Dimensional decoupling at continuous quantum critical Mott transitions
aa r X i v : . [ c ond - m a t . s t r- e l ] J u l Dimensional decoupling at continuous quantum critical Mott transitions
Liujun Zou and T. Senthil Department of Physics, Harvard University, Cambridge, MA 02138, USA Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Dated: May 7, 2018)For continuous Mott metal-insulator transitions in layered two dimensional systems, we demon-strate the phenomenon of dimensional decoupling: the system behaves as a three-dimensional metalin the Fermi liquid side but as a stack of decoupled two-dimensional layers in the Mott insulator. Weshow that the dimensional decoupling happens at the Mott quantum critical point itself. We derivethe temperature dependence of the interlayer electric conductivity in various crossover regimes nearsuch a continuous Mott transition, and discuss experimental implications.
I. Introduction
Despite its great success in describing many metals,Fermi liquid theory fails to characterize many stronglycorrelated metals. Examples of “non-Fermi liquid” met-als found experimentally include heavy fermion com-pounds, the cuprates, and some organic salts .The central conceptual building block in Fermi liquidtheory is the existence of long-lived electronic quasipar-ticles near a sharply defined Fermi surface in momentumspace. The electron spectral function A ( ~k, ω ) near theFermi surface takes the form A ( ~k, ω ) = Zδ ( ω − ǫ ( ~k )) (1)where Z , the quasiparticle residue, measures the overlapbetween the wave function of a quasiparticle and that ofthe original electron, and ǫ ( ~k ) is the (gapless) dispersionof quasiparticles. Metals in which this building blockbreaks down will show non-Fermi liquid properties in anumber of experimental probes.Our concern in this paper is on non-Fermi liquid met-als in layered quasi-two dimensional systems near Mottmetal-insulator (and other closely related) phase transi-tions. Specifically we will focus on metals at or near con-tinuous ( i.e quantum critical) Mott transitions in suchsystems. Previous work has demonstrated, for a singleisolated two dimensional (2 d ) layer, the possibility of asecond order quantum phase transition from a Fermi liq-uid metal to a quantum spin liquid Mott insulator witha Fermi surface of charge neutral spin-1 / .These studies were motivated by the phenomenology ofthe quasi-2 d triangular lattice organic materials and thecuprate metals , and they have obtained support fromnumerical studies . Here we study the effect of weakinterlayer coupling on the fate of such continuous Motttransitions in the physical three dimensional material.In the specific context of the cuprate metals it has longbeen appreciated that the combination of their layeredquasi-two dimensional structure and their possible non-Fermi liquid properties could lead to peculiar interlayertransport. Specifically interlayer transport was arguedto be related to the single particle spectrum in such lay-ered materials, and hence probe very different physicsfrom intralayer transport . Interlayer transport is thus a very useful spectroscopic probe of a correlated quasi-two dimensional metal, particularly in situations whereother direct probes like photoemission or tunneling is notfeasible .We show that, in the spin liquid Mott insulating phase,different layers decouple from each other so that the sys-tem behaves as a stack of two dimensional layers. We callthis ‘dimensional decoupling’. In contrast, in the Fermiliquid, different layers recouple to form a coherent threedimensional Fermi surface (see Fig. 1). The transitionis thus between a three dimensional Fermi liquid and adimensionally decoupled stack of two dimensional insu-lators. We show that this phase transition is continuous,and further that the dimensional decoupling happens al-ready at the quantum critical point. In other words, uni-versal critical properties are correctly obtained from apurely two dimensional theory though the metallic phaseon one side is (at low energies) three dimensional. We dis-cuss the implications for physical properties as the tran-sition is approached from the metallic side. In particularwe show that the interlayer conductivity as a function ofdecreasing temperature in the nearly critical metal has a‘coherence’ peak at a crossover scale determined by thedistance from the quantum critical point, and determinethe detailed universal temperature dependence in variousregimes.Very recent experiments have studied interlayer trans-port in the doped triangular lattice κ − ET organicmetals , and our results should be a useful guide to theirinterpretation. Ref. 7 found the intralayer transport ap-pears to be non-Fermi liquid like at ambient pressure andbecomes Fermi liquid like under high enough pressure.As the temperature is decreased in the non-Fermi liquidregime, the interlayer resistivity first increases when thetemperature is high. But it starts to decrease when thetemperature is low enough, so it has a peak as temper-ature changes. This ‘interlayer coherence’ peak becomesbroadened and is shifted to higher temperatures as thepressure becomes higher. Further this peak seems to oc-cur in a regime where the intralayer transport is non-Fermi liquid like. It is thus timely to study issues relatedto interlayer coupling near continuous Mott transitions.In the cuprate context previous theoretical work ondoped Mott insulators identified, within a slave bosonframework, an “Incoherent Fermi Liquid” (IFL) regimethat has some phenomenological appeal as a descriptionof the strange metal normal state . Our results describethe interlayer transport of this Incoherent Fermi Liquid.Our results also carry over straightforwardly to Kondobreakdown transitions (of the kind studied in Refs. 13)in Kondo lattice models in layered systems.We emphasize that dimensional decoupling is not guar-anteed to describe all two dimensional non-Fermi liquids.To illustrate this we consider a class of non-Fermi liquidsthat develop at metallic quantum critical points associ-ated with onset of broken symmetry. A good and topicalexample is the onset of Ising nematic ordering from asymmetry preserving metal. In Appendix A we show,within the existing theory of this transition, that at thequantum critical point there is no dimensional decou-pling. We provide arguments that this is likely the caseat all Landauesque quantum critical points driven by bro-ken symmetry order parameter fluctuations (coupled tothe metallic electrons).In contrast continuous Mott transitions or the Kondobreakdown transitions are driven by electronic structurefluctuations which cannot be captured through Landauorder parameters. On approaching this kind of quan-tum critical point from the metallic Fermi liquid sidethe quasiparticle residue Z is expected to vanish con-tinuously. Right at the critical point the quasiparticleis thus destroyed everywhere. Despite this, Ref. 14 ar-gued that the quantum critical point is characterized bya sharply defined Fermi surface (dubbed a “critical Fermisurface”). Concrete examples which illustrate the generalarguments of Ref. 14 are in Refs. 8, 13, and 15 based onslave-particle gauge theories.Clearly such phase transitions driven by electronicstructure fluctuations require a different conceptualframework from more conventional order parameterdriven ones. The corresponding quantum critical phe-nomenology will also be very different. Our results add tothe growing list of distinctions between these two classesof metallic quantum critical phenomena. FIG. 1. Dimensional decoupling across the phase transition.On the side with g > g c we have a 3D Fermi liquid, and atthe quantum critical point and on the other side we have astate that behaves as a stack of many decoupled layers at lowenergies. To address the possibility of dimensional decouplingin these systems, it is necessary to examine the ef-fects of all possible interlayer interactions. The sim-plest amongst these is electron tunneling between dif-ferent layers. Other potentially important couplings isthat between slowly fluctuating bosonic order parame-ters obtained as bilinears made out of the underlyingelectrons. The phenomenon of dimensional decouplingrequires that, in the renormalization group (RG) sense,none of the possible interlayer interactions is relevant atthe decoupled fixed point with no interlayer coupling. Forthe order parameter driven phase transitions, the cou-pling between order parameter fluctuations in differentlayers becomes relevant and destabilizes the decoupledfixed point. However this does not happen at the contin-uous Mott transition.In passing, we note that our starting point is that thecoupling between different layers is weak at the latticelevel, so it is legitimate to first consider the system as astack of decoupled layers and then study the effects ofthe interlayer couplings. In contrast, other works, suchas Ref. 16, study the situation where the interlayer in-teractions are as strong as intralayer interactions, and inthis case considering a 3D system from the beginning ismore appropriate.The rest of this paper is organized as follows. We beginwith a brief review of some general (semi-) quantitativeaspects of a critical Fermi surface and continuous Motttransitions in Sec. II. Before diving into the more com-plicated case of continuous Mott transitions, in Sec. IIIwe will warm up by illustrating the phenomenon of di-mensional decoupling in a simpler context: a quantumphase transition between a Fermi liquid and an orthog-onal metal. We will first consider a single 2D layer andreview the nature of the orthogonal metal state and thephase transition in 2D, then we consider a stack of manysuch 2D layers and examine the effects of interlayer in-teractions. Following the same strategy, we will go intothe case of continuous Mott transitions and demonstratedimensional decoupling thereof in Sec. IV. After this, inSec. V we will calculate the interlayer electric conductiv-ity induced by electron tunneling between different layersin the QC regimes of the various cases of interests. Be-cause of the rich crossover structure predicted by the the-ory of continuous Mott transitions, in order to identifysome of its experimental signatures, in Sec. VI we studythe interlayer electric conductivity as the system crossesover from its QC regimes to the Fermi liquid regime. Fi-nally, we conclude with some discussions on experimentsin Sec. VII.
II. Preliminaries
In this section we collect together some previous resultson critical Fermi surfaces and continuous Mott transi-tions that will be used extensively in the rest of this pa-per.
