Mott insulating states with competing orders in the triangular lattice Hubbard model
Alexander Wietek, Riccardo Rossi, Fedor ?imkovic IV, Marcel Klett, Philipp Hansmann, Michel Ferrero, E. Miles Stoudenmire, Thomas Schäfer, Antoine Georges
MMott insulating states with competing orders in the triangular lattice Hubbard model
Alexander Wietek, ∗ Riccardo Rossi,
1, 2
Fedor ˇSimkovic IV,
3, 4
Marcel Klett, Philipp Hansmann, Michel Ferrero,
3, 4
E. Miles Stoudenmire, Thomas Sch¨afer, and Antoine Georges
4, 1, 3, 7 Center for Computational Quantum Physics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA Institute of Physics, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland CPHT, CNRS, ´Ecole Polytechnique, IP Paris, F-91128 Palaiseau, France Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstraße 1, 70569 Stuttgart, Germany Department of Physics, University of Erlangen-N¨urnberg, 91058, Erlangen, Germany DQMP, Universit´e de Gen`eve, 24 quai Ernest Ansermet, CH-1211 Gen`eve, Suisse (Dated: February 26, 2021)The physics of the triangular lattice Hubbard model exhibits a rich phenomenology, ranging froma metal-insulator transition, intriguing thermodynamic behavior, and a putative spin liquid phaseat intermediate coupling, ultimately becoming a magnetic insulator at strong coupling. In thismulti-method study, we combine a finite-temperature tensor network method, minimally entangledthermal typical states (METTS), with two Green function-based methods, connected-determinantdiagrammatic Monte Carlo (DiagMC) and cellular dynamical mean-field theory (CDMFT), to es-tablish several aspects of this model. We elucidate the evolution from the metallic to the insulatingregime from the complementary perspectives brought by these different methods. We compute thefull thermodynamics of the model on a width-4 cylinder using METTS in the intermediate to strongcoupling regime. We find that the insulating state hosts a large entropy at intermediate tempera-tures, which increases with the strength of the coupling. Correspondingly, and consistently with athermodynamic Maxwell relation, the double occupancy has a minimum as a function of temper-ature which is the manifestation of the Pomeranchuk effect of increased localisation upon heating.The intermediate coupling regime is found to exhibit both pronounced chiral as well as stripy an-tiferromagnetic spin correlations. We propose a scenario in which time-reversal symmetry brokenstates compete with nematic, lattice rotational symmetry breaking orders at lowest temperatures.
I. INTRODUCTION
The interplay between strong electronic interactionsand geometric frustration gives rise to a plethora of in-triguing physical phenomena. It also raises fundamentalquestions that are still largely open, such as how insu-lating spin liquids transition into a metallic or supercon-ducting phase upon reducing the interaction strength orintroducing doped charge carriers.Several classes of experimental platforms are availablein which these questions can be explored. The most re-cent one is the rapidly developing field of twisted moir´eheterostructures of two-dimensional materials, such asgraphene [1–3] or transition-metal dichalcogenides [4, 5].Recent work has demonstrated that these heterostruc-tures provide a versatile platform for quantum materi-als design in which a broad range of lattice and bandstructures can be engineered [6, 7]. A triangular latticestructure, which is the focus of the present paper, can berealized in this context for WSe /WS moir´e superlat-tices [8, 9], twisted WSe double bilayers [10] as well astwisted bilayer Boron Nitride [11, 12]. The observationof a Mott insulating state in e.g. the WSe /WS moir´esuperlattice system [8] provides direct experimental evi-dence of the importance of strong electronic correlationsin these materials. We also note that the triangular su- ∗ awietek@flatironinstitute.org perlattice dichalcogenide 1T-TaS has been proposed tohost a spin-liquid state [13–16].Besides moir´e materials, strong electronic correlationsin the context of (anisotropic) triangular lattice struc-tures are also directly relevant to the two-dimensionalmolecular materials of the κ -ET family [17]. This classof materials has been the subject of intense experimentalresearch and displays a diversity of remarkable phenom-ena (for reviews, see e.g.[18, 19]). Among those are Mottinsulating phases with either magnetic long-range orderor spin liquid behavior as in e.g. κ (ET) Cu (CN) ,a pressure-induced metal-insulator transition (MIT), aswell as superconductivity with a critical temperaturereaching ∼
14 K [20–30]. Finally, transition metal oxidessuch as the layered superconductor Li x NbO also formtriangular lattices, with structural similarities to some ofthe dichalcogenides [31–33].While the triangular lattice Hubbard model is directlyrelevant to this wide variety of materials, it is also aparadigmatic model of strongly correlated electrons sub-ject to geometric frustration and has therefore been sub-ject to intense computational and theoretical research.However, due to the high complexity of the problem, onlya partial understanding of its physics has been reached.The model is defined by the Hamiltonian:ˆ H = − t (cid:88) (cid:104) i,j (cid:105) ,σ (cid:16) ˆ c † iσ ˆ c jσ + ˆ c † jσ ˆ c iσ (cid:17) + U (cid:88) i ˆ n i ↑ ˆ n i ↓ , (1)where ˆ c † iσ , ˆ c iσ denote the fermionic creation and an- a r X i v : . [ c ond - m a t . s t r- e l ] F e b nihilation operators on site i with fermion spin σ ,ˆ n iσ = ˆ c † iσ ˆ c iσ , and (cid:104) i, j (cid:105) denotes summation over nearest-neighbor bonds of the triangular lattice.At half-filling ( (cid:104) ˆ n i ↑ + ˆ n i ↓ (cid:105) = 1) the model has a metal-lic phase for small U/t , while it is an insulator withlong-range magnetic order in the large
U/t limit for T = 0. It has been suggested early on that an inter-mediate insulating phase without magnetic long-rangeorder exists between these two phases, at intermediate U/t [34][35]. The existence of this intermediate phasehas been corroborated by several different computationalmethods [36–43]. Recent density matrix renormalizationgroup (DMRG) studies showed strong evidence that theintermediate phase ground state realizes a gapped chi-ral spin liquid (CSL) [43–45]. This elusive state of mat-ter has been proposed in the late 1980s [46, 47] and inthe last years has been found to be stabilized in sev-eral frustrated spin systems [48–54], including extendedtriangular lattice spin-1 / D max (Appendix A), the main limitation is in the finitetransverse size.As the triangular-lattice Hubbard model is afflicted bythe fermionic sign problem, we cannot use traditionalQuantum Monte Carlo techniques [85]. DiagrammaticMonte Carlo can work directly in the thermodynamiclimit and is therefore immune from the sign problem,while being numerically exact: it is possible to computequantities with arbitrary precision given enough com-putational time, and the convergence can be checkedby comparing results from different expansion orders.Reaching the strong-coupling regime can, however, behindered by the increased difficulty of