The transition from a bad metal to a Mott insulator: the case of organic charge transfer salts Ross McKenzie condensedconcepts.blogspot.com

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condensedconcepts.blogspot.com

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Collaborators

•  Theory: Jaime Merino (Madrid) Ben Powell (UQ) Rep. Prog. Phys. 74, 056501 (2011) Jure Kokalj (UQ -> Slovenia) •  Experiment: Martin Dressel (Stuttgart)

Outline •  Organic charge transfer salts are model tuneable systems with a bad metal – Fermi liquid crossover. •  Signatures of a bad metal. •  Numerical study (Finite Temperature Lanczos Method) of the Hubbard model on the anisotropic triangular lattice exhibits a transition between a bad metal and Mott insulator. •  FTLM results consistent with physical picture of Dynamical Mean Field Theory. J. Kokalj & RHM, Phys. Rev. Lett. 110, 206402 (2013)

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Signatures of a bad metal •  Resistivity of order ha/e2 ~ mΩ-cm •  Non-monotonic temperature dependence of resistivity, Hall resistance, and thermopower •  Thermopower, S ~ kB/e ~ 80 µV/K •  Absence of Drude peak in ac conductivity σ(ω) •  Low coherence temperature Tcoh << TF0 , noninteracting Fermi temperature. Merino & RHM, Phys. Rev. B 61, 7996 (2000) condensedconcepts.blogspot.com

Bad metals are ubiquitious Merino & RHM, Phys. Rev. B 61, 799 (2000) •  Transition metal oxides, e.g., V2O3 •  High-Tc cuprate superconductors •  Superconducting organic charge transfer salts, κ-(BEDT-TTF)2Cu[N(CN)2]Br •  Intercalated buckyballs, K3C60 •  Heavy fermion compounds, UPt3 •  Iron pnictide superconductors, LaO1−xFxFeAs

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When good metals turn bad •  In many strongly correlated electron materials there is a smooth crossover with increasing temperature from a Fermi liquid metal with welldefined quasi-particles to a bad metal. •  Crossover occurs at a coherence temperature Tcoh much less than the Fermi degeneracy temperature in the absence of electron-electron interactions. •  Described well by a Dynamical Mean-Field Theory (DMFT) treatment of the Hubbard model. Merino & RHM, Phys. Rev. B 61, 799 (2000)

Temperature dependence of thermopower exhibits Fermi liquid to bad metal crossover S(T) ~ T at low T S ~ kB/e at high T Tcoh ~ 50 K TF0 ~ 1000 K κ-(BEDT-TTF)2Cu[N(CN)2]Br Yu et al, PRB 44, 6932 (1991)

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Temperature dependence of thermopower exhibits Fermi liquid to bad metal crossover in heavy fermion compounds

S(T) ~ T at low T S ~ kB/e at high T

E.D. Mun et al., PRB 86, 115110 (2012)

Superconducting molecular charge transfer salts, β-ET2I3 Sulphur

Carbon

Hydrogen

Iodine

B. Powell and RHM, Rep. Prog. Phys. 74, 056501 (2011)

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Organic charge transfer salts are tuneable Phase diagram: Temperature vs. pressure X=1

X=0

κ-ET2Cu[N(CN)2]BrxCl1-x

metal

B. Powell and RHM, Rep. Prog. Phys. 74, 056501 (2011)

Minimal model Hamiltonian for κ-(ET)2X

Hubbard model on anisotropic triangular lattice at half filling

ˆ =H ˆ +H ˆ H t U Hˆ t = t ∑ cˆi+σ cˆ jσ + tʹ′ ∑ cˆi+σ cˆ jσ − µ ∑ cˆi+σ cˆiσ {ij}σ

ij σ

iσ

Hˆ U = U ∑ cˆi+↑cˆi↑cˆi+↓ cˆi↓ i

Each site corresponds to the bonding orbital of an ET dimer

t

t’

Kino and Fukuyama, 1996; RHM, 1998: B. Powell and RHM, Rep. Prog. Phys. 74, 056501 (2011)

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Experiment: conductivity vs. frequency

Collapse of Drude peak above Tcoh ~ 50 K

J. Merino et al., Phys. Rev. Lett. 100, 0864040 (2008)

Conductivity from Dynamical meanfield theory (DMFT)

J. Merino et al., Phys. Rev. Lett. 100, 0864040 (2008)

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Numerical method: Finite temperature Lanczos method Jacklic & Prelovsek, Adv. Phys. 43, 1 (2000) •  Hubbard model with 16 electrons on 16 sites (half filling). •  Dimension of Hilbert space ~ billion states •  Exact diagonalisation of Hamiltonian to obtain lowlying eigenstates. •  Average over twisted boundary conditions to reduce finite-size effects. •  Thermal sums accelerated by averaging over random states. •  Unreliable at low temperatures (T < t/30).

