commit 040c8fb8c22a1883384f31a4076f9b81e7da64af parent 20d98c1504d1fc68c9c2f2e501001f65f0bd80ee Author: Miroslav Urbanek <mu@miroslavurbanek.com> Date: Thu, 29 Jun 2017 01:49:35 +0200 Add a simulation of many-body localization The system of 2 particles in a 4 x 4 lattice is too small to observe a crossover from thermalization to localization. The calculation serves as a simple example. Diffstat:
| README | | | 73 | +++++++++++++++++++++++++++++++++++++++++++++++-------------------------- |
| contrib/openblas-0.2.6.patch | | | 57 | --------------------------------------------------------- |
| examples/mbl.lisp | | | 104 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| inputs/conv.lisp | | | 57 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| inputs/gsl.lisp | | | 46 | ++++++++++++++++++++++++++++++++++++++++++++++ |
| inputs/mbl-w-4-h-4.pot | | | 2 | ++ |
| inputs/mbl.lisp | | | 118 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
| inputs/printpot.lisp | | | 23 | +++++++++++++++++++++++ |
| outputs/mbl | | | 916 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ |
9 files changed, 1313 insertions(+), 83 deletions(-)
diff --git a/README b/README @@ -1,41 +1,44 @@ 1. INTRODUCTION -TEBDOL is a program for calculating time evolution of a system of -ultracold atoms in an optical lattice. It utilizes the time-evolving -block decimation algorithm (TEBD). It is written in Common Lisp and -supports parallelization using MPI. +TEBDOL [1] is a program for calculating time evolution of a system of +ultracold atoms in one-dimensional and two-dimensional optical +lattices. It utilizes the time-evolving block decimation algorithm +(TEBD). One-dimensional version is parallelized using MPI. 2. INSTALLATION -TEBDOL has following dependencies: +TEBDOL is written in Common Lisp and has the following dependencies: -* SBCL +* Steel Bank Common Lisp (SBCL) SBCL is an ANSI Common Lisp compiler. Source and binaries are - available at http://sbcl.org/. TEBDOL was tested with version 1.2.3. + available at http://sbcl.org/. TEBDOL was tested with version 1.2.4. -* BLAS and LAPACK +* Basic Linear Algebra Subprograms (BLAS) and Linear Algebra Package + (LAPACK) BLAS and LAPACK libraries provide vector and matrix routines. There are several vendor-provided implementations available. TEBDOL was - tested with OpenBLAS 0.2.6, which can be download at - http://www.openblas.net/. Due to a numerical bug in LAPACK, a patch - in the file "contrib/openblas-0.2.6.patch" has to be applied to this - version of OpenBLAS. TEBDOL should also work with Intel MKL. + tested with OpenBLAS 0.2.12, which can be download at + http://www.openblas.net/. It also works with Intel MKL. -* MPI +* Message Passing Interface (MPI) MPI provides support for parallelization. TEBDOL was tested with - OpenMPI 1.4.3. + version 1.6.5. -After installing SBCL, BLAS/LAPACK, and MPI, you'll need to edit the -parameters in the script "run" in the top-level directory: +* GNU Scientific Library (GSL) -* LD_LIBRARY_PATH Path to directories with BLAS/LAPACK and MPI - libraries. -* SBCL Path to the SBCL binary. -* DSS Maximum amount of memory allocated for a single - SBCL process. + GSL is used for random number generation. TEBDOL was tested with + version 1.16. + +After installing SBCL, BLAS/LAPACK, MPI, and GSL, you'll need to edit +the parameters in the script "run" in the top-level directory: + +* LD_LIBRARY_PATH Paths to directories containing the libraries. +* SBCL Path to the SBCL binary. +* DSS Maximum amount of memory allocated for a single + SBCL process. You'll probably also need to edit the file "conf.lisp", where you can specify the name of the BLAS/LAPACK library and switch MPI @@ -76,12 +79,20 @@ There are several examples in the directory "examples": * Equilibration example ("trotzky.lisp"). - This example simulates time evolution of a model from an article - [1]. Usage is similar to previous examples: + This example simulates time evolution of a model from article + [2]. Usage is similar to previous examples: $ ./run inputs/trotzky.lisp $ ./run examples/trotzky.lisp +* Many-body localization in two dimensions ("mbl.lisp"). + + This example is inspired by article [3]. Usage is similar to + previous examples: + + $ ./run inputs/mbl.lisp + $ ./run examples/mbl.lisp + The sample outputs are in the directory "outputs". 