resumming theperturbative series beyond their radius of convergence[74], and many orders must be computed, which in itselfcan present computational challenges. In this work, us-ing the recent computational advances of the connected-determinant version [73], we are able to compute up to10 orders of the perturbative expansion at fixed density;this is achieved [86] by renormalizing the chemical po-tential in the spirit of [76]. Thanks to the high ordersreached, we manage to get converged results with con-trolled errorbars at temperatures T /t = 0 . U/t = 10.CDMFT also works directly in the thermodynamiclimit for the lattice, but it retains only a finite num-ber of real-space components of the self-energy organizedaccording to spatial locality and approximated by theirvalue on a (self-consistent) cluster of finite size N c . Themethod is controlled in the sense that it converges tothe exact solution in the limit N c → ∞ , but in practice,this convergence can only be reached in specific parame-ter regimes. Here, CDMFT is used as an approximationwith N c = 7, and on-site and nearest-neighbor compo-nents of the self-energy are taken into account, besidesall temporal (quantum) correlations already present insingle-site DMFT.This article is organized as follows. We discuss thetransition from a metallic state at weak-coupling to an in-sulating state at strong coupling in Sec. II. There, we per-form a critical comparison between our numerical meth-ods and propose that, in the accessible range of temper-atures, a smooth crossover between these states is foundrather than a first-order phase transition. In Sec. III wediscuss the locality of the electronic self-energy by com-paring results from CDMFT and DiagMC. In Sec. IV weinvestigate the basic thermodynamic properties of thesystem and firmly establish an order-by-disorder effect,where increasing temperature decreases the double oc-cupancy. We relate this effect to an increase in entropyupon increasing the interaction strength via a Maxwellrelation. Sec. V discusses competing (magnetic) ordersas a function of temperature and interaction strength. Inparticular, we investigate magnetic structure factors andthe chiral susceptibility to propose a scenario where chi-ral and stripy antiferromagnetic spin correlations coexistat low temperatures. Finally, we summarize and discussour findings in Sec. VI. II. METAL-INSULATOR CROSSOVER
We begin by investigating the evolution from a metal-lic state at weak coupling to a Mott insulating state atstrong coupling. At high enough temperature, this is acrossover. Whether it remains a crossover down to lowesttemperatures or whether a phase transition also exists atlow but finite temperature is discussed at the end of thissection.In order to identify this crossover, we consider twocomplementary observables, which are accessible withinthe CDMFT and METTS frameworks respectively. Thefirst one is the zero-frequency value of the local (on-site)electronic spectral function: A c ( ω = 0) = − π Im G c ( iω n → i + ) . (2)This quantity is evaluated by considering the central siteof the cluster within the center-focused CDMFT method(see App. C and [83]) — hence the subscript in A c . Theextrapolation to zero frequency is obtained from a fitof the Matsubara frequency Green’s function G ( iω n ).We have also calculated within CDMFT the local andnearest-neighbor components of the self-energy and canextract the low-frequency slope: Z c = (cid:20) − ∂ Σ c ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω → (cid:21) − , (3)which is also a good indicator of the MIT. The non-localcomponents of the self-energy are found to be quite smallfor weak to intermediate U/t (see Sec. III for more de-tails), hence Z c is a reasonable approximation in thisregime to the spectral weight of quasiparticles. A c (0) isplotted in Fig. 1(a) as a function of U/t for several valuesof temperature, along with Z c at T /t = 0 . T , A c (0) undergoes a markeddrop as U/t is increased, signalling a crossover from ametal with a large value of the zero-frequency spectral . . . . . . A c ( ) (a) U/t . . . Z c T/t=0.10
T/t = 0 . T/t = 0 . T/t = 0 . T/t = 0 . T/t = 0 . U/t . . . . λ (b) T/t = 0 . T/t = 0 . T/t = 0 . L = 8 L = 16 L = 24 FIG. 1. Metal-Insulator crossover at finite-temperature fromCDMFT and METTS. (a) Spectral weight at the Fermi levelfrom CDMFT. The inset shows the quasiparticle renormaliza-tion factor for the central site in CDMFT (squares) as well asfor PM-restricted DMFT (dots). A drop of the spectral weightat the Fermi level is observed. (b) Normalized localizationlength as a function of temperature and system size as ob-tained from METTS on the YC4 cylinder. Simulations havebeen performed with maximal bond dimension D max = 4000.The normalized localization length attains a finite value inthe insulating regime and increases in the linear system size L in the metallic regime, where higher temperatures decreasethe localization length. density to an insulator with a small one (but as expectedstill finite at non-zero T ). Being a crossover there is a cer-tain arbitrariness in defining its location precisely but itis apparent that, at the lowest temperatures, it occurs for8 (cid:46) U/t (cid:46)
9. Correspondingly, Z c drops rapidly as U/t is increased. At still larger values of
U/t , the CDMFTself-energy has the characteristic divergent low-frequencybehavior of an insulator, see Fig. 14 in Appendix C. Im-portantly, we see that A c (0) increases upon cooling for U/t (cid:46)
9, while it decreases upon cooling for
U/t (cid:38) λ , which on open bound-ary conditions as employed by METTS is defined as [89–91], λ = 1 N (cid:16) (cid:104) X (cid:105) − (cid:104) X (cid:105) (cid:17) . (4)Here, X = (cid:80) i r i n i denotes the position operator, where − k x /π . . . . S c ( k x ) (a) U/t = 6 − k x /π . . . . . S c ( k x ) (b) U/t = 10
U/t . . . . α , β (c) T/t = 0 . T/t = 0 . T/t = 0 . T/t = 0 . T/t = 0 . FIG. 2. Static charge structure factor S c ( k ) for k y = 0 at var-ious temperatures on the 16 × D max = 4000. (a) In the metal-lic regime at U/t = 6, the density structure factor behaves as S c ( k x ) ≈ α | k x | . (b) In the insulating regime at U/t = 10 weobserve S c ( k x ) ≈ βk x . (c) Optimal fit parameters α, β for theansatz S c ( k x ) = α | k x | + βk x of S c ( k x ) close to k x = 0. α (resp. β ) is shown as triangles (resp. circles). The crossover inter-action strength U c /t can be defined by the intersection of α and β . We observe U c /t shifting towards weaker interactionsat higher temperatures. r i denote the coordinates of the lattice, n i the local den-sity operators, and N the total number of sites.At zero temperature, the localization length λ is di-rectly related to the real part of the conductivity σ ( ω )by [92] λ = (cid:126) πe n (cid:90) ∞ dωω Re σ ( ω ) , ( T = 0) (5)where e denotes the electron charge and n the averagedensity. The integral on the right-hand side, also re-ferred to as Souza-Wilkens-Martin integral [89, 91, 92],diverges in the metallic regime and attains a finite valuein the insulating regime for N → ∞ . The behavior of λ for temperatures T /t = 0 . , . , .
