory with a prescribed winding along this decimated groupmanifold.As another example, consider the U (1) theory (regulated to look like Z N with very large N )and perform one decimation. Any state in the original theory will now be a mix of two states livingon circles half the size of the original one; one of these two states will have trivial winding aroundthe new circle, while the other state will pick up a factor of g = g N/ = − r decimations, each state in the original theory will become a mix of 2 r states, each of which with adefinite winding g kr = e πik/ r around the decimated target space.The reduced density matrix after r decimations thus takes the form ρ r = r − (cid:77) k =0 p k (cid:37) k . (6)Here, (cid:37) k are the 2 r diagonal blocks of the original density matrix in the L eigenbasis, normalizedso that each has unit trace in its N r -dimensional subspace. The normalization factors p k form aprobability distribution on the space of sectors. They are given by the expectations of the centerof A r , and it is straightforward to check that p k = 12 r r − (cid:88) l =0 (cid:68) L lr (cid:69) g − klr . (7)Another useful way to express these probabilities is via the eigenstates | (cid:96) (cid:105) of the shift operator L .These states satisfy L | (cid:96) (cid:105) = g (cid:96) | (cid:96) (cid:105) , (cid:96) = 0 , . . . , N −
1, and for a given state | Ψ (cid:105) the classical probabilities8re p k = (cid:88) (cid:96) = k +2 r k (cid:48) k (cid:48) =0 , , ..., N r − |(cid:104) (cid:96) | Ψ (cid:105)| . (8)We already see from this form that in the deep IR, when 2 r = N , the sector probabilities reduceto the classical probabilities of eigenstates of operators in the remaining Abelian algebra. Wealso see that p k is the momentum-space analogue of Wigner, Husimi, and other quasiprobabilitydistributions that are often the main tools in studying eigenstate statistics, as in [10, 11, 26, 27] andreferences therein. However, we stress that in our case the states (cid:96) are determined by the groupstructure of the target space.For any initial pure state, the matrices (cid:37) k all describe pure states. The only mixing thathappens is between different sectors, and the entropy of this mixing is the Shannon entropy S r = − (cid:88) k p k log p k . (9)This is a nondecreasing function of r for any starting state so it can be identified as a c -function forour RG, though it is generally time-dependent. As we will discuss next, there is no time dependencefor energy eigenstates of the UV Hamiltonian, and we may define the average IR entropy of theHamiltonian as ¯ S r ≡ N (cid:88) n S ( n ) r , (10)where | n (cid:105) are the N energy eigenstates and S ( n ) r are their entropies after r decimations. The averageIR entropy is bounded by r log 2, and after each decimation it cannot rise by more than log 2. Thisin turn implies that in the deep IR, at r = log N , the IR entropy is bounded by¯ S IR ≡ ¯ S log N ≤ log N. (11)In the examples we will find that this entropy is connected to the mixing properties of the underlyingclassical system — or, more precisely, that it measures the quantum ergodicity of the theory. In the UV, dynamics is given by ρ t = e iHt ρ e − iHt . In the IR, things are no longer as simple.If the UV Hamiltonian contains no operators that are lost after r decimations, i.e. if it is block- An educated reader might guess that the (cid:37) k should be thermal states, at least for eigenstates of chaotic theories,as per the eigenstate thermalization hypothesis. This is not the case in quantum mechanics. L r eigenbasis with H = (cid:76) k H k , then the appropriate probability distribution p k ofany state will be time-independent. The time evolution of any state will then just be captured by (cid:37) tk = e iH k t (cid:37) k e − iH k t .However, as long as the Hamiltonian is not the identity matrix, eventually the decimation willmake us lose operators that govern time evolution. Once this happens, no