4. FILES @@ -104,10 +115,20 @@ tebdol/serial.lisp: Serialization routines. tebdol/tebd.lisp: Time-evolving block decimation algorithm. tebdol/tebdol.asd: System definition. tebdol/tensor.lisp: Symmetric tensor routines. +tebdol/ttns.lisp: Routines for working with tree tensor network states. tebdol/util.lisp: Miscellaneous utilities. 5. REFERENCES -[1] Trotzky S. et al. Probing the relaxation towards equilibrium in an -isolated strongly correlated one-dimensional Bose gas. Nature Phys. 8, -325–330 (2012). doi:10.1038/nphys2232 +[1] M. Urbanek and P. Soldán, "Parallel implementation of the +time-evolving block decimation algorithm for the Bose–Hubbard model", +Comput. Phys. Commun. 199, 170–177 +(2016). doi:10.1016/j.cpc.2015.10.016 + +[2] Trotzky S. et al., "Probing the relaxation towards equilibrium in +an isolated strongly correlated one-dimensional Bose gas", Nature +Physics 8, 325–330 (2012). doi:10.1038/nphys2232 + +[3] Choi J.-y. et al., "Exploring the many-body localization +transition in two dimension", Science 352, 1547–1552 +(2016). doi:10.1126/science.aaf8834 diff --git a/contrib/openblas-0.2.6.patch b/contrib/openblas-0.2.6.patch @@ -1,57 +0,0 @@ -diff --git a/Makefile.rule b/Makefile.rule -index 4e23857..8ea235e 100644 ---- a/Makefile.rule -+++ b/Makefile.rule -@@ -105,7 +105,7 @@ VERSION = 0.2.6 - # If any gemm arguement m, n or k is less or equal this threshold, gemm will be execute - # with single thread. You can use this flag to avoid the overhead of multi-threading - # in small matrix sizes. The default value is 50. --# GEMM_MULTITHREAD_THRESHOLD = 50 -+# GEMM_MULTITHREAD_THRESHOLD = 4 - - # If you need santy check by comparing reference BLAS. It'll be very - # slow (Not implemented yet). -diff --git a/Makefile.system b/Makefile.system -index fc89d2d..ab9ead0 100644 ---- a/Makefile.system -+++ b/Makefile.system -@@ -53,7 +53,7 @@ GETARCH_FLAGS += -DUSE64BITINT - endif - - ifndef GEMM_MULTITHREAD_THRESHOLD --GEMM_MULTITHREAD_THRESHOLD=50 -+GEMM_MULTITHREAD_THRESHOLD=4 - endif - GETARCH_FLAGS += -DGEMM_MULTITHREAD_THRESHOLD=$(GEMM_MULTITHREAD_THRESHOLD) - -diff --git a/c_check b/c_check -index 4d82237..f26fce2 100644 ---- a/c_check -+++ b/c_check -@@ -241,7 +241,7 @@ print CONFFILE "#define FUNDERSCORE\t$need_fu\n" if $need_fu ne ""; - - if ($os eq "LINUX") { - -- @pthread = split(/\s+/, `nm /lib/libpthread.so* | grep _pthread_create`); -+ @pthread = split(/\s+/, `nm /lib/*-linux-gnu/libpthread.so* | grep _pthread_create`); - - if ($pthread[2] ne "") { - print CONFFILE "#define PTHREAD_CREATE_FUNC $pthread[2]\n"; -diff --git a/kernel/x86/lsame.S b/kernel/x86/lsame.S -index d7e48ad..3ac7a73 100644 ---- a/kernel/x86/lsame.S -+++ b/kernel/x86/lsame.S -@@ -74,11 +74,11 @@ - #else - movl %eax, %ecx - subl $32, %ecx -- cmovg %ecx, %eax -+ cmovge %ecx, %eax - - movl %edx, %ecx - subl $32, %ecx -- cmovg %ecx, %edx -+ cmovge %ecx, %edx - - subl %eax, %edx - movl $0, %eax diff --git a/examples/mbl.lisp b/examples/mbl.lisp @@ -0,0 +1,104 @@ +;; many-body localization in a two-dimensional optical lattice + +;; parameters + +(defparameter *potential* #p"inputs/mbl-w-4-h-4.pot") +(defparameter *diameter* 2) +(defparameter *u* 24.4d0) ;; *j* = 1 + +(defparameter *dimension* 7) +(defparameter *step* (/ 