300 and YC4cylinder lengths L = 8 , ,
24 computed by METTS isshown in Fig. 1. The metallic and insulating regimes canbe coarsely distinguished by the behavior of λ . Whereasin the insulating regime, λ is almost constant as a func-tion of the cylinder length L and temperature, it increaseswith L in the metallic regime. We also observe that athigher temperature, such as T /t = 0 .
300 in Fig. 1, thelocalization length decreases, indicating increased local-ization of the system. Furthermore, we study the behavior of the chargestructure factor given by, S c ( k ) = 1 N N (cid:88) l,m =1 e i k · ( r l − r m ) (cid:104) ( n l − (cid:104) n l (cid:105) )( n m − (cid:104) n m (cid:105) ) (cid:105) , (6)where n l denotes the local density at site l . When usingMETTS we are working in the canonical ensemble withzero global charge fluctuation, which implies S c (0) = 0at any temperature. The behavior of S c ( k ) around k = 0is indicative of metallic or insulating behavior. While ametallic state at T = 0 is characterized by a linear chargedispersion [93–96], S c ( k ) ≈ α | k x | , (7)an insulating state exhibits a quadratic dispersion, S c ( k ) ≈ βk x . (8)The behavior of the charge structure factor on the 16 × U/t =6 in (a) and the insulating regime at
U/t = 10 in (b).The two regimes clearly exhibit the expected linear (resp.quadratic) behavior close to k . We observe a rather mildtemperature dependence. To quantify this behavior, wemake the following ansatz close to k x = 0,˜ S c ( k ) = α | k x | + βk x . (9)The value of α (resp. β ) can be interpreted as a metallic(resp. insulating) weight. We fit this ansatz to the nu-merical data shown in Fig. 2(a,b) for the seven k -pointsclosest to the origin. Results for a range from U/t = 4to
U/t = 12 are shown in Fig. 2(c). At low
U/t in themetallic regime, we observe that increasing temperaturedecreases the metallic weight α while increasing the in-sulating weight. In the insulating regime, we observeonly a weak temperature dependence of α and β . Wecan define a crossover interaction strength U c /t by theintersection of α ( U ) and β ( U ). We observe that U c /t shifts towards weaker interaction strengths when increas-ing temperatures. From this we estimate U c /t ≈ . T /t = 0 . U c /t ≈ . T /t = 0 . U c /t ≈ . T /t = 0 . U/t inthat range, the system undergoes increased localisationupon heating.To further study the metal to insulator crossover weinvestigate the potential energy, E pot = U (cid:88) i (cid:104) ˆ n i ↑ ˆ n i ↓ (cid:105) , (10)and the kinetic energy, E kin = − t (cid:88) (cid:104) i,j (cid:105) ,σ (cid:104) ˆ c † iσ ˆ c jσ + ˆ c † jσ ˆ c iσ (cid:105) . (11)Since these quantities are accessible with all our methods,we show a direct comparison in Fig. 3 to assess the effects U/t . . . . . E p o t / N T /t = 0 .
100 (a)
DiagMCMETTSCDMFTDMFTDMFT (PM)0 2 4 6 8 10 12
U/t − . − . − . − . E k i n / N (b) U/t . . ∂ E k i n / ∂ U U/t . . . . E p o t / N T /t = 0 . T /t = 0 . T /t = 0 . T /t = 0 . (c) U/t . . . . E p o t / N T /t = 0 . T /t = 0 . T /t = 0 . T /t = 0 . (d) L = 8 L = 16 L = 24 L = 32 FIG. 3. Comparison of energies from different computationalmethods at
T /t = 0 .
1. METTS results are obtained on a16 × D max = 4000, cluster DMFT isperformed on a 7-site cluster, while DiagMC is numerically-exact within the estimated errorbar. (a) Potential energy E pot . METTS and cluster-DMFT agree with the DiagMCresults up to U/t ≈ .
5. Beyond this point we find excellentagreements between cluster DMFT and METTS. (b) Kineticenergy E pot . We observe good agreement between all meth-ods up to U/t = 8. The inset shows the derivative of thekinetic energy w.r.t. the coupling strength U . (c,d) The po-tential energy density from METTS as a function of cylinderlength L at temperatures T /t = 0 .
025 and
T /t = 0 . of finite cluster size in CDMFT and finite cylinder sizein METTS. We focus on a temperature of T /t = 0 . U/t = 10 .
5, which are used as benchmark.Remarkably, results from all methods agree within errorbars up to an interaction strength of
U/t = 8. Beyondthis point, we still observe that the potential energy from METTS compares well with CDMFT up to the strongcoupling regime
U/t = 12. A key difference betweenMETTS and CDMFT is seen in the kinetic energy, whichis lower for METTS in the strong coupling regime. Wealso note that the CDMFT kinetic energy exhibits a slopediscontinuity around
U/t ∼ T /t (cid:39) .
1, all methods agree that the passagefrom the metal to the insulator is a smooth crossover [97].Furthermore, it is clear from previous work [36–39, 43,44, 98], that at T = 0 a metal-insulator phase transition(MIT) takes place. We now discuss whether our dataallow us to settle whether a sharp MIT also exists at lowbut finite temperature or whether a smooth crossoverapplies for any non-zero temparature.Let us first recall what the situation is in the single-siteDMFT approximation. When solving the DMFT equa-tions constrained to solutions without long-range mag-netic order, one indeed finds that a first-order MIT de-velops for T < T
DMFT c (cid:39) . t ( U DMFT c /t (cid:39)
11) as pre-viously demonstrated by several authors and also shownin Fig. 10 of Appendix B. Since there is no symmetrydistinction at finite temperature between a metal andan insulator with no broken symmetries, a first-ordertransition line ending at a second-order critical endpoint( U c , T c ) (analogous to a liquid-gas transition) is a pri-ori possible. This is what happens in DMFT [80, 99–107], as well as in cluster extensions of DMFT [108–113]when restricted to non-magnetic solutions. This transi-tion is the reason for the cusp in the kinetic energy foundwith these methods, as apparent on Fig. 3(b) around U CDMFT c /t (cid:39)
9. However it should be emphasizedthat, when allowing for spin and translational symme-try breaking, the single-site DMFT approximation yieldsa solution with 120 ◦ N´eel ordering for
U/t (cid:38) . T MIT in paramagneticDMFT/CDMFT is a hint that a similar phenomenonmight take place in our model. On a qualitative level,frustration appears as a favorable factor by further sup-pressing ordering. The ET-organic materials with an(anisotropic) triangular structure do display such a tran-sition experimentally [19].Our METTS and DiagMC results do not provide ev-idence for such a first-order MIT or liquid-gas criticalendpoint at finite temperature. In the range of tempera-tures that we could investigate, the kinetic and potentialenergy displayed in Fig. 3 do not appear to have a sin-gularity as a function of
U/t . However, we acknowledgethat limitations of our computational methods preventus to reach a definitive conclusion about this issue. OurMETTS results yield a smooth crossover between themetallic and the insulating regime for temperatures downto
T /t = 0 . T /t = 0 .
