The quantum cluster (QCT) code : User's manual
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1. by Potthoff and the integrand of the wavevector integral is evaluated by the diagonalization of a full M x M matrix M being the number of iterations in the band Lanczos method This method is actually consider ably slower than the one used by default in this code the band Lanczos method followed by a numerical integration over frequencies It is provided for sake of comparison only cdmft Uses the CDMFT self consistency loop instead of a minimization of the grand potential Uses a fixed grid for frequencies and wavevector Reads the variational parameters like for the option nr below The self consistency loop stops when the relative change in the distance function is less than CDMFT_REL_ACCUR defined in cdmft C cf Uses the continued fraction method to calculate the cluster Green function This was the only method in used before August 2006 but is much slower than the band Lanczos method which is now the default It is however recommended when the dimension of the Hilbert space is small say less than ten times the number of orbitals cg Optimizes the SEF with the conjugate gradient method Since the SEF is not a minimum in general by a saddle point and the conjugate gradient method only finds minima or maxima this options uses in alternance distinct conjugate gradient procedures for the variables for which the SEF is a minimum i e the Weiss fields and for the chemical potential for which it is a maximum This option2. is slower than nr but more robust and can be used as a first stage of search It stops when the position of the variational parameters changes by less than accur_cg set in the parameters file cluster_data Writes to a file the cluster ground state expectation value of the number of electrons and spin at each sites as well as the corresponding correlations dos Calculates the density of states and puts it in 5 column format in the file output _dos Range of frequencies specified as in option s below The five columns are 1 frequencies 2 lattice DOS for spin up normalized to one 3 lattice DOS for spin down normalized to one 4 cluster DOS for spin up normalized to one 5 cluster DOS for spin down normalized to one full In constructing the hamiltonian builds a full sparse matrix for H instead of using tensor factors for the off diagonal terms Not useful in practice takes more memory and slower contrary to initial expectation grad Calculates the gradient of the SEF at the end of a CDMFT run ksym Symmetrizes the Green function over a point group This option needs an argument R or D e R means rotational symmetry by 7 2 and exchange x gt y so that 8 terms are combined e D means diagonal symmetry x gt y only two terms are combined loop Does a loop over variational parameters and writes the grand potential or the distance function to a file The variational parameter
3. The quantum cluster QCT code User s manual David S n chal version August 7 2006 1 General principles The qct code applies quantum cluster methods to the Hubbard model and performs various types of calcu lations such as 1 Ordinary CPT spectral functions for energy distribution curves EDCs and momentum distribution curves MDCs 2 The VCPT grand potential Q and the search for saddle points CDMFT calculations The order parameters associated with the various terms in the Hamiltonian OR cape Oe Systematic loops over variational parameters in calculating Q or the CDMFT distance function It is written in C and borrows many utility routines integration etc from other sources The command line is qct and requires options as well as input files Input files required The code requires a parameter file called para dat by default The format of this file is described below It also requires a cluster description file usually with the suffix clus whose precise name is specified in the parameter file 2 Command line options If the command qct is entered without any option a list of available options with a short description is printed to the screen The same is true if the option h is typed aaph Calculated the self energy functional Q SEF with the method explained in the paper by Aichhorn Ar rigoni Potthoff and Hanke cond mat 0607271 the frequency integral is calculated analytically formula
4. and indices omitted if nband 1 The link direction e g AR 1 0 for a NN link in the x direction in 2D e A multiplier e g 1 for all hopping terms by convention The multiplier can be opposite for singlet pairing operators in the x and y directions for instance so that a d wave pairing is obtained if those two parameters are equal Local operators Local operators are defined by specifying the explicit terms in the hopping matrix without explicit reference to an inter cluster extension The list of operators must be contained between the keyword OPERATORS and an asterisk Here is an example OPERATORS tbiu 4 h n 1 1 1 2 2 1 3 3 1 4 4 1 eblu 4 b n 1 1 1 2 2 1 3 3 1 4 4 1 d1 8 b s 1 22 1 3 4 1 1 3 1 2 4 1 5 6 1 T 8 1 5 7 1 6 8 1 M 8 c s 1 1 1 gas T 1 2 2 1 2 24 1 3 3 1 3 3 1 4 4 1 4 4 1 The first line of each operator descriptions specifies 1 the name of the parameter 2 the number of lines in the list of terms that follows 3 whether the operator is based on the cluster c on the bath b or is a hybridization term between the two h and 4 whether the parameter is of N type n or of S type s see the mvcpt option above for explanations Then a number of lines follows for each operator giving 1 the label of the first site with its spin or 2 the label of the second
