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A Numerical Renormalization Group for Continuum One-Dimensional Systems Robert M. Konik and Yury Adamov CMPMS Department, Brookhaven National Laboratory, Upton, NY 11973 (Dated: February 6, 2008) We present a renormalization group (RG) procedure which works naturally on a wide class of interactingone-dimensionmodelsbasedonperturbed(possiblystrongly)continuumconformal and 7 integrable models. This procedure integrates Kenneth Wilson’s numerical renormalization group 0 with Al. B. Zamolodchikov’s truncated conformal spectrum approach. Key to the method is that 0 such theories provide a set of completely understood eigenstates for which matrix elements can 2 be exactly computed. In this procedure the RG flow of physical observables can be studied both n numerically and analytically. To demonstrate the approach, we study the spectrum of a pair of a coupled quantum Ising chains and correlation functions in a single quantum Ising chain in the J presenceof a magnetic field. 4 PACSnumbers: 2 ] l The numerical renormalization group (NRG), as de- E E e veloped by Kenneth Wilson [1], is a tremendously suc- 5 ~1/R - tr cessful technique for the study of generic quantum im- E4 s purityproblems,systemswhereinteractionsareconfined E . t to a single point. But the NRG as such is not directly 3 a E m generalizable to systems where interactions are present 2 - in the bulk. The natural generalization of the NRG in E d realspacetreatsboundaryconditionsbetweenRGblocks R 1 n inadequately, leading to qualitatively inaccurate results. o To overcome this difficulty, Steven White developed the FIG. 1: A schematic of the finite sized system, both in real c space and in terms of energy levels, analyzed in the TSA [ density matrix renormalizationgroup(DMRG) [2]. This procedure. tool is now ubiquitous in the study of low dimensional 1 strongly correlated lattice models and can access both v static and dynamic quantities [3]. 5 at the start, dramatically lessening the computational 0 In this letter we offer a distinct generalization realiz- burden. In one of the TSA’s first applications, Al. B. 6 ingarenormalizationgroupprocedureforawiderangeof Zamolodchikov studied a critical Ising chain in a mag- 1 0 stronglyinteractingcontinuumone-dimensionalsystems. netic field [5], a continuum version of the lattice model 7 Itcantreatanymodelwhichisrepresentableasaconfor- 0 mal or integrable field theory (CFT/IFT), with Hamil- H0Ising=−J (σizσiz+1+σix); Hpert=−h σiz, (1) / X X t tonian, H0, plus a relevant[4] perturbation (of arbitrary i i a strength), H . Beyond this there is no real constraint m pert where σa are the standard Pauli matrices and i indexes onH orH . Inparticular,thefulltheory,H +H , i - 0 pert 0 pert the sitesofthe lattice. The continuummodel,itselfinte- d need notbe integrableor conformal. Thus the technique grable,hasacomplicatedspectrumofeightfundamental n canhandle a standardarrayofmodelsofperturbedLut- excitations. The TSA was able to produce the gaps of o tingerliquidsorMottinsulators. Italsocapableoftreat- the first five excitations within 2% of the analytic, infi- c ing disordered systems, either by envisioning H as a : pert nitevolumevaluesbydiagonalizingamere39x39matrix. v random field or, equally well, considering a non-unitary To see how remarkablethis is, consider that ina compu- Xi supersymmetric CFT, H0 arising from disorder averag- tationally equivalent exact diagonalization of the lattice ing a system with quenched disorder [6]. This technique r model,onewouldbelimitedtostudyingafivesitechain. a alsoallowsthe studyofcoupledCFT/IFTs,allowingthe The TSA begins by taking the model to be studied, studyofsystemsbetweenoneandtwodimensions. Inall H = H +H , and placing it on a finite ring of cir- 0 pert cases,the lowenergyspectrumandcorrelationfunctions cumference, R. Doing so makes the