Continuous-time quantum Monte Carlo algorithms for impurity problems

Transcription

1 Continuous-time quantum Monte Carlo algorithms for impurity problems Michel Ferrero Centre de Physique Théorique Ecole Polytechnique, France Quantum Monte Carlo methods at work for novel phases of matter Trieste, February 2-3, 2012

2 Outline Quantum impurity problems Continuous-time quantum Monte Carlo (CT-QMC) methods Generic introduction to Monte Carlo The interaction-expansion algorithm (CT-INT) The hybridization-expansion algorithm (CT-HYB) Some applications of CT-QMC solvers TRIQS: a Toolbox for Research on Quantum Interacting Systems Implementation details for the CT-INT

3 References Quantum impurity problems: A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press CT-QMC solvers: E. Gull et al., RMP (2011) The interaction-expansion algorithm Rubtsov et al., PRB (2005) and Rubtsov and Lichtenstein, JETP Lett. (2004) The hybridization-expansion algorithm Werner and Millis, PRB (2006) and Werner et al., PRL (2006) Dynamical mean-field theory: A. Georges et al., RMP (1996) Open source codes with CT-QMC solvers: TRIQS: ALPS:

4 Quantum impurity problems They generically describe the behavior of a magnetic impurity embedded in an electronic host The impurity is a set of orbitals carrying local many-body interactions and localized spins. It can exchange electrons with an uncorrelated fermionic bath e «Atomic» problem with several orbitals H loc e Electronic bath

5 The Kondo problem Experiments have shown that magnetic impurities in a metallic host (e.g. iron in gold) induce a minimum in the resistivity versus temperature curve The reason is that at low temperatures the conduction electrons screen the magnetic impurities: this is the Kondo effect

6 The Kondo effect: general idea At high temperature, conduction electrons pass by the impurity magnetic moment and only experience a small antiferromagnetic correlation impurity d y ¾ v F c y k¾ conduction electron At low temperature, the impurity magnetic moment and conduction electrons bind to form an overall non-magnetic state impurity d y ¾ c y k¾ conduction electron

7 The Anderson model A very successful model to understand magnetic impurities in a metallic host is the Anderson model Hamiltonian: ½(²) d y ¾ U ² 0 V k¾ Z = Tr exp( H) H = X c y k¾ ¾=";# ² 0 d y ¾d ¾ + Un d" n d# + X k;¾ V k¾ c y k¾ d ¾ + h:c: + X k;¾ ² ² k¾ c y k¾ c k¾ local many-body interaction Hybridization to the bath free bath states

8 Action for the Anderson model After integrating out the fermionic bath: Z Z = D[d y ; d]e S Z S = X d d 0 d y ¾( )G 1 0¾ ( 0 )d ¾ ( 0 ) + d Un d" ( )n d# ( ) ¾ 0 0 G 1 0¾ (i! n) = i! n ² 0 ¾ (i! n ) non-interacting Green s function ¾ (i! n ) = X k Remember: G 0¾ Generic case: jv k¾ j 2 i! n ² k¾ Z hybridization function: describes the transition between the bath and the orbital contains the information about the structure of the bath! S = X a;b Z 0 d d 0 d y a( )G 1 0;ab ( 0 )d b ( 0 ) + Z 0 d H int (fd y a; d a g)( ) G 1 0;ab (i! n) = i! n ± ab + t ab ab (i! n )

9 A difficult problem but a rich toolbox of methods The Anderson model is a many-body (correlated) problem with an infinite number of degrees of freedom! It has attracted a lot of interest and many techniques have been developed: Analytical methods (Bethe Ansatz, BCFT) Controllable algorithms: Exact diagonalization Numerical renormalization group Density matrix renormalization group Continuous-time quantum Monte Carlo algorithms Approximate solvers (e.g. NCA) All have pros and cons!

10 Impurity model in many contexts Quantum impurity problems appear in many different contexts magnetic impurity in metallic host nanostructures, quantum dots within the DMFT: pnictides, cuprates, cold-atomic systems,

11 A renewed interest within DMFT There are very efficient Monte Carlo algorithms for bosons For fermions, the situation is different and approximation methods are needed dynamical mean-field theory (DMFT) DMFT has brought a renewed interest in quantum impurity models in the late 90 s but also emphasized the limitations of existing algorithms and asked for the development of new tools

12 Dynamical mean-field theory The dynamical mean-field theory makes an approximation of a lattice model using an auxiliary quantum impurity problem It has been used a lot to find an approximate solution of the Hubbard model: H = X k ² k c y k c k + U X i n i" n i# free electrons on a lattice many-body interaction on every lattice site The DMFT has been used in many different contexts: realistic electronic structure calculations, models for superconductivity, cold-atomic systems, etc.