operator in the remainingobservable algebra will be able to determine how the sector probabilities evolve, unless the UV statewas an energy eigenstate to begin with. This follows easily from eq. (7), withd p k d t = − i r (cid:88) l (cid:68) [ H, L lr ] (cid:69) g − klr ; (12)any operator in H that is also in A r will necessarily commute with L lr , and hence d p k / d t will bedetermined by the UV expectation values alone. From the point of view of the IR, the sectors allcouple to each other as they evolve in time, and in addition this evolution has external parameterswhose time dependence is specified by the UV state and Hamiltonian.The above conclusions are very general; we could have started from any other algebraic decima-tion scheme, and the IR time-dependence would have still been encoded in the UV data. Assumingthat algebraic decimation is the only controlled way of performing RG, this means that there is no effective IR theory that governs the dynamics of a QM system. Said another way, time evolution inQM is always sensitive to UV physics because of the nontrivial center. The only exception to thisrule justifies the lore stated in the Introduction: only eigenstates of the Hamiltonian, due to theirtrivial time-dependence, have a UV-insensitive time evolution at long time scales. Therefore, theonly IR physics that one can meaningfully talk about is obtained by decimating energy eigenstates. We now work out some examples in increasing order of complexity. Our goal is to study theaverage IR entropy. Where available, we perform computations analytically; otherwise, we performpreliminary numerical studies and leave a more detailed analysis for future work. We will alwayswork with the class of Hamiltonians with a quadratic kinetic term, H = α ( L + L − ) + (cid:88) n β n ( U n + U − n ) (13)for some constants α and β n . Note that the continuum limit is reached by letting N → ∞ whilescaling α ∼ / d θ ∼ N and β n ∼ N . 10 .1 Free particle on a circle Let us start with a free particle moving on a Z N group in the limit of large N . The Hamiltonian is H = α ( L + L − ) N →∞ −→ − µ ∂ ∂θ . (14)(Note that we drop constant terms in H when going to the continuum.) Eigenstates and eigenvaluesof the Hamiltonian are ψ n ( θ ) = 1 √ π e inθ , E n = n µ , n ∈ Z . (15)Sector probabilities for each state ψ n are easily calculated from (7) or (8), and they are p k = δ ( k + n ≡ r ) . (16)Being eigenstates of L r , the states ψ n ( θ ) retain their purity during decimation. For each n , there isa unique k such that n + k ∈ r Z , and this is the sector the state is in. Note that this means that theclassical probability is a δ -function for each eigenstate, independent of the number of decimations.The average IR entropy is thus ¯ S IR = 0 . (17)The fact that the average entropy is independent of N is one possible signature of quantum ergod-icity, as we will discuss later.The Hamiltonian commutes with all of the decimated operators in this theory, and thereforeall the p k ’s for all the states will stay constant. In fact, they can be found explicitly for any stateΨ( θ ) = √ π (cid:80) n Ψ n e inθ . Sector probabilities are given by (8), p k = (cid:88) n + k ≡ r | Ψ n | . (18)Note that once all of the U ’s are decimated, we get the famous result p k = | Ψ k | . This is theaforementioned decoherence in the IR. Further decimation now coarse-grains the Abelian algebraof shift operators. This can be interpreted as going to ever smaller σ -algebras of events in classicalprobability. It is only because of this decimation in the deep IR that the free theory is not, strictlyspeaking, a fixed point of our procedure. Due to the degeneracy of ψ n and ψ − n , we could have chosen different eigenstates, and this would have led todifferent sector probabilities. This is not generic; typically we will deal with completely nondegenerate systems. Evenif we encounter small degeneracies, they will only affect the IR entropies by finite amounts, and this will not changeany of our conclusions. .2 SHO on a circle Now we add the simplest possible potential to the free particle: H = α ( L + L − ) + β ( U + U − ) N →∞ −→ − µ ∂ ∂θ + µω cos θ. (19)This is the potential of the quantum rotor. The eigenstates are solutions to the Mathieu equationand can be manipulated numerically.Before proceeding, let us notice that potentials U n + U − n , for any n ≤ N/