5. er cluster hopping or pairing More specifically for a normal one body term defined as h 5o Sij a Hc 1 950 the corresponding order parameter is defined as 2 dw is A 79 ff a D a 4 where is the matrix s and depends on K the reduced wavevector is it corresponds to a hopping term with inter cluster hopping The traces includes a sum over spins In the case of anomalous terms a similar form is used in practice the Nambu formalism is used so there is no big difference between a hopping term and an anomalous term The frequency integral is taken along a contour C that half circles all we hope negative axis poles of the Green function The output is appended to a file called prop dat Two lines are written each time a commented one for the description of parameters with column numbers and the following for their values The first four columns give 1 the name of the cluster description file 2 the value of U 3 the value of the density of electrons on the cluster 4 the value of the grand potential The remaining columns give the values of the various parameters defined in the cluster description file as well as their expectation value If inter cluster values are different from their default they are also given qn In VCPT uses the quasi Newton method instead of the Newton Raphson method see option rn above Takes less time when a large number of variational parameters is used reinit Reinitializes the
6. eter The converged values of the variational parameters found for each value of the loop parameter are used as initial values for the next run that is the whole point of this option nmin Performs a super loop over chemical potential in order to achieve a preset value of density n Does a o at each value of mu and uses the zbrent routine to find the preset value of n The following values are read in the parameter file e target_n target density followed par required accuracy see option n loop above e mui the lower bound for p e mu2 the upper bound for p nr Looks for an extremum point of 2 in the space of variational parameters Uses the Newton Ralphson method The initial point must be set with common sense The variational parameters are specified in the parameter file between the keyword VARIATIONAL_PARAMETERS and the next asterisk Each variational parameter is specified on a separate line with i its name ii the starting value iii the minimum value and iv maximum value the parameter can take acceptable range and v the initial incremental step the parameter should be given in the search Example VARIATIONAL_PARAMETERS mu 0 5 5 5 0 05 tx 0 25 3 3 0 02 The procedure stops when the calculated gradient is inferior to the parameter accur_grad set in the param eters file Each time the grand potential is evaluated in the optimization procedure the values of the parameters and grand
7. parameters each time when doing a loop over a lattice parameter ploop instead of starting from the variational parameters found for the previous value of the parameter 75S Computes the CPT spectral function EDCs and places the output in the file output _sp The first column is for freqencies the others for the various wavevectors sweep specified by wavevectors in the parameter file The wavevectors are written on the first commented line of the file The following values are read in the parameter file e wavevectors the wavevector sweep see section below e eta the imaginary part of the frequency e wmin the lower bound of the plot e wmax the upper bound of the plot e step the frequency step e period_option the type of periodization used 0 by default seed Sets the seed for the random number generator used in setting the initial state in the Lanczos algorithm Needs the seed as an argument Useful to check whether a strange result can be attributed to a degeneracy of the ground state sef Calculates the SEF i e the grand potential Q for the set of parameters specified in the parameter file Appends the result in output sef self_dist When doing CDMFT uses a distance function weighed by the norm of the self energy H w instead of a flat constant weight sym Uses the cluster symmetries to shorten the calculation The cluster symmetries need to be specified in the cluster descri
8. pe N parameters as varia tional setting the other ones to zero in other words it looks for the normal state solution Then it considers each S type parameter in turn varying it together with all the N type parameters If it fails to find a non trivial solution it tries again with a larger starting value or smaller if the procedure led to a divergent value At last it varies all parameters together using the previous converged values as starting points This is the best option to use if one wants a more complete picture of the variational space nloop Performs a loop over the chemical potential u and another parameter typically U so as to find and then stay at the conditions for a preset filling Syntax targetn n An n loop param start end increment Ap where n is the target density e g n 1 for half filling An is the required precision for n e g 0 001 param is the name of the parameter to be varied e g U start end increment are the starting end and increment values of the parameter and Ay is the step in yz to be taken to progress towards half filling The advantage of this method over the nmin option below is that the parameters evolve in a smooth way thus allowing the solutions to be found in a continuous way from the previous point s solution This option must be specified in conjunction with either cdmft nr cg or prop to specify the type of action taking place at each value of the param
9. pes i interaction terms ii link based operators and iii local operators Interaction terms If nband 1 the interaction is taken as the usual Hubbard interaction U for each site on the cluster bath excluded Nothing to write For multi band models the interactions must be specified explicitly for instance by INTERACTION U i 1 1 0 1 Up i 2 2 0 1 V i 1 2 0 1 each line describes an interaction parameter in this order 1 its name 2 the keyword i for interaction 3 the two band indices concerned 4 the displacement vector always zero for an interaction and 5 the multiplier usually unity Link based operators Link based operators are hopping terms normal or anomalous that are normally defined for the whole lattice but are restricted to the cluster If an operator like NN hopping is defined using links its average value computed with the lattice Green function will take into account the inter cluster link as well Lattice hopping terms must be defined like this A typical definition of link operators in the cluster description file would be LINK_OPERATORS tx h by b2 AR multiplier Dx s by be AR multiplier px t by b2 AR multiplier where the columns are The name of the parameter the type of link h hopping term s singlet superconducting pairing term t triplet superconducting pairing term x off diagonal spin pairing term if no_Nambu is set The b