16d0)) + +(defparameter *maxdim* + (list (* (expt 2 32) double-float-epsilon) nil 128)) + +;; libraries + +(load "conf.lisp") +(require :tebdol) + +(use-package '(:blas :ttns :tebd :bhm)) +(setf *print-level* 2) + +;; functions + +(defun make-half-circle-ttns (width height diameter dimension) + (loop + with p = (make-vacuum-ttns width height dimension) + with xshift = (ceiling (- width diameter) 2) + with yshift = (floor (- height diameter) 2) + for i below (ceiling diameter 2) + do + (loop + for j below diameter + when (and (<= + (sqrt + (+ (expt (- i (/ (1- diameter) 2)) 2) + (expt (- j (/ (1- diameter) 2)) 2))) + (/ diameter 2)) + (>= (+ i xshift) 0) + (>= (+ j yshift) 0) + (< (+ i xshift) width) + (< (+ j yshift) height)) + do + (ttns-create-particle p (+ i xshift) (+ j yshift) dimension)) + finally + (compress-ttns p nil) + (return p))) + +(defun imbalance (pn) + (loop + with w = (array-dimension pn 0) + with h = (array-dimension pn 1) + for i below w + for s = (loop + for j below h + sum (aref pn i j)) + when (< i (/ w 2)) + sum s into l + else + sum s into r + finally + (return (/ (- l r) (+ l r))))) + +(defun load-potential (pathname) + (with-standard-io-syntax + (with-open-file (s pathname) + (read s)))) + +;; initialization + +(defparameter *potarray* (load-potential *potential*)) +(defparameter *width* (array-dimension *potarray* 0)) +(defparameter *height* (array-dimension *potarray* 1)) + +(defparameter *tbhsh* (bose-hubbard-site-hamiltonian *dimension* 1d0 0d0 0d0)) +(defparameter *ssps* (make-ssps *width* *height* *dimension* *u* *potarray* (/ *step* 2))) +(defparameter *dsps* (make-dsps *width* *height* *dimension* *u* *potarray* (/ *step* 2))) +(defparameter *dspy* (make-double-site-propagator *tbhsh* *tbhsh* (/ *step* 2))) + +(defparameter *ttns* (make-half-circle-ttns *width* *height* *diameter* *dimension*)) +(defparameter *pn* (all-pn-operators *width* *height* *dimension*)) + +(format t "# potential = ~A~%" *potential*) +(format t "# width = ~A~%" *width*) +(format t "# height = ~A~%" *height*) +(format t "# diameter = ~A~%" *diameter*) +(format t "# u = ~A~%" *u*) +(format t "# step = ~A~%" *step*) +(format t "# maxdim = ~A~%" *maxdim*) +(format t "~%# time imbalance norm~%# ----~%# density~%# ----~%~%") + +;; main loop + +(loop + for i to 128 + do + (unless (zerop i) + (ttns-tebd-evolve *ttns* *ssps* *dsps* *dspy* *maxdim*)) + (let* ((n (realpart (ttns-overlap *ttns* *ttns*))) + (p (all-pn *ttns* n *pn*))) + (format t "~,4f ~F ~F~%" (* i *step*) (imbalance p) n) + (format t "----~%") + (print-pn p) + (format t "----~%"))) diff --git a/inputs/conv.lisp b/inputs/conv.lisp @@ -0,0 +1,57 @@ +(defpackage :conv + (:use :common-lisp) + (:export :normalized-gaussian + :convolve)) + +(in-package :conv) + +(defun gaussian (width height sigma) + (loop + with a = (make-array (list width height)) + for i below width + do (loop + for j below height + do (setf (aref a i j) + (exp (/ (+ (expt (- i (/ (1- width) 2)) 2) + (expt (- j (/ (1- height) 2)) 2)) + (* -2 (expt sigma 2)))))) + finally (return a))) + +(defun sum-array (array) + (loop + for i below (array-total-size array) + sum (row-major-aref array i))) + +(defun normalize-array (array) + (loop + with a = (make-array (array-dimensions array)) + with s = (sum-array array) + for i below (array-total-size a) + do (setf (row-major-aref a i) (/ (row-major-aref array i) s)) + finally (return a))) + +(defun normalized-gaussian (width height sigma) + (normalize-array (gaussian width height sigma))) + +(defun convolve (array kernel) + (loop + with a = (make-array (array-dimensions array)) + with aw = (array-dimension array 0) + with ah = (array-dimension array 1) + with kw = (array-dimension kernel 0) + with kh = (array-dimension kernel 1) + for i below aw + do (loop + for j below