A. Critical Fermi surface
A continuous Mott transition from a Fermi liquid metalrequires a sudden death of the the metallic Fermi surface.The transition to the Mott insulator occurs without thevanishing of the free carrier density, and hence the Fermisurface cannot simply shrink to zero. Thus continuousMott transitions necessarily involve the death of an en-tire Fermi surface of some fixed size. Ref. 14 argued thatat the corresponding quantum critical point there will bea sharp critical Fermi surface but without well-definedLandau quasiparticles. To discuss interlayer coupling ef-fects in different systems in a unified manner, we willneed some very general scaling properties of the electronspectral function at such a critical Fermi surface whichwe now summarize.Suppose one can access such a quantum critical pointby tuning a parameter g . It is natural that at tempera-ture T , the electron spectral function around the criticalFermi surface near such a critical point has the scalingform: A ( ~K, ω ; g, T ) ∼ c | ω | αz F (cid:18) ωT , c k k T z , k k ξ (cid:19) (2)where α and z are two universal exponents, and F is auniversal function. k k is the deviation of momentum ~K from the Fermi surface. Constants c and c are non-universal. ξ is the correlation length of the system atthe same g but at zero temperature, and it is a measureof the deviation of g from the quantum critical point g c since ξ − ∼ | g − g c | ν , with a universal zero temperaturecorrelation length exponent ν . This scaling form of thespectral function applies to each patch of the Fermi sur-face, and, in general, α , z , ν , c and c can all depend onthe position of the patch on the Fermi surface where (2)is applied. We will take the convention that the Fermiliquid is on the g > g c side through out this paper.According to (2), the number of fermions with a certainmomentum ~k is n ( ~k ) = Z −∞ dω A c ( ~k, ω ) ∼ | k k | z − α (3)Because the fermion number is upper bounded, we musthave z > α (4)Again, we notice that (2) and (4) apply to each patchof the Fermi surface, and in general the exponents candepend on the position of patch. The same scaling con-siderations apply to all systems with a critical Fermi sur-face. B. Theory of a continuous d Mott transition
We consider two types of continuous Mott transitions:chemical potential tuned and bandwidth tuned, for which in a single 2 d layer there are well developed theories.Theoretically, it is expected that these Mott transitionscan be realized by the Hubbard-type Hamiltonian on atriangular lattice H = − t X ij,σ ( c † iσ c jσ + h . c . )+ U X i n i ↑ n i ↓ − µ X i,σ n iσ (5)where c iσ annihilates an electron with spin σ at site i ,and n iσ = c † iσ c iσ is the number operator of the fermion.There is numerical evidence for a regime described by aMott insulator with a spinon Fermi surface . Exper-imentally the organic compound κ − ( ET ) Cu ( CN ) ,which is believed to be well described by a one-bandHubbard model on a triangular lattice, also exhibits sig-natures of these transitions .The transition is conveniently accessed by formallywriting the electron operator c iσ as c iσ = f iσ · b i (6)where the fermionic spinon f carries spin- , and the bo-son carries unit physical charge. The physical electronoperator is invariant under a local U (1) gauge transfor-mation f iσ → f iσ e iφ i and b i → b i e − iφ i , which leads toan emergent U (1) gauge field at low energies. Considera situation where spinons form a Fermi surface. If theboson b condenses, the gauge field is Higgsed out and weget a Fermi liquid of the original electron. If b is gapped,we get a spin liquid with a spinon Fermi surface coupledto a U (1) gauge field. The longitudinal component of thegauge field is screened by the spinon Fermi surface, butthe transverse components remain strongly interactingwith the spinons. The Mott transition from the Fermiliquid is driven by losing the condensate of b .For both the chemical potential and bandwidth tunedtransitions, Ref. 8 showed that the quantum critical fluc-tuations of b (at T = 0) are dynamically decoupled fromthe spinon-gauge system. However, the dynamics of thelatter is affected by the criticality of b . This leads to atractable theory of these continuous Mott transitions anda number of universal physical properties have been com-puted. An interesting feature shared by both transitionsis that the crossover from the quantum critical metal tothe Landau Fermi liquid on the metallic side occurs intwo stages. The charge sector crosses over at an energyscale parametrically larger than the spin sector. At in-termediate energies a non-Fermi liquid metallic regimeis reached which is distinct from the quantum criticalnon-Fermi liquid.We will study the effects of interlayer coupling on thesetransitions, and determine the nature of interlayer trans-port both in the quantum critical non-Fermi liquid andin the intermediate energy non-Fermi liquid that existsbefore the emergence of the fully coherent Landau Fermiliquid.Notice in principle this U (1) gauge field should betaken to be compact, but as pointed out in Ref. 21,the effect of instantons of this compact U (1) gauge fieldwill be suppressed by the spinon Fermi surface, so itis adequate to just consider a noncompact U (1) gaugefield. It is also sufficient for our purposes to treat thisnon-compact U (1) gauge field under a random-phase-approximation (RPA), which can be formally justifiedby a controlled expansion in its leading order . III. Warm-up: Dimensional decoupling in aquantum phase transition between a Fermiliquid and an orthogonal metal
To examine effects of interlayer coupling at continuousMott transitions we need to confront the full theory of thethe critical b fluctuations and the spinon-gauge system.In this section we warm up to this task by considering adifferent problem which has some of the same ingredients.Rather than studying the transition from a Fermi liquidto a Mott insulator, we study the transition to a differentphase dubbed the orthogonal metal. We demonstrate thephenomenon of dimensional decoupling at this quantumphase transition.An orthogonal metal is a state where the electron isfractionalized into a fermion f iσ that carries both thecharge and spin of the electron and a discrete degree offreedom s i that is gapped . This fractinonalization is ac-companied by a deconfined discrete gauge field to whichboth f iσ and s i are coupled. The f iσ forms a Fermisurface, and the system has metallic charge/spin trans-port. However the gapless f iσ have zero overlap with thephysical electrons (they are orthogonal) . Thus despitethe metallic charge transport single partilce-probes liketunneling or photoemission will see insulating behavior.The orthogonal metal is the simplest non-Fermi liquid inspatial dimension d ≥
2, and its universal properties areeasily computed. Some lattice models that can realizethis state are provided in Ref. 15.Formally we write the electron operator c iσ as c iσ = f iσ · s i (7)Assuming the fermions f are in their Fermi liquid phase,if s condenses, the resulting state is a Fermi liquid of theoriginal electrons. However, if s is gapped, we get anorthogonal metal, whose nature will be determined bythe nature of the discrete variable s .In passing, we remark that all systems discussed inthis paper have a Fermi surface, either of the physicalelectrons or of some emergent fractionalized fermions. Itis well-known that a Fermi surface may potentially sufferfrom an instability towards Cooper pairing, but becausethis instability occurs only at very low temperatures, wewill assume our systems are in a regime free of pairinginstability through out this paper. We also notice thatin some systems with a Fermi surface coupled to a gaugefield, the pairing instability is suppressed .Now consider a single 2D layer first. One can drive atransition from a Fermi liquid of electrons, a condensate of s , to an orthogonal metal by destroying the conden-sate. As discussed in Ref. 15, in 2D the transition toa Z orthogonal metal in a lattice needs fine-tuning ofthe parameters in the Hamiltonian, and the simplest ex-ample where a generic second-order phase transition canoccur in a 2D lattice is between an electronic Fermi liquidand a Z orthogonal metal. The critical theory for sucha transition is described by the following Lagrangian L = b ∗ ∂ τ b + 12 m b |∇ b | + t | b | + u | b | + v (cid:2) b + ( b ∗ ) (cid:3) (8)where the b transforms as b → ib under the Z trans-formation. Notice this is not a symmetry-breaking phasetransition because b itself is not gauge invariant. Instead,this is a transition associated with electron fractionaliza-tion. FIG. 2. The schematic phase diagram and crossover structureof an orthogonal metal transition. When g > g c we have aFermi liquid (FL), while we have an orthogonal metal (OM)when g < g c . At finite temperature, there is a quantumcritical regime (QC) where the system is non-Fermi liquidlike. At the critical point, the quasiparticle residue of theelectron vanishes and the excitations are incoherent. Thespectral function of the Z spins at the critical pointis A s ( ~q, Ω) ∼ δ (Ω − ~q m b ). Since the spinons are intheir Fermi liquid phase, they have spectral function A f ( ~k, ω ) ∼ δ ( ω − v F k k ). Convolving them we get theelectron spectral function at the quantum critical point: A c ( ~k, ω ) = Z ~q Z ω d Ω A s ( ~q, Ω) A f ( ~k − ~q, ω − Ω) ∼ ω − k k m b ! θ ω − k k m b ! (9)This has the form of (2), and we see that this orthog-onal metal transition has a critical Fermi surface with α = − z = 2. As we