Cut cucumber in half lengthwise and cut each half lengthwise into 3 spears. Cut spears crosswise into 1/4-inch-thick pieces and transfer to a large bowl. Peel jicama and cut enough into 1/4-inch dice to measure 1 cup. Cut bell pepper and celery into 1/4-inch dice and add with diced jicama to cucumber. Mince garlic and toss with cucumber mixture. Stir in remaining ingredients and season soup with salt and pepper.
In a blender puree half of soup until smooth and return to bowl, stirring to combine. Chill soup, covered, until cold, at least 1 hour, and up to 2 days. Season soup and serve garnished with chives.