Rigorous analysis of discontinuous phase transitions via mean-field bounds

被引:30
作者
Biskup, M [1 ]
Chayes, L [1 ]
机构
[1] Univ Calif Los Angeles, Dept Math, Los Angeles, CA 90095 USA
基金
美国国家科学基金会;
关键词
D O I
10.1007/s00220-003-0828-2
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
We consider a variety of nearest-neighbor spin models defined on the d-dimensional hypercubic lattice Z(d). Our essential assumption is that these models satisfy the condition of reflection positivity. We prove that whenever the associated mean-field theory predicts a discontinuous transition, the actual model also undergoes a discontinuous transition (which occurs near the mean-field transition temperature), provided the dimension is sufficiently large or the first-order transition in the mean- field model is sufficiently strong. As an application of our general theory, we show that for d sufficiently large, the 3-state Potts ferromagnet on Z(d) undergoes a first-order phase transition as the temperature varies. Similar results are established for all q-state Potts models with qgreater than or equal to3, the r-component cubic models with rgreater than or equal to4 and the O(N)-nematic liquid-crystal models with Ngreater than or equal to3.
引用
收藏
页码:53 / 93
页数:41
相关论文
共 54 条
[1]   DISCONTINUITY OF THE MAGNETIZATION IN ONE-DIMENSIONAL 1/[X-Y]2 ISING AND POTTS MODELS [J].
AIZENMAN, M ;
CHAYES, JT ;
CHAYES, L ;
NEWMAN, CM .
JOURNAL OF STATISTICAL PHYSICS, 1988, 50 (1-2) :1-40
[2]   GEOMETRIC ANALYSIS OF PHI-4 FIELDS AND ISING-MODELS .1.2. [J].
AIZENMAN, M .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1982, 86 (01) :1-48
[3]   ON THE CRITICAL-BEHAVIOR OF THE MAGNETIZATION IN HIGH-DIMENSIONAL ISING-MODELS [J].
AIZENMAN, M ;
FERNANDEZ, R .
JOURNAL OF STATISTICAL PHYSICS, 1986, 44 (3-4) :393-454
[4]   THE PHASE-TRANSITION IN A GENERAL-CLASS OF ISING-TYPE MODELS IS SHARP [J].
AIZENMAN, M ;
BARSKY, DJ ;
FERNANDEZ, R .
JOURNAL OF STATISTICAL PHYSICS, 1987, 47 (3-4) :343-374
[5]   Non-perturbative criteria for Gibbsian uniqueness [J].
Alexander, KS ;
Chayes, L .
COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1997, 189 (02) :447-464
[6]   A LATTICE MODEL OF LIQUID-CRYSTALS WITH MATRIX ORDER PARAMETER [J].
ANGELESCU, N ;
ZAGREBNOV, VA .
JOURNAL OF PHYSICS A-MATHEMATICAL AND GENERAL, 1982, 15 (11) :L639-L643
[7]  
[Anonymous], 2011, APPL MATH
[8]  
[Anonymous], 1992, RANDOM WALKS CRITICA, DOI DOI 10.1007/978-3-662-02866-7
[9]   Reflection positivity of the random-cluster measure invalidated for noninteger q [J].
Biskup, M .
JOURNAL OF STATISTICAL PHYSICS, 1998, 92 (3-4) :369-375
[10]  
BISKUP M, UNPUB MEAN FIELD DRI