Entropy accumulation near quantum critical points: effects beyond hyperscaling
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Entropy accumulation near quantum critical points:effects beyond hyperscaling
Jianda Wu , Lijun Zhu and Qimiao Si Department of Physics & Astronomy, Rice University, Houston, Texas 77005, USA Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, LosAlamos, New Mexico 87545, USAE-mail: [email protected]
Abstract.
Entropy accumulation near a quantum critical point was expected based on generalscaling arguments, and has recently been explicitly observed. We explore this issue further intwo canonical models for quantum criticality, with particular attention paid to the potentialeffects beyond hyperscaling. In the case of a one-dimensional transverse field Ising model, wederive the specific scaling form of the free energy. It follows from this scaling form that thesingular temperature dependence at the critical field has a vanishing prefactor but the singularfield dependence at zero temperature is realized. For the spin-density-wave model above itsupper critical dimension, we show that the dangerously irrelevant quartic coupling comes intothe free energy in a delicate way but in the end yields only subleading contributions beyondhyperscaling. We conclude that entropy accumulation near quantum critical point is a robustproperty of both models.
1. Introduction
Quantum critical points (QCPs) have been extensively studied in heavy fermion metals andrelated systems [1]. They occur as a non-thermal control parameter is tuned to a second-orderphase transition at zero temperature. For thermodynamics, it was shown [2] based on scalingconsiderations that the thermal expansion [ α = (1 /V )( ∂V /∂T ) p,N ∝ ∂S/∂p , the variation ofentropy S with pressure p ] is more singular than the specific heat [ c p = ( T /N )( ∂S/∂T ) p ], inthe general case when the tuning parameter is linearly coupled to pressure. Correspondingly,the Gr¨uneisen ratio, Γ = α/c p , diverges at the QCP, and the entropy is maximized. Thesame conclusions apply to a field-tuned QCP, where the magnetic Gr¨uneisen ratio is themagnetocaloric effect. The predicted divergence of the Gr¨uneisen ratio is by now widely observedin quantum critical heavy-fermion metals [3, 4], and the entropy enhancement has recently beenexplicitly observed in a quantum critical ruthenate [5].The scaling arguments proceed as follows. Near a QCP, the critical part of the free energytakes the hyperscaling form F = F T d/z +1 f ( r/T / ( νz ) ), where r = p − p c , d is the spatialdimension and ν , z are respectively the correlation-length and dynamic exponents. The universalfunction f ( x ) has different asymptotic behaviors in the x → x → ±∞ limits, respectivelycorresponding to the quantum critical and quantum disordered/renormalized classical regimes.The divergence of Γ can be readily derived, in universal forms as 1 /r for | r | ≫ T and as 1 /T /νz for | r | ≪ T [2]. The 1 /r divergence in the low-temperature limit also amounts to a sign changeof Γ across the QCP. As seen in Fig. 1, this corresponds to an entropy maximization at the QCPn the low-temperature limit. The extension of this sign change to the finite temperature phasetransitions has also been discussed [6].The hyperscaling form for the free energy is based on the existence of a single critical energyscale, i.e. , the gap for the quantum critical excitations ∆ ∼ r νz . In this paper we explore howthe scaling form of the free energy arises in some specific models for QCPs, paying particularattention to the possible effects that go beyond hyperscaling. Figure 1.
Divergence of the Gr¨uneisen ratio and the accumulation of entropy near QCPs.
2. One-dimensional transverse field Ising model
Consider first the one-dimensional transverse field Ising model (1D TFIM), defined by theHamiltonian [7]: H I = − J X i (cid:0) g ˆ σ xi + ˆ σ zi ˆ σ zi +1 (cid:1) , (1)where J > gJ is thetransverse magnetic field. With the tuning of g , there exists a quantum phase transition from aferromagnet to quantum paramagnet at g c = 1 [7, 8]. Using a Jordan-Wigner transformation, wecan represent the Hamiltonian in the thermodynamic limit in terms of free fermions and the freeenergy can be written as F = − N k B T (cid:2) ln 2 + π R π d k ln cosh( ε k / k B T ) (cid:3) , with the dispersion ε k = 2 J (cid:0) g − g cos k (cid:1) / . The normalized free energy density is f ≡ FN J k B = f ( g ) − tπ Z π dk ln (cid:0) e − A (cid:1) ; A = q (1 − g ) /t + 4( g/t )sin ( k/
2) (2)where t = k B T /J , and f ( g ) is the ground state energy. Our focus will be on the T -dependentpart, f ( g, T ) = f ( g, T ) − f ( g ).Consider first the quantum critical regime, where | g − | /t ≪
1. At temperatures smallcompared to J , we can introduce a momentum cutoff k c ≈ t , below which | sin ( k/ | /t ≪ A ≪
1. The free energy can be approximated by the contributions from k < k c , whichs f ≈ − tπ " (cid:18) − gt (cid:19) k c + gt k c k c ln 2 k c ≈ t ≈ − t π (cid:16) g (cid:17) + 12 (cid:18) − gt (cid:19) ! . (3)While Eq. (3) contains approximations for the regular pieces, it reveals an important point.Compared with the general hyperscaling form, this expression is particular in that the linearterm | − g | /t in the expansion vanishes. As a result, the magnetic Gr¨uneisen ratio as a functionof temperature at the QCP stays finite.Consider next the low-temperature regions where | − g | /t ≫
1. Here A ≫ e − A ) as e − A and obtain f ≈ − t √ πg (cid:18) | − g | t (cid:19) / e − t | − g | . (4)We can further replace 1 / √ g by 1 / √ g c without missing any singularity. The free energy has theexact hyperscaling form with z = 1. We obtain the magnetic Gr¨uneisen ratio to beΓ = 1 g − , (5)which is indeed divergent at the QCP, g c = 1. The accumulation of entropy immediately follows(Fig. 1).