2, all lead to theSchr¨odinger equation of a Mathieu type. At n = N/ L and all its powers commute with this Hamiltonian).Here the theory is H = α ( L + L − ) + βU N/ . (20)To control the computation we must keep thinking about the discrete target space, as the potentialoscillates with a wavelength equal to the shortest distance in the problem.This system splits into N/ L . Eachsector is spanned by two states, | odd , n (cid:105) ∝ g n | g (cid:105) + g n | g (cid:105) + . . . and | even , n (cid:105) ∝ | (cid:105) + g n | g (cid:105) + . . . with a sector Hamiltonian H n = β α ( g n + g − n ) α ( g n + g − n ) − β . (21)Here n runs from 0 to N/ −
1. The eigenstates of the full Hamiltonian are thus states | n ± (cid:105) ∝ α ( g n + g − n ) | odd , n (cid:105) + (cid:16) β ± (cid:112) α ( g n + g − n ) + β (cid:17) | even , n (cid:105) (22)with eigenvalues E n ± = ± (cid:112) α ( g n + g − n ) + β . (23)These are eigenstates of L and all other powers of L . In particular, these states are eigenstatesof L r up until the last decimation, when r = log N . Hence, until the last decimation, eigenstatesstay pure. At the last decimation, the entropy of each eigenstate rises by a term bounded by log 2.Going back to the U + U − potential (19), we numerically investigate how periodic Mathieufunctions (eigenstates in the continuum limit) behave under decimation. These are functions ψ Sn ( θ )and ψ Cn ( θ ) that, as ωµ →
0, reduce to ordinary sines and cosines. For a fixed ωµ , the states12ith n (cid:28) ωµ are oscillations localized at θ = π , the minimum of the potential; at n (cid:29) ωµ , thewavefunctions approach ordinary sines and cosines spread out over the entire circle.We have enough control over these functions to immediately bound their IR entropy ¯ S r at themaximal number of decimations, r = N log 2. To do this, it is enough to use (8) and calculate theoverlaps of ψ S/Cn ( θ ) with the plane waves √ π e i(cid:96)θ . At n (cid:29) ωµ , only two sectors ( (cid:96) = ± n ) have anontrivial probability, leading to an entropy of log 2 for each high-momentum state. At n (cid:46) ωµ ,there are more sectors with nontrivial probabilities, but they are all located at | (cid:96) | (cid:46) ωµn , and theentropy for these states can be very roughly bounded by log ωµn . At ωµ (cid:29)
1, the contribution fromall the low-energy states is then approximately bounded by (cid:80) ωµn =1 log ωµn ∼ ωµ , and the average IRentropy is ¯ S IR (cid:46) ωµ + ( N − ωµ ) log 2 N N →∞ −→ log 2 . (24)The average IR entropy is finite in the continuum limit, and approximately satisfies the same boundas the one we analytically derived for the U N/ potential. As before, the finiteness of the IR entropyis a signature of the system not being ergodic.It is worthwhile pointing out that the IR entropy increases with ωµ . This increase in entropycomes from the low-energy states that are more and more localized by the deep cosine potential at θ = π . This low-energy sector very much resembles the spectrum of a particle in a box at ωµ (cid:29) A particle in a box does can be modeled by an infinitely repulsive δ -function potential on a circle, H = α ( L + L − ) + β (cid:88) n U n N →∞ −→ − µ ∂ ∂θ + βδ ( θ ) , β (cid:29) N = δ (0) . (25)There will then be N − δ -function — the position eigenstate | (cid:105) . The energy of the latter willbe O ( βN ) while the standing waves will have energies O (1); therefore we may think of the standingwaves as an effective low-energy description of the δ -function potential on the circle.Coarse-graining will