Quantum oscillations in insulators with neutral Fermi surfaces

Inti Sodemann, Debanjan Chowdhury, and T. SenthilDepartment of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Dated: August 23, 2017)

We develop a theory of quantum oscillations in insulators with an emergent fermi sea of neutralfermions minimally coupled to an emergent U(1) gauge field. As pointed out by Motrunich [1], in thepresence of a physical magnetic field the emergent magnetic field develops a non-zero value leadingto Landau quantization for the neutral fermions. We focus on the magnetic field and temperaturedependence of the analogue of the de Haas-van Alphen effect in two- and three-dimensions. Attemperatures above the effective cyclotron energy, the magnetization oscillations behave similarlyto those of an ordinary metal, albeit in a field of a strength that differs from the physical magneticfield. At low temperatures the oscillations evolve into a series of phase transitions. We provideanalytical expressions for the amplitude and period of the oscillations in both of these regimes andsimple extrapolations that capture well their crossover. We also describe oscillations in the electricalresistivity of these systems that are expected to be superimposed with the activated temperaturebehavior characteristic of their insulating nature and discuss suitable experimental conditions forthe observation of these effects in mixed-valence insulators and triangular lattice organic materials.

I. INTRODUCTION

In the past few years a number of materials that arein close proximity to the metal to Mott insulator tran-sition have come to the forefront as prime candidatesto harbor spin liquid phases, notably the triangular lat-tice organic materials κ−(BEDT-TTF)2Cu2(CN)3 andEtMe3Sb[Pd(dmit)2]2 [2–5]. These materials are chargeinsulators that lack spin order down to the lowest tem-peratures. Theoretically, such phases of matter can beunderstood to arise when the electron is splintered apartinto fractionalized excitations that carry its charge andspin separately [6, 7]. The precise nature of the spinliquid realized in these materials is still contended, butremarkably, the dmit compound remains a thermal con-ductor with a finite intercept of the ratio of heat con-ductivity to temperature down to the lowest measurabletemperatures [4], in spite of displaying clear charge in-sulating behavior. It is believed that a good startingpoint to describe the phenomenology of these organicspin liquids is a state with a Fermi surface of emergentneutral spin-1/2 fermions (dubbed spinons) [8, 9].

In other fascinating recent developments, mixed va-lence insulators such as samarium hexaboride (SmB6)have been seen to display de Haas-van Alphen (dHvA)oscillations in their magnetization in an applied ex-ternal magnetic field. Initially these oscillations wereattributed to the metallic surface state that SmB6 isknown to possess [10, 11]. However subsequent exper-iments have raised the dramatic possibility that thesequantum oscillations are a bulk effect in this electricalinsulator [12, 13], possibly related to other mysteriouslow-temperature anomalies in the thermodynamic andoptical properties [10–12, 14–16]. Inspired by this sit-uation, we recently described a new phase of matter- dubbed composite exciton Fermi liquid - in a mixedvalence insulator with neutral fermionic quasiparticles(coupled to a dynamical U(1) gauge field) that form

a Fermi surface [17]. The composite exciton Fermi liq-uid is sharply distinct from other proposals which eitherposited a Fermi surface of Majorana fermions [18, 19],nearly gapless bosonic excitons [20], or magnetic break-down mechanisms in inverted band insulators [21–23] asdescriptions of the phenomena in SmB6.

In conventional phases of matter that have a con-served charge, charge neutral quasiparticles are neces-sarily bosons. However, in the presence of fractionaliza-tion, neutral fermions can emerge and they can in turnform a Fermi surface under suitable conditions. Thespinon Fermi-surface state is one such example. Anotherclassic example is the half-filled Landau level where acharge neutral fermion (i.e. the composite fermion1)emerges, albeit in a metallic phase [26]. The fermioniccomposite-exciton, recently proposed by us for corre-lated mixed-valence insulators [17] is yet another exam-ple of this phenomenon.

Can electronic solids with a neutral Fermi surface inan external magnetic field show a de Haas-van Alpheneffect? Can they display quantum oscillations in otherproperties (such as the resistivity at a non-zero temper-ature)? This would be remarkable because conventionalinsulating phases respond in a rather innocuous way toapplied magnetic fields: a magnetization that is linearin the external field usually develops typically with anopposite direction to try to screen it (diamagnetism)2,and the resistivity displays a smooth temperature acti-vated behavior. Such behavior is in stark contrast tothe situation in ordinary metals which at low temper-atures display oscillations of magnetization and resis-tivity as a function of the magnetic fields with a fre-quency that diverges as 1/B at low fields, rendering theresponse a non-analytic function of B. Such behavior

1 For arguments on neutrality of composite fermions see [24, 25].2 We are imagining a non-magnetic insulating state.

arX

iv:1

708.

0635

4v1

[co

nd-m

at.s

tr-e

l] 2

1 A

ug 2

017

2

is a fingerprint of the non-perturbative modification inthe low energy spectrum of the metal associated withLandau quantization. It would therefore be striking torealize insulating phases of matter displaying such quan-tum oscillations at weak magnetic fields, which are oftenthought to be fingerprints of metallic behavior.

The possibility of observing quantum oscillations ina Mott-insulator with a spinon Fermi-surface was firststudied in a pioneering work by Motrunich [1], inthe context of the organic material κ-(ET)2Cu2(CN)3.Motrunich emphasized that generically in the presenceof an external magnetic field, an internal magnetic fieldof the emergent gauge field will develop leading to Lan-dau quantization of the spinons. Within this model,Motrunich found, quite remarkably, that the strengthof such an effective magnetic field experienced by thespinons could even be larger than the one experiencedby bare electrons3. Another important property empha-sized in Motrunich’s work was the softening of the stiff-ness of the emergent magnetic field as one goes deeperinto the insulating phase. As we will see, this effectmakes the magnetization response of fractionalized neu-tral fermi seas differ qualitatively from those of met-als once the temperature is lower than the effective cy-clotron energy of the neutral fermions. In particular,this allows for the average value of the emergent mag-netic field to self-consistently adjust itself to lower theenergy. One of the concerns of Motrunich’s work wasthat such reduced stiffness would lead to an enhancedtendency to form non-uniform states analogous to Con-don domains observed in metals [27–33]. As we willargue, we believe that this will not preempt the obser-vation of quantum oscillations in neutral fermi seas inthe semiclassical regime, to the same extent that it doesnot preempt the observation of quantum oscillations inmetals. However, in practice it will change the preciseshape of the oscillatory component of the magnetiza-tion as a function of the external magnetic field. Wewill also show that quantum oscillations will occur inthe finite temperature resistivity but may be hard toobserve except under certain suitable conditions thatwe will describe.

Our study is constructed around a minimal low energyeffective field theory for the neutral fermi sea, and wehave deliberately attempted to keep our results as uni-versally applicable as possible. One of our focus, whichcomplements that of Motrunich, is that we have devel-oped a detailed quantitative theory of the temperaturedependence of the quantum oscillations. In particular,we study how the high temperature regime, that closelyresembles that of a metal, evolves into the low tempera-

3 The strength can be defined operationally in a gauge invariantfashion by considering the amount of Aharonov-Bohm phasethat the spinon acquires in a loop enclosing some given smallarea as compared to that acquired by the electron.

ture regime previously identified by Motrunich. We willshow that at low temperatures the quantum oscillationscan be viewed as an infinite sequence of phase transi-tions between states in which the emergent magneticfield takes different values, and we will see how this ten-dency evolves into the more conventional form of quan-tum oscillations as the temperature is raised. Our re-sults are widely applicable to fractionalized phases withfermi surfaces of neutral fermions with a charge-gap,i.e. for insulating states4. These include the conven-tional U(1) spin liquids with spinon fermi surface andthe composite exciton Fermi liquid in mixed valence in-sulators [17].

Our paper is organized as follows: In Section II wediscuss the general setup and principles needed to com-pute the magnetization of fractionalized neutral fermiseas. In Section III we study quantum oscillations in themagnetization of two-dimensional fractionalized neutralfermi seas, and will show the interesting property thattheir period at low temperatures will in general be dif-ferent from their period at higher temperatures. In Sec-tion IV we will develop the theory of quantum oscilla-tions in three dimensional fractionalized fermi seas, suchas that for the composite exciton Fermi liquid proposedto arise in mixed valence insulators [17]. In contrastto the two-dimensional case, the period of the oscilla-tions in three dimensions does not change with temper-ature. In Section V we will show that fractionalizedfermi seas can display also a form of Shubnikov-de Haasoscillations in the resistivity at finite temperature su-perimposed with the activated behavior characteristicof charge insulators. We close in Section VI with a sum-mary of our results and a discussion on their implica-tions on current and future experiments both in organicspin liquids and in mixed valence insulators. We sum-marize the main formulas for the amplitude and periodof the oscillations in Table I.

II. GENERAL SETUP

A. Low energy field theory

We are considering phases of matter which can emergeout of a Hilbert space where the local degrees of free-dom are electrons that couple minimally to the staticexternal gauge-field A = (A0,A). We assume that thetotal electron number, Q ∈ Z, is a good quantum num-ber and we will refer to this quantum number as thecharge. We are interested in phases with an emergentfermion with no physical charge, coupled to an emer-gent U(1) gauge field, which we denote a = (a0,a), and

4 They are not applicable to the half-filled Landau level which ismetallic.

3

a gapped boson that carries the physical charge Q = 15.Let us denote the neutral fermionic creation and an-nihilation operators by ψ†, ψ (where they satisfy theusual anticommutation algebra) and the bosonic opera-tors by ϕ†, ϕ. Suppose now that the physical electronis represented in terms of these emergent fractionalizedquasiparticles as: c† = ψ†ϕ†. These fractionalized parti-cles must necessarily carry gauge charge under a, whichwe denote as q, because they are non-local 6. Moreover,these excitations carry opposite gauge charge under a,as demanded by locality of the physical electron. With-out loss of generality, we take them to be qψ = 1 andqϕ = −1.

In the above representation, we recover the ordinarymetallic phase when the boson is condensed, 〈ϕ〉 6= 0,while an insulating state with a Fermi-surface of theneutral fermion can emerge when the boson is gapped,〈ϕ〉 = 0. For concreteness, let us write down a La-grangian describing the low energy physics as follows:

L = ψ†(i∂t − a0 −

(p− a)2

2mψ

)ψ

+ |(i∂µ + aµ −Aµ)ϕ|2 − g|ϕ|2 − u

2|ϕ|4 + · · · (1)

where u, g,mψ are effective parameters, and any otherterms that are invariant under gauge transformationsof a are also allowed7. Here the fermions have a finitedensity, and our interest is to describe the response ofthe system in the fractionalized phase to the presenceof an external magnetic field B = ∇ × A. All of thediscussion that follows in this paper can be viewed asa mean field treatment of the physics contained withinthis Lagrangian.

