Robert Konik, Yury Adamov Brookhaven National Laboratory Jean-Sébastien Caux

1. A Numerical RenormalizationGroup Approach to Strongly Correlated Systems: From Semiconducting Carbon Nanotubes to Atomic Bose Gases Robert Konik, Yury Adamov Brookhaven National Laboratory Jean-Sébastien Caux Universiteit van Amsterdam INSTANS 2nd Annual Conference September 12, 2008

2. Overview We will present a numerical renormalization group method able to compute the spectrum and correlation functions of arbitrary one dimensional systems with a relativistic continuum representation It is then the analog to DMRG for lattice systems In certain cases the method can be extended to two dimensional systems and one dimensional systems not relativistically invariant

3. Outline • Starting point: truncated conformal spectrum • approach • Problems with technique as a general method • Solution: employ a numerical renormalization • group • -two coupled Ising chains • iv) Semiconducting carbon nanotubes • Extension to 2D arrays of coupled chains • Atomic Bose gases in a trap

4. Overview of Truncated Spectrum Approach (TSA) for One Dimensional Systems Basic idea is to study a known (i.e. integrable or conformal) continuum system together with some perturbation: Consider the model on a finite sized ring of circumference, R i.e. metallic carbon nanotube gap term R E Spectrum of Hknown then becomes discrete: E5 ~1/R E4 E3 E2 E1

5. Input of strongly correlated information in the form of matrix elements: HKnown Truncate Hilbert space, making it finite dimensional. This allows one to write full Hamiltonian as a finite dimensional matrix En+1 E . En . . H = E3 E2 E1 Diagonalize H numerically and extract spectrum

6. Example of the TSA: Quantum Critical Ising Chain in a Magnetic Field Hamiltonian: Hknown Φpert continuum limit Model is exactly solvable (A. Zamolodchikov) and has a spectrum with 8 excitations Φpert Hknown Yurov and Zamolodchikov, 1991 TSA Results keeping 39 states TSA Exact (infinite volume) Ratios of spectral gaps Equivalent Exact Diagonalization Computation Chain with only five sites

7. Matrix elements using TCSA Good agreement with exact value (Delfino and Mussardo, 1995) at high but not low energies Obtain different energies parametrically by varying system size

8. Problems with the TSA as a General Methodology • 1. Matrix elements (response functions) converge slowly • Even matrix elements of a single Ising chain 2. Slow convergence of energy spectrum to actual value as a function of truncation energy • 3. This problem plagues models with complicated Hilbert spaces • Semi-conducting carbon nanotubes

9. N N N N+Δ N+Δ N+Δ N+2Δ N+2Δ N+2Δ Numerical Renormalization Group and the TSA R. K. and Y. Adamov, PRL 98, 147205 (2007) We have handled truncation in the crudest possible fashion: how to improve on this Numerical Renormalization Group (in the same spirit K. Wilson used it to study the Kondo problem) States with large energy only marginally influence the low-energy physics Fundamental observation: conduction electrons far from the impurity influence the physics only marginally NRG Recipe: 1) Take first N+Δ states (blue) of the theory 2) Compute the Hamiltonian and numerically diagonalize 3) Form a new basis of states using first N eigenstates (purple) plus next Δ states (blue) in original basis 4) Recompute Hamiltonian and numerically diagonalize 5) Repeat E

10. NRG and Two Ising Chains As a test case we study a solvable model where convergence issues will be present: two coupled critical quantum Ising chains: As shown by Leclair, Ludwig, and Mussardo (Nucl. Phys. B 512 (1998) 523), this model has a spectrum identical to the sine-Gordon model at β2=π (two solitons and six breathers -- bound states of the solitons)

11. Gap, D2, Flow Under NRG Each point represents gap of second breather at a particular RG step Gap converges to infinite volume analytic result Δ2 = 10.6: TSA Δ2= 10.2: NRG Δ2 = 9.73: exact

