Effect of magnetic field on spontaneous Fermi surface symmetry breaking
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Effect of magnetic field on spontaneous Fermi surface symmetrybreaking
Hiroyuki Yamase
Max-Planck-Institute for Solid State Research,Heisenbergstrasse 1, D-70569 Stuttgart, Germany (Dated: October 26, 2018)
Abstract
We study magnetic field effects on spontaneous Fermi surface symmetry breaking with d -wavesymmetry, the so-called d -wave “Pomeranchuk instability”. We use a mean-field model of electronswith a pure forward scattering interaction on a square lattice. When either the majority or theminority spin band is tuned close to the van Hove filling by a magnetic field, the Fermi surfacesymmetry breaking occurs in both bands, but with a different magnitude of the order parameter.The transition is typically of second order at high temperature and changes to first order at lowtemperature; the end points of the second order line are tricritical points. This qualitative picturedoes not change even in the limit of a large magnetic field, although the magnetic field substantiallysuppresses the transition temperature at the van Hove filling. The field produces neither a quantumcritical point nor a quantum critical end point in our model. In the weak coupling limit, typicalquantities characterizing the phase diagram have a field-independent single energy scale while itsdimensionless coefficient varies with the field. The field-induced Fermi surface symmetry breakingis a promising scenario for the bilayer ruthenate Sr Ru O , and future issues are discussed toestablish such a scenario. PACS numbers: 71.18.+y, 75.30.Kz, 74.70.Pq, 71.10.Fd . INTRODUCTION Usually the Fermi surface (FS) respects the point-group symmetry of the underlying lat-tice structure. However, recently a symmetry-breaking Fermi surface deformation with a d -wave order parameter, where the FS expands along the k x direction and shrinks along the k y direction, or vice versa, was discussed in various two-dimensional interacting electron modelon a square lattice, the t - J ,[1, 2, 3] Hubbard,[4, 5, 6] and extended Hubbard[7] model. This d -wave type Fermi surface deformation ( d FSD) is often called d -wave Pomeranchuk instabil-ity, referring to Pomeranchuk’s stability criterion for isotropic Fermi liquids.[8] However, theFermi surface symmetry breaking can happen even without breaking such a criterion, sincethe instability is usually of first order at low temperature.[9, 10] Moreover, the new conceptof the Fermi surface symmetry breaking is applicable also to strongly correlated electronsystems such as those described by the t - J model.[1, 2, 3] The d FSD instability is drivenby forward scattering processes of electrons close to van Hove points in the two-dimensionalBrillouin zone. The instability is thus purely electronic and the lattice does not play a role.As a result, symmetry of an electronic state is reduced from C v to C v , while the latticeretains C v symmetry as long as no electron-phonon coupling is considered.The d FSD competes with superconductivity. Several analyses of the Hubbard[11, 12]and t - J [1, 3] model showed that the d -wave superconductivity becomes a leading instabilityand the spontaneous d FSD does not happen. However, appreciable correlations of the d FSD remain.[13] As a result, the system becomes very sensitive to a small external xy -anisotropy and shows a giant response to it, leading to a strongly deformed FS. This ideawas invoked for high-temperature cuprate superconductors.[1] In particular, the recentlyobserved anisotropy of magnetic excitations in YBa Cu O x [14] has been well understoodin terms of d FSD correlations in the t - J model.[15] Although the reduced symmetry dueto the d FSD is the same as the electronic nematic phase proposed in the context of theso-called spin-charge stripes,[16] the underlying physics is very different. The Fermi surfacesymmetry breaking does not require the assumption of charge stripes, but is driven byforward scattering processes of electrons. The d FSD provides an essentially different routeto the nematic phase.Although a spontaneous d FSD has not been proposed for cuprates because of the com-petition with the d -wave singlet pairing,[1, 3] the material Sr Ru O turned out to have an2nteresting possibility of a spontaneous d FSD.[17] Sr Ru O is the bilayered ruthenate withtwo metallic RuO planes where Ru-ions form a square lattice. Unlike the correspondingsingle-layered material Sr RuO , a well-known triplet superconductor,[18] Sr Ru O has aparamagnetic ground state.[19] However, the material is close to a ferromagnetic transi-tion, which was suggested by the strongly enhanced uniform magnetic susceptibility with alarge Wilson ratio,[20] uniaxial-pressure-induced ferromagnetic transition,[21] inelastic neu-tron scattering,[22] and band structure calculations.[23, 24] By applying a magnetic field h ,Sr Ru O shows a metamagnetic transition at h = h c , around which non-Fermi liquid behav-ior was observed in various quantities: resistivity,[25, 26] specific heat,[26, 27, 28] thermalexpansion,[29] and nuclear spin-lattice relaxation rate.[30] This non-Fermi liquid behaviorwas frequently discussed in terms of a putative metamagnetic quantum critical end point(QCEP), and in fact Sr Ru O was often referred to as a system with a metamagneticQCEP.[31] However, after improving sample quality, the hypothetical QCEP turned out tobe hidden by a dome-shaped transition line of some ordered phase around h c .[17] Whilea second order transition was speculated to occur around the center of the dome, a firstorder transition was confirmed at the edges of the transition line and was accompanied by ametamagnetic transition. Grigera et al. [17] associated this instability with the spontaneous d FSD, which turned out to be consistent with a large magnetoresistive anisotropy recentlyobserved inside the dome-shaped transition line.[32]Any first order transition as a function of a magnetic field is generically accompanied by ametamagnetic transition, which follows from the stability of the thermodynamic potential.This was demonstrated in the case of the first order d FSD transition in connection withSr Ru O .[33] Quite recently we showed that the most salient features observed in Sr Ru O ,not only the metamagnetic transition but also the phase diagram and the non-Fermi liquidlike behavior of the uniform magnetic susceptibility and the specific heat coefficient, are wellcaptured in terms of the d FSD instability near the van Hove singularity without invokinga putative QCEP.[34] We also predicted anomalies associated with the d FSD instability inthe temperature dependence of the magnetic susceptibility and the specific heat.[34]The main purpose of this paper is to expand our previous paper about the d FSD in-stability in the presence of a magnetic field.[34] It is particularly interesting to perform acomprehensive analysis of magnetic field effects on the d FSD instability, since a magneticfield is often employed as a tuning parameter of a quantum phase transition. A naive ques-3ion may be whether a QCEP is realized for the d FSD instability as for the ferromagneticinstability.[35, 36] The phase diagram of the d FSD is known to be characterized by severaluniversal numbers,[10] which can be compared directly with experimental data. It is theninteresting also how the universal numbers evolve in the presence of a magnetic field.We analyze the d FSD instability in the charge channel in a one-band model on a squarelattice. The model describes electrons interacting via a pure forward scattering interactiondriving the d FSD in the presence of a magnetic field (Sec. II). We solve this model numeri-cally in Sec. III and investigate the weak coupling limit analytically in Sec. IV. In Sec. V, wediscuss the reported phase diagram for Sr Ru O [17] as well as relations to other scenariossuch as a QCEP,[31, 35, 37] phase separation,[38] and magnetic domain formation.[39] Sec-tion VI is the conclusion. In Appendix A, detailed features of the d FSD phase diagram arepresented. In Appendix B, we analyze the d FSD instability in the spin channel, often called spin-dependent Pomeranchuk instability , which shows exactly the same phase diagram as inthe charge channel.
