Journal of the Korean Physical Society, Vol. 52, February2008, pp. S158S161
Correlation Functions at Quantum Criticality under Broken Time-Reversal Symmetry
Min-Chul
Cha
Department of Applied Physics, Hanyang University, Ansan, 426-791
Gerardo
Ortiz
Department of Physics, Indiana University, Bloomington, IN 47405, U.S.A.
We investigate correlation functions in critical phenomena at zero temperature. We nd that the equal-time correlation function of a quantum system at criticality does not have a diverging correlation length in the absence of time-reversal symmetry. We illustrate our main results analytically in the quantum spherical model as well as more explicitly by Monte Carlo calculations in the two-dimensional quantum rotor model.
PACS numbers: 05.70.Jk, 73.43.Nq, 05.70.Fh, 05.30.Jp Keywords: Quantum phase transition, Critical phenomena, Correlation function I. INTRODUCTION
The presence of long-range correlations in a physical system lead to singular properties in the system's observables, such as the divergence of an appropriate susceptibility [1]. In the study of critical phenomena universal properties, independent of any microscopic details, appear because of a diverging correlation length governing long-wavelength uctuations. Quantum critical phenomena deal with static and dynamic aspects of correlations on an equal footing, so that the correlation time diverges along the temporal direction at the same time as the correlation length diverges so along a spatial direction [2], thus de ning the dynamical critical exponent z as z . The divergence of the correlation length near criticality is a direct consequence of the peculiar behavior of the system's correlation functions. The latter are the basic quantities used to describe the response of the system to an external eld. Long-range correlation often leads to a power-law behavior of the correlation functions, typically G(r) r (d 2+) in classical critical phenomena, where r is distance, d is spatial dimensionality of the system, and is the anomalous critical exponent. Quantum critical phenomena [3{6], on the other hand, are frequently approached by mapping the quantum system to an equivalent classical model which includes uctuations in the temporal direction. Formally, in the language of Feynman's path-integrals, quantum many-body (or quantum eld theory) problems in d dimensions are transformed into (d + 1)-dimensional clas
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sical statistical models, where the extra dimension is the temporal dimension, whose e ective dimensionality is z . This suggests that we might borrow the framework of classical critical phenomena to describe quantum criticality just by replacing d by d + z , resulting in the form of correlation functions G(r; = 0) r (d+z 2+) and G(r = 0; ) (d+z 2+)=z in quantum critical phenomena, where is distance (or time) in the temporal dimension [3]. This simple dimensional extension allows us to construct a scaling ansatz of the free-energy density [7], whose validity is well con rmed in many di erent situations. Furthermore, for the special case that z is equal to unity, the equivalence between the spatial and the temporal directions readily leads to the conclusion that quantum criticality is nothing but (d + z )-dimensional classical criticality. However, if there is a term breaking the time-reversal symmetry in the quantum problem, the validity of the simple dimensional extension above should be reconsidered, since this has no counterpart in classical models. In particular, it is quite plausible that the asymmetry in time will modify the behavior of the correlation functions. The purpose of this paper is to study a general form of correlation function at quantum criticality in a model with broken time-reversal symmetry. We nd that in the quantum rotor model with broken particle-hole symmetry, which can be described in the Ginzburg-LandauWilson scheme with a term breaking the time-reversal symmetry, the equal-time two-point correlation function does not show power-law behavior at the critical point. Instead, it is characterized by a nite length which is inversely proportional to the coupling parameter of the symmetry-breaking term. We derive these results ana-
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Correlation Functions at Quantum Criticality under { Min-Chul Cha and Gerardo
lytically in the spherical approximation and numerically by Monte Carlo simulations in the two-dimensional quantum rotor model. We also argue that these results do not contradict the divergence of the susceptibility at criticality.