3. Spin-density-wave quantum critical point
Quantum phase transitions in itinerant magnets are traditionally described in terms of a T = 0spin-density-wave (SDW) transition and formulated as a quantum Ginzburg-Landau theorywith dynamic exponent z > i.e. , with d = 3 and z = 2. The effective dimension of the Ginzburg-Landau theoryis d + z , which is above the upper critical dimension 4. Correspondingly, the quartic coupling u ,appearing in the action as u R φ , is irrelevant in the renormalization-group (RG) sense. Underthe RG transformation, u renormalizes to zero as the fixed point is reached, leaving only theGaussian (quadratic) part. However, u is dangerously irrelevant [10, 11]. For instance, in thequantum critical regime, the correlation length is determine as ξ − ∼ r ( T ) = r + cuT / (where c is a constant, 4( n + 2)(1 / − ln 2 √ / (3 π )), with n being the number of components of the orderparameter). One can consider u as introducing a new energy scale, and the scaling functions forgeneric physical quantities are expressed in terms of two variables f ( r/T, uT / /r ).Within the RG approach [10], it is natural to focus on the free energy associated with theGaussian term, F G . The expression for F G in the RG approach can be shown to be equal toa Gaussian free energy with r replaced by renormalized r ( T ). We will focus on the quantumcritical regime, and write F G = F (1) G + F (2) G , where F (1) G = − nV Z Λ0 d q (2 π ) Z Γ q dε π (cid:16) coth ε T − (cid:17) tan − ε/ Γ q r ( T ) + ( q/ Λ) , (6)and F (2) G = − nV Z Λ0 d q (2 π ) Z Γ q dε π tan − ε/ Γ q r ( T ) + ( q/ Λ) . (7)Here, Γ q = Γ q z − , while Λ and Γ are respectively the ultraviolet momentum andenergy cutoffs. The leading temperature-dependent contribution from F (2) G turns out to be nV Λ Γ π (cid:16) √ π + ln 2 − √ ln( √ (cid:17) c ( u/ Γ / ) T / , which is more singular than that from F (1) G .his Gaussian form, however, is incomplete. The quartic coupling also introduces anexplicitly linear in u term ( t ≡ T / Γ ), which takes the form F u = − n ( n + 2) N Γ u Z ∞ e − x (cid:18)Z d q (2 π ) Z dεπ coth ε te x εε + ( re x + q ) (cid:19) f u dx ,f u ≡ π Z dεπ coth ε te x εε + ( re x + 1) + 2 π Z d q (2 π ) coth 12 te x
11 + ( re x + q ) . (8)We find that, to the linear order in u , the leading temperature-dependent contribution from F u exactly cancels that of F (2) G , leaving the singular contribution to the total free energy to be F tot ≈ F (1) G , (9)which is given by F tot ∼ − T Γ + 2 πT / Γ / r + cuT / / Γ / T / Γ ! . (10)We can trace the singular terms included in Eq. (6) to the contributions of the low-energydegrees of freedom, and those included in Eq. (7) to a unphysical origin from the degress offreedom at high energies and short distances near the cutoff scales. Eqs. (6,9) are thereforeexpected to be the regularization prescription that is valid to all orders in u . The explicit resultgiven in Eq. (10) is consistent with those often quoted in the literature. The T term representsthe background Fermi liquid contribution. The critical part of the free energy is consistent withthe scaling form T / (1 + r/T ). The contribution of the renormalized quartic coupling uT / tothe free energy is subleading; the dangerously irrelevant parameter does not modify the leadingsingular part of the free energy from its hyperscaling form, thereby preserving the divergence ofthe Gr¨uneisen ratio and the accompanied entropy accumulation.
4. Summary
We have considered the thermodynamic properties of two models of quantum criticality. Wehave analyzed the potential effects beyond the most general hyperscaling forms in both models,and established that entropy accumulation is a robust property of both models.
Acknowledgments
We acknowledge the support of NSF Grant No. DMR-1006985, the Robert A. Welch FoundationGrant No. C-1411 (JW and QS), and the U.S. DOE through the LDRD program at LANL (LZ).
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