Mott metal-insulator transition on compressible lattices Markus Garst Universität zu Köln T p in collaboration with : Mario Zacharias (Köln) Lorenz Bartosch (Frankfurt) T c Mott insulator p c T metal pressure p
Outline Introduction: Mott transition Universality of the Mott endpoint Experiments Theory: Mott endpoint on compressible lattices Summary
Introduction: Mott transition
Mott metal-insulator transition Hubbard model: H = t X ij c i c j + U X i c i c i c i c i hopping on a lattice simplification: on-site repulsion (screened Coulomb interaction)
Mott metal-insulator transition Hubbard model: H = t X ij c i c j + U X i c i c i c i c i hopping on a lattice simplification: on-site repulsion (screened Coulomb interaction) t hopping: gain of kinetic energy t
Mott metal-insulator transition Hubbard model: H = t X ij c i c j + U X i c i c i c i c i hopping on a lattice simplification: on-site repulsion (screened Coulomb interaction) hopping: gain of kinetic energy t
Mott metal-insulator transition Hubbard model: H = t X ij c i c j + U X i c i c i c i c i hopping on a lattice simplification: on-site repulsion (screened Coulomb interaction)
Mott metal-insulator transition Hubbard model: H = t X ij c i c j + U X i c i c i c i c i hopping on a lattice simplification: on-site repulsion (screened Coulomb interaction) t hopping: gain of kinetic energy t
Mott metal-insulator transition Hubbard model: H = t X ij c i c j + U X i c i c i c i c i hopping on a lattice simplification: on-site repulsion (screened Coulomb interaction) U hopping: gain of kinetic energy t doubly occupied site: cost of energy U
Mott metal-insulator transition Hubbard model: H = t X ij c i c j + U X i c i c i c i c i hopping on a lattice simplification: on-site repulsion (screened Coulomb interaction) at half-filling: strong competition between kinetic energy and on-site repulsion metallic for t >> U (without nesting) insulating for t << U metal-insulator transition at a critical ratio t/u
Phase diagram temperature line of 1 st order Mott transition Mott endpoint Mott insulator metal 0 (t/u)c t/u
Phase diagram temperature Mott endpoint Mott insulator metal 0 AF insulator t/u to release spin entropy: magnetic ordering at low T
Tuning by pressure applying pressure reduces lattice constant increases overlap of electron wavefunctions, enhancement of hopping amplitude t
Phase diagram with pressure tuning temperature Mott endpoint Mott insulator metal 0 AF insulator pressure
Phase diagram of V2O3 McWhan et al PRB (1973)
Universality of the Mott end point
Mott critical endpoint T Mott endpoint Mott insulator metal 0 p What are the critical properties of the Mott endpoint?
Liquid-gas end point line of first-order liquid-gas transitions terminates at second-order end point pressure solid liquid end point gas temperature line defines an emergent mirror symmetry Ising symmetry
Liquid-gas end point line of first-order liquid-gas transitions terminates at second-order end point pressure solid end point gas temperature line defines an emergent mirror symmetry Ising symmetry
Ising criticality Ising magnet in a field: L = T T c 2T c M 2 +( M) 2 + u 4! M 4 HM H end point 0 Tc T
Ising criticality Ising magnet in a field: L = T T c 2T c M 2 +( M) 2 + u 4! M 4 HM H hmi > 0 1 st order transition 2 nd order end point 0 Tc T hmi < 0
Ising criticality Ising magnet in a field: H L = T T c 2T c M 2 +( M) 2 + u 4! M 4 HM Ginzburg criterion 0 Ising criticality T Landau mean-field criticality critical behavior: non-trivial Ising exponents close to endpoint
Universality of the Mott end point: Experiments
Mott critical endpoint T Mott endpoint pressure Mott insulator metal solid liquid end point gas 0 p temperature Is the Mott endpoint analogous to the liquid-gas endpoint? Is it in the Ising universality class?
Mott end point of Cr-doped V2O3
Mott end point of Cr-doped V2O3
Mott end point of Cr-doped V2O3 Landau mean-field exponents indications for Ising exponents close to the endpoint? non-trivial assumption of the scaling analysis: conductivity is the Ising order parameter!
Mott end point of κ-(bedt-ttf)2x conducting 2D layers of BEDT-TTF
Mott end point of κ-(bedt-ttf)2x
Mott end point of κ-(bedt-ttf)2x exponents neither Landau nor Ising? unconventional universality class?