Charge susceptibility vs. temperature t’/t=1

Metallic phase (small U): c - smaller than for U=0 - weak T dependence

Insulating phase (large U): - strongly suppressed at low T (i) - activated behavior c =

ae

c /T

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Mott metal-insulator transition Charge susceptibility vs. U Charge gap vs. U

Is transition Continuous or First-order? J. Kokalj & RHM, Phys. Rev. Lett. 110, 206402 (2013)

Phase diagram at zero temperature: interactions vs. frustration •  x Consistent with other numerical methods: e.g., Variational QMC, Path integral RG

J. Kokalj & RHM, Phys. Rev. Lett. 110, 206402 (2013)

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Specific heat vs. temperature t’=t

In metallic phase, CV linear in T for T < Tcoh~ 50 K << t ~TF(U=0), consistent with experiment J. Kokalj & RHM, Phys. Rev. Lett. 110, 206402 (2013)

Entropy vs. temperature •  Low-T entropy increased due to U •  approaches ln(2):

U=12t

U=0

t’=t

s = kB

1 [ln N

⇥H⇤ ] T

•  Large in metallic phase (U≈Uc) •  signaling development of local moments •  Smooth transition from metal to insulator

= Tr(e

(H E0 )

)

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Spin susceptibility vs. temperature (Sz )2 ⇥ s NT •  Does not show Pauli paramagnetism (weak T dependence). =

t’=t

•  Close to Curie-Weiss behavior s

=

TCW

1 1 4 T + TCW = J + J /2

Mean-field for Heisenberg antiferromagnet

-  Well formed local moment already for U≈Uc -  Longer range spin correlations important only at low T < 0.4t (0.1t) -  Supports bad metal behavior with short range spin correlations

Signatures of the bad metal •  •  •  •  • 

Small charge compressibility Large entropy per electron ~ kB ln(2) Large spin susceptibility Low coherence temperature, Tcoh ~ 50 K Qualitatively consistent with Dynamical Mean-Field Theory (DMFT)

J. Kokalj & RHM, Phys. Rev. Lett. 110, 206402 (2013)

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)

of the DMFT era some 20 years ago, yet, surprisingly, theoretical method that is most reliable precisely at high some of its basic features have remained ill understood and even confusing. Most studies focused on characterizing temperatures. the low-temperature behavior, where a strongly correlated Fermi liquid (FL) forms on the metallic side of the Mott Model and DMFT solution.—We consider a single-band transition. At low temperatures, this FL phase is separated from the Mott insulator by an intervening phase coexistence Hubbard model at half filling region (see Fig. 1), and the associated first-order transition line (FOTL) terminating at the critical end point (CEP) at X X T = T . The behavior in the immediate vicinity of the CEP H¼% tij ðcyi% cj% þ c:c:Þ þ Uni" ni# ; (1) has attracted much recent attention but, unsurprisingly (as 18

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c

where cyi% and ci% are the electron creation and annihilation operators, respectively, ni% ¼ cyi% ci% , tij is the hopping amplitude, and U is the repulsion between two electrons on the same site. We use a semicircular density of states, and the corresponding half-bandwidth D is set to be our energy unit. We focus on the paramagnetic DMFT 10

δU>0

ρ/ρc

10

T

0.2

Classical critical

Mott insulator

Coexistence

FIG. 1. (Color online) Phase diagram of the half-filled maximally frustrated Hubbard model. The background is an actual color map of the resistivity obtained using the IPT impurity solver (see the text): Blue, small resistivity; red, large resistivity.

displaying all features expected of quantum crit resistivity around this line exhibits a characte shaped” form, surprisingly similar to experimen in several systems,1,20,21,25–27 reflecting gradua from metallic to insulating transport. The scalin in this high-temperature crossover regime was t to encapsulate the universal features of finite-t transport near the metal-insulator transition. The work of Ref. 17 focused on behavior c “instability line” and the associated quantum crit regime around it. It should be noted, however, t other finite-temperature crossover lines have been by other authors16,24,28–30 to characterize the met region. The exact relationship between these diff and approaches—for the same model—thus re open and rather confusing issue that needs to b investigated and understood. This important task subject of this paper, where we present a detaile precise characterization of all the crossover regime entire phase diagram for the maximally frustrate model at half filling, within the paramagnetic dynamical mean-field theory. We carefully chara relevant crossover lines employing all the variou criteria used for their definitions. Two fundamenta crossover regions are identified: one referring to destruction of long-lived quasiparticles and th the gradual opening of the Mott gap. The insta as previously determined from a thermodynamic belongs to the latter region, and is found to lie v the line of inflection points in the resistivity curve The scaling of resistivity curves found around bo lines is analyzed and discussed from the perspectiv quantum criticality and its experimental observa end, we outline the generalized concept of the W and argue that they gain a new fundamental m the context of quantum-phase transitions, which avenue to put our results into a more general framework.

