Probing a Bose Metal via Electrons:Inescapable non-Fermi liquid scattering and pseudogap physics

Xinlei Yue (岳辛磊),1, 2 Anthony Hegg,2, 3 Xiang Li (李翔),1, 2 and Wei Ku (顧威)2, 3, 4, ∗

1Zhiyuan College, Shanghai Jiao Tong University, Shanghai 200240, China2School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

3Tsung-Dao Lee Institute, Shanghai 200240, China4Key Laboratory of Artificial Structures and Quantum Control (Ministry of Education), Shanghai 200240, China

(Dated: February 16, 2022)

Non-Fermi liquid behavior and pseudogap formation are among the most well-known examples of exoticspectral features observed in several strongly correlated materials such as the hole-doped cuprates, nickelates,iridates, ruthenates, ferropnictides, doped Mott organics, transition metal dichalcogenides, heavy fermions, d-and f -electron metals, etc. We demonstrate that these features are inevitable consequences when fermions cou-ple to an unconventional Bose metal [1] mean field consisting of lower-dimensional coherence. Not only dowe find both exotic phenomena, but also a host of other features that have been observed e.g. in the cupratesincluding nodal anti-nodal dichotomy and pseudogap asymmetry(symmetry) in momentum(real) space. Obtain-ing these exotic and heretofore mysterious phenomena via a mean field offers a simple, universal, and thereforewidely applicable explanation for their ubiquitous empirical appearance.

Strongly correlated materials present some of the mostmystifying phenomena in physics. The reason for this is two-fold. First, these materials have a tendency toward exoticproperties that defy conventional theory. Second, and perhapsmore importantly, many of these exotic observations exhibituniversality in that they can be found in a range of materi-als that seem completely unrelated at the microscopic level.The combination of their exotic nature and the mathematicaldifficulty in theoretical treatment has generated an enormousamount of research [2–5] over many decades with little signof stagnation, and this research has been a wellspring of noveland exciting physics. Yet, comparatively little progress hasbeen made in producing theory that measures up to the empir-ical universality.

Within electronic spectral observations, non-Fermi liquid(NFL) scattering and pseudogap (PG) formation are twosuch exotic and universal phenomena. A phenomenon is la-beled as NFL scattering when the corresponding scatteringrate of quasiparticle excitation violates T 2 and ω2 depen-dence [6] near the chemical potential. NFL scattering hasbeen observed in cuprates [7–10], ruthenates [11], ferrop-nictides [12, 13], transition metal dichalcogenides [14], andheavy fermions [15, 16]. A PG is said to form when a gapscale appears in the quasiparticle excitation spectrum, but asignificant number of states fill the gap even at low tem-perature. PG behavior is widely found in cuprates [3, 17–26], nickelates [27], irridates [28], ruthenates [29], ferrop-nictides [30, 31], Mott organics [32], and transition metaldichalcogenides [14, 33, 34].

The names of these two phenomena highlight both the vi-olation of and the empirical universality they share with thecorresponding conventional theory. In a Fermi liquid (FL) thevanishing phase space near the chemical potential suppressesscattering and preserves the quasiparticle nature. The result-ing energy and temperature dependence of the single particle

spectrum is highly constrained and very robust. As for gapformation, perfectly coherent scattering (from a mean fieldfor example) always opens a clean gap. As long as the gapscale remains dominant, states only fill it at best exponentiallyslowly. The robust universal nature of these conventional phe-nomena strongly suggests that their violation would be diffi-cult and rare, yet empirical evidence directly contradicts thispicture. Instead, NFL and PG phenomena have been foundin many seemingly unrelated materials suggesting, in analogywith the conventional physics above, that they harbor an as-yet unidentified robust universal nature of their own.