13 Dynamical mean-field theory cont d Lattice Hubbard model A. Georges and G. Kotliar, PRB (1992) A. Georges et al., RMP (1996) Anderson impurity model Bath G 0 Lattice Hubbard model is mapped onto a single-site model with an effective action: S = X ¾ Z 0 d d 0 d y ¾( )G 1 0¾ ( 0 )d ¾ ( 0 ) + Z 0 d Un d" ( )n d# ( ) The bath has to be set self-consistently

14 DMFT equations and limitations S e = Z 0 Z c y ¾( )G 1 0 ( 0 )c ¾ ( 0 ) + 0 (i! n ) = G 1 0 (i! n) G 1 (i! n ) lattice (i! n ; k) Ã (i! n ) Un " ( )n # ( ) G(i! n ) = X k G lattice (i! n ; k) G 1 0 (i! n) = ³ X k (i!n ) i! n + ¹ ² k (i! n ) In DMFT the self-energy of the lattice is local: Z, m*, coherence temperature, lifetimes are constant along the Fermi surface

15 Example: the Mott transition The DMFT is good at describing transfer of spectral weight due to correlation effects Example: the destruction of coherent quasiparticles (small QP coherence scale, short lifetime) near Mott transition Metal Mott transition U Insulator

16 What s the challenge? The impurity solver must compute the local Green s function Bath can have a rich structure, be gapped (insulators, superconductors) Structures appear at all scales (transfer of spectral weight in the Mott transition) The impurity solver must be able to treat many orbitals (e.g. pnictides) The interaction Hamiltonian must be generic (pair-hopping, spin flip terms) The model is studied in different temperature regimes One would like to be able to have real-frequency spectra

17 Outline Quantum impurity problems Continuous-time quantum Monte Carlo (CT-QMC) methods Generic introduction to Monte Carlo The interaction-expansion algorithm (CT-INT) The hybridization-expansion algorithm (CT-HYB) Some applications of CT-QMC solvers TRIQS: a Toolbox for Research on Quantum Interacting Systems Implementation details for the CT-INT

18 Continuous-time quantum Monte Carlo methods They have been a small revolution! There are different flavors: CT-INT: Interaction expansion CT-HYB: Hybridization expansion CT-AUX: Auxiliary-field formulation The underlying principle is the same for all these algorithms They stem from earlier work on diagrammatic Monte Carlo Prokof ev and Svistunov, PRL (1998) Prokof ev et al., Sov. Phys. JETP (1998)

19 Some notable results DMFT used together with electronic structure calculations Rotationally invariant multi-orbital problems Kondo lattice Cluster DMFT (e.g. for cuprate superconductors) Fermionic cold-atomic systems 4-point correlators for extensions of DMFT (dual-fermions) First steps in non-equilibrium

20 Diagrammatic Monte Carlo Write the partition function in a perturbation theory Z = Generate stochastically diagrams so that the statistical weight of a diagram is the absolute value of its contribution in the partition function expansion (Monte Carlo) This is possible even if the diagrams depend on continuous variables Use these diagrams to compute the partition function, Green s functions, etc.

21 Outline Quantum impurity problems Continuous-time quantum Monte Carlo (CT-QMC) methods Generic introduction to Monte Carlo The interaction-expansion algorithm (CT-INT) The hybridization-expansion algorithm (CT-HYB) Some applications of CT-QMC solvers TRIQS: a Toolbox for Research on Quantum Interacting Systems Implementation details for the CT-INT

22 Monte Carlo: a quick summary The Monte Carlo is a method to do sums: X X p(x)f(x); where p(x) > 0; p(x) = 1 x x The idea is to generate stochastically a large set of such that the probability to find a given is p(x n ) The original sum is replaced by an average over the set fx i g X p(x)f(x)» 1 N x NX f(x i ) Note that x should be understood as a set of variables x = ( ; ¾; : : :) One can think of x as a configuration in a phase space i=1 x n x i

23 Markov chain How do we generate configurations with the correct distribution? We don t sample independent configurations, but generate a Markov chain: x 1! x 2! x 3! x 4! : : : The transition probability W x;y to go from configuration x to a configuration y must satisfy: Normalization: X W x;y = 1 y Ergodicity: one must be able to reach any configuration Stationary distribution, balance condition: X d dt p(x) = 0 p(y)w y;x X p(x)w x;y = 0 y6=x y6=x Detailed balance (sufficient but not necessary): W x;y = p(y) W y;x p(x)

24 Metropolis algorithm How do we get a transition probability satisfying these criteria? Proposal rejection scheme: Propose a change with a chosen proposal rate Accept this proposal with a probability Otherwise don t change the configuration The total transition rate is Metropolis algorithm: A x;y W x;y = A x;y P x;y h A x;y = min 1; P i y;xp(y) P x;y p(x) P x;y It satisfies the detailed balance and therefore the Markov chain will be distributed according to p(x)

25 The fermionic sign problem Imagine we want to compute this average: P x hfi = P w(x)f(x) x w(x) We would like to use w(x) as a probability. But what if it can be negative? We must use the absolute value instead: hfi = P x jw(x)jf(x)sign(w(x)) Px jw(x)jsign(w(x))» P N i=1 f(x i)sign(w(x i )) P N i=1 sign(w(x i)) If signs alternate the denominator is very small and there is a big variance! Gets worse at low temperatures, big systems Fermionic problems almost always suffer this sign problem!

26 A Monte Carlo algorithm propose update x! y compute P x;y ; P y;x ; p(y) Metropolis h A x;y = min 1; P i y;xp(y) P x;y p(x) NO x! x initialize simulation leave configuration unchanged measures observables YES x! y update configuration

27 A couple of things to be careful about The measurements must start when the Markov chain has reached a stationary distribution. One must make sure to let the system thermalize! Successive points in the Markov chain are clearly correlated. These autocorrelation effects must be taken into account when computing the statistical errors: Binning analysis Jackknife Ideally one should measure only when configurations have become uncorrelated There are tools around to do these analysis!