can also see, (4) is indeedsatisfied. The large and negative value of α means in thequantum critical regime (QC) above the quantum criti-cal point, the system is highly non-Fermi liquid like. If g = g c , starting from the QC regime, as the temperatureis decreased, the system crosses over to a Fermi liquidmetal or an orthogonal metal, depending on the relativemagnitude of g compared to g c (see Fig. 2).Now consider a stack of such 2D systems, with eachlayer going through a phase transition between a Fermiliquid and a Z orthogonal metal. When the interlayerinteractions are absent, in terms of RG, the critical pointis described by a fixed point where each layer correspondsto an individual fixed point and all these fixed points aredecoupled. We will call such a fixed point a “decoupledfixed point”. Let us examine the effect of all possible in-terlayer interactions on this decoupled fixed point. Notethat the decoupled fixed point has separate conservationof physical electric charge in different layers correspond-ing to independent global U (1) symmetry rotations ineach layer. In writing (7), we have introduced a Z gaugeredundancy on each layer, so the interlayer interactionsshould be invariant under a local Z gauge transforma-tion within each layer. The most obvious physical cou-pling is simply electron tunneling between different layers- this breaks the infinite number of U (1) symmetries as-sociated with conservation of electric charge separatelyin each layer to a single common global U (1). We willfirst however focus on interlayer couplings that preservethis infinite U (1) symmetry.The most important such interactions consistent withgauge invariance and global symmetries are the couplingbetween energy densities of different layers, which is ofthe form δ L = X α Z dτ d xg αβ | b α | | b β | (10)and the coupling between the energy density of one layerand the collective excitation around the spinon Fermisurface of the other layer, which is of the form δ L = X αβ,σ Z dτ d xg αβ | b α | f † βσ f βσ (11)Here we use α and β to index the layers.Perturbing the decoupled fixed point with δ L , we getits RG equations dg αβ dl = −C g αβ (12)with a constant C > δ L is (marginally) irrelevant.On the other hand, to consider the effect of δ L on thedecoupled fixed point, it is convenient to integrate outthe degrees of freedom from the spinons, and the mostrelevant interaction generated from this procedure hasthe following Landau damping form X α g ′ αβ Z ω,~q Π( ~q, ω ) · | b α | ( ~q, ω ) · | b β | ( ~q, ω ) (13)with Π( ~q, ω ) ∼ | ω || ~q | (14) for small frequencies and momenta. Because the decou-pled fixed point has dynamical exponent z = 2, the de-pendence of Π on the frequency and momentum | ω || ~q | ∼ q makes this interaction irrelevant.As for other interactions, one should in principle con-sider the couplings between charge densities, spin den-sities, charge currents and spin currents, and the mostrelevant ones of these couplings in this case are of theform of four-fermion interactions of the spinon f . It iswell-known that in the presence of a Fermi surface, mostof the four-fermion interactions are strongly irrelevant,except for the forward scattering and the BCS scattering.The former is marginal and will not modify the physicalproperties of the system qualitatively, and they can bedescribed by a set of Landau parameters, while the latteris marginally irrelevant and can induce pairing instabilityat very low temperatures . In our case, most of theseinterlayer four-fermion interactions are also strongly ir-relevant due to the kinematic constraint of the Fermisurface, and the analog of forward scattering will renor-malize the in-plane Landau parameters, but they will notgive rise to any qualitative change of the physics. Theanalog of BCS scattering can in principle induce inter-layer pairing at very low temperatures, but as declaredbefore, we will ignore it.We now return to the important effect of interlayerelectron tunneling. To discuss these we will use a scalingargument that is somewhat different from the one above.Consider a general action for the interlayer electronhopping. At low energies it is appropriate to work withelectronic modes near the critical Fermi surface. Thehopping term can then be written as δS tunneling = − Z dωdk k dθ X γδ t γδ ( θ ) · (cid:0) c † γ ( k k , θ, ω ) c δ ( k k , θ, ω ) + h . c . (cid:1) (15)Now for any critical Fermi surface consider a scalingtransformation that renormalizes toward the Fermi sur-face. We let k k → k ′k = k k s , and ω → ω ′ = ωs z . The twopoint correlation function of c ( k k , θ, ω ) satisfies h c ( k k , θ, ω ) c † ( k k , θ,ω ) i = δ ( k k − k k ) δ ( ω − ω ) G ( k k , θ, ω )If the electron spectral function for a 2D layer has thescaling form (2), then the the electron operator trans-forms as c ( k k , θ, ω ) → c ′ ( k ′k , θ, ω ′ ) = s − α + z +12 c ( k k , θ, ω ).It follows that the scaling of the hopping parameter is t ′ ( θ ) = t ( θ ) s α . If α <
0, we expect that the interlayerelectron hopping is irrelevant.As discussed above the electron spectral function fora 2D layer at the critical point for the Fermi liquid toorthogonal metal transition satisfies the scaling form ofEqn. 2 with α = − z = 2., and thus the interelectron hopping indeed scales to zero at low energy.We point out a caveat in this scaling analysis. In theexample of orthogonal metal transition discussed in thissection (and the chemical potential tuned Mott transi-tion discussed later), the boson sector has dynamical ex-ponent z b = 2 while the fermion sector has dynamicalexponent z f = 1, and the electron Green’s function hasdynamical exponent z c = 2. So when we do a scalinganalysis, how should we scale space and time? Noticethe above dynamical exponents indicate the importantdynamical regions are ω ∼ k and ω ∼ k , and the formergoverns the important electron dynamics. Moreover, theformer is a slower regime compared to the latter. There-fore, to consider the lowest energy physics which is alsopertinent to the electrons, we choose z = 2 in the abovescaling analysis.Therefore, none of the interlayer interactions is rele-vant at the decoupled fixed point, and different layersdo decouple at the quantum critical point. If we go intothe orthogonal metal side, since b is gapped there, theabove interactions that involve b will be more irrelevantand the other interactions stay as irrelevant as they areat the critical point. So as long as the system leaves theFermi liquid phase, it behaves as a stack of many de-coupled layers, as shown in Fig. 1. This is our simplestexample of dimensional decoupling.Before moving on, we emphasize the important roleof the emergent gauge invariance in obtaining dimen-sional decoupling. Gauge invariance strongly constrainsthe possible form of the interlayer interactions, and inthe absence of this constraint there will be relevant in-teractions in general (see Appendix A). IV. Dimensional decoupling in continuous Motttransitions
Warmed up with the example of the orthogonal metaltransition, now we are ready to deal with the more com-plicated problem of continuous Mott transitions.
A. Chemical potential tuned continuous Motttransition
If the system is not at half-filling and the transition isaccessed by changing the electron doping, or equivalently,by tuning the chemical potential, the low-energy effectiveLagrangian of this transition is L = L b + L f + L gauge (16)with L b = ¯ b " ∂ τ − ia − µ − ( ~ ∇ − i~a ) m b b + V ( | b | ) L f = ¯ f " ∂ τ + ia − ( ~ ∇ + i~a ) m f − µ f f L gauge = 14 e ( ǫ µνλ ∂ ν a λ ) (17) The potential V ( | b | ) can be taken to be of the conven-tional form r | b | + g | b | . Notice we did not include a directinteraction between bosons and spinons, since this can beshown to be irrelevant . This model has been extensivelystudied. Closely related models appear in theories of thecuprates and of the Kondo breakdown phenomenon inKondo lattice systems . FIG. 3. Phase diagram and crossover structure of a chemi-cal potential tuned Mott transition. The critical point cor-responds to µ = 0 and T = 0. The µ < µ > There are several interesting features associated withthis transition . First of all, it is shown that at the quan-tum critical point the bosons are dynamically decoupledfrom the spinon-gauge-field sector. Therefore, the transi-tion is in the universality class of a dilute (nonrelativistic)bose gas. Using this, one can calculate the electron spec-tral function at the critical point and show there is indeeda critical Fermi surface, and A ( ~K, ω ) in this case has thesame form as (9). Since the details of this calculationwere not given in Ref. 8, we present it in Appendix C.As shown in Fig. 3, there is a QC regime above the zerotemperature quantum critical point, where the system isstrikingly non-Fermi liquid like, just as the QC regime ofthe Z orthogonal metal transition.Moreover, when the system crosses over out from theQC regime to the Fermi liquid, the system has to first gothrough an intermediate regime. In particular, we canchoose the boson phase stiffness ρ s as the characteristicenergy scale on the Fermi liquid side. The system is inits quantum critical (QC) regime if T ≫ ρ s . If we de-crease the temperature from QC so that T ≪ ρ s , thebosons appear to condense. However, the spinon-gauge-field sector has not yet felt the Higgs effect and it behavesas if it is still in its own quantum critical regime. Thesystem in this regime is also a non-Fermi liquid, and itis dubbed “incoherent Fermi liquid” (IFL) . When thetemperature is further decreased