II. MODEL AND FORMALISM
We investigate the d FSD instability in the charge channel under a magnetic field on asquare lattice. The minimal model reads H = X k ,σ ( ǫ k − µ ) n k σ + 12 N X k ,σ, k ′ ,σ ′ f kk ′ n k σ n k ′ σ ′ − h X k ,σ σn k σ (1)where n k σ = c † k σ c k σ counts the electron number with momentum k and spin σ ; c † k σ ( c k σ ) isan electron creation (annihilation) operator; µ is the chemical potential; N is the numberof lattice sites; h is an effective magnetic field and is defined as h = g µ B H where g is a g -factor, µ B is Bohr magneton, and H is a magnetic field. For hopping amplitudes t and t ′ between nearest and next-nearest neighbors on the square lattice, respectively, the baredispersion relation is given by ǫ k = − t (cos k x + cos k y ) − t ′ cos k x cos k y . (2)The forward scattering interaction driving the spontaneous d FSD has the form f kk ′ = − g d k d k ′ , (3)4ith a coupling constant g ≥ d -wave form factor d k = cos k x − cos k y . This ansatzmimics the structure of the effective interaction in the forward scattering channel as obtainedfor the t - J ,[1] Hubbard,[4] and extended Hubbard[7] model. For h = 0, this model and asimilar model were studied in Refs. 10 and 9, respectively.We decouple the interaction by introducing a spin-dependent mean field η σ = − gN X k d k h n k σ i , (4)which becomes finite when the system breaks orientational symmetry and is thus the orderparameter of the d FSD. The mean-field Hamiltonian reads H MF = X k ,σ ξ k σ n k σ + N g η , (5)where ξ k σ = ǫ k + ηd k − µ σ . (6)Here the σ -summed mean filed η = P σ η σ enters ξ k σ , and thus a finite η σ in general inducesa finite η − σ ; the magnetic field is absorbed completely in the effective chemical potential µ σ = µ + σh . The grand canonical potential per lattice site is given by ω = − TN X k σ log(1 + e − ξ k σ /T ) + η g . (7)By minimizing Eq. (7) with respect to η , we obtain a self-consistency equation η = − gN X k ,σ d k f ( ξ k σ ) . (8)We consider the solution with η ≥
0, since the free energy Eq. (7) is an even function withrespect to η . The self-consistency equation is written also as η σ = − gN X k d k f ( ξ k σ ) . (9)Note that our Hamiltonian (1) does not allow momentum transfer, and thus the mean-fieldtheory solves our model exactly in the thermodynamic limit. III. NUMERICAL RESULTS
LDA band calculations[23, 24] for Sr Ru O without a magnetic field showed that theelectronic structure is similar to that for the single-layered material Sr RuO except that5here are six FSs because of the bilayered structure. Since the d FSD instability is drivenby electrons near the van Hove points on a square lattice, we focus on the FS closest to k = ( π,
0) and (0 , π ), and mimic such a FS by choosing t ′ /t = 0 .
35. For h = 0 the baredispersion has the van Hove energy at 4 t ′ = 1 . t , from which we measure the chemicalpotential µ . We take g/t = 1 for numerical convenience, but the result for g/t = 0 . t = 1 so that all quantities with dimensionof energy are in units of t , and consider a region of h ≥ h → − h and σ → − σ .Figure 1(a) shows a phase diagram for µ = − . h and temperature T ; the dotted line at h = h vH = 0 . d FSD transition is of second order for high T andchanges to first order for low T ; the end points of the second order line are tricritical points.Hence the transition line forms a domed shape around the van Hove energy. Figure 1(b)shows the h dependence of the order parameter η for low T together with η ↑ and η ↓ . Wesee that although the phase transition happens around the van Hove energy of the up-spinband, both η ↑ and η ↓ show a jump at the first order point; the η ↑ has the same sign as η ↓ , butwith a different magnitude. The FSs at T = 0 .
01 are shown in Fig. 1(c) and (d) for h = 0 . .