28 Partition function: Diagrammatic Monte Carlo Z = Propose diagrams in a Markov chain: Z C w(c)» XMC C sign(w(c)) Accept these proposals with a rate (Metropolis) such that the diagrams appear with probability density w(c) From the generated configurations, compute the observables you are interested in hfi = 1 Z w(c)f(c)» 1 XMC f(c) sign(w(c)) Z C Z The different versions of continuous-time Monte Carlo solvers correspond to different choices of writing the partition function C

29 Getting a series expansion for the partition function Write the action in two parts S = S A + S B Express the partition function as Z Z Z = D[d y ; d]e S A e S B = D[d y ; d]e S A Using that hxi = 1 Z D[d y ; d]e S X Z X n ( 1) n n! S n B we get Z = Z A X n ( 1) n n! D E T SB n A This is an average over the states described by the action A. It generally involves sums and integrals over imaginary time and can have a diagrammatical representation.

30 Outline Quantum impurity problems Continuous-time quantum Monte Carlo (CT-QMC) methods Generic introduction to Monte Carlo The interaction-expansion algorithm (CT-INT) The hybridization-expansion algorithm (CT-HYB) Some applications of CT-QMC solvers TRIQS: a Toolbox for Research on Quantum Interacting Systems Implementation details for the CT-INT

31 Interaction-expansion CT-QMC We focus on the simplest Anderson model (can be generalized to density-density multi-orbital problems) We want to derive an expansion around the non-interacting limit (expansion in the interaction) ½(²) d y ¾ U ² 0 V k¾ c y k¾ H = X ¾=";# ² 0 d y ¾d ¾ + Un d" n d# + X k;¾ V k¾ c y k¾ d ¾ + h:c: + X k;¾ ² ² k¾ c y k¾ c k¾

32 Interaction expansion cont d We work in the imaginary-time formalism Z Z = D[d y ; d]e S hai = 1 Z D[d y ; d]e S A Z The action for the Anderson model: S = X ¾ Z 0 d d 0 d y ¾( )G 1 0¾ ( 0 )d ¾ ( 0 ) + Z 0 d Un d" ( )n d# ( ) The partition function can be written as Z = Z D[d y ; d]e S 0 ³ Z exp 0 d ( U)n d" ( )n d# ( ) this will produce a time-ordered average over the noninteracting state

33 Interaction expansion cont d We write a series expansion for the exponential ³ Z = Z 0 DT Z exp = Z 0 D T X = Z 0 X n n Z 0 0 ( U) n n! E d ( U)n d" ( )n d# ( ) 0 ³ Z ne d n d" ( )n d# ( ) 0 d 1 d n ( U) n n! 0 D E T n d" ( 1 )n d# ( 1 ) n d" ( n )n d# ( n ) 0 At this stage we have a perturbation expansion for the partition function of the type: Z XMC Z = w(c)» sign(w(c)) However there is an obvious sign problem! C C

34 Interaction expansion cont d The trick is to rewrite the interaction differently Un " n # = U X ³n " s" ³n # s# + U ³n " + n # 2 2 s=";# + const s¾ = (2± s¾ 1)± we absorb this term in the chemical potential We eventually get (up and down spins decouple) Z = Z 0 X n ( U) n n! Z d i 1 2 n X s i D E T (n d" ( 1 ) s1 ") (n d" ( n ) sn ") 0 D E T (n d# ( 1 ) s1 #) (n d# ( n ) sn #) By tuning ± we can improve the sign problem a lot! 0

35 Z = Z C w(c)» XMC C Interaction expansion cont d In the end we have X Z X X Z = Z 0 d 1 d n n s 1 s n ( U) n D E T n!2 n (n d" ( 1 ) s1 ") (n d" ( n ) sn ") Product of two non-interacting averages sign(w(c)) hfi = 1 Z Now we need to find a way to compute the averages Z Sum over many (some continuous) variables D T (n d# ( 1 ) s1 #) (n d# ( n ) sn #) C w(c)f(c)» 1 Z XMC C 0 E 0 f(c) sign(w(c))

36 Computing the averages The averages are on a non-interacting state. Therefore we can use Wick s theorem With this definition of the Green s function G 0¾ ( ) = T hc ¾ ( )c y ¾(0)i 0 we get D E (n d¾ ( 1 ) s1 ¾) (n d¾ ( n ) sn ¾) = det Dn ¾ = 0 G 0¾ (0 ) s1 ¾ G 0¾ ( 1 2 ) G 0¾ ( 2 1 ) G 0¾ (0 ) s2 ¾ G 0¾ ( 2 3 ) G 0¾ ( n n 1 ) G 0¾ (0 ) sn ¾ and X Z Z = Z 0 n X X d 1 d n s 1 s n ( U)n n!2 n det D" n det D # n

37 MC sum: Monte Carlo elements X Z X X d 1 d n n 1 > > n s 1 s n The configurations are diagrams of the perturbation expansion. They can be seen as a set of interaction vertices at different imaginary times with an auxiliary spin at every vertex. s i n The weight of every diagram is given by ³ U n det D " 2 n det D n #

38 Generating diagrams We need to create a Markov chain of diagrams We can propose any changes to go from one diagram to another. A simple solution is to use two moves : An insertion of a vertex: we pick a random imaginary time and insert a vertex with a spin randomly up or down (A) A removal of a vertex: pick a random vertex and remove it (B) (B) (A) 1 2 i n 0