so that T ≪ ρ s , theHiggs effect is manifested and the spinon-gauge-field sec-tor also crosses over out of its quantum critical regime,and the system appears as a Fermi liquid. These resultscan be shown, for example, by calculating the electronspectral function in various regimes.On the Fermi liquid side, the propagator of the trans-verse components of the gauge field in Coulomb gaugeunder RPA is D c ( ~q, i Ω) = 1 k | Ω || ~q | + χ d q + ρ s (18)where k is of the order of a typical Fermi momentumof the spinon Fermi surface, and χ d q is the diamagneticterm. Upon approaching the quantum critical point fromthe Fermi liquid side, the superfluid density vanishes as ρ s ∼ ξ − z ∼ ( g − g c ) zν up to a logarithmic correction thatwe will ignore , with z = 2 and ν = in the universalityclass of a dilute bose gas. After the system passes thecritical point and enters the spin liquid Mott insulatorphase, the RPA gauge field propagator in Coulomb gaugebecomes D c ( ~q, i Ω) = 1 k | Ω || ~q | + (cid:0) χ d + (cid:1) q (19)where ∆ is of the order of the boson gap.Now let us examine the effects of interlayer interac-tions. Similar to the warm-up example in Sec. III, theinterlayer interactions should also obey the gauge invari-ance within each layer. In the present case the gauge fieldstructure is U (1). Apart from electron tunneling betweenlayers, the other most important interlayer interactionshere include the coupling between energy densities of dif-ferent layers of the form (10) and the coupling betweenthe energy density of one layer and the collective exci-tation of the spinon Fermi surface of the other layer ofthe form (11). Since the chemical potential tuned Motttransition is in the universality class of a dilute bose gas,the interaction (10) is already seen to be irrelevant dueto similar reasons as discussed in Sec. III. Also, althoughthe spinons are in their own non-Fermi liquid state at thecritical point, the factor Π( ~q, ω ) obtained by integratingthe spinons out still has the form given by (14) . Again,because the dynamical exponent z = 2, this interactionis also irrelevant.However, due to the presence of a gapless U (1) gaugefield, there is a coupling of the form δ L = X αβ Z dτ d xg αβ ( ∇ × ~a α ) · ( ∇ × ~a β ) (20)In momentum space, this coupling has the form δ L = P αβ R dωd qg αβ q ~a ( ~q, iω ) · ~a ( − ~q, − iω ). At the criticalpoint, ρ s = 0, and comparing (18) and the momentum-space representation of δ L , we see this coupling ismarginal. As for any physics that only involves quan-tities within the same layer, the effect of this coupling ismerely to modify the effective diamagnetic susceptibility.In particular, under the assumption that the interlayerinteractions are weak, this coupling is not able to changethe sign of χ d , so it will not modify the physics quali-tatively. Its most important effect on physics involving different layers may be that it can potentially enhancethe interlayer pairing (see Appendix D). Since in this pa-per we consistently assume we are in a regime away fromany pairing instability, we will also ignore it here and (20)will not modify the physics.One should in principle also consider the couplingsbetween electric currents, spin currents and spin densi-ties between different layers. The most relevant couplingbetween electric currents contains two more derivativescompared to (10), so it is even more irrelevant than thelatter. The most relevant couplings between spin cur-rents and spin densities are all of the form of four-fermioninteraction, which is not relevant for similar reasons asin the case of the orthogonal metal transition.Of course potentially the most important interlayercoupling is through electron tunneling between the dif-ferent layers. We can discuss it within the frameworkintroduced to analyze the same issue for the orthogonalmetal transition. As the exponent α < B. Bandwidth controlled continuous Motttransition
If the electron filling is fixed to be at half, we can ac-cess the bandwidth controlled Mott transition by tuningthe ratio of the electron bandwidth to the interactionstrength, which can be done, for example, by tuning thepressure. The effective field theory of this phase transi-tion is similar to (16), except that now the bosons arerelativistic and described by the following Lagrangian L b = | ( ∂ µ − ia µ ) b | + V ( | b | ) (21)because of the emergent particle-hole symmetry of theboson. This difference has been realized since the studyof the transition between a bosonic Mott insulator and asuperfluid .As in the case of the chemical potential tuned Motttransition, the boson sector is again dynamically decou-pled from the spinon-gauge-field sector, and the transi-tion is therefore of the universality class of the 3D XYmodel. Using this, it is shown that at the quantum criti- FIG. 4. Phase diagram and crossover structure of the band-width controlled Mott transition. The g < g c side correspondsto a spin liquid Mott insulator (MI), and the g > g c side cor-responds to a Fermi liquid (FL) metal. The quantum critical(QC) regime is highly non-Fermi liquid like. In crossing overto the Fermi liquid, the system has to go through an interme-diate marginal Fermi liquid regime (MFL). In crossing over tothe Mott insulator, the system has to go through a marginalspinon liquid (MSL) regime, but this regime will not be dis-cussed in this paper. cal point the system is non-Fermi liquid like and electronspectral function is A c ( ~k, ω ) ∼ ω η ln Λ ω f ω ln Λ ω v F k k ! (22)where η is the anomalous dimension of 3D XY model andthe universal function f is f ( x ) = (cid:18) − x (cid:19) η θ ( x −
1) (23)Fitting this spectral function into the form (2), we find α = − η and z = 1 + , with the understanding that expres-sion such as ω z should be interpreted as ω ln ω . Again,we see (4) is satisfied.In addition, the crossover out of QC regime tothe Fermi liquid regime again involves an intermediateregime. Choosing the boson phase stiffness ρ s as thecharacteristic energy scale of the Fermi liquid, when thetemperature is such that T ≫ ρ s , the system is in its QCregime and displays non-Fermi liquid behaviors. Whenthe temperature is decreased so that T ≪ ρ s , the bosonsbehave as if they already condensed ( i.e Higgsed) , butthe spinon-gauge-field sector does not feel the Higgs ef-fect until the temperature is further lowered to the orderof ρ s . In this intermediate regime, the system behavesas a marginal Fermi liquid (MFL) state that was origi-nally proposed by Varma et al. to describe the optimallydoped cuprates . Only when the temperature is furtherlowered so that T ≪ ρ s does the spinon-gauge system no-tice the the boson condensation and the system behavesas a Fermi liquid. The difference in the powers of ρ s below which scale a Fermi liquid results in the chemicalpotential tuned and bandwidth tuned Mott transitionsreflects that the two transitions are in different univer-sality classes. In this case on the Fermi liquid side the RPA gaugefield propagator in Coulomb gauge is D b ( ~q, i Ω) = 1 k | Ω || ~q | + σ p Ω + ~q P (cid:18) √ Ω + ~q ρ s (cid:19) (24)where the universal function P ( x ) satisfieslim x → P ( x ) ∼ x and lim x →∞ P ( x ) = 1, and σ ∼ e h isa universal conductance. On the Mott insulator side, itbecomes D b ( ~q, i Ω) = 1 k | Ω || ~q | + σ p Ω + ~q Q (cid:18) √ Ω + q ∆ (cid:19) (25)where ∆ is on the order of the boson gap and theuniversal function Q ( x ) satisfies lim x → Q ( x ) ∼ x andlim x →∞ Q ( x ) = 1.Suppose we have a stack of such 2D layers, let us nowexamine the effects of interlayer interactions on the de-coupled fixed point. These interlayer interactions mustbe invariant under the local U (1) gauge transformationwithin each layer. We will follow closely the strategyused in previous sections. Interlayer electron tunnelingis potentially the most important coupling. Because inthis case α = − η <
0, according to the similar argu-ments in the previous sections it is irrelevant. The otherimportant interlayer interactions are again the couplingbetween energy densities on different layers and the cou-pling between the energy density in one layer and thecollective excitations of the spinon Fermi surface in an-other layer, which still has the form of (10) and (11),respectively. However, because this transition is in an-other universality class, the previous argument should bemodified. In particular, because the energy density | b | has scaling dimension 3 − ν , the scaling dimension of g αβ is ν −
3. It is known that ν > for the 3D XY model,so g αβ has negative scaling dimension and (10) is irrel-evant. On the other hand, integrating out the degrees offreedom from the spinons in (11), the most relevant in-teraction we get is again of the form (13). Since we havedynamical exponent z = 1 in this case, the scaling di-mension of coupling constant of this resulting interactionis the same as that of the coupling between energy den-sities, which is negative as discussed above. Therefore,this interaction is also irrelevant.Notice in this case, the coupling of the form (20) issimply irrelevant at the critical point, because there thegauge field propagator (24) is D b ( ~q, i Ω) = 1 k | Ω || ~q | + σ p Ω + ~q (26)For the gauge field the most important fluctuations arethe modes with Ω ∼ q ≪ q , so we can further approxi-mate (26) as D b ( ~q, i Ω) = 1 k | Ω || ~q | + σ | ~q | (27)The coupling of the form (20) contains two spatial deriva-tives, so it is irrelevant. However, because of the struc-ture of (25), as in the case of the chemical potential tunedMott transition, (20) is marginal deep in the Mott insu-lator side. But similar arguments as there indicates thiscoupling does not modify the physics.For similar reasons as in the case of chemical potentialtuned Mott transition, the couplings between charge cur-rents, spin currents and spin densities in different layersare also not relevant. As for interlayer electron tunneling,the critical Fermi surface in this problem has α = − η < V. Interlayer conductivity