5, respectively, together with the FSs for g = 0; the outer (inner) FS corresponds tothe up-spin (down-spin) electron band; the splitting of the FSs is due to the Zeeman energy.The FS instability drives a deformation of both FSs and typically leads to an open outerFS. Electron density for each spin also shows a jump at the first order transition point, butthe size of the jump is different [Fig. 1(e)]. This difference yields a metamagnetic transitionas shown in Fig. 1(f). This is a generic consequence of the concavity of the grand canonicalpotential when a first order transition occurs as a function of h , the canonical conjugatequantity to the magnetization m . The metamagnetic transition disappears for high T wherethe d FSD transition becomes second order. The T dependence of the magnetization is shownin Fig. 1(g) for several choices of h . The magnetization shows a kink at the second ordertransition. Roughly speaking, compared with the non-interacting case, the magnetization isenhanced (suppressed) by the d FSD transition for h . h vH ( h & h vH ). Since the transitionis of first order for low T as a function of h , there appears a phase separation as a functionof the magnetization [Fig. 1(h)]. The width of the phase separated region corresponds to amagnitude of a jump of m at the first order transition point seen in Fig. 1(f).6he d FSD instability has a dome-shaped transition line around the van Hove energy ofthe up-spin band for the chemical potential ( µ <
0) as shown in Fig. 1(a). When one invokesmuch larger µ ( > d FSD transition then happens around the van Hove energy of thedown-spin band. When µ is around zero, the second order line extends down to h = 0 anda first order line appears only on the larger h side for low T as shown in Fig. 2. For somespecial values of µ , see Appendix A.Although a metamagnetic transition generically accompanies a first order d FSD phasetransition, the metamagnetic transition can also occur inside the symmetry-broken phasebecause of a level crossing between two local minima of the free energy. This is actually thecase in the present model as demonstrated in Fig. 3, where m is plotted as a function of h for µ = − .
2; the h dependence of the order parameter is shown in the inset. In additionto a jump associated with the first order d FSD transition at h ≈ .
34 (see also Fig. 2),another metamagnetic transition appears at h ≈ .
28 in the symmetry-broken phase. Thismetamagnetic transition is a weak feature in the sense that it is smeared out by thermalbroadening for a relatively low temperature while the metamagnetic transition associatedwith the d FSD instability from the symmetric state is robust up to T = T tri c .Figure 4(a) summarizes numerical results in the plane of ( µ, h ) at low T = 0 .
01; thesolid circles denote a first order transition to the symmetry-broken phase, where the orderparameter η shows a jump from zero to a finite value, accompanied by a metamagnetictransition as a function of h ; the cross symbols are positions, where a first order transitionoccurs in the symmetry-broken phase due to a level crossing of local minima of the freeenergy, yielding an additional metamagnetic transition; the dotted line represents positionsof the van Hove energy for the up-spin band ( µ <
0) and the down-spin band ( µ > d FSD phase is stabilized around the van Hove energy of each spin band as afunction of h for a given µ . With respect to the axis µ = 0, the phase diagram is nearlysymmetric and becomes fully symmetric when t ′ is set to be zero. When we take a smaller g ,the symmetry-broken phase is stabilized closer to the van Hove energy, but the qualitativefeatures of Fig. 4(a) do not change except that the energy scale is substantially decreased.We define T vH c as T c at the van Hove energy, namely at h = | µ | , and show its h dependencein Figs. 4(b) and (c) for µ < µ >
0, respectively. The T vH c is suppressed with increasingmagnetic field. In particular, for a smaller g , a relatively small h suppresses T vH c drastically.However, the suppression saturates for larger h , leading to a finite T vH c , where we obtain a7hase diagram similar to Fig. 1(a). That is, neither a quantum critical point (QCP) nor aQCEP of the d FSD is realized by the magnetic field, which we will further discuss in Sec. IV.
IV. WEAK COUPLING LIMIT
The d FSD instability occurs around the van Hove filling and thus the transition is dom-inated by states with momentum near the saddle points of ǫ k . In the weak coupling limit,therefore, the mean-field equations can be treated analytically by focusing on the state nearthe saddle points, similar to the analysis in Ref.10 in the absence of a magnetic field. Sincethe magnetic field is absorbed completely in the effective chemical potential µ σ = µ + σh , wecan extend such an analysis to the present case by allowing the σ dependence of the chemicalpotential. The chemical potential µ is measured from the van Hove energy at h = 0 so that µ σ = 0 indicates that the σ -spin band is at the van Hove filling. We first determine a zerotemperature phase diagram in the plane of ( µ, h ), and then investigate T c suppression bya magnetic field, µ dependence of typical quantities characterizing the phase diagram, andthe limit h → ∞ . A. Zero temperature phase diagram