Existing theoretical treatments of NFL behavior typicallystart with a FL and enhance scattering channels at low energyto disrupt the quasiparticle nature. Several examples includemarginal Fermi liquid [35], multichannel Kondo physics [36],orbital fluctuations [37], holographic methods [38], and quan-tum critical fluctuations [39–49]. However, since a systematictheoretical treatment of unconventional features over a widerange of experiments is lacking, a correspondingly systematicunderstanding of these relatively universal characteristics re-mains elusive.

Theories exhibiting PG behavior typically start from theformation of a clean gap and then introduce long-range fluc-tuations to refill it. Some examples include preformed pairsscenarios [50, 51] and models with spin correlations [52–54].These methods rely on coherent order parameters and then in-troduce fluctuations to weaken them so they vanish at longdistances. Such fluctuations require proximity to a quantumcritical point or phase boundary, but experiments often findPG formation over wide parameter ranges even at very lowtemperatures [55]. Perhaps these scenarios could be realizedin nature, but such restrictive constraints eliminate any chanceto account for the observed universality yet again.

Since the theories above are highly specialized to exhibitone exotic phenomenon each, it is perhaps not surprising that

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FIG. 1. (a) Illustration of the Bose metal one-body phase structure(XY plane). The bond-centered bosons, shown by ellipses, form acheckerboard lattice. Ellipse color indicates local phase of the domi-nant mode. Each color represents a fixed randomly chosen phase fac-tor. (b) Illustration of the spectral weight for several ratios of bindingenergy with bandwidth. Unbound and bound quasiparticle density ofstates (DOS) are given in green and purple respectively. The chemi-cal potential is marked by blue dotted lines. The region we study isrepresented by panel II marked with a red arrow. Comparison couldbe made with the numerical results [56].

NFL and PG theoretical understanding have essentially nooverlap. Describing one exotic phenomenon in one material isdifficult enough, let alone two seemingly unrelated such mys-teries. Yet many materials exhibit both NFL scattering and PGformation in the same experimental parameter range. They arealso both properties of the single particle excitation spectrum,so when they occur simultaneously over a broad region of thephase diagram of a material, it is natural to assume a universalmechanism would exhibit both phenomena.

Here we identify a robust paradigm where the exotic phe-nomena of non-Fermi liquid scattering rates and pseudogapformation inevitably result. The crucial component is an un-conventional effective mean field derived from a recently pro-posed Bose metal [1]. The mean field studied here only ex-hibits lower-dimensional coherence leaving it a measure ofincoherence, and it is this incoherence that defeats the con-ventional framework. To demonstrate the qualitative effects ofthis idea, we couple this mean field to simple non-interactingfermions so that all the characteristic features originate onlyfrom the mean field. Not only do we find NFL and PG behav-ior but also a range of additional phenomena all of which havebeen observed together in nature [17, 18, 20, 22, 25, 26]. Pro-ducing these phenomena from a single mean field provides theframework for a systematic comparison of these unconven-tional features and contains the wide applicability necessaryto account for existing and future observations.

To generate NFL and PG behavior we must break the highlyconstrained nature of Fermi quasiparticles. The former byopening up a significant scattering channel at low-energy andlow-temperature. The latter by coupling to incomplete coher-ent structure. These conditions can be met by coupling to arecently proposed Bose metal [1]. This Bose metal is a sta-ble phase of matter that is neither superfluid nor insulator andit is comprised of real (conserved) bosons, so it is not con-fined to the vicinity of any critical transition. In one example,

the ground state of the Bose metal consists of coherent 2Dplanes that are effectively independent of each other to one-body level in the low-energy limit. This should be contrastedwith the typical long-range isotropic coherence found in, forexample, superfluidity. The dominant one-body phase struc-ture for this example is illustrated in Fig.1 (a) in which eachcoherent region is represented by a single color.