spectrum discrete of the model are computable. (see Fig. 1). We nonetheless expect to be able to ob- Ourstartingpointisthetruncatedspectrumapproach taininfinitevolumeresultsprovidedweworkinaregime (TSA) pioneered by Al. B. Zamolodchikov. The TSA where R∆ ≫ 1 with ∆ a characteristic energy scale of was developed to treat perturbations of simple confor- the system. In the discrete system, the spectrum can mal field theories. While straightforward in conception, then be ordered in energy, |1i,|2i,.... Non-perturbative it has an advantage over other numerical techniques in information is input in the next step of the TSA where that it analytically embeds strongly correlated physics the matrix elements, H =hi|H |ji, are computed pertij pert 2 exactly. It is important to stress this is always a practi- which at each step a finite lattice is expanded by one cal possibility. If H is a CFT, the attendant Virasoro site,the modeldiagonalized,andhighenergyeigenstates 0 algebra (or, equally good, some more involved algebraic thrownaway. Itisthisiterativeprocedurethatwemimic. structure such as a current or a W-algebra) permits the computation of such matrix elements straightforwardly. 11.5 If H is instead an IFT, such matrix elements are avail- 0 able through the form-factor bootstrap programme [11]. NRG In an IFT, the matrix elements which we will want to RG Fit 11 focus on involve states, |ii, with few excitations and, as TSA such, are readily computable. 2 With the matrix elements in hand, one can then ex- ∆ 10.5 press the full Hamiltonian as a matrix. The penultimate stepintheprocedureistotruncatethespectrumatsome RG Fit Region energy, E , making the matrix finite. This matrix trunc 10 is then numerically diagonalized from which the spec- Exact Value of ∆ 2 trum andcorrelationsfunctions canbe extracted. When H represents a theory with a relatively simple set of 12 16 20 24 28 0 E eigenstates,thisprocedure,evenwithacrudetruncation trunc of states, works remarkable well in extracting the spec- itsrummo.reHcoowmevpelircwatheedn(tshaeysataCrtFinTgbpaosiendtHonamaiSltUon(2ia)n,cHu0r-, FtrIuGn.ca3t:ioPnloetnsesrhgoyw,iEntgrutnhce(bfoerhaRvi=or5oλ˜f−∆1)2.asafunctionofthe k rent algebra such as would be encountered in the study LetusdenotetheinitialbasisofferedbyH as{|ii} . 0 ∞i=1 of spin chains), or one is interested in computing corre- We begin by keeping a certain number, say N + ∆N, lationfunctions inthe fulltheory,H0+Hpert,thesimple of the lowest energy states,{|ii}N+∆N (in blue in Fig. i=1 truncation scheme ceases to produce accurate results at 2). We diagonalize the problem so extracting an ini- reasonable numerical cost. For example, errors in an ex- tial spectrum and set of eigenvectors,{|˜ii}N+∆N (in red i=1 citation gap, ∆, introduced by the truncation of states in Fig 2). We then toss away a certain number, ∆N, behave as power laws, δ∆ ∼ Et−ruαnc (α ∼ 1). Thus in- of the eigenvectors corresponding to the highest energy, creasingEc doesnotdramaticallyreduceδQwhileatthe i.e. {|˜ii}iN=+N∆+N1 . A new basis is then formed, consist- same time greatly increasing the cost of the exact diag- ing of the remaining eigenvectors together with the first onalization routine which should scale similarly to the ∆N states of {|ii} that we had previously ignored, ∞i=1 partition of integers, i.e. as eβ√Etrunc/Etrunc. It is the i.e. {|˜ii}N ∪{|ii}N+2∆N , and the procedure is re- i=1 i=N+∆N+1 aim of this letter to outline an RG technique offering a peated. We present the technique schematically in Fig. dramatic improvement on this truncation scheme. 