B. Thermodynamics of magnetic systems

Let us briefly review the thermodynamics of magneticfields in matter for the sake of completeness. Thermody-namic quantities that are well behaved and that do notundergo spontaneous symmetry-breaking typically fallinto two categories: conserved quantities and param-eters of the Hamiltonian that can be macroscopically

5 This is one of the simplest patterns of fractionalization thatallows for the emergence of a neutral fermion in a Hilbert spaceof microscopic electrons.

6 Namely ψ† and b† cannot be written in terms of of a finitenumber of electron creation/annihilation operators acting overa finite region of space.

7 Strictly speaking this Lagrangian is appropriate near the crit-ical point associated with boson condensation at fixed bosonnumber. Otherwise a linear in time derivative term for the bo-son would dominate over the relativistic quadratic term at lowenergies. However these details will not affect the mean fieldtreatment that we employ here as the only requirement is thatthe bosons are gapped.

controlled. Faraday’s law implies that one can considerthe average physical net magnetic flux in any directionas a conserved quantity:

Bi =1

V

ˆddx Bi(x). (2)

Thermodynamics of any system in the presence ofmagnetic fields can be described in a microcanonicalmagnetic ensemble, by specifying Bi, or in a magneticcanonical ensemble, by specifying an associated conju-gate variable to Bi, which is Hi. In this paper, we willperform our calculations in the microcanical ensembleand specify Bi. The conceptual advantage of viewing Bias a conserved quantity is that one can consider systemsplaced on strictly periodic geometries with a non-zeroaverage Bi and describe their thermodynamics withoutboundaries, entirely bypassing questions related to theexternal sources of magnetic fields and boundary cur-rents. If u is the energy of the system per unit volume(including the “vaccum” magnetic field energy), s its en-tropy per unit volume, then f , its Helmholtz free energyper unit volume, is:

f(T,B, {λ}) = u− Ts, (3)

where {λ} denotes the set of other non-magnetic ther-modynamic variables. The stability of the thermody-namic state requires the following to be a positive defi-nite matrix:

∂2f

∂Bi∂Bj

∣∣∣∣T,{λ}

. (4)

If we use units in which the vacuum magnetic field en-ergy density is B2/2 (i.e. the vacuum permeability isset to 1), the physical magnetization of a system is givenby:

4πM = B − ∂f

∂B. (5)

Therefore, the task of describing the static magnetic re-sponse of a system reduces to finding the form of the freeenergy as a function of the average magnetic field. Theconjugate field Hi can be obtained as Hi = ∂f/∂Bi

8.

8 In experiments one can only control the external magnetic fieldB0 in which the sample is placed and this generally differs fromthe field B inside the sample. B will ultimately be determinedby details of the sample geometry. In the case of an ellipsoidalsample one controls the following linear combination B0 = (1−n)H+nB, where n ∈ [0, 1] is the demagnetization factor. ThusB0 interpolates between H, at n = 0, in the limit in which theellipsoid becomes a cylindrical rod, and B, at n = 1, in thelimit in which the ellipsoid becomes a flat pancake [27].

4

III. MAGNETIZATION OF TWODIMENSIONAL FERMI SEA

In metals, at low temperatures, f(B) can have a neg-ative second derivative, implying the absence of stablehomogeneous states. Since the average of B is con-served, the system will phase separate into regions oflocally different B, a phenomenon known as Condon-Domain formation [27, 28]. As we will see these in-stabilities are not typical at low temperatures for thefractionalized fermi seas of our interest, in spite of themdisplaying an analogue of dHvA oscillations. Instabili-ties of this sort will be present however over a region offinite temperatures as described in section IV B.

A. Two dimensional metals

Let us briefly recapitulate the magnetostatics of met-als with a single fermi surface in a magnetic field B. Atzero temperature their energy density is:

u(n,B) = nεe(n,B) +χ

2B2, (6)

where n is the electron density, εe the kinetic energy perelectron, and χ the magnetic susceptibility of all back-ground gapped matter including the magnetic energy ofvacuum9. For simplicity we consider spinless electrons.If fermion-fermion interactions can be ignored, and as-suming a parabolic dispersion, we have (~ = e = c = 1):

εe(n,B) = εe(n, 0)

(1 +

νf (1− νf )

ν2

), (7)

εe(n, 0) =πn

me, (8)

where ν = N/Nφ = nΦ0/B = 2πn/B = S/(2πB) isthe filling of the Landau level spectrum and we writeas ν = νi + νf , where νi,f are the integer and fractionalparts of ν, and S = πk2F is the area of the Fermi surface.εe is depicted in Fig. 1(a). Therefore, the electroniccontribution to the magnetization is:

4πMe ≡ −n∂εe∂B

= 2µenνi(1 + νi)

ν− µen(1 + 2νi), (9)

where µe = 1/2me is the Bohr magneton. The jumpin the magnetization, which can be associated with theamplitude of the magnetic oscillations at zero tempera-ture, is given by:

9 Note that in 2D the vacuum contribution contains a prefactorof the film thickness d.

2 4 6 8ν

-1.0

-0.5

0.5

1.0

4 π Men μe

a)

b)0.4 0.6 0.8 1.0 1.2 1.4

2 π BS

1.1

1.2

1.3

1.4

1.5

ϵe (B)ϵe (0)

FIG. 1: (Color online) Electronic contribution to (a) energyand (b) magnetization for a two-dimensional electron gas atzero temperature.

4πδMe ≡4πMe(ν → (n+ 1)+)− 4πMe(ν → (n+ 1)−)

=2µen.

(10)

Notice that the second derivative of the energy is:

∂2εe(n,B)

∂B2= −2εe(n, 0)

(nϕ0)2νi(1 + νi), ν /∈ Z. (11)

This implies that for:

νi(1 + νi) > 2πmeχ =χ

12χe, (12)

the metal will be thermodynamically unstable except atthe discrete values of B for which the Landau levels arecompletely filled ν ∈ Z, since the energy curve has ainfinite second derivative and is locally concave aroundthese singular points as depicted in Fig. 1. In the aboveexpression, χe ≡ 1/(24πme) is the Landau diamagneticsusceptibility of a spinless two-dimensional metal withparabolic dispersion.

B. Two dimensional neutral Fermi sea at T = 0

Since the neutral Fermi sea is a uniform state, at amean field level, we can consider the internal magnetic

5

a)

0.15 0.20 0.25 0.302 π BS

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

u

χϕS2 π

2

0.10 0.15 0.20 0.25 0.30B

0.15

0.20

0.25

0.30

0.35

0.40b

4 5 6 7 8 9 10S

2 π B

-0.04

-0.02

0.02

0.04

4 π Mψχϕ S2 π

b) c)

FIG. 2: (Color online) (a) Contribution of the neutral fermion and the gapped boson in fractionalized neutral fermi sea toenergy. (b) Emergent magnetic field as a function of phyisical magnetic field (horizontal and vertical dashed lines indicatevalues obtained from Eq. (15) and Eq. (17) respectively.). (c) Contribution to net magnetization (orange and green envelopescorrespond to the amplitude obtained from Eq (21)). All curves are at zero temperature and in the two-dimensional case.

field, b = ∇×a, to be uniform. Unlike the physical mag-netic field B, however, the average value of the internalfield b is not an independent thermodynamic quantity10.At small B, the mean field energy of the system is:

u = nεψ(n, b) +χϕ2

(b−B)2 +χ

2B2. (13)

Here εψ is the kinetic energy per particle of the neutralfermions, n is their density, b is the internal magneticfield, χϕ is the magnetic susceptibility of the gappedbosons, and χ is the susceptibility of all backgroundtrivial gapped matter. The form above follows simplyfrom adding the energy of the boson,

χϕ2 (b−B)2, which

has a quadratic form at small fields because the boson isgapped, and the kinetic energy of the neutral fermions.The essential point is that since b is a dynamical de-gree of freedom it will be self-consistently adjusted tominimize the energy, unlike B which is a fixed thermo-dynamic parameter. Therefore, operationally, the taskis to find the minimum of the energy in Eq. (13) withrespect to b at any given B.

Let us define the neutral fermion filling factor withrespect to b as νψ = 2πn/b = S/(2πb), and similarly de-note νψi,ψf the integer and fractional parts of νψ. Noticethat under the approximation of a parabolic dispersionfor the neutral fermions, εψ has the same form as εegiven in Eq. (7) except that me → mψ and ν → νψ. Atsmall enough fields the following condition is satisfied:

νψi(1 + νψi) > 2πmψχϕ =χϕ

12χψ, (14)

10 Strictly speaking b is a conserved quantity at low energies.This is because monopole fluctuations of b are irrelevant inthe infrared [34, 35]. However these fluctuations will alwaysbe present and lead to relaxation of global value of b. We willignore the more subtle issue that this relaxation process mightbe slow due to irrelevance of monopole fluctuations and assumethat on sufficiently long time scales the system relaxes the spa-tial average of b to the global minimum of the free energy.

and when this happens, the local minima of Eq. (13)are always achieved at integer filings of the Landau lev-els of ψ. This remarkable tendency, first identified inMotrunich’s work [1], can be traced back to the exis-tence of cusps in the energy per particle at integer fill-ing as a function of particle number (i.e the cyclotrongaps, see Fig. 2(a)). It tells us that the internal magneticfield, b, will remain locked at a constant value for a finiterange of B so that the neutral fermions fill an integernumber of Landau levels within such range, as depictedin Fig. 2(b). Interestingly, any integer filling at a givenexternal B, describes a metastable state because it isa local minima of the free energy. The ground state isselected from these metastable states as the one havingthe absolute lowest energy. These different metastablestates can thus be labeled by an integer p:

bp = 2πn/p, p ∈ Z, (15)

and their energy is:

up(n,B) = nεψ(n, 0) +χϕ2

(bp −B)2 +χ

2B2, (16)

where we have used the special property that the kineticenergy of any number of completely filled Landau levelsis identical to the kinetic energy at zero field for Galileanfermions. The critical fields at which the ground state ofthe system transitions from filling νψ = p+ 1 to νψ = pis given by:

Bp = πn

(1

p+

1

p+ 1

). (17)

For Bp+1 ≤ B ≤ Bp the fermion filling factor is νψ =p+1. For this range of magnetic fields, the contributionto the magnetization from the neutral fermi sea andthe gapped boson is locally a decreasing function of thephysical magnetic field:

4πM = −χϕ(B − 2πn

p+ 1

). (18)

6

The behavior of this magnetization is depicted inFig. 2(c). Remarkably, the period of the oscillationsis controlled by the bare magnetic field. In other wordsthe oscillations are periodic in 1/B with a period givenby:

∆

(1

B

)=

1

2πn=

2π

S, B � S, T = 0, (19)

where S is the area of the Fermi surface. Let us thus de-note by ν = 2πn/B. Therefore, to find the integer fillingdescribing the ground state, p(ν), at a given magneticfield B, we need to solve the following inequality:

2p(p+ 1)

2p+ 1≤ ν ≤ 2(p+ 1)(p+ 2)

2p+ 3, p ∈ Z, ν ∈ R. (20)

For p � 1 the inequality reduces to: p + O(1/p) ≤ν − 1/2 ≤ p + 1 + O(1/p), and thus can be solved byp ≈ [ν−1/2]. The size of the jump of the magnetizationat the transition from νψ = p+ 1 to νψ = p is therefore:

4πδM ≡4πM(B → B−p )− 4πM(B → B+p ) =

2πnχϕp(p+ 1)

≈ 2πnχϕ

(B

2πn

)2

, p� 1.