025 in Fig. 3(c) andfor
T /t = 0 .
100 in Fig. 3(d). The smooth behavior at allsystem sizes does not exhibit any tendency to develop adiscontinuity for L → ∞ . However, it is possible thatthe chosen cluster geometry or the finite precision weachieve conceal potential singularities developing in theinfinite-volume limit. The data points shown in Fig. 1are spaced by ∆ U = 0 . t where the maximal absolutestatistical error is of size ε = 5 · − . The DiagMC re-sults, while dealing with the infinite system, are limitedin the present work to T /t (cid:38) . U/t (cid:46)
10. This islargely due to difficulties in computing enough expansioncoefficients with small enough error bars to allow for con-trolled resummations of the perturbative series beyondthe aforementioned values of U . III. (NON-)LOCALITY OF CORRELATIONS:SELF-ENERGIES
Although the electronic Coulomb interaction is mod-elled as a purely local repulsion in the Hubbard modelin Eq. (1), the correlations it generates can be non-local.To assess in which part of the phase diagram non-localcorrelations become sizable in comparison to local ones,we calculate the local and nearest-neighbor (n.n.) self-energy in DiagMC and CDMFT in real space on theMatsubara axis. Fig. 4 displays the self-energy at thefirst Matsubara frequency Σ( iω = iπT ) calculated byDiagMC (crosses) and CDMFT (squares) for two differ-ent temperatures (left panels: imaginary part, right pan-els: real part). The results of the calculations from bothmethods agree within error bars for both local and n.n.components.At high T /t = 0 .
40 the correlations are mostly local andcontinuously increase from small to large U . However,there is an onset of non-locality already visible in the in-crease of the n.n. component at the largest interactionsshown. The non-local correlations remain very small atlower T /t = 0 .
10 (close to the critical temperature of theMIT in CDMFT), until quite close to
U/t ≈ .
25 at whichthe MIT takes place in CDMFT. Hence, through most ofthe metallic regime except close to the MIT, the self- − I m Σ ( i ω ) T /t = 0 . R e Σ ( i ω ) T /t = 0 . U/t − − − I m Σ ( i ω ) T /t = 0 . U/t R e Σ ( i ω ) T /t = 0 . CDMFT localCDMFT n.n. DiagMC localDiagMC n.n.
FIG. 4. Imaginary (left panels) and real (right panels) partsof the local (blue) and nearest neighbor (orange) self-energyat its lowest Matsubara frequency calculated by DiagMC(crosses) and CDMFT (squares) as a function of
U/t fortwo different temperatures
T /t = 0 . T /t = 0 . U/t ≈ .
25) larger than the one of increased localcorrelations (
U/t ≈ energy in this temperature range is local to a good ap-proximation in this frustrated system.Upon entering the insulating regime, non-local correla-tions continuously increase. These non-local correlationssignal increasing magnetic fluctuations. The onset of amagnetically ordered phase in DMFT (see App. B) un-derpins this interpretation. In the true solution of thesystem, of course, the Mermin-Wagner theorem [114, 115]prohibits magnetic ordering at finite temperature but thecorresponding magnetic fluctuations are responsible forthe increase of the (non-local) correlations. We note thatthe effect of magnetic fluctuations beyond DMFT usingthe dual fermion approximation has been investigated forthis model in Refs. [38, 97, 116] and that the implicationsof non-local effects for transport has been investigated inRef. [113]. IV. THERMODYNAMICS
We now turn to discussing the thermodynamic proper-ties of the system for a range of interactions from
U/t = 6to
U/t = 12. Figure 5 displays the specific heat, thermalentropy and double occupancy as a function of tempera- . . . . . . . . . . C / N (a) . . . . . . . . . S / N (b) U = 6 U = 8 U = 9 U = 10 U = 12 .
00 0 .
25 0 .
50 0 .
75 1 .
00 1 . T /t . . D (c) DMFTMETTSCDMFT
FIG. 5. Thermodynamics of the 16 × D max = 3000 (a) Spe-cific heat C . At U/t = 9 , ,
12 we observe a broad continuumat higher temperatures with a small peak at
T /t ≈ .
05. (b)Thermal entropy S . We observe an increase in entropy whenincreasing U/t at low temperature. (c) Double occupancy D .At low temperatures, increasing T /t decreases the double oc-cupancy. This order-by-disorder phenomenon is related to theincrease in thermal entropy with
U/t by the Maxwell relation ∂S/∂U = − ∂D/∂T . We compare our data to results from(cluster) DMFT and find the CDMFT data closely matchingthe METTS results. ture. Results for the specific heat, C = ∂E∂T , (12)of the 16 × U/t are shown in Fig. 5(a). At intermediate andlarge interaction strengths
U/t = 9 , ,
12 the specificheat exhibits a large mostly featureless plateau down totemperatures of
T /t ≈ .
1. For
U/t = 10 ,
12 a small peakdevelops at
T /t ≈ .
