Rapidly converging methods for the location of quantum critical points from finite size data

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1 Rapidly converging methods for the location of quantum critical points from finite size data CNISM Cristian Degli Esposti Boschi CNR, Unità di ricerca CNISM di Bologna and Dipartimento di Fisica, Università di Bologna Marco Roncaglia Max Planck Institute of Quantum Optics, Garching Lorenzo Campos Venuti Fondazione ISI, Torino 2008

2 Quantum phase transitions (QPT's) in a nutshell! They occur ideally at zero temperature when some other parameter (pressure, doping, field, etc.) is varied Driven solely by quantum fluctuations Not academic: The signature of the QCP at T = 0 is experimentally relevant for the physics of a quantum critical region at T > 0 (Sachdev's scenario) S. Sachdev, Quantum phase transitions (1999) M. Vojta's website and Rep. Prog. Phys. 66, 2069 (2003) (borrowed from cond-mat/ )

3 QPT's are still an open problem in quantum physics, at least from the experimental and numerical points of view Theoretical rule of thumb: QPT's in d spatial dimensions are equivalent to classical phase transitions in (d+ ) spatial dimensions dynamic exponent g g c energy gap correlation length To be used with care: granted for thermodynamics and universal features, but not necessarily for dynamics

4 Why: Limits in numerical simulations The spatial dimensions are necessarily of finite extension and, for a lattice system with L sites, the overall dimension of the Hilbert space grows exponentially dim H=q L q=dim H site Methods (low-energy levels and correlations) Lanczos algorithm Virtually exact, max ~ 30 sites DMRG [RMP 77, 259 (2005)] Very accurate in 1D,~ 1000 sites QMC Only choice in 2D or 3D, sign problem with fermions? Hybrid: (SR )MPS [Sandvik, arxiv: ], Strings [Schuch et al, PRL 100, (2008)]

5 How: Finite size scaling (FSS) issues Useful also for real finite systems in experiments The first problem is to locate the critical point, if it is not known a priori thanks to symmetry, duality,... Phenomenological renormalisation group (PRG): using the excited levels Using the ground-state energy and its derivatives w.r.t. to the parameter g: Maxima of specific heat, subsceptibility,... Finite-size crossing method Binder ratios in QMC: using moments of observables (magnetisation,...) Other model-specific tricks (e.g. level spectroscopy)

6 H =H 0 gw General setting e g = H /V b g W /V = g e g Not an order parameter in general t g g c z t L 1/ Privman-Fisher hypotesis e sing g t 2 =2 d e g c =e g c CL d e g =e, reg g L d [ z C g ] O L d ~z 2, z 1 ~z 2, z 1 Casimir-like term (all dims are finite) C= c v/6 0 due to 1 st irrel/marg term ( in CFT) b g =b,reg g L d [ sgn g g c L 1/ ' z C ' g ]

7 Phenomenological Renormalisation Group Close to criticality ~ L or ~ L G L L L L L L L =0 from the FSS ansatz L g =L z O L 0 =2 v x ~z, z 1 ~ 0 1 z 2 z 2, z 1 ( in CFT) The zeroes of G L converge as g L g c ~L PRG PRG = 1 Curiously no attention has been paid to the points of local minima or maxima that scale as g L g c ~L m Better when and while 1 =0 PRG =1/ m =2 PRG

8 Finite size crossing method Campos Venuti, DEB, Roncaglia & Scaramucci Phys. Rev. A 73, (R) (2006) Near the critical point the expectation value of the term driving the transition, at successive values of L cross with slope ~ in a sequence of points L 2 / d g L g c ~L FSCM, FSCM =2/ The shift exponents depend on the boundary conditions and it is generally believed that 1 Slow convrgence for cases with large values of (extreme case: Berezinskii-Kosterlitz-Thouless transition with exp. small gap = ) The convergence would be more rapid if we could eliminate the part coming from the Casimir-like term

9 A homogeneity criterion z 1 b g =b,reg g L d [ sgn g g c k L 2/ t C ' g ] First an L-derivative (finite difference between L and L+ L) eliminates the, reg term At t = 0 the dominant part is a homogeneous function of L of degree (d+ +1) {L L [ L b g, L ] d 1 [ L b g, L ]} g=gc =0 When we plug the expression above into this condition we find a larger shift exponent fast =2/ The same behaviour is found if we look for the suitable * =C'(g)/C(g) such that = e b has no Casimir term and use its crossing points

10 First check: XY spin 1/2 chain with transverse field The model can be solved exactly (Jordan-Wigner + Bogolioubov transformations): =d= =1, =2 FSCM: Homogeneity condition: PRG: H = j 1 x x j j 1 1 y y z j j 1 h j h L 1 L 2 2 /6 h L 1 L h L 1 L Note: For = 2/(d+ ) one has to include (ln L) terms in the ansatze

11 Nonintegrable example Spin-1, d = 1, H = j S j S j 1 1 S z z j S j 1 D S z j 2 DMRG with 3^7 states; c = 1 transition ( =1) at = =? Campos Venuti et al. Eur. Phys. J. B 53, 11 (2006)

12 Nonintegrable example (cont'd) Spin-1, d = 1, H = j S j S j 1 1 S z z j S j 1 D S z j 2 DMRG with 3^7 states; c = 1 transition ( =1) at = 0.5 homogeneity b= S z 2 PRG 2.38 =?

13 Homogeneity criterion for BKT transitions (d= =1) exp a t With the following ansatz (n Z) e g =e, reg g L 2 [ K at ln L n / C g ] O L 2 the homogeneity condition { L [ L 3 L b g, L ]} g=gc =0 provides a sequence of points that converge to the BKT critical point with shift exponent BKT = / n 1 Note: In order to work properly the homogeneity approach requires that the finite differences in L are adjusted properly to cancel exactly the L d term. For istance with =d=1 and uniform step L b ' L b L L b L L 2 L L 3 b' ' g, L [3 L 2 L 2 ]b' g, L =0 b ' ' L b L L 2 b L b L L L 2

14 Heisenberg spin 1/2 with frustration H = j J 1 j j 1 J 2 j j 2 DMRG with 1024 states; c = 1 BKT transition ( =1) at J 2 = (J 1 = 1) Okamoto & Nomura, Phys. Lett. A 169, 433 (1992) with level spectoscopy Location of BKT with GS data only (non model specific)

15 In summary, we have found a way to improve both the FSCM and the PRG with a larger shift exponent =2/. In particular the homogeneity criterion is valid also for BKT transitions. The only thing to be known is the dynamic exponent. We hope to move to 2D systems with QMC soon For more informations about our activities cristian.degliesposti@unibo.it This work: Roncaglia et al., Phys. Rev. B 77, (2008) DMRG simulations were performed on a cluster of Linux machines at the Bologna section of the INFN