(21)

Notice that the neutral fermi sea displays genericallya concave energy as a function of B except at dis-crete points, as illustrated in Fig. 2(a). Therefore, thespinon fermi surface does not suffer from the genericlow-temperature thermodynamic instabilities that weencountered in a metal (compare Figs. 1(a) and 2(a)).The physical picture for why this happens is that the in-ternal magnetic field can adjust itself until the neutralfermions completely fill an integer number of Landaulevels, and when this happens the system enjoys thestability of being able to form a uniform fully gappedstate11.

Since the magnetization of the neutral fermions is lo-cally a decreasing function of B (Fig. 2(c)), one couldsay that the neutral fermion system tends to be differ-entially diamagnetic, unlike electrons which tend to bedifferentially paramagnetic at low temperatures. Thisdistinction will be washed out at higher temperatures.Notice, however, that the magnetization of the neutralfermion is not a strictly decreasing function of B be-cause the curve is piece-wise discontinuous (Fig. 2(c)).

11 This state is technically a chiral spin liquid. It is also easy toargue that this state is stable beyond the simple mean field pic-ture we describe in the main text, because the gauge field fluc-tuations will be gapped at long wavelengths due to the Chern-Simons term induced by the neutral fermions forming an integerquantum Hall state.

C. Two-dimensional neutral Fermi sea at finite T

In this section, we consider the magnetic responseof the system at a finite temperature when the neu-tral fermion cyclotron spacing is much smaller than thetemperature broadening but the temperature is muchsmaller than the fermi energy of the neutral fermionand any other scale associated with the gap of the bo-son, specifically n/mψ � T � b/mψ. To leading orderin small b and B the free energy of the system is:

f = f0(B) +χψ2b2 +

χϕ2

(B − b)2, (22)

where χψ characterizes the diamagnetic coefficient ofthe neutral fermi sea that expresses the energy cost forb to be non-zero (i.e. the analogue of the Landau dia-magnetic susceptibility of a metal) and it is given by(spinless fermions): χψ = 1/(24πmψ), but we will keepit as an unspecified coefficient for generality, and f0 in-cludes the contribution of the background trivial gappedmatter and the magnetic energy of vacuum.

bB =χϕ

χψ + χϕB ≡ αB. (23)

B − bB =χψ

χψ + χϕB = (1− α)B. (24)

Therefore, for |χψ| � |χϕ| the internal field experiencedby the fermions tends to track the external field (α→ 1),while in the opposite limit the neutral fermions experi-ence just a small fraction of the field while the bosons ex-perience almost the full external field (α→ 0). The lat-ter is typically what one expects if the bosons are deep inan insulating phase, as their energy would hardly changein the presence of an effective field, B−b, implying thatχϕ → 0 as we move deeper into the insulator. The oppo-site behavior, χϕ →∞, is what one expects if the bosonis condensed, developing a Meissner effect for the effec-tive field, B − b. Across a continuous phase transitionbetween the insulator and the metal [36, 37], assumingthe transition is of the kind that occurs at fixed bosondensity, one expects that χϕ diverges as one approachesthe metal from the insulating side because in the metalthe boson has condensed. This implies that in the phasetransition between an ordinary metal and the fraction-alized neutral fermi sea, the coefficient α that relatesthe effective field experienced by the neutral fermions,b, to the physical magnetic field, B, starts continuouslydecreasing from 1 at the critical point (see Fig. 3)12.

12 It is worth bearing in mind that the value of this coefficient inthe fractionalized phase is not universal and can be even largerthan 1 under special circumstances, as found by Motrunich dueto the proximity to the uniform flux phase of the spinon fermisea that he considered [1].

7

metal fractionalized fermi sea

↵

1

g

FIG. 3: (Color online) Qualitative behavior of the parame-ter α that controls the effective magnetic field experiencedby the neutral fermions in response to a physical magneticfield across the transition from metal to fractionalized neu-tral fermi surface, driven by a parameter g which controlswhether the boson is gapped (g > 0) or condensed (g < 0).

At small fields and at temperatures higher than the cy-clotron energy, the contribution to the physical magne-tization of the neutral fermi sea depends linearly on theexternal field:

4πM = −

(1

χ−1ϕ + χ−1ψ

)B ≡ −χ0B. (25)

The physical content of this expression is that whicheverof the two components (i.e. the fermions or the bosons)is less responsive in changing its energy in the presenceof an effective magnetic field, will dominate the magne-tization response to external fields. Notice that this isopposite to the case of “stacking” two physical systemsfor which the respective χ’s add up and hence the moreresponsive component dominates. This result is an ex-ample of the well-known Ioffe-Larkin rule for additionof electromagnetic response functions for partons [38].

D. Magnetic oscillations of two-dimensionalneutral fermi sea at finite T

The oscillatory part of the free energy of the neutralfermi sea can be estimated at a mean field level by usingthe classic result from two-dimensional metals [28]:

fosc =χoscb

2

2

∞∑k=1

(−1)k

k2

kτS2πb

sinh(kτS2πb )cos

(kS

b

). (26)

Here S is the area of the neutral Fermi surface, S =πk2F = 4π2n, τ = 2π2T/εF , εF = k2F /(2mψ) and χosc

is a parameter controlling the amplitude of the oscil-lations, which for spinless parabolic fermions is χosc =1/(2π3mψ). This form also holds for arbitrary disper-sions within an effective mass approximation around the

Fermi surface, with corrections from non-parabolicityappearing as higher powers in (b/S). The task is to min-imize the sum of the energies from Eqs. (22) and (26) asa function of b. At low temperatures, namely when thetemperature is lower than the effective cyclotron energyof the neutral fermions (T � ωψ ≡ b/mψ), and at lowb (b � S), the minima of the free energy will be dom-inated by the oscillatory part, because this term willdominate the derivative of the energy due to the diver-gence of its oscillation frequency as 1/b for b→ 0. Thisallows to recover the result found in Section III B, thatthe minima will be pinned at a fixed value of b withina range of B. These correspond to the minima of foscwhich are given by:

bp =S

2πp=

2πn

p, p ∈ Z. (27)

One thus recovers the result of Eq. (15). Therefore thefree energy of the p−th metastable state can be approx-imated at low temperature as:

fp ≈f0 +χψ + χϕ

2(bp − bB)2 +

χosc2b2p

( ∞∑k=1

(−1)k

k2

),

=f0 +χψ + χϕ

2(bp − bB)2 − π2χosc

24b2p,

(28)

where bB was defined in Eq. (23) and f0 = f0 +χ0B2/2

includes the linear in B contribution to magnetizationfrom Eq. (25). Following a similar line of reasoning asin Section III B, one finds the following behavior for theamplitude of the oscillatory part of the magnetizationat low temperatures:

4πδMosc ≈χ3ϕ

(χϕ + χψ − π2χosc/12)2S

2π

(2πB

S

)2

,

=2πnχϕ

(B

2πn

)2

, B � S, T � ωψ.

(29)

At low temperatures (T � ωψ) the period of the oscil-lations can be estimated to be:

∆

(1

B

)≈(

χϕχψ + χϕ − π2χosc/12

)2π

S,

=2π

S, B � S

(30)

where in the second line of the two preceding equationswe have used the relation χψ = π2χosc/12, valid forparabolic fermions, and we have recovered the resultsof Eq. (21) and Eq. (19). Now, at higher tempera-ture (T & ωψ ≡ b/mψ) the oscillatory contribution to

8

the free energy is negligible compared to the smoothpart from Eq. (22). In this case, the minimum of thefree energy will be achieved when b = bB = αB, andgenerically there is a single solution and no multiplemetastable states. In this regime the free energy of theequilibrium state can therefore be approximated as:

f ≈f0 + fosc(b→ bB). (31)

Therefore the functional form of the magnetic oscilla-tions of the neutral fermi sea at higher temperature areessentially the same as that of a metal but in an effectivefield given by bB . Therefore the leading contribution tothe oscillatory magnetization at small b (bB � S) is:

4πMosc ≈χoscαS

2

∞∑k=1

(−1)k+1

k

kSτ2πbB

sinh(kSτ2πbB

) sin

(kS

bB

)bB � S, εF � T & ωψ.

(32)

An interesting feature of the oscillations in two dimen-sions is that the period of the oscillations has differentbehavior at low temperatures (T � ωψ) than at hightemperatures (εF � T & ωψ). While at low temper-atures we found a period given in Eq. (30), at highertemperatures we have

∆

(1

B

)=

2πα

S⊥, B � S, T & ωψ. (33)

On the other hand, at high temperature, the amplitudeof the oscillations is therefore given by,

4πδMosc ≈χoscαS

∞∑k=1

(−1)k+1

k

kSτ2πbB

sinh(kSτ2πbB

)bB � S, εF � T & ωψ.