We demonstrate the effects of scattering against our Bosemetal by coupling it to a simple tight binding Fermi gas

H = Hkinetic +Hscattering (1)

where Hkinetic contains the bare fermion dispersion andHscattering describes the coupling between these fermions to thedominant bosonic mode, chosen to be Bogoliubov-like

Hscattering = ∑(i, j)∈NN

V b†i jc jci +h.c. (2)

Here V is the effective coupling strength and NN denotesnearest neighbors in the XY -plane. This choice for the cou-pling is not required, but it represents a scenario [57] wherethe bosons are formed from binding the fermions into localpairs at high energy and are therefore indistinguishable fromeach other. Suppose we are in the regime as shown by thered arrow in Fig.1 (b) where most of the fermion weight isin a tightly bound state (in purple) represented by a boson,yet some weight remains in the unbound fermionic state (ingreen). Assuming the boson density is much higher than thatof the fermions, the influence of the fermions on the Bosemetal structure is negligible. For convenience, we further as-sume a scale V ∼ t of the order of the nearest neighbor hop-ping strength t.

Since the dominant bosonic energy scale is very low com-pared to the fermionic bandwidth, we replace our Bose metalwith a well-justified (c.f. Appendix 1) static effective meanfield at low temperature

Hscattering→ ∑kxky

{∑q

Vy(q)ckx,kyc−kx,−ky+q

+∑q

Vx(q)ckx,ky c−kx+q,−ky

}+h.c.,

(3)

where the notation kz = 0 has been dropped given the perfectin-phase coherence along the z-direction in our Bose metalstructure shown in Fig.1(a) (c.f. Appendix 2). As shown be-low, the resulting spectral features are very broad in energyand momentum and therefore insensitive to fine-tuned param-eters.

Fig.2 summarizes the main features of our results using thetight binding parameters of the cuprates [24] as an example.Panels (a) and (b) correspond to the resulting spectral func-tions A(k,ω) along the momentum cuts I-VII illustrated inpanel (c) before and after coupling to the Bose metal respec-tively. The fermionic chemical potential is chosen separately

3

FIG. 2. Spectral function A(k,ω) at doping δ ≈ 0.11 (a) without and (b) with coupling to the Bose metal for selected lines in k-space denotedin (c). Red lines indicate the momentum nearest to the Fermi vector kF. (d) the spectral function near the chemical potential reveals ’Fermiarc’ structure.

FIG. 3. Energy contours of the band structure at δ ≈ 0.11 doping.The white line indicates the Fermi surface. States with crystal mo-menta near P= (π/2,π/2) (X=(π,0)), marked by the green circle(blue disk), couple with points along the green dashed lines(bluesolid lines) through the Bose metal mean field. This large couplingspace leads to scattering in general broadening all peaks, but the cou-pling near X involves many states in a small energy window (near avan Hove singularity) resulting in scattering so strong the quasiparti-cle peaks are destroyed completely.

for each case in order to fix the particle number. Panel (d)gives the momentum dependent spectral weight at the chemi-cal potential, which clearly shows a dichotomy between the’nodal’ region near P= (π/2,π/2) and the ’anti-nodal’ re-gion near X= (π,0) reminiscent of the infamous ’Fermi arc’

phenomenon[58].Near the nodal region (panels I-III) A(ω) acquires a rather

large scattering rate (width denoted by green bar in I) nearthe chemical potential even at the Fermi wave vector kF(red curve). In a clear departure from Fermi liquid theory,this corresponds to a finite imaginary part of the self-energyImΣ(ω,T ) at the chemical potential. Having a finite scatter-ing rate at the chemical potential even in the low-temperaturelimit is of profound significance. This implies anomalous dis-sipation in a clean quantum system of fermions, unimaginablewithin the standard lore.

The origin of this large scattering rate can be understoodclearly from the structure of the Bose metal effective meanfield coupling as illustrated in Fig.3. Due to the lack of trans-lational symmetry of the Bose metal mean field, each bosoncouples a fermion state into every state along a line. For ex-ample, a state at P would couple to every state along the verti-cal and horizontal blue lines through scattering against bosonsthat move horizontally and vertically respectively in Fig.1(a).This significantly opens up the phase space for scattering ascompared to bosons with well-defined momenta. Since thestates along the blue lines cover a broad range of energy,the fermionic poles are always broadened even down to thechemical potential. This enhanced phase space for scatteringis even more effective than the typical NFL scenario, whichinvolves scattering against bosons of well-defined momen-tum [59–61]. Furthermore, since the Bose metal is a stablephase composed of real bosons (with fixed density), the effectof coupling is actually strengthened in the low-temperaturelimit or far away from a phase boundary.