Mott end point of κ-(bedt-ttf)2x exponents neither Landau nor Ising? unconventional universality class? S. Papanikolaou et al. PRL (2008): NO! conductivity NOT necessarily the Ising order parameter! instead: conductivity ~ energy density (cf. Fisher-Langer scaling) consistency with Ising criticality!
Mott end point of κ-(bedt-ttf)2x Thermodynamics easier to interpret than transport: thermal expansion as a function of T: consistent with 2d Ising critical behavior M. de Souza, A. Brühl, Ch. Strack, B. Wolf, D. Schweitzer, and M. Lang, PRL (2007) L. Bartosch, M. de Souza, and M. Lang, PRL (2010)
Universality of the Mott end point: Theory
Mott critical endpoint T Mott endpoint pressure Mott insulator metal solid liquid end point gas 0 p temperature Is the Mott endpoint analogous to the liquid-gas endpoint? Is it really in the Ising universality class?
Mott transition on incompressible lattices
Mott transition on incompressible lattices YES! Mott endpoint liquid-gas endpoint at least for the Hubbard model. Subtler: long-range Coulomb T Mott endpoint eg. optical lattice Mott insulator metal 0 t/u Castellani et al. PRL (1979); Kotliar et al. PRL (2000); Papanikolaoou et al. PRL (2008)
Mott transition on compressible lattices
Mott transition on compressible lattices NO! Mott endpoint liquid-gas endpoint i.e. for basically all solid state realizations! T Mott endpoint Mott insulator metal 0 pressure
Mott transition on compressible lattices T Mott endpoint 1 st order transition V 0 at the first order transition: pressure jump of conjugate quantity, i.e. volume solid-to-solid isostructural transition V Mott endpoint solid-solid endpoint
Mott transition on compressible lattices Mott endpoint solid-solid endpoint liquid-gas endpoint Difference between a solid and a liquid:
Mott transition on compressible lattices Mott endpoint solid-solid endpoint liquid-gas endpoint Difference between a solid and a liquid: finite shear modulus G = Fl A x wikipedia
Liquid-gas critical point pressure order parameter: change in density solid liquid end point compressibility = 1 V V p = 1 p gas diverges at the endpoint temperature speed of sound vanishes c 2 sound = p = 1 vanishing sound velocity at the liquid-gas endpoint
Solid-solid critical point T Mott endpoint order parameter: change in volume V 0 1 st order transition Isotropic solid: V p longitudinal sound waves shear sound waves compressibility = 1 V diverges at the endpoint c 2 l = K + 4 3 G c 2 s = G V p = 1 K remain finite for K 0 due to shear modulus G! finite sound velocity at the solid-solid endpoint
Macroscopic instability of a solid elasticity theory: L = 1 2 ij C ijkl kl solid becomes unstable if a eigenvalue of Cijkl vanishes: det C ijkl =0 phonon velocities are determined by a different matrix: D ij = C ijkl ˆq k ˆq l BUT: det C ijkl =0 ; detd ij =0 phonon velocities generally stay finite!
Universality classes p Liquid-gas critical point: solid liquid vanishing sound velocity critical microscopic fluctuations gas T Ising criticality T Solid-solid critical point: Mott insulator metal isostructural transition, no change in crystal symmetry finite sound velocity absence of critical fluctuations 0 p Landau (mean-field) criticality applies to all crystal symmetries: Cowley, PRB (1976), Folk, Iro, Schwabl, Z Physik B (1976)
Mott critical endpoint Landau criticality Two scenarios: weak Mott-elastic coupling: T Mott-Ising Mott-Landau critical Mott insulator metal crossover from Mott-Landau to Mott-Ising to elastic Landau criticality p
Mott critical endpoint Landau criticality Two scenarios: weak Mott-elastic coupling: T Mott-Ising elastic- Landau Mott-Landau critical Mott insulator metal crossover from Mott-Landau to Mott-Ising to elastic Landau criticality p
Mott critical endpoint Landau criticality Two scenarios: weak Mott-elastic coupling: strong Mott-elastic coupling: T Mott-Ising elastic- Landau Mott-Landau critical T elastic Landau critical Mott Landau critical Mott insulator metal Mott insulator metal crossover from Mott-Landau to Mott-Ising to elastic Landau criticality p single crossover from Mott- to elastic Landau criticality p
Theory and relation to experiments
Theory for the Mott endpoint Assumption: Mott endpoint without coupling to the lattice = Ising critical How does the Ising order parameter couple to strain? L int = 1" + 1 2 2 " 2 ": singlet irred. representation of crystal group linear coupling allowed as Ising symmetry is emergent! Linear coupling only to singlet " no crystal symmetry breaking isostructural instability! similar to critical piezoelectric ferroelectrics Levanyuk and Sobyanin JETP Lett. (1970), Villain, SSC (1970)
Theory for the Mott endpoint How does the Ising order parameter couple to strain? L int = 1" + 1 2 2 " 2 Effective theory: neglect non-critical phonons effective potential for macroscopic strain singlet V( ) = K 0 2 2 p + f sing (t 0 + 2,h 0 + 1 ) T Mott-Ising bare modulus pressure free energy density of the Ising model p weak Mott-elastic coupling
Theory for the Mott endpoint effective potential: V( ) = K 0 2 2 p + f sing (t 0 + 2,h 0 + 1 ) without coupling to crystal elasticity: γ1= γ2=0 with coupling to crystal elasticity: γ1 0; γ2=0 Mott-Ising endpoint preempted by isostructural instability mean-field exponents!