(34)

Notice that if we were to naively extrapolate the ampli-tude of the oscillations to the low temperature regime,we would obtain the following amplitude: 4πδMnaive

osc ≈χoscαS log(2), B � S, T � bB/mψ, which would over-estimate the correct result given in Eq. (29). Similarly ifwe extrapolate the low temperature result to high tem-peratures we would overestimate the result of Eq. (32).This motivates a simple functional form that interpo-lates between the two regimes and that captures rea-sonably well the crossover region:

1

4πδMosc≈

1

4πδMosc(T � ωψ)+

1

4πδMosc(T & ωψ),

(35)

a)

b)

0.06 0.08 0.10 0.12 0.142 π bBS

0.06

0.08

0.10

0.12

0.14

2 π bS

0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.202 π bBS

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

u(B)u0

FIG. 4: (Color online) (a) Energy of three-dimensional frac-tionalized neutral fermi sea as a function of magnetic field(the scale is u0 = (χϕ + χψ)(S⊥/2π)2) (b) Internal field bas function of the rescaled external field bB = αB. Dashedhorizontal and vertical lines correspond to the values ex-pected from Eqs. (40) and (42) respectively. Both curveswere computed from direct numerical minimization of themodel described in Section IV B, and parameters were chosenas χosc/(χϕ + χψ) = 0.1 and effective temperatures τ = 0.1(green) and τ = 0.5 (blue). In the energy, the smooth back-ground term associated with Eq. (25) is not included.

where the formulas for the right hand side are given inEqs. (29) and (32) respectively. A simple approximationis to keep the leading harmonic in the oscillatory part,as is done in the celebrated Lifshitz-Kosevich formulafor metals, to obtain

1

4πδMosc≈ 1

χϕS2π

(2πBS

)2 +1

χoscαSSτ

2πbB

sinh(

Sτ2πbB

) . (36)

IV. MAGNETIZATION OF THREEDIMENSIONAL FERMI SEA

There are several important qualitative differencesbetween three and two-dimensions that we now wishto emphasize. First of all, gauge-field fluctuations areweaker in three-dimensions and hence one expects thekind of mean field treatment that we have outlined tobe more reliable. On the other hand, in three dimen-sions the spectrum of the fermions in a magnetic field

9

is that of Landau bands that disperse along the di-rection of the magnetic field instead of Landau levels,which leads to important quantitative differences withrespect to the two-dimensional case. In spite of thisdifference, the analysis in three-dimensions parallels inseveral ways that in two dimensions. For example, wewill find in three dimensions the same tendency for theinternal magnetic field to remain pinned at certain dis-crete values at ultra-low temperatures, in spite of thepresence of Landau bands rather than Landau levels.Hence, the ultra-low temperature quantum oscillationswill become a series of phase transitions between differ-ent metastable states, similar to two-dimensions. Simi-larly, the high temperature behavior of the oscillationswill again resemble those of an ordinary metal experi-encing an effective magnetic field bB = αB; an impor-tant qualitative difference with two-dimensions is thatthe period of the oscillations will not change with tem-perature.

A. Three dimensional neutral Fermi sea at finite T

Consider a three dimensional fractionalized Fermi sea.For generality, we assume an anisotropic dispersion witheffective mass mz along the z−axis and m⊥ in thexy−plane, and take the magnetic field to act along thez−direction. Now at temperatures that are higher thanthe transverse cyclotron energy (εF � T � b/m⊥), thefree energy has the same form as in Eq. (22)

f = f0 +χψ2b2 +

χϕ2

(B − b)2, (37)

except that the coefficients are different (for spinlessparabolic fermions χψ = νF /(12m2

⊥) = n/(8m2⊥εF ),

νF = 3n/(2εF ), n = ((2εF )3mzm2⊥)1/2/(6π2)). Once

again, the minimum of the energy is achieved at thesame field and the contribution to the magnetizationfrom the partons has the same form,

bB =χϕ

χψ + χϕB ≡ αB,

4πM = −

(1

χ−1ϕ + χ−1ψ

)B ≡ −χ0B.

(38)

Now at lower temperatures, the oscillatory part of thefree energy becomes important and we will have an extracontribution of the form [27, 28],

fosc =χoscb

2

2

(2πb

S⊥

)1/2

×

∞∑k=1

(−1)k

k5/2

kτS⊥2πb

sinh(kτS⊥2πb )

cos

(kS⊥b∓ π

4

),

(39)

where χosc = (mzεF )1/2/(4π2m⊥), S⊥ is the area ofthe cross-section of the Fermi surface perpendicular tothe magnetic field, and the −(+) signature is for anelectron-like (hole-like) dispersion in the direction of thefield, namely for mz > 0 (mz < 0). Much like in metals,to derive this expression one does not need to assumethat the dispersion of the fermions is exactly parabolic.More generally, the dominant contribution to the fielddependence of the free energy at small fields (b � S⊥)arises from the cross sections of extremal area of thefermi surface that are orthogonal to the applied mag-netic field. One assumption is that there are no non-trivial Berry surface terms, but otherwise the formulacan be applied if mz is understood to arise from thecurvature of the dispersion near a given extremal crosssection and m⊥ is understood as the cyclotron mass forstates near such a cross section. The mathematical de-tails of the derivation in this more general case parallelidentically those for a conventional metal provided inRef. [27].

Similar to the situation in tw dimensions, at low tem-peratures (T � b/m⊥) and small fields (b � S⊥) thederivative of the free energy will be dominated by the os-cillatory part, due to the diverging oscillation frequencyas 1/b. Therefore the minima of the free energy willbe controlled by the oscillatory part, which is indepen-dent of B. Therefore we encounter again a multiplicityof metastable states associated with the different localminima of the oscillatory part of the free energy, whichare approximately given by:

b ≈ bp ≡S⊥

2πp± π/4, p ∈ Z. (40)

This staircase behavior of the internal field is depicted inFig 4(b) where we show the explicit numerical solutionof a simplified model that includes a single harmonic ofthe oscillatory free energy described in Sec. IV B. On agiven metastable state, the free energy can therefore beapproximated as,

fp ≈f0 +χψ + χϕ

2(bp − bB)2 − cχosc

2b2p

(2πbpS⊥

)1/2

,

(41)

where c is a pure number which corresponds to theabsolute value of the minimum of the periodic func-

tion∑∞k=1

(−1)kk5/2 cos

(kS⊥b ∓

π4

), which is approximately

c ≈ 0.87. The critical field at which the system switchesfrom the (p+ 1) to the p metastable state can be foundby solving for fp = fp+1, and is found to be,

bB(p) ≈ bp + bp+1

2−cχosc

(2πS⊥

)1/2χψ + χϕ

b5/2p − b5/2p+1

2(bp − bp+1). (42)

10

a)

b)

12 14 16 18 20 22 24

S2 π bB

-2.0

-1.5

-1.0

-0.5

0.5

1.0f ′′osc(B)

24 26 28 30 32

S2 π bB

-0.00001

-5.×10-6

5.×10-6

0.00001

4 π M

5 10 15 20 25 30

S2 π bB

-0.015

-0.010

-0.005

0.005

0.010

0.015

0.020

4 π M

S

2⇡bB

FIG. 5: (Color online) (a) Oscillatory magnetization of athree-dimensional fractionalized neutral fermi sea as a func-tion of magnetic field (the scale is 4πM0 = χϕ(S⊥/2π)). Thegreen and yellow envelopes correspond to the amplitude ex-pected from the simplified formula of Eq. (36). The insetis a zoom-in to the small field region illustrating the goodagreement with the formula. (b) The second derivative ofthe oscillatory component of the free energy with respect tothe effective field bB = αB. The curve shows clearly how thedelta function-like spikes in the second derivative, occurringat the discontinuous jumps of the magnetization, transforminto the regions where the second derivative is smooth butnegative where instabilities might appear (see discussion inSection IV B). Both curves were computed from direct nu-merical minimization of model described in Section IV B,and parameters were chosen as χosc/(χϕ+χψ) = 0.1 and ef-fective temperature τ = 0.5. The smooth background termassociated with Eq. (25) is not included.

For bB(p) < bB < bB(p− 1), the internal magnetic fieldwill remain pinned at the value b = bp, and the contri-bution to the physical magnetization coming from theboson and the neutral fermion, including their linearcontribution described in Eq. (25), is

4πMp ≈ χϕ(bp −B). (43)

It is interesting to note that the differential susceptibil-ity of the Fermi sea plus the charged gapped-boson isdiamagnetic and coincides with the susceptibility of thecharged boson in the bare external field B. The physicalpicture behind this phenomenon in the low temperatureregime is that when the system is in the p-th metastable

a)

b)

0.5 1.0 1.5 2.0τ

0.05

0.10

0.15

0.20

4 π δMosc0.5 1.0 1.5 2.0

τ

-15

-10

-5

log(4 π δMosc)

0.5 1.0 1.5 2.0τ

0.002

0.004

0.006

0.008

0.010

4 π δMosc

FIG. 6: (Color online) (a) Amplitude of the magnetizationoscillations in fractionalized neutral fermi surfaces as a func-tion of dimensionless temperature τ . The blue dots corre-spond to direct result from numerical study of model de-scribed in Section (IV B) and the red line is the behaviorexpected from the Eq.(36). (b) Comparison of the ampli-tude of the fractionalized neutral fermi surface (blue) from(a) with that expected for a conventional metal (orange)with the same effective mass and subjected to the same ef-fective magnetic field strength (to generate the orange plotwe simply use the formula valid for T � bB/m⊥ but plot itover the entire temperature range). Parameters were chosenas χosc/(χϕ + χψ) = 0.1 and effective field 2πbB/S⊥ = 0.1.

state, the internal field and the fermionic energy becomepinned and do not change with small changes of the ex-ternal magnetic field, leaving the energy of the boson asthe only changing part in response to the external mag-netic field that contributes to the magnetization. Thebehavior of the magnetization is depicted in Fig 5(a),obtained from an explicit numerical solution of a sim-plified model described in Sec. IV B. The amplitude ofthe magnetization jump between adjacent metastablestates (i.e. amplitude of oscillations) can be estimatedto be

4πδM ≈ χϕS⊥2π

(2πbBS⊥

)2

, bB � S⊥, T � bB/m⊥,(44)

which is similar to that found in Eq. (29) in the two-dimensional case. Interestingly, and in sharp contrastto the two dimensional case, the period of the oscilla-

11

tions at low temperatures is still governed by the aver-age field bB , the same that as we shall see governs theoscillations at higher temperature. The period of theoscillations can be obtained from Eq. (42), by comput-ing the range of 1/B over which the solution remains inthe p-th metastable state. Following this, at low tem-peratures, one finds that the oscillatory portion of themagnetization will be periodic as a function of 1/B witha period given by

∆

(1

B

)=

2πα

S⊥, bB � S⊥, T � bB/m⊥, (45)

namely the periodicity is that of a metal but in a fieldreduced by the factor α = χϕ/(χψ + χϕ).

Let us now consider the behavior of the oscillationsat higher temperatures (εF � T & bB/m⊥). Herethe contribution to the free energy from the oscillatorycomponent is negligible, and therefore the energetics ofthe internal field is dominated by the smooth part ofthe free energy given in Eq. (37). As a consequence,there is generically a single equilibrium state and notthe plethora of metastable states realized at lower tem-peratures. Similar to the two dimensional case, the freeenergy of the equilibrium state can therefore be approx-imated as

f ≈f0 + fosc(b→ bB). (46)

Therefore the functional form of the magnetic oscilla-tions of the neutral fermi sea at higher temperatures areessentially the same as that of a metal but in an effectivefield given by bB . Therefore the leading contribution tothe oscillatory magnetization at small b (bB � S) is

4πMosc ≈χoscαS⊥

2

(2πbBS⊥

)1/2

×

∞∑k=1

(−1)k+1

k3/2

kSτ2πbB

sinh(kSτ2πbB

) sin

(kS

bB∓ π

4

),

bB � S, εF � T & ωψ.