Following the analysis in Ref.10, the self-consistency equation Eq. (8) is written as η = ¯ g X σ (cid:2) ( µ σ − η ) log | µ σ − η | − ( µ σ + η ) log | µ σ + η | + 2 η (1 + log ǫ Λ ) (cid:3) , (10)where ¯ g = 2 mg/π is the dimensionless coupling and ǫ Λ = Λ / (2 m ) is a cutoff energy; m isthe effective mass near the van Hove energy and is related to the hopping integrals t and t ′ .The grand canonical potential is then given by ω ( η ; µ, h ) = 2 mπ (cid:26)(cid:20) g − − (1 + log ǫ Λ ) (cid:21) η + 12 X σ (cid:20)
12 ( µ σ + η ) log | µ σ + η | + 12 ( µ σ − η ) log | µ σ − η | (cid:21)(cid:27) + const . (11)It is not difficult to see that Eqs. (10) and (11) are symmetric with respect to interchange of h and µ , that is, the magnetic field h plays exactly the same role as the chemical potential µ .We focus on the region 0 ≤ µ < h and introduce rescaled variables ˜ η = η/h and ˜ µ σ = ˜ µ + σ µ = µ/h . Equation (10) then reads˜ η ˜ g = 12 X σ [(˜ µ σ − ˜ η ) log | ˜ µ σ − ˜ η | − (˜ µ σ + ˜ η ) log | ˜ µ σ + ˜ η | ] (12)with a renormalized coupling constant˜ g − = ¯ g − + 2 log h − ǫ Λ ) . (13)Similarly Eq. (11) is written as ω ( η ; µ, h ) = mπ h ˜ ω (˜ η ; ˜ µ ), where˜ ω (˜ η ; ˜ µ ) = (cid:18) g − (cid:19) ˜ η − X σ ˜ µ σ log | ˜ µ σ | + 12 X σ (cid:20)
12 (˜ µ σ + ˜ η ) log | ˜ µ σ + ˜ η | + 12 (˜ µ σ − ˜ η ) log | ˜ µ σ − ˜ η | (cid:21) (14)and the energy is shifted such that ˜ ω (˜ η = 0; ˜ µ ) = 0.At zero temperature, the d FSD transition is usually of first order as we have seen inFigs. 1(a) and 2. The first order transition is determined by solving Eq. (12) and ˜ ω (˜ η ; ˜ µ ) = 0numerically for a given ˜ µ , yielding a solution e η and e g . A corresponding magnetic field h is then obtained from Eq. (13) h = exp (cid:18) e g (˜ µ ) (cid:19) ǫ Λ e − / (2¯ g ) , (15)and thus the chemical potential and the order parameter are | µ | = ˜ µh and η = e η h ,respectively. All quantities, h , µ , and η , are determined by a single energy scale ǫ Λ e − / (2¯ g ) and the magnetic field just changes its dimensionless coefficient. We plot ( µ , h ) in Fig. 5;the phase diagram is symmetric with respect to the µ = 0 axis and the h = | µ | axis;such symmetry is not seen in our numerical result [Fig. 4(a)], since a finite t ′ breaks thesymmetry when g is not in the weak coupling limit. The first order transition line has akink at | µ | ≈ . ǫ Λ e − / (2¯ g ) and h ≈ . ǫ Λ e − / (2¯ g ) (see the inset of Fig. 5). At this point, thesolution of ˜ ω (˜ η ; ˜ µ ) = 0 [Eq. (14)] shows a jump, indicating double local minima of the freeenergy ˜ ω (˜ η ) away from ˜ η = 0, which then may yield an additional first order transition inthe symmetry-broken phase as seen in Figs. 3 and 4(a). B. T c suppression by a magnetic field A magnetic field suppresses the d FSD transition temperature [Fig. 4(b)]. Here we clarifykey factors of this suppression. 9ince the d FSD transition is of second order as a function of T at the van Hove filling,the T c is obtained by linearizing the right-hand side of Eq. (8) with respect to η , namely1 = gN ( µ, h, T c ) , (16)where we introduce N p ( µ, h, T ) = − N X k ,σ d p k f ′ ( ǫ k − µ σ ) , (17)a weighted density of states averaged over an energy interval of order T around µ σ ; p is aneven integer; f ′ is a first derivative of Fermi distribution function with respect to ǫ k . In theweak coupling limit[10], Eq. (17) reads N p ( µ, h, T ) = − mπ X σ Z ǫ Λ − ǫ Λ d ǫ log ǫ Λ | ǫ | f ′ ( ǫ − µ σ ) . (18)No p dependence appears on the right-hand side, since we have redefined d k = (cos k x − cos k y ) in the present analysis in the weak coupling limit[10] so that | d k | = 1 at k = ( π, , π ), namely at the saddle points of ǫ k . The van Hove filling of the σ spin electronband is set by choosing µ σ = µ + σh = 0 , µ − σ = µ − σh = − σh . (19)We consider the σ = ↓ case, namely for µ > h >
0. Since − Z ǫ Λ − ǫ Λ d ǫ log ǫ Λ | ǫ | f ′ ( ǫ − h ) ǫ Λ /T →∞ −−−−−→ log ǫ Λ + a ( h, T ) , (20)where a ( h, T ) = Z ∞−∞ d x log | T x + 2 h | ∂∂x x + 1 , (21)we obtain N ( h, T ) = 2 mπ (cid:20) ǫ Λ e γ πT + a ( h, T ) − a (0 , T ) (cid:21) . (22)where a (0 , T ) = log[2e γ / ( πT )] with Euler constant γ ≈ . h T = h/T , we may write ζ (˜ h T ) = a (0 , T ) − a ( h, T )= log 2e γ π − Z ∞−∞ d x log | x + 2˜ h T | ∂∂x x + 1 , (23)10here ζ (˜ h T ) increases monotonically with ˜ h T , and ζ (0) = 0 and ζ (˜ h T ) = ζ ( − ˜ h T ). Substi-tuting Eq. (22) into Eq. (16), we obtain T vH c at the van Hove filling for a given ˜ h T T vH c = 2e γ π ǫ Λ e − / (2¯ g ) e − ζ (˜ h T ) (24)= T vH c (0)e − ζ (˜ h T ) (25)where T vH c (0) = 2e γ π ǫ Λ e − / (2¯ g ) (26)is the critical temperature for h = 0; the corresponding magnetic field is obtained as h =˜ h T T vH c . We evaluate ζ (˜ h T ) numerically and show in Fig. 6 the h dependence of T vH c . The T vH c is suppressed with h , as we have seen in Fig. 4(b). Defining a magnetic field h / , atwhich T vH c is suppressed down to a half value of T vH c (0), we read off h / = 1 . T vH c (0) fromFig. 