From classical to quantum criticality

From classical to quantum criticality Daniel Podolsky,1 Efrat Shimshoni,2 Pietro Silvi,3 Simone Montangero,3 Tommaso Calarco,3 Giovanna Morigi,4 and Shmuel Fishman1 1 Department of Physics, Technion, Haifa 32000, Israel Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel 3 Institut f¨ ur Quanteninformationsverarbeitung, Universit¨ at Ulm, D-89069 Ulm, Germany 4 Theoretische Physik, Universit¨ at des Saarlandes, D-66123 Saarbr¨ ucken, Germany (Dated: March 12, 2014)

arXiv:1403.2422v1 [cond-mat.stat-mech] 10 Mar 2014

2

We study the crossover from classical to quantum phase transitions at zero temperature within the framework of φ4 theory. The classical transition at zero temperature can be described by the Landau theory, turning into a quantum Ising transition with the addition of quantum fluctuations. We perform a calculation of the transition line in the regime where the quantum fluctuations are weak. The calculation is based on a renormalization group analysis of the crossover between classical and quantum transitions, and is well controlled even for space-time dimensionality D below 4. In particular, for D = 2 we obtain an analytic expression for the transition line which is valid for a wide range of parameters, as confirmed by numerical calculations based on the Density Matrix Renormalization Group. This behavior could be tested by measuring the phase diagram of the linear-zigzag instability in systems of trapped ions or repulsively-interacting dipoles.

I.

INTRODUCTION

One of the fascinating aspects of critical phenomena in statistical mechanics is the universality that characterizes the behavior of thermodynamic functions close to a second order phase transition1–3 . This behavior was first modeled phenomenologically by Landau, whose theory can be microscopically derived using a mean-field theoretical approach4 . The classical mean-field theory predicts the critical exponents of thermodynamic quantities for several systems undergoing a second-order phase transition. It predicts, however, wrong exponents and scaling functions for dimensions D lower than 4 due to the enhanced role of fluctuations at lower dimensionality. The renormalization group is required to find their correct values for5 D < 4. At T = 0, in particular, quantum fluctuations become relevant near quantum critical points at low dimensions. Here, the critical behavior is usually described in a field theoretical framework whose dimension D is related to the spatial dimension d by D = d + z, where z is the dynamical critical exponent6 . Existing studies of quantum phase transitions in D = 1+1 dimensions are usually either based on mapping to well-known critical models that can be performed by employing conformal field theory, and/or analyzed with full numerical simulations3,6,7 . In this paper we connect classical mean-field and quantum critical behaviors in D = d + 1 in a systematic fashion (i.e. we restrict ourselves to models with z = 1). ˜ To this end we introduce a dimensionless parameter ~, which quantifies the strength of quantum fluctuations. ˜ Quite generally, the strength of quantum fluctuations ~ can be expressed in terms of a ratio between typical kinetic and potential energy scales, respectively UK and UP , such that8,9 r ˜ ∼ UK . ~ (1.1) UP

We then analyze how the location of the critical point in ˜ from parameter space varies when we smoothly tune ~ ˜ = 0, where the classical mean-field description holds, ~ to a small finite value. This tuning may be performed by changing physical parameters, e.g. the linear density in a chain of interacting particles such as trapped ions10 . The theory is captured by the zero temperature partition function Z, which can be expressed in the form of a path integral6 : Z ˜ ˜ Z = Dφ e−S[φ]/~ , (1.2) where φ is a real field and S˜ is the dimensionless action, which is defined on the D-dimensional Euclidean spacetime (D = d + 1) as   Z 1 ε 2 S˜ = dD r (∂µ φ) − φ2 + gφ4 . (1.3) 2 2 Here µ ∈ {1 . . . D} and g > 0. Moreover, the components of the space-time vector r have been rescaled in order to make the action dimensionless and to fix the speed of sound to unity. The action is defined with an implicit cutoff at short distances corresponding to an ultraviolet cutoff Λ in momentum space. For an array of interacting particles with inter-particle distance a, for instance, Λ = π/a. The action Eq. (1.3) undergoes an Ising phase transition at a critical value εc 6,8,11 . For ε < εc the system is disordered, hφi = 0, corresponding to a paramagnetic phase, whereas for ε > εc the system orders and hφi = 6 0, corresponding to a ferromagnetic phase. The value of εc depends on the strength of the quantum fluctuations: ˜ = 0 the partition function Eq. (1.2) is solved exFor ~ actly by a saddle-point evaluation, yielding a classical ˜ > 0, on the other mean-field transition at εc = 0. For ~ hand, the transition is shifted to εc > 0, and becomes of the quantum Ising universality class. Our purpose is to

2 study the crossover between classical and quantum phase ˜ transitions for small values of ~. This paper is organized as follows. In Sec. II we present the derivation of a scaling relation for εc (~) employing a renormalization group (RG) procedure. The result is verified numerically for D = 2 in Sec. III by means of a Density Matrix Renormalization Group (DMRG) calculation. In Sec. IV we summarize our main results and conclusions. In particular, we discuss possible experimental systems, which can serve as testbeds for these predictions. II.