ah + do (setf (aref a i j) + (loop + for k below kw + for m = (+ i (- k (floor kw 2))) + when (and (>= m 0) (< m aw)) + sum (loop + for l below kh + for n = (+ j (- l (floor kh 2))) + when (and (>= n 0) (< n ah)) + sum (* (aref array m n) + (aref kernel k l)))))) + finally (return a))) diff --git a/inputs/gsl.lisp b/inputs/gsl.lisp @@ -0,0 +1,46 @@ +(defpackage :gsl + (:use :common-lisp :sb-alien) + (:export :gsl-rng-type + :gsl-rng + :+gsl-rng-ranlxd2+ + :*gsl-rng-default* + :*gsl-rng-default-seed* + :gsl-rng-alloc + :gsl-rng-set + :gsl-rng-free + :gsl-rng-env-setup + :gsl-ran-binomial)) + +(in-package :gsl) + +(load-shared-object "libgslcblas.so.0") +(load-shared-object "libgsl.so.0") + +(define-alien-type gsl-rng-type (struct nil)) +(define-alien-type gsl-rng (struct nil)) + +(define-alien-variable ("gsl_rng_ranlxd2" +gsl-rng-ranlxd2+) + (* gsl-rng-type)) + +(define-alien-variable ("gsl_rng_default" *gsl-rng-default*) + (* gsl-rng-type)) + +(define-alien-variable ("gsl_rng_default_seed" *gsl-rng-default-seed*) + unsigned-long) + +(define-alien-routine gsl-rng-alloc (* gsl-rng) + (type (* gsl-rng-type))) + +(define-alien-routine gsl-rng-set void + (r (* gsl-rng)) + (s unsigned-long)) + +(define-alien-routine gsl-rng-free void + (r (* gsl-rng))) + +(define-alien-routine gsl-rng-env-setup (* gsl-rng-type)) + +(define-alien-routine gsl-ran-binomial unsigned-int + (r (* gsl-rng)) + (p double) + (n unsigned-int)) diff --git a/inputs/mbl-w-4-h-4.pot b/inputs/mbl-w-4-h-4.pot @@ -0,0 +1,2 @@ +;; width = 4, height = 4, diameter = 2, epp = 0.14d0, fwhm = 1.0d0, seed = 0 +#2A((29.651652249358083d0 16.567252253892597d0 17.860483916475815d0 31.17380679705555d0) (16.052735710640196d0 3.2574307872105224d0 3.5240962865932723d0 18.307746347439696d0) (16.92775153011918d0 3.479964227027768d0 3.051200719491302d0 17.40290802941907d0) (30.935305770523602d0 17.062230185470135d0 17.04918426851492d0 30.630766615907486d0)) diff --git a/inputs/mbl.lisp b/inputs/mbl.lisp @@ -0,0 +1,118 @@ +;; random lattice potential + +;; parameters + +(defparameter *width* 4) +(defparameter *height* 4) +(defparameter *diameter* 2) +(defparameter *epp* 1.4d-1) +(defparameter *fwhm* 1d0) + +;; fixed seed +(defparameter *seed* 0) + +;; random seed, uncomment to get a real random potential +;; (defparameter *seed* (random #x100000000 (make-random-state t))) + +;; libraries + +(load (merge-pathnames "gsl.lisp" *load-truename*)) +(load (merge-pathnames "conv.lisp" *load-truename*)) +(use-package '(:gsl :conv)) + +;; main + +(defun harmonic-potential (width height) + (loop + with a = (make-array (list width height)) + for i below width + do (loop + for j below height + do (setf (aref a i j) + (+ (expt (- i (/ (1- width) 2)) 2) + (expt (- j (/ (1- height) 2)) 2)))) + finally (return a))) + +(defun half-circle-mean-energy (potential diameter) + (loop + with m = (ceiling (- (array-dimension potential 0) diameter) 2) + with n = (floor (- (array-dimension potential 1) diameter) 2) + with p = 0 + with e = 0 + for i below (/ diameter 2) + do (loop + for j below diameter + when (<= (sqrt (+ (expt (- i (/ (1- diameter) 2)) 2) + (expt (- j (/ (1- diameter) 2)) 2))) + (/ diameter 2)) + do + (incf p) + (incf e (aref potential (+ i m) (+ j n)))) + finally (return (/ e p)))) + +(defun scaled-harmonic-potential (width height diameter epp) + (loop + with a = (harmonic-potential width height) + with c = (/ (* 24.4d0 epp) (half-circle-mean-energy a diameter)) + for i below (array-total-size a) + do (setf (row-major-aref a i) (* c (row-major-aref a i))) + finally (return a))) + +(defparameter *gsl-rng* (gsl-rng-alloc +gsl-rng-ranlxd2+)) + +(gsl-rng-set *gsl-rng* *seed*) + +(defun binomial-variate (p n) + (gsl-ran-binomial *gsl-rng* p n)) + +(defun squared-binomial-mean (p n) + (/ (+ p (* (expt p 