Anomalies and Renormalization of Impure States in Quantum Theories
aa r X i v : . [ h e p - t h ] J u l Anomalies and Renormalization of Impure States in Quantum Theories
Kumar S. Gupta
1, 2, ∗ and Amilcar Queiroz † Theory Division, Saha Institute of Nuclear Physics, International Institute of Physics (IIP-UFRN)Av. Odilon Gomes de Lima 1722, 59078-400 Natal, Brazil Instituto de Fisica, Universidade de Brasilia, Caixa Postal 04455, 70919-970, Brasilia, DF, Brazil
In a Hamiltonian approach to anomalies parity and time reversal symmetries can be restored byintroducing suitable impure (or mixed) states. However, the expectation values of observables suchas the Hamiltonian diverges in such impure states. Here we show that such divergent expectationvalues can be treated within a renormalization group framework, leading to a set of β -functions in themoduli space of the operators representing the observables. This leads to well defined expectationvalues of the Hamiltonian in a phase where the impure state restores the P and T symmetry. Wealso show that this RG procedure leads to a mass gap in the spectrum. Such a framework may berelevant for long wavelength descriptions of condensed matter systems such as the quantum spinHall effect. Anomalies occur when a classical symmetry is brokendue to quantization [1]. In quantum field theories, theywere first discovered within the perturbative approach[2–4]. It was soon realized that anomalies are intrinsi-cally non-perturbative in nature [5, 6], characterized bygeometric and topological structures of the theory [7].This point of view has led to a much wider realization ofanomalies in diverse areas including particle physics [8],black holes [9, 10] and important condensed matter sys-tems such as quantum Hall systems and the associatededge states [11–15].One way to understand anomalies involves the studyof unbounded operators [16] in quantum theories such asthe momentum and the Hamiltonian. The complete defi-nition of such unbounded operators requires the specifica-tion of appropriate boundary conditions or equivalentlytheir domains so that they become self-adjoint opera-tors. A symmetry can be implemented in the quantumtheory if the corresponding generators leave the domainof the Hamiltonian invariant and then if they commutewith the Hamiltonian. Sometimes a symmetry genera-tor does not preserve the domain of the Hamiltonian. Inthis case, the symmetry is broken due to quantizationbeing thus anomalous. This approach to anomalies [17–19] is very general and it allows its study in a large classof quantum systems such as molecular physics [20, 21],condensed matter systems [22], integrable models [23–25]and black hole physics [20, 26, 27]. This approach canalso be adapted to quantum field theories [28], where themode expansion encodes the information about the ap-propriate boundary conditions.In recent papers [28, 29], it has been shown that whilea system might admit an anomaly when in a pure state,it is possible to restore the anomalous symmetry whenthe system is in a suitable impure (or mixed) state. This ∗ On leave from SINP, India; [email protected] † [email protected] naturally leads to the question as to what are the ex-pectation values of the observables of the system in suchimpure state.The aim of this Letter is to provide an answer to thisquestion. In particular, we focus our attention to thecase of the expectation values of the Hamiltonian. If theimpure state is obtained by an appropriate summationof pure states each corresponding to different domains,it is well known that the expectation values of observ-ables such as the Hamiltonian diverge in such an impurestate. In this Letter we show that such divergences can behandled by using renormalization group (RG) techniques[18, 30]. The corresponding beta function provides a flowin the moduli space of the operators and the choice of theassociated impure state depends on the energy scale ofthe problem. Therefore, the RG flow removes the diver-gence and even more interestingly it makes the quantumtheory scale dependent. Physically it says that if we de-mand anomalous symmetry restoration, then the appro-priate boundary conditions depend on the energy scale atwhich the system is being probed. To our knowledge thisis the first attempt that divergences arising from suchimpure states are analysed using non-perturbative RGtechniques.We will present these ideas below in the problem ofa particle on a circle and its anomalous parity P andtime-reversal T symmetries. We first recall how impurestates with a non-trivial von Neumann