(47)

As in two-dimensions, we can propose an interpolationbetween the low and intermediate temperature regimesfor the amplitude of magnetic oscillations following theformula (35). A simple approximation is to keep theleading harmonic in the oscillatory part, as is done forthe Lifshitz-Kosevich formula of metals, to obtain

1

4πδMosc≈

1

χϕS⊥2π

(2πbBS⊥

)2 +1

χoscαS⊥

(2πbBS⊥

)1/2 τS⊥2πbB

sinh(τS⊥2πbB

) .

(48)

In Fig. 6 we illustrate the behavior of the amplitude ob-tained from an explicit numerical solution of the modelwith a free energy with a single harmonic described insection IV B. We see that Eq. (48) captures the lowand high-temperature behavior accurately and providesa good match for the crossover region as well.

Interestingly, in three dimensions, the period of theoscillations at higher temperatures coincides with thatat low temperatures (Eq. (45)), and can be read off fromEq. (47) to be

∆

(1

B

)=

2πα

S⊥, bB � S⊥. (49)

We would like to conclude this section by briefly dis-cussing the impact of disorder. The impact of weakshort range impurities in the amplitude of oscillations isqualitatively similar to that of temperature [27]. Colli-sions off impurities will broaden the energy levels of thefermions on a scale of the order of the scattering rate∼ 1/τimp. Therefore, at a mean field level, heuristicallythe impact of disorder can be captured by replacing thetemperature in all of our discussion with ∼ T + 1/τimp.More specifically, the impact of impurities can be cap-tured by adding an additional suppression to the ampli-tude of oscillations in the form of a Dingle-type factor,

∼ exp

(−2πm⊥bτimp

), (50)

where τimp is the quantum lifetime of the neutralfermion as a result of scattering off impurities. Thusa simple way to estimate the impact of disorder is byabsorbing this Dingle factor into the coefficient χosc →χosc exp (−2πm⊥/(bBτimp)) that controls the amplitudeof the oscillations in our previous discussions. One in-teresting consequence of including this factor occurs inthe two-dimensional systems discussed in Section III D.Following Eq. (30), we expect that the period of oscilla-tions at low temperatures in two-dimensions can dependon the strength of disorder and magnetic field via thisfactor.

There is an interesting consequence of disorder be-yond the mean field level in two-dimensions. As evenfeatureless disorder is expected to couple like a ran-dom field to the energy density, following general argu-ments [39–41], the first order transitions that we haveencountered at low temperatures in two-dimensions willbe rounded into continuous transitions. However, if dis-order is weak, the rounding will be mild and the transi-tions could in practice appear to be essentially discon-tinuous.

12

a) b) c)

0.0 0.2 0.4 0.6 0.8 1.0α

45

50

55

60

65

70

S2 π bB

0.2 0.4 0.6 0.8 1.0α

10

20

30

40

50

60

70

S2 π B

20 30 40 50 60 70

S2 π bB

0.2

0.4

0.6

0.8

τFully Stablecontinuous

Fully Stablecontinuous

Generically Stablediscontinuous

unstableunstable

Generically Stablediscontinuous

Generically Stablediscontinuous

Fully Stablecontinuousunstable

FIG. 7: (Color online) Three kinds of behavior: “generically stable/discontinuous” regions have discontinous magnetizationas function of B field (see Fig. 5(a)) but the only thermodynamic instabilities occur at the isolated values of magneticfield associated with discontinuities; “fully stable continuous” regions have continuous magnetization and no thermodynamicinstabilities; “unstable” regions have thermodynamic instabilities over finite ranges of magnetic fields (although not for everymagnetic field, as stable and unstable regions alternate as the magnetic field is swept). (a) Regions at fixed temperatureτ = 0.2 as a function of the parameter α = χϕ/(χϕ + χψ) controlling the effective field experienced by the neutral fermionsbB = αB. (b) same as (a) but plotted as a function of the physical magnetic field B instead of bB . (c) Regions as a functionof effective temperature τ and bB for fixed α ≈ 0.9. All plots are generated using the ratio χosc/χψ = 3

√2 for spinless

parabolic fermions and in the limit χ → 0 to allow visualization of the unstable regions. In the more realistic limit χ → ∞the unstable region will shrink to become an increasingly thin sliver, but the generically stable/discontinuous region willremain largely unchanged.

B. Intermediate temperature instabilities inneutral fermi seas

Metals in two and three dimensions have low tem-perature instabilities towards phase separation. As wehave seen in the previous two sections, at low tem-peratures fractionalized neutral Fermi seas find a wayto circumvent these generic instabilities by adjustingtheir internal magnetic fields to find local minima ofthe free energy which remain thermodynamically sta-ble. At elevated temperatures neither metals nor frac-tionalized Fermi seas will display instabilities simply be-cause the oscillatory component of the magnetizationwill be exponentially suppressed. There remains to beexplored the intermediate temperature regime at whichthe crossover between these two kind of behaviors hap-pens. This will be the subject of discussion in this sec-tion.

We will begin describing a direct numerical analysisof this intermediate regime and will later provide ana-lytic results. We consider a simplified model with onlythe leading harmonic of the oscillatory part of the freeenergy in three dimensions. We therefore use the fol-lowing total free energy of the system (including thebackground matter and vacuum energies):

f ≈(χ+

χϕχψχϕ + χψ

)B2

2+χψ + χϕ

2(b− bB)2

− χoscb2

2

(2πb

S⊥

)1/2 τS⊥2πb

sinh( τS⊥2πb )

cos

(S⊥b∓ π

4

).

(51)

Here the first term is the explicit expression for f0,

which includes the energy of the vacuum and back-ground matter in the term proportional to χ. The formabove is enough to capture the essential physics even atlow temperatures, but more so at intermediate temper-atures since the exponential temperature suppression ismore pronounced for the higher harmonics.

The task is to numerically minimize Eq. (51) as afunction of b for fixed B. Typical numerical solutionsare depicted in Figures 4 and 5. Figure 4(b) shows thetypical staircase behavior of the internal magnetic fieldas a function of external field. As the temperature is in-creased this staircase smoothens and transforms into astraight line as evidenced in the low field portion of thegreen curve in Fig. 4(b). Figure 5(b) displays the be-havior of the magnetization that results from this stair-case. The magnetization is a discontinuous function ofexternal field at low temperatures and transforms intoa continuous function as temperature is increased. Thelow temperature discontinuities can be thought of asfirst order phase transitions. In particular Fig. 5(b) dis-plays the second derivative of the free energy withoutthe f0 smooth background (see description of f0 belowEq. (28)). We notice that the second derivative has neg-ative delta-function-like spikes at the fields for which themagnetization jumps discontinuously. Beyond a criticalfield (or temperature) these delta function spikes disap-pear and become regions with negative second deriva-tives of the free energy (Fig. 5(b)). As discussed in theintroduction, a negative second derivative of the freeenergy indicates a thermodynamic instability towardsphase separation.

Thus we see that the fields at which the first ordertransitions happen are expected to turn into the regionsat which thermodynamic instabilities occur over a finite

13

a)

b)

Fully Stablecontinuous

Generically Stablediscontinuous

unstable

unstable

Generically Stablediscontinuous

Fully Stablecontinuous

0.2 0.4 0.6 0.8 1.0α

0.05

0.10

0.15

0.20

0.25

τ

0.2 0.4 0.6 0.8 1.0α

0.05

0.10

0.15

0.20

0.25

τ

FIG. 8: (Color online) (a) Dimensionless temperature τ andα = χϕ/(χϕ + χψ) phase diagram of different regions for a

constant average internal effective field 2πS⊥bB

= 55. The dots

joined by solid lines correspond to the regions determinedfrom direct numerical study. The blue and orange dashedlines are the boundaries obtained from solving exactly theinequalities appearing in Eqs. (53) and (56) and the greenand red dotted lines are obtained from the approximationsdescribed in Eqs. (55) and (58). (b) Analogous to figure (a)

but holding the physical magnetic field fixed at 2πS⊥B

= 45instead of the effective field bB . All plots are generated usingthe ratio χosc/χψ = 3

√2.

region of fields, as parameters are varied (i.e field isdecreased or temperature is increased). In order for athermodynamic instability to occur, the full free energymust have a negative second derivative with respect tothe external field, including the magnetic energy of vac-uum. Thus, the criterion for thermodynamic stabilityis:

∂2f

∂B2> 0, (52)

where we imagine that we have explicitly solved for b asa function of B while taking the derivatives. Notice thatthis second derivative will contain χ, which accounts forthe energy of vacuum as well as the background trivialmatter. Therefore, the regions over which instabilitiesoccur are dictated by the somewhat extrinsic parameterχ.

To facilitate numerical identification, we perform the

numerical analysis in the limit χ → 0, so that thereis no background helping stabilize the uniform state13,and will discuss later on the more realistic case of largeχ. Phase diagrams depicting the behavior as a functionof the different dimensionless parameters of the prob-lem are presented in Figs. 7 and 8 in the limit of χ→ 0.Figure 8 shows the three kinds of regimes present in theproblem. At high temperatures we have the fully sta-ble quantum oscillations regime that resembles that ofa metal but in a reduced effective field bB = αB. Inthis regime the magnetization curve is continuous andthe stability criterion in Eq.(52) is always satisfied. Welabel this regime as “fully stable continuous” behaviorin the plots. At low temperatures we have a regime inwhich the internal gauge field remains “pinned” at cer-tain values as the external field is swept giving rise tothe staircase pattern described in Section IV. In this lowtemperature regime instabilities occur at isolated valuesof the field bB for which the system transitions betweendifferent metastable states, giving rise to discontinuousjumps in the magnetization. We label this region in theplot as “generically stable discontinuous” region. Fi-nally there is the intermediate region between these tworegimes where the stability condition in Eq. (52) is vi-olated over finite ranges of magnetic field bB , at whichphase separation will occur. Notice that generically thesystem will have an alternation of regions that are sta-ble and unstable as the magnetic field is swept in thisregime. We label this region “unstable” in the plots.