2. ConvergenceoftheprocedureintheKondoproblemis promotedbythesmallmatrixelementsinvolvingsitesfar N N+∆N N+2∆N from the impurity. While matrix elements in the proce- 1) Take the first N+∆N states of the theory N N+∆N N+2∆N dure just described growprogressivelysmaller under the 2) Compute Hamiltonian and diagonalize NRG(scalingas1/E ),herenumericalconvergenceis trunc 3) Form a new basis of states using the first N eigenstates + the next ∆ N states in the original basis N N+∆N N+2∆N not necessarily the goal. Rather we aim to merely bring the quantity into a regime where its flow is governed by 4) Recompute Hamiltonian and numerically diagonalize a simple flow equation. 5) Return to 3) The algorithm just described implements a Wilsonian RG in reverse. It does so at all loop orders and so the FIG. 2: An outline of theNRGalgorithm RG flow it describes is exact. As the flow proceeds it, however,evolves closer and closer to a flow described by Our framework hews closely to the original Wilsonian a one-loopequation. Analytically, the equation is nearly conception of the NRG. In developing the NRG for the trivial as it is given solely in terms of the anomalous di- Kondo model, Wilson transformed the original Kondo mension, α , of the flowing quantity, Q (whether it be Q Hamiltonian using the “Kondo basis” to a lattice model an energy eigenvalue or a matrix element). More specif- of an impurity situated at the end of a infinite half line ically, with sites far from the end characterized by rapidly di- d∆Q minishing matrix elements. We, in a sense, start in this =−g(α )∆Q, (2) Q dlnE position. The ordering in energy of the states provided trunc by H is in direct analogy to the half-line on which the where ∆Q = Q(E )−Q(E = ∞) describes the 0 trunc trunc impurity lives in Wilson’s Kondo work. The next step deviation of the quantity as a function of the trunca- in Wilson’s NRG is an iterative numerical procedure by tion energy from its ’true’ value (i.e. the value where 3 the cutoff in energy is taken to infinity). The function g(α ) can be determined exactly using high energy per- Q exc. Exact TSA (10) NRG RGImproved turbation theory, well-controlled provided H is rele- pert ∆s1 11.2206 11.92/12.67 11.32/11.54 11.17(2)/11.15(5) vant. For example, if Q is some energy eigenvalue, then g(α )=α =1. ∆s2 11.2206 11.92/12.66 11.32/11.54 11.17(2)/11.16(6) Q Q ∆1 4.9936 5.29/5.61 5.03/5.12 4.97(1)/4.97(2) ThevirtueofEqn.(2)isthatitallowsustoruntheRG in two steps. We first implement the NRG as described ∆2 9.7369 10.69/11.55 9.89/10.24 9.70(3)/9.7(1) aboveuntilwereachatruncationenergyplacingussafely ∆3 13.9918 15.58/16.65 14.33/14.84 14.02(5)/14.20(5) in the one-loop regime. We then continue the RG by ∆4 17.5452 19.672/20.923 18.69/18.03 17.6(1)/17.7(1) merelyintegratingtheaboveequationallowingustofully ∆5 20.2188 23.64/24.64 20.80/21.62 20.2(2)/20.5(2) eliminate the effects of truncation. ∆6 21.8785 23.65/25.28 22.39/23.08 21.8(1)/21.8(2) We now consider two examples using this RG procedure, one where we compute the spectrum of a model TABLEI:TheexcitationenergiesfortwocoupledIsingchains at valuesof R=4/5λ˜−1. (in units of λ˜=λ4/7/(2π)3/7). and one where we analyze correlation functions. Both examples are chosenso that a straightapplicationof the TSA leads to poor results. Spectrum: In the first example, we consider a pair of state, ∆2, in the spectrum. We first show the results quantum critical Ising chains coupled together: of a straight TSA analysis (red squares) as a function of increasing truncationenergies (given in units of 1/R). H =HIsing 1+HIsing 2; H =−λ dxσzσz. (3) While the gap,∆ , is convergingtowardsits infinite vol- 0 0 0 pert Z 1 2 2 ume value, 9.7368648...