It is important to note that the fixed boson number inour theory is essential to sustain large scattering in the

4

𝜔𝜔 [100𝑚𝑚𝑚𝑚𝑚𝑚] 𝜔𝜔 [100𝑚𝑚𝑚𝑚𝑚𝑚]-2 -1 -1-20 0

𝐴𝐴𝜔𝜔

[0.5𝑚𝑚𝑚𝑚𝑚𝑚

−1 ]

𝐴𝐴𝜔𝜔

[𝑎𝑎.𝑢𝑢

.]

FIG. 4. Theoretical(a)(b) and experimental(c)(d) spectral functionsnear the anti-nodal region following the arrow shown in the insetfor the high temperature phase(a)(c) and the low-temperature pseu-dogap regime(b)(d). Experimental plots are inferred from ARPESdata for optimally doped Bi2201 [22, 25] (Tc ≈ 34K). Notice thatthe gap minimum does not occur at the bare Fermi vector(red line),and instead is found(green line) in the back-bending band(green cir-cles) generated through the formation of the pseudogap. Also noticethe dispersion of the gap center (yellow vertical bars) defined by theminimum of A(ω) in the gap.

low-temperature limit. In contrast, the typical scenario inwhich fermions scatter against thermally excited bosons (e.g.phonons, magnons, spinons, holons, phasons, acoustic plas-mons, orbitons, excitons, etc.) finds diminishing scatteringat low-temperature. This limitation forces the existing at-tempts to account NFL behavior to resort to critical fluc-tuations, which allow for a macroscopic number of criticalbosons [39, 43, 47, 49]. However, this overconstrained sce-nario is inconsistent with observations of NFL behavior overa broad region of the phase diagram, for example in the pseu-dogap regime of the cuprates, whereas our anomalous dissi-pation scenario involves a stable phase compatible with theseexperiments. Incidentally, the stability of the Bose metal asa phase of matter and the qualitative agreement between ourresults and experiments strongly suggests that the pseudogapregime of the cuprates is a finite temperature phase of a Bosemetal.

Our large scattering rate is unimaginable to-date since manytheories [35, 39–49, 62] have willingly explored extreme ex-otic fermions but strictly maintain dissipationless behaviorwhen the system is clean. On the other hand, having acceptedthis new possibility, any smooth analytic scattering rate canshow NFL behavior ImΣ(ω,T )≈ c+aω +bT + · · · . Inciden-tally, experiments [7, 12] observe the same smooth analyticbehavior that is always finite at low-energy, which only trans-forms into an exotic non-analytic mystery if c is assumed toarise from disorder [63] and subsequently removed.

Moving away from the nodal region toward the anti-nodalregion (Fig.2 V-VII) we find qualitatively different behav-ior. Here at low-energy A(ω) loses its quasiparticle peak-likestructure. Instead, we find wide but distinct peaks above and

(a) (b)

(c) (d)

𝐴𝐴𝜔𝜔

[0.05𝑚𝑚

𝑚𝑚𝑉𝑉−1 ]

𝐷𝐷𝐷𝐷𝐷𝐷

[𝑎𝑎.𝑢𝑢

.]

𝐴𝐴𝜔𝜔

[𝑎𝑎.𝑢𝑢

.]

𝑑𝑑𝑑𝑑/𝑑𝑑𝑉𝑉

[𝐺𝐺Ω−1 ]

?