Based on Dynamical Mean-Field Theory [DMFT] of Hubbard model at half-filling

075143-2

δU<0

0.1

Quantum critical

Interaction

1

1

0.05

Bad metal

Fermi liquid

(b) IPT CTQMC

-0.2<δU<+0.2

100

20,21

Temperature

Quantum critical transport near the Mott transition (a)

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any other finite-temperature CEP), it display scaling behavior of the standard classical liquid-gas (Ising) universality class.19 Indeed, several experiment reporting transport in this regime have successfully been interpreted22 using these classical models.

i

hi;ji%

ρ/ρMott

similar transport features [1–3]. Here, the family ssumes a characteristic a)], reflecting a gradual lating transport. At the ty depends only weakly tration of charge carriers T is lowered, the system and rapidly converges insulating state. Since toff scale for the metalor is precisely what one ]. In some cases [1], the eautiful scaling behavior, metry’’ of the relevant which microscopic conscaling phenomenology? force for the transitions? sic physics questions recontroversy and debate. d-driven Anderson localons is at present rather scaling formulation [5] mple of a T ¼ 0 quantum d, a considerable number de compelling evidence —some form of Mott minant mechanism [6]. very different transport ure? Is the paradigm of ul language to describe d the Mott point? These address, because convenmply cannot be utilized ure incoherent regime. estion in the framework

0.1

0.3

0.6

1

3

T/T0

Vucicevic, H. Terletska, D. Tanaskovic, and V. Dobrosavljevic FIG. J. 1 (color online). (a) DMFT resistivity curves as a function of temperature along(2011); differentPRB trajectories %0:2 ' "U ' PRL 107, 026401 88, 075143 (2013) þ0:2 with respect to the instability line "U ¼ 0 (black dashed line; see the text). Data are obtained by using IPT impurity solver. (b) Resistivity scaling; essentially identical scaling functions are found from CTQMC (open symbols) and from IPT (closed symbols).

026401-1

! 2011 American Physical Society

Open questions & future work •  Nature of fluctuating local magnetic moments in the bad metal phase? •  Is Mott transition first or second order? •  Is there a pseudogap? •  Thermal expansion anomalies •  Calculate transport properties with FTLM •  Detailed comparison of FTLM with DMFT, esp. quantum critical transport.

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Conclusions •  Organic charge transfer salts are model tuneable systems with a bad metal – Fermi liquid crossover. •  Numerical study (FTLM) of Hubbard model on the anisotropic triangular lattice exhibits a transition between a bad metal and Mott insulator. •  FTLM captures low coherence temperature. •  FTLM consistent with Dynamical Mean Field Theory picture. J. Kokalj & RHM, Phys. Rev. Lett. 110, 206402 (2013)

condensedconcepts.blogspot.com

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Heisenberg model on anisotropic triangular lattice: what is ground state? Wavevector (q,q) for magnetic order vs. J’/J quantum classical

Series expansions Zheng, Singh, RHM PRB (1999)

Heisenberg model on anisotropic triangular lattice: frustration destroys magnetic order Ordered moment vs. J’/J

Valence bond solid

Spin gap vs. J’/J

Zheng et al., PRB 1999

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Small change in U/t and t’/t changes ground state RVB theory of Hubbard-Heisenberg model on the anisotropic triangular lattice at half-filling t t

t’ J’

J

J Powell and RHM, Phys. Rev. Lett, 2007.

Numerical method Finite temperature Lanczos method •  Random vector is decomposed on to the approximate Lanczos eigenstates M

|r⇥

ri | i ⇥

•  Extremal Lanczos eigenstates converge H|⇥i ⇥ i |⇥i ⇥ rapidly (ground state for M≈50) 2 •  |ri | on average roughly proportional to the density of at •  states Thermodynamic sum (example) H Nst R M r = Tr e |ri |2 e ⇥i R r=1 i=1 i=1

i

J.Jaklič, P.Prelovšek, Adv. Phys. 49, 1 (2000).

University of Queensland, Brisbane, Australia

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Lanczos method •  expansion of arbitrary vector in Hilbert space into approximate “Lanczos” eigenstates M

|r⇥

|r

|

i

H|⇥i ⇥

i |⇥i ⇥

University of Queensland, Brisbane, Australia

i=1

ri | i ⇥

•  errors are smaller for larger number of Lanczos vectors (M) •  extremal Lanczos “eigenstates” converge more rapidly with M (ground state for M≈50) •  calculation of dynamics and finite T properties •  limitations from size of the Hilbert space (memory and CPU time)

Test of FTLM System size Twisted boundary conditions

University of Queensland, Brisbane, Australia

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Comparison of expt. and DMFT: Resistivity vs. temperature

P. Limelette et al. PRL 91, 016401 (2003)

Why might DMFT be relevant to a 2D triangular lattice? •  DMFT is exact in the limit of infinite dimensions, but 2<<∞. •  Frustration significantly reduces spatial correlations, enhancing the validity of a single site approximation. •  For the antiferromagnetic Heisenberg model on a triangular lattice at T=J/3, the spin correlation length is about a one lattice constant versus 50a for the square [Elstner et al. PRB ’93, JAP ‘94]. •  Curie-Weiss law (which results from a local approximation) holds to a much lower T for triangular lattice than on square lattice [Zheng et al. PRB ’05].

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