Theory for the Mott endpoint effective potential: V( ) = K 0 2 2 p + f sing (t 0 + 2,h 0 + 1 ) T Mott-Ising away from the critical point (blue regime): perturbative minimization: p/k 0 p F pert = p 2 2K 0 + f sing (t 0 + 2 p/k 0,h 0 + 1 p/k 0 ) used by Bartosch et al. PRL (2010) pressure dependence of tuning parameters
Theory for the Mott endpoint yellow non-perturbative regime: elastic Landau-critical T Mott-Ising expansion in " = " " V( ) K 2 2 (p p)+ u 4! 4 + f sing ( t, h) with and " chosen such that cubic term vanishes h = h 0 + 1 and t = t 0 + 2 p renormalized modulus: K = K 0 2 1 h h 2 1 2 h t 2 2 t t most singular correction due to linear coupling: For K 0 minimization yields: h h = 2 hf sing ( t, h) diverges and drives bulk modulus to zero! = (6(p p)/u) 1/3 breakdown of Hooke s law! with mean-field exponent
Application to κ-(bedt-ttf)2x breakdown of Hooke s law: smoking-gun criterion for elastic Landau critical regime with non-perturbative Mott-elastic coupling T T c Mott insulator p T metal p c pressure p non-linear strain-stress relation width of the elastic Landau regime: p 45 bar = 4.5MPa T 2.5K
Application to κ-(bedt-ttf)2x breakdown of Hooke s law: smoking-gun criterion for elastic Landau critical regime with non-perturbative Mott-elastic coupling T T c Mott insulator p T metal p c pressure p non-linear strain-stress relation width of the elastic Landau regime: p 45 bar = 4.5MPa T 2.5K
Application to κ-(bedt-ttf)2x using fitting parameters of Bartosch et al. PRL (2010): dotted line: perturbative; solid line: full solution width of the elastic Landau regime: p 45 bar = 4.5MPa T 2.5K
Mott end point of Cr-doped V2O3 What is the critical singlet irred. representation of strain for V2O3? determined elastic constant matrix by measureing various sound velocities trigonal crystal symmetry 0 C = B @ C 11 C 12 C 13 C 14 0 0 C 12 C 11 C 13 C 14 0 0 C 13 C 13 C 33 0 0 0 C 14 C 14 0 C 44 0 0 0 0 0 0 C 44 C 14 C 0 0 0 0 C 11 C 12 14 2 1 C A phonon velocities related to eigenvalues of D ij = C ijkl ˆq k ˆq l
Mott end point of Cr-doped V2O3 What is the critical singlet irred. representation of strain for V2O3? trigonal crystal symmetry 0 C = B @ C 11 C 12 C 13 C 14 0 0 C 12 C 11 C 13 C 14 0 0 C 13 C 13 C 33 0 0 0 C 14 C 14 0 C 44 0 0 0 0 0 0 C 44 C 14 C 0 0 0 0 C 11 C 12 14 2 1 C A crystal stability: eigenvalues must be positive strong softening! Singlet!
Mott end point of Cr-doped V2O3 strong softening close to critical temperature already in undoped V2O3! probably no regime with non-trivial Ising criticality! T elastic Landau critical Mott Landau critical Mott insulator metal strong Mott-elastic coupling p
Summary Phys. Rev. Lett. 109, 176401 (2012) Mott endpoint liquid-gas endpoint Mott endpoint = solid-solid endpoint Mott endpoint criticality: Landau (mean-field) not Ising critical elasticity regime: breakdown of Hooke s law two scenarios: T Mott-Ising T! p κ-(bedt-ttf)2x estimate for κ-(bedt-ttf)2x: p 45 bar = 4.5MPa T 2.5K V2O3 p