The limit χ→ 0 makes the size of the unstable regionas large as possible; however this exaggerates its size incomparison to the expected realistic behavior. We willnow provide a quantitative description of the instabilityboundaries that can be used for more realistic estimates.The internal magnetic field experienced by the neutralfermions b will be pinned whenever the second derivativeof the smooth part with respect to b is smaller than thesecond derivative of the oscillatory part. Thus discon-tinuous behavior can be estimated to occur wheneverthe following inequality is satisfied (assuming low fieldsb� S⊥),

χϕ + χψ < 2π2χosc

(S⊥2πb

)3/2 τS⊥2πb

sinh( τS⊥2πb )

. (53)

The region bounded by this inequality is depicted inFig. 8 as a dashed blue line. For the above to be satis-fied, one needs that the following number be less thanO(1):

13 We are restricting to the natural case of χ being strictly positiveotherwise the background “vacuum” would have instabilities ofits own.

14

χϕ + χψ2π2χosc

(2πbBS⊥

)3/2

< O(1). (54)

One can then replace xsinh x ∼ e−x at x & 1 to obtain a

simple closed form expression in order to estimate thecritical temperature at which discontinuities appear,

T1 ≈bB

2π2m⊥log

(max

[1,

2π2χoscχϕ + χψ

(S⊥

2πbB

)3/2])

,(55)

which is depicted as a dotted green line in Fig. 8 thatcaptures qualitatively well the trends and values of thenumerical solution. The second temperature scale isthat at which the second derivative with respect to Bbecomes negative and generic instabilities appear when-ever the following inequality is satisfied:

χ+χϕχψχϕ + χψ

<

2π2α2χosc

(S⊥

2πbB

)3/2 τS⊥2πbB

sinh( τS⊥2πbB

).

(56)

To obtain a solution to this inequality one needs thatthe following is satisfied:

χ+χϕχψχϕ+χψ

2π2α2χosc

(2πbBS⊥

)3/2

< O(1). (57)

Using a similar approximation scheme, one obtains:

T2 ≈bB

2π2m⊥max{T1,

log

(max

[1,

2π2α2χoscχ+

χϕχψχϕ+χψ

(S⊥

2πbB

)3/2])}.

(58)

The appearance of T1 in the above inequality comesfrom the fact that in order to use the approximationb ≈ bB = αB one has to assume that the critical temper-ature derived from the inequality (56) is larger than T1.Therefore, T2 should be taken as the largest value be-tween T1 obtained from inequality (53) and the one ob-tained from the inequality (56). As one approaches themetal to insulator transition χϕ → ∞, α → 1, bB → Band other quantities such as χψ and χosc can be assumedto remain non divergent within mean field approxima-tion. In this limit T1 → 0 whereas T2 becomes the valuethat dictates the temperature for the Condon-domainformation instability in ordinary metals:

limα→1

T2 ≈

B

2π2m⊥log

(max

[1,

2π2χoscχ+ χψ

(S⊥

2πB

)3/2])

.

(59)

This occurs only for fields satisfying

χ+ χψ2π2χosc

(2πB

S⊥

)3/2

< O(1). (60)

The maximum as a function of B of Eq. (59) is achievedwhenever

Bmax ≈S⊥2πe

(2π2χoscχ+ χψ

)2/3

∼ 6− 30 Tesla, (61)

where e is Euler’s constant, and where we used thetypical values of the diamagnetic response of metalsχosc ∼ χψ ∼ 10−6χ and typical metallic densitiesn ∼ 1028−1029m−3. This leads to the following estimatefor the maximum temperature for the phase separationinstability in a typical metal

limα→1

Tmax2 ≈ 3εF4π2e

(2π2χoscχ+ χψ

)2/3

∼ 10−5εF ∼ 0.4− 3 Kelvin.

(62)

for εF ∼ 2 − 10 eV. These numbers are in good agree-ment with the parameters at which the Condon domaininstabilities are typically seen [29–33].

Unlike the physical magnetic field in metals, the frac-tionalized neutral Fermi surfaces can generically relaxthe global internal gauge field, reducing the region ofinstabilities to isolated values of the magnetic field atlow temperatures, as we have described. One of themain differences between the phenomena in fractional-ized states and that in metals is that the range of ef-fective fields bB over which the discontinuous behavioroccurs will become wider as we enter further into theinsulating regime (α → 0) as indicated by Eq. (54).In both cases however the typical temperature for theappearance of instabilities is controlled by the effectivecyclotron frequency T ∼ bB/m⊥ of the fermions up tologarithmic corrections. Phase diagrams as a function ofdimensionless temperature τ and α are shown in Fig. 8in the limit χ → 0. This exaggerates the region forinstabilities but allows to visualize it more easily.

Let us discuss the more realistic limit χ→∞ in frac-tionalized neutral Fermi surface states. The boundarygiven by T1 will remain essentially unchanged becauseit is independent of this parameter. On the other handthe boundary given by T2 will shrink and become athin sliver surrounding the region given by T1. The re-gion bounded by T1 corresponds to the behavior of thefractionalized Fermi sea that we have labeled “gener-ically stable discontinuous” in the plots. The regionof effective fields bB for this regime is expected to beconsiderably larger than that of the conventional Con-don Domain formation in ordinary three dimensionalmetals. The critical temperature for a given field isstill controlled by the effective cyclotron frequency of

15

the neutral fermions T1 ∼ bB/m⊥, up to logarithmiccorrections. Unlike metals, the region of instability inmagnetic fields can even extend into the deep quantizedregime. To appreciate this more directly we can re-cast the inequality (54) that describes the field regionof “generically stable discontinuous” behavior as follows

B .S⊥2π

(1− α)2/3

α

(2π2χoscχψ

)2/3

. (63)

Since the ratio χosc/χψ is expected to be of order 1,this extends well into the deep quantized regime of B ∼S⊥/α as we move deeper into the insulating limit, α→0 (contrast this range of fields with that in metals inEq. (60) and (61)). In this ultra-quantum regime thecritical temperatures for the appearance of genericallystable but discontinuous behavior are on the order of theFermi temperature. In fact the maximum temperature,T1, is achieved at a field on the order of that in Eq. (63)and has typical values of the form

Tmax1 ≈ 3εF4π2e

(2π2χoscχϕ + χψ

)2/3

∼ 0.5εF ,

(64)

where in the second line we did the estimate consideringthe limit χϕ → 0. We caution however that to achievethe ultra-quantum regime of the neutral fermions be-comes harder as one goes deeper into the insulator,since the effective field experienced by the fermions isreduced by the factor α. Moreover, our theory is de-veloped under the assumption of small fields, bB � S⊥,so strictly speaking we are naively extrapolating outsidethe regime of control of the theory, but the qualitativepoint that we wish to emphasize is the trend that thecritical temperatures separating the fully stable uniformphases from the phase separation instability regimes willbecome typically much higher in fractionalized Fermiseas as we move deeper into the insulator.

We close this section by noting that Condon domainformation in metals occurs at temperatures of the orderof Kelvins which are well above those for which the de-Haas van-Alphen effect is still visible. Thus in practicethe domain formation does not affect the observationof the de-Haas van-Alphen effect in metals, but changesthe harmonic content of the shape of the measured mag-netization as a function of field. In the semiclassicalregime, we expect that the same will largely hold infractionalized Fermi seas, namely the domain formationwill not prevent the observation of magnetization os-cillations in systems with neutral Fermi surfaces, butwill simply change the precise shape of the magnetiza-tion curve, especially its higher harmonics, just like inmetals. Perhaps the fact that this phenomenon is ex-pected to survive to higher temperatures and magneticfields might facilitate the observation of these domains

in fractionalized Fermi seas with a sizable value of theparameter α.

V. RESISTIVITY OSCILLATIONS OFFRACTIONALIZED FERMI SEA

In this section, we will consider the possibility thatelectrical insulators with emergent neutral Fermi sur-faces display quantum oscillations in their resistivity atfinite temperature. As we will see, these oscillations willbe superimposed on the activated temperature depen-dence, characteristic of the insulating behavior of suchphases. We will discuss suitable conditions that willmake more likely their observation in actual materials.

Even though the phase under consideration is an in-sulator, it will generically have a finite resistivity at fi-nite temperature. The Ioffe-Larkin rule [38] states thatthe resistivities, rather than the conductivities, of theFermionic and bosonic partons add up to produce thephysical net resistivity of the system,

ρ = ρψ + ρϕ. (65)

Since the boson is gapped one expects an Arrhenius-type activated behavior of its resistivity with an energyscale controlled by the charge gap (∆)

ρϕ ≈ ρ0ϕe∆T . (66)

The fermion, however, experiences an effective magneticfield and hence will develop an oscillatory component inthe resistivity. Detailed theories of quantum oscillationof resistivities are rather involved. Here we will followa simple approach that captures correctly the order ofmagnitude of the effect in metals. In metals the main ef-fect of the magnetic field is to change the scattering rateoff impurities. Such rate is proportional to the densityof states and hence oscillates as a result of the Landauquantization [27]. Following the analogous derivationfor a metal detailed in Ref. [27], one obtains that theresistivity will have an oscillatory component given by,

δρoscψρ0ψ

∼ 1

νF

(m⊥b

S⊥

)2∂2fosc∂b2

≈

χosc

24χψ

(2πb

S⊥

)1/2 ∞∑k=1

(−1)k+1

k1/2

kSτ2πb

sinh(kSτ2πb

) cos

(kS

b∓ π

4

),

(67)

where ρ0ψ is the resistivity of the fermions at zero ef-fective field, δρoscψ is the oscillatory component of the

resistivity, and fosc is given in Eq. (39). As we haveseen in the previous section, the internal magnetic fieldexperienced by the Fermions is bB = αB, in the hightemperature regime T & ωψ. However, apart from the

16

TABLE I: Summary of the behavior of the period ∆(1/B) (in units of 2π/S⊥) and amplitude 4πδMosc (in units of χϕS⊥/(2π))of the magnetization oscillations in two-dimensional and three-dimensional fractionalized fermi seas. Here α = χϕ/(χψ +χϕ)and ωψ = αB/m⊥ (units e = c = ~ = 1). χosc controls the amplitude of the oscillations; in the clean limit of spinless Galileanfermions, it is given by χosc = 12χψ/π

2 in two-dimensions and χosc = 3√

2χψ in three-dimensions. The impact of a short-range disorder potential can be captured by adding a Dingle factor into this coefficient χosc → χosc exp (−2πm⊥/(αBτimp)).