39 What is the acceptance rate? Insertion of a vertex h A x;y = min 1; P i y;x½(y) P x;y ½(x) 1 2 n x; n 1 2 n n+1 0 y; n + 1 P x;y ½(x) = d n+1 P y;x ½(y) = n + 1 ³ U 2 Accept move with: ³ U 2 n det D " n det D # n ny d i i=1 n+1 n+1 det D " n+1 det Y D# n+1 d i i=1 h A x;y = min 1; U n + 1 det D" n+1 D# i n+1 det D nd " n #

40 What is the acceptance rate? Removal of a vertex h A x;y = min 1; P i y;x½(y) P x;y ½(x) 1 2 n x; n 1 n 1 0 y; n 1 P x;y ½(x) = n ³ U 2 P y;x ½(y) = d n ³ U 2 Accept move with: n det D " n det D # n ny d i i=1 n 1 det D " n 1 det D# n 1 n 1 Y i=1 d i d 0 i h A x;y = min 1; n U det D" n 1 D# i n 1 det D nd " n #

41 Getting the Green s function We know how to generate a distribution corresponding to the terms in the partition function Now we just need to find how to measure the Green s function from this distribution G( ) = T hd( )d y (0)i = 1 Z D[d y ; d]e S d y (0)d( ) Z S = X ¾ Z 0 Z d d 0 d y ¾( )G 1 0¾ ( 0 )d ¾ ( 0 ) + 0 d Un d" ( )n d# ( ) G 1 0¾ ( 0 ) = (@ 0 ² 0 )±( 0 ) ¾ ( 0 ) We finally see that G ¾ ( ) = 1 ± ln Z ± ¾ ( )

42 Getting the Green s function cont d We have Z = Z 0 Z C ³ U 2 n det D " n det D # n The functional derivative eventually gives: G ¾ (i! n ) = G 0¾ (i! n ) 1 G2 0¾(i! n ) Z X ³ [Dn ¾ ] 1 ij ei! n( i j ) U 2 C ij n det D " n det D # n So we see that we need to compute the following Monte Carlo average to get the Green s function G ¾ (i! n )» G 0¾ (i! n ) 1 Z G2 0¾ (i! n) XMC C X ij [D ¾ n ] 1 ij ei! n( i j ) sign(w(c))

43 Computational effort The effort comes from the calculation of the determinants and of the inverse matrix (needed for the Green s function measure) Z ³ U n Z = Z 0 det D " C 2 n det D n # G ¾ (i! n )» G 0¾ (i! n ) 1 Z G2 0¾ (i! n) XMC It would be very slow to calculate them from scratch at every move They can be updated quickly using the Sherman-Morrison formula The computational effort grows in O(n 3 ) C X ij [D ¾ n ] 1 ij ei! n( i j ) sign(w(c))

44 Outline Quantum impurity problems Continuous-time quantum Monte Carlo (CT-QMC) methods Generic introduction to Monte Carlo The interaction-expansion algorithm (CT-INT) The hybridization-expansion algorithm (CT-HYB) Some applications of CT-QMC solvers TRIQS: a Toolbox for Research on Quantum Interacting Systems Implementation details for the CT-INT

45 Hybridization-expansion CT-QMC We focus on the simplest Anderson model (can easily? be generalized to generic multi-orbital problems) We want to derive an expansion around the atomic limit (expansion in the hybridization) ½(²) d y ¾ U ² 0 V k¾ c y k¾ H = X ¾=";# ² 0 d y ¾d ¾ + Un d" n d# + X k;¾ V k¾ c y k¾ d ¾ + h:c: + X k;¾ ² ² k¾ c y k¾ c k¾

46 Hybridization expansion cont d We work in the imaginary-time formalism Z Z = D[d y ; d]e S hai = 1 Z D[d y ; d]e S A Z The standard action for the Anderson model: Z S = X d d 0 d y ¾( )G 1 0¾ ( 0 )d ¾ ( 0 ) + d Un d" ( )n d# ( ) ¾ 0 0 G 1 0¾ (i! n) = i! n ² 0 ¾ (i! n ) ¾ (i! n ) = X jv k¾ j 2 i! n ² k¾ k Rewrite the action as the sum of an atomic term containing all local terms U; ² 0 and a term with the hybridization to the bath: Z S = S loc + X ¾ Z 0 d d 0 d y ¾( ) ¾ ( 0 )d ¾ ( 0 ) = S loc + X ¾ S ¾ hyb

47 Z = Z = Z = Z Hybridization expansion cont d P Z Y h X 1 D[d y ; d]e S loc ¾ S¾ hyb = D[d y ; d]e S ( 1) n i loc S ¾ n n! hyb ¾ n=0 1X Y ( 1) DT n ¾ E n¾ ^S¾ n ¾! hyb loc n " ;n # =0 ¾ 1X Z d " 1 n " ;n # =0 0 Trace involving both spin up and down operators : : : d 0" n " Z Y ¾ 0 ( 1) n ¾ n ¾! d # 1 Yn ¾ i=1 : : : d 0# n # n Y " Tr he H loc T ¾ ( ¾ i 0¾ i ) i=1 n# d y " ( " i )d "( 0" i ) Y Sum over many (continuous) variables i=1 Product of hybridization functions d y # ( # i )d #( 0# i ) i Z = Z C w(c)» XMC C sign(w(c)) hfi = 1 Z Z C w(c)f(c)» 1 Z XMC C f(c) sign(w(c))