6. Therefore from Eq. (26) we obtain h / ∝ ǫ Λ e − / (2¯ g ) , (27)that is, the suppression of T vH c is controlled by the dimensionless coupling constant ¯ g =2 mg/π . When g becomes smaller, h / gets smaller exponentially, leading to a strongsuppression of T vH c by the magnetic field. Similarly, the smaller effective mass suppresses T c substantially with a magnetic field. C. µ dependence of characteristic quantities of the phase diagram We have derived the analytic expressions for h [Eq. (15)] and T vH c [Eq. (24)]. Thereis another important quantity characterizing the d FSD phase diagram, the tricritical point( T tri c , h tri ), which we first calculate for a given µ . We then summarize how h , T vH c , T tri c , and h tri evolve as a function of µ . Since all these quantities are scaled by a single energy, ratiosof these different quantities become universal, whose µ dependence is also clarified.At the tricritical point, both the quadratic and quartic coefficient of the Landau freeenergy, ω ( η ), vanish simultaneously. This condition is determined by Eq. (16) and ∂ ∂µ N ( µ, h, T ) = 0 (28)where N is defined in Eq. (17). While we have analyzed Eq. (16) at the van Hove point[Eq. (19)] in the previous section, we here consider Eq. (16) for general µ σ and obtain T tri c = e α ǫ Λ e − / (2¯ g ) , (29)11here α (˜ µ T , ˜ h T ) = 12 X σ Z ∞−∞ d x log | x + ˜ µ T + σ ˜ h T | ∂∂x x + 1 (30)with ˜ µ T = µ/T . Similarly Eq. (28) is written in the weak coupling limit as12 X σ Z ∞−∞ d x log | x + ˜ µ T + σ ˜ h T | ∂ ∂x x + 1 = 0 . (31)It is easy to observe that Eqs. (30) and (31) are symmetric with respect to ˜ µ T ⇆ ˜ h T andthus h plays exactly the same role as µ . We solve Eq. (31) numerically for a given ˜ µ T anddetermine ˜ h T . The tricritical temperature T tri c is then obtained from Eqs. (29) and (30);the original µ and h are determined as h tri = ˜ h T T tri c for a given µ = ˜ µ T T tri c . Figure 7(a)summarizes T vH c , T tri c , h tri , and h as a function of µ . We see that all quantities are suppressedwith increasing | µ | , but do not reach zero; in fact they approach certain asymptotic valuesas we show in the next subsection. Since all quantities are scaled by the single energy, weobtain universal numbers by taking ratios of the different quantities. In Fig. 7(b), we plotrepresentative universal ratios T tri c /T vH c and T tri c / | h tri − h vH | , where h vH is the van Hoveenergy and is given by h vH = | µ | . The strong µ dependence appears for relatively small µ and the variation is within a factor of 1.5.Although universal ratios are obtained in the weak coupling limit, these numbers alsocharacterize well the phase diagram for a relatively large g . For example in Fig. 1(a), T tri c / | h tri − h vH | ≈ . − . T tri c /T vH c ≈ . − .
75, which are comparable values to thosein Fig. 7(b).
D. Large h limit For a larger µ , each quantity in Fig. 7(a) approaches a certain asymptotic value. Inaddition, the universal ratios in Fig. 7(b) converge to the same values as those at µ = 0. Tounderstand such asymptotic behavior, we consider T vH c in the limit µ ↑ → ∞ while keeping µ ↓ = 0 [see Eq. (19)], namely, the limit of µ → ∞ and h → ∞ . Since the up-spin electronband becomes fully occupied, we have f ′ ( ǫ − µ ↑ ) = 0 for | ǫ | < ǫ Λ . Hence from Eq. (18) wehave N ( T ) = − mπ Z ǫ Λ − ǫ Λ d ǫ log ǫ Λ | ǫ | f ′ ( ǫ ) (32) ǫ Λ /T →∞ −−−−−→ mπ log 2 ǫ Λ e γ πT . (33)12he gap equation Eq. (16) then yields T vH c = 2e γ π ǫ Λ e − / ¯ g (34)= T vH c (0)e − / (2¯ g ) . (35)That is, the d FSD transition occurs even for h → ∞ under the condition of h = | µ | , although T vH c is suppressed a factor of e − / (2¯ g ) compared to the case of h = 0. Since the other spinband is fully occupied (empty) for µ → + ∞ ( µ → −∞ ), only one spin band is subject to the d FSD instability. Therefore in the h → ∞ limit, our model is reduced to a “spinless” fermionmodel, and thus the same results as those for h = 0 (Sec. V in Ref. 10) are obtained exceptthat the energy scale is replaced by ǫ Λ e − / ¯ g . Hence various ratios of different quantitiescharacterizing the phase diagram show exactly the same universal numbers as those for h = 0. The magnetic field just reduces the energy scale and cannot produce a QCP nor aQCEP of the d FSD instability.
V. RELEVANCE TO Sr Ru O We have shown that when either the up- or the down-spin electron band is tuned by amagnetic field close to the van Hove filling, the d FSD transition occurs in both bands butwith a different magnitude of the order parameter. The field-induced d FSD instability is apromising scenario for Sr Ru O . In fact, several important features observed experimentallysuch as the phase diagram, the metamagnetic transition, and the non-Fermi liquid likebehavior of the magnetic susceptibility and the specific heat coefficient are well captured interms of the d FSD instability around the van Hove filling as we have shown in Ref. 34.On the basis of the present results, we discuss in more detail the reported phase diagramfor Sr Ru O .[17] Since no experimental evidence of a symmetry-broken phase was obtainedat h = 0 and LDA band calculations[23, 24] showed that the van Hove energy is locatedabove the Fermi energy, corresponding to µ <
0, we expect that the chemical potential µ is away from the van Hove energy, but rather close to it, for example µ ≈ − . d FSD instability, which is very similar to the reported phase diagram.[17] Themaximal T c in the experiment is about 1 K, which is much smaller than the energy scale inFig. 1(a). The coupling constant g in Sr Ru O is thus expected to be very small. In the weak13oupling limit (Sec. IV), the phase diagram is characterized by a single energy scale ǫ Λ e − / g and thus various ratios among T vH c , T tri c , h tri , and h become universal numbers [Fig. 7(b)].The universal ratios are compared directly with experimental data. The data by Grigera etal. [17] provide T vH c ≈ T tri c ∼ . h tri ≈ h = g µ B H with H ≈ . . h vH = g µ B H vH with H vH ≈ .