2D 3D

period T � ωψχϕ

χψ+χϕ−π2χosc/12α

period T & ωψ α α

Amplitude T � ωψ(

χϕχϕ+χψ−π2χosc/12

2πBS

)2 (2παBS⊥

)2

Amplitude T & ωψ2παχoscχϕ

2π2Tωψ

sinh

(2π2Tωψ

) 2παχoscχϕ

(2παBS⊥

)1/2 2π2Tωψ

sinh

(2π2Tωψ

)

rescaling of the effective field, the functional from of theoscillatory component of the resistivity is essentially thesame as that in metals. At lower temperatures the inter-nal magnetic field will follow a stair-case type behavioraround the average value bB = αB, due to the pinningof the internal field at the local minima of the free en-ergy described in Section. IV. This will typically lead toa form of the resistivity oscillations that is more anhar-monic than that of metals at low temperatures. In spiteof this, at the mean field level, the period and the am-plitude of the oscillations is expected to have the samefunctional form as that in metals, even at low temper-atures (T � ωψ). Therefore the amplitude of the oscil-latory component of the resistivity is expected to followan expression analogous to the Lifshitz-Kosevich form,

δρoscψρ∼

ρ0ψ

ρ0ϕe∆T

(2πbBS⊥

)1/2 S⊥τ2πbB

sinh(S⊥τ2πbB

)e− 2πm⊥bBτimp . (68)

Here we have added a Dingle factor to capture the im-pact of disorder on the amplitude of resistivity oscilla-tions. Notice that the oscillatory component is expo-nentially small relative to the total resistance due tothe temperature activated resistive background. In thissense, we expect that the effect will be hard to detectin strongly insulating materials with a large charge gap.In addition the Dingle factor highlights that disorderwill contribute further to reduce the amplitude of theoscillatory component, eventually suppressing the effectat strong disorder. Therefore the ideal materials forthe observation of this effect will lie in a “sweet spot”that corresponds to those which have a relatively smallcharge gap and thus are not strongly insulating, whilebeing sufficiently clean for the amplitude of the oscilla-tions to be visible.

However, we wish to emphasize that the above is thedependence based on the mean field theory under theassumption that the effective magnetic field only af-fects the fermion scattering by changing their densityof states. As we have discussed in Section IV B thereis a large region in which the solutions with uniform

effective field b are unstable to states with inhomoge-neous values of b. We believe that, in the semiclassicalregime b � S⊥, this will not have a substantial effecton the major features of the oscillations because thefluctuations of the effective field are between the valuesassociated with consecutive minima of the free energy,and therefore should not change dramatically the disor-der landscape. Thus we expect that this effect will nothinder the observation of the resistivity oscillations butwill change details such as the precise harmonic contentof the oscillations and perhaps even enhance the scat-tering off domain walls leading to larger amplitude ofthe resistivity oscillations. This expectation is partlybased on the rule of thumb that such domain formationin metals does not affect the essential observation of theShubnikov-de Haas effect.

We would like to mention a caveat that applies to themixed valence insulators for which we have conjecturedthe presence of the composite exciton Fermi liquid [17].These materials are known to possess a metallic sur-face [10, 12] which has even been argued to have a topo-logical origin [42]. The net resistance of these materialscan be modeled as arising from two resistors in parallelaccounting for the surface and the bulk conduction,

Rtot =RsurfRbulk

Rsurf +Rbulk. (69)

At temperatures well below the crossover value at whichthe resistivity ceases to have the activated Arrhenius-type behavior and saturates into a metallic surface dom-inated regime, most of the electrical current will flowthrough the surface because in this regime Rbulk �Rsurf . The bulk resistance can then be split as the sumof the resistance of the boson and the fermions, follow-ing the Ioffe-Larkin rule, as Rbulk = Rϕ + Rψ. Takingthe limit Rϕ → ∞ in Eq. (70) leads to the followingleading behavior of the net resistance,

Rtot ≈ Rsurf − e−∆TR2

surf

R0ϕ

+ e−2∆TRψR

2surf +R3

surf

(R0ϕ)2

,(70)

17

where we have used an Arrhenius form for the bosonresistance Rϕ = R0

ϕe∆T to highlight its temperature de-

pendence. Therefore we see that the contribution fromthe bulk quantum oscillations of the neutral fermions iseffectively exponentially suppressed when the tempera-ture is well below the value for which the resistance sat-urates, simply because the electrical current is shuntedthrough surface of the material. Therefore, the effectwe have mind should be extracted from temperatureshigher than this crossover temperature, which in princi-ple, can be pushed to lower temperatures by increasingthe size of the sample as this enhances its volume tosurface ratio.

To summarize this section, we have seen that thefractionalized Fermi sea will behave as an insulator inthe sense that the resistivity will increase as the tem-perature is lowered, but, unlike conventional insulators,this smooth rise will be accompanied by an oscillatorycomponent that resembles that of a metal. The phys-ical origin of this effect can be traced back to the factthat the finite amount of thermally induced bosonic car-riers will self-consistently set an internal electric fieldwhen a steady current flows, which will itself act on thefermions. This effect might be most visible in systemswhere the charge gap of the insulator is not so large soas to completely overwhelm the oscillatory componentarising from the neutral Fermi sea.

VI. SUMMARY AND DISCUSSION

We have developed a quantitative theory of quantumoscillations in the magnetization of fractionalized neu-tral Fermi seas in response to external magnetic fields,with special focus on the temperature dependence ofthese oscillations. Table I summarizes some of our mainformulae describing magnetization oscillations in two-and three-dimensional fractionalized neutral Fermi seas.In these systems, the neutral fermions experience an in-ternal magnetic field of the emergent gauge field that isset in self-consistently due to the diamagnetism of thegapped charge carrying degrees of freedom.

Quite generally, the oscillations display distinct be-havior depending on whether the temperature is loweror higher than the effective cyclotron energy of the neu-tral fermions. At high temperatures the oscillations re-semble those of a metal, albeit in an effective field thathas a different strength from the physical magnetic fieldinside the sample. At low temperatures the system dis-plays a multiplicity of metastable states and quantumoscillations can be viewed as a series of phase transi-tions between these states. This phenomenon resemblesCondon domain formation in ordinary metals, but it isdistinct from it because of the ability of the internalgauge field to adjust itself to minimize the free energy.As a result, the instabilities at low temperatures occuronly at isolated values of the external magnetic field,rather than over finite ranges as is the case in ordinary

Compensated semi-metal

Composite exciton fermi liquid

↵ < 1 ↵ = 1

Fermi surface of neutral fermions

gc

gelectrical insulator electrical conductor

Fermi surface of electrons and holes

FIG. 9: (Color online) The composite exciton fermi liquidstate can have a direct metal to insulator transtion to an or-dinary compensated semimetal. The proximity of a materialwith the composite exciton fermi liquid phase to such a crit-ical point will enhance the possibility to observe quantumoscillations by enhancing the effective field of the fermionsand reducing the activated resistive background of the in-sulator. An analogous statement can be made for a systemwith a spinon fermi surface state which is in proximity to ametal to insulator transition quantum critical point.

metals. We believe that even if experiments reach suchultra-low temperature regime this will not hinder theessential observation of quantum oscillations but willaffect details, such as the precise amplitude of differentharmonics, as is the case in metals.

We have also discussed an analogue of the Shubnikov-de Haas oscillations of the resistivity in metals. In thepresent case this is possible because of the thermal ac-tivation of current carrying quasiparticles. Followingthe Ioffe-Larkin rule that dictates the addition of theresistivities of the partons, we conclude that the frac-tionalized neutral Fermi sea will display an oscillatoryresistivity superimposed with the more conventional ac-tivated resistivity behavior that characterizes insulators.

A. Connections to materials

As we have seen, the composite exciton Fermi liquidphase that we have proposed to arise in mixed valenceinsulators [17] supports quantum oscillations as a resultof the fact that an external magnetic field induces a fi-nite value for the emergent magnetic field to which theneutral fermions couple. The parameter α that controlsthe strength of the internal magnetic field experiencedby the fermions is expected to approach 1 as the systemapproaches a critical point between the insulator and acompensated semi-metal (see Fig. 9). Certain mixed va-lence insulators might lie closer to such a critical point,enhancing the field experienced by the neutral fermionand thus enhancing the possibility to observe quantumoscillations. In fact, more recent measurements on amixed valence insulator compound different from SmB6,display clear bulk quantum oscillations and have a finite

18

intercept for the ratio of κxx/T [43] down to the lowestmeasurable temperatures, in a clear indication of theformation of a Fermi surface of neutral fermions. As wehave discussed, the observation of resistivity oscillationswould require to meet certain suitable conditions— thematerial should not be strongly insulating or else it willbe hard to detect the oscillations on top of a large ac-tivated resistive background and at the same time itshould be sufficiently clean as for the oscillations to re-main sizable. Therefore the proximity to a critical pointseparating the composite exciton fermi liquid from acompensated semimetal (see Fig. 9) will also enhancethe chance to observe resistance oscillations because ofthe reduction of the insulating charge gap. One impor-tant caveat for mixed valence insulators with metallicsurfaces is that the resistivity measurements should beperformed above the temperature where the resistivitysaturates due to surface dominated transport. Observa-tion of resistance oscillations that meet these conditionswould provide independent evidence of the presence ofbulk oscillations in these materials.

Resistivity oscillations would also be an importanttool in the elucidation of the nature of the quantum spinliquid phases in the organic materials. Importantly, inthese materials the metal to insulator transition can bedriven in a clean fashion by applying pressure [44–47].A natural question that arises is: what is the fate ofShubnikov-de Haas effect that is present in the metal asone tunes the pressure to drive the metal to insulatortransition? Given that the metal-insulator transitionis either continuous or very weakly first order [47], we

expect that the Shubnikov-de Haas oscillations will per-sist into the insulating phase. Previous experimentalattempts to detect quantum oscillations in these mate-rials were at ambient pressure and studied the magneti-zation. However no oscillations were detected [48]. Thissuggests that the parameter α may be too small at ambi-ent pressure. Based on the considerations in this paperwe advocate searching for quantum oscillations in theresistivity under pressure close to the metal-insulatortransition where α will approach 1. We expect that evenon the insulating side there will be quantum oscillationssuperimposed on the activated resistivity as describedin this paper. The pressure induced tunability of thecharge gap and the high quality of the organic materi-als might facilitate finding the “sweet spot” to observethe resistivity oscillations that we have described.

Acknowledgments

We thank Piers Coleman, Lu Li, Yuji Matsuda, OlexeiMotrunich, Suchitra Sebastian, and Takasada Shibauchifor many stimulating discussions. D.C. is supported bya postdoctoral fellowship from the Gordon and BettyMoore Foundation, under the EPiQS initiative, GrantGBMF-4303, at MIT. I.S. is supported by the MIT Pap-palardo Fellowship. T.S. is supported by a US Depart-ment of Energy grant DE-SC0008739, and in part by aSimons Investigator award from the Simons Foundation.

[1] O. I. Motrunich, “Orbital magnetic field effects in spinliquid with spinon fermi sea: Possible application toκ−(ET)2cu2(CN)3,” Phys. Rev. B 73, 155115 (2006).

[2] Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, andG. Saito, “Spin liquid state in an organic mott insulatorwith a triangular lattice,” Phys. Rev. Lett. 91, 107001(2003).