12. RG Improved TSA Computations The behaviour of a quantity, Q, as the truncation energy is increased obeys an RG equation. At leading order it takes the form: R. K. and Y. Adamov, PRL 98, 147205 (2007) G. Feverati, K. Graham, P. Pearce, G. Toth, G. Watts, cond-mat/0612203 • ΔQ = Q(ETrunc.) - Q(ETrunc.=∞), Q is the quantity • of concern • α = dimension α = 1 for excitation gaps α = related to dim. of operator (for matrix elements)

13. RG Improved TSA Gaps Employing the flow equation improves the error from 4-5% to 1% Δ2 = 9.73: exact Δ2 = 9.7±.1: RG flow

14. e(k) Focus on a single subband of a semiconducting carbon nanotube with gap Δ Δbare K K´ divide tube Hamiltonian into two pieces so that we can use the TSA kinetic term gap term e-e interactions Four bosons with the charge boson having a strongly renormalized Luttinger parameter gap term Computing Spectrum of Semiconducting Carbon Nanotubes Non-Perturbatively 2nd SB Δbare 1ST SB

15. Singlet Exciton Energies for Two Lowest Subbands of (10,9) Nanotube • Results are consistent with • two generic features of the • optical spectra of semi • -conducting carbon • nanotubes: Kane and Mele, • PRL 93, 197402 (2004) • i) Gaps are blue shifted • relative to their non • -interacting values ii) Ratio of gaps between first two subbands scale as Moreover only a weak dependence on Kc is observed. E1exc/E2exc = 24/(5-Kc). Egger and Gogolin, 1998

16. Single Particle Excitation Energies of (10,9) Nanotube Single particle excitation energies are much more strongly renormalized than exciton energies Energies are multiples of their non-interacting values Comparatively strong dependence on Kc is seen First subband

17. Density Matrix Renormalization Group: Large Coupled Arrays of Chains DMRG is a real space RG technique -It is ideally suited for single 1D chains where a chain of length N is built up two sites at a time in an iterative fashion RG Step RG Step TSA, as it offers a compact representation for each chain, allows DMRG to work in the traditional form: RG Step RG Step

18. J⊥ J⊥ . . . array of quantum Ising chains Methodology Able to Describe Ordering Transition of (60) Coupled Disordered Quantum Ising Chains . . . R. K. and Y. Adamov, arXiv:0707.1160 • ~(J⊥c-J⊥)ν ν = 0.622±.019 J⊥c = 0.184±.0025 for 3D Ising ν= 0.6300±.0002

19. Good Analytical Control of Finite Size Effects The ground state energy in finite volume is modified by the production of virtual excitations This modification is controlled by the gap of the lower lying excitations and so can be estimated

20. 1D Atomic Bose Gas in a Trap We now want to consider extension of numerical renormalization group approach to non-relativistic integrable systems: 1D Bose gas in a trap Matrix elements of density operator, , are available via the algebraic Bethe ansatz

21. Problems to be tackled: i) Look at regimes where local density approximation (LDA) is not valid ii) Study dynamics of gas under sudden changes of trap strength iii) Study optical lattices Potential difficulties: i) Excitation spectrum is vastly more complicated than typical relativistic field theory ii) Finite size scaling is not under the same control

22. LDA vs TSA Analysis of Trapped Bose Gas TSA Trap frequency, ω, is same order as Fermi energy and so LDA is expected to be invalid

23. Conclusions We have demonstrated a numerical renormalization technique able to analyze arbitrary perturbations of arbitrary continuum relativistic one dimensional systems - can be thought of as the counterpart to DMRG for such systems We believe it can be extended to 2D systems as well as systems without relativistic invariance

24. RG Improved Matrix Elements Flow of the two excitation matrix element, for a single chain in a magnetic field We see that the behavior of matrix elements also satisfies the RG flow equation Energy, w, is varied parametrically by varying R, length of chain ( )2