with
In order to investigate correlation functions explicitly in a model with broken time-reversal symmetry, let us consider the quantum rotor model with an o set. The Hamiltonian of the model is given by H=
UX
(nj
n )2
2J
X
cos(i j ); (1) 2 j where j denotes the phase angle of a planar rotor at the j -th site, J is the coupling energy between neighboring rotors, and U represents the rotational energy scale of rotors. Note that quantized rotations are represented by an operator nj = 1i @@ j , whose eigenvalues are 0; 1; 2; , and n is a number denoting a rotational energy o set. The quantum rotor model is often adopted to describe interacting boson systems [8,9], where nj represents particle numbers, U and J denote interaction and hopping energies, respectively, and U n = is a chemical potential. In this language, the model has particle-hole symmetry (nj $ nj ) when n = 0 (or = 0). The latter situation represents time-reversal symmetry in the quantum rotor model language since clockwise and counterclockwise rotations of rotors are equivalent. We can identify the symmetry properties more easily in the corresponding classical action S [ ], which determines the partition function Z = Trfg e H Z Y = D[ ( )]D[ ( )] (j j ( )j2 1)e S [ ] (2) j;
with S[ ] =
+
X
i;j
Z
0
d [K
X
i
Jij i ( ) j ( )];
i ( )(@
)2 i ( )
(3)
where j ( ) = e ij ( ) , K = 1=(2U ), is the inverse temperature, and Jij denotes a nearest-neighbor hopping matrix element. This action clearly shows that a nite ( 6= 0) breaks the time-reversal symmetry. The correlation function of the quantum rotor model can be calculated in the spherical approximation [10]. By introducing a Lagrange multiplier that takes into account the constraint in Eq. (2), the spherical approximation constructs a partition function Z Z0 = D[ ( )]D[ ( )]e S0 [ ] (4)
0
X
i;j II. MODEL
Z
S0 [ ] =
Ortiz
d [K
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X
i
i ( )(@
)2 i ( )
Jij ) i ( ) j ( )]:
(ij
(5)
The determination of in the quantum spherical model by the self-consistent condition 1 = h j ( ) j ( )i0 , where h i0 represents the average over S0 , completes the approximation scheme. In this work, we focus on the correlation function in the case with an o set (n = =U 6= 0). The correlation function between R~ i and R~ j , distant by in time, can be evaluated from Eq. (5) to yield ~i G(jR
~ j j; ) R
1
= d L
p p ( =K +)
~ i R~ j ) ( )( q~=K +) X ei~q(R e q~
p
p
2 Kq~
1 e
q ~ e ( =K ) p + ; 1 e ( q~=K )
q ~
(6) (7)
q~ where P q = (2=L)n (n = 1; 2; ; L) and = 2J cos q in a system of linear size L. From this expression, we have the equal-time correlation function q q~ 1 X ei~q~r p coth G(r; 0) = d ( =K + ) L q~ 4 Kq~ 2
q
+ coth ( q~=K 2
)]:
(8)
III. RESULTS
Figure 1 shows the equal-time correlation functions in Eq. (8) along a spatial direction, say x, at the critical points in two and three dimensions for n = 0:1 and n = 0:3. The critical points are determined by the nitesize scaling of the super uid sti ness. Clearly, they do not show power-law behavior, contrary to the common lore. Instead, the correlation functions decay rapidly with a nite length scale. By tting the curves with the functional form G(x; 0) = A
e x=`
e (L x)=`
+C (9) + xd=2 (L x)d=2 for 1 x L (L = 200), we have A = 0:26, ` = 6:11 and C = 1:17 10 4 for n = 0:1 and A = 0:29, ` = 1:87, and C = 7:22 10 5 for n = 0:3, respectively, in two dimensions (d = 2). Similarly, in three dimensions (d = 3), we have A = 0:10, ` = 3:94 and C = 4:66 10 6 for n = 0:1 and A = 0:19, ` = 1:37 and C = 3:82 10 6 for n = 0:3, respectively.
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Journal of the Korean Physical Society, Vol. 52, February2008
=
e (1 d) (d 1)=2 K(d 1)=2 () (2)(d+1)=2
(12)
for d 1, where 2 2 + 2 , = (r2 + 2 )1=2 =, and Kn () is the modi ed spherical Bessel function of the third kind, whose asymptotic form is Kn ()
(
1=2 e 1; 2 (n) n 1; 2 2
(13)
where (n) is the Gamma function. Therefore, for 6= 0, the long-range behavior of the correlation function at the critical point ( 2 ! 0) is given by p e ( r2 + 2 ) G(r; ) ; (14) (r2 + 2 )d=4 while, for = 0, G(r; ) 1=(r2 + 2 )(d 1)=2 : Fig. 1. Equal-time correlation functions along the x-axis at the critical point for n = 0:1 and n = 0:3 in two and three dimensions. The curves, contrary to the common lore, do not show the power-law behavior, decaying with a nite length scale.
The decaying length scale ` is a nite number, insensitive to the size of the system. This means that the above feature of the equal-time correlation functions that we observe is not a nite-size e ect [11]. Di erent aspect ratios =Lz yield the same feature. Only very small aspect ratios blur the rapid decaying feature, probably due to nite-temperature e ects, but still no power-law behavior occurs. We can obtain the long-distance properties of the equal-time correlation function more clearly in the asymptotic form of its continuum limit. Under periodic boundary conditions along the time axis, one can Fourier-transform the elds as r 1 X i(q~R~ j ! ) ~ e (~q; !); (10) j ( ) = d L q~;!
with Matsubara frequencies ! = 2n= (n = 0; 1; 2; ). Then, in the long-wavelength limit (q ! 0), Eq. (5) becomes S0 =
X
q~;!