48 Hybridization expansion cont d MC sum: 1X Z n " ;n # =0 0 d " 1 : : : d 0" n " Z 0 d # 1 : : : d 0# n # Diagrams: Weight: ¾ =" ¾ =# n Y " Tr he H loc T " 1 0" 1 " 2 0" 2 i=1 n# d y " ( " i )d "( 0" i ) Y Y ¾ # 1 0# 1 ( 1) n ¾ Unfortunately these diagrams have alternating signs problems! n ¾! n ¾ Y i=1 i=1 d y # ( # i )d #( 0# i ) i ¾ ( ¾ i 0¾ i ) 0

49 Trick: resumming diagrams The idea is to resum diagrams into a determinant. We start from a diagram where " 1 > : : : > n " " and sum all the permutations of f " i g ¾ =" " 1 0" 1 " 2 0" 1 + " 2 0" 2 " 1 0" 2 " 1 > " 2 " 1 $ " 2 h i Tr e H loc d y " ( " 1 )d "( 0" 1 )dy " ( " 2 )d "( 0" 2 ) h i ( 1)Tr e H loc d y " ( " 1 )d "( 0" 1 )dy " ( " 2 )d "( 0" 2 ) "( " 1 0" 1 ) "( " 2 0" 2 ) 1 2 "( " 2 0" 1 ) "( " 1 0" 2 ) = h i Tr e H loc d y " ( " 1 )d "( 0" 1 )dy " ( " 2 )d "( 0" 2 ) 1 det 2 "( " 1 k;l 2 k 0" l )

50 MC sum: Hybridization expansion Monte Carlo 1X n " ;n # =0 Z d " " 1 >:::> " 1 n 0" 0" 1 >:::> n : : : d 0" n Z d # # 1 >:::> # 1 n 0# 0# 1 >:::> n : : : d 0# n # Diagrams: ¾ =" " 1 0" 1 " 2 0" 2 ¾ =# # 1 0# 1 0 Weight: n Y " Tr he H loc T i=1 n# d y " ( " i )d "( 0" i ) Y i=1 d y # ( # i )d #( 0# i ) i ( 1) n "+n # det " ( " k 0" l ) det # ( # k 0# l ) 1 k;l n " 1 k;l n #

51 Generating diagrams New diagrams are generated with two moves : Insertion of an (anti)-link: chose a spin flavor and pick two random imaginary times such that there is no operator between them. Either construct a link (A) or an anti-link (B) Removal of a link: chose a spin flavor and remove a random link (C) " 1 0" 1 " 2 0" 2 (A) ¾ =" (C) " 1 0" 1 " 2 0" 2 " 3 0" 3 (B) " 1 0" 1 " 2 0" 2 " 1 0" 1 " 2 0" 2 " 3 0" 3 0

52 Insertion of an (anti)-link What is the acceptance rate for this move? h A x;y = min 1; P i y;x½(y) P x;y ½(x) ¾ =" " 1 0" 1 " 2 0" 2 " 1 0" 1 " 2 0" 2 " 3 0" 3 x; n y; n + 1 P x;y ½(x) = 1 2 d n+1 d 0 n+1 l max TrC x det Cx P y;x ½(y) = n + 1 TrC y det Cy n+1 Y i=1 ny d i d i 0 i=1 d i d 0 i 0 Accept with probability: h A x;y = min 1; l max n + 1 TrC y det i Cy TrC x det Cx

53 Removal of a link What is the acceptance rate for this move? h A x;y = min 1; P i y;x½(y) P x;y ½(x) ¾ =" " 1 0" 1 " 2 0" 2 " 1 0" 1 n Y 0 x; n y; n 1 P x;y ½(x) = n TrC x det Cx P y;x ½(y) = 1 2 d n i=1 d i d 0 i d 0 n l max TrC y det Cy n 1 Y i=1 d i d 0 i Accept with probability: h A x;y = min 1; n TrC y det i Cy l max TrC x det Cx

54 Measuring the Green s function We know how to sample diagrams with weights corresponding to their contribution in the partition function. Z = Z C How do we get the Green s function? G " ( ) = 1 Z G " ( ) = 1 Z ( 1) n "+n # det "C det #C TrC = Z Z C C G ¾ ( ) = 1 ± log Z ± ¾ ( ) ± det "C ± " ( ) ( 1)n "+n # det #C TrC X ±( " k 0" l + )[ 1 "C ] k;l ( 1) n "+n # det "C det #C TrC k;l Z C w(c)» XMC C sign(w(c)) w(c) Measure: G ¾ ( )» 1 Z XMC C X k;l ±( ¾ k 0¾ l + ) [ 1 ¾C ] k;l sign(w(c))

55 Measuring the Green s function cont d Each configuration give contributions for a discrete set of imaginary times: G ¾ ( )» 1 XMC X ±( k ¾ l 0¾ + ) [ 1 ¾C Z ] k;l sign(w(c)) C k;l ¾ =" " 1 0" 1 " 2 0" 2 " 2 0" 2 0 These contribution can be binned on a very fine imaginary-time grid. This induces high frequency noise in Matsubara frequencies

56 Measuring using Legendre polynomials Legendre polynomials are a basis to express function defined over an interval L. Boehnke et al., PRB (2011) We can express the imaginarytime Green s function in this basis p 2l + 1 G( ) = X l 0 P l [x( )]G l The coefficients G l in this basis decay very quickly

57 Legendre basis acting as a noise filter The noise in the Matsubara frequencies can be reduced by truncating the Legendre coefficients that are zero within their error bars A typical outcome of this procedure:

58 Computational effort Can the contribution of a diagram be computed quickly? Determinants can be updated quickly (Sherman-Morrison) For simple Hamiltonians, the trace is very easy ¾ =" " 1 0" 1 " 2 0" 2 > l " > 0 ¾ =# # 1 0# 1 0 > l # > 0 overlap: l 1 l 2 Tr C = e (l "+l # )¹ U(l 1 +l 2 ) Computational effort grows in O(n 3 )

59 What about non density-density Hamiltonians? The hybridization expansion algorithm can be modified for generic Hamiltonians Configurations are a set of creation / destruction operators of different flavor on a single imaginary-time line " 1 0" 1 " 2 " 0" 3 0" 2 3 0" 4 " 4 0 The main drawback is that there is no longer a quick way to compute the trace Operators are matrices that must be multiplied and traced over all atomic states The number of these atomic states quickly becomes huge with several orbitals

60 CT-INT versus CT-HYB CT-INT & CT-AUX: series in the interaction Many orbitals, weak coupling, high temperatures density-density Hamiltonians Average perturbation order» U CT-HYB: series in the hybridization function Good at low temperatures, strong coupling Can treat generic Hamiltonians Hard to treat many orbitals Average perturbation order is the kinetic energy

61 Pros and cons of the CT-QMC algorithms Pros: Faster than before Monte Carlo can easily be parallelized Flexible Hamiltonians (not CT-INT) Good scaling with number of orbitals if density-density Cons: Many orbitals difficult with generic Hamiltonian They are mainly in imaginary time, so one needs to do analytical continuation, and this is a very delicate procedure! Note: real-time algorithms have been developed. However the sign problem is a lot more present there. Sign problem

62 Outline Quantum impurity problems Continuous-time quantum Monte Carlo (CT-QMC) methods Generic introduction to Monte Carlo The interaction-expansion algorithm (CT-INT) The hybridization-expansion algorithm (CT-HYB) Some applications of CT-QMC solvers TRIQS: a Toolbox for Research on Quantum Interacting Systems Implementation details for the CT-INT

63 What did the CT-QMC algorithms change? In simple model studies, CT-QMC algorithms allow to: Reach much lower temperatures Acquire high-quality imaginary-time data and do more reliable analytical continuations In the study of realistic materials, they allow to treat the full SU(2) exchange in multi-orbital problems (e.g. pnictides, nickelates, ruthenates, ) More involved models can be studied Cluster extensions of DMFT Dual-fermion approaches Holstein-Hubbard model, out-of-equilibrium

64 State of the art: NRG versus CT-QMC Direct comparison between NRG and CT-QMC for a half-filled Anderson impurity model (flat Wilson bath) Parameters are: W = 1; U = 0:5; U=¼ = 2:5 Analytical continuation of the CT-QMC selfenergy using Padé approximants NRG: Rok Zitko CT-QMC: Jernej Mravlje

65 Detailed structures of self-energy Even fine details of the self-energy are captured correctly! Single-band Bethe lattice DMFT calculation show very good agreement, where only small temperature effect induce a difference between NRG and CT-QMC Im (!)!

66 Mott transition in the Hubbard model Single-site DMFT studies have shown that the transition is first order at finite temperature, but second order at zero temperature DMFT cluster extensions have been considered (more later) Werner et al., PRB (2007)

67 Not only single-site DMFT for the Hubbard model Holstein-Hubbard model H = X ² k c y k¾ c k¾ + X Un i" n i# k¾ i + X (b y i + b X i)(n i 1) +! 0 b y i b i Kondo lattice i i H = X k¾ ² k c y k¾ c k¾ + J X i ~S i ~¾ i Extensions of DMFT to include spatial correlations Cluster extensions that ask for multi-impurity models Dual-fermion approaches that require for higher correlators

68 Cuprate superconductors A(k i ;!) ARPES A(k i ;!) Kaminski et al., PRB (2005) A(k;! = 0) (0,0) (, ) Spectral properties are isotropic along the Fermi surface! (, )! Fermi arcs Strong variations over the Brillouin zone! (0,0)

69 k-space variations are a problem for single-site DMFT S e = Z 0 Z c y ¾( )G 1 0 ( 0 )c ¾ ( 0 ) + 0 (i! n ) = G 1 0 (i! n) G 1 (i! n ) lattice (i! n ; k) Ã (i! n ) Un " ( )n # ( ) G(i! n ) = X k G lattice (i! n ; k) G 1 0 (i! n) = ³ X k (i!n ) i! n + ¹ ² k (i! n ) In DMFT the self-energy of the lattice is local: Z, m*, coherence temperature, lifetimes are constant along the Fermi surface

70 Including a k-dependence Dynamical cluster approximation (DCA) M. H. Hettler et al., PRB (1998) T. Maier et al., RMP (2005) Bath G 0 Anderson multiimpurity model Cluster self-energies for every cluster momenta: (K) There are ways to improve DMFT, but then you need to solve a multi-impurity Anderson model!

