Rapidly­converging methods for the location of quantum critical 

points from finite­size dataCristian Degli Esposti BoschiCNR, Unità di ricerca CNISM di Bologna andDipartimento di Fisica, Università di Bologna

Marco RoncagliaMax Planck Institute of Quantum Optics, Garching

Lorenzo Campos VenutiFondazione ISI, Torino

CNISM

 2008

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 andRep. Prog. Phys. 66, 2069 (2003)

(borrowed from cond-mat/0010285)

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

To be used with care: granted for thermodynamics and universal features, but not necessarily for dynamics

dynamic exponent ∝−

∝∣g−gc∣−

energy gap correlation length

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

Methods (low-energy levels and correlations)

Why: Limits in numerical simulations 

dim H=qL

q=dim H site

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:0710.3362], Strings 

[Schuch et al, PRL 100, 040501 (2008)]

How: Finite­size 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)

General setting

H=H 0gW t≡∣g−gc∣eg=⟨H ⟩/V

b g≡⟨W ⟩ /V=∂g eg

esing g∝t2− =2−d egc=e∞gc−CL

−d−

eg=e∞ , reggL−d− [ z−C g ]O L−d−−

z≡t L1 /

Privman-Fisher hypotesis

Casimir-like term(all dims are finite)

0 due to 1st irrel/marg term~z

2−, z≫1 ~z

2, z≪1

b g=b∞ ,reg gL−d− [ sgng−gcL

1 / ' z−C ' g]

( in CFT)C=c v /6

Not an order parameter in general

Phenomenological Renormalisation Group 

Close to criticality ~ L or ~ L–

from the FSS ansatz

The zeroes of GL converge as

Curiously no attention has been paid to the points of local minima or maxima that scale as Better when and while

GL≡LL−LL

L L=0

L g=L−zO L−−

~z, z≫1 ~01 z2 z

2 , z≪1

0=2v x( in CFT)

∣gL−gc∣~L−PRG

PRG=1/∣gL−gc∣~L

−m

m=2PRG

PRG=−1

1=0

Finite­size crossing method

Campos Venuti, DEB, Roncaglia & ScaramucciPhys. Rev. A 73, 010103(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

The shift exponents depend on the boundary conditions and it is generally believed that

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

L2 /−d

gL−gc~L−FSCM , FSCM=2 /

∝−1

A homogeneity criterion

b g=b∞ ,reg gL−d− [ sgng−gck L

2 /t−C ' g ]

z≪1

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)

When we plug the expression above into this condition we find a larger shift exponent

The same behaviour is found if we look for the suitable *=C'(g)/C(g) such that has no Casimir term and use its crossing points

{L ∂L [∂L b g , L ]d1 [∂L b g , L ]}g=gc=0

fast=2 /

=∗e−b

First check: XY spin­1/2 chain with transverse field

The model can be solved exactly (Jordan-Wigner + Bogolioubov transformations): =d==1, =2

FSCM:

Homogeneity condition:

PRG:

Note: For = 2/(d+) one has to include (ln L) terms in the ansatze

H=−∑j1 j

x j1

x1− j

y j1

y−h j

z

hL≃1L−2

2/6

hL≃1L−4 742

2−3

7202

hL≃1L−3

3

48∣∣4

2−3

Nonintegrable example

H=∑jS j⋅S j1−1S j

zS j1

zD S j

z2Spin-1, d = 1,

DMRG with 3^7 states; c = 1 transition (=1) at = 0.5

≃2.38

Campos Venuti et al.Eur. Phys. J. B 53, 11 (2006)

=?

Nonintegrable example (cont'd)

H=∑jS j⋅S j1−1S j

zS j1

zD S j

z2Spin-1, d = 1,

DMRG with 3^7 states; c = 1 transition (=1) at = 0.5

b=⟨S z2 ⟩

PRG

homogeneity

≃2.38

=?

Homogeneity criterion for BKT transitions (d==1)

∝exp a t−

eg=e∞ , reggL−2 [K at−−ln L −n /−C g ]O L−2−

With the following ansatz (n ∈ ℤ)

the homogeneity condition

provides a sequence of points that converge to the BKT critical point with shift exponent

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

{∂L [L3∂L b g , L ]}g=gc

=0

BKT=/n−1

L3b ' ' g , L[3L2−L2]b ' g , L=0

b ' L ≡b L L −b L− L

2 Lb ' ' L ≡

b L L −2 b L b L−L

L 2

Heisenberg spin­1/2 with frustration

H=∑jJ1 j⋅ j1J 2 j⋅ j2

DMRG with 1024 states; c = 1 BKT transition (=1) at J

2 = 0.2411 (J

1 = 1) Okamoto & Nomura, Phys. Lett. A 169, 433 (1992)

with level spectoscopy

Location ofBKT with GSdata only(non modelspecific) 

✔ In summary, we have found a way to improve both the FSCM and the PRG with a larger shift exponent . 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

http://www.df.unibo.it/fismat/theory/

This work: Roncaglia et al., Phys. Rev. B 77, 155413 (2008)

DMRG simulations were performed on a cluster of Linux machines at the

Bologna section of the INFN

=2 /