05 before the specific heat tends to-wards zero at T = 0.For U/t = 6 , T -linear behavior of C at lowtemperature, consistent with a metallic phase with gap-less excitations. We note that the low- T slope for U/t = 8is approximately three times larger than that at
U/t = 6: this is qualitatively consistent with Z c (inset of Fig. 1)being approximately three times smaller. For a metalin which the self-energy can be approximated as local,the quasiparticle effective mass enhancement which con-trols the slope of C is related to the quasiparticle weight Z by m ∗ /m = 1 /Z . Indeed, as shown in the previoussection, the non-local components of the self-energy aresmall through most of the metallic regime. Our findingsfor Z c and the slope of C are thus consistent with quasi-particles developing a rather heavy mass as the insula-tor is approached. Although this is difficult to ascertainfrom our data, we find no evidence for a divergence of theeffective mass when approaching the MIT however, con-sistent with the increasing non-locality of the self-energyin this regime (see Sec. III). We also observe that for U/t = 6, the specific heat appears to have another quasi-linear regime for
T /t (cid:38) . C also appears to be linear in T . However,given the few data points in this regime, it is difficult todiscern this behavior from other scenarios. We note thatthe specific heat of the YC4 cylinder closely resembles thespecific heat that has been obtained on smaller clustersusing the finite-temperature Lanczos method [117].We also investigate the thermodynamic entropy, S = log( Z ) + ET = S + (cid:90) T dΘ C (Θ)Θ , (13)where Z denotes the partition function and E the inter-nal energy. S denotes a residual entropy at zero tem-perature. The entropy is obtained by integrating thespecific heat as in Eq. (13) from T = 0, where the inter-nal (ground state) energy is computed by DMRG and weassume S = 0, i.e. we assume a unique ground state onthe finite size cylinder.Our results from METTS on the YC4 cylinder areshown in Fig. 5(b). Interestingly, we find that the en-tropy (at fixed T ) increases rapidly with increasing in-teraction strength from the metallic regime at U/t = 6to the insulating regime beyond
U/t = 9. The in-crease in thermal entropy as a function of interactionstrength has previously also been observed using thefinite-temperature Lanczos method on smaller cluster ge-ometries [117]. Naively, one would expect a decrease inentropy when the system is localizing. However, we findthe exact opposite behavior, i.e. ∂S/∂U >
0. This be-havior is also reflected in the temperature dependence ofthe double occupancy, D = 1 N N (cid:88) i =1 (cid:104) n i ↑ n i ↓ (cid:105) , (14)shown in Fig. 5(c). The Maxwell relation ∂S∂U (cid:12)(cid:12)(cid:12)(cid:12) T = − ∂D∂T (cid:12)(cid:12)(cid:12)(cid:12) U (15)relates the increase in entropy as function of U to a de-crease in double occupancy as a function of T . Indeed,as shown on Fig. 5(c), we observe a decrease of D at low- T upon heating, for all displayed values of U/t up to atemperature
T /t ≈ . V. MAGNETISM
We study the magnetic properties as a function of tem-perature and interaction strength of the system on theYC4 cylinder using the METTS algorithm. Results onmagnetic ordering from (dynamical) mean-field theorycan be found in Appendix B. To distinguish differentkinds of orderings, we investigate the magnetic structurefactor, S m ( k ) = 1 N N (cid:88) l,m =1 e i k · ( r l − r m ) (cid:104) (cid:126)S l · (cid:126)S m (cid:105) . (16)The momenta k resolved by the YC4 cylindrical geom-etry are shown in the inset of Fig. 6(a). Magnetic 120 ◦ N´eel order can be detected by observing a peak in thestructure factor at the K point in the Brillouin zone. Onthe YC4 cylinder, the K point is not exactly resolved,which is why we resort to the closest point K (cid:48) , shownin Fig. 6(a), to indicate 120 ◦ N´eel order. A peak at theM-point can indicate the following two kinds of magneticcorrelations:(i) A collinear ‘stripy’ antiferromagnetic ordering ischaracterized by breaking both spin and discrete C lattice rotation symmetry. This kind of ordering ischaracterized by the spins being aligned ferromag-netically along one direction of the triangular lat-tice and antiferromagnetically along the other two.Note that we use here the term ‘stripy’ in relationto spin degrees of freedom - we find no indicationof a charge stripe density modulation.(ii) Non-coplanar tetrahedral order, on the other hand,is formed when spins in a 2 × . . . S m ( K ) (a) M K (b) . . X (c) (d) U/t . . . . S m ( M ) (e) T/t (f)
T/t = 0 . T/t = 0 . T/t = 0 . T/t = 0 . T/t = 0 . U/t = 6
U/t = 8
U/t = 9
U/t = 10
U/t = 12
FIG. 6. Magnetic ordering as a function of interactionstrength
U/t (left) and temperature
T /t (right) on the 16 × D max = 3000. (a,b) Mag-netic structure factor S m (K (cid:48) ) indicating 120 ◦ N´eel order. Theinset in (a) shows the momenta resolved by the YC4 cylinder,and the position of the ordering vectors K (cid:48) and M (c,d) chiralsusceptibility X as defined in Eq. (17). Chiral correlationsbuild up in both the intermediate as well as the strongly cou-pled regime at lower temperature. (e,f) Magnetic structurefactor S m (M) indicative of collinear stripy antiferromagneticorder. Several recent DMRG studies [43–45] have demonstratedthat the triangular Hubbard model in the intermediateregime is susceptible to time-reversal symmetry breaking,which is indicated by a non-zero expectation value of thescalar chirality operator in the thermodynamic limit. Tostudy such a scenario, we compute the chiral susceptibil-ity X = 1 N (cid:88) µ,ν ∈(cid:52) (cid:104) χ µ χ ν (cid:105) , (17)where the scalar chirality operator on a triangle µ = ( l, m, n ) is given by χ µ = (cid:126)S l · ( (cid:126)S m × (cid:126)S n ) . (18)The sum in Eq. (17) extends over all pairs of elementarytriangles. In the case of spontaneous time reversal break-ing, we expect long range chiral correlations indicated by FIG. 7. Snapshots of METTS states | ψ i (cid:105) at temperature T /t = 0 . × U/t = 10 (a) and in the strong coupling regime at
U/t = 12 (b). The length of the arrows is proportional to the spincorrelation (cid:104) (cid:126)S · (cid:126)S i (cid:105) , where the black cross marks the reference site. Hexagons and arrows which are blue indicate positive andred indicate negative spin correlations. The color of the triangles indicates the magnitude of the chiral correlation (cid:104) χ χ µ (cid:105) , wherethe reference triangle is indicated in gray just below the reference site. Nearest-neighbor spin correlations (cid:104) (cid:126)S i · (cid:126)S j (cid:105) are indicatedas the width and color of the bonds. We observe collinear stripy spin correlations at this temperature in the intermediate regimeat U/t = 10. a large value of the chiral susceptibility X . Since oursimulations are working with real-valued wave functions,the expectation values of the scalar chirality operators, (cid:104) χ µ (cid:105) , are exactly zero.The behavior of the above quantities as both a func-tion of U/t and temperature
T /t is shown in Fig. 6. Weobserve three distinct regimes as a function of
U/t . For
U/t (cid:46) . . (cid:46) U/t (cid:46) . K (cid:48) and M points Fig. 6(b,f). But below the temper-ature T /t = 0 . U/t = 10, the K (cid:48) -point structure fac-tor begins to decrease, while at the M -point it increasessharply beginning around T /t = 0 .