In this section we derive the interlayer electric conduc-tivity σ in perturbation theory in the electron tunneling.As is well known, this leads to a formula for the interlayerconductivity in terms of the electron spectral function ineach layer. We then we apply the formula to the QCregimes of the various phase transitions discussed above,and show in all three cases σ → T →
0. This isof course consistent with that the interlayer tunneling isirrelevant at the decoupled fixed point.Suppose the total Hamiltonian of the layered three di-mensional system is H = X a H a + H tunneling (28)where H a is the Hamiltonian for the a th layer that maytake the form of (5), and the tunneling Hamiltonian H tunneling takes the following specific form H tunneling = − t X a,i (cid:16) c † i,a +1 c i,a + h . c . (cid:17) (29)where a labels the layer and i labels the position of thesite on a given layer.As shown in Appendix E, the interlayer DC electricconductivity to the second order of interlayer tunnelingamplitude t is σ = − πN ( ted ) X k Z dω (cid:16) A ( ~k, ω ) (cid:17) dfdω (30) where N is the number of layers, d is the interlayer spac-ing, A ( ~k, ω ) is the electron spectral function on eachlayer, and f ( x ) = e βx +1 is the Fermi-Dirac function.For systems with a critical Fermi surface, in the quan-tum critical regime where the zero temperature correla-tion length ξ is so large that it drops out from the univer-sal function, we can write the scaling form of the spectralfunction (2) as A ( ~k, ω ) = c | ω | αz F c (cid:18) ωT , c k k T z (cid:19) (31)with another universal function F c simply related tothe original universal function F in (2) via F c ( x, y ) = F ( x, y, ∞ ). Applying (31)to (30), we get σ ∼ Z dθ c ( θ ) c ( θ ) T − α ( θ ) z ( θ ) (32)where θ denotes the angular position of a patch on theFermi surface, and the integral is over all these patches.Therefore, as long as α < for each patch, the interlayerelectric conductivity vanishes as the temperature goesto zero. Notice the condition for the interlayer electronhopping to be irrelevant, i.e. α <
0, is stronger than thiscondition, which is of course expected. Both conditionsare satisfied for all three cases discussed above, and thisconfirms our previous assertion that σ → T → − αz > z − > − σ . T (34)Now we apply (32) to the examples discussed in theprevious sections. For all examples, the exponents α and z do not depend on its angular position θ (even thoughthe non-universal constants c and c can depend on θ ingeneral), and (32) simplifies to σ ∼ T − αz (35)For the QC regimes of the chemical potential tunedMott transition (and the related problem of the orthog-onal metal transition) , we have α = − z = 2.Therefore σ ∼ T (36)For the QC regime of the bandwidth controlled Motttransition, we have α = − η and z = 1 + , so up to loga-rithms we have σ ∼ T η (37)0Before continuing, we comment on the in-plane electricconductivity in these QC regimes. For orthogonal metals,as discussed before, because the fermion f carries chargeand it is in its Fermi liquid state, the in-plane electricconductivity is Fermi liquid like. In the QC regime ofthe chemical potential tuned Mott transition, the behav-ior of the in-plane conductivity is non-Fermi liquid like.In particular, this conductivity behaves as ln (cid:0) T (cid:1) at thelowest temperatures according to the conventional slave-particle theories . As for the QC regime in the band-width controlled Mott transition, if g > g c , the in-plane resistivity has a universal crossover from a finite valueto the impurity-induced residual resistivity of the Fermiliquid . VI. Crossovers out of criticalities
As reviewed in Sec. IV, a very interesting feature ofthe continuous Mott transitions is that the crossoversout from the QC regimes to the Fermi liquid regime in-volve an intermediate regime: there is an IFL regime forchemical potential tuned Mott transition and an MFLregime for bandwidth controlled Mott transition. It isclearly interesting to study the interlayer electric trans-port properties in these regimes.As long as at the lattice level the interlayer interac-tions are much weaker than the intralayer interactions,it may still be legitimate to apply (30) to calculate theinterlayer electric conductivity. However, unlike that inthe quantum critical regime where the zero temperaturecorrelation length ξ drops out from the universal functionand we can get the scaling behavior of the conductivitysimply by using the scaling form of the electron spectralfunction A ( ~k, ω ), here we no longer have a simple scalingform for it and we need to calculate it explicitly.This is done by recalling that the physical electron op-erator is written as c iσ = f iσ b i (38)In both IFL and MFL regimes, despite the Mermin-Wagner theorem (stateing that there is no true long-range order at any finite temperature in 2D), as long as T ≪ ρ s , the correlation length and correlation time arestill extremely large, and the bosons behave as if theyalready condense. Therefore, to calculate the electronspectral function, we only need to calculate the spinonspectral function and multiply it by the condensate mag-nitude |h b i| ∼ ( g − g c ) β , where β is the order parameterexponent of the transition.The spinon spectral function is determined by theimaginary part of the spinon Green’s function A f ( ~k, ω ) = − π Im G f ( ~k, iω → ω + iη ) (39)Writing the (real frequency) spinon Green’s function in terms of spinon self-energy Σ f ( ~k, ω ), we get G f ( ~k, ω ) = 1 iω − ǫ f ( ~k ) − Σ f ( ~k, ω ) (40)with the bare spinon dispersion ǫ f ( ~k ) = v F k k + κk ⊥ (41)where v F is a (finite) bare Fermi velocity, and κ is de-termined by the Fermi surface curvature. So the spinonspectral function is given by A f ( ~k, ω ) = − π Σ ′′ f ( ~k, ω ) (cid:16) ω − ǫ f ( ~k ) − Σ ′ f ( ~k, ω ) (cid:17) + (cid:16) Σ ′′ f ( ~k, ω ) (cid:17) (42)where Σ ′ f ( ~k, ω ) and Σ ′′ f ( ~k, ω ) are the real and imaginaryparts of Σ f ( ~k, ω ), respectively. A. IFL in a chemical potential tuned Motttransition
According to (42), to calculate the spinon spectralfunction, we need to calculate the spinon self-energy. Inthe IFL regime, the imaginary frequency self-energy ofthe spinons near a certain patch on the spinon Fermisurface is given by (see Fig. 5)Σ f ( ~k, iω )= v F T X Ω n Z ~q D c ( ~q, i Ω) G (0) f ( ~k − ~q, i ( ω − Ω))(43)where the gauge field propagator D c ( ~q, i Ω n ) takes theform of (18), and the bare spinon propagator G (0) f ( ~k, iω )is given by G (0) f ( ~k, iω ) = 1 iω − ǫ f ( ~k ) (44)The summation is over the bosonic Matsubara frequen-cies Ω = 2 πnT , n = 0 , ± , ± , · · · . FIG. 5. Spinon self-energy. The wavy line represents a gaugeboson with momentum and frequency ( ~q, i
Ω), and the dashedline represents a spinon with momentum and frequency ( ~k − ~q, i ( ω − Ω)).
Due to the structure of the bare spinon propagator,the important region of integral involves q k ∼ q ⊥ ≪ q ⊥ ,so we can ignore the q k dependence in the gauge field1propagator D c ( ~q, i Ω). After this, the integral over q k canbe done and we find the spinon self-energy does not havea singular dependence on k k . This implies, because of theform of the spinon spectral function (42) and the form ofthe interlayer electric conductivity (30), having Σ ′′ f ( ~k, ω )is sufficient to obtain σ .We calculate Σ ′′ f ( ~k, ω ) and find its leading singular partis (see Appendix F)Σ ′′ f ( ~k, ω ) ∼ ( − T √ ρ s , | ω | . T −| ω | , | ω | & T (45)Plugging this result into (42) and (30), we find the lead-ing dependence of σ on the temperature in IFL regimeto be σ ∼ t |h b i| T − (46)This result means in the IFL regime, if the tempera-ture is fixed, when we get closer and closer to the QCregime, the interlayer electric conductivity decreases as( g − g c ) β . On the other hand, if we fix g , which exper-imentally corresponds to fixing the doping, as the tem-perature is lowered, the interlayer electric conductivityincreases as T − , so the out-of-plane transport appearsto be conducting. As discussed in Ref. 12, the in-planeconductivity of the IFL regime is non-Fermi liquid like,which behaves as T − . Combining these and previousresults for the QC regime, this implies on the g > g c side, in the QC regime, the in-plane conductivity is non-Fermi liquid like, while the out-of-plane electric trans-port appears to be insulating. When the temperatureis decreased and the system enters the IFL regime, thein-plane conductivity is still non-Fermi liquid like, butthe out-of-plane electric transport already seems to beconducting but non-Fermi liquid like. If the temperatureis further decreased, both the in-plane and out-of-planeelectric transports behave like a Fermi liquid (see Fig.6). This phenomenon can be viewed as an experimentalsignature of the crossover structure displayed in Fig. 3. FIG. 6. Schematic crossover behaviors of the resistivity withrespect to temperature for chemical potential tuned Motttransition (left) and for bandwidth controlled Mott transition(right). The solid line represents the out-of-plane conductiv-ity and the dashed line represents the in-plane conductivity.