DEPENDENCE OF THE CRITICAL POINT ON QUANTUM FLUCTUATIONS

Our aim is to determine the boundary between the disordered and the ordered phase in parameter space for the model described by Eq. (1.3) and at T = 0, as ˜ is increased from the effect of quantum fluctuations, ~, ˜ as a function zero. In this section we calculate εc (~) ˜ of ~. We first obtain a crude result using naive scaling and then obtain a more accurate result with the help of the Renormalization Group (RG), using Wilson’s momentum shell integration2,5 . This is a standard problem of crossover12,13 , but our problem is simple enough that it can be analyzed in more detail. For any space-time dimensionality D = d + 1, the ˜ = 0 and ε = 0. model in Eq. (1.3) has a fixed point at ~ This fixed point leads to scaling that is mean-field in nature. To see this, consider a generalization of the model to include high order anharmonicities, such as φ6 , φ8 , etc, which were implicitly ignored in Eq. (1.3) and which are present in realistic systems (e.g. for chains of interacting particles14 ):  Z 1˜ 1 ε 1 2 S= (∂µ φ) − φ2 + gφ4 dD r ˜ ˜ 2 2 ~ ~  +g6 φ6 + g8 φ8 + . . . . (2.1) ˜ → 0, the partition function can be evaluated using As ~ a saddle-point approximation. Then, the phase transition occurs at ε = 0, and near the phase transition the p value of the order parameter is small, of order |ε|, such that high anharmonicities such as φ6 and higher can be neglected (provided that g > 0). Thispcan also be seen ˜ by introducing a rescaled field ϕ ≡ φ/ ~,  Z 1˜ 1 ε 2 ˜ 4 S = dD r (∂µ ϕ) − ϕ2 + g ~ϕ ˜ 2 2 ~ i ˜2 ϕ6 + g8 ~ ˜ 3 ϕ8 + . . . . +g6 ~ (2.2) In this form it is clear that all anharmonicities rescale ˜ → 0, with higher anharmonicities to zero in the limit ~ more strongly suppressed. Therefore, for small but pos˜ (and g > 0), it is sufficient to consider quartic itive ~ anharmonicities only, as done in Eq. (1.3) and through the rest of this paper. This is in contrast with scaling

near the tricritical point, where the quartic term strictly vanishes and one must keep the sixth order term3 . ˜ fluctuations tend to extend the size of For positive ~, the disordered phase, thus shifting the phase transition to positive values of ε. Provided 2 ≤ D < 4 the resulting fixed point is in the Ising universality class. However, for ˜ the shape of the phase transition line, εc (~), ˜ is small ~ expected to be controlled by scaling near the mean-field ˜ = 0 = ε. We will use this fact to first fixed point at ~ ˜ and then improve on this give a naive derivation of εc (~) estimate using the RG. A.

Naive scaling and fluctuation corrections

We proceed with a simple analysis based on spatial rescaling. For this purpose we introduce the scaling factor b > 1 and perform the following rescaling of the spacetime coordinate r as r0 = r/b ,

(2.3)

which leads to the rescaling of the correlation length ξ 0 = ξ/b .

(2.4)

Using the critical scaling of the correlation length ξ and of the order parameter φ near the mean-field fixed point ˜ = 0 and ε = 0, at ~ ξ ∼ ε−ν , φ ∼ εβ ,

(2.5) (2.6)

one obtains ε0 = ε b1/ν ,

φ0 = φ bβ/ν ,

(2.7)

where the second equality directly follows from the first. The critical exponents appearing in these expressions are the mean field exponents β = ν = 21 . Then, all terms in the action [Eq. (1.3)] are rescaled by the factor bD−4 , S˜ = bD−4 S˜0 ,

(2.8)

where S˜0 is obtained from S˜ by replacing the various variables with the corresponding primed ones. In the rescaling process, the expression for the partition function is unaffected, only the overall value of the action is changed. ˜ This can be summarized by rescaling the effective ~, ˜0 = ~b ˜ 4−D , ~