05. The chiral corre-lations in Fig. 6(d) also increase below this scale, and thespecific heat simultaneously develops a small maximumthen rapidly decreases as shown in Fig. 5(a). The de-velopment of low-T chiral correlations is consistent withprevious DMRG results [43, 45], which proposed thatat T = 0 the system spontaneously breaks time-reversalsymmetry and forms a chiral spin liquids. However, wealso observe that the chiral correlations similarly buildup beyond U/t (cid:38) . Z e − βH = (cid:88) i p i | ψ i (cid:105) (cid:104) ψ i | , (19)where p i ≥ | ψ i (cid:105) are the so-called METTS wave functions, and Z denotesthe partition function. The pure states | ψ i (cid:105) are sampledwith probability p i . We show the properties of a typ-ical METTS wave function sampled in our simulationsat T /t = 0 . U/t = 10 in Fig. 7(a). The stripyspin correlations are clearly pronounced for this METTSstate. We also observe sizeable chiral correlations (cid:104) χ χ µ (cid:105) which are indicated by the color of the inner triangles.When comparing the snapshot at U/t = 10 in Fig. 7(a)to the snapshot at a larger
U/t = 12 in Fig. 7(b) weobserve that the nearest-neighbor spin correlations aremore strongly pronounced along the short direction ofthe cylinder at
U/t = 10, whereas for
U/t = 12 the spincorrelations on the other two directions are enhanced.One last notable aspect of the intermediate regime wefind is that increasing the temperature from T = 0 notonly suppresses double occupancy, as shown previously in0 U/t . . . S m ( K ) (a) T/t = 0 . T/t = 0 . T/t = 0 . U/t . . . X (b) U/t . . . . S m ( M ) (c) FIG. 8. Size dependence of magnetic structure factor S m ( k )and the chiral susceptibility X for three different tempera-tures. We compare YC4 cylinders of length L = 16 , , D max = 4000 (a) Magnetic structure factor S m (K (cid:48) ) (b)Chiral susceptibility X (c) Magnetic structure factor S m (M) Fig. 5, but also increases the 120 ◦ N´eel order correlations.Therefore, the effect of increasing temperature is similarto the effect of further increasing the coupling strength
U/t , which also both localizes the system and favors 120 ◦ N´eel order for
U/t ≥ . U/t (cid:38) . S m (K (cid:48) ) at lower temperatures, shownin Fig. 6(a,b). This is indicative of 120 ◦ N´eel order inthe ground state. We observe strong antiferromagneticcorrelations for
U/t = 12 setting in at a temperaturebelow
T /t = 0 .
05, which again coincides with the smallmaximum in the specific heat observed in Fig. 5(a). Wefind that the behavior of the chiral correlations is sim-ilar to the intermediate coupling regime. In particular, X as shown in Fig. 5(d) is rather comparable between U/t = 10 and
U/t = 12.As our results in Fig. 6 are obtained on a par-ticular system size, we investigate the dependence ofthe magnetic observables on the cylinder length L inFig. 8. We compare simulations at temperatures T /t =0 . , . , .
300 for 6 ≤ U/t ≤
12. We find that outresults only weakly depend on the cylinder length L andare, therefore, expected to be robust in the limit L → ∞ . VI. DISCUSSION
The physics of the triangular lattice Hubbard model athalf-filling is coarsely organized in three different regimesas a function of the coupling strength,
U/t : a metallicregime is followed by an intriguing insulating regime atintermediate coupling regime whose nature is currentlyhotly debated. At large interaction strength the systementers a magnetic insulating regime, where coplanar 120 ◦ N´eel order is stabilized in the ground state. Evidence forthe existence of an intermediate non-magnetic insulatingregime is ample in the literature [34, 36–43, 45, 116] andclearly confirmed by several of our findings using multiplenumerical methods.We firmly establish the order-by-disorder effect at in-termediate coupling
U/t , where increasing temperatureparadoxically leads to increased localization, as apparentin the double occupancy shown in Fig. 5(c). As discussedin Sec. IV, this effect is similar to the Pomeranchuk ef-fect [118] observed when liquid Helium 3 solidifies uponheating, and previously found to occur for the Hubbardmodel in DMFT studies [99, 100, 119, 120, 122]. The de-crease in double occupancy upon heating is confirmed byboth our METTS and cluster DMFT results, where wefound good quantitative agreement between these twovery different numerical techniques. This observationsuggests that localized excitations carry a large thermalentropy at low temperatures. This is consistent with theMaxwell relation Eq. (15) relating decreasing double oc-cupancy in temperature to an increasing entropy withinteraction strength U , which we confirm by computingthe thermal entropy from METTS in Fig. 5(b), as alsopreviously observed using the finite-temperature Lanczosmethod on smaller cluster geometries [117].This order-by-disorder effect naturally gives rise to thequestion about the nature of the proliferating excitationscausing the localization at finite temperature. Let us firstdiscuss the intermediate coupling regime at U/t = 10. Aswe have shown in Fig. 6, both the chiral correlations aswell as the magnetic structure factor at the M point de-velop a maximum towards T = 0 at U/t = 10 on the16 × S m (K (cid:48) ) at T /t = 0 . ◦ spin correlations. It is interesting to note, that in-creasing temperature appears to have the same effect asincreasing interaction strength, which also favors 120 ◦ order. As previously explained, both increasing temper-ature (cf. Fig. 5(c)) as well as increasing the interactionstrength U/t localizes the system, which appears to beenergetically favorable for 120 ◦ spin correlations.Let us now turn to discussing the orders which maydevelop in the intermediate regime 8 . (cid:46) U/t (cid:46) . S m (M) is not necessar-ily related to the formation of a CSL. While a peak at theM point could in principle be indicative of non-coplanartetrahedral order [125], we find that in the present geome-try this peak is related to the formation of nematic, stripyantiferromagnetic correlations. This finding is backed upby a recent ground state DMRG study [45], which analo-gously found a peak in S m (M) in the intermediate regime,although the authors found these correlations to be onlyshort-ranged at T = 0.The occurrence of stripy spin correlations is remarkablegiven that most known instances of chiral spin liquidsare stabilized in close proximity to non-coplanar mag-netically ordered states [48, 50–52, 55, 56]. The meltingof non-coplanar magnetic ordering has even been sug-gested as a guiding principle to understand the forma-tion of CSLs [126]. This seems to be rather differentin the present case, where nematic collinear correlationsand chiral correlations both develop as the temperatureis decreased. In this context, the variational study ofthe triangular lattice Heisenberg model with an addi-tional ring-exchange term performed in Ref. [62] is partic-ularly interesting. The model with ring exchange can bethought of an approximate low-energy effective Hamil-tonian for the intermediate coupling regime [42, 59].The authors compared variational energies of severalGutzwiller projected ansatz wave functions, including anansatz for a gapped chiral spin liquid and a gapless ne-matic spin liquid, breaking rotational symmetry. Whileboth of these wave functions have been shown to havea comparable, competitive energy, the gapless nematicstate had the lower variational energy for this particu-lar model. A more recent variational Monte Carlo studyhas similarly suggested the stabilization of a gapless ne-matic spin liquid in the context of the half-filled trian-gular Hubbard model, albeit upon adding further secondnearest-neighbor interaction [65]. Remarkably, our finite-temperature METTS simulations