𝜔𝜔 [10𝑚𝑚𝑚𝑚𝑉𝑉] 𝜔𝜔 [100𝑚𝑚𝑚𝑚𝑉𝑉]-5 0 0 1 2-1-2

𝛼𝛼filling up

FIG. 5. Seemingly contradictory experimental spectral functions ofunderdoped BSCCO with asymmetric/symmetric pseudogap struc-ture in (c) momentum [21] and (d) real [18] space. Similar fea-tures are observed to be compatible in our theoretical results (a) and(b). The effect of thermal reduction α = Φ(T )/Φ(0) of the orderparameter(Φ) is shown in panel (a) as well displaying characteristicsof gap filling as opposed to gap closing.

below the chemical potential defining a clear gap scale. No-tice that there is substantial weight within this gap, a featuretypically regarded as a pseudogap.

Despite the dichotomy between nodal region near P and theanti-nodal region near X, our pseudogap can be understoodby examining the coupling to the Bose metal mean field onceagain. In both regions we couple to states over a broad energyrange and the difference between these regions changes verysmoothly. But the coupling along the green dotted lines inFig.3 now includes quantitatively more states of similar en-ergy circled in yellow, which makes the coupling effectiveenough to open a gap scale. However, the incomplete coher-ence of our mean field cause the gap to be filled with states asopposed to the case where complete coherence opens a cleangap.

Surprisingly, our pseudogap exhibits many of the symp-toms typically associated with pseudogap observations. InFig.4 we exhibit a back-bending band traced in green cir-cles and accompanied by a shadow band traced in purplecircles in panel (b) in agreement with experiments in panel(d). The back-bending band and the gap center betweenthese bands (outlined in vertical yellow bars) develop disper-sion [4, 21, 22, 25], and correspondingly kF (red line) is nolonger the gap minimum (green lines). The diagnosis is clear:we couple to many states of different energy thereby breakingone of the conditions required to make kF the gap minimum.

A more important characteristic of our pseudogap is howit dissolves with increasing temperature. Fig.5(a) illustratesthat upon thermal reduction of the mean field by a factor α =Φ(T )/Φ(0)∼ 0.5, the gap gets filled up by more states fasterthan it closes in agreement with experiments [22, 23, 25]. Ourpseudogap becomes indistinguishable from the continuum at

5

a temperature T ∗ long before the Bose metal mean field (gapscale) is depleted at a higher temperature TBM.

Perhaps the most significant feature of our pseudogap, asillustrated in Fig.5(a), is a strong asymmetry of the peak en-ergies above and below the Fermi energy. This asymmetryis also observed experimentally as in Fig.5(c) and has led topreference toward fluctuating charge density wave, spin den-sity wave [52–54] or pairing density wave [4, 64] over a Bo-goliubov (preformed pair). However, we were able to obtainthis asymmetry using a Bogoliubov coupling since our Bosemetal mean field couples fermion states in a line and thereforewe do not expect symmetry in general.

Surprisingly, in real space our pseudogap appears to beperfectly symmetric in energy as illustrated in Fig.5(b) de-spite the strong asymmetry in momentum space consideringthe former is the sum over the latter. It is precisely thissum that loses the information about partial coherence andleaves only the symmetry of our Bogoliubov coupling. Infact, this apparent contradiction between real and momen-tum space has been haunting the experimental communityin the cuprates for decades between scanning tunneling mi-croscopy (STM) [18, 20] and angle-resolved photoemissionspectroscopy (ARPES) experiments. To our knowledge, ourtheory provides the first natural resolution to this big puzzle.

In essence, our theory is successful because it encapsu-lates simple ingredients required for NFL and PG phenom-ena. Through the interplay between coherence and incoher-ence, we introduce just enough scattering to generate a non-Fermi liquid scattering rate. On top of that, when assisted bya large amount of phase space, for example near a van Hovesingularity, the system will further generate a pseudogap dueto the combination of partial coherence (gap scale) and inco-herence (states inside). To our knowledge, no theory prior toours has managed to explain both NFL and PG at the sametime.