95 Tesla. We thus obtain h tri /h ≈ T tri c /T vH c ∼ . T tri c / | h tri − h vH | ∼ ( k B · . / ( g µ B · . ≈ g − with Boltzmann constant k B . Thefirst one is consistent with our result Fig. 7(a); the value of the second one is comparablewith Fig. 7(b); as for the last one, however, the discrepancy is by a factor of 10 if we assume g = 2.While the field-induced d FSD instability is well analyzed in the present model, furtherstudies are necessary to make more quantitative comparison with experiments. (i) Theinteraction in our model retains SU(2) symmetry and thus the present theory cannot addressthe issue why the possibility of the d FSD instability was clearly observed only over a narrowregion of applied field angle to the RuO plane.[40] One possible origin of such a fieldangle dependence lies in a magnetic anisotropy, which we expect to originate mainly from arelatively strong spin-orbit coupling connected with the heavy Ru-ion. Our model should beextended by taking the magnetic anisotropy into account for a more quantitative study. (ii)Inclusion of a magnetic interaction in our model may also be necessary, since Sr Ru O isexpected to be close to a ferromagnetic transition.[20, 21, 22, 23, 24] This is suggested alsoby comparing Fig. 1(g) with the experimental data.[17] In Fig. 1(g) the d FSD instabilityproduces a kink in the T dependence of m at the transition temperature T c . This kinkis actually observed in the experiment[17] and our result for h & .
35 is similar to theexperiment. But the present theory cannot reproduce the observed upward curvature ofthe T dependence of m for T > T c ,[17] which may come from a magnetic interaction. (iii)Sr Ru O is a bilayered material. Since the d FSD instability is driven by forward scatteringprocesses of quasi-particles near k = ( π,
0) and (0 , π ), a form of bilayer coupling can beimportant if the bilayer coupling gives rise to k z dispersion around k = ( π,
0) and (0 , π ).Insights into the k z dispersion will be obtained from further detailed LDA calculations.[23,24]The d FSD instability was discussed in basic lattice models such as two-dimensional t - J ,[1, 2, 3] Hubbard,[4, 5, 6] and extended Hubbard[7] model, and can be a generic tendencynear van Hove filling in correlated electron systems.[41] Hence the d FSD is an interesting14ossibility when the Fermi energy is tuned close to the van Hove energy in other materialsalso such as Sr − y La y RuO [42] where La-substitution introduces electron carriers and makesthe FS closer to the van Hove point. However, La introduces some disorder in the RuO plane; its effects should be considered carefully, since the physics near the van Hove singu-larity may in general depend strongly on sample purity. In fact, the specific heat coefficientfor low T in Sr − y La y RuO [42] shows different behavior from that in Sr Ru O althoughboth systems are expected to be nicely tuned close to the van Hove filling. It is interest-ing to investigate impurity effects on the d FSD instability by chemical (electron) doping toSr Ru O . According to our phase diagram [Figs. 4(a) and 5, see also Appendix A], the d FSD instability can occur for a smaller magnetic field and even without the field if theimpurity effect by the electron doping is not serious.Around the van Hove filling, various ordering tendencies such as antiferromagnetism,ferromagnetism, superconductivity, and d -density wave develop,[11, 12] and compete withthe d FSD instability. Since the d FSD instability is suppressed by a magnetic field and itssuppression is controlled by g ∝ mg [see Eq. (27)], the absolute values of the effective massand the coupling constant are crucial to the possible d FSD instability over other instabilities.In this sense, microscopic derivation of m and g as well as magnetic field dependences ofother instabilities are important future issues.Some order competing with the d FSD instability is in fact expected in Sr Ru O . Whilethe experimental phase diagram[17] is very similar to our result Fig. 1(a), the closer com-parison between them reveals a difference of slope of the first-order-transition line. In theexperiment, the edges of the first order line are shifted to the center of the phase diagram sothat the d FSD state is stabilized in a narrower region for lower T . This can be interpretedas development of some ordering tendency for lower T in Sr Ru O , which then suppressesthe d FSD instability. Although a different interpretation was given in Ref. 17, a theoreticalresult consistent with this interpretation was indeed obtained in the case of competition ofthe d FSD and superconductivity.[43]It should be kept in mind that even if the d FSD instability does not become a leadinginstability, the system can still keep appreciable correlations of the d FSD.[13] As a result,small external perturbations such as an anisotropic strain and a lattice anisotropy can drivesizable d -wave type FS deformations. This idea was proposed for high-temperature cupratesuperconductors.[1] 15r Ru O is frequently refereed to as a material with a metamagnetic QCEP.[31] Thecompelling evidence for this was the systematic decrease of a metamagnetic critical endpoint by rotating the magnetic field out of the plane.