2) (1- n))) n)) + +(defconstant +squared-binomial-fwhm+ 1.06d-1) ;; measured value + +(defun squared-binomial-potential (width height fwhm) + (loop + with c = (/ fwhm +squared-binomial-fwhm+) + with m = (squared-binomial-mean 5d-1 49) + with a = (make-array (list width height)) + for i below (array-total-size a) + do (setf (row-major-aref a i) + (* c (- (expt (/ (binomial-variate 5d-1 49) 49) 2) m))) + finally (return a))) + +(defun convolved-potential (width height fwhm) + (loop + with p = (squared-binomial-potential (+ width 8) (+ height 8) fwhm) + with g = (normalized-gaussian 9 9 5d-1) + with a = (convolve p g) + with b = (make-array (list width height)) + for i below width + do (loop + for j below height + do (setf (aref b i j) + (aref a (+ i 4) (+ j 4)))) + finally (return b))) + +(defun full-potential (width height diameter epp fwhm) + (loop + with a = (scaled-harmonic-potential width height diameter epp) + with b = (convolved-potential width height fwhm) + for i below (array-total-size a) + do (incf (row-major-aref a i) (row-major-aref b i)) + finally (return a))) + +(defun save-potential (potential pathname) + (when (or (not (probe-file pathname)) (y-or-n-p "Overwrite potential ~S?" pathname)) + (with-standard-io-syntax + (with-open-file (s pathname :direction :output :if-exists :supersede) + (format s ";; width = ~A, height = ~A, diameter = ~A, epp = ~A, fwhm = ~A, seed = ~A~%" + *width* *height* *diameter* *epp* *fwhm* *seed*) + (prin1 potential s) + (terpri s))))) + +(save-potential + (full-potential *width* *height* *diameter* *epp* *fwhm*) + (pathname + (format nil "inputs/mbl-w-~A-h-~A.pot" *width* *height*))) + +(gsl-rng-free *gsl-rng*) diff --git a/inputs/printpot.lisp b/inputs/printpot.lisp @@ -0,0 +1,23 @@ +;; print potential + +(unless (= (length *posix-argv*) 2) + (format t "usage: printpot filename~%") + (exit :code 1)) + +(defun read-potential (pathname) + (with-standard-io-syntax + (with-open-file (s pathname) + (read s)))) + +(defun print-potential (potential) + (loop + with w = (array-dimension potential 0) + with h = (array-dimension potential 1) + for j from (1- h) downto 0 + do (loop + for i below w + do (format t "~6,3F~:[ ~;~%~]" + (aref potential i j) + (eql i (1- w)))))) + +(print-potential (read-potential (pathname (second *posix-argv*)))) diff --git a/outputs/mbl b/outputs/mbl @@ -0,0 +1,916 @@ +# potential = inputs/mbl-w-4-h-4.pot +# width = 4 +# height = 4 +# diameter = 2 +# u = 24.4d0 +# step = 0.0625d0 +# maxdim = (4.768371582031251d-7 NIL 128) + +# time imbalance norm +# ---- +# density +# ---- + +0.0000 1.0 1.0 +---- +0.0 0.0 0.0 0.0 +0.0 1.0 0.0 0.0 +0.0 1.0 0.0 0.0 +0.0 0.0 0.0 0.0 +---- +0.0625 0.9922418850847765 0.9999998297813255 +---- +0.000011568924772761605 0.003099318434514578 0.000011556715763915659 0.0 +0.0036696620656758105 0.9892125918480463 0.0038625218028969913 0.0000037757773706609104 +0.003698386874444776 0.9892562447151854 0.0038642580807774084 0.000003771631798138162 +0.000012075708426906586 0.003282036513709711 0.000012218144595733203 0.000000012762020625683063 +---- +0.1250 0.969444930412789 0.9999969719129144 +---- +0.00012400265494990405 0.009706178168958432 0.00014653587708548835 0.0 +0.011733777215919207 0.9626358525457173 0.0150765320683529 0.000048130322888661976 +0.012189537973648001 0.9620447422912766 0.01507406595579576 0.00004777972334106194 +0.00013450340727393721 0.010876336155045845 0.00016174679280364995 0.0000002788469435081509 +---- +0.1875 0.9330229053989856 0.9999944467381765 +---- +0.00031432039525705456 0.014291676586032327 0.0004934110828400442 0.000001316953314827173 +0.018047770271392045 0.9330963313656053 0.03280360268759707 0.00018860359846899829 +0.019855934668119815 0.9292773004836973 0.03271125556629923 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