now reveal exactly thiscompetition between between chiral and stripy spin cor-relations at finite, but low temperatures. While the re-cent DMRG studies [43, 45] provided strong evidence,that ultimately at T = 0 a CSL is formed on the in-vestigated geometries, we now propose that stripy spincorrelations become relevant immediately at finite tem-peratures. We would like to point out, that an estimatefor the gap ∆ of the CSL has been stated in Ref. [43]to be ∆ ≈ . t , which is below the lowest temper-ature T /t = 0 . × S m (K (cid:48) ) whenincreasing temperatures in Fig. 6(a), or the decrease ofthe double occupancy in Fig. 5(c).This raises the question about whether a (rotationally-symmetric) perturbation could stabilize nematic stripy(quasi-)order at T = 0. In particular, it would be inter-esting to find out whether indeed a nematic gapless spinliquid with algebraic spin correlations can be realized and to study its transition to the CSL. However, such a statewill likely have larger quantum entanglement than thegapped CSL, rendering accurate DMRG computationsmore difficult.We expect the balance between chiral or nematic spincorrelations to be strongly dependent on the finite sizegeometry. As the DMRG studies on the YC3-6 and XC4cylinders have shown [43–45], the precise nature of theground state still has rather strong dependence on theexact shape of the cylindrical geometry. A more detailedcomparison of our finite-temperature METTS data fordifferent geometries is therefore highly desirable. How-ever, due to computational resource requirements this iscurrently beyond the scope of this manuscript and willbe subject of a future study. At this point, we wouldlike to comment that finite-size effects are expected tobecome less severe at higher temperatures, since corre-lation lengths typically decrease. It remains to be seendown to which temperature scale the finite-size cylinderscan fully capture the two-dimensional limit.Finally, an outstanding question is the occurrence ofsuperconductivity in the present model. While generalarguments suggest that the metallic phase studied herehosts a low-temperature superconducting instability atweak coupling (see e.g. [127]), the possible occurrenceof an unconventional superconducting phase near themetal-insulator phase boundary [36] or upon doping theinsulating phase [58] are intriguing questions for futurecomputational studies. ACKNOWLEDGEMENTS
We are indebted to Nils Wentzell, Steven White,Michael Zaletel, and Sabine Andergassen for insightfuldiscussions and support. We thank Elio K¨onig for valu-able comments on the manuscript. We thank the com-puter service facility of the MPI-FKF and the Scien-tific Computing Core of the Flatiron Institute for theirhelp. METTS results were obtained using the ITensorLibrary (C++ version) [128] and CDMFT computationsused the TRIQS library[129]. This work was grantedaccess to the HPC resources of TGCC and IDRIS un-der the allocations A0090510609 attributed by GENCI(Grand Equipement National de Calcul Intensif). Thepresent work was supported by the Austrian ScienceFund (FWF) through the Erwin-Schr¨odinger FellowshipJ 4266 - “
Superconductivity in the vicinity of Mott insu-lators ” (SuMo, T.S.). It also has been supported by theSimons Foundation within the Many Electron Collabora-tion framework. A.G. also acknowledges the support ofthe European Research Council (ERC-QMAC-319286).The Flatiron Institute is a division of the Simons Foun-dation.2 . . D (a) T /t = 0 . (b) T /t = 0 . . . S m ( M ) (c) (d) S m ( K ) (e) (f) U/t . . X (g) U/t (h) D max = 1000 D max = 2000 D max = 3000 D max = 4000 FIG. 9. Convergence of METTS results on the 16 × D max . Wecompare results from simulations performed with D max =1000 , , , D max = 2000 , , D max = 1000 deviate slightly. Comparisons are performed at T /t = 0 .
025 (left) and
T /t = 0 . D (a,b), the magnetic structure factors evaluatedat M (c,d) and K (cid:48) (e,f), and the chiral susceptibility X (g,h). APPENDIXAppendix A: Convergence of METTS simulations
We employ the METTS algorithm as described inRef. [84]. Thermal expectation values of an operator O are evaluated as, (cid:104)O(cid:105) = (cid:104) ψ i |O| ψ i (cid:105) , (A1)where the minimally-entangled typical thermal states, | ψ i (cid:105) = e − βH/ | σ i (cid:105) , (A2)are imaginary-time evolved product states | σ i (cid:105) . Here, · · · denotes statistical averaging over a series of subsequentMETTS. As such, the METTS algorithm is subject tostatistical sampling uncertainty, which can be reduced bycomputing more samples and whose size can be estimatedusing standard time series analysis. The imaginary-time U/t T / t QCPDMFTQCPMFT T
DMFTN T MFTN T DMFTN T DMFT ( PM ) c T CDMFT ( PM ) c FIG. 10. Magnetic phase diagram of the isotropic triangu-lar Hubbard model calculated by MFT (red dashed line) andDMFT (red circles and solid line). The red lines mark thesecond order transition from a paramagnet (white) to a 120 ◦ N´eel ordered phase (red shaded). Also shown are the criticaltemperatures and interactions of the (PM restricted) DMFT(orange triangle) and CDMFT (blue square). Please note thatthe CDMFT solution has been restricted to its paramagneticphase. evolution is performed by using the time-dependent vari-ational principle (TDVP) algorithm for matrix productstates [130–132]. In Ref. [84], some of us showed thatthe maximal bond dimension d of the matrix productstate representation of the METTS serves as a controlparameter to achieve accurate and controlled computa-tions on finite size cylinders. Here, we performed exten-sive comparisons between simulations at different bonddimensions. Results on the 16 × T /t = 0 .
025 and
T /t = 0 . U/t are shown in Fig. 9. Simulations have been performed upto a maximal bond dimension of D max = 4000. We findthat all quantities of interest are converged within error-bars already at D max = 2000. Results shown in the mainmanuscript have all been attained from simulations at D max = 2000. Appendix B: Magnetic phase transition in(dynamical) mean-field theory
In this Appendix we give an overview of the mag-netic properties of the Hubbard model on the isotropictriangular lattice calculated by means of the dynamicalmean-field theory (DMFT). By the inclusion of all tem-poral correlations present in the Hubbard model Eq. (1),DMFT has proven to provide a good starting point forthe application of more sophisticated techniques, whichaim at including spatial correlations on top (see, e.g.,[38, 69, 116]).The main panel of Fig. 10 shows the N´eel tempera-ture calculated in DMFT T DMFT N (red line and circles)3 k x / k y / T/t=0.40, U/t=2.0 k x / k y / T/t=0.10, U/t=2.0 k x / k y / T/t=0.40, U/t=8.0 k x / k y / T/t=0.10, U/t=8.0 k x / k y / T/t=0.40, U/t=10.0 k x / k y / T/t=0.10, U/t=9.5 Re DMFTm ( k , i n = 0) FIG. 11. Momentum dependence of the DMFT magneticsusceptibility at zero frequency for several temperatures andinteractions. The leading contribution is (centered around) k = K . T R e m , D M F T ( k = K , i n = ) U/t=8U/t=12
FIG. 12. Temperature dependence of the (inverse) magneticsusceptibility at k = K and zero Matsubara frequency for twodifferent interaction values calculated by DMFT. For U/t = 12the linear fit to determine T DMFT N is shown. and static mean-field theory (MFT, dashed line), sep-arating the paramagnetic from a magnetically ordered phase [133]. In contrast to the case of the Hubbard modelon a square lattice where due to the nesting propertiesof the Fermi surface the order appears for every finite U at low enough temperatures, here a quantum criticalpoint separates a Fermi liquid from a magnetically or-dered ground state at U DMFTQCP /t ≈ . T DMFT N increases steeply with a maximum of T DMFT N, max /t ≈ .