eventually make any state become a mix that includes the high-energylocalized state. From the point of view of the low-energy theory in the box, this means that aftereach coarse-graining, there are fewer and fewer states that do not mix with the high-energy degreeof freedom. In this sense, the low-energy theory upon decimation loses degrees of freedom and hasto be modeled with Hilbert spaces of ever lower dimensionality. This corresponds to the walls of13he box closing in on the particle until the particle has no low-energy degrees of freedom left.The localized state always decimates to a maximally mixed state, as can be readily checkedby representing its density matrix in momentum basis and noticing that all diagonal elements areequal to 1 /N . Its entropy is thus log N in the deep IR, but since there is only one such state, itscontribution to the average IR entropy is negligible.The standing waves also contribute nontrivially to the IR entropy. Compared to the propagatingwaves of the free particle, a standing wave entropy is higher because the δ -function forces it to havea node at θ = 0, so after enough decimations it will necessarily become mixed.The wave with p humps corresponds to the state | p (cid:105) = (cid:88) n g pn/ − g − pn/ i √ N | g n (cid:105) N →∞ −→ √ π (cid:90) d θ sin p θ | e iθ (cid:105) . (26)This makes sense when 0 < p < N . The overlaps in (8) can be computed to be |(cid:104) (cid:96) | p (cid:105)| = 12 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − g Np/ − g − (cid:96) + p/ − − g − Np/ − g − (cid:96) − p/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (27)When p = 2 q (cid:54) = 0, we simply get |(cid:104) (cid:96) | q (cid:105)| = 12 ( δ (cid:96), q + δ (cid:96), − q ) . (28)On the other hand, when p = 2 q + 1, we get |(cid:104) (cid:96) | q + 1 (cid:105)| = 2 N (cid:12)(cid:12)(cid:12)(cid:12) − g − (cid:96) + q +1 / − − g − (cid:96) − q − / (cid:12)(cid:12)(cid:12)(cid:12) . (29)For even harmonics, the entropy in the IR is simply log 2. For odd harmonics, the formulaeare slightly more complicated, but the sector probabilities in the deep IR are still highly peaked at (cid:96) = ± ( q + 1) and (cid:96) = ± ( q − S IR ≈
32 log 2 . (30)As expected, this result is not very interesting, as it just agrees with the intuition that there isno mixing in d = 1. To make matters more interesting, we must go to higher dimensions. Theequivalent of a particle on a circle with a δ -function potential will be the famous Sinai billiard, andindeed we will see signatures of chaos there. 14
50 100 150 200 250 - - - - - - - - Figure 1:
Representative energy eigenfunctions in the delocalized (above) and localized (below) phase on256 sites in the target space. Localization of all eigenstates happens at β (cid:38) . ∼ /N in our model. Before leaving the comfortable confines of d = 1, let us study a classically ergodic model and seehow it qualitatively differs from the examples above. We consider a variant of the Aubry-Andr´emodel [29], i.e. a particle in a weak quasiperiodic potential, H = α ( L + L − ) + (cid:88) n (cid:2) β n ( U n + U − n ) + iγ n ( U n − U − n ) (cid:3) N →∞ −→ − µ ∂ ∂θ + β [cos( ξθ ) + sin( ζθ )] . (31)To ensure quasiperiodicity, ξ and ζ are taken to be irrational, and we include both the sine andcosine in order to eliminate the parity symmetry and make the spectrum nondegenerate. In thediscrete model, the couplings are β n = ( − n β π ξ sin( πξ ) ξ − n , γ n = ( − n β π n sin( πζ ) ζ − n . (32)This model is very similar to one in which the couplings are randomly chosen [30]. The wavefunctionsare localized in position space whenever the coupling is greater than the scale set by the systemsize (1 /N ). The original Aubry-Andr´e model is more sophisticated and, for almost all irrationalperiods, features a localization/delocalization transition in the continuum limit. I thank Wen Wei Ho for extensive discussions on this issue. b S IR b S IR Figure 2:
Dependence of the average IR entropy on the coupling β for N = 128 (blue, lower) and N = 256(purple, upper). The second graph zooms in to the area around the crossover to localization, where ¯ S IR isfound to smoothly interpolate between the two phases. The difference between the two entropies asymptotesto log 2, and each entropy asymptotes to log N , its maximal possible value. We exactly diagonalize the discrete model with up to N = 256 sites. The onset of delocalizationdue to finite size effects can be seen on Fig. 1, where typical eigenstates are shown at values of β below and above 1 /N . In the localized regime the energy eigenstates are expected to have nontrivialIR entropy, just like how in the presence of the δ -function the odd-numbered standing waves receivedextra IR entropy due to the node at θ = 0. The increase in IR entropy is much more significanthere as eigenstates are localized much more brutally than at just one point in position space.The average IR entropy is straightforward to evaluate using (8). The main finding is that in thedelocalized regime, the IR entropy does not grow with N , just like in the previous examples, whilein the localized regime it grows as ¯ S IR = − s ( β ) + s ( β ) log N, (33)with s ( β ) >
0. The parameters s ( β ) and s ( β ) approximately saturate the entropy bound, s ∗ ≈ s ∗ ≈ β (cid:29) /N . Examples of this behavior are shown on Fig. 2. In the next Section we willargue that this behavior of ¯ S IR corresponds to the lack of mixing in the underlying classical model(or, more precisely, to the lack of quantum ergodicity in the quantum model). Our final example is the Sinai billiard: a particle moving on a two-dimensional torus with a circular,totally reflecting obstacle. We will model this system the same way we modelled a particle in abox, by studying motion on Z N ⊗ Z N with several impenetrable δ -functions scattered across thetorus surface. A single obstacle would be enough to make the very excited states ergodic [10], butthis will provide only a O (1 /N ) contribution to the IR entropy. The entropy will increase the more16
10 20 30 40 50 60 R S IR R S IR Figure 3:
Average IR entropies of standing waves on a two-torus with R scatterers, with N = 64 (left)and N = 256 (right). The fitted functions have form s + s log R , and the best-fit parameters are s ∼ . s ∼ .
7. This is just an illustrative sample; the parameters were not obtained from statistical averaging overdifferent scatterer configurations. scatterers we add, especially if we break symmetries and lift degeneracies while adding them. TheHamiltonian is H = − µ (cid:18) ∂ ∂θ + ∂ ∂θ (cid:19) + β R (cid:88) i =1 δ ( (cid:126)θ − (cid:126)ϑ i ) , (34)where as before β (cid:29) N and (cid:126)ϑ i are random points on the torus.To study the influence of the scatterers, we compute the IR entropy for different numbers R ofrandomly sprinkled δ -functions on a two-torus. Each obstacle will contain a localized state whichwill contribute a log N to the sum over entropies. However, we are interested in the entropycontained in the standing waves only, as these are the eigenstates one gets when studying theLaplacian on the surface bounded by the obstacles. This is why the IR entropy we present in whatfollows contains an average only over these standing wave entropies. (For the particle in a d = 1box, this was irrelevant, as there was only one localized state; here we may have O ( N ) localizedstates.)We find that, for R scatterers, the average IR entropy of the remaining N − R eigenstates growsas ¯ S IR = s + s log R. (35)These results for N = 64 and N = 256 are shown on Fig. 3. The number R can be construedas the size of the scattering obstacle, and we thus see that the entropy of each standing wave witha macroscopic obstacle ( R ∼ N ) is of the order log N without saturating the bound log N until,approximately, the obstacle size fills up the entire torus at R = N . This is a striking qualitativedeviation from all the previous results, where the average entropy was either O (1) or essentiallyequal to its theoretical maximum. As we will argue in the next Section, this is the signature ofquantum ergodicity. 17 Infrared entropy and quantum ergodicity