[44] In fact, several theoretical scenariosbased on a metamagnetic QCEP were proposed.[35, 37] However, recent data for ultrapurecrystals showed that the critical end point does not reach zero.[40] Quite recently we havefound that several important features are already well captured even without the putativeQCEP.[34] The non-Fermi liquid like behavior of the magnetic susceptibility[20] and specificheat[26, 27, 28] observed in Sr Ru O can originate from the van Hove singularity of thedensity of states. Moreover, while the first d FSD transition as a function of a magneticfield is generically accompanied by a metamagnetic transition [Fig. 1(a) and Ref. 33], the d FSD transition does not lead to a metamagnetic QCEP in the present model. It remainsto be studied whether a concept of a metamagnetic QCEP can be a good basis to discusselectronic properties in Sr Ru O and how the putative QCEP can be related to the d FSDinstability and the van Hove singularity. In this sense, it is important to clarify whether theanomalous T dependence of the resistivity observed around the metamagnetic transition[26]can be explained in terms of d FSD fluctuations and the van Hove singularity or whether wehave to invoke quantum fluctuations originating from some QCEP.Different scenarios from the d FSD and the QCEP were proposed for Sr Ru O , micro-scopic phase separation due to Coulomb energy[38] and magnetic domain formation due tolong-range dipolar interactions.[39] In our analysis, we employ the ground canonical ensem-ble and a first order phase transition occurs as a function of a magnetic field or the chemicalpotential for low T . Such a transition in turn appears as a magnetic phase separation oran electronic phase separation as a function of its canonical conjugate variable, namely themagnetization [see Fig. 1(h)] or the electron density [see Fig. 1(g) in Ref. 10]. Thereforeinvoking the dipolar interaction or Coulomb force, we expect some inhomogeneous states inline with Refs. 38 and 39. However our possible inhomogeneous state may replace only phaseseparated regions in Fig. 1(h) and thus is realized only near a metamagnetic transition, incontrast to Refs. 38 and 39, where the inhomogeneous state is stabilized in the entire regionof the phase diagram. 16 I. CONCLUSION
We have performed a comprehensive analysis of magnetic field effects on the d FSD in-stability in a one-band mean-field model with a pure forward scattering interaction. In theplane of µ and h ( > d FSD instability occurs around the axis of µ = ± h , namelyaround the van Hove filling of the majority band ( µ <
0) and the minority band ( µ > h → ∞ . The magnetic field, however, suppresses substantially the energyscale of the d FSD instability. The d FSD instability occurs through a second order transitionat high T and typically changes to a first order transition at low T ; the end points of thesecond order line are tricritical points. Neither a QCP nor a QCEP of the d FSD is realizedby the magnetic field in the present model. In the weak coupling limit, typical quantitiescharacterizing the phase diagram have the field-independent single energy scale, while itsdimensionless coefficient varies with the magnetic field. The magnetic field-induced d FSDinstability is a promising scenario for Sr Ru O and we have discussed various future issuesto establish such a scenario. Acknowledgments
The author is grateful to A. A. Katanin for collaboration on a related work and fruitfuldiscussions. He also thanks C. Honerkamp, G. Khaliullin, A. P. Mackenzie, W. Metzner,and R. S. Perry for helpful discussions, and R. Zeyher for critical reading of the manuscript.
APPENDIX A: PHASE DIAGRAM FOR µ = − .
35 AND − . Figures 1(a) and 2 are two typical phase diagrams as a function of h in the present model.These phase diagrams, however, are not connected smoothly by changing µ . In fact, twoother types of the phase diagram appear in a very limited µ region as shown in Fig. 8. For µ = − .
35 [Fig. 8(a)], the dome-shaped transition line is realized on the relatively large h side. In addition, the region surrounded by a first order transition line appear around( h, T ) = (0 , T = 0, therefore, there is reentrant behavior as a function of h and thesymmetric phase appears in the intermediate h region (0 . . h . . µ = − . T near h = 0, accompanied by a tricritical point. As a17esult, a first order transition happens as a function of T in a sizable h region. This is a veryspecial case in our model since a first order transition as a function of T is usually realizedin a very limited h region as seen in Figs. 1(a) and 2.These peculiar types of the phase diagrams are understood from Fig. 5(a) or similarly fromFig. 4(a). The first order transition line in Fig. 5(a) is almost straight near | µ | ≈ . ǫ Λ e − / (2¯ g ) [ µ ≈ − .
34 in Fig. 4(b)]. Therefore we can have an extended h region of the first ordertransition as seen in Fig. 8(b). When we look at closely the region near | µ | ≈ . ǫ Λ e − / (2¯ g ) and h . . ǫ Λ e − / (2¯ g ) (inset of Fig. 5), the first order transition line turns out to have asmall inward curvature. This is why a symmetric phase is intervened between the two d FSDphases in Fig. 8(a).