B. MFL in a bandwidth controlled Mott transition
Similar to the chemical potential tuned Mott transi-tion, we again just need the imaginary part of the spinonself-energy to calculate σ . The difference between thesetwo cases is that now the gauge field propagator is givenby (24). Plugging (24) into (43), we find the leadingsingular part of Σ ′′ f ( ~k, ω )Σ ′′ f ( ~k, ω ) ∼ ( − T ln Tρ s , | ω | . T − (cid:16) | ω | + λT ln | ω | ρ s (cid:17) , | ω | & T (47)with λ ∼ O (1) a non-universal constant. To the best ofour knowledge, the result for | ω | . T is new. Combiningthis result, (42) and (30), we get the leading dependenceof σ on the temperature in MFL regime to be σ ∼ t |h b i| (cid:18) T ln (cid:18) Tρ s (cid:19)(cid:19) − (48)If we ignore the logarithmic correction in the aboveresult, just as the case of the IFL regime in the chem-ical potential tuned Mott transition, when the systemgets closer to the QC regime from the MFL regime inthe bandwidth controlled Mott transition with temper-ature fixed, the interlayer electric conductivity becomessmaller and smaller as if it vanishes as ( g − g c ) β . And ifwe decrease the temperature with g fixed, which exper-imentally corresponds to fixing the pressure, the inter-layer electric conductivity increases as T − , so the out-of-plane transport appears to be conducting. As shownin Appendix F, the in-plane electric conductivity in theMFL regime behaves as T − , which is Fermi liquid like.Combining these results and the discussion on the QCregime, we see on the g > g c side, in the QC regime thein-plane resistivity has a universal jump, while the out-of-plane electric transport seems insulating. When thetemperature is lowered and the system enters the MFLregime, the in-plane conductivity already behaves as thatof a Fermi liquid, while the out-of-plane conductivity isconducting but non-Fermi liquid like. If the temperatureis further lowered so that the system is in the Fermi liquidregime, the temperature dependence of the in-plane con-ductivity does not change qualitatively, and the out-of-plane conductivity eventually becomes Fermi liquid like(see Fig. 6). This phenomenon serves as an experimentalsignature of the crossover structure displayed in Fig. 4.We end this section by commenting on the validity ofthe perturbative calculation of σ . We first discuss thecase of the IFL regime. Notice (30) is derived perturba-tively up to the second order of the interlayer tunnelingamplitude t , and (46) shows σ → ∞ as T →
0, so onemay wonder whether the higher order terms in t shouldbe included. However, in the IFL regime, we are workingin a temperature regime T ≫ ρ s and we will not go intoan arbitrarily low temperature. As argued in AppendixG, in this regime the higher order contributions are ex-pected to be indeed small compared to this leading ordercontribution, so the perturbative result (46) is valid.2Now we turn to the MFL regime. Again, in the MFLregime T ≫ ρ s and we will not go to an arbitrarily lowtemperature. As argued in Appendix G, in most param-eter regimes of experimental interests, the perturbativecalculation is expected to be valid as long as the interlayerelectron tunneling amplitude t is small. If the system isextremely close to the quantum critical point, there maybe a narrow window where the perturbative calculationbreaks down, and the temperature dependence of σ is ex-pected to be more singular than (48), but it should notbe more singular than the Fermi liquid form T − . Sincethis window is very narrow, it may not be too significantexperimentally. VII. Discussion
In this paper, we have demonstrated the phenomenonof dimensional decoupling at continuous Mott transitionsbetween a Fermi liquid metal and a Mott insulator in amultilayered quasi-2D system. At low energies, in theMott insulating phase as well as right at the quantumcritical point, the system behaves as a stack of manydecoupled 2D layers, while it behaves as a 3D Fermi liq-uid in the metallic side. Experimentally, for example,this implies the interlayer electric transport will becomeinsulating in the quantum critical regimes of these tran-sitions. We emphasize the importance role that electronfractionalization plays in obtaining dimensional decou-pling, and we also point out, under reasonable assump-tions, this phenomenon cannot occur in a non-Fermi liq-uid obtained near a conventional quantum critical pointthat is associated with a spontaneous breaking of internalsymmetries.By calculating the temperature dependence of the in-terlayer electric conductivity σ induced by electron tun-neling between different layers, we have systematicallyexplored the crossover behavior of the interlayer trans-port in Sec. VI. We find for these continuous Mott tran-sitions the interlayer conductivity vanishes as the temper-ature goes to zero in the QC regimes, which is consistentwith that the interlayer electron tunneling is irrelevantat low energies. Intuitively, this is because an electronwill be split into a boson and a fermion across the tran-sitions. To have an electron tunnel from one layer to theother, both the boson and the fermion have to tunnel col-lectively. However, since the boson is to become gappedacross the transitions, this process is suppressed. One ofthe interesting features of these continuous Mott tran-sitions is the existence of an intermediate regime whenthe system crosses over from the QC regime to the Fermiliquid, and we also derived the scaling behavior of σ inthese regimes and showed σ increases as temperature de-creases. In particular, in the IFL regime of the chemicalpotential tuned Mott transition we find σ ∼ T − , and inthe MFL regime of the bandwidth controlled Mott tran-sition we find σ ∼ (cid:0) T ln T (cid:1) − . This metallic behavior ofinterlayer transport is because of the partial recombina- tion of the boson and the fermion. When the temperatureis low enough so that the system enters the Fermi liquidregime, the boson and the fermion will fully combine andbecome the original electron, so the system behaves as acoherent 3D metal where both the intralayer and inter-layer conductivities behave as T − . Therefore, there willbe a coherence peak in the temperature dependence ofthe interlayer resistivity (see Fig. 6). The position of thepeak is around where the system crosses out from the QCregimes into the intermediate regimes, so the deeper thesystem is in the Fermi liquid side, the higher the tem-perature of this peak. These predictions serve as usefulguidance to compare this theory to experiments.Recently out-of-plane transport of a layered quasi-2D doped organic compound has been studiedexperimentally . This material is suggested to bea candidate of a doped spin liquid, in the sense thatcharge and spin will be separated in the absence ofdoping. As reviewed in Sec. I, this material showsnon-Fermi liquid like transport behavior at ambientpressure, and becomes Fermi liquid like under highenough pressure. So pressure can be viewed to drive thiscrossover from non-Fermi liquid to Fermi liquid, andthe higher the pressure is, the deeper the system is inthe Fermi liquid side. It is found that there is indeed aregime where the in-plane transport is non-Fermi liquidlike, and the out-of-plane transport behaves insulating athigh temperatures but becomes metallic at low temper-atures. The coherence peak of the interlayer resistivityseems to occur at a temperature that is small comparedto the relevant lattice energy scales, and it shifts tohigher temperatures as the pressure is increased. Allthese features agree with the predictions of our theoryon the metallic side qualitatively. Ref. 7 also suggested,similar to the considerations in this paper, that themetallic behavior of out-of-plane transport when thein-plane transport is non-Fermi liquid like is due to therecombination of charge and spin.Finally, we note, as discussed in Sec. V, for a layeredquasi-2D non-Fermi liquid, as long as the electron spec-tral function is singular enough ( α > ), the out-of-planetransport may be metallic. This scenario can occur innon-Fermi liquids obtained near a conventional quantumcritical point where the transition is driven by fluctu-ating local order parameter, and a possible example isdiscussed in Appendix A. However, in this scenario thereis not expected to be a peak in the temperature depen-dence of the interlayer resistivity as long as the systemis in the scaling regime of the transition. Therefore, theobservation made in Ref. 7 is unlikely to fall into thisscenario. VIII. Acknowledgement
We acknowledge helpful discussions with Deban-jan Chowdhury, K. Kanoda, Samuel Lederer, AdamNahum, Michael Pretko and Chong Wang. This work3was supported by a US Department of Energy grantde-sc0008739. TS was also partially supported by a Si-mons Investigator award from the Simons Foundation.
A. An example that does not display dimensionaldecoupling: non-Fermi liquid near anIsing-nematic critical point
In this appendix we provide an example of non-Fermiliquid that does not display dimensional decoupling: thenon-Fermi liquid near an Ising-nematic critical point.Consider an electronic system that has an instabilitytowards the formation of an Ising ferromagnet, whichbreaks the Z symmetry of spin flips. Such a phase tran-sition can viewed as a realization of the “Ising-nematic”transition . Theoretically, the universal physics of theIsing-Nematic transition in 2+1 dimension is believed tobe described by focusing on a pair of patches on the Fermisurface that are parallel to each other. The low-energyeffective Lagrangian is L IN = X s = ± ψ † s (cid:18) η ∂∂τ − is ∂∂x − ∂ ∂y − λφ (cid:19) ψ s + ( ∂ y φ ) + rφ (A1)where ψ s with s = ± are the fermion operators on thetwo parallel patches, and φ is the Ising-nematic order pa-rameter that flips sign under the Z transformation .It is found in the QC regime of this transition, thesystem is very non-Fermi liquid like, and the electronspectral function near this critical point also has the formof (2). As calculated in the framework of a controlledexpansion, in this case α ≈ . z = 3 .Now consider we have a stack of many layers of suchsystems and examine the effects of interlayer interactionson the decoupled fixed point. First consider the interlayerelectric conductivity σ induced by interlayer electron tun-neling. Because α ≈ .