25 around
U/t = 11 before slowly decreas-ing again.Two points are particularly noteworthy:(i) as in the case of the Hubbard model on a squarelattice the critical end point of the Mott MIT (or-ange triangle) visible in the paramagnetically re-stricted DMFT is shadowed , i.e. preempted, by themagnetic phase transition of DMFT (with the mag-netically order phase being the thermodynamicallystable phase of DMFT) and(ii) the CDMFT critical end point (blue square) liesclose to the phase boundary of the magnetic phase.Please note that we did not calculate the mag-netic phase diagram in case of CDMFT, which isrestricted to its paramagnetic solution.Comparing the magnetic phase diagram of DMFT tothe self-energies of CDMFT at
T /t = 0 .
10 presented inFig. 4 of the main text, one can observe that the nearest-neighbor component of the CDMFT self-energy starts toincrease at the interaction value
U/t = 9 .
5, where theDMFT orders magnetically. In other words the spatialmean-field approximation reflects the increase of non-local fluctuations by entering an ordered phase.For the determination of the DMFT phase boundary,we calculated the momentum-dependent magnetic sus-ceptibility χ DMFT m ( k , i Ω n = 0) at zero Matsubara fre-quency by means of the solution of the Bethe-Salpeterequations with the irreducible vertex extracted from theself-consistently determined Anderson impurity model[80], using the continuous time quantum Monte Carlosolver in its interaction expansion (CT-INT) and the tprfframework [134] of TRIQS [129]. For the vertex we usedup to N iω = 50 positive fermionic Matsubara frequenciesand extrapolated the value of the physical susceptibilityto N iω → ∞ with χ ∼ a + b/N iω (see, e.g., SupplementalMaterial of [135]).Due to the second order nature of the phase transi-tion, approaching the phase boundary χ DMFT m ( k , i Ω n = 0)diverges at the ordering vector k = Q . Fig. 11 shows χ DMFT m ( k , i Ω n = 0) at T /t = 0 .
40 (left column) and
T /t =0 .
10 (right column) for several interaction values. Onecan see that the leading contribution always stems frommomentum vectors centered around k = K . Approachingthe transition, at T /t = 0 . χ DMFT m ( k , i Ω n = 0) con-tinuously grows before it eventually diverges at k = K .The temperature dependence of the inverse susceptibility χ − m, DMFT ( k = K, i Ω n = 0) is shown in Fig. 12 for two dif-ferent values of the interaction. At U/t = 8 (with a Fermiliquid ground state present in DMFT) it exhibits Pauli-4 ~a ~a tt t FIG. 13. Cluster geometry with N c = 7 used in CDMFT.The cluster consists of a central site (marked in red) and sixequivalent outer sites arranged on a ring. The translationvectors are (cid:126)a = (5 / , √ /
2) and (cid:126)a = (5 / , − √ / like behavior, i.e. approaching a constant at low temper-atures. At U/t = 10 (with a magnetically ordered groundstate) the susceptibility diverges as χ ∼ (cid:12)(cid:12) T − T DMFT N (cid:12)(cid:12) − γ T at T DMFT N /t ≈ .
22 with γ T = 1 being the susceptibility’smean-field critical exponent. Appendix C: Cellular dynamical mean-field theory:cluster geometry, Matsubara data andcomputational details
For the CDMFT calculations performed in this workwe used N c = 7 sites which are arranged according toFig. 13 with a central site and six equivalent sites thatform an outer ring. We restrict the CDMFT to its para-magnetic solution. Due to the previously found obser-vation [83] that the self-energy obtained from a clustercenter focused extrapolation converges faster with thecluster size than the periodization schemes previously in-troduced in the literature, for single-particle observables(like the spectral function shown in panel (a) of Fig. 1)and potential energies (double occupancies) we show val-ues for the central site. Similarly for the self-energiesshown in Fig. 4 we took the central site as representativefor its local component and as its nearest-neighbor com-ponent the values from central site to one of the (equiva-lent) outer ring sites. This results in a remarkable goodagreement with the results from numerically exact Di-agMC in the regimes where DiagMC can be controllablyresummed.For completeness and reference in Fig. 14 we show theMatsubara frequency dependence of the single-particleproperties spectral function A c as expressed by theGreen function (upper panel), the imaginary part (cen-tral panel) and real part (lower panel) of the self-energy. These quantities are shown for T /t = 0 . U/t . For the self-energy weshow both the central site and nearest neighbor values. n A c = I m G c ( i n ) T/t=0.1
U=1.0U=2.0U=3.0U=4.0U=6.0U=7.5U=8.0 U=8.25U=8.5U=8.75U=9.0U=9.25U=9.75U=10.5 n I m c ( i n ) U=1.0U=2.0U=3.0U=4.0U=6.0U=7.5U=8.0 U=8.25U=8.5U=8.75U=9.0U=9.25U=9.75U=10.5 n I m n . n . ( i n ) U=1.0U=2.0U=3.0U=4.0U=6.0U=7.5U=8.0 U=8.25U=8.5U=8.75U=9.0U=9.25U=9.75U=10.5 n R e c ( i n ) U n / U=1.0U=2.0U=3.0U=4.0U=6.0U=7.5U=8.0 U=8.25U=8.5U=8.75U=9.0U=9.25U=9.75U=10.5 n R e n . n . ( i n ) U=1.0U=2.0U=3.0U=4.0U=6.0U=7.5 U=8.0U=8.25U=8.5U=8.75U=9.0
FIG. 14. (Upper row) Spectral weight A c expressed with theimaginary part of the Green function on the central site ofCDMFT at T /t = 0 . For the real part at the central site we have subtractedthe respective Hartree term. The MIT is clearly visiblebetween
U/t = 9 and 9 .
25 as (i) a suppression of the spec-tral weight at low frequencies and (ii) a change of slope[136, 137]) and eventually divergence of the imaginarypart of the self-energy on the central site with increas-ing
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