71 k-dependence of the self-energy The DCA approach has been tested against a direct Monte Carlo evaluation of the Hubbard lattice problem The solution scales nicely with the number of sites and has converged for 32 sites Kozik et al., EPL (2010)

72 Back to cuprates with a 2-site extension M. Ferrero et al., PRB (2009) 1 2 (, ) electron exchange (0,0) 0% pseudogap in antinodal orbital metallic, protected nodal orbital Singlet state dominates Ordinary Fermi liquid metal 16% Hole doping Selective Mott transition

73 Can one describe the phenomenology of the cuprates? ARPES intensity maps (, ) 8% 12% 25% (0,0) Tunneling conductance (STM) 8%

74 Optical conductivity Interplane charge dynamics Resistivity underdoped overdoped

75 Confirmation of the results on larger clusters E. Gull et al., PRB (2010) Computations on clusters up to 16 sites have been performed Results show that the main features of the above results are robust The approach to the Mott insulator is characterized by successive Mott transitions in reciprocal space The regions close to the antinode get insulating first, while regions close to the node remain metallic down to the Mott insulating state

76 Dual-fermion calculations for the Hubbard model Dual-fermion approaches are another way to include spatial correlations. They require the knowledge of 4-point vertex functions These methods have benefitted a lot from the new CT-QMC Spectral intensity map of the Hubbard model at 14% doping Dynamical susceptibility at half-filling with a visible magnon spectrum in the paramagnetic state Rubtsov et al., PRB, (2009) Hafermann et al., PRL, (2009)

77 Stabilization of a Sarma phase in cold-atomic systems Dao et al., PRL (2008) Cold-atomic systems allow to study superfluid phases with great precision The situation is rather clear when there is an attraction between two balanced populations (BCS BEC crossover) But what happens as we introduce a population imbalance and the Fermi surface mismatch? Could we obtain a Sarma phase which is both superfluid and polarized? (¼; ¼) Sarma has shown that it is not stable at weak coupling (0; 0) But what about strong coupling?

78 A study of a negative-u Hubbard model shows that: At weak coupling a phase separated state is stabilized at low temperature At larger coupling a polarized superfluid is stabilized down to very low temperatures The presence of two Fermi surfaces could be observed in cold-atomic experiments Sarma phase cont d

79 Electronic structure calculations There has been a large effort to treat fully many-body effects within density-functional-based (DFT) electronic structure calculations One of the approaches is DFT+DMFT where the many-body effects are computed by DMFT electron exchange Unit cell hybridized to a bath Because realistic systems often involve many bands and because it is sometimes important to consider a full SU(2) Hamiltonian, new solvers have been extremely useful

80 As-p, O-p Fe-d La Correlations in pnictides superconductors LaFeAsO M. Aichhorn et al., PRB (2009) - moderate correlations - m* lifetime effects - shifted bands A(k;!) LDA bands Density of states Correlation effects seen by LDA+DMFT

81 Interaction quench in the Hubbard model Eckstein et al., PRB (2010) What happens it we suddenly turn on an interaction in the Hubbard model? Does it reach a steady state? Is this steady state a thermal state? A real-time CT-QMC algorithm has been used to study this problem within non-equilibrium DMFT 0 < U < 3:2V 3:2V < U

82 Summary Continuous-time quantum Monte Carlo algorithms have allowed for progress in computing the properties of strongly-correlated systems Lower temperatures Generic Hamiltonians, new models Larger number of orbitals / sites The idea of the algorithms is to sample stochastically the diagrams of a series expansion of the partition function According to one s need, different expansions can be used There are still limitations (sign problem, speed, ) and more work has to be done!

83 Outline Quantum impurity problems Continuous-time quantum Monte Carlo (CT-QMC) methods Generic introduction to Monte Carlo The interaction-expansion algorithm (CT-INT) The hybridization-expansion algorithm (CT-HYB) Some applications of CT-QMC solvers TRIQS: a Toolbox for Research on Quantum Interacting Systems Implementation details for the CT-INT

84 A need for tools to do many-body calculations Solving a quantum impurity problem is usually just one step of a larger calculation For example, a typical DMFT calculation might require to Manipulate Green s functions Solve a quantum impurity problem Sum quantities over the Brillouin zone of a lattice Analyze, archive, plot data One might also have to use codes from different origins and glue results together It is therefore useful to have a set of flexible tools to work with the objects of a many-body calculations

85 Illustration: a DFT+DMFT calculation Wien2k Converged? ½ LDA+DMFT! V KS New density Nºº k = X G 0 ºº 0(k; i! n )e i! n0 + n ½ DMFT (r) = X hrjª kº inºº k hª 0 kº 0jri k;ºº 0 DFT Step V KS! ² kº ; ª kº (r) Projectors Wien2k dmftproj P mº (k) = h! km jª kº i Gmm 0 1 = ³ 0 mm 0 + G loc 1 mm 0 Green s functions manipulations Lattice Green s function G 1 ºº 0 (k; i! n ) = (i! n + ¹ ² kº )± ºº 0 ºº 0(k; i! n ) G loc mm 0 (i! n ) = X kºº 0 P mº (k)g ºº 0(k; i! n )P º 0 m 0 (k) Anderson Impurity Model G 0 mm 0 (i! n )! mm 0(i! n ) CT-QMC

86 Toolbox for Research on Interacting Quantum Systems Was born from the need to have a versatile environment to study interacting quantum systems in different contexts Started at Ecole Polytechnique and IPhT Saclay (with O. Parcollet, M. Aichhorn, L. Pourovskii, V. Vildosola, C. Martins, L. Boehnke, ) It is an open-source (GPL license) library

87 What is in the package? Python libraries: Lattice tools Data analysis tools Green s functions Plotting interface Quantum impurity solvers Ready-to-use applications: Wien2TRIQS C++ libraries for developers: Monte Carlo Array library Determinant manipulations

88 Example: Bethe lattice DMFT

89 Example: Bethe lattice DMFT

90 Making codes open source Probably the best way to learn about Monte Carlo methods is to implement one s own algorithm In the hands-on we try to code an interaction-expansion continuous-time algorithm Note however that in general it is not always a good idea to reinvent the wheel! There are good open-source tools around and one should never hesitate to use them Making one s codes available for the rest of the scientific community is a very good idea!