10. potential is appended to output _sef The value of the extremum found its properties as well as the values of the first and second derivatives for each variable are written to a file whose name contains the names of the variational parameters prefixed by var_ At the end of the procedure the properties of the extremum are calculated i e the prop option is run except when the procedure fails to converge to a value within bounds See also mvcpt ploop Does a loop over a lattice parameter e g chemical potential or U Syntax p loop param loop start end increment or p loop param list N list of N numbers Examples ploop mu loop 2 4 3 2 0 05 or p loop mu list 7 2 4 2 5 2 6 2 7 2 8 2 9 3 0 This option must be specified in conjunction with either sc vcpt sc_vcpt or prop to specify the type of action taking place at each value of the parameter The converged values of the variational parameters found for each value of the loop parameter are used as initial values for the next run unless the option reinit is specified prop Calculates the order parameters associated with each term of the Hamiltonian or properties Basically for a diagonal hopping or anomalous term in the cluster Hamiltonian the expectation value of the corresponding operator is calculated If the hopping or anomalous term is defined with a link type see below then the corresponding expectation value takes into account int
11. ption file In addition the symmetries are checked against the particular parameters used and only those that are really present in the matrix t of the code are really used this is much safer than in previous versions This option can save a lot of time on larger clusters and is heavily recommended test Used for debugging Action may depend on version and stage of development Not for general use 3 The cluster description file The cluster description file is specified following the keyword cluster_file in the parameter file The best way to understand the structure of that file is to look at an example Here is a description of the generic parameters specified in that file e dim the spatial dimension of the model 1 2 or 3 e L the number of sites L of the cluster e nb the number of bath sites zero if omitted e nband the number of bands in the model set to unity if omitted e The optional keyword no_Nambu which means that the Nambu formalism will not be used and that the model can involve the operators Sy and Sy can be defined but without anomalous terms e sites the x y positions of the sites as integers preceded by the number label of each site from 1 to L For dim gt 1 only e neighbors the basis vectors of the superlattice of clusters again in x y format For dim gt 1 only Following these generic parameters various one body terms in the Hamiltonian are specified These are of three ty
12. s are specified in the parameter file between the keyword LOOP_PARAMETERS and the next asterisk Each variational parameter is specified on a separate line with i its name ii the starting value iii the end value and iv the increment A maximum of 3 nested loops is allowed By default it calculates the grand potential Q for each value and appends a line in the file output _sef Otherwise if the option cdmft is specified it is the corresponding distance function that is calculated mdc Calculates the CPT spectral function as a momentum distribution curve MDC The following values are read in the parameter file e freq the real part of the frequency e k_eta the imaginary part of the frequency for broadening e k_step the step in wavevector Depending on the ksym option the range of wavevectors is either the top half of the Brillouin zone case D or the first quadrant case R Example freq 0 k_eta 0 05 k_step 0 02 mvcpt Applies the VCPT procedure Newton Raphson or Quasi Newton to an increasingly large body of param eters Variational parameters are either of the symmetry breaking type S or of the normal type N Examples of N type parameters are the chemical potential or a hopping term Examples of S type parame ters are the AF Weiss field M of the various dSC Weiss fields This option first reads the list of variational parameters see the vcpt option Then in a first procedure only treats the ty
13. should not normally have a nonzero lattice value and are necessarily specified here Example CLUSTER_PARAMETERS mu 1 1 tx 0 3 M 0 14521 Dx 0 3721 Variational and loop parameters are specified the like with the keywords CLUSTER_PARAMETERS and LOOP_PARAMETERS See the corresponding options in the first section of this document
14. site with its spin or the value of a multiplier Important notes e Hermitian conjugates are taken care of automatically and must not be specified in order to avoid double counting e site labels are counted from 1 to L for cluster sites and from 1 to ny for bath sites In the hybridization case h cluster sites are specified in the first column bath sites in the second column e A space must be kept between the site label and the spin label i e 1 and not 1 4 The parameter file Each parameter defined in the cluster description file has a cluster value and a lattice value The values of the lattice parameters are read in the parameter file between the keyword LATTICE_PARAMETERS and the next asterisk Each parameter is specified on its own line by its name followed by its numerical value Example LATTICE_PARAMETERS U 2 If a parameter is not specified its value is assumed to be zero unless this parameter has a default parameter in which case its value is taken to be the same as that of the default parameter If a parameter is specified that does not exist in the model a simple warning is issued on the screen but the program is otherwise unaffected The values of the cluster parameters is by default the same as that of the lattice parameters specified The default values are overridden by the list specified between the keyword CLUSTER_PARAMETERS and the next asterisk Some parameters like the Weiss fields
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