7, the arguments in the main textindicates it is relevant. Furthermore, according to (32),the electrons are not incoherent enough and σ actuallydiverges as the temperature goes to zero.We can also consider the coupling between the orderparameters between different layers. This coupling issymmetric under a global Z transformation for all lay-ers and is thus an allowed perturbation, and it does notinvolve interlayer electron tunneling. It has the followingform X αβ g αβ Z dτ dxdyφ α ( x, y, τ ) φ β ( x, y, τ ) (A2)Following the RG analysis in Ref. 34, the scaling di-mensions of various quantities of interests are [ y ] = − x ] = −
2, [ τ ] = − φ ] = 2, from which we can de-duce that [ g αβ ] = 2. This means this coupling is stronglyrelevant.These results means that the decoupled fixed point isunstable to interlayer interactions in this example, so it does not display the phenomenon of dimensional decou-pling.In fact, under reasonable assumptions, it can be ar-gued that for any non-Fermi liquid obtained near a con-ventional quantum critical point associated with a phasetransition that involves internal symmetry breaking, thecoupling between the order parameters on different lay-ers is relevant. To see this, consider a single 2D layer anddenote the order parameter associated with this transi-tion by O . It is reasonable to assume that at the criticalpoint the susceptibility of the order parameter diverges,which implies the scaling form, hO ( ~k, iω ) O ( ~k, iω ) i ∼ k δ · h (cid:16) ωk z (cid:17) (A3)has δ >
0, where h is a universal function. In terms of δ , the scaling dimension of the order parameter is [ O ] = D − δ , where D is the total scaling dimension of the space-time.Now consider a stack of many such 2D layers, the in-terlayer coupling of order parameters R dτ dxdy O i O j hasscaling dimension − δ <
0, so it is relevant and pre-vents dimensional decoupling. Notice to get this resultwe have assumed the susceptibility of the order param-eter diverges at the critical point, but we cannot provethis must hold for all symmetry-breaking transition in ametallic environment, neither can we find any counterex-ample.Before ending this appendix, we note this statementdoes not imply the theoretic works that treat these sys-tems as a single 2D layer are all incorrect. In fact, as longas the interlayer interactions are much weaker than theintralayer interactions at the lattice level, this treatmentis valid unless one goes to extremely low energies.
B. RG equation of perturbation δ L In this appendix we derive the RG equation of theperturbation δ L , (12). FIG. 7. Feynman diagrams that contribute to the RG equa-tion of δ L to the one-loop order. At tree-level, this perturbation is marginal in two spa-tial dimensions. Up to one-loop order, there are threedistinct Feynman diagrams that contribute to the RGequation of δ L , as shown in Fig. 7. To obtain the RGequation, we will carry out a Wilsonian procedure byintegrating out the fast modes with momenta betweenthe old and new cutoff scales, Λ and Λ e − l , respectively,4where e − l is a scaling factor. We do not put any cutofffor the frequency. The first two diagrams vanish after thefrequency integral is carried out, which, physically speak-ing, is because for non-relativistic bosons there need tobe at least two particles to have any interaction at alland the ground state at the unperturbed critical point isa vacuum with no particle. The third diagram does notvanish, and different g ab ’s do not mix from this diagram.For the purpose of obtaining the RG equation, we can setthe external momenta and frequencies to be zero, thenthe third diagram gives − g αβ (2 π ) Z ΛΛ e − l d k Z ∞−∞ dω iω − k m b − iω − k m b = −C g αβ · l (B1)with C = m b π . Therefore, the one-loop RG equation of g ab is dg αβ dl = −C g αβ (B2)In fact, this RG equation is valid up to arbitrary orderof perturbation, which is again related to the fact thatin this model there need to be at least two particles sothat there is interaction at all and the ground state atthe critical point is the vacuum with no particle . C. Electron spectral function at criticality of thechemical potential tuned Mott transition
In this appendix we calculate the electron spectralfunction at the critical point of the chemical potentialtuned Mott transition.Because the electron vertex is not singularlyenhanced , we can get the electron spectral function byconvolving the boson spectral function A b ( ~k, ω ) ∼ δ (cid:18) ω − ~q m b (cid:19) (C1)and the spinon spectral function A f ( ~k, ω ) ∼ ω (cid:16) λ sin π ω − ǫ f~k (cid:17) + λ cos π ω (C2)where spinon dispersion is taken to be in the form of (41).The electron spectral function is then A ( ~k, ω ) = Z ω d Ω Z ~q A b ( ~q, ω − Ω) A f ( ~k − ~q, Ω) (C3)From the expressions of spectral functions of bosonand spinon, we see in the low-energy and low-momentumregime the important region of integral is ω ∼ Ω ∼ q k ∼ q ⊥ ∼ k k . Then in the spinon spectral function, k k − q k is much larger than all other terms, and we can approxi-mate it by a delta function A f ( ~k − ~q, Ω) ∼ δ ( k k − q k ) (C4) Now plug (C1) and (C4) into (C3), we can do the integraland get A ( ~k, ω ) ∼| ω | f m b ωk k ! (C5)with f ( x ) = (cid:18) − x (cid:19) θ ( x −
1) (C6)Fitting into the scaling form (2), we have α = − z = 2. Notice the electron spectral function (C5) has thesame form as (9), the electron spectral function at thecritical point of a Z orthogonal metal transition. D. Effects on the interlayer pairing of (20)
In this appendix we discuss the effects on the inter-layer pairing of the coupling (20). First consider a cir-cular Fermi surface in a single layer, then the dominatepairing occurs between opposite patches on the Fermisurface. Because these opposite patches carry oppositeelectric currents, the Amperean force between them is re-pulsive, which suppresses pairing . Now suppose wehave many layers coupled by (20), the dominate inter-layer pairing is among patches on different layers thathave parallel norms. If g αβ is positive (negative), themagnetic fields on the α -th layer and on the β -th layerwill favor to have opposite (same) signs. When theyhave opposite (same) signs, the Amperean force betweenpatches on the two layers with parallel norms will be at-tractive (repulsive). Therefore, (20) will enhance inter-layer pairing if g αβ > g αβ < In the rest of this appendix, we will apply an RG anal-ysis to justify the above physical picture. It is convenientto write (20) in the frequency-momentum space δ L = X αβ Z ~q,ω g αβ q · a α ( ~q, iω ) a β ( − ~q, − iω ) (D1)and introduce the momentum modes along the z direc-tion for the gauge fields: a q z ( ~q, iω ) = 1 √ N X α a α ( ~q, iω ) e − iq z z α (D2)Then at the critical point of the chemical potential tunedMott transition or in the spinon Fermi surface phase, thegauge field propagator can be diagonalized in the q z basisas D q z ( ~q, iω ) = 1 k | ω || q | + q ( χ d + E ( q z )) (D3)with the dispersion along the z direction E ( q z ) = P αβ g αβ e iq z ( z α − z β ) .5The coupling between the gauge field and the spinonson the α -th layer should also be properly written in the q z basis as a α ¯ f α f α = √ N X q z a q z ( ~q, iω ) e iq z z α ! ¯ f α f α (D4)As we can see, each q z mode of the gauge field and thespinons on each layer are all coupled, but there is a fac-tor e iq z z α in the coupling constant. Due to the oscillat-ing nature of this factor, different layers may appear tohave opposite “charges” for a given q z mode, which canpotentially induce attractive interlayer four-fermion in-teraction and enhance the interlayer pairing instability.We note that similar phenomenon has been studied inthe context of bilayer composite Fermi liquids .An RG approach that is suitable to study the effect ofpairing in the presence of a gauge field in 2D has been re-cently developed in Ref. 23. It is pointed out, the renor-malization of the BCS channel scattering amplitude, V ,comes from two aspects. The first is the conventionalcontribution arising from renormalizing the thickness ofthe momentum shell around the Fermi surface being con-sidered to zero, which contributes to the beta function of V by an amount − V . The second is from the interac-tions among different patches of the Fermi surface gen-erated by integrating the high momentum modes of thegauge field as the sizes of the patches are renormalizedto zero, and this contributes a positive constant to thebeta function.This RG procedure can be straightforwardly general-ized into our multilayered system. Denote the interlayerBCS scattering amplitude of the fermions on the α -thand the β -th layers by V αβ , the aforementioned first typeof contribution remains to be − V αβ , but for the secondtype of contribution we need to add all q z modes witheach of them multiplied by the oscillating exponentialfactor e iq z ( z α − z β ) . Borrowing the result from Ref. 23,the beta function of V in the weak interlayer couplingregime can be calculated to be dV αβ dl = − ǫ g αβ χ d − V αβ (D5)where ǫ is a positive constant.From this equation we see if g αβ <
0, it suppressesthe interlayer pairing instability, while it enhances thisinstability if g αβ >
0. This confirms the physical pic-ture at the beginning of this appendix. But since g αβ is assumed to be very small, the potential pairing insta-bility will only occur at extremely low temperatures andhence we ignore it. E. Derivation of the formula for interlayer electricconductivity
In this appendix we sketch the derivation of the for-mula for interlayer electric conductivity (30). Starting from the Hamiltonian (28), we replace t by te ieAd where A is the external field. The partition func-tion of the system in the presence of the external fieldis Z [ A ]= Z [ Dc ] e − P α S α + R dτ P iα t ( e ieAd c † i,α +1 c i,α +h . c . ) (E1)where S a is the Euclidean action of the a th layer and thesecond term in the exponent is the action correspondingto interlayer electron tunneling.Suppose this external field induces current j , going tothe frequency-momentum space, the conductivity in thelinear response regime is σ ( ω ) = 1 iω j ( ω ) A ( ω ) = 1 iω δ δA ( ω ) ln (cid:18) Z [ A ] Z [0] (cid:19) (E2)Expanding ln (cid:16) Z [ A ] Z [0] (cid:17) in terms of A , the quadratic termis − t ( ed ) Z dτ A ( τ ) X iα h c † i,α +1 c i,α + h . c . i − ( ted ) Z dτ · Z dτ ′ A ( τ ) A ( τ ′ ) X iα,i ′ α ′ D (cid:16) c † i,α +1 ( τ ) c i,α ( τ ) − h . c . (cid:17) · (cid:16) c † i ′ ,α ′ +1 ( τ ′ ) c i ′ ,α ′ ( τ ′ ) − h . c . (cid:17) E (E3)Assuming different layers are decoupled and expanding tothe leading order of t , the above expression in frequency-momentum space becomes N ( ted ) β X kω ω | A ( iω ) | · G ( ~k, iω ) (cid:16) G ( ~k, iω ) − G ( ~k, i ( ω + ω )) (cid:17) (E4)with the electron Green’s function on each layer G ( ~k, iω ) = Z d xdτ h T { c i ( τ ) c (0) }i e − i~k · ~r i + iωτ (E5)By (E2), σ ( ω ) = K ( iω n → ω + iη ) iω (E6)with K ( iω n ) = N ( ted ) β · X ~kω m G ( ~k, iω m ) (cid:16) G ( ~k, iω n ) − G ( ~k, i ( ω n + ω m )) (cid:17) (E7)Plugging spectral decomposition G ( ~k, iω ) = Z d Ω A ( ~k, Ω) iω − Ω (E8)6into (E6) and taking its real part, we get the interlayerelectric conductivity at frequency ω : σ ( ω ) = − πN ( ted ) · X k Z d Ω A ( ~k, Ω) A ( ~k, ω + Ω) f ( ω + Ω) − f (Ω) ω (E9)Taking the limit ω →
0, we get (30) as the DC interlayerelectric conductivity.