We now want to make connections between the calculated IR entropies and the quantum ergodicityof theories in question. Recall that a system is called quantum ergodic if, roughly, its energyeigenstates are randomly distributed, or if there is “eigenstate thermalization.” More precisely, thiscondition is fulfilled if for almost any operator, its expectation value in an energy E eigenstate isequal to the average of the classical function representing the operator over the constant energy E surface in classical phase space [31, 32]. This happens when the underlying classical system ischaotic.The entropy of the probability distribution (8) measures how much the eigenvectors of H deviatefrom an “ordered” set of eigenvectors — the eigenstates of the free particle on the underlying group.In this sense, one would expect that the larger the IR entropy, the more random the eigenstate.However, there is a duality at work here: if an eigenstate’s entropy reaches its theoretical maximumlog N (with N the Hilbert space dimension), this state is very atypical — it is localized in positionspace. Indeed, we saw in the Aubry-Andr´e model that the localized phase has an average IR entropythat approximately saturates its maximal value. In addition, even with the cosine potential, we sawthat the entropy was a monotonically increasing function of µω . If we took this coupling to infinity,all eigenstates would become localized in the bottom of the cosine potential, and indeed we wouldretrieve ¯ S IR = log N .This situation can be compared to that in quantum field theories on a lattice when both sidesof a strong-weak coupling duality are tractable. Examples include the Ising chain in a transversefield and the Kitaev toric code in two spatial dimensions. The strong coupling regimes in thesetwo theories would be called “ordered” and “confined,” respectively; at weak coupling the regimesare disordered and deconfined/topological, which are just order and confinement in the dual basisof states. At strong coupling, the ground state entanglement entropy of a region is zero; at weakcoupling, the entropy is the maximal possible one consistent with locality and global constraints.We see that in QM the same story plays out, with “ordered” replaced by “free” (or “localized inmomentum space”) and “disordered” replaced by “localized in position space,” and with the IRentropy playing the role of the ground state entanglement entropy.We are thus led to expect that a theory is quantum ergodic when its average IR entropy divergesin the continuum/thermodynamic limit ( N → ∞ ) — but only when it does not diverge maximallyfast. In particular, we may speculate that the most chaotic classical theories give ¯ S IR = s log N upon quantization, with s being a potentially universal number near 1 /
2. Deviations from this18onjectural baseline would then reflect the existence of nonergodic states like quantum scars [33].Ideas like these are likely not new to the cognoscenti of quantum chaos. The novelty of ourapproach is the exclusive usage of groups as target spaces. One benefit of this new approach is thatfree particle eigenstates (both the position and the momentum ones) can be defined with referencejust to the group structure, and thus provide a privileged set of eigenstates with respect to whichthe overlaps in (8) are to be measured.A different but complementary benefit is that the notion of algebraic decimation naturally givesrise to the entropy of sector probabilities as a measure of long-time behavior (and, therefore, ofquantum ergodicity). Moreover, performing a few decimations already allows us to compute anentropy function via eq. (7), giving us a natural coarsening of the microscopic overlap function thatmay still be a useful diagnostic.