91 Outline Quantum impurity problems Continuous-time quantum Monte Carlo (CT-QMC) methods Generic introduction to Monte Carlo The interaction-expansion algorithm (CT-INT) The hybridization-expansion algorithm (CT-HYB) Some applications of CT-QMC solvers TRIQS: a Toolbox for Research on Quantum Interacting Systems Implementation details for the CT-INT

92 Reminder: A Monte Carlo algorithm propose update x! y compute P x;y ; P y;x ; p(y) Metropolis h A x;y = min 1; P i y;xp(y) P x;y p(x) NO x! x initialize simulation leave configuration unchanged measures observables YES x! y update configuration

93 Implementing the CT-INT algorithm in practice Monte Carlo loop Configuration 1 2 n 0 P y;x p(y) P x;y p(x) Moves acting on the configuration (insert/remove) Measures on the configuration Z; G Results

94 Implementation cont d Good practice to try to separate the algorithm in pieces The Monte Carlo loop does not need to know about the system. It just calls moves and expects back a Metropolis ratio to decide if the move is accepted or rejected What we need to do is implement insertions and removals in the configuration and compute the ratio 1 2 n 0 P y;x p(y) P x;y p(x) This ratio involves a determinant! This is the tricky part!

95 Reminder The weight of a Monte Carlo configuration is p(c n ) = ³ U 2 n det D " n det D # n And the matrix in the determinant has this definition det Dn ¾ = G 0¾ (0 ) s1 ¾ G 0¾ ( 1 2 ) G 0¾ ( 2 1 ) G 0¾ (0 ) s2 ¾ G 0¾ ( 2 3 ) G 0¾ ( n n 1 ) G 0¾ (0 ) sn ¾ with s¾ = (2± s¾ 1)±

96 Inserting a new vertex Inserting a vertex adds a line and column to the matrix h A x;y = min 1; U 1 2 i n + 1 det D" n+1 D# i n+1 n det D nd " n # 0 det D ¾ n = G 0¾ (0 ) s1 ¾ G 0¾ ( 1 2 ) G 0¾ ( 1 3 ) G 0¾ ( 2 1 ) G 0¾ (0 ) s2 ¾ G 0¾ ( 2 3 ) G 0¾ ( 3 1 ) G 0¾ ( 3 2 ) G 0¾ (0 ) s3 ¾ det Dn+1 ¾ = G 0¾ (0 ) s1 ¾ G 0¾ ( 1 2 ) G 0¾ ( 1 3 ) G 0¾ ( 1 4 ) G 0¾ ( 2 1 ) G 0¾ (0 ) s2 ¾ G 0¾ ( 2 3 ) G 0¾ ( 2 4 ) G 0¾ ( 3 1 ) G 0¾ ( 3 2 ) G 0¾ (0 ) s3 ¾ G 0¾ ( 3 4 ) G 0¾ ( 4 1 ) G 0¾ ( 4 2 ) G 0¾ ( 4 3 ) G 0¾ (0 ) s4 ¾

97 Removing a vertex Removing a vertex removes a line and column in the matrix n G 0¾ (0 ) s1 ¾ G 0¾ ( 1 2 ) G 0¾ ( 1 3 ) G 0¾ ( 1 4 ) G 0¾ ( 2 1 ) G 0¾ (0 ) s2 ¾ G 0¾ ( 2 3 ) G 0¾ ( 2 4 ) G 0¾ ( 3 1 ) G 0¾ ( 3 2 ) G 0¾ (0 ) s3 ¾ G 0¾ ( 3 4 ) G 0¾ ( 4 1 ) G 0¾ ( 4 2 ) G 0¾ ( 4 3 ) G 0¾ (0 ) s4 ¾ i h A x;y = min 1; n U det D" n 1 D# i n 1 det D nd " n #

98 No need to recompute a full determinant We see that we need to compute ratios of determinant of matrices that have an added / removed line and column This can be done faster that by computing the full determinant of the matrix before and after the move The trick is to keep a copy of both the matrix and its inverse and to update them Indeed: D 1 ij = ( 1) i+j 1 det D det D[j; i] Means the matrix with line j and column i removed

99 Fast updates for determinant and inverse matrix This is a special case of the Sherman-Morrison formula µ A B C d µ U V W x = µ AU + BW AV + xb CU + dw CV + xd = µ Insertion Removal V = xa 1 B x 1 = d CA 1 B µ A B det C d det A = x 1 A 1 = U x 1 V W W = xca 1 U = A 1 + x 1 V W det A µ = x A B det C d

100 Fast updates cont d One can find the determinant ratios and the new inverse matrix in operations of order n 2 Here we have shown insertions and removals happening at the very end of the matrix. If the insertions or removals are somewhere in the middle, one must add a correction ( 1) i+j An important thing to note is that the determinant ratio can be obtained before actually computing the inverse matrix. We can therefore decide to accept or reject the move and update the inverse matrix only if the move is accepted! Another good thing about have the inverse matrix is that we need it anyway to compute the Green s function These are all the ingredients to write an efficient algorithm!