F. Calculations of the imaginary parts of thespinon self-energies in the IFL and MFLregimes, and of the in-plane conductivity in theMFL regime
In this appendix we first provide more details of thecalculations of the imaginary parts of the spinon self-energies in the IFL and MFL regimes, namely, (45) and(47), respetively. Then we also sketch the calculation ofthe in-plane conductivity in the MFL regime.Both self-energies are calculated from an integral of theform Σ f ( ~k, iω )= v F T X Ω Z ~q D ( ~q, i Ω) G (0) f ( ~k − ~q, i ( ω − Ω)) (F1)Whether the result is for chemical potential tuned Motttransition or bandwidth controlled Mott transition de-pends on whether the gauge field propagator is of theform (18) or (24).We first use spectral decomposition to write the aboveintegral asΣ f ( ~k, iω ) = − v F Tπ X Ω Z ~q · Z d Ω d Ω Im D ( ~q, Ω ) A (0) f ( ~k − ~q, Ω )( i Ω − Ω )( i ( ω − Ω) − Ω ) (F2)Now we can carry out the Mastubara frequency summa-tion and getΣ f ( ~k, iω ) = v F π Z d Ω d Ω [ n (Ω ) + 1 − f (Ω )] · Z ~q Im D ( ~q, Ω ) A (0) f ( ~k − ~q, Ω ) iω − Ω − Ω (F3)Performing the analytic continuation to real frequency iω → ω + iη and taking the imaginary part, we getΣ ′′ f ( ~k, ω ) = − v F Z d Ω[ n (Ω) + 1 − f ( ω − Ω)] · Z ~q Im D ( ~q, Ω) A (0) f ( ~k − ~q, ω − Ω) (F4)Now we need to carry out the integral over ~q . To do that,we need to specify the gauge field propagator.
1. Calculation of the imaginary part of the spinonself-energy in the IFL regime
In the IFL regime in the chemical potential tuned Motttransition, the gauge field propagator is given by (18). Asnoted before, the important region of integral involves q k ∼ q ⊥ ∼ Ω ∼ ω ≪ q k , so we can ignore the q k depen-dence of the gauge field propagator. Now the integralover q k can be carried out and we getΣ ′′ f ( ~k, ω ) = − v F π Z d Ω cosh (cid:16) βω (cid:17) sinh (cid:0) β (cid:0) Ω − ω (cid:1)(cid:1) + sinh (cid:16) βω (cid:17) · Z dq ⊥ k | q ⊥ | (cid:16) k q ⊥ (cid:17) + ( χ d q ⊥ + ρ s ) (F5)The integral over q ⊥ is Z dq ⊥ k | q ⊥ | (cid:16) k q ⊥ (cid:17) + ( χ d q ⊥ + ρ s ) = k Ω ρ s ln (cid:16) Ω a | Ω | (cid:17) , | Ω | . Ω c k Ω χ d ( k Ω) , | Ω | & Ω c (F6)where the characteristic frequency scale Ω c = a ( c ) ρ s k √ χ d witha constant a ( c ) that is on the order of unity. Plugging thisresult in (F5), performing the integral over Ω and usingthat in the IFL regime T ≫ Ω c , we get (45).
2. Calculation of the imaginary part of the spinonself-energy in the MFL regime
In the MFL regime of the bandwidth controlled Motttransition, the gauge field propagator is given by (24).Again, the important region of integral involves q k ∼ q ⊥ ∼ Ω ∼ ω ≪ q k , so we can ignore the q k dependenceof the gauge field propagator. After carrying out theintegral over q k and q ⊥ , we getΣ ′′ f ( ~k, ω ) = − v F k π σ ρ s Z d Ω cosh (cid:16) βω (cid:17) sinh (cid:0) β (cid:0) Ω − ω (cid:1)(cid:1) + sinh (cid:16) βω (cid:17) · " ln (cid:18) Ω b Ω (cid:19) ! + Ω b | Ω | (cid:18) π − arctan (cid:18) Ω b | Ω | (cid:19)(cid:19) (F7)where the characteristic frequency scale Ω b = a ( b ) σ ρ s k with another constant a ( b ) on the order of unity. Finally,performing the integral over Ω and using that in the MFLregime T ≫ Ω b , we get (47).7
3. Calculation of the in-plane conductivity in theMFL regime
In order to calculate the in-plane conductivity, weneed to calculate the transport scattering rate of spinons,which is the imaginary part of the spinon self-energy mul-tiplied by an additional factor q ⊥ /k . So from (F4), wesee the transport scattering rate of spinons is γ tr = v F k Z d Ω[ n (Ω) + f (Ω)] · Z ~q q ⊥ Im D ( ~q, Ω) A (0) f ( ~k − ~q, ω − Ω) (F8)As before, because the important region of integral in-volves q k ∼ q ⊥ Ω ∼ ω ≪ q k , we will ignore the q k de-pendence of the gauge field propagator (24). Then theintegral can be done and we find the dominant contribu-tion is γ tr ∼ T (F9)This implies the in-plane conductivity behaves as T − (F10) G. Validity of the perturbative calculations of theinterlayer electric conductivity
In this appendix we discuss the validity of the pertur-bative calculation of the interlayer electric conductivity,which only keeps the leading order terms in t , the in-terlayer electron tunneling amplitude. Notice in generalone needs to consider all types of interlayer interactionsdiscussed in the main text, but here, for simplicity, weonly consider interlayer electron tunneling specified bythe model (28).According to (E3), schematically, each higher orderterm in the perturbative expansion picks up one more t |h b i| G f factor compared to the previous term. Thisimplies, due to the structure of (42), the n th term in theperturbative expansion of σ has the structure t n |h b i| n Σ ′′ n − f (G1)So the ratio of the n + 1th term to the n th term can beschematically written as t |h b i| Σ ′′ f (G2) For the IFL regime of the chemical potential tunedMott transition, we expect (G2) is of the order of t |h b i| /T ∼ t ( g − g c ) β /T . Because T ≫ ρ s ∼ ( g − g c ) ν in the IFL regime, we have t ( g − g c ) β /T ≪ t ( g − g c ) β − ν ) . In the universality class of 2D dilute bosegas β = ν = , so this ratio is much smaller than unity aslong as the system is close to the quantum critical pointand t is small, which means the higher order terms in theperturbative expansion is much smaller than the lead-ing order contribution. Moreover, as shown in Ref. 12,the in-plane electric conductivity scales as T − , so theinterlayer electric conductivity in (46) is much smallerthan the in-plane one as long as the temperature is lowenough. Therefore, it is expected that the perturbativeseries in the IFL regime of the chemical potential tunedMott transition will converge.The issue is a little more subtle in the MFL regime ofthe bandwidth controlled Mott transition, which lies inthe temperature regime ( g − g c ) β η ≪ T ≪ ( g − g c ) β η ,where we have used ρ s ∼ ( g − g c ) ν and the scaling re-lation β = ν (1+ η )2 in the universality class of 3D XYmodel. (G2) in this case is expected to be of the order of t |h b i| / (cid:16) T ln Tρ s (cid:17) . Ignoring the logarithmic correction,the condition under which the perturbative calculation isvalid is that T & t |h b i| ∼ t ( g − g c ) β . So long as the inter-layer electron tunneling amplitude t is very small, we canstill have ( g − g c ) β η > t ( g − g c ) β , then the higher orderterms in the perturbative series are small compared to theleading order one. 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