[3] S. Yamashita, Y. Nakazawa, M. Oguni, Y. Oshima,H. Nojiri, Y. Shimizu, K. Miyagawa, and K. Kanoda,“Thermodynamic properties of a spin-1/2 spin-liquidstate in a [kappa]-type organic salt,” Nat Phys 4, 459(2008).

[4] M. Yamashita, N. Nakata, Y. Senshu, M. Na-gata, H. M. Yamamoto, R. Kato, T. Shibauchi,and Y. Matsuda, “Highly mobile gapless exci-tations in a two-dimensional candidate quan-tum spin liquid,” Science 328, 1246 (2010),http://science.sciencemag.org/content/328/5983/1246.full.pdf.

[5] S. Yamashita, T. Yamamoto, Y. Nakazawa, M. Tamura,and R. Kato, “Gapless spin liquid of an organic trian-gular compound evidenced by thermodynamic measure-ments,” Nature Communications 2, 275 EP (2011).

[6] S.-S. Lee and P. A. Lee, “U(1) gauge theory of the hub-bard model: Spin liquid states and possible application

to κ−(BEDT−TTF)2cu2(CN)3,” Phys. Rev. Lett. 95,036403 (2005).

[7] O. I. Motrunich, “Variational study of triangular latticespin-1/2 model with ring exchanges and spin liquid statein κ−(ET)2cu2(CN)3,” Phys. Rev. B 72, 045105 (2005).

[8] S.-S. Lee and P. A. Lee, “U(1) gauge theory of the hub-bard model: Spin liquid states and possible applicationto κ−(BEDT−TTF)2cu2(CN)3,” Phys. Rev. Lett. 95,036403 (2005).

[9] O. I. Motrunich, “Variational study of triangular latticespin-1?2 model with ring exchanges and spin liquid statein κ−(ET)2cu2(CN)3,” Phys. Rev. B 72, 045105 (2005).

[10] G. Li, Z. Xiang, F. Yu, T. Asaba, B. Lawson,P. Cai, C. Tinsman, A. Berkley, S. Wolgast,Y. S. Eo, D.-J. Kim, C. Kurdak, J. W. Allen,K. Sun, X. H. Chen, Y. Y. Wang, Z. Fisk,and L. Li, “Two-dimensional fermi surfaces inkondo insulator smb6,” Science 346, 1208 (2014),http://science.sciencemag.org/content/346/6214/1208.full.pdf.

[11] J. D. Denlinger, S. Jang, G. Li, L. Chen, B. J. Law-son, T. Asaba, C. Tinsman, F. Yu, K. Sun, J. W.Allen, C. Kurdak, D.-J. Kim, Z. Fisk, and L. Li, “Con-sistency of Photoemission and Quantum Oscillationsfor Surface States of SmB6,” ArXiv e-prints (2016),

19

arXiv:1601.07408 [cond-mat.str-el] .[12] B. S. Tan, Y.-T. Hsu, B. Zeng, M. C. Hatnean,

N. Harrison, Z. Zhu, M. Hartstein, M. Kiourlappou,A. Srivastava, M. D. Johannes, T. P. Murphy, J.-H.Park, L. Balicas, G. G. Lonzarich, G. Balakrish-nan, and S. E. Sebastian, “Unconventional fermisurface in an insulating state,” Science 349, 287 (2015),http://science.sciencemag.org/content/349/6245/287.full.pdf.

[13] M. Hartstein, W. Toews, Y.-T. Hsu, and et al., Unpub-lished; S. Sebastian, private communication (2017).

[14] K. Flachbart, S. Gabani, K. Neumaier, Y. Paderno,V. Pavlık, E. Schuberth, and N. Shitsevalova, “Spe-cific heat of SmB 6 at very low temperatures,” PhysicaB Condensed Matter 378, 610 (2006).

[15] N. Wakeham, P. F. S. Rosa, Y. Q. Wang, M. Kang,Z. Fisk, F. Ronning, and J. D. Thompson, “Low-temperature conducting state in two candidate topolog-ical kondo insulators: smb6 and ce3bi4pt3,” Phys. Rev.B 94, 035127 (2016).

[16] N. J. Laurita, C. M. Morris, S. M. Koohpayeh, P. F. S.Rosa, W. A. Phelan, Z. Fisk, T. M. McQueen, and N. P.Armitage, “Anomalous three-dimensional bulk ac con-duction within the kondo gap of smb6 single crystals,”Phys. Rev. B 94, 165154 (2016).

[17] D. Chowdhury, I. Sodemann, and T. Senthil, “Mixed-valence insulators with neutral Fermi-surfaces,” ArXive-prints (2017), arXiv:1706.00418 [cond-mat.str-el] .

[18] G. Baskaran, “Majorana Fermi Sea in InsulatingSmB6: A proposal and a Theory of Quantum Oscil-lations in Kondo Insulators,” ArXiv e-prints (2015),arXiv:1507.03477 [cond-mat.str-el] .

[19] O. Erten, P.-Y. Chang, P. Coleman, and A. M. Tsvelik,“Skyrme insulators: insulators at the brink of super-conductivity,” ArXiv e-prints (2017), arXiv:1701.06582[cond-mat.str-el] .

[20] J. Knolle and N. R. Cooper, “Excitons in topologicalkondo insulators: Theory of thermodynamic and trans-port anomalies in smb6,” Phys. Rev. Lett. 118, 096604(2017).

[21] J. Knolle and N. R. Cooper, “Quantum oscillationswithout a fermi surface and the anomalous de haas˘vanalphen effect,” Phys. Rev. Lett. 115, 146401 (2015).

[22] L. Zhang, X.-Y. Song, and F. Wang, “Quantum oscil-lation in narrow-gap topological insulators,” Phys. Rev.Lett. 116, 046404 (2016).

[23] P. Ram and B. Kumar, “Theory of quantum oscillationsof magnetization in kondo insulators,” Phys. Rev. B 96,075115 (2017).

[24] N. Read, “Theory of the half-filled landau level,” Semi-conductor Science and Technology 9, 1859 (1994).

[25] N. Read, “Lowest-landau-level theory of the quantumhall effect: The fermi-liquid-like state of bosons at fillingfactor one,” Phys. Rev. B 58, 16262 (1998).

[26] B. I. Halperin, P. A. Lee, and N. Read, “Theory of thehalf-filled landau level,” Phys. Rev. B 47, 7312 (1993).

[27] A. A. Abrikosov and A. Beknazarov,Fundamentals of the Theory of Metals, Vol. 1 (North-Holland Amsterdam, 1988).

[28] D. Shoenberg, Magnetic oscillations in metals (Cam-bridge University Press, 2009).

[29] G. Solt, C. Baines, V. S. Egorov, D. Herlach, andU. Zimmermann, “Diamagnetic domains in beryllium

observed by muon-spin-rotation spectroscopy,” Phys.Rev. B 59, 6834 (1999).

[30] A. Gordon, M. A. Itskovsky, and P. Wyder, “Quantiz-ing field-induced magnetic phase in a three-dimensionalelectron gas,” Phys. Rev. B 59, 10864 (1999).

[31] A. Gordon, I. Vagner, and P. Wyder, “Magnetic do-mains in non-ferromagnetic metals: the non-linear dehaas-van alphen effect,” Advances in Physics 52, 385(2003).

[32] R. B. G. Kramer, V. S. Egorov, V. A. Gasparov,A. G. M. Jansen, and W. Joss, “Direct observationof condon domains in silver by hall probes,” Phys. Rev.Lett. 95, 267209 (2005).

[33] V. S. Egorov, “Condon domains-these non-magneticdiamagnetic domains,” arXiv preprint cond-mat/0505415 (2005).

[34] M. Hermele, T. Senthil, M. P. A. Fisher, P. A. Lee,N. Nagaosa, and X.-G. Wen, “Stability of u(1) spinliquids in two dimensions,” Phys. Rev. B 70, 214437(2004).

[35] S.-S. Lee, “Stability of the u(1) spin liquid with a spinonfermi surface in 2 + 1 dimensions,” Phys. Rev. B 78,085129 (2008).

[36] S. Florens and A. Georges, “Slave-rotor mean-field the-ories of strongly correlated systems and the mott tran-sition in finite dimensions,” Phys. Rev. B 70, 035114(2004).

[37] T. Senthil, “Theory of a continuous mott transition intwo dimensions,” Phys. Rev. B 78, 045109 (2008).

[38] L. B. Ioffe and A. I. Larkin, “Gapless fermions andgauge fields in dielectrics,” Phys. Rev. B 39, 8988(1989).

[39] Y. Imry and S.-k. Ma, “Random-field instability of theordered state of continuous symmetry,” Phys. Rev. Lett.35, 1399 (1975).

[40] Y. Imry and M. Wortis, “Influence of quenched impuri-ties on first-order phase transitions,” Phys. Rev. B 19,3580 (1979).

[41] M. Aizenman and J. Wehr, “Rounding of first-orderphase transitions in systems with quenched disorder,”Phys. Rev. Lett. 62, 2503 (1989).

[42] M. Dzero, K. Sun, V. Galitski, and P. Coleman, “Topo-logical kondo insulators,” Phys. Rev. Lett. 104, 106408(2010).

[43] Y. Matsuda, L. Li, and T. Shibauchi, private commu-nication (2017).

[44] T. Komatsu, N. Matsukawa, T. Inoue, and G. Saito,“Realization of superconductivity at ambient pressureby band-filling control in - (bedt-ttf)2cu2(cn)3,” Jour-nal of the Physical Society of Japan 65, 1340 (1996),http://dx.doi.org/10.1143/JPSJ.65.1340 .

[45] Y. Kurosaki, Y. Shimizu, K. Miyagawa, K. Kanoda,and G. Saito, “Mott transition from a spin liquid toa fermi liquid in the spin-frustrated organic conduc-tor κ−(ET)2cu2(CN)3,” Phys. Rev. Lett. 95, 177001(2005).

[46] T. Furukawa, K. Miyagawa, H. Taniguchi, R. Kato, andK. Kanoda, “Quantum criticality of mott transition inorganic materials,” Nature Physics 11, 221 (2015).

[47] T. Furukawa, K. Kobashi, Y. Kurosaki, K. Miya-gawa, and K. Kanoda, “Quasi-continuous transitionfrom a fermi liquid to a spin liquid,” arXiv preprintarXiv:1707.05586 (2017).

20

[48] D. Watanabe, M. Yamashita, S. Tonegawa, Y. Os-hima, H. Yamamoto, R. Kato, I. Sheikin, K. Behnia,T. Terashima, S. Uji, et al., “Novel pauli-paramagnetic

quantum phase in a mott insulator,” Nature Communi-cations 3, 1090 (2012).