[(!
i)2 + q 2 + 2 + 2 ]j ~(~q; ! )j2 ;
(11)
where we have rescaled ~q, !, and other quantities to set parameters to be unity. is the correlation length, diverging at criticality. With this correlation length , the correlation function is given by Z dd qd! ei(q~~r ! ) G(r; ) = d +1 (2) (! i)2 + q2 + 2 + 2
(15)
Note that the powers in the denominator in Eqs. (14) and (15) are di erent, since we take leading powers in di erent limits of Eq. (13). Thus, for 6= 0, the equal-time spatial correlation is characterized by a length 1=, not by the diverging , while the temporal correlation does display long-range behavior, G( ) 1= d=2 . The long-range correlation associated with the divergence of , however, leads to divergence of the susceptibility Z
dd rd G(r; ) ! 1;
(16)
as ! 1, since (! i)2 + q2 + 2 + 2 ! 0 in the longwavelength (q ! 0) and low-frequency (! ! 0) limit. Therefore, the range of the zero-frequency correlation function is characterized by a diverging in the form G(r; ! = 0)
exp( r= )=r(d 1=rd 2
1)=2 r=
1; (17)
r= 1;
which, near criticality, is usually represented as G(r; ! = 0) exp( r= )=rd 2 . This form is the same as that of d-dimensional classical criticality. One may note that the broken time-reversal symmetry is directly related to the long-range behavior of the equal-time correlation function. In the model of Eq. (11), a non-vanishing brings in an imaginary term in the action linearly coupled to !. In the absence of this term, quantum uctuations along the temporal and spatial directions are isotropic, which means that the dynamical critical exponent z is unity and the simple mapping of the d-dimensional quantum phase transition to a (d + z )dimensional classical phase transition is well justi ed. When 6= 0, the term breaking the time-reversal (or particle-hole) symmetry introduces an asymmetry in the temporal direction leading to z = 2. Note that this case has no analogous counterpart in any known classical critical phenomena [12].
Correlation Functions at Quantum Criticality under { Min-Chul Cha and Gerardo
Ortiz
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strongly supports the fact that the equal-time correlation function has a short-range behavior, characterized by a nite rather than a diverging correlation length. It has been observed that the length scale ` at the critical point is inversely proportional to , and ` remains nite except at integer llings ( = 0) where the time-reversal symmetry is restored [15].
IV. SUMMARY
Fig. 2. Equal-time correlation functions of the quantum c = 0:2978 for n = 0:2 in rotor model at the critical point K lattices of size 16 16 64; 20 20 100; 24 24 144; 28 28 196; 32 32 256 and 40 40 400, via quantum Monte Carlo calculations.
It will be useful to con rm the above results by a Monte Carlo calculation of the quantum rotor model in Eq. (1). We performed Monte Carlo calculations of the total correlation function G(r; ) in a square lattice of size LLL , with periodic boundary conditions, where L is the size in a spatial direction and L in the temporal direction. For the purpose of quantum Monte Carlo calculations, we represent the quantum rotor model in the form of a classical (d + 1)-dimensional action [13] J=0 1 rX Jx2(r; ) + Jy2(r; ) + (J (r; ) n )2 ; S [J] = 2K (r; )
p
(18)
where K 2J=U is the tuning parameter controlling the quantum uctuations, J 's are integers, and r J = 0 represents the current conservation condition at each site. A recently developed worm algorithm [14] is used to update the current con gurations in Eq. (18) and measure the correlation functions. Figure 2 shows the equal-time correlation function of the two-dimensional quantum rotor model as a function of x in lattices of di erent sizes for n = 0:20 at the critical point K c = 0:2978, which is determined by nite-size scaling of the super uid sti ness. Again we use the same tting function of Eq. (9) and obtain ` 1:4 1:9 for di erent sizes, while C , a quantity related to the condensate fraction, depends sensitively on K . This result
In summary, we have observed that the equal-time correlation function at a quantum critical point is shortranged when the time-reversal (or particle-hole) symmetry is broken. We have shown this result not only analytically in the quantum spherical model but also via Monte Carlo simulations of the planar quantum rotor model. We notice that this feature of the correlation function does not violate the divergence of the susceptibility due to a diverging correlation length at criticality, since the zero-frequency correlation function does involve long-range critical uctuations.
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