APPENDIX B: SPIN-DEPENDENT d -WAVE FERMI SURFACE DEFORMA-TION We have analyzed the d -wave Fermi surface symmetry breaking in the charge channel.From the point of view of Landau Fermi liquids, we can consider Fermi surface instabilityalso in the spin channel. This possibility was pursued in several references[45, 46, 47, 48] inthe context of a general Landau Fermi liquid theory,[46, 47, 48] and a possible relation to ahidden order in URu Si .[45] Here we clarify the relation between the charge-channel d FSDand the spin-channel d FSD, and discuss its relevance to Sr Ru O .The minimal model for the spin-dependent d FSD reads H = X k ,σ ( ǫ k − µ ) n k σ + 12 N X k ,σ, k ′ ,σ ′ f a kk ′ S k · S k ′ − h X k ,σ σn k σ , (B1)where S k = P α,β c † k α σ αβ c k β , f a kk ′ = − g a d k d k ′ , and the rest of notation is the same as themodel (1). Since the interaction has SU(2) symmetry and the magnetic field is assumedto be applied along the z direction, we assume that the S z component can have a finiteexpectation value. Defining a mean field η a = − g a N X k d k h S z k i , (B2)we obtain the mean-field Hamiltonian H MF = X k ,σ ξ a k σ n k σ + N g a ( η a ) (B3)18here ξ a k σ = ǫ k + ση a d k − µ σ . The grand canonical potential per lattice site thus reads ω = − TN X k ,σ log(1 + e − ξ a k σ /T ) + ( η a ) g a (B4)= − TN X k ,σ log(1 + e − ξ a ′ k σ /T ) + ( η a ) g a . (B5)In the second line, we have introduced ξ a ′ k σ = ǫ k + 12 η a d k − µ σ , (B6)noting that d k changes its sign with respect to k x ⇆ k y so that σd k in the original ξ a k σ canbe written as d k in Eq. (B6).Comparing Eqs. (B5) and (B6) with Eqs. (7) and (6), respectively, we see that the freeenergy becomes exactly the same under the following mapping,12 η a ⇆ η , g a ⇆ g . (B7)Hence the thermodynamics in the spin channel of the d FSD is the same as that in the chargechannel in the sense that we obtain the exactly the same results as Figs. 1(a) and (e)-(h)under the mapping (B7). The difference appears in the “internal” structure of the orderparameter and in a deformation of the FS. In the spin channel, we can write η a = P σ ση aσ ,where 12 η aσ = − g a N X k d k h n k σ i (B8)= − g a N P k d k f ( ξ a ′ k ↑ ) for σ = ↑ + g a N P k d k f ( ξ a ′ k ↓ ) for σ = ↓ . (B9)Comparing with Eq. (9), we see the relation under the mapping (B7)12 η a ↑ ⇆ η ↑ , − η a ↓ ⇆ η ↓ . (B10)As we have actually seen in the numerical result in Sec. III, η ↑ in general has the same signas η ↓ in the charge channel. Therefore η a ↑ has an opposite sign of η a ↓ as seen in Fig. 9(a).While we have introduced ξ a ′ k σ [Eq. (B6)], the Fermi surface itself is defined by ξ a k σ = 0.Since ξ a k σ contains a factor ση a d k , a Fermi surface deformation in the spin channel occursalways in the opposite direction between the up-spin and the down-spin electron band as19hown in Fig. 9(b); note that the deformation is determined by η a , not by η aσ . As a result,the net deformation of the band is partially compensated. This is a crucial difference fromthe d FSD instability in the charge channel.The recent experiment by Borzi et al. [32] showed a strong xy -anisotropy of the magne-toresistivity by applying an additional small magnetic field to the RuO plane. This stronganisotropy may be discussed more naturally in terms of the d FSD instability in the chargechannel rather than the spin channel. A conclusive discussion on which channel is moredominant would be to study microscopic deviation of the d FSD attractive interaction inboth charge and spin channel in the context of Sr Ru O and to compare the strength ofthe each channel. 20
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4. (a) h - T phase diagram;the transition line contains a second order line, T c , for high T and two first order line, T c , forlow T ; the solid circles are tricritical points; the dotted line denotes the van Hove energy of theup-spin band. (b) h dependence of the order parameter at T = 0 .
01; note that η = η ↑ + η ↓ . (c)and (d) FSs for h = 0 . . T = 0 .
01; the solid lines (gray lines) are FSs for g = 1 ( g = 0);the deformation of the inner FS in (d) is hardly visible. (e) h dependence of n σ at T = 0 .
01 for g = 1 (solid line) and 0 (dotted line). (f) h dependence of the magnetization at T = 0 .
01 for g = 1(solid line) and 0 (dotted line). (g) T dependence of the magnetization for several choices of h ; thedotted line represents the result for g = 0. (h) m - T phase diagram; the system undergoes phaseseparation in the shaded regions surrounded by T PS c ; the other notation is the same as that in (a). IG. 2: h - T phase diagram for t ′ /t = 0 . g = 1, and µ = − .
2; the notation is the same as thatin Fig. 1(a).FIG. 3: h dependence of m at µ = − . T = 0 .
01 for g = 1 (solid line) and 0 (dotted line).The inset shows the h dependence of the order parameter. IG. 4: (Color online) (a) The first order d FSD transition line in the plane of µ and h at T = 0 . g = 1; the symmetry-broken phase is stabilized in the colored (shaded) area; for notation, seethe text. (b) and (c) h dependence of T c along the van Hove energy, namely along the dotted linein (a) for µ < µ >
0, respectively; the solid (dotted) line is the result for g = 1 (0 . T vH c for g = 0 . d FSD transition line in the weak coupling limit in the planeof µ and h at T = 0; µ and h are scaled by the energy ǫ Λ e − / (2¯ g ) ; the symmetry-broken phaseis stabilized in the colored (shaded) area; the dotted line represents the van Hove energy of theup-spin band ( µ <
0) and the down-spin band ( µ > µ ≈ . IG. 6: T c along the van Hove energy, namely along the line of h = | µ | , in the weak coupling limit; T c and h are scaled by T vH c (0).FIG. 7: (Color online) (a) µ dependence of T vH c , T tri c , h tri − h vH , and h − h vH , where h vH = | µ | ; all quantities are scaled by the energy ǫ Λ e − / (2¯ g ) . (b) µ dependence of the universal ratios T tri c / | h tri − h vH | and T tri c /T vH c . IG. 8: h - T phase diagram for µ = − .
35 (a) and µ = − .
34 (b) for t ′ /t = 0 .
35 and g = 1; thenotation is the same as that in Fig. 1(a).FIG. 9: The mean-field solution in the spin-channel d FSD instability for t ′ /t = 0 . µ = − . g a = 4. (a) h dependence of the order parameter at T = 0 .
01. (b) FSs at h = 0 . g a = 0.= 0.