λ˜, it is doing so only slowly and Thismodelisknowntobeintegrableandtohaveaspec- atexponentiallyincreasingnumericalcost. Wehaveper- trum equivalent to the sine-Gordon model at β2 = π formedthestraightTSAanalysisuptolevel15(i.e. keep- [10], that is, a spectrum with a pair ofsolitons with gap, ingstateswithenergieslessthan15/R)wheretheHilbert ∆ =∆ , together with a set of six bound states with spacecontains≈3500states. Atthispoint,theTSApro- s1 s2 gaps, ∆ = 2∆ sin(πk/14), k = 1,···,6. By compar- duces a result deviating by ≈20% from the exact value. k s ingconformalperturbationtheorywithathermodynamic We also plot the value of ∆ as given by the NRG algo- 2 Bethe ansatz analysis, ∆ can be expressed in terms rithm as it iterates through states of ever higher energy. s1,2 of the coupling constant, λ: ∆ = 11.2205920λ˜ with Here we have run the algorithm so as to take into ac- s1,2 λ˜ =λ4/7/(2π)3/7 [7]. count states up to level 25 (in total ≈ 150000/sector). The underlying finite volume Hilbert space of H is We see, reassuringly, that where the TSA results exists, 0 considerably more complicated than that of single Ising theNRGalgorithmproducesmatchingresults(atafrac- chain. Inasinglechaintherearefourpotentialsectorsof tionofthe numericalcost). The NRGalgorithmendsup theHilbertspace[5]: asector,I,composedofevennum- producingavalueof∆2 witha5%error. Finallyweplot bersofhalf-integerfermionicmodesactingoveraunique the result of fitting Eqn.(2) to the NRG results between vacuum, |Ii; a sector, F, composed of odd numbers of level 20 and 25 (where we believe a one-loop RG equa- half-integer modes over|Ii; and finally two sectors,σ/µ, tion describes the NRG flow). Extrapolating the fit to composedofevennumbers ofboth rightandleft moving Etrunc =∞gives∆2 =9.70λ˜,avaluedeviatingfromthe integerfermionicmodesoverdegeneratevacuua,|σi/|µi. exact result by 0.5%. Thesectorsσ andµareconnectedbyapplyingaproduct In Table 1 we present the results of our RG analysis ofanoddnumberofevenmodefermionicoperators. Un- on the complete spectrum of the two coupled chains. In der periodic boundary conditions, however, the Hilbert the first column we provide the exact value of spectrum spaceofasinglechainisreducedtotwosectors,I andσ. as determined by integrability and the TBA analysis of Inthe twochaincase,this is no longertrue. Notonly do Ref. [7]. In the second column we give the values of wehavesectorsofthe formI⊗I,I⊗σ, σ⊗I,andσ⊗σ the spectrumattwo differentsystemsizes (R=4/5λ˜ 1) − (such tensor products arising naturally from considering computed using a straight TSA analysis truncating at twochains),butF⊗F,F⊗µ,µ⊗F,andµ⊗µ. Unlikea level10 (i.e. keeping ≈600states in eachof the relevant single chain, allpossible sectorsare consistentwith peri- sectors). We see that the disagreement with the exact odicboundaryconditions. TheHilbertspacethatresults resultrangesupto20%. Inthe thirdcolumnwegivethe is thus much largerand applying the TSA with a simple results coming from applying the NRG algorithm (again truncationschemeleadstopoorresultsforthespectrum. iteratinguntilwehavereachedlevel25). Weseeamarked Computing the spectrum of this model is thus an ideal improvement over the TSA analysis, but still we obtain testing ground for our proposed RG procedure. resultswitherrorsrangingupto5%. Inthefinalcolumn, In Figure 3 we outline the procedure by which we we give the results for the spectrum arrivedat by fitting extract the values of the spectrum of the two coupled the one-loop RG equation (Eqn.(2)) to the NRG data. Ising chains focusing for specificity on the second bound We see that our errors are now less than 1% . 4 We are able to compute the necessary infinite volume 0.04 Exact matrix element over a continuous range of energies by 〉) RG Improved studying a single matrix element in finite volume where p - NRG the spectrumisdiscrete. We dosoby continuouslyvary- (10.03 TSA (Level 10) A ingthesystemsize,R. Undersuchvariations,theenergy 0.03 p) ω = 2(p2+m2)1/2 of the state, |A (p)A (−p)i, changes ( 1 1 1 A10.02 0.025 continuouslyduetothequantizationconditionofthemo-  mentum, p, (i.e. p=2πn/R+δ(p,−p) where δ(p,−p) is ) 0.02 0 ( a two-body scattering phase). σ 0.015  0.01 In Figure 4 by varying R we parametrically plot the 0 〈 0.01 resultsofourcomputationsoff2(p)vsitsexactvalue[9]. 