or entanglemententropy can be used to restore such anomalous symme-tries. Then we discuss how to obtain expectation valuesof the observables such as the Hamiltonian in the impurestate making explicit the appearance of divergences. Wethen show how the application of RG techniques lead tofinite answer for the expectation values of observables inthe impure state. We next provide general arguments tothe emergence of such impure states restoring anomaloussymmetries. Finally we discuss how this approach can beapplied to system with gapped bulk and edge states thatare modelled by quantum states of a particle on a circle.The Hamiltonian for a particle on a circle of radius R is given by H = − R ∂ ∂ϕ . (1)The Hamiltonian H is self-adjointin the family of domains D θ = (cid:8) L ( S , dϕ ) : ψ (2 π ) = e iθ ψ (0) , ψ ′ (2 π ) = e iθ ψ ′ (0) (cid:9) ,with θ ∈ [0 , π ). The parameter θ may be associated tothe magnetic flux through the circle. In simpler terms,this parameter labels the allowed family of boundaryconditions for which the Hamiltonian H is self-adjoint.The dimension of the Hamiltonian is [length] − .For a fixed domain D θ , the eigenvalues and eigenfunc-tions of H are given by ǫ θ,n = 1 R ( n + θ π ) ≡ n θ R , ψ θ,n = e in θ ϕ √ π , (2)with n ∈ Z .The action of parity P on the wave-function is givenby [ P ψ θ ]( ϕ ) = ψ − θ ( ϕ ). Indeed, since [ P ψ ]( ϕ ) = ψ (2 π − ϕ ), we see that [ P ψ θ ](2 π ) = ψ θ (0) = e − iθ ψ θ (2 π ) = e − iθ [ P ψ θ ](0). The parity operator therefore changes theboundary condition of the Hamiltonian. Thus, if θ = 0 , π ,parity is broken due to quantization being thus anoma-lous [28]. Also, since time-reversal T is a anti-unitarytransformation, then for θ = 0 , π it also changes thedomain of the Hamiltonian, [ P ψ θ ]( ϕ ) = ψ − θ ( ϕ ), beingtherefore anomalous.The eigenstates of ( H, D θ ) are pure states. They aremapped to the domain D − θ by parity P and time reversal T . In [28], it was proposed that if the system is in a phasedescribed by the impure state of the form ρ = 12 ( ρ θ + ρ − θ ) , with ρ θ = | ψ θ ih ψ θ | , (3)then parity P and time-reversal T are no longer anoma-lous. Indeed, parity P and time reversal T leave theimpure state ρ invariant. Indeed, under these operations θ
7→ − θ , but since the impure state ρ is obtained by av-eraging over these values of θ , it is invariant under P or T .Eigenstates of the self-adjoint operator H form a ba-sis in the Hilbert space. Therefore we may expand anystate in an arbitrary domain in terms of the eigenstatesin the original domain of the Hamiltonian. Let ψ θ,n bean eigenstate of H belonging to the domain D θ . Then χ θ,n = P ψ θ,n ∈ D − θ , with θ = 0 , π . We may then ex-pand χ θ,n in terms of the eigenstates { ψ θ,k } ⊂ D θ as χ n,θ = X k b nk ψ θ,k , (4)where b nk = Z π dϕ π e − i ( k + θ π ) ϕ e i ( n − θ π ) ϕ = 12 π e − iθ − n − k − θπ . The expectation value of H ≡ ( H, D θ ) in the impurestate (3) associated to the n -th energy level, that is, ρ n =(1 / ρ θ,n + ρ − θ,n ), is h H i n = Tr ρ n H = 12 ǫ θ,n + X k | b nk | ǫ θ,k ! , (5)where | b nk | = 12 π − cos 2 θ (cid:0) n − k − θπ (cid:1) . (6)The second term in the RHS of (5) diverges in the UVregion, that is, when k → ∞ . To see this, we first convert P k → R dk , so that X k | b nk | ǫ k → I ( n, θ ) = Z ∞−∞ dk | b nk | ǫ k . (7)We then use (6) together with (2), to get I ( n, θ ) = 1 − cos 2 θ π R Z K − K dk k θ ( n − θ − k θ ) (8)= 2 K − Kn − θ K − n − θ + 2 n − θ log (cid:12)(cid:12)(cid:12)(cid:12) K − n − θ K + n − θ (cid:12)(cid:12)(cid:12)(cid:12) , where we have introduced a UV cut-off K . We observethat the first term in the RHS is linearly divergent whilethe remaining terms vanish as K → ∞ . Therefore (8)becomes effectively I ( n, θ ) = 1 π lim K →∞ − cos 2 θR K. (9)The expectation value of the Hamiltonian is thus h H i n = 12 R (cid:20) n θ + lim K →∞ − cos 2 θπ K (cid:21) . (10)Observe that the second term on the RHS of (10),which is divergent, does not depend on the level n . In-deed, the UV divergence occurs solely due to the pro-jection of a state in the domain D − θ in terms of thedomain D θ . Thus even though the impure state removesthe P and T anomaly, expectation values of the observ-ables such as the energy calculated in the impure stateappear to diverge. This is the reason why until now nomeaningful result has been obtained using a impure stateformed out of states belonging to different domains of aself-adjoint operator.We now show that it is indeed possible to control thedivergence using RG techniques leading to finite expec-tation values of the observables in the impure state.We observe first that the expectation value of theHamiltonian in (10) depends on the self-adjoint parame-ter θ , the ultraviolet cut-off K and the radius of the circle R . From (10), we observe that the fixed n -th level energy E n = h H i n diverges as K → ∞ or R →