Not only is our renormalization group of quantum mechanics not a group, it also does not renor-malize any infinity. Nevertheless, we have argued that it is still a very useful operation to consider:it formalizes the Wilsonian idea of flowing to longer times and allows us to understand the fixedpoints of quantum mechanics in the deep infrared. We have shown that for any quantum mechanicsdefined on a flat group manifold, the fixed point is not another quantum mechanics — instead, itis a classical theory of probability. The entropy of this infrared probability distribution is a naturaldiagnostic of quantum ergodicity while naturally sharing many parallels with other entropies fromQFT.The following points are worth emphasizing at the close of this paper:1. The group structure was crucial to defining and interpreting the algebraic decimation. Manyquantum mechanical problems lack an apparent underlying group structure, including someintegrable ones like the Calogero model. Perhaps a group structure can be uncovered in eachsuch problem, just how we have interpreted a particle in a box as a particle on a torus witha totally reflecting barrier.2. We have not addressed the infrared fate of particles moving on nonabelian groups. For posi-tively curved groups, this is a straightforward, if tedious, extension of the methods presentedhere. For hyperbolic spaces (which are of great interest on their own), a few more tools seemto be needed, and this analysis will be presented in a separate publication [34].19. One of our main results is that there exists no effective quantum mechanics that governs theevolution in the infrared. Our analysis was valid for all Hilbert space dimensions. However,perhaps when this dimensionality is large, an approximate, statistical description of the IRevolution can be defined. Understanding this possibility would be a valuable goal.4. Our results were all derived for Hilbert spaces whose dimensions were powers of two. Extendingthese to other powers requires a straightforward modification of the decimation procedure soas to involve different powers of original position generators. Amusingly, if N is prime, there isonly one decimation needed before reaching the classical probability regime — independentlyon the size of N !5. A single Majorana fermion is an operator χ that satisfies χ = . (Its fermionic nature isonly apparent when other fermions are around.) In the context of our work, we can recog-nize the Majorana algebra as the Abelian algebra that quantum mechanics on Z reducesto after one decimation. In other words, a single Majorana fermion shoud be viewed as anIsing spin for which we can only measure spin along one direction. Its natural Hilbert spaceis two-dimensional, and the possible set of states of one Majorana are all classical proba-bility distributions on two elements. Generalizations of Majorana fermions are obtained bydecimating any Z N group when N is prime.6. Degeneracy of the spectrum can lead to ambiguity in the infrared entropy. We manually fixedthis issue when it arose, but a more systematic treatment might be possible.7. We end by simply emphasizing the most important question that this paper fails to address:the origin of random matrix universality in quantum chaotic systems. We have argued thatgoing to long times does not give rise to an effective Hamiltonian that is a random matrix. Adifferent kind of flow and universality potentially await to be uncovered. Acknowledgments