1.01 1.02 1.03 1.04 1.05 1.06 AstraightapplicationoftheTSA(withalevel10trunca- 01 1.25 1.5 1.75 2 tion) produces acceptable results at higher energies but ω/2∆ doespoorlyatenergiesaroundthreshold,2∆ . Atlarger 1 1 values of R (and so smaller energies), the TSA breaks down. The TSA curve in this region is then double val- FIG. 4: A plot of the matrix element, h0|σ(0)|A1(p)A1(−p)i as a function of energy,ω=2(p2+m21)1/2. ued. ComputingthesamematrixelementwiththeNRG algorithmleadstoaconsiderableimprovementbutatthe lowestenergiesadeviationfromthe exactresultremains Correlation Functions: We now turn to the compu- (see inset to Fig. 4). RG improving the computation of tation of correlationfunctions using the above described f (p) largely removes this discrepancy even at energies RG methodology. For simplicity we consider only T =0 2 next to threshold. (In applying Eqn. (2) to Q = f (p), response functions although a generalization to finite 2 perturbation theory yields, g(α ) = 2(1− 1/8) where temperature multi-point functions is readily realizable. f2 1/8is the anomalousdimensionof the spin operator,σ). At zero temperature, the imaginary piece of a retarded In conclusion we have presented an RG scheme by correlation function, G (x,t) = hO(x,t)O(0)i , has a ret ret spectral decomposition, S (x,ω) = −1ImG (x,ω > which a large number of one-dimensional continuum OO π ret models can be studied with quantitative accuracy. With 0) equal to [12] this methodology, both the spectrum and spectral func- S (x,ω)= eiPsnx|h0|O(0,0)|n;s i|2δ(ω−E ),(4) tions of a model can be determined. n sn OO nX,sn RMK andYA acknowledgesupportfromthe US DOE (DE-AC02-98CH10886)togetherwithusefuldiscussions where |n;s i is an eigenstate of the system with en- n with A. Tsvelik. ergy/momentumE /P builtoutofnfundamentalsin- sn sn gle particle excitations carrying internal quantum numbers s . Thus the computation of any response func- n tionisequivalenttothe computationofanumberofma- trix elements ofthe formh0|O(x,0)|n;sni. Ostensibly to [1] K. Wilson, Rev.Mod. Phys. 47, 773 (1975). compute the response function fully one would need to [2] S.R. White, Phys.Rev.Lett. 69, 2863 (1992). compute an infinite number of such matrix elements. In [3] T.D. Ku¨hner and S.R. White, Phys. Rev. B 60, 335 practice if one is interested in the response function at (1999). [4] Further studies need to be done on the efficacy of the low energies only a small finite number of such matrix method for marginal perturbations. elements need be computed [12]. We will illustrate the [5] V.P.YurovandAl.B.Zamolodchikov,Int.J.Mod.Phys computationundertheRGofonesuchnon-trivialmatrix A 6, 4557 (1991). element for a critical Ising chain in a magnetic field (i.e. [6] K. Efetov, Adv.Phys. 32, 53 (1983). Eqn.(1)). Whilethespectrumofthis theoryrapidlycon- [7] V. A.Fateev, Phys. Lett. B 324 45 (1994). verges upon the increase of E , the matrix elements [8] A. B. Zamolodchikov, Adv. Studies in Pure Math. 19, trunc are less well behaved. And unlike the two-chain case, 641 (1989). [9] G. Delfino and G. Mussardo, Nucl. Phys. B 455, 724 analytical results are available for comparison [9]. Thus (1995). this computation is a good test of our RG methodology. [10] A. Leclair, A. Ludwig, and G. Mussardo, Nucl. Phys. B We specifically compute the two excitation contribu- 512, 523 (1998). tion f2(ω) = h0|σ(0)|A1(p)A1(−p)|0i with p = (ω2 − [11] F. Smirnov, “Form Factors in Completely Integrable ∆2 ), to the spin-spin correlator, S (p = 0,ω > 0). ModelsofQuantumFieldTheory”,WorldScientific,Sin- 1I σσ (Here ∆ is the gap of the lowest lying single particle gapore (1992). 1I excitation,A ,inanIsingchaininamagneticfield.) This [12] F. Essler and R. M. Konik in “From Fields to Strings: 1 Circumnavigating Theoretical Physics”, ed. by M. Shif- contribution takes the form man, A. Vainshtein, and J. Wheater, World Scientific, δS (ω)=π 1ω(ω2−4∆2 )1/2Θ(ω−2∆ )|f (ω)|2. (5) Singapore (2005). σσ − 1I 1I 2