0, or both. Wenext consider two of these cases separately. R fixed while K → ∞ : This situation correspondsto the particle on a circle of fixed radius R . In this case,as the UV cut-off is removed, that is, K → ∞ , the ex-pectation value of the Hamiltonian in the impure stateconstructed out of the level n = 0, given by h H i = lim K →∞ R (cid:20) θ π + 1 − cos 2 θπ K (cid:21) , (11)clearly diverges. In any physical system we must have afinite expectation value of the Hamiltonian in the groundstate. In order to achieve that we now propose a RGflow in the parameter θ . Therefore we promote the self-adjoint parameter θ to a function of the ultraviolet cut-off K . We then demand that as K → ∞ , the expectationvalue of the Hamiltonian in the ground state becomesindependent of the cut-off [18, 30]. This leads to thecondition lim K →∞ d h H i dK = 0 . (12)We find from (11) that the corresponding β -function is β K ( θ ) ≡ K dθdK = 2 K cos 2 θ − θ + 4 K sin 2 θ . (13)For small θ , such that sin 2 θ/ θ → θ → − θ ,this equation can be integrated to give θ = C √ K ≈ C √ K , (14)with C a constant of integration. The constant C is notpredicted by the theory but must be obtained empirically.For example, if the value of the self-adjoint extensionparameter θ can be measured to have the value θ at agiven scale K = K , then C = √ K θ .The above RG flow, for small values of θ , renders theexpectation value of the Hamiltonian in the impure statecomprising of the n = 0 level finite, as the UV cut-off K is taken to infinity while R is fixed. Indeed, substituting(14) into (12), then taking the limit K → ∞ , we obtain h H i = C π R . (15)Under this RG flow, the ratio of the expectation valuesof the Hamiltonian in the excited state to that in theground state also remains finite, that is, h H i n h H i = 1 + 4 π n C . (16)In our approach, the expectation value of the Hamilto-nian in the ground state is traded in favour of the RGflow given by the β -function and the expectation valuesin the excited states are measured in terms of that in theground state energy. R → and K → ∞ : Let us now consider the doublelimit K → ∞ and R → K → ∞ is natural, since it was introducedas a regulator and the physical quantities should finally be independent of such regulators. The limit R → ∞ stands in a different footing. Indeed, the problem of aparticle on a circle does not require this limit and forthat problem, we are free to choose any fixed value of R .We can nevertheless study a different physical problem.For example, consider an infinitesimally thin topologicaldefect such as a flux tube, carrying magnetic flux passingperpendicular to a plane. In order to study dynamics ofparticles at the edge of this defect, we first regularize theproblem, by considering a flux tube of finite radius R .We study the dynamics in the region beyond the radius R and finally take the limit R →
0. For this problem,the radius R serves now also as a regulator and as suchwe need to remove it at the end of the analysis. Observehowever that in real system no defect is infinitely thinbut has a finite radius depending on the scale at whichit is being probed. We now see that a suitable RG flowwith respect to the quantity R allows us to study thissituation.In this case, the expectation value of the Hamiltonianin the impure state constructed from n = 0 level states,which is given by (11), now diverges as both R → K → ∞ . In the same spirit as before, we can now con-sider the self-adjoint extension parameter θ as a functionof both K and R , that is, θ ≡ θ ( K, R ) and demand thatlim K →∞ ∂ h H i ∂K = 0 , lim R → ∂ h H i ∂R = 0 . (17)We again assume that θ is small and also that it canbe factorized as θ ( K, R ) = θ ( K ) θ ( R ). This gives thecorresponding β -functions β K ( θ ) = K ∂θ ( K ) ∂K = − θ , (18) β R ( θ ) = − R ∂θ ( R ) ∂R = − θ , (19)which can be integrated to θ ( K ) = C √ K , θ ( R ) = C √ R. (20)As before the constants C and C have to be determinedempirically. If at a given K = K and R = R , wehave θ = θ ( K ) , θ = θ ( R ), we find that C = θ ( K ) √ K and C = θ ( R ) / √ R .These flows of the self-adjoint extension parameter, inthe limit of K → ∞ and R →
0, render h H i finite, thatis, h H i = C C π . (21)The ratio of the expectation values of the Hamiltonianin the excited state to that in the ground state is givenby h H i n h H i = 1 + 4 π C C n R . (22)We observe that at any value R , the system is gappedand the gap increases as R →
0. This may be physicallyinterpreted as follows. Suppose there exists a topologicaldefect such as a vertex operator or a flux tube passingperpendicular to a plane. Suppose further that we aimat analyse the dynamics of parity P and time-reversal T invariant excitations at the boundary of these defects. In-sistence on these symmetries leads us to the use of impurestates. Naively the energies associated to such impurestates would be infinite as the UV cut-off K is removed.Our renormalization procedure gives a completely well-defined answer to this problem when K → ∞ . Now, inaddition we may analyse also the nature of the such ex-citations as the radius of the topological defect changes,that is, it is governed by the RG flow associated to theradius R . For any radius R , these “edge” excitations aregapped and the gap increases as R → A generated by operators exp ( ip ϕ ˆ ϕ )and exp ( iϕ ˆ p ϕ ). A state ω on this algebra is a non-negative linear functional compatible with adjoint conju-gation and normalizable to 1. For any observable O ∈ A ,one may associate the expectation value ω ( O ) ≡ hOi ρ =Tr ρ O , where ρ is some density matrix.Under time-reversal T (or parity P ), the algebra A maybe written as A + ⊕ A − , where A ± = { a ∈ A , T aT = ± a } . The subalgebra of T -even observables is A even = A + ⊕ CP − , where P − is an orthonormal projector ( P − = P − ) to A − such that = P + + P − and C denotes complexnumbers.Consider a state ω θ such that under T it becomes ω − θ ,that is, the associated density matrix ρ θ satisfies T ρ θ T = ρ − θ . Then the restriction of such state to the subalgebraof T -even observables A even is equivalent to consideringthe impure state ω = ω θ + ω − θ on the algebra A . Indeed,if O odd / ∈ A even , then ω ( O ) = ( ω θ + ω − θ ) ( O odd ) = 0.The Hamiltonian (1) is only formally T - and P -invariant. As we have seen before, a quasi-periodicboundary condition defining the domain of H breaksthese symmetries. Therefore the self-adjoint Hamiltonian H is neither T - or P -even nor T - or P -odd. Nevertheless, H being an observable in A may be decomposed as an T -even and T -odd components, that is, H = H even + H odd .To see this, note that H θ = 1 R (cid:18) p ϕ + θ π (cid:19) = 1 R (cid:18) p ϕ + θ π + θπ p ϕ (cid:19) . (23) The even and odd components of this Hamiltonian are H even = 1 R (cid:18) p ϕ + θ π (cid:19) , H odd = 1 R θπ p ϕ . (24)Under T -even partial observations, only the first compo-nent H even gives a non-zero contribution. Thus, the ex-pectation value of this Hamiltonian under a T -even stategives ω ( P even H P even ) = 1 R (cid:18) ω (cid:0) p ϕ (cid:1) + θ π ω ( P even ) (cid:19) . (25)Therefore, from ρ constructed out of a n -level eigenstateof p ϕ leads to h H even i n = 1 R (cid:18) n + θ π h P even i n (cid:19) . (26)This equation is equivalent to equation (10) a part froma θ/ π -translation in n , that is, n n + θ/ π .The mechanism of anomaly restoration discussed inthis work may have applications to time-reversal and/orparity invariant edge states. Edge states appear in con-densed matter systems with boundaries where the bulkis gapped. There is an intimate relationship betweenanomalies and edge states. The quantum Hall effect [22]is a well known example of this relationship, where the T symmetry is broken. A natural question is whether sim-ilar anomaly/edge states relationship can be realized insystems with T and/or P invariant edge states. For in-stance, quantum spin Hall (QSH) samples [32] are in theuniversality class of systems with gapped bulk and T in-variant edge states. The quantum states of a particle ona circle may serve to model certain classes of edge statesin such systems. In particular, the partial observationof T -even observables would induce the emergence of T even impure states in the edge. The associated entangle-ment entropy would provide evidence for the mixed state.Such partial observations may be a result of externallychoices of what one is attempting to measure in suchsamples. Nevertheless, it may also be argued that somecouplings among the microscopic degrees of freedom maynaturally reproduce such partial observations. Examplesof such couplings seem to be the spin-orbit couplings [32]or coupling leading to instanton-like terms in the effectiveHamiltonian [33]. ACKNOWLEDGEMENT
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