It is a great pleasure to thank Jaume Gomis, Guy Gur-Ari, Wen Wei Ho, Raghu Mahajan, Xiao-Liang Qi, Grant Salton, Steve Shenker, and Sho Yaida for useful conversations. The author wassupported by a Stanford Graduate Fellowship and has benefited from the hospitality of PerimeterInstitute while a part of this work was done. Research at Perimeter Institute is supported bythe Government of Canada through Industry Canada and by the Province of Ontario through theMinistry of Economic Development & Innovation.20 eferences [1] Wigner, E. P., & Dirac, P. A. M. 1951, Proceedings of the Cambridge Philosophical Society,47, 790.[2] F. J. Dyson, “Correlations between the eigenvalues of a random matrix,” Commun. Math.Phys. , no. 3, 235 (1970). doi:10.1007/BF01646824[3] M. V. Berry, “Some quantum-to-classical asymptotics,” in Les Houches Lecture Series LII(1989), eds. M.-J. Giannoni, A. Voros and J. Zinn-Justin, North-Holland, Amsterdam, 251-304.[4] M. Gutzwiller, “Chaos in classical and quantum mechanics,” Springer (1990).[5] R. Nandkishore and D. A. Huse, “Many body localization and thermalization in quantumstatistical mechanics,” Ann. Rev. Condensed Matter Phys. , 15 (2015) doi:10.1146/annurev-conmatphys-031214-014726 [arXiv:1404.0686 [cond-mat.stat-mech]].[6] S. H. Shenker and D. Stanford, “Black holes and the butterfly effect,” JHEP , 067 (2014)doi:10.1007/JHEP03(2014)067 [arXiv:1306.0622 [hep-th]].[7] J. Maldacena, S. H. Shenker and D. Stanford, “A bound on chaos,” arXiv:1503.01409 [hep-th].[8] M. V. Berry, “Riemann’s zeta function: a model for quantum chaos?” in Quantum chaosand statistical nuclear physics, eds. T H Seligman and H Nishioka, Springer Lecture Notes inPhysics No. 263, 1-17 (1986).[9] M. C. Gutzwiller, “Periodic Orbits and Classical Quantization Conditions,” J. Math. Phys. 12,343 (1971).[10] M. V. Berry, “Regular and irregular semiclassical wave functions,” J. Phys. A, 10, 2083-91(1977).[11] E. B. Bogomolny, “Smoothed wave functions of chaotic quantum systems,” Phys. D, 31, 2(1988).[12] M. Sieber, K. Richter, “Correlations between Periodic Orbits and their Role in SpectralStatistics,” PhyS, T90, 1 (2001).[13] S. M¨uller, S. Heusler, P. Braun, F. Haake, A. Altland, “Periodic-orbit theory of universalityin quantum chaos,” Phys. Rev. E, 72, 4 (2005).2114] O. Bohigas, M. J. Giannoni and C. Schmit, “Characterization of chaotic quantumspectra and universality of level fluctuation laws,” Phys. Rev. Lett. , 1 (1984).doi:10.1103/PhysRevLett.52.1[15] S. M¨uller, S. Heusler, A. Altland, P. Braun, F. Haake, “Periodic-orbit theory of universal levelcorrelations in quantum chaos,” New J. Phys. 11, 103025 (2009).[16] L. P. Kadanoff, “Scaling laws for Ising models near T(c),” Physics , 263 (1966).[17] K. G. Wilson, “Renormalization group and critical phenomena. 1. Renormalization group andthe Kadanoff scaling picture,” Phys. Rev. B , 3174 (1971). doi:10.1103/PhysRevB.4.3174[18] J. Polonyi, “Renormalization group in quantum mechanics,” Annals Phys. , 300 (1996)doi:10.1006/aphy.1996.0133 [hep-th/9409004].[19] V. de Alfaro, S. Fubini and G. Furlan, “Conformal Invariance in Quantum Mechanics,” NuovoCim. A , 569 (1976). doi:10.1007/BF02785666[20] C. Chamon, R. Jackiw, S. Y. Pi and L. Santos, “Conformal quantum mechanics as theCFT dual to AdS ,” Phys. Lett. B , 503 (2011) doi:10.1016/j.physletb.2011.06.023[arXiv:1106.0726 [hep-th]].[21] J. Polchinski and V. Rosenhaus, “The Spectrum in the Sachdev-Ye-Kitaev Model,” JHEP , 001 (2016) doi:10.1007/JHEP04(2016)001 [arXiv:1601.06768 [hep-th]].[22] J. Maldacena and D. Stanford, “Comments on the Sachdev-Ye-Kitaev model,”arXiv:1604.07818 [hep-th].[23] H. Casini, M. Huerta and J. A. Rosabal, “Remarks on entanglement entropy for gauge fields,”Phys. Rev. D , no. 8, 085012 (2014) doi:10.1103/PhysRevD.89.085012 [arXiv:1312.1183[hep-th]].[24] D. Radiˇcevi´c, “Notes on Entanglement in Abelian Gauge Theories,” arXiv:1404.1391 [hep-th].[25] S. Ghosh, R. M. Soni and S. P. Trivedi, “On The Entanglement Entropy For Gauge Theories,”JHEP , 069 (2015) doi:10.1007/JHEP09(2015)069 [arXiv:1501.02593 [hep-th]].[26] S. Nonnenmacher, “Anatomy of quantum chaotic eigenstates,” Poincar´e seminar 2010, [27] B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky. “Quantum chaos: localization vs.ergodicity.” Physica D: Nonlinear Phenomena 33, 1 (1988).2228] D. Radiˇcevi´c, “Entanglement Entropy and Duality,” arXiv:1605.09396 [hep-th].[29] S. Aubry, G. Andr´e. Ann. Israel Phys. Soc. 3, 133 (1980).[30] P. W. Anderson, “Absence of Diffusion in Certain Random Lattices,” Phys. Rev.109