(2.9)

and shows that, by continuing the process, one moves farther away from the mean-field critical point. (As a side remark, note that D = 4 is a marginal case of the rescaling and that for D > 4 quantum fluctuations become irrelevant.) ˜∗ ) lying on the Let us consider a reference point (ε∗ , ~ phase transition line. We can then locate other points on the transition line that lie closer to the mean-field critical point and which reach the reference point after ˜ be one an iterative repetition of this rescaling. Let (ε, ~)

3 such point. Assuming that it takes l rescaling steps to ˜∗ = ~b ˜ (4−D)l and reach the reference point, this yields ~ ∗ 2l ε = εb , where in the last equation we used Eq. (2.5) with ν = 1/2. This implies that ˜ ∼ ε(4−D)/2 . ~

(a)

(b)

(2.10)

Let us now explore the effects of fluctuations. For small ˜ the leading correction to equation Eq. (2.10) values of ~, comes from the diagram in Fig. 1.(a), which leads to a self-consistent equation for the renormalization of ε: Z 1 dD q ˜ εren = ε − 12g ~ , (2.11) (2π)D q 2 − εren |q| 2, the integral above is convergent at εren =0, giving Z AD ˜ dD q 1 ˜ = g~ (2.12) εc = 12g ~ (2π)D q 2 D−2 |q| 2, εc is linear in ˜ This result seems to contradict Eq. (2.10). This can be ~. understood from the fact that the critical value εc itself is not a universal number – the correction in Eq. (2.12) constitutes an analytic shift in the critical point by a non-universal amount. However, as we will show in the following, if we introduce a shifted variable R, defined by R≡ε−

AD ˜ g~ D−2

(2.14)

˜ satisfy the naive scaling for 2 < D < 4, then R and ~ namely, ˜2/(4−D) . Rc ∼ ~

FIG. 1. (a) and (c) Lowest order Feynman diagrams contributing to the renormalization of ε and (b) to the renormalization of g.

we introduce the Fourier transform φ(q) of the real field φ(r), such that Z dD q φ(q) exp(−i(q · r)) . (2.16) φ(r) = (2π)D |q| (q), where φ< and φ> only have support outside the shell (for small momenta) and within the momentum shell (for large momenta), respectively. In detail,  φ(q) for |q| < Λ/b φ< (q) = (2.18) 0 otherwise R

and 

(2.15)

On the other hand, for D = 2 the integral in Eq. (2.11) is logarithmically divergent as εren → 0. In this case the infrared fluctuations renormalize ε in a more fundamental way, requiring a more careful treatment. B.

(c)

Renormalization group

In the following fluctuations will be taken into account using the Renormalization Group (RG), by implementing a momentum-shell integration2,5 . For this purpose,

φ> (q) =

φ(q) for Λ/b < |q| < Λ 0 otherwise

(2.19)

In terms of these fields, the partition function can be written schematically as Z ˜ ˜ (2.20) Z = Dφ< Dφ> e−S[φ< ,φ> ]/~ Z   1 (q 2 − ε) φ2< + φ2> (2.21) S˜ = 2 q Z  4  +g φ< + 4φ3< φ> + 6φ2< φ2> + 4φ< φ3> + φ4> . q1 ,q2 ,q3

4 where the cross term φ< φ> vanishes due to momentum conservation since the two fields do not have common support in the Fourier domain. The RG procedure consists of two steps. In the first step, the high-frequency field φ> is integrated out, yielding an effective action S˜eff [φ< ] involving only the small momentum field, which is defined by the equation Z ˜ ˜ ˜eff [φ< ]/~ ˜ −S = Dφ> e−S[φ< ,φ> ]/~ . (2.22) e The resulting field theory has a reduced cutoff Λ/b. In the second step, all length scales are rescaled by a factor of b following Eq. (2.3), such that the cutoff is returned to its original value Λ. ˜ the RG is controlled by Feynman diagrams For small ~ with few loops. We will work to one loop order, taking into account the diagrams in Fig. 1.(a) and (b). These diagrams lead to renormalization of ε and g, respectively. The diagram in Fig. 1.(c), contributing to the field renor˜ and will not be taken malization, is of higher order in ~ into account in this analysis. Let us first consider diagram 1.(a). It describes fluctuations that originate from the φ2< φ2> interaction. Upon integrating out φ> , this leads to a renormalization of ε, Z dD q 1 ˜ εeff = ε − 12g ~ (2.23) (2π)D q 2 − ε Λ/b