For those interested in the cuprates, our successful repro-duction of NFL scattering, PG formation, and many relatedsymptoms found in the cuprates leads one to wonder whetherthe pseudogap regime of the cuprate phase diagram belowT ∗ is actually contained within the finite temperature phaseof a Bose metal below a higher temperature TBM. In fact,the Bose metal mean field used here directly results from thephase frustrated regime [1, 65] of the emergent Bose liquidtheory [57, 66] of the cuprates. Assuming this is true, then it isno wonder that the observations from single particle fermionprobes such as ARPES and STM are so difficult to explain.Such experiments would be probing a probe (dilute fermionsystem) of the dominant physical state (a Bose metal) andtherefore adding an extra barrier to trace from observation tomicroscopic picture.

In conclusion, we obtain non-Fermi liquid scattering andpseudogap formation by coupling a simple tight bindingFermi gas to an effective mean field for a recently proposedBose metal with one-body sub-dimensional coherence. Thisserendipitous result generates a new paradigm where a genericfermionic system couples to a generic real bosonic field of in-

complete coherence to produce unconventional physics. Re-lying on real bosons that do not vanish in the low temperaturelimit, we find a finite scattering rate at zero frequency and zerotemperature. It is this non-trivial scattering rate at the chemi-cal potential that is ultimately responsible for both non-Fermiliquid scattering and pseudogap formation. This couplingproduces additional qualitative characteristics including nodalanti-nodal dichotomy (resulting in Fermi arcs) and particle-hole asymmetry(symmetry) in momentum(real) space as wellas other interesting features consistent with experiments. Abeneficial byproduct of our study is that it appears to suggestan explanation to the infamous pseudogap of the cuprates. Ourmethod is universal and should therefore be widely applicableto many materials exhibiting these exotic phenomena.

We thank Dong Qian for discussion regarding ARPES ex-periments. This work is supported by National Natural Sci-ence Foundation of China (NSFC) #11674220 and 11745006and Ministry of Science and Technology #2016YFA0300500and 2016YFA0300501.

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8

Appendix

1. Justification of mean-field approximation

In this appendix we justify the use of the effective mean-field approximation in our study. The key points are that the influenceof slow bosonic modes on the typical dynamics of fast fermionic modes can be well-approximated by a static field with anaveraged momentum p0. As explained in the text, we treat the bosons as approximately unaffected by the fermions because ofthe relatively small population of the latter. The effect onto the fermionic system can be expressed via

H f = H0 +Hcoupling. (4)

Here H0 is the fermion Hamiltonian (for example a simple tight-binding Fermi gas) modified by coupling to the predefinedbosonic modes b†

p of lattice momentum p via

Hcoupling = ∑p,p

Vpb†pcpcp−p +h.c., (5)

as shown in Fig.S1(a). Without loss of generality, we illustrate with a local coupling such that the coupling strength |Vp|=V isp-independent.

(a) (b)

FIG. S1. (a) Illustration of the coupling between boson and fermion, here only bc†c† is shown. Straight line with single line arrow correspond-ing to the fermion, the wiggled line corresponding to boson. (b) Illustration of the lowest order correction to the normal green function.

First, it is understandable that objects of very different time scales (frequencies) are insensitive to minute differences due todispersion. For example, when the fermionic modes ξp are much higher energy than the bosonic modes ξp� εp ≡ ε0 +∆, thescattering of the fermion by such bosons can be approximated as εp ≈ ε0.

Further, at small enough temperature, virtually all the bosons are located in a energy window small compared to the bandwidthnear the dispersion minimum p0. Let p = p0 + δ , in this window δ << 2π and the scattering against the bosons will have amomentum transfer dominated by roughly p0, and ε0 ≈ 0. In the presence of condensation this result is even more pronounced.

As an illustration, the leading contribution to the fermionic self-energy due to the coupling is then given in Figure S1(b),

Σ(p, iηm) =−1β

∑p

∑iωn

|V |2D(p, iωn)G(0)(p− p, i(ωn−ηm)), (6)

where D(p, iωn) denotes the finite-temperature time-ordered Green functions for the bosons and G(0)(p, iηm) for the fermions.For a representative bosonic Green function with a pole near εp, Eq.6 is roughly

Σ(p, iηm) = ∑p|V |2 1−iηm−ξp−p + ε p

nB(εp), (7)

given fermionic Green functions with poles near ξp. It is straightforward to confirm that the integration over momentum gives

Σ(p, iηm)≈ |V |2N

−iηm−ξp0−p + εp0

≈ |V |2 N−iηm−ξp0−p

(8)

corresponding to the diagram in Fig.S2(a), where N denotes the number of bosons and we have made use of the large energyseparation (ξp � εp) discussed above. Altogether, this is equivalent to replacing the original coupling in Fig.S1(a) by a staticaverage field in Fig.S2(b), or equivalently replacing Hcoupling in Eq.5 with

9

(a) (b)

FIG. S2. (a) The lowest order correction to the normal Green function. Straight dashed line corresponds to the static bosonic field, 〈b〉. (b)Coupling between a fermion and the static bosonic field, here only the c†c† interaction is shown.

Hcoupling ≈∑p

V√

Ncpcp0−p +h.c. (9)

This example clearly demonstrates that, as long as the dynamics of the bosons are much slower than that of the fermions, onecan approximate the effect on the fermions as coming from a static average with an approximate momentum p0 as representedin Eq.9. Such simplification is even better justified when the physical broadening, as illustrated in Fig.2, is significant. In ourcase, such boradening occurs due to the lack of translational symmetry in the phase structure of the bosonic field, which couplesall momenta in a line that spans the Brillouin zone.

2. Bose Metal Effective Mean Field

In the previous appendix, we identified a simple average static field form to approximate the coupling to the bosons abovegiven by Eq.(9) and illustrated in Fig.S2(b). In this appendix, we reproduce an explicit form of that average static field for a Bosemetal as outlined in a previously published work[1]. For completeness, we include relatively tedious details due to the mismatchbetween the boson and fermion lattice that do not affect the more general ideas of this work.

The effective mean field is given by the form of the boson dispersion minimum. For the Bose metal[1] this is simply

< b†x+ 1

2 ,y ,z>=< bx+ 1

2 ,y ,z >=

√δ

4e i [θy+π x ], (10)

for horizontal bonds and

< b†x ,y+ 1

2 ,z>=< bx ,y+ 1

2 ,z >=

√δ

4e i [θx+π y ], (11)

for vertical bonds, where δ is the doping fraction and b†x+ 1

2 ,y

(b†

x,y+ 12

)creates a boson coherent along the x(y) direction. Here

θx(θy) must be chosen at random from 0 to 2π , which reflects the lack of coherence due to geometric frustration in this Bosemetal[1].

The convention used in the main text, bi j, represents a boson between sites i and j. These two representations are related via

b(x ,y ,z)(x ,y+1 ,z) = (−1)ybx ,y+ 12 ,z (12)

b(x ,y ,z)(x+1 ,y ,z) = (−1)xbx+ 12 ,y ,z, (13)

where the sign change is due to fermionic commutation relations (see the supplementary material of [59] for details). For ourmean-field-like treatment we find

Hcoupling = ∑kxky

(∑q

Vy(q)ckx,kyc−kx,−ky+q +∑q

Vx(q)ckx,kyc−kx+q,−ky

)+h.c. (14)

where

Vy(q) =1N

√δ

4V ∑

ye i [θy+aqy ] (15)

10

Vx(q) =1N

√δ

4V ∑

xe i [θx+aqx ]. (16)

Note that although the static field denoted by Eqs.(15,16) has a non-trivial form, it has no tunable parameters. Its phasestructure simply represents the static structure of the dispersion minimum for